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@@ -0,0 +1,1208 @@
+CLIP the Gap: A Single Domain Generalization Approach for Object Detection
+Vidit Vidit1 Martin Engilberge1 Mathieu Salzmann1,2
+CVLab, EPFL1, ClearSpace SA2
+[email protected]
+Abstract
+Single Domain Generalization (SDG) tackles the prob-
+lem of training a model on a single source domain so that
+it generalizes to any unseen target domain. While this has
+been well studied for image classification, the literature on
+SDG object detection remains almost non-existent. To ad-
+dress the challenges of simultaneously learning robust ob-
+ject localization and representation, we propose to leverage
+a pre-trained vision-language model to introduce semantic
+domain concepts via textual prompts. We achieve this via
+a semantic augmentation strategy acting on the features ex-
+tracted by the detector backbone, as well as a text-based
+classification loss. Our experiments evidence the benefits of
+our approach, outperforming by 10% the only existing SDG
+object detection method, Single-DGOD [49], on their own
+diverse weather-driving benchmark.
+1. Introduction
+As for most machine learning models, the performance
+of object detectors degrades when the test data distribu-
+tion deviates from the training data one.
+Domain adap-
+tation techniques [3, 5, 8, 30, 41, 43] try to alleviate this
+problem by learning domain invariant features between a
+source and a known target domain. In practice, however,
+it is not always possible to obtain target data, even un-
+labeled, precluding the use of such techniques.
+Domain
+generalization tackles this by seeking to learn representa-
+tions that generalize to any target domain.
+While early
+approaches [1, 10, 25, 26, 28, 47, 57] focused on the sce-
+nario where multiple source domains are available during
+training, many recent methods tackle the more challenging,
+yet more realistic, case of Single Domain Generalization
+(SDG), aiming to learn to generalize from a single source
+dataset. While this has been well studied for image clas-
+sification [13, 35, 45, 48, 56], it remains a nascent topic in
+object detection. To the best of our knowledge, a single ex-
+isting approach, Single-DGOD [49], uses disentanglement
+and self-distillation [22] to learn domain-invariant features.
+In this paper, we introduce a fundamentally different ap-
+Figure 1. Semantic Augmentation: We compare the PCA pro-
+jections of CLIP [36] image embeddings obtained in two different
+manners: (Top) The embeddings were directly obtained from the
+real images from 5 domains corresponding to different weather
+conditions. (Bottom) The embeddings were obtained from the day
+images only and modified with our semantic augmentation strat-
+egy based on text prompts to reflect the other 4 domains. Note that
+the relative positions of the clusters in the bottom plot resembles
+that of the top one, showing that our augmentations let us gener-
+alize to different target domains. The principal components used
+are the same for both the figures.
+proach to SDG for object detection. To this end, we build
+on two observations: (i) Unsupervised/self-supervised pre-
+training facilitates the transfer of a model to new tasks [2,
+1
+arXiv:2301.05499v1  [cs.CV]  13 Jan 2023
+
+Imageday
+Imagenight
+Imagefoggy
+Image rainyday
+Image rainy night
+Imageday
+Semanticaugmentationnight
+Semantic augmentation foggy
+Semanticaugmentationrainy day
+Semantic augmentation rainy night4, 18]; (ii) Exploiting language supervision to train vision
+models allows them to generalize more easily to new cat-
+egories and concepts [9, 36]. Inspired by this, we there-
+fore propose to leverage a self-supervised vision-language
+model, CLIP [36], to guide the training of an object detec-
+tor so that it generalizes to unseen target domains. Since the
+visual CLIP representation has been jointly learned with the
+textual one, we transfer text-based domain variations to the
+image representation during training, thus increasing the di-
+versity of the source data.
+Specifically, we define textual prompts describing po-
+tential target domain concepts, such as weather and day-
+time variations for road scene understanding, and use these
+prompts to perform semantic augmentations of the images.
+These augmentations, however, are done in feature space,
+not in image space, which is facilitated by the joint image-
+text CLIP latent space. This is illustrated in Fig. 1, which
+shows that, even though we did not use any target data
+for semantic augmentation, the resulting augmented embed-
+dings reflect the distributions of the true image embeddings
+from different target domains.
+We show the effectiveness of our method on the SDG
+driving dataset of [49], which reflects a practical scenario
+where the training (source) images were captured on a
+clear day whereas the test (target) ones were acquired in
+rainy, foggy, night, and dusk conditions. Our experiments
+demonstrate the benefits of our approach over the Single-
+DGOD [49] one.
+To summarize our contributions, we employ a vision-
+language model to improve the generalizability of an object
+detector; during training, we introduce domain concepts via
+text-prompts to augment the diversity of the learned image
+features and make them more robust to an unseen target do-
+main. This enables us to achieve state-of-the-art results on
+the diverse weather SDG driving benchmark of [49].
+2. Related Work
+Domain Adaptation for Object Detection.
+Domain
+adaptation methods seek to align the source domain distri-
+bution to a particular target domain. To bridge the global
+and instance-level domain gaps, [3, 5, 41, 43] learn feature
+alignment via [15] adversarial training; [58] and [46] utilize
+category-level centroids and attention maps, respectively, to
+better align instances in the two domains; [8, 30] generate
+pseudo-labels in the target domain and use them for target-
+aware training. Domain adaptation, however, assumes that
+images from the target domain are available during training.
+In contrast, domain generalization aims to learn models that
+generalize to domains that were not seen at all during train-
+ing. Below, we focus on the domain generalization methods
+that, as us, use a single source domain to do so.
+Single Domain Generalization (SDG).
+Several image
+classification works [13,35,45,48,56] have proposed strate-
+gies to improve the performance on unseen domains while
+training on a single source domain. In particular, [35,45,48]
+introduce data augmentation strategies where diverse input
+images are generated via adversarial training; [13, 56] pro-
+pose normalization techniques to adapt the feature distri-
+bution to unseen domains. While SDG has been reason-
+ably well studied for image classification, the case of ob-
+ject detection remains largely unexplored, and poses addi-
+tional challenges related to the need to further localize the
+objects of interest. This was recently tackled by Single-
+DGOD [49] with an approach relying on learning domain-
+specific and domain-invariant features.
+Specifically, this
+was achieved by exploiting contrastive learning to disentan-
+gle the features and self-distillation [22] to further improve
+the network’s generalizability. Here, we introduce a fun-
+damentally different approach that leverages the CLIP [36]
+pre-trained model and semantically augments the data us-
+ing textual prompts. As will be shown by our results, our
+method outperforms the state-of-the-art Single-DGOD [49].
+Vision-Language Models.
+Jointly learning a representa-
+tion of images and text has been studied in many works [9,
+11,12,14,24,27,36,55]. They use image-text pairs to train
+visual-semantic embeddings which can be used not only
+for image classification, captioning or retrieval but also for
+zero-shot prediction on unseen labels. VirTex [9] relies on
+image-caption-based pre-training to learn a rich visual em-
+bedding from a small amount of data. CLIP [36] proposes a
+scalable contrastive pre-training method for joint text and
+image feature learning. CLIP leverages a corpus of 400
+million image-text pairs and a large language model [37] to
+learn a joint embedding space, which was shown to have su-
+perior zero-shot learning ability on classification tasks. The
+image-text-based training is also useful for Open Vocabu-
+lary Detection (OVD) [53], where the objects are detected
+using arbitrary textual descriptions. To address this task,
+[53] train their own visual-semantic representation, whereas
+[16, 39] employ CLIP embeddings. Recently, [29, 54] in-
+troduced a phrase-grounding-based pre-training for better
+OVD and zero-shot object detection. In contrast to these
+works, whose objective is to generalize to novel categories
+or objects, we seek to generalize to new domains depicting
+the same object categories as the source one.
+3. Method
+Let us now introduce our approach to exploiting a vision-
+language model for single-domain generalization in object
+detection. Below, we first present our semantic augmenta-
+tion strategy aiming to facilitate generalization to new do-
+mains. We then describe the architecture and training strat-
+egy for our object detector.
+2
+
+Figure 2. Our Approach: (Left) We first estimate a set of semantic augmentations A using a set of textual domain prompts {Pt, ps}
+and source domain images. The goal of these semantic augmentations is to translate source domain image embeddings to the domain
+specified by the prompts. We can do this because of the CLIP’s joint embedding space and its ability to encode semantic relationships via
+algebraic operations. Lopt is minimized w.r.t A over random image crops of the same size as CLIP [36]. (Right) The optimized semantic
+augmentations are used to train our modified detector which minimizes a text-based classification loss Lclip�t. Here, we train with the full
+image and add a randomly sampled Aj after average pooling. This pooling operation allows us to use A on extracted feature maps of the
+arbitrary-sized image. We initialize the detector with the pre-trained CLIP [36] V and T encoders to leverage their general representations.
+3.1. Semantic Augmentation
+In SDG, we have access to images from only a single
+domain. To enable generalization, we seek to learn object
+representations that are robust to domain shifts. Here, we
+do so by introducing such shifts while training the model
+on the source data. Specifically, we exploit CLIP’s joint
+representation to estimate shifts in the visual domain using
+textual prompts, as illustrated in Fig. 1. This corresponds to
+the optimization step shown in the left portion of Fig. 2.
+Formally, let T denote CLIP’s text encoder and V its im-
+age one. For reasons that will become clear later, we further
+split V into a feature extractor Va and a projector to the em-
+bedding space Vb. The CLIP [36] model is trained to bring
+image features closer to their textual captions. In essence,
+this means that, for an image I and a corresponding prompt
+p, it seeks to minimize the distance between Vb(Va(I)) and
+T (p).
+A useful property of the text embedding space is that
+algebraic operations can be used to estimate semantically
+related concepts. Word2Vec [31] had demonstrated such a
+learned relationship (e.g. king-man+woman approaches the
+word representation of queen). Such a relationship exists
+with CLIP embeddings as well [38].
+To exploit this for SDG, we define a generic textual
+prompt ps related to the source domain, such as An image
+taken during the day, and a set of prompts Pt =
+{pt
+j}M
+1
+encompassing variations that can be expected to
+occur in different target domains, e.g, describing different
+weather conditions or times of the day. Our objective then
+is to define augmentations {Aj} of the features extracted
+from a source image such that the shift incurred by Aj cor-
+responds to the semantic difference between ps and pt
+j.
+To achieve this, we first compute the embeddings qs =
+T (ps) and qt
+j = T (pt
+j) of the textual prompt. We then take
+multiple random crops from a source image. For each such
+crop Icrop, we create a target image embedding
+z∗
+j = z +
+qt
+j − qs
+∥qt
+j − qs∥2
+,
+(1)
+where z = V(Icrop). We then search for an augmentation
+Aj ∈ RH×W ×C such that
+¯zj = Vb(Va(Icrop) + Aj)
+(2)
+is as similar as possible to z∗
+j , which we measure with the
+cosine similarity. Ultimately, we estimate the augmenta-
+tions {Aj}M
+1
+through an optimization process using only
+source domain images. Specifically, we minimize the loss
+function
+Lopt =
+�
+Icrop
+�
+j
+D(z∗
+j , ¯zj) + ∥¯zj − z∥1 ,
+(3)
+where
+D(a, b) = 1 −
+a − b
+∥a − b∥2
+(4)
+is the cosine distance. The loss also includes an l1 regu-
+larizer that prevents the embeddings from deviating too far
+from their initial values, so as to preserve the image content.
+As the objective is to estimate the meaningful fea-
+ture augmentation while preserving the original CLIP pre-
+training, we keep the image crop size the same as the orig-
+inal CLIP training. Note that the optimization of the aug-
+mentations is done once in an offline stage, and we then use
+the resulting augmentations to train our detector.
+3
+
+Semantic Augmentations
+A = [A1,A2, .., AM]
+RPN
+va
++)
+ROI
+ROI
++
+vb
+Align
+head
+Avgpool
+2
+zj
+Random Crops
+A; = Sample(A)
+Lclip-t
+q
+qf
+K classes
+CLIP Init.
+Car
+ Source Domain prompt
+Bus
+CLIP Frozen
+a photo of
+Person
+Domain prompts
+pt
+pt =- (pi, p2,...PM]
+Random Init.
+Truck
+ CLIP Frozen
+Optimization Step
+Training StepFigure 3. Diverse Weather Dataset [49]: Day-Clear acts as our source domain while the other weather condition are our target domains.
+In these domains, the objects’ appearance drastically changes from the Day-Clear scenario. As we do not utilize any target domain images,
+learning generalizable features on source images is crucial for the SDG task.
+3.2. Architecture
+Let us now describe our detector architecture. As shown
+in the right portion of Fig. 2, it follows a standard Faster-
+RCNN [40] structure but departs from it in two ways. First,
+to exploit the augmentations optimized as discussed in the
+previous section, we initialize the blocks before and af-
+ter the ROI align one with the corresponding Va and Vb
+modules of the ResNet-based trained CLIP model. Second,
+to further leverage the vision-language model, we incorpo-
+rate a text-based classifier in our model’s head. Note that,
+in contrast to OVD [16, 39] where a text-based classifier
+is used to handle novel categories, we employ it to keep
+the image features close to the pre-trained joint embedding
+space.
+Specifically, we define textual prompts that represent the
+individual categories we seek to detect, and extract corre-
+sponding embeddings Q ∈ R(K+1)×Dclip, for K categories
+and the background class, using the text encoder T . For
+a candidate image region r proposed by the Region Pro-
+posal Network(RPN) [40], we then compute the cosine sim-
+ilarities between the text embeddings Q and the features
+Fr ∈ RDclip obtained by projection to the embedding space
+using Vb after ROI-Align [19] and the text embeddings Q.
+These cosine similarities, sim(Fr, Q) ∈ RK+1, act as log-
+its to the softmax based cross-entropy loss
+Lclip�t =
+�
+r
+LCE
+�
+esim(Fr,Qk)
+�K
+k=0 esim(Fr,Qk)
+�
+.
+(5)
+Similarly to [36], we formulate prompts of the form a
+photo of a {category name} to obtain our text
+embeddings.
+3.3. Training with Augmentation
+Following the standard detector training [40], we use the
+full image as our input. This subsequently increases the
+output feature map size of Va, hence we use average pool-
+ing operation and obtain channel-wise augmentations which
+can work for arbitrary-sized feature maps. The training of
+our modified object detector with the semantic augmenta-
+tions is as follows, first, we randomly sample an augmenta-
+tion Aj from the full set and collapse its spatial dimension
+using average pooling. We then add the resulting vector to
+every element in the feature map extracted by Va. In prac-
+tice, we apply augmentations to a batch with a probability
+θ.
+The detector is then trained with the loss
+Ldet = Lrpn + Lreg + Lclip�t ,
+(6)
+which combines the Lclip�t loss of Eq. (5) with the standard
+RPN and regression losses [40]. During inference, we use
+the detector without any augmentation of the feature maps.
+4. Experiments
+4.1. Experimental setup
+Datasets.
+To evaluate our model, we use the same
+datasets as [49]. They include five sets, each containing
+images with different weather conditions: daytime sunny,
+night clear, dusk rainy, night rainy, and daytime foggy.
+The images have been selected from three primary datasets,
+Berkeley Deep Drive 100K (BBD-100K) [52], Cityscapes
+[7] and Adverse-Weather [17]. Additionally, rainy images
+are rendered by [50], and some of the foggy images are syn-
+thetically generated from [42]. Our model is trained on the
+daytime sunny scenes, consisting of 19,395 training images,
+the remaining 8,313 daytime sunny images are used for val-
+idation and model selection. The four other weather condi-
+tions are only used during testing. They consist of 26,158
+images of clear night scenes, 3501 images of rainy scenes
+at dusk, 2494 images of rainy scenes at night, and 3775 im-
+ages of foggy scenes during daytime. All the datasets con-
+tain bounding box annotations for the objects bus, bike, car,
+motorbike, person, rider and truck. Fig. 3 shows examples
+from this dataset.
+Metric.
+In all our experiments, we use the Mean Average
+Precision (mAP) as our metric. Specifically, following [49],
+we report the [email protected], which considers a prediction as a
+true positive if it matches the ground-truth label and has an
+intersection over union (IOU) score of more than 0.5 with
+the ground-truth bounding box.
+4
+
+Day - Clear
+Day - Foggy
+Dusk-Rainy
+Night - Clear
+Night - RainyFigure 4. Qualitative Results. We visualize the predictions of the detectors trained only with day-clear images. (Top) FasterRCNN [40]
+predictions. (Bottom) The predictions with our approach. Night-Clear and Night-Rainy contain scenes that are taken under low light
+conditions. Due to this, the appearance of the object is obscure and deviates from the daytime case. FasterRCNN fails to detect most of
+the objects. As shown in the Night-Clear, it misclassifies a car to bus. By contrast, we can still detect car under such a big shift. For
+Dusk-Rainy scenes, the rain pattern on the windscreen and the wet ground causes an appearance shift. As shown FasterRCNN fails to
+detect several cars and misclassifies person on the bottom-left.
+Figure 5. Qualitative Results. In the foggy scenes, the objects
+further away w.r.t the camera are more obscure than the near ones.
+Due to this FasterRCNN (Top) struggles to detect them. car and
+person missed by FasterRCNN are successfully recovered by our
+approach (Bottom).
+4.2. Implementation Details
+We use the Detectron2 [51] implementation of Faster-
+RCNN with a ResNet101 [20] backbone. We initialize the
+detector with CLIP [36] pre-trained weights, where ResNet
+convolution blocks 1-3 act as Va, and block-4 along with
+the CLIP attention pooling act as Vb. This follows from the
+standard FasterRCNN implementation with ResNet back-
+bone.
+Optimization Step.
+As the benchmark dataset evalu-
+ates the method on different weather conditions, we cu-
+rated a list of domain prompts Pt matching the concept
+weather.
+To this end, we take all the hyponyms of the
+term weather from WordNet [44] and generate their text
+embeddings using the CLIP text encoder T .
+We prune
+away the words whose cosine similarity with the term
+weather is lower than 0.5. Additionally, we filter out the
+words that are not in the top 10k frequent words in GloVe
+wordlist [34]. After combining the synonyms, we get to
+a list of six words: snow, fog, cloudy, rain, stormy, sun-
+shine. We remove sunshine as it corresponds to our source
+domain concept.
+Furthermore, we consider three times
+of the day: day, night, evening.
+This lets us generate
+M = 15 prompts using the template an image taken
+on a {weather} {time of the day}. We use an
+image taken during the day as the source do-
+main prompt ps. We provide more details in our supple-
+mentary material.
+To optimize the augmentations with these prompts, we
+generated random crops from the source images and re-
+sized them to 224 × 224 pixels. The resulting output fea-
+ture map of Va and Aj are in R14×14×1024. We initial-
+ize Aj ∀ 1 ≥ j ≥ M with zeros and train it using the
+Adam [23] optimizer while keeping the CLIP encoder, V
+and T , frozen. Optimization was done for 1000 iterations
+with a learning rate of 0.01.
+Detector Training with Augmentation.
+When training
+the detector, the input image is resized to 600 × 1067 and V
+and T are initialized with CLIP pre-trained weights. While
+T is kept frozen during the training, the ResNet blocks 3-
+4 and attention pooling of V, along with the other Faster-
+RCNN learnable blocks, are trained with Stochastic Gra-
+dient Descent (SGD) for 100k iterations. We train with a
+learning rate of 1e−3, scaled down by a factor of 0.1 after
+40k iterations. We use a batch size of 4 and apply Aj to
+the features with probability θ = 0.5. We also use random
+5
+
+Night-Clear
+Dusk-RainyDay-Foggy
+Day-FoggymAP
+Method
+Day
+Clear
+Night
+Clear
+Dusk
+Rainy
+Night
+Rainy
+Day
+Foggy
+FR [40]
+48.1
+34.4
+26.0
+12.4
+32.0
+SW [33]
+50.6
+33.4
+26.3
+13.7
+30.8
+IBN-Net [32]
+49.7
+32.1
+26.1
+14.3
+29.6
+IterNorm [21]
+43.9
+29.6
+22.8
+12.6
+28.4
+ISW [6]
+51.3
+33.2
+25.9
+14.1
+31.8
+S-DGOD [49]
+56.1
+36.6
+28.2
+16.6
+33.5
+Ours
+51.3
+36.9
+32.3
+18.7
+38.5
+Table 1. Single domain generalization results. We show consis-
+tent improvements across all the target domains. S-DGOD boosts
+the source domain results, but at the cost of reduced generalization
+ability. By contrast, our approach is robust to domain changes.
+The numbers for S-DGOD, SW, IBN-Net, IterNorm, ISW are
+taken from [49].
+horizontal flipping augmentation as in Single-DGOD [49].
+Dclip is set to 512 as in [36] and background class is initial-
+ized by zeros in Q. All of our training was done on a single
+NVIDIA A100 GPU. Our code will be made public upon
+acceptance.
+4.3. Comparison with the State of the Art
+We compare our method trained with semantic augmen-
+tations against the state-of-the-art Single-DGOD [49]. Sim-
+ilar to them, we also show comparisons with feature nor-
+malization methods, SW [33], IBN-Net [32], IterNorm [21],
+and ISW [6]. These methods improve network generaliza-
+tion by using better feature normalization. We addition-
+ally report the performance of FasterRCNN (FR) initialized
+with ImageNet pre-trained weights. For the SDG task, we
+evaluate the generalization performance on unseen target
+domains, hence we compare the mAP scores on the out-
+of-domain datasets: day-foggy, night-rainy, dusk-rainy, and
+night-clear.
+Our approach of combining CLIP pre-training and se-
+mantic augmentation outperforms the baselines on all of the
+target domains. Tab. 1 shows a consistent improvement in
+all domains with close to 15% improvement on day-foggy
+and dusk-rainy compared to Single-DGOD. In the challeng-
+ing scenario with Night conditions, we improve by 12.6%
+on night-rainy while being comparable with Single-DGOD
+on night-clear. On the source domain, both our method and
+Single-DGOD are better than the FR baseline. However,
+while Single-DGOD gains improvement at the cost of los-
+AP
+mAP
+Method
+Bus Bike Car Motor Person Rider Truck
+All
+FR [40] 28.1 29.7 49.7
+26.3
+33.2
+35.5
+21.5
+32.0
+S-DGOD [49] 32.9 28.0 48.8
+29.8
+32.5
+38.2
+24.1
+33.5
+Ours 36.1 34.3 58.0
+33.1
+39.0
+43.9
+25.1
+38.5
+Table 2. Per-class results on Daytime Clear to Day Foggy. Our
+method consistently performs better on all categories for the dif-
+ficult foggy domain. This shows that CLIP initialization and our
+semantic augmentations improve the detector’s generalizability.
+AP
+mAP
+Method
+Bus Bike Car Motor Person Rider Truck
+All
+FR [40] 28.5 20.3 58.2
+6.5
+23.4
+11.3
+33.9
+26.0
+S-DGOD [49] 37.1 19.6 50.9
+13.4
+19.7
+16.3
+40.7
+28.2
+Ours 37.8 22.8 60.7
+16.8
+26.8
+18.7
+42.4
+32.3
+Table 3. Per-class results on Daytime Clear to Dusk Rainy.
+Our approach generalizes to rainy road conditions along with the
+low light conditions of the dusk hours. The car category sees the
+biggest improvement, but we nonetheless also boost the perfor-
+mance of all the other classes.
+ing out for domain generalization, we improve on both the
+source and target domains. The failure of feature normal-
+ization baselines suggests a large domain gap between the
+source and target domains. Fig. 4 and Fig. 5 provide a qual-
+itative results on different weather-datasets.
+In the remainder of this section, we discuss the per-class
+results on the individual target domains.
+Daytime Clear to Day Foggy.
+The object appearance
+drastically changes in the foggy images compared to the
+day-clear scenario. As shown in Tab. 2, our method brings
+in a large improvement for the car, person, and bike cat-
+egories, while still being consistently better than Single-
+DGOD and FR on the others.
+Daytime Clear to Dusk Rainy.
+Dusk Rainy scenes re-
+flect a low light condition and along with the rainy pat-
+tern.
+The image distribution is thus further away from
+the daytime clear images.
+As shown in Tab. 3, our
+method improves the AP of each class, with the biggest
+improvement in the car and person categories. Since we
+leverage CLIP pre-training and bring in concepts such as
+rain/cloudy/stormy and evening/night hours through our se-
+mantic augmentation, the learnt detector generalizes better.
+6
+
+AP
+mAP
+Method
+Bus Bike Car Motor Person Rider Truck
+All
+FR [40] 34.7 32.0 56.6
+13.6
+37.4
+27.6
+38.6
+34.4
+S-DGOD [49] 40.6 35.1 50.7
+19.7
+34.7
+32.1
+43.4
+36.6
+Ours 37.7 34.3 58.0
+19.2
+37.6
+28.5
+42.9
+36.9
+Table 4. Per-class results on Daytime Clear to Night Clear.
+While being comparable to S-DGOD on most of the categories,
+we improve on car and person.
+AP
+mAP
+Method
+Bus Bike Car Motor Person Rider Truck
+All
+FR [40] 16.8
+6.9
+26.3
+0.6
+11.6
+9.4
+15.4
+12.4
+S-DGOD [49] 24.4 11.6 29.5
+9.8
+10.5
+11.4
+19.2
+16.6
+Ours 28.6 12.1 36.1
+9.2
+12.3
+9.6
+22.9
+18.7
+Table 5. Per-class results on Daytime Clear to Night Rainy.
+This dataset presents the most challenging scenario, where the low
+light and rainy conditions obscure the objects. We still perform
+better than the baseline on most of the categories.
+Daytime Clear to Night Clear.
+The Night Clear dataset
+shows a challenging night driving scene under severe low-
+light conditions. In Tab. 4, we show that while being com-
+parable to Single-DGOD, we bring in a larger improvement
+in the car and person categories. Night scenes are partic-
+ularly challenging as the low light condition leads to more
+confusion among visually closer categories such as bus and
+truck.
+Daytime Clear to Night Rainy.
+This is the most chal-
+lenging scenario where dark night conditions are exacer-
+bated by patterns occurring due to rain. Tab. 5 shows consis-
+tent improvement by our approach for most of the classes.
+The car class sees the biggest improvement with an increase
+in AP of more than 22% compared to Single-DGOD. The
+lower performance of the class rider can be attributed to an
+increase in the confusion between the visually similar per-
+son and rider classes under adverse conditions.
+4.4. Ablation Study
+To understand how each element of the proposed method
+contributes to the overall performance, we conduct an ab-
+lation study.
+We test five individual components of our
+model. Specifically, we remove semantic augmentation, re-
+place CLIP attention pooling in Vb with average pooling,
+replace Lclip�t with the FasterRCNN classification loss, and
+change the weight initialization from the CLIP model to
+an ImageNet classification model.
+Removing those five
+components turns our model back into the standard Faster-
+RCNN. The ablation study results are provided in Tab. 6
+and discussed below.
+CLIP initialization.
+When the FasterRCNN backbone
+V is initialized with CLIP pre-trained weights, the model
+performance consistently increases both in the in-domain
+and out-of-domain scenarios, as shown in the second row
+of Tab. 6. This setting itself already outperforms Single-
+DGOD (penultimate row of Tab. 1). This goes to show that,
+for the generalization task, model weight initialization plays
+a crucial role. We further improve this performance with se-
+mantic augmentations.
+Attention pooling and Lclip�t.
+Next we test the impact
+of the text-embedding-based loss Lclip�t for classification.
+As visible in the third row of Tab. 6, when combined with
+CLIP initialization, it improves the generalization perfor-
+mance for the rainy scenarios, but degrades it for the other
+ones. Replacing average pooling in Vb with CLIP attention
+pooling helps to mitigate the detrimental effect of Lclip�t
+and exhibits consistent improvement on all datasets.
+Semantic augmentation.
+Finally, adding semantic aug-
+mentation gives us the best results, as shown in the last row
+of Tab. 6. Exposing the visual encoder V to targeted seman-
+tic augmentations helps the overall model to better gener-
+alize when exposed to new domains sharing similarity with
+the augmentations.
+4.5. Additional Analyses
+Study of semantic augmentation.
+Our proposed method
+involves translating feature maps by semantic augmenta-
+tions learned using plausible domain prompts. To further
+study the utility of our approach, we replace the augmen-
+tation strategy in our training pipeline with (a) no-aug: no
+augmentation; (b) random: A is initialized with a normal
+distribution; (c) clip-random: we define Pt with concepts
+that are not specific to weather. We generate prompts with
+a template an image of {word}, where the words are
+desert, ocean, forest, and mountain. Tab. 7 illustrates the
+importance of the semantics in our augmentation strategy.
+The random augmentation performs worse than the no-aug
+strategy. clip-random is comparable to no-aug and doesn’t
+show any consistent trend but is mostly better than random.
+Our semantic augmentation strategy provides a consistent
+improvement over no-aug because the translations are per-
+formed with prompts from the relevant weather concept.
+7
+
+Model Component
+mAP
+Source
+Target
+CLIP init
+Lclip�t
+Attn. Pool
+Sem. Aug
+Day
+Clear
+Night
+Clear
+Dusk
+Rainy
+Night
+Rainy
+Day
+Foggy
+48.1
+34.4
+26.0
+12.4
+32.0
+✓
+51.2
+37.0
+31.0
+15.7
+37.5
+✓
+✓
+50.7
+36.0
+31.3
+16.3
+36.9
+✓
+✓
+✓
+51.0
+35.9
+31.3
+16.7
+37.7
+✓
+✓
+✓
+✓
+51.3
+36.9
+32.3
+18.7
+38.5
+Table 6. Ablation study. We study the influence of five different components of our approach: the backbone weight initialization strategy,
+the classification loss, the attention pooling, and the semantic augmentation. When those five components are removed (first row of the
+table) the model is equivalent to the standard FasterRCNN. Initializing the detector with CLIP weights (second row) largely improves the
+generalization performance; on its own it already outperforms Single-DGOD (penultimate row of Tab. 1) on most of the datasets, hence
+suggesting that CLIP has better generalizability than ImageNet pre-trained weights. Combining this with the text embedding-based loss
+Lclip�t (third row) improves the results on the challenging scenarios of dusk rainy and night rainy, but has a detrimental effect for the other
+weather conditions. Adding attention pooling to the architecture (fourth row) helps to mitigate these detrimental effects as it brings the
+visual features closer to the joint embedding space. Finally, the best results are obtained when the semantic augmentation is added (last
+row), greatly helping with adverse weather, rainy and foggy, scenarios.
+mAP
+Aug. Type
+Day
+Clear
+Night
+Clear
+Dusk
+Rainy
+Night
+Rainy
+Day
+Foggy
+no-aug.
+51.0
+35.9
+31.3
+16.7
+37.7
+random
+51.2
+36.0
+30.4
+15.3
+37.3
+clip-random
+51.5
+36.4
+30.2
+15.9
+37.9
+Ours w/ seg.aug
+51.3
+36.9
+32.3
+18.7
+38.5
+Table 7. Semantic Augmentation. Our semantic augmentation
+consistently outperforms other augmentation strategies.
+While
+random augmentations are worse than no-aug., clip-random is
+comparable to no-aug.. Only when we give relevant prompts, there
+is a consistent improvement across datasets.
+5. Limitations
+Our method augments visual features using textual
+prompts. To generate these prompts, it is assumed that some
+information about the domain gap is known. In our experi-
+ments, we assumed that the domain gap was due to changes
+in weather and daytime conditions. In practice, we only
+used the word weather and time of the day to derive all the
+prompts used in our augmentation; nonetheless, some extra
+information was used. In most applications, however, the
+domain gap can be known in advance, and providing a few
+keywords characterizing it shouldn’t be an issue. In the rare
+cases where no information can be known, our approach
+still has the potential to be used by using multiple broad
+concept keywords such as weather, ambiance, or location.
+6. Conclusion
+We have proposed an approach to improving the gener-
+alization of object detectors on unseen target domains. Our
+approach fundamentally departs from existing method by
+leveraging a pre-trained vision-language model, CLIP, to
+help the detector to generalize. Specifically, we have ex-
+ploited textual prompts to develop a semantic augmentation
+strategy that alters image embeddings so that they reflect
+potential target domains, and to design a text-based image
+classifier. We have shown that our approach outperforms
+the state of the art on four adverse-weather target datasets.
+In future work, we plan to extend our approach to learning
+the prompts to further improve generalization.
+8
+
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+

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+arXiv:2301.03062v1  [cs.LG]  8 Jan 2023
+AnycostFL: Efficient On-Demand Federated
+Learning over Heterogeneous Edge Devices
+Peichun Li∗,†, Guoliang Cheng∗, Xumin Huang∗,†, Jiawen Kang∗, Rong Yu∗, Yuan Wu†, and Miao Pan‡
+∗School of Automation, Guangdong University of Technology, Guangzhou, China
+†State Key Laboratory of Internet of Things for Smart City, University of Macau, Macau, China
+‡Department of Electrical and Computer Engineering, University of Houston, Houston, USA
+Email: [email protected], guoliang [email protected], huangxu [email protected],
+{kavinkang, yurong}@gdut.edu.cn, [email protected], [email protected]
+Abstract—In this work, we investigate the challenging prob-
+lem of on-demand federated learning (FL) over heterogeneous
+edge devices with diverse resource constraints. We propose a
+cost-adjustable FL framework, named AnycostFL, that enables
+diverse edge devices to efficiently perform local updates under
+a wide range of efficiency constraints. To this end, we design
+the model shrinking to support local model training with elastic
+computation cost, and the gradient compression to allow param-
+eter transmission with dynamic communication overhead. An
+enhanced parameter aggregation is conducted in an element-wise
+manner to improve the model performance. Focusing on Any-
+costFL, we further propose an optimization design to minimize
+the global training loss with personalized latency and energy
+constraints. By revealing the theoretical insights of the conver-
+gence analysis, personalized training strategies are deduced for
+different devices to match their locally available resources. Ex-
+periment results indicate that, when compared to the state-of-the-
+art efficient FL algorithms, our learning framework can reduce
+up to 1.9 times of the training latency and energy consumption
+for realizing a reasonable global testing accuracy. Moreover,
+the results also demonstrate that, our approach significantly
+improves the converged global accuracy.
+Index Terms—Federated learning, edge intelligence, mobile
+computing, resource management.
+I. INTRODUCTION
+Federated learning (FL) is an emerging distributed learning
+paradigm that enables multiple edge devices to train a common
+global model without sharing individual data [1]. This privacy-
+friendly data analytics technique over massive devices is envi-
+sioned as a promising solution to realize pervasive intelligence
+[2]. However, in many real-world application areas, mobile de-
+vices are often equipped with different local resources, which
+raises the emerging challenges for locally on-demand training
+[3]. Given different local resources status (e.g., computing
+capability and communication channel state) and personalized
+efficiency constraints (e.g., latency and energy), it is crucial to
+customize training strategies for heterogeneous edge devices.
+We perform an in-depth analysis on the time delay and the
+energy consumption for performing the local model updates at
+edge devices. Specifically, we evaluate and record the cost of
+local training on three different NVIDIA Jetson family plat-
+forms (i.e., Nano, NX AGX, and Xavier AGX) under different
+channel states (i.e., good, medium, and poor). On the one hand,
+we observe that the learning efficiency differs significantly
+Xavier-Good
+NX-Medium
+Nano-Poor
+0
+1
+2
+3
+4
+5
+6
+7
+8
+time delay (in second) of single-round local update
+local model training
+parameter transmission
+4.0 times
+0.7 times
+Xavier-Good
+NX-Medium
+Nano-Poor
+0
+2
+4
+6
+8
+10
+energy consumption (in joule) of single-round local update
+local model training
+parameter transmission
+Fig. 1.
+The time delay (top) and energy consumption (bottom) of single-
+round local update on different hardware platforms with varying communica-
+tion conditions.
+with diverse learning scenarios. As shown in Fig. 1, the single-
+epoch training on Nano with poor communication condition
+consumes about 4.0 times training latency than that of Xavier
+AGX with good communication condition, while its energy
+consumption is about 0.7 times less than the latter one’s. On
+the other hand, we observe that the bottlenecks of latency and
+energy are induced by parameter transmission and local model
+training, respectively.
+The above observations provide insights for a proper design
+of the on-demand FL system. To handle the resource hetero-
+geneity, it is suggested to alleviate the energy and the latency
+cost of the local device. More importantly, the computation
+and communication costs should be jointly reduced to achieve
+efficient local training. In the literature, most existing studies
+either employ resource allocation and device scheduling to
+mitigate the system cost [4]–[10], or design gradient com-
+pression to accelerate the parameter transmission procedure
+[11]–[17]. The former method inherits the ideas of traditional
+design for mobile edge systems and takes no account of the
+optimization for neural networks, while the latter overlooks
+the computation cost of local model training.
+In this paper, we propose “anycost” FL, named AnycostFL,
+to break the latency and energy bottlenecks for on-demand
+distributed training over heterogeneous edge devices. Our goal
+is to develop a cost-adjustable FL framework that enables
+edge devices to perform local updates under diverse learning
+scenarios. To this end, we first design the model shrinking and
+
+gradient compression to enable adaptive local updates with
+different computation and communication costs. Meanwhile,
+an enhanced parameter aggregation scheme is proposed to
+fuse the knowledge of the local updates. Following that,
+we investigate the on-demand learning of AnycostFL by
+regulating the local model structure, gradient compression
+policy and computing frequency under personalized latency
+and energy constraints. However, customizing training strategy
+for different learning scenarios is a non-trivial task, since how
+the global accuracy is affected by the local model structure and
+compression rate is still unknown. To address this issue, we
+theoretically reveal the convergence insights of our framework,
+which are further leveraged to guide optimization analysis.
+Finally, the optimal training strategy is derived for each device
+according to its locally available resource.
+Our main contributions are summarized as follows.
+• We propose a novel FL framework, named AnycostFL,
+that enables the local updates with elastic computation
+cost and communication overhead.
+• We theoretically present the optimal aggregation scheme
+and convergence analysis for AnycostFL.
+• We investigate the on-demand training problem of Any-
+costFL, and the optimal training strategy is devised to
+adapt the locally available resource.
+• Extensive experiments indicate that the proposed Any-
+costFL outperforms the state-of-the-art efficient FL meth-
+ods in terms of resource utilization and learning accuracy.
+The remainder of this paper is organized as follows. Section
+II describes related studies. In Section III, we detail the main
+operations of AnycostFL to fulfill the single-round training.
+The problem formulation, theoretical analysis and the corre-
+sponding solution are provided in Section IV. The experiment
+evaluations are presented in Section V, and we finally conclude
+the paper in Section VI and discuss the future directions.
+II. RELATED WORK
+Resource Management Methods. Resource management
+methods aim to reduce the FL system cost by arranging
+the local and system resources. Resource allocation methods
+employ frequency scheduling [18], transmission power control
+[19], and bandwidth allocation [20] to balance the cost of local
+training. Recent device selection methods directly exclude
+those weak devices with poor computation or communication
+capabilities to accelerate the convergence time [21]–[23].
+Besides, topology-aware management is another very effective
+method to mitigate the network throughput [18], [24], [25].
+However, these methods inherit the ideas of the efficient design
+for traditional mobile systems and overlook the optimization
+of neural networks.
+Neuron-aware Techniques. Neuron-aware techniques focus
+on revealing the black box of neural networks to improve the
+training efficiency of the FL system. Early gradient compres-
+sion utilizes sparsification [11], [26], and quantization [14],
+[27], [28] to reduce the transmission cost of FL system. In
+addition, feature maps fusion and knowledge distillation can be
+carried out to improve the information aggregation [29], [30].
+Besides, FedMask proposes to train a personalized mask for
+each device to improve the test accuracy on the local dataset
+[31]. Recently, model structure pruning enables multiple de-
+vices with different model architectures to train a shared global
+model [32], [33]. Such methods can reduce the cost of local
+training, but how to customize optimal training strategies (e.g.,
+gradient compression and model pruning policy) for different
+learning scenarios is still unknown.
+III. TRAINING WITH ANYCOSTFL
+In this section, we first outline the overall design of Any-
+costFL. Next, we detail the key techniques of our framework,
+including elastic model shrinking (EMS), flexible gradient
+compression (FGC), and all-in-one aggregation (AIO).
+A. Outline of AnycostFL
+We consider a generic application scenario of FL with a set
+of I edge devices I = {1, 2, · · · , I}. We use Di to denote the
+local training data of the device i, and D = ∪I
+i=1Di indicates
+the global data. Let Fi(w) = ℓ(w, Di) represent the local
+training loss of device i with respect to model weight w, where
+ℓ(·, ·) is the predetermined loss function. The objective of the
+FL system is to minimize the following global loss function
+F(w)
+∆=
+I
+�
+i=1
+|Di|
+|D| Fi(w),
+(1)
+where |Di| is the size of Di. Given the specified learning task,
+the original training workload of single sample W and the data
+size of uncompressed gradient S can be empirically measured.
+As shown in Fig. 2(a), to reduce the computational com-
+plexity of the local model training and the communication cost
+of gradient update transmission, we propose AnycostFL with
+two device-side techniques, i.e., model shrinking and gradient
+compression. At the t-th global iteration of AnycostFL, the
+device i is enabled to adjust its training workload and gradient
+size as Wt,i = αt,iW and St,i = βt,iS, respectively. Here,
+αt,i ∈ (0, 1] and βt,i ∈ (0, 1] are defined as the model shrink-
+ing factor and the gradient compression rate, respectively. The
+training procedure of AnycostFL is summarized as follows.
+1) Elastic local training: At the t-th global round, the
+device i downloads the latest global model wt from the pa-
+rameter server. With the pre-calculated model shrinking factor
+αt,i, the specialized sub-model wα
+t,i = shrink(wt, αt,i) can
+be efficiently derived, where function shrink(·, ·) indicates
+the operations for model shrinking. Then, the local training
+is conducted with sub-model wα
+t,i and local data Di, and the
+updated local sub-model wα
+t+1,i is obtained. Furthermore, the
+local gradient update can be acquired as ut,i = wα
+t,i −wα
+t+1,i.
+2) Flexible gradient upload: To further reduce the uplink
+traffic, the local device i is motivated to compress the gradient
+update ut,i before the parameter transmission. With the given
+compression rate βt,i, the compressed gradient update ˜ut,i =
+cmprs(ut,i, βt,i) is uploaded to the server, where cmprs(·, ·)
+is the function for gradient compression.
+
+computation capacity
+communication capacity
+device C
+device A
+device B
+local data
+compressed update
+comp. capacity
+comm. capacity
+the neural structure becomes larger
+the data size of local update becomes larger
+C
+A
+B
+param. server
+global model
+aggregation
+model distribution
+param. upload
+power
+the size of each hidden layer is reduced by half, and 
+the training complexity is reduced by ¼ approximately.
+. . . 
+1 0 0 0 0 0 1 0
+0 0 0 1 0 1 0 0
+0 0 1 1 0 0 0 0
+… … … … … … … …
+. . .
+sparsification
+binary mask
+quantization
+entropy 
+encoding
+Golomb 
+encoding
+compressed 
+update
+an example of gradient compression (single layer)
+(b) optimization for the training strategy
+(c) model shrinking & gradient compression 
+(a) outline of AnycostFL
+…
+…
+global model
+sub-model
+16
+32
+64
+8
+16
+32
+size: 16x8x3x3
+encoding
+an example of model shrinking
+Fig. 2. left: AnycostFL over heterogeneous edge devices. middle: the neural structure and gradient compression strategies are customized for diverse devices
+according to their locally available resources; the darker color indicates the higher computing complexity for training and the larger marker size denotes the
+larger data size of the local update. right: illustrations of the model shrinking for the local model and the gradient compression for the local update.
+3) Parameter aggregation: The server collects the com-
+pressed local updates {˜ut,i}∀i with different shrinking factors
+{αt,i}∀i and compression rates {βt,i}∀i. After that, the global
+update is calculated by ˜ut
+= aioagg({˜ut,i}∀i), where
+aioagg(·) is the server-side all-in-one aggregation. Then, the
+updated global model is computed as wt+1 = wt − ˜ut.
+After the T -round training of the above three-step iterations,
+the final global model wT is obtained. Before introducing how
+to customize the values of {αt,i}∀i and {βt,i}∀i in Section
+IV, we illustrate the details of model shrinking, gradient
+compression and update aggregation in the rest of this section.
+B. Elastic Model Shrinking
+We aim to derive the sub-model wα
+t,i with training complex-
+ity of αt,iW from global model wt by reducing the width of
+the global model. The shrinking operations work as follows.
+1) Server-side channel sorting: To avoid incurring extra
+memory cost for the edge devices, the server first sorts
+the channels of the latest global model before the model
+distribution. Given one layer of the weight of the global
+model, the server sorts the output channels in the current
+layer in descending order according to their values of L2
+norm, and meanwhile, the input channels of the next layer
+should be sorted accordingly in the same order to maintain
+the permutation invariance of the whole model [34].
+2) Layer-wise uniform shrinking: Next, the server broad-
+casts the weight of each layer of the global model in a channel-
+by-channel manner. Instead of downloading the full global
+model, each device only receives those important parameters
+from the global model to assemble the local sub-model. Here,
+we utilize the fixed shrinking ratio for each layer in the same
+sub-model. Empirically, given model shrinking factor αt,i, we
+can reduce the size of the hidden layer by √αt,i to acquire
+the sub-model. For example, as shown in Fig. 2(c), when
+shrinking a global model with hidden sizes of {16, 32, 64}
+under αt,i =
+1
+4, we approximately reduce the size of each
+hidden layer by half as {8, 16, 32} to form the sub-model.
+At the beginning of the t-th global round, all device ini-
+tialize their local sub-models {wα
+t,i}∀i by choosing the most
+important channels from the global model wt. In this way, the
+training complexity is significantly reduced while maintaining
+the performance of local sub-models. After that, the local
+training of device k is conducted with sub-model wα
+t,i, which
+produces the local gradient ut,i with data size of αt,iS.
+C. Flexible Gradient Compression
+Given the local update ut,i with the desired compression
+rate βt,i, we aim to obtain the compressed update ˜ut,i with
+data size of αt,iβt,iS. Let ρt,i and Lt,i denote the sparsity
+rate and the number of quantization levels, respectively. The
+gradient compression scheme works as follows.
+1) Kernel-wise sparsification: Without loss of generality,
+we take the convolution neural network (CNN) as an example
+to illustrate the sparsification procedure. We aim to acquire the
+sparse update ˆut,i from ut,i. Let ut,i[k] denote the k-th kernel
+of ut,i, and ut,i = {ut,i[k]}∀k. We measure the importance
+of each kernel and obtain N = {∥ut,i[k]∥2}∀k, where ∥ · ∥2
+denotes the L2 norm operation. Next, by selecting the ⌈ρt,iK⌉-
+th largest value in N as the threshold Π, the kernel-wise
+sparsification is expressed as
+ˆut,i[k] =
+�
+0
+if ∥ut,i[k]∥2 < Π,
+ut,i[k]
+otherwise.
+(2)
+Meanwhile, the binary mask of ˆut,i is denoted as mt,i.
+2) Probabilistic quantization: Motivated by the studies
+in [35], [36], we aim to obtain the quantized update ˜ut,i
+with the given sparse ˆut,i and the quantization level Lt,i.
+Let u ∈ ˆut,i be a scalar value. To begin with, we first
+calculate the magnitude range of the non-zero elements of
+ˆut,i, denoted as [umin, umax], where umin = min{|u|}∀u̸=0,
+and umax = max{|u|}∀u̸=0. Next, let Q = {Ql}Lt,i
+l=1 denote
+the set of quantization points, where Ql is computed by
+Ql = l (umax − umin)
+Lt,i
++ umin.
+(3)
+
+1
+1
+1
+2
+2
+3
+2 1
+1
+1
+3
+1
+2
+3
+1
+1
+2
+1
+1
+1
+2
+2
+3
+1
+2
+1
+2
+model structure 
+update of each layer 
+2 2
+2
+2
+2 2
+2
+2
+2
+2
+2 2 2
+2 2
+3 3
+3
+3 3
+3
+3 3
+3 3
+1 1 1
+1 1
+1 1
+1 1
+1
+1
+1
+1
+1
+1 1
+1 1
+1
+1
+local compressed updates
+global update
+legend
+normal update
+zero update
+1
+2
+3
+1
+2
+1
+3
+2
+3
+not existing
+zero update
+update by device 1
+update by all devices
+update by devices 1&2
+update by device 3
+update by devices 1&3
+update by devices 2&3
+update by device 2
+�ut,1
+�ut,2
+�ut,3
+�ut
+Fig. 3. An illustration of the all-in-one aggregation.
+For any u ∈ ˆut,i and u ̸= 0, we can always find a quantization
+interval [Ql, Ql+1] such that Ql ≤ |u| ≤ Ql+1, and its
+corresponding quantized value ˜u is further computed by
+˜u =
+�
+sgn(u) · Ql
+with probability Ql+1−|u|
+Ql+1−Ql ,
+sgn(u) · Ql+1
+otherwise,
+(4)
+where sgn(·) calculates the sign of the given scalar. Fur-
+thermore, the set of the quantization indices of all ˜u ∈ ˜ut,i
+is denoted as Lt,i = {l, Ql = ˜u}∀˜u̸=0. Now, ˜ut,i can be
+represented by a tuple of {umin, umax, Lt,i, mt,i, Lt,i}.
+3) Lossless encoding: Due to the distribution characteristics
+of Lt,i that smaller indices may occur more frequently, we
+apply entropy coding to reduce the data size [14], [37].
+Besides, the sparse binary matrix mt,i can be compressed by
+Golomb encoding [11], [38].
+After determining the compression scheme, we can vary
+the combinations of {ρt,i, Lt,i} and record the corresponding
+compression rates. Based on the results, we can build a
+piecewise linear function to predict the compression strategy
+{ρt,i, Lt,i} with the given βt,i. Notably, this function can be
+efficiently fitted by the server with a rather small amount of
+public training data (e.g., 16 samples) in an offline manner.
+D. All-in-One Aggregation
+After all the devices upload their encoded updates, the
+server receives, decodes and then reconstructs the compressed
+local updates {˜ut,i}∀i. Our goal is to obtain the global
+update ˜ut by aggregating {˜ut,i}∀i. However, the aggregation
+of local updates in our framework cannot be supported by
+conventional FedAvg [1], since the local updates are produced
+by different model structures with different levels of precision
+(i.e., different quantization levels and sparsity).
+To tackle the above challenge, we propose an all-in-one
+aggregation scheme that fuses the local updates in an element-
+wise manner. Let the set {1, 2, · · · , J} index elements of the
+global update ˜ut, and ˜u[j]
+t
+denote the j-th element of ˜ut. To
+accomplish the aggregation for ˜u[j]
+t , we first determine the
+subset of devices Ij ⊆ I whose local model structure also
+contains the j-th element. Then, we have
+˜u[j]
+t
+=
+
+
+
+
+
+
+
+0
+if �
+i∈Ij
+m[j]
+t,i = 0,
+1
+�
+i∈Ij
+pt,im[j]
+t,i
+�
+i∈Ij
+pt,im[j]
+t,iu[j]
+t,i
+otherwise,
+(5)
+where pt,i is the aggregation coefficient for the j-th device at
+the t-th global round. The optimal values of {pt,i}∀i will be
+further analyzed in Section IV. Fig. 3 gives an example to illus-
+trate the aggregation details. Specifically, different elements in
+the global update are updated by different subsets of devices,
+and more important elements will “absorb” knowledge from
+more devices. When the j-th element is zeroed out by all the
+devices in Ij, we have ˜u[j]
+t
+= 0.
+IV. THEORETICAL ANALYSIS AND OPTIMIZATION
+In this section, we focus on the optimization of our frame-
+work by customizing the training strategies for diverse devices.
+We first formulate the on-demand training problem of Any-
+costFL. Then, we derive the upper bound of the convergence
+rate and reveal the key insights to improve the performance of
+AnycostFL. Based on the analysis, the optimization problem is
+transformed into a tractable form, and the closed-form solution
+is derived.
+A. AnycostFL over Wireless Networks
+In this subsection, we formulate the computation and com-
+munication models for our framework. After that, we build
+up an on-demand learning problem that minimizes the global
+training loss with given delay and energy constraints.
+1) Computation model: For the device i at the t-th global
+round, given the model shrinking factor αt,i and computing
+frequency ft,i, the time consumption of local model training
+can be measured by
+T cmp
+t,i
+= τ|Di|αt,iW
+ft,i
+,
+(6)
+where τ denotes the number of local epochs. Meanwhile, the
+corresponding energy consumption can be given by
+Ecmp
+t,i = ǫif 2
+t,iτ|Di|αt,iW,
+(7)
+where ǫi is the hardware energy coefficient of the device i.
+2) Communication model: We consider the frequency divi-
+sion multiple access (FDMA) scheme for the transmission of
+the local gradient update. For the device i at the t-th global
+round, the achievable transmitting rate can be estimated by
+rt,i = bilog2
+�
+1 + |ht,i|P com
+t,i
+N0bi
+�
+,
+(8)
+where P com
+t,i
+is the transmitting power; bi is the achievable
+bandwidth; |ht,i| denotes the path loss of wireless channel; N0
+is the power spectral density of the additive white Gaussian
+noise. For the device i at t-th global round, given the update
+˜ut,i generated by the local model with a shrinking factor
+of αt,i and compression rate of βt,i, the required time T com
+t,i
+and energy consumption Ecom
+t,i
+of uplink transmission can be
+respectively measured by
+T com
+t,i
+= αt,iβt,iS
+rt,i
+, and Ecom
+t,i
+= T com
+t,i P com
+t,i .
+(9)
+With the above computation and communication models,
+we next focus on the optimization problem of AnycostFL.
+
+3) Problem formulation: To optimize AnycostFL, we study
+an on-demand training problem. Specifically, the shared max-
+imal latency for each round T max is determined by the server.
+The local energy consumption budget for each round Emax
+t,i
+is
+customized by the device itself. Given multiple devices with
+diverse local resources (e.g., computation, communication and
+data), our goal is to customize the training strategy for each
+device to minimize the global training loss with personalized
+constraints (e.g., latency and energy). To sum up, at the t-th
+global round, we aim to optimize the following problem.
+(P1)
+min F
+�
+wt; {αt,i}∀i, {βt,i}∀i
+�
+(10)
+subject to:
+T cmp
+t,i + T com
+t,i
+≤ T max, ∀i,
+(10a)
+Ecmp
+t,i + Ecom
+t,i ≤ Emax
+t,i , ∀i,
+(10b)
+αmin ≤ αt,i ≤ 1, ∀i,
+(10c)
+0 ≤ βt,i ≤ βmax, ∀i,
+(10d)
+f min
+i
+≤ ft,i ≤ f max
+i
+, ∀i,
+(10e)
+variables:
+{αt,i, βt,i, ft,i}∀i,
+where F
+�
+wt; {αt,i}∀i, {βt,i}∀i
+�
+denotes the global loss of the
+t-th round with given the global model weight wt under the
+training strategies of {αt,i}∀i and {βt,i}∀i. In the rest of this
+section, we analyze the relationship between training loss and
+training strategies. After that, Problem (P1) is further solved
+based on the theoretical insights.
+B. Assumptions and Key Lemmas
+Being in line with the studies in [5], [39], we make the
+following assumptions for the local loss function Fi, ∀i.
+Assumption 1. Fi is λ-Lipschitz: ∥Fi(w) − Fi(w′)∥
+≤
+λ ∥w − w′∥, where λ > 0.
+Assumption 2. Fi is ν-strongly convex: Fi(w) ≥ Fi(w′) +
+(w − w′)⊤∇Fi(w′) + ν
+2 ∥w − w′∥2.
+Assumption 3. Fi is twice-continuously differentiable. Based
+on Assumptions 1 and 2, we have νI ⪯ ∇2Fi(w) ⪯ λI.
+Assumption 4. The ratios between the norms of ∇Fi(w) and
+∇F(w) are bounded: ∥∇Fi(w)∥2 ≤ ε ∥∇F(w)∥2, where ε ≥
+0 is a positive constant.
+Assumption 5. For the moderate shrinking factor α ≥ αmin,
+the first-shrinking-then-training can be approximated as first-
+training-then-shrinking: ∇Fi(wα) = [∇Fi(w)]α. Here, we
+use [∇Fi(w)]α to denote the shrinking operation for ∇Fi(w).
+Next, we give the following two definitions.
+Definition 1 (Local gradient divergence). The local gradient
+divergence δt,i is defined as the difference between ut,i and
+˜ut,i, which is given by δt,i = ∥ut,i − ˜ut,i∥.
+Definition 2 (Global gradient divergence). The global gra-
+dient divergence ∆t is defined as the difference between
+ut and ˜ut, which is measured by ∆t = ∥ut − ˜ut∥ =
+���
+I�
+i=1
+pt,iut,i −
+I�
+i=1
+pt,i˜ut,i
+���.
+Notably, in Definition 1, ut,i and ˜ut,i may have different
+dimensions. We pad the missing elements in ˜ut,i with zeros
+before the arithmetic operation. Next, we are interested in how
+the training strategies {αt,i, βt,i}∀i affect {δt,i}∀i and ∆t. We
+derive the following two lemmas.
+Lemma 1. For the local training with the model shrinking
+factor αt,i and compression rate βt,i. The square of the local
+gradient divergence is bounded by
+E∥δt,i∥2 ≤
+�
+1 − αt,i(2 − αt,i)
+�
+βt,i
+�2E∥ut,i∥2.
+(11)
+Proof. See Appendix A.
+Lemma 2. For the local update {˜ut,i, ∀i} with the corre-
+sponding training strategies {αt,i, βt,i}∀i and aggregation co-
+efficients {pt,i}∀i, the square of the global gradient divergence
+is bounded by
+E∥∆t∥2 ≤ Iεη2
+I
+�
+i=1
+p2
+t,i
+�
+1 − αt,i(2 − αt,i)
+�
+βt,i
+�2E∥∇F(wt)∥2.
+(12)
+Proof. See Appendix B.
+C. Optimal Aggregation Scheme and Convergence Analysis
+Intuitively, the local update ut,i generated with larger
+{αt,i, βt,i} may carry more accurate information, and thus a
+larger pt,i should be assigned during the aggregation. Based
+on Lemma 2, we deduce the following theorem.
+Theorem 1 (Optimal aggregation scheme). Given the lo-
+cal updates {˜ut,i}∀i with corresponding training strategies
+{αt,i, βt,i}∀i, the optimal aggregation coefficients are
+p∗
+t,i =
+1
+�
+1−αt,i(2−αt,i)√
+βt,i
+�2
+�
+i
+1
+�
+1−αt,i(2−αt,i)√
+βt,i
+�2
+, ∀i.
+(13)
+Proof. Based on Lemma 2, we study the following optimiza-
+tion problem to minimize the global gradient divergence.
+(P2)
+min
+{pt,i}∀i
+I
+�
+i=1
+p2
+t,i
+�
+1 − αt,i(2 − αt,i)
+�
+βt,i
+�2
+(14)
+subject to:
+pt,i ≥0, ∀i,
+(14a)
+I
+�
+i=1
+pt,i = 1.
+(14b)
+It can be verified that Problem (P2) is a convex op-
+timization problem. We further solve the problem by the
+Karush–Kuhn–Tucker (KKT) conditions. Let {̟}∀i and θ
+be the Lagrange multipliers for Constraints (14a) and (14b),
+respectively. Then, we obtain
+̟i ≥ 0, ̟ipt,i = 0, pt,i ≥ 0,
+I
+�
+i=1
+pt,i = 1,
+2pt,i
+�
+1 − αt,i(2 − αt,i)
+�
+βt,i
+�2 − ̟i + θ = 0, ∀i.
+(15)
+Being in line with the study in [40], we can obtain
+pt,i = −
+θ
+2
+�
+1 − αt,i(2 − αt,i)
+�
+βt,i
+�2 .
+(16)
+By putting Eqn. (16) into Eqn. (14b), we obtain
+θ = −
+2
+�
+k
+1
+�
+1−αt,i(2−αt,i)√
+βt,i
+�2
+.
+(17)
+Putting Eqn. (17) into Eqn. (16) completes the proof.
+
+With the optimal aggregation scheme, we investigate the
+upper bound of the convergence rate of AnycostFL.
+Definition 3 (Local and global learning gains). The local
+and global learning gains are defined as gt,i = α4
+t,iβt,i and
+gt = �
+i gt,i/I, respectively. Specifically, the local and global
+learning gains (i.e., gt,i ∈ [0, 1] and gt ∈ [0, 1]) measure the
+amount of effective information carried in the local and global
+updates, respectively.
+Theorem 2 (Convergence rate of AnycostFL). Let gmin =
+min{gt}∀t be the minimal global learning gain over the T -
+round training. The upper bound of the convergence rate of
+AnycostFL satisfies
+E
+�
+F(wT ) − F(w∗)
+�
+≤ ZT −1E
+�
+F(w0) − F(w∗)
+�
+,
+(18)
+where Z = 1 − ν
+λ
+�
+1 − ε(1 − gmin)
+�
+. Recall that parameters
+ν, λ and ǫ are defined in Assumptions 1 to 4 before.
+Proof. See Appendix C.
+Based on Definition 3 and Theorem 2, we derive the
+following proposition.
+Proposition 1. The key to minimizing the training loss of
+AnycostFL is to maximize the learning gain gt for each global
+round. If gt = 1 ∀t, AnycostFL degrades to conventional FL
+without model shrinking and gradient compression.
+D. Solution for Problem (P1)
+Based on Theorem 2 and Proposition 1, Problem (P1) can
+be transformed into the following problem.
+(P3)
+max 1
+I
+I
+�
+t=1
+α4
+t,iβt,i
+(19)
+subject to:
+Constrains (10a) to (10e),
+variables:
+{αt,i, βt,i, ft,i}∀i.
+Based on Constraints (10a) and (10b) for the training latency
+and energy, we obtain the following lemma.
+Lemma 3. The equality will always hold for Constraints
+(10a) and (10b) when confirming the optimal training strategy
+{α∗
+t,i, β∗
+t,i, f ∗
+t,i}∀i, and thus T ∗
+t,i = T max and Et,i = E∗
+t,i ∀i.
+Proof. The lemma can be proved by showing the contradic-
+tion. Suppose that there exists i0 such that T ∗
+t,i0 < T max.
+We can find a new solution {α′
+t,i0, β∗
+t,i0, f ′
+t,i0} for device i0
+and α′
+t,i0 > α∗
+t,i0, f ′
+t,i0 < f ∗
+t,i0, such that T ′
+t,i0 = T max
+and E′
+t,t0 = Emax
+t,i . Since the global learning gain increases
+with the increase of αt,i0, we have g′
+t > g∗
+t . Likewise, the
+contradiction also appears when E∗
+t,i0 < Emax
+t,i0 , and thus we
+complete the proof.
+Based on Lemma 3, we employ two intermediate variables
+(i.e., φt,i and ϕt,i) for each device to reparameterize Problem
+(P3). Specifically, φt,i ∈ [0, 1] and ϕt,i ∈ [0, 1] are the splitting
+factors for latency and energy, respectively, such that
+T cmp
+t,i = φt,iT max, T com
+t,i = (1 − φt,i)T max,
+Ecmp
+t,i = ϕt,iEmax
+t,i , Ecom
+t,i = (1 − ϕt,i)Emax
+t,i , ∀i.
+(20)
+By combining Eqns (6) and (20), the local learning gain of
+the device i at the t-th round can be rewritten as
+gt,i(φt,i) = κt,i
+�
+Emax
+t,i
+− (1 − φt,i)T maxP com
+t,i
+�
+(φ2
+t,i − φ3
+t,i), (21)
+where κt,i = rt,i
+Sǫi
+� T max
+τ|Di|W
+�3.
+Note that Problem (P3) can be transformed into I sub-
+problems because the decision-making procedure of each
+device is independent. Based on Eqn. (21), the i-th sub-
+problem can be expressed as a single-variable optimization
+problem with respect to φt,i as follows.
+(P4)
+max
+φt,i
+gt,i
+�
+φt,i
+�
+(22)
+subject to:
+φmin
+t,i
+≤ φt,i ≤φmax
+t,i ,
+where the lower and upper limits of φt,i can be acquired by
+φmin
+t,i
+= max
+�αminτ |Di| W
+f max
+i
+T max
+, 1 − βmaxS
+rt,iT max
+�
+,
+φmax
+t,i
+= min
+� τ |Di| W
+f min
+i
+T max , 1 − αminβminS
+rt,iT max
+�
+.
+(23)
+Based on the first-order optimality condition ∂gt,i/φt,i = 0,
+we obtain the stationary points as
+φs1
+t,i =
+�
+ψt,i − 3Emax
+t,i
+8P com
+t,i T max
++ 3
+4, φs2
+t,i = −
+�
+ψt,i + 3Emax
+t,i
+8P com
+t,i T max
+− 3
+4, (24)
+where ψt,i = 4(P com
+t,i T max)2 − 4Emax
+t,i P com
+t,i T max + 9(Emax
+t,i )2.
+Let St,i = {φmin
+t,i , φmax
+t,i , φs1
+t,i, φs2
+t,i} denote the union of the
+stationary points and the boundary points for Problem (P4).
+Then, S′
+t,i = {φt,i|φt,i ∈ [φmin
+t,i , φmax
+t,i ], φt,i ∈ St,i} is the set
+of the feasible solutions of St,i. The optimal solution for
+Problem (P4) can be acquired by
+φ∗
+t,i = arg max
+φt,i∈S′
+t,i
+gt,i(φt,i).
+(25)
+Furthermore, we obtain the optimal solution for device i at the
+t-th global round by putting φ∗
+t,i into the following equations.
+ϕ∗
+t,i = 1 − (1 − φ∗
+t,i)T maxP com
+t,i
+Emax
+t,i
+, α∗
+t,i =
+3
+�
+(φ∗
+t,iT max)2ϕ∗
+t,iEmax
+t,i
+ǫi(τ |Di| W )3
+,
+β∗
+t,i = rt,i(1 − φ∗
+t,i)T max
+α∗
+t,iS
+, f ∗
+t,i = α∗
+t,iτ |Di| W
+φ∗
+t,iT max
+.
+(26)
+Notably, the decision-making process of each device does
+not involve the auxiliary information of the resource status
+from other devices. At the beginning of each global round,
+each device can determine its training strategy locally.
+V. EXPERIMENT EVALUATIONS
+A. Experiment Settings
+1) Setup for FL training: We consider the FL application
+with image classification on Fashion-MNIST and CIFAR-
+10 datasets [41], [42]. For Fashion-MNIST, we use a small
+convolutional neural network (CNN) with data size of model
+update as 53.22Mb [1]. For the CIFAR-10 dataset, we employ
+VGG-9 with data size of model update as 111.7Mb [43].
+For IID and non-IID data settings, we follow the dataset
+partition strategy in [34]. For the learning hyper-parameters,
+the learning rate, batch size and local epoch are set as {0.01,
+32, 1} for Fashion-MNIST and {0.08, 64, 1} for CIFAR-10
+dataset. The maximal latency is set as T max = 10 seconds
+
+0
+5
+10
+15
+20
+25
+30
+80
+82
+84
+86
+88
+90
+92
+0
+5
+10
+15
+20
+25
+30
+80
+82
+84
+86
+88
+90
+0
+10
+20
+30
+40
+50
+60
+30
+40
+50
+60
+70
+80
+90
+0
+10
+20
+30
+40
+50
+60
+30
+40
+50
+60
+70
+80
+14
+16
+89
+90
+16
+18
+20
+88
+89
+Test accuracy (%)
+Time consumption (min)
+Time consumption (min)
+STC
+QSGD
+UVeQFed
+HeteroFL
+FedHQ
+AnycostFL
+(a) FMNIST IID
+(b) FMNIST non-IID
+(c) CIFAR-10 IID
+(d) CIFAR-10 non-IID
+Energy consumption (KJ)
+Energy consumption (KJ)
+Fig. 4. Performance on various network architectures and datasets. ((a-b): global accuracy vs. time consumption with Fashion MNIST on 2-layer CNN; (c-d):
+global accuracy vs. energy consumption with CIFAR-10 on VGG-9.)
+TABLE I
+PERFORMANCE COMPARISON BETWEEN ANYCOSTFL AND OTHER METHODS ON FASHION-MNIST AND CIFAR-10 DATASETS.
+IID
+non-IID
+Dataset
+Method
+#Round
+Energy
+(KJ)
+Latency
+(min)
+Comp.
+(TFLOPs)
+Comm.
+(GB)
+Best Acc.
+(%)
+#Round
+Energy
+(KJ)
+Latency
+(min)
+Comp.
+(TFLOPs)
+Comm.
+(GB)
+Best Acc.
+(%)
+FMNIST
+{90%, 89%}∗
+STC
+305 (1.7×) 10.94 (1.4×) 25.42 (1.7×)
+152.71
+0.71
+90.28±0.18 283 (1.3×) 10.17 (1.1×) 23.56 (1.3×)
+141.53
+0.66
+89.47±0.16
+QSGD
+283 (1.6×) 11.40 (1.4×) 23.56 (1.6×)
+141.53
+0.80
+90.39±0.04 279 (1.3×) 11.27 (1.2×) 23.28 (1.3×)
+139.86
+0.79
+89.49±0.07
+UVeQFed
+247 (1.4×) 11.36 (1.4×) 20.58 (1.4×)
+123.67
+0.72
+90.44±0.10 266 (1.2×) 12.21 (1.3×) 22.14 (1.2×)
+133.01
+0.77
+89.64±0.16
+HeteroFL
+233 (1.3×) 12.03 (1.5×) 21.78 (1.5×)
+92.21
+0.57
+90.43±0.13 242 (1.1×) 12.51 (1.3×) 22.62 (1.3×)
+95.77
+0.59
+89.42±0.10
+FedHQ
+288 (1.6×) 13.89 (1.7×) 24.03 (1.6×)
+144.36
+0.86
+90.21±0.07 313 (1.5×) 14.96 (1.6×) 26.06 (1.5×)
+156.55
+0.93
+89.27±0.19
+AnycostFL 179 (1.0×) 8.07 (1.0×) 14.94 (1.0×)
+67.49
+0.35
+91.20±0.09 214 (1.0×) 9.63 (1.0×) 17.83 (1.0×)
+80.51
+0.42
+90.32��0.14
+CIFAR-10
+{82%, 80%}∗
+STC
+341 (1.2×) 35.39 (1.3×) 56.83 (1.2×)
+4160.56
+1.78
+85.38±0.29 412 (1.1×) 42.39 (1.3×) 68.67 (1.1×)
+5026.84
+2.15
+83.09±0.53
+QSGD
+337 (1.2×) 39.82 (1.5×) 56.17 (1.1×)
+4111.76
+2.14
+84.83±0.54 430 (1.2×) 50.29 (1.5×) 71.61 (1.2×)
+5242.39
+2.73
+81.94±0.13
+UVeQFed
+296 (1.0×) 40.77 (1.5×) 49.28 (1.0×)
+3607.45
+2.12
+85.09±0.16 377 (1.0×) 51.59 (1.5×) 62.89 (1.0×)
+4603.87
+2.71
+82.30±0.28
+HeteroFL
+332 (1.1×) 50.07 (1.9×) 69.14 (1.4×)
+3222.26
+1.65
+83.75±0.55 413 (1.1×) 62.88 (1.9×) 85.78 (1.4×)
+3990.49
+2.05
+80.68±0.45
+FedHQ
+340 (1.2×) 48.95 (1.9×) 56.67 (1.2×)
+4148.36
+2.32
+84.02±0.22 435 (1.2×) 61.99 (1.9×) 72.44 (1.2×)
+5303.40
+2.96
+81.00±0.41
+AnycostFL 294 (1.0×) 26.43 (1.0×) 48.94 (1.0×)
+2459.92
+1.56
+87.72±0.23 372 (1.0×) 33.51 (1.0×) 62.06 (1.0×)
+3118.60
+1.98
+84.91±0.51
+*{x, y}: x and y denote the target global model accuracy under IID and non-IID data settings, respectively.
+and the energy budget is set as Emax
+t,i
+∼ U[3, 9] joules for the
+CIFAR-10 dataset, and the corresponding hyper-parameters for
+the FMNIST dataset are halved by default. Additionally, we
+set αmin = 1/4 and βmax = 1/15.
+2) Setup for mobile system: We investigate a mobile system
+with I = 60 devices located within a circle cell with a
+radius of 550 meters, and a base station is situated at the
+center. To simulate the mobility, the position of each device
+is refreshed randomly at the beginning of each round [44].
+For the computation, the energy coefficient is set as ǫi ∼
+U[5 × 10−27, 1 × 1−26]. For communication, the bandwidth
+is set as 1MHz equally for each device, and the path loss
+exponent is 3.76. The transmission power is set as 0.1W, and
+N0 is set as −114dBm/MHz.
+B. Performance Comparisons
+We compare the proposed AnycostFL with the following
+efficient FL algorithms with three different random seeds.
+• STC. The sparse ternary compression (STC) is adapted
+to reduce the cost of uplink parameter transmission [11].
+• QSGD. The TopK sparsification and probabilistic quanti-
+zation are combined to compress the local gradient [36].
+• UVeQFed. The TopK sparsification and universal vector
+quantization are used to compress the local gradient [14].
+• HeteroFL. Each device trains the local sub-model in
+different widths to match its computation capacity [32].
+• FedHQ. Each device uses different quantization levels to
+compress the gradient according to its channel state [40].
+Fig. 4 shows the performance of the global model over
+time consumption and energy consumption under the IID and
+the non-IID data setting. With the same training efficiency
+(i.e., time and energy consumption), the proposed AnycostFL
+consistently outperforms the baseline schemes to improve the
+test accuracy of the global model. Meanwhile, Table I provides
+the best accuracy and required system cost for achieving
+the specified test accuracy. Particularly, when compared with
+HeterFL and FedHQ, AnycostFL can reduce up to 1.9 times
+the energy consumption to reach the test accuracy of 82%
+on CIFAR-10 dataset under the IID setting. When compared
+with STC, AnycostFL can reduce up to 1.7 times the time
+consumption to reach the test accuracy of 90% on FMNIST
+dataset under the IID setting. Moreover, our framework can
+significantly improve the best accuracy of the global model
+by 2.33% and 1.82% on CIFAR-10 dataset under the IID and
+the non-IID settings, respectively.
+C. Impact of Key Mechanisms and Hyper-parameters
+Fig. 5(a) verifies the advantages of the main techniques of
+AnycostFL. We gradually remove the elastic model shrinking
+(w/o EMS), the flexible gradient compression (w/o FGC) and
+the all-in-one aggregation (w/o AIO), and record the required
+system cost to achieve 80% test accuracy with CIFAR-10
+dataset under the IID setting. We observe that the proposed
+EMS and FGC can significantly save the energy consumption
+and training time, respectively. Besides, AIO contributes to
+saving both energy and time.
+
+AnycostFL w/o EMS w/o FGC
+w/o AIO
+20
+30
+40
+50
+60
+Energy consumption
+Required time
+Energy consumption (KJ)
+2.4x
+1.3x
+1.5x
+1.3x
+35
+40
+45
+50
+55
+60
+65
+70
+75
+Required time (min)
+0
+2
+4
+6
+8
+10
+45
+60
+75
+90
+105
+120
+6
+7
+8
+9
+70
+80
+Average time consumption (min)
+Level of communication heterogeneity
+STC
+QSGD
+UVeQFed
+HeteroFL
+FedHQ
+AnycostFL
+0
+2
+4
+6
+8
+10
+12
+24
+32
+40
+48
+56
+Average energy consumption (KJ)
+Level of computation heterogeneity
+STC
+QSGD
+UVeQFed
+HeteroFL
+FedHQ
+AnycostFL
+0.06
+0.09
+0.12
+78
+81
+84
+87
+90
+STC
+HeteroFL
+AnycostFL
+Global test accuracy (%)
+Computational complexity (GFLOPs)
+(a) Impact of key mechanisms
+(b) Impact of comm. heterogeneity
+(c) Impact of comp. heterogeneity
+(d) Performance of sub-models
+Fig. 5. The main advantages of AnycostFL. ((a): the impact of key mechanisms; (b-c): the impact of system heterogeneity; (d): the performance of sub-models.)
+We next evaluate the impact of resource heterogeneity on
+the training efficiency in Fig. 5(b-c). We set the average energy
+coefficient ǫi as 7.5×10−27 and the average distance between
+the base station and edge devices as 400 meters, and then
+change their variances to simulate the computation and com-
+munication heterogeneity, respectively. The larger variance
+indicates a higher level of system heterogeneity. As we expect,
+the proposed AnycostFL shows more resilience than other
+baselines to tackle the high level of system heterogeneity.
+We also evaluate the performance of sub-models in different
+widths in Fig. 5(d). Specifically, We compare AnycostFL with
+HeteroFL (i.e., local training with different widths) and STC
+(i.e., the best-performing compression-only method). The sub-
+models are derived from the well-trained global model without
+further re-training. Surprisingly, the sub-models of the global
+model trained by AnycostFL can still maintain satisfactory test
+accuracy, which provides dynamic inference for diverse edge
+devices after the training time.
+VI. CONCLUSION
+In this paper, we proposed AnycostFL, a joint computation
+and communication efficient framework for FL, that enables
+edge devices with diverse resources to train a shared global
+model. We aimed to minimize the global training loss under
+given personalized latency and energy constraints. By leverag-
+ing the theoretical insight of AnycostFL, we decomposed the
+optimization problem into multiple sub-problems. Following
+that, the optimal training strategy is derived for each de-
+vice according to its locally available resource. Experiments
+demonstrate the advantage of our framework in improving the
+system efficiency and model performance compared to the
+state-of-the-art methods.
+ACKNOWLEDGMENT
+Rong Yu and Yuan Wu are the corresponding authors. This
+work was supported in part by National Key R&D Program
+of China under Grant 2020YFB1807802, in part by National
+Natural Science Foundation of China under Grants 61971148,
+62102099, U22A2054 and 62001125, in part by Science and
+Technology Development Fund of Macau SAR under Grant
+0162/2019/A3, in part by FDCT-MOST Joint Project under
+Grant 0066/2019/AMJ, in part by the Guangdong Basic and
+Applied Basic Research Foundation (2022A1515011287), and
+in part by US National Science Foundation under grant CNS-
+2107057.
+APPENDIX A
+PROOF OF LEMMA 1
+Proof. For the given local gradient ˜ut,i with shrinking factor
+αt,i and gradient compression rate βt,i, we aim to capture the
+divergence between ˜ut,i and ut,i. Suppose that the absolute
+value of the element in ut,i follows uniform distribution |u| ∼
+U(0, umax), and umax = max{|u|}∀u∈ut,i.
+For clear notation, we sort the element-wise absolute
+value of ut,i in ascending order. Then, we obtain ut,i =
+[u[1]
+t,i, . . . , u[j]
+t,i, . . . , u[J]
+t,i ]⊤ and |u[j]
+t,i| ≤ |u[j+1]
+t,i
+|. Thus, we have
+E∥ut,i∥2 = E
+J
+�
+j=1
+|u[j]
+t,i|2 = JE|u[j]
+t,i|2 = Ju2
+max
+3
+.
+(27)
+Based on Assumption 5, the update generated from lo-
+cal training with wα
+t,i is equal to shrink(ut,i, αt,i). The
+operation of model shrinking on ut,i with αt,i can be
+viewed as removing (1 − αt,i)J elements with the least
+value from ut,i. Then, we obtain shrink(ut,i, αt,i)
+=
+[0, . . . , 0, u[(1−αt,i)J+1]
+t,i
+, . . . , u[J]
+t,i ]⊤. Thus, we have
+E∥ut,i − shrink(ut,i, αt,i)∥2 = E
+(1−αt,i)J
+�
+j=1
+|u[j]
+t,i|2
+= J(1 − αt,i)3u2
+max/3 = (1 − αt,i)3E∥ut,i∥2.
+(28)
+We next focus on the gradient compression. The operation
+of gradient sparsification on ut,i with sparsity of ρt,i can
+be viewed as removing ρt,iJ elements with the least value
+from ut,i. Then, the quantization is conducted on the non-
+zero elements of ˆut,i, and we obtain cmprs(ut,i, βt,i) =
+[0, . . . , 0, ˜u[ρt,iJ+1]
+t,i
+, . . . , ˜u[J]
+t,i ]⊤. Furthermore, we have
+E∥ut,i − cmprs(ut,i, βt,i)∥2
+= E
+ρt,iJ
+�
+j=1
+|u[j]
+t,i|2
+�
+��
+�
+(A)
++ E
+J
+�
+j=ρt,iJ+1
+|u[j]
+t,i − ˜u[j]
+t,i|2
+�
+��
+�
+(B)
+.
+(29)
+Likewise to Eqn. (28), we have (A) = ρ3
+t,iE∥ut,i∥2. Based on
+Eqn. (4) and the statistical feature of ut,i, we obtain (B) =
+(1 − ρt,i)3E∥ut,i∥2/(2L2
+t,i).
+Given plain update ut,i in 32-bit floating point and the
+desired compression rate βt,i, we can set ρt,i = 1−
+�
+βt,i and
+Lt,i = 232√
+βt,i for the analysis. In this way, the operations
+
+of sparsification and quantization contribute equally to the
+gradient compression. Furthermore, we have
+E∥ut,i − cmprs(ut,i, βt,i)∥2 ≤ (1 − βt,i)2E∥ut,i∥2.
+(30)
+Next, we focus on the local divergence δt,i with respect to
+αt,i and βt,i. According to the Definition 1, we have
+E∥δt,i∥2 = E∥ut,i − cmprs([ut,i]α, βt,i)∥2
+= E∥ut,i − [ut,i]α∥2 + E∥[ut,i]α − cmprs([ut,i]α, βt,i)∥2
++ 2 < ut,i − [ut,i]α, [ut,i]α − cmprs([ut,i]α, βt,i) >
+�
+��
+�
+(C)
+. (31)
+It can be verified that the two vectors in term (C) are
+orthogonal, and we obtain (C) = 0. According to Eqns (28)
+and (30), we further obtain
+E∥δt,i∥2 ≤ (1 − αt,i)3E∥ut,i∥2 + (1 −
+�
+βt,i)2E∥[ut,i]α∥2
+(a)
+≤ (1 − αt,i)3E∥ut,i∥2
++ (1 −
+�
+βt,i)2αt,i(α2
+t,i − 3αt,i + 3)E∥ut,i∥2
+(b)
+≤
+�
+1 − αt,i(2 − αt,i)
+�
+βt,i
+�2E∥ut,i∥2.
+(32)
+Likewise to Eqn. (28), inequality (a) stems from the fact that
+E∥[ut,i]α∥2 = αt,i(α2
+t,i−3αt,i+3)E∥ut,i∥2. Besides, inequal-
+ity (b) holds for all αt,i ∈ [αmin, 1] and βt,i ∈ [0, βmax]. Thus,
+we complete the proof.
+APPENDIX B
+PROOF OF LEMMA 2
+Proof. Based on Definition 2 and Lemma 1, we have
+E∥∆t∥2 = E
+���
+I
+�
+i=1
+pt,iut,i −
+I
+�
+i=1
+pt,i˜ut,i
+���
+2
+≤ E
+� I
+�
+i=1
+pt,i
+�
+1 − αt,i(2 − αt,i)
+�
+βt,i
+�
+∥ut,i∥
+�2
+.
+(33)
+We use η to denote the learning rate, and ut,i = η∇Fi(wt).
+Based on Assumption 4, we obtain
+E∥∆t∥2 ≤ εη2�
+I
+�
+i=1
+pt,i
+�
+1 − αt,i(2 − αt,i)
+�
+βt,i
+��2
+E∥∇F(wt)∥2.
+(34)
+According to Cauchy–Schwarz inequality, we obtain
+E∥∆t∥2 ≤ Iεη2
+I
+�
+i=1
+p2
+t,i
+�
+1 − αt,i(2 − αt,i)
+�
+βt,i
+�2E∥∇F(wt)∥2.
+(35)
+Thus, we complete the proof.
+APPENDIX C
+ON THE CONVERGENCE OF ANYCOSTFL
+Proof. Inspired by the studies in [5], [39], we deduce the
+convergence analysis of AnycostFL. According to Taylor
+expansion and Assumption 3, we have
+F(wt+1) ≤ F(wt) + (wt+1 − wt)⊤∇F(wt) + λ
+2 ∥wt+1 − wt∥2
+= F(wt) − ˜u⊤
+t ∇F(wt) + λ
+2
+��˜ut
+��2.
+(36)
+By using learning rate η = 1
+λ, we obtain
+E
+�
+F(wt+1)
+�
+≤ E
+�
+F(wt) − λ (ut − ∆t)⊤ut + λ
+2 ∥ut − ∆t∥2�
+= E
+�
+F(wt) − 1
+2λ∥∇F(wt)∥2 + λ
+2 ∥∆t∥2�
+.
+(37)
+We now pay attention to the upper bound of ∥∆t∥2. Based on
+Jensen’s inequality and Eqn. (34), we obtain
+E∥∆t∥2 ≤ εη2
+I
+�
+i=1
+pt,i
+�
+1 − αt,i(2 − αt,i)
+�
+βt,i
+�2
+�
+��
+�
+(D)
+E∥∇F(wt)∥2.
+(38)
+By putting Eqn. (13) into (A), we have
+E∥D∥ ≤ E
+��������
+I
+I�
+i=1
+1
+(1−αt,i(2−αt,i)√
+βt,i)
+2
+��������
+(c)
+≤E
+��������
+I
+I�
+i=1
+1
+1−α4
+t,iβt,i
+��������
+,
+(39)
+where (c) always holds for αt,i ∈ [0, 1] and βt,i ∈ [0, 1].
+According to Definition 3, we have gt,i = α4
+t,iβt,i and gt =
+�
+i gt,i/I. Since 1/
+��
+i
+1
+1−gt,i
+�
+is a concave function with
+respect to gt,i, based on Jensen’s inequality, we obtain
+E∥A∥ ≤
+I
+�
+i
+1
+1−E(α4
+t,iβt,i)
+= 1 − gt.
+(40)
+Since the training strategies of each device and the norm of
+the gradient of global data ∥∇F(wt)∥ are independent, by
+putting Eqn. (40) back to Eqn. (38), we obtain
+E∥∆t∥2 ≤ E
+�
+εη2�
+1 − gt
+�
+∥∇F(wt)∥2�
+.
+(41)
+Next, by putting Eqn. (41) back to Eqn. (37), we have
+E
+�
+F(wt+1)
+�
+≤ E
+�
+F(wt) − 1 + ε
+�
+gt − 1
+�
+2λ
+∥∇F(wt)∥2�
+. (42)
+Subtracting F(w∗) in both sides of Eqn. (42) yields
+E
+�
+F(wt+1 − F(w∗)
+�
+≤ E
+�
+F(wt) − 1 + ε(gt − 1)
+2λ
+∥∇F(wt)∥2 − F(w∗)
+�
+.
+(43)
+Based on Assumptions 2 and 3, we have [5], [45]
+∥∇F(wt)∥2 ≥ 2ν
+�
+F(wt) − F(w∗)
+�
+.
+(44)
+Plugging Eqn. (44) into Eqn. (43), we have
+E
+�
+F(wt+1) − F(w∗)
+�
+≤ ZtE
+�
+F(wt) − F(w∗)
+�
+,
+(45)
+where Zt = 1 − ν
+λ (1 − ε(1 − gt)).
+Let gmin = min{gt}∀t be the minimal global learning
+gain over T global rounds. By recursively applying the above
+inequality from iteration round 0 to T , we can obtain
+E
+�
+F(wT ) − F(w∗)
+�
+≤ ZT −1E
+�
+F(w0) − F(w∗)
+�
+,
+(46)
+where Z = 1 − ν
+λ
+�
+1 − ε(1 − gmin)
+�
+. Thus, we complete the
+proof.
+
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+Astronomy & Astrophysics manuscript no. 45358arxaa
+©ESO 2023
+January 3, 2023
+Letter to the Editor
+Analysis of the first infrared spectrum of quasi-bound
+H2 line emission in Herbig-Haro 7
+E. Roueff1, M. G. Burton2, T. R. Geballe3, and H. Abgrall1
+1 Sorbonne Université, Observatoire de Paris, PSL University, CNRS, LERMA, F-92190, Meudon, France
+e-mail: [email protected]
+2 Armagh Observatory and Planetarium, College Hill, Armagh, BT61 9DB, Northern Ireland
+e-mail: [email protected]
+3 Gemini Obsevatory/NSF’s NOIRLab, 670 N. A’ohoku Place, Hilo, HI 96720, USA
+e-mail: [email protected]
+Accepted in Astronomy Astrophysics Letters on december 22, 2022
+ABSTRACT
+Context. Highly excited molecular hydrogen (H2) has been observed in many regions of shocked molecular gas. A recently published
+K-band spectrum of Herbig-Haro 7 (HH7) contains several vibration-rotation lines of H2 from highly excited energy levels that
+have not been detected elsewhere, including a line at 2.179 µm identified as arising from the v=2 J=29 level, which lies above the
+dissociation limit of H2. One emission line at 2.104 µm in this spectrum was unidentified.
+Aims. We aim to complete the analysis of the spectrum of HH7 by including previously missing molecular data that have been recently
+computed.
+Methods. We re-analysed the K-band spectrum, emphasising the physics of quasi-bound upper levels that can produce infrared
+emission lines in the K band.
+Results. We confirm the identification of the 2 − 1 S (27) line at 2.1785 µm and identify the line at 2.1042 µm as due to the 1-0 S (29)
+transition of H2, whose upper level energy is also higher than the dissociation limit. This latter identification, its column density, and
+the energy of its upper level further substantiate the existence of a hot thermal component at 5000 K in the HH7 environment.
+Conclusions. The presence of the newly identified 1 − 0 S (29) line, whose quasi-bound upper level (v=1, J=31) has a significant
+spontaneous dissociation probability, shows that dissociation of H2 is occurring. The mechanism by which virtually all of the H2 in
+levels with energies from 20,000 K to 53,000 K is maintained in local thermodynamic equilibrium at a single temperature of ∼5,000
+K remains to be understood.
+Key words. molecular hydrogen – interstellar medium – shocks
+1. Introduction
+The interaction of the collimated outflow from the protostar
+SSV13 (Strom et al. 1976) and the molecular cloud out of which
+it formed has produced a collection of Herbig-Haro (HH) ob-
+jects, HH7-HH11, in a more or less linear arrangement on the
+sky. The most distant of these from SSV13, HH7, has a classic
+bow shock shape. It is bright in line emission from shock-excited
+vibrational states of molecular hydrogen (H2), first observed in
+the v = 1 − 0 S (1) transition by Zealey et al. (1984), Harti-
+gan et al. (1989), and Garden et al. (1990) and subsequently in
+vibrational levels 0 − 4 and rotational levels 1 − 15 by Burton
+et al. (1989) and Fernandes & Brand (1995). HH7 also emits
+strongly in pure rotational lines of H2 and CO (Neufeld et al.
+2006; Yuan & Neufeld 2011; Neufeld et al. 2019; Molinari et al.
+2000) as well as in [OI]63µ (Sperling et al. 2020), Hα, [OI]λ6300,
+and [SII]λ6716 (Hartigan et al. 2019). The vibrationally excited
+H2 lines, observed mainly in the 2.0 − 2.5 µm interval, are emit-
+ted predominantly in the hottest shock-heated gas, while the pure
+rotational low-J transitions of H2 and the pure rotational transi-
+tions of CO, observed in the mid- and far-infrared, arise in a
+somewhat cooler gas downstream.
+Much more highly vibrationally and rotationally excited
+molecular hydrogen was found by Pike et al. (2016, hereafter
+P16) in a 3′′× 3′′region near the tip of the HH7 bow shock, in
+K-band spectra they obtained at a resolving power, R, of 5000.
+Figure 6 of their paper shows the 2.01 − 2.45 µm spectrum of a
+0′′.6×0′′.9 area in that region. Their paper demonstrated the ex-
+istence in HH7 of a small percentage (1.5%) of the emitting
+H2 at a temperature of ∼5,000 K. Subsequently, Geballe et al.
+(2017) discovered the presence of small percentages of 5,000 K
+H2 in shocked gas at several locations in the Orion Molecular
+Cloud. Giannini et al. (2015) detected a similar phenomenon in
+another bright HH object, HH1, at a somewhat higher tempera-
+ture, ∼6300 K.
+P16 identified a weak emission line at 2.179 µm in the HH7
+spectrum as the 2 − 1 S (27) transition of H2, which arises from
+the upper level, v = 2, J = 29, whose energy is above the
+dissociation limit of the ground state of H2; this corresponds
+to 51,965.84 K, using the latest measurements (Hölsch et al.
+2019) and the Committee on Data for Science and Technology
+(CODATA) definition of fundamental constants (Tiesinga et al.
+2021). The column density of H2 in the upper level could only be
+crudely estimated by P16, as the Einstein A coefficient for that
+transition was not known. P16 also reported the detection of a
+faint line near 2.104 µm, which they were unable to identify.
+Roueff & Abgrall (2022) have recently proposed a simple
+and efficient method for computing the emission spectrum pro-
+Article number, page 1 of 5
+arXiv:2301.00741v1  [astro-ph.GA]  2 Jan 2023
+
+A&A proofs: manuscript no. 45358arxaa
+duced by quasi-bound levels, providing accurate wavenumbers
+and Einstein emission coefficients. The application to H2 al-
+lowed them to calculate the Einstein coefficient of the 2-1 S (27)
+transition and suggested that the line at 2.104 µm in HH7 is the
+1 − 0 S (29) transition of H2, whose upper-state energy also lies
+above the dissociation limit.
+The present paper analyses these two high excitation lines in
+the light of the new theoretical developments. Section 2 revisits
+the observations of HH7, Sect. 3 summarises the recent theoret-
+ical achievements, and Sect. 4 contains the resulting extended
+observational analysis of the H2 line emission in HH7. We pro-
+vide a discussion of our results in Sect. 5.
+2. Observations
+A detailed description of the observations of HH7 and a re-
+duction of the spectral data have been given by P16. In brief,
+the Gemini facility integral field spectrometer, the Near Infrared
+Field Spectrometer (NIFS; McGregor et al. 2003), was used at
+the Frederick C. Gillett Gemini North Telescope on Maunakea,
+Hawai’i, to obtain spectra of a 3′′× 3′′region near the tip of the
+HH7 bow shock, for program GN-2007B-Q-47. The angular res-
+olution of the spectra was 0′′.35. Within this 3′′× 3′′region, the
+spectra showed H2 ro-vibrational line emission from upper-state
+levels covering a wide range of energies, including a dozen in
+the range 40, 000 − 50, 000 K. Because some rotational energies
+and associated rotational quantum numbers of the upper levels of
+these lines are high (J ≳ 15), collisions rather than the absorp-
+tion of ultraviolet (UV) photons are probably the main producer
+of the populations in those rotational levels. Somewhat lower
+values of J associated with high vibrational quantum numbers
+are commonly found in dense photon-dominated regions (PDRs)
+such as NGC 2023, the Orion Bar, S140, and IC63. (Burton et al.
+1992; McCartney et al. 1999; Kaplan et al. 2021). H2 is excited
+in PDRs by UV pumping, which is followed by electronic fluo-
+rescence, but the ∆J = ±1 selection rule for electronic transitions
+maintains J at values below ∼ 13.1
+P16 concentrated their analysis on the spectrum of the 0′′.6
+× 0′′.9 area shown in their Fig. 2; the spectrum is plotted in their
+Fig. 6. The upper two panels of our Fig. 1 show in more de-
+tail two 0.01 µm wide portions of that spectrum, each contain-
+ing one of the two highly shock-excited H2 lines discussed in
+the Introduction. Wavelength calibration employed the spectrum
+of an argon lamp and is accurate to ∼ 0.00002 µm. The hor-
+izontal scales are vacuum laboratory wavelengths and as such
+can be directly compared with the theoretically calculated wave-
+lengths (see Sect. 3).The uppermost panel contains the previ-
+ously unidentified line at 2.1042 µm along with the nearby 4 − 3
+S (7) line. Similarly, the middle panel contains the previously
+identified 2−1 S (27) line and the adjacent 5−4 S (15) line. Spec-
+tral images of the four lines, extracted from the NIFS data cube,
+are shown in the bottom panel of the figure and demonstrate
+that, to within the limits imposed by the noise levels, the four
+emission lines have identical morphologies, which also match
+the morphology of the strong 1 − 0 S (1) line shown in Fig. 2 of
+P16. Based on the fluctuations in the baseline, we estimate the
+confidence of the detection of the 1 − 0 S (29) line to be 3.5σ.
+The wavelengths of these two weak lines are slightly different
+than those reported by P16 and are more accurate.
+1 We contacted K. Kaplan to check if the two transitions at 2.1785 µm
+and 2.1042 µm were present in his PDR spectra obtained with the Im-
+mersion Grating INfrared Spectrometer (IGRINS), and they were not.
+Fig. 1. Observational data showing highly excited H2 lines in HH7. Top
+two panels: Spectra of a 0′′.6 × 0′′.9 area of HH7 in two narrow wave-
+length intervals, each containing a line of H2 from a quasi-bound energy
+level and one adjacent line, from Fig. 6 of P16. Vertical dashed lines are
+the line wavelengths calculated as described in Sect. 3. Bottom: Spec-
+tral images of the four lines shown above, extracted from the NIFS data
+cube. The field of view is 2′′.5 × 2′′.5 and corresponds to the left part of
+Fig. 2 of P16; the field centre corresponds to RA = 3:29:08.42, Dec =
++31:15:27:45 (J2000), with an estimated uncertainty of 0′′.25.
+Article number, page 2 of 5
+
+2HH
+1
+Density
+Flux
+0.5
+Rel.
+res.
+4-3 S(7)
+-0S(29)
+2.098
+2.1
+2.102
+2.104
+2.106
+LabVacuumWavelength(um)
+2HH
+res.
+Rel.
+4 S(15)
+2.176
+2.178
+2.18
+2.182
+2.184
+Lab Vacuum Wavelength (um)
+S(15)E. Roueff et al.: Analysis of the first infrared spectrum of quasi-bound H2 line emission in Herbig-Haro 7
+3. Theoretical aspects
+Molecular quasi-bound levels correspond to states whose ener-
+gies lie above the dissociation limit of the ground state of the
+molecule but well below the dissociation energy of the electron-
+ically excited molecule. For H2, the Schrödinger equation rele-
+vant to excited rotational levels is
+− ℏ2
+2µ · d2 fv,J(R)
+dR2
++ Vmod
+e f f (R, J) fv,J(R) = Ev,J fv,J(R),
+(1)
+where Vef f (R, J) = V(R) + ℏ2J (J+1)
+2µR2
+, with V(R) the ground state
+electronic molecular potential of H2 and µ = Mp/2 the nuclear
+reduced mass of H2.
+Fig. 2. H2 molecular potentials in eV as a function of the interatomic
+distance, R, expressed in atomic units. The zero value corresponds to
+photo-dissociated H2. The black curve is the electronic potential of the
+X 1Σ+
+g ground state from Czachorowski et al. (2018) expressed in eV.
+The blue and red curves denote the effective potentials with J= 29 and
+J= 31, respectively. The quasi-bound levels v = 2, J = 29 and v =
+1, J = 31 are also displayed in blue and red, respectively, in the allowed
+ranges of interatomic distances.
+Figure 2 displays the electronic molecular potential of the
+X1Σ+
+g ground state of H2 as well as the effective potentials cor-
+responding to J = 29 and J = 31, the two quasi-bound lev-
+els (sometimes referred to as shape resonances) previously men-
+tioned. The presence of the centrifugal potential, ℏ2J (J+1)
+2µR2
+, signif-
+icantly modifies the shape of the electronic contribution, V(R),
+by reducing the potential well, shifting the minima to larger in-
+teratomic distances and exhibiting broad bump maxima above
+the dissociation limit, peaking near 4.5 atomic units.
+Figure 2 also displays the resonant quasi-bound eigenval-
+ues, Er, which are located above the dissociation limit and are
+trapped inside the centrifugal barrier. The associated wave func-
+tion for each level has a non-vanishing probability in the inter-
+atomic range displayed, becomes vanishingly small after the sec-
+ond turning point when Er ≤ Vef f (R), and has an oscillatory be-
+haviour for large R when Er becomes larger than Vef f (R). The
+associated quasi-discrete stationary states have complex energy
+eigenvalues, E = Er − (i Γ/2), where Er is the energy at reso-
+nance and Γ characterises the width of the level and determines
+its lifetime against dissociation, τ = ℏ/Γ, due to tunnelling from
+the quasi-bound to the continuum oscillatory dissociating state at
+large interatomic distances. Roueff & Abgrall (2022) computed
+the various resonance energy level positions of H2 and the corre-
+sponding emission spectrum arising from these levels by using
+the recent highly accurate molecular potential of the H2 ground
+state of Czachorowski et al. (2018) and extending the effective
+potential by a constant value from the maximum value of the po-
+tential function. This method allows one to use a standard numer-
+ical integration of the Schrödinger equation applied to strictly
+bound levels and has been demonstrated to be very precise for
+determining the resonant energy level positions and the emission
+rates. However, it does not allow a derivation of the widths or
+the dissociation lifetimes. Those are obtained through different
+methods based on scattering properties (Schwenke 1988; Selg
+2010).
+These computations predict wavelengths of 2.1785 µm for
+the 2 − 1 S (27) transition and 2.1042 µm for the 1 − 0 S (29)
+transition. The predicted wavelengths for the two stronger lines
+in Fig. 1 are 2.10043 µm for 4−3 S (7) and 2.18179 µm for 5−4
+S (15). As can be seen in the figure, all are in excellent agreement
+with the observed wavelengths. Therefore, we are confident in
+the previous identification of the 2 − 1 S (27) line by P16 and in
+our identification of the weak and previously unidentified feature
+at 2.1042 µm as the 1 − 0 S (29) line. These two transitions are
+the only lines in the 2.01 - 2.45 µm interval from quasi-bound
+levels that would have been detectable in our data. (We note in
+Table 1 the small Einstein A coefficient of the 2 − 0 Q(29) line
+at 2.4007 µm.)
+4. Column density analysis
+The analysis undertaken here follows that described in P16 for
+the H2 line emission from HH7 reported in that paper, with the
+addition of the 1–0 S (29) and 2–1 S (27) lines presented here. A
+two-component Boltzmann distribution with temperatures Thot
+and Twarm was fitted to the column densities obtained from the
+de-reddened line intensities,
+Ni = Ni,hot + Ni,warm,
+(2)
+with each component described by a Boltzmann distribution at
+the corresponding temperatures, as per P16. This is shown in
+Fig. 3.
+We obtain Twarm = 1, 783 ± 20 K and Thot = 5, 133 ± 17 K,
+with 98.5% of the total column of excited H2 gas in the warm
+component of the gas and 1.5% in the hot component. This com-
+pares to values of Twarm = 1, 803±12 K and Thot = 5, 200±12 K
+found without these two extra lines included in the analysis2.
+The additional lever arm provided by the two higher excitation
+energy levels has only led to a marginal decrease in the derived
+temperatures; in other words, the result is essentially the same.
+We conclude that the two quasi-bound H2 lines are well mod-
+elled by the same hot local thermodynamic equilibrium (LTE)
+component as per all lines measured in HH7 arising from energy
+levels ≥15,000 K. The level populations for the two quasi-bound
+lines are ∼ 10−5 times that of the v = 1, J = 3 upper level of the
+brightest H2 emission line, 1 − 0 S (1).
+5. Discussion
+Table 1 summarises the present knowledge available for the two
+quasi-bound levels of H2 v = 2, J = 29 and v = 1, J = 31, that
+2 The errors quoted here are the formal errors derived from the least
+squares fit.
+Article number, page 3 of 5
+
+J=29
+J=31
+0
+dissociation limit
+-1
+-2
+-3
+-4
+g
+-5
+2
+6
+8
+10
+12
+0
+4
+R (au)A&A proofs: manuscript no. 45358arxaa
+Table 1. Properties of the two detected quasi-bound levels, v = 2, J = 29 and v = 1, J = 31, of H2 and their emission spectrum.
+Transition
+˜ν
+λ
+A
+Ar
+τd
+Eqb
+upper
+label
+cm−1
+µm
+s−1
+s−1
+s
+K
+2 − 0 O(31)
+1387.04
+7.2096
+2.704E-11
+5.482E-06
+8.130E12
+676.0
+2 − 0 Q(29)
+4165.40
+2.4007
+4.592E-08
+5.482E-06
+8.130E12
+676.0
+2 − 0 S (27)
+7034.45
+1.4216
+2.240E-07
+5.482E-06
+8.130E12
+676.0
+2 − 1 Q(29)
+1944.67
+5.1423
+3.947E-07
+5.482E-06
+8.130E12
+676.0
+2 − 1 S (27)
+4590.29
+2.1785
+2.944E-06
+5.482E-06
+8.130E12
+676.0
+2 − 2 S (27)
+2399.19
+4.1681
+1.873E-06
+5.482E-06
+8.130E12
+676.0
+2 − 3 S (27)
+489.60
+20.4248
+2.582E-10
+5.482E-06
+8.130E12
+676.0
+1 − 0 Q(31)
+1974.02
+5.0658
+2.452E-07
+5.219E-06
+4.083E06
+1520.5
+1 − 0 S (29)
+4752.38
+2.1042
+2.101E-06
+5.219E-06
+4.083E06
+1520.5
+1 − 1 S (29)
+2531.65
+3.9500
+2.872E-06
+5.219E-06
+4.083E06
+1520.5
+1 − 2 S (29)
+586.98
+17.0364
+4.963E-10
+5.219E-06
+4.083E06
+1520.5
+Notes. ˜ν is the computed transition frequency; λ is the corresponding vacuum wavelength. A is the sum of the electric quadrupole and the magnetic
+dipole contributions to the Einstein radiative emission coefficients of the transition from Roueff & Abgrall (2022). Ar is the total radiative decay
+probability, and τd is the dissociation lifetime of the upper level. Eqb
+upper is the quasi-bound upper level energy expressed in K above the dissociation
+limit.
+Fig. 3. Level column densities, divided by their degeneracies, Ni/gi,
+plotted as a function of level energy, Ti, for the H2 lines measured in
+HH7. They are normalised to unity for the (v, J) = (1, 3) upper-state
+level at 6,952 K, which emits the 1–0 S (1) line. The two blue points (in
+the lower right) are for the newly analysed 2–1 S (27) and 1–0 S (29)
+lines. The dashed red line shows the best two-temperature LTE fit, as
+described in Sect. 4.
+have been detected. The upper level involved in the 2 − 1 S (27)
+transition at 2.1785 µm, 676 K above the dissociation energy of
+the ground state, is very stable against dissociation, whereas that
+of the 1 − 0 S (29) transition at 2.1042 µm, located 845 K higher,
+has a dissociation probability of approximately five percent and
+a dissociative lifetime, τd = ℏ/Γd, resulting from quantum tun-
+nelling through the centrifugal barrier (see Fig. 2) of 4.083 × 106
+s, corresponding to less than two months. This indicates that the
+shock wave in HH7 is partially dissociative.
+As shown in Fig. 3, the 5,000 K component represents a
+small percentage of the line-emitting H2 in HH7. As noted pre-
+viously, similar small percentages have been observed in HH1
+and in the Orion molecular outflow. Figure 3 also shows that H2
+in energy levels greater than ∼ 20,000 K above the ground state
+are populated only by this component. In the case of HH1, Gi-
+annini et al. (2015) observed a wide range of neutral and ionised
+species emitting in close proximity to the H2, many at opti-
+cal wavelengths. Their analysis yields a temperature range of
+8, 000 − 80, 000 K to account for the emission. They further find
+that neutral and fully ionised regions coexist inside the shock.
+However, for the heavily extincted H2 line emission from HH7
+(AV = 12−28 mag; P16), the species producing the optical emis-
+sion lines observed by Solf & Boehm (1987), Hartigan et al.
+(1989), and Hartigan et al. (2019) cannot be mixed with the H2.
+In view of the detections by Giannini et al. (2015), P16, and
+Geballe et al. (2017) of 5, 000 − 6, 000 K H2 in diverse envi-
+ronments, it seems likely that a small percentage of H2 existing
+at those temperatures is a common occurrence in collisionally
+shocked molecular gas, at least in cases where collisions between
+outflows and ambient molecular material occur at velocities of
+many tens of km s−1, as is the case for HH1, HH7, and the Orion
+Molecular Cloud. In addition, although transitions emitted from
+quasi-bound levels have only been detected towards HH7, we
+expect that they are present in HH1 and OMC-1 at roughly the
+same intensities relative to the stronger H2 lines, as in HH7.
+It is generally accepted that the maximum temperatures of
+nearly all of the vibrationally excited H2 in each of the above
+shocked clouds and in many others are suppressed by continu-
+ous shocks, in which the collisional acceleration of the ambient
+clouds and deceleration of the colliding outflows from the proto-
+stars are sufficiently gradual to heat the H2 only to temperatures
+of ∼2,000 K and prevent its dissociation (for more details, see
+Sect. I of P16 and references therein). The existence of H2 at a
+range of lower temperatures in gas cooling behind the contin-
+uous shocks, which has been demonstrated by observations of
+pure rotational lines (e.g. Neufeld et al. 2019), is also unsurpris-
+ing. However, it seems remarkable that virtually all of the highly
+ro-vibrationally excited H2 in levels with energies from 20,000
+K to 53,000 K is maintained in LTE at a single temperature of
+∼5,000 K, and that there is virtually no H2 at temperatures be-
+tween 2,000 K and 5,000 K. The mechanism that produces this
+bimodal temperature distribution is unclear.
+The location and morphology of the 5,000 K gas also is un-
+clear. The gas could be located in thin (currently unresolvable)
+sheets where the molecular cloud is being collisionally acceler-
+ated, the wind is being collisionally decelerated, or both. Its line
+emission could alternatively also be occurring in small clumps
+Article number, page 4 of 5
+
+100
+N[T,
+1 = 98.5%
+warm
+N[Thot] = 1.5%
+10-2
+10-4
+10-5
+10-6
+0
+1×10°
+2×10°
+3×10°
+4×10°
+5×104
+6×104
+Energy Level (K)E. Roueff et al.: Analysis of the first infrared spectrum of quasi-bound H2 line emission in Herbig-Haro 7
+of unusually hot and/or unusually dense gas scattered along the
+shock front. Comparisons of the velocity profiles of lines origi-
+nating from levels whose populations are dominated by the gas
+at 5,000 K with those from levels dominated by the 2,000 K
+component, at higher spectral resolution than has been employed
+to date, might reveal small differences and constrain the rela-
+tive locations of the two components. The good fit of the v=1,
+J = 31 column density to the fit to the population-energy di-
+agram (Fig. 3) indicates that dissociation is taking place in the
+5,000 K gas.
+One can consider if the short lifetime of the v=1, J=31 quasi-
+bound level indicates a significant continuous reformation of
+molecular hydrogen in the gas phase at high temperatures. We
+have estimated the formation rate of H2 through radiative asso-
+ciation via that resonance level, i, H + H ↔ H2i → H2 + hν,
+following the theory of Bain & Bardsley (1972), to be
+αres
+i
+=
+� 2πℏ2
+MkT
+�3/2
+(2I + 1)(2Ji + 1)
+Ai
+r Ad
+Air + Ad
+exp(−Ei/kT),
+(3)
+where Ad = 1/τ and M is the reduced mass of the colliding
+atoms. The contribution of v = 1, J = 31 with I = 1 and similar
+values of Ar and Ad is the most efficient by orders of magnitude.
+However, the derived value for its contribution at 5000K is 1.37
+× 10−30 cm3 s−1, which is negligible.
+Although it is difficult to assess the direct implication of the
+measurable presence of these quasi-bound H2 states for shock
+chemistry, their detections confirm the predictability of theoreti-
+cal computations based on highly accurate potential curves. The
+physical conditions associated with the astrophysical environ-
+ments in which their lines are emitted may not be reproducible
+in the laboratory due to their very large rotational quantum num-
+bers. Thus, they probably offer the only way to probe these lev-
+els. Martin et al. (1996) introduced quasi-bound levels of H2 in
+their master equation studies of collisional excitation of H2 by
+H and specifically mentioned the v = 2, J = 29, and v = 1,
+J = 31 quasi-bound levels detected here. However, they find
+that the highly excited rotational levels are not thermally popu-
+lated for the range of physical conditions that they considered,
+in contrast to what astronomical observations have revealed. Fi-
+nally, we note that the contribution of quasi-bound levels to the
+partition function of H2 and its isotopologues has been recently
+computed by Zúñiga et al. (2021) using the same potential as us.
+Acknowledgements. We thank K. Kaplan for having searched the two transi-
+tions in his IGRINS spectra of various PDRs. We are grateful to the referee for
+helpful comments. E.R. and H.A. acknowledge support by the Programme Na-
+tional de Physique et de Chimie du Milieu Interstellaire (PCMI) of CNRS/INSU
+with INC/INP co-funded by CEA and CNES.This research is based in large part
+on observations obtained at the international Gemini Observatory, a program of
+NSF’s NOIRLab, which is managed by the Association of Universities for Re-
+search in Astronomy (AURA) under a cooperative agreement with the National
+Science Foundation, on behalf of the Gemini Observatory partnership: the Na-
+tional Science Foundation (United States), National Research Council (Canada),
+Agencia Nacional de Investigación y Desarrollo (Chile), Ministerio de Ciencia,
+Tecnología e Innovación (Argentina), Ministério da Ciência, Tecnologia, Ino-
+vações e Comunicações (Brazil), and Korea Astronomy and Space Science In-
+stitute (Republic of Korea).
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+

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+arXiv:2301.02863v1  [math.OC]  7 Jan 2023
+Noname manuscript No.
+(will be inserted by the editor)
+A Regularized Limited Memory Subspace Minimization Conjugate
+Gradient Method for Unconstrained Optimization
+Wumei Sun1 · Hongwei Liu1 · Zexian Liu2
+Received: date / Accepted: date
+Abstract In this paper, based on the limited memory techniques and subspace minimization conjugate gra-
+dient (SMCG) methods, a regularized limited memory subspace minimization conjugate gradient method is
+proposed, which contains two types of iterations. In SMCG iteration, we obtain the search direction by min-
+imizing the approximate quadratic model or approximate regularization model. In RQN iteration, combined
+with regularization technique and BFGS method, a modified regularized quasi-Newton method is used in
+the subspace to improve the orthogonality. Moreover, some simple acceleration criteria and an improved
+tactic for selecting the initial stepsize to enhance the efficiency of the algorithm are designed. Additionally,
+an generalized nonmonotone line search is utilized and the global convergence of our proposed algorithm
+is established under mild conditions. Finally, numerical results show that, the proposed algorithm has a
+significant improvement over ASMCG PR and is superior to the particularly well-known limited memory
+conjugate gradient software packages CG DESCENT (6.8) and CGOPT(2.0) for the CUTEr library.
+Keywords Limited memory · Subspace minimization conjugate gradient method · Orthogonality ·
+Regularization model · Quasi-Newton method
+Mathematics Subject Classification (2010) 49M37 · 65K05 · 90C30
+1 Introduction
+Consider problem
+min
+x∈Rn f(x),
+(1)
+where f : Rn → R is a continuously differentiable nonlinear function.
+Wumei Sun
+E-mail: [email protected]
+Hongwei Liu �
+E-mail: [email protected]
+Zexian Liu
+E-mail: [email protected]
+1 School of Mathematics and Statistics, Xidian University, Xi’an 710126, China
+2 School of Mathematics and Statistics, Guizhou University, Guiyang 550025, China
+
+2
+Wumei Sun1 et al.
+Throughout the article, we use the following notations. sk−1 = xk − xk−1, fk = f(xk), gk = g(xk),
+yk−1 = gk−gk−1, ∥·∥ represents the Euclidean norm and λmax denotes the maximum eigenvalue. Moreover,
+dist{x, S} = inf{∥y − x∥, y ∈ S}, where x ∈ Rn and S ∈ Rn.
+Nonlinear conjugate gradient(CG) method is a well-known method for solving the problem (1), which
+main iteration is
+xk+1 = xk + αkdk, k = 0, 1, 2, · · · ,
+(2)
+where xk is the kth iteration point, αk > 0 is the stepsize and dk is the search direction obtained by
+d0 = −g0, dk = −gk + βkdk−1, k ≥ 1,
+(3)
+where gk is the gradient of f(xk) and βk is the conjugate parameter.
+It is shown in theory that the convergence and numerical performance variation of different CG meth-
+ods depend on the selection of conjugate parameters. Some very classical choices of the conjugate param-
+eter βk are Fletcher-Reeves(FR) [9], Polak-Ribi`ere-Polyak(PRP) [30,31], Dai-Yuan(DY) [7] and Hestenes-
+Stiefel(HS) [16], and are given by
+βF R
+k
+= ∥gk+1∥2
+∥gk∥2 ,
+βP RP
+k
+= gT
+k+1yk
+∥gk∥2 ,
+βDY
+k
+= ∥gk+1∥2
+dT
+k yk
+,
+βHS
+k
+= gT
+k+1yk
+dT
+k yk
+.
+CG algorithms have evolved considerably, and some well-known CG packages such as CG DESCENT [12,
+14] and CGOPT [5] have been proposed in recent years. Other recent related studies on nonlinear CG
+algorithms can be found in [4,13].
+The subspace minimization conjugate gradient (SMCG) algorithm, as a generalization of the CG algo-
+rithm, has received much attention from scholars [1,37], which can be traced back to the work of Yuan and
+Stoer [39]. The search direction of SMCG method is obtained by minimizing the following problem:
+min
+d∈Ωk
+gT
+k d + 1
+2dT Bkd,
+(4)
+where Ωk is a subspace spanned by the vectors gk and sk−1, i.e., Ωk = Span{gk, sk−1}, and Bk ∈ Rn×n is
+an approximation of Hessian matrix, which is positive definite and symmetric. Then the search direction d
+is given by
+d = ugk + vsk−1,
+(5)
+where u and v are both real parameters. Substituting (5) to (4) and combined with the standard secant
+equation Bksk−1 = yk−1, formula (4) is reorganized as follows:
+min
+u,v∈R
+
+ ∥gk∥2
+gT
+k sk−1
+
+
+T 
+ u
+v
+
+ + 1
+2
+
+ u
+v
+
+
+T 
+
+ρk
+gT
+k yk−1
+gT
+k yk−1 sk−1yk−1
+
+
+
+ u
+v
+
+ .
+(6)
+where ρk ≈ gT
+k Bkgk.
+On the basis of the Barzilai-Borwein(BB) method [2], Dai and Kou [6] proposed an effective BBCG3
+method for strictly convex quadratic minimization problem. Afterwards, based on BBCG3 method, Liu and
+Liu [26] proposed SMCG BB method for solving general unconstrained optimization problems. Motivated
+by SMCG BB method, some efficient SMCG methods [20,21,36,42] were later proposed, among which
+
+Title Suppressed Due to Excessive Length
+3
+the method based on the regularization model presented by Zhao et al. [42] is the best in the numerical
+performance.
+The nonlinear CG method is very effective for unconstrained optimization problems. However, the
+convergence of the algorithm can be very slow for some ill-posed problems and even for quadratic problems
+with very small dimensions, which may be due to the loss of orthogonality [15]. Hager and Zhang [15] pointed
+out theoretically that the generated successive gradients either in the CG method or the L-BFGS method
+for the quadratic test problem should be orthogonal. Yet, Hager and Zhang [15] observed that, when solving
+the quadratic strictly convex minimization problem PALMER1C in the CUTEr library [10], the CG method
+loses orthogonality due to the rounding errors, while L-BFGS method preserves the orthogonality. In view
+of this, they developed the limited memory CG method (CG DESCENT(6.8)) to correct the possible loss
+of orthogonality in ill conditioned optimization problems. For the test problems in the CUTEr library [10],
+their performance results indicated that CG DESCENT(6.8) has an significant improvement over their
+previously proposed package CG DESCENT(5.3).
+Although CG DESCENT(6.8) [15] is an efficient method for unconstrained optimization, it still suffers
+from the following shortcomings:
+(i) In the numerical implementation, the AWolfe line search [14] utilized in the algorithm CG DESCENT(6.8)
+does not guarantee global convergence.
+(ii) CG DESCENT(6.8) contains the following three pre-conditioners, corresponding to three different it-
+erations:
+Pk = I, Pk = Zk ˆB−1
+k+1ZT
+k , Pk = Zk ˆB−1
+k+1ZT
+k + σk ¯Zk ¯ZT
+k ,
+(7)
+where σk is determined by (4.2) of [15], ˆBk+1, Zk and ¯Zk are given by the matrices in literature [15]. These
+three pre-conditioners make the algorithm CG DESCENT(6.8) look complex.
+(iii) In the convergence analysis, the algorithm CG DESCENT(6.8) needs to impose the following assump-
+tions on the pre-conditioners:
+∥Pk∥ ≤ γ0, gT
+k+1Pkgk+1 ≥ γ1∥gk+1∥2, dT
+k P −1
+k
+dk ≥ γ2∥dk∥2,
+(8)
+where γ0 > 0, γ1 > 0 and γ2 > 0. These assumptions are comparatively strict and difficult to be verified in
+actual practice.
+To address the above-mentioned shortcomings, Liu et al. [27] presented an improved Dai¨CKou CG
+algorithm called CGOPT(2.0), which combines limited memory technology and Dai-Kou CG method.
+In CGOPT(2.0) [27], they utilized a modified quasi-Newton method to restore the lost orthogonality, and
+established the convergence of CGOPT(2.0) with fewer assumptions. Some numerical experiments indicated
+that CGOPT(2.0) is better than the famous CG software package CG DESCENT(6.8) [15].
+In view of the above discussion, a regularized limited memory subspace minimization conjugate gradient
+method on the basis of SMCG method and limited memory technique is studied in this paper. To recover
+orthogonality, we propose a modified regularized quasi-Newton method. The major contributions of this
+paper are the following.
+1. A regularized limited memory subspace minimization conjugate gradient algorithm is proposed, which
+combines limited memory technology and SMCG method.
+
+4
+Wumei Sun1 et al.
+2. Based on the idea of regularization and BFGS method, an improved regularized quasi-Newton method
+is exploited to improve orthogonality.
+3. Some simple acceleration criteria and an improved initial stepsize selection strategy are designed to
+enhance the efficiency of the algorithm. Additionally, an generalized nonmonotone line search condition
+is presented, which may be regarded as an extension of the Zhang-Hager’s [41] nonmonotone line search.
+4. The convergence of the method is built under mild conditions and the corresponding numerical perfor-
+mance shows that the new method is much more effective than the existing methods.
+The structure of the paper is as follows. In Section 2, we describe the detail of the regularized limited
+memory subspace minimization conjugate gradient algorithm, including the direction selection of SMCG
+iteration and regularized Quasi-Newton iteration and an effective acceleration technique. Moreover, the
+decision of the initial step size and the generalized nonmonotone Wolfe line search are also given in this
+section. In Section 3, some important properties of the search direction are analyzed and the global con-
+vergence of the proposed algorithm is established. Numerical experiments for algorithm comparison are
+showed in Section 4. Conclusions are given in the last section.
+2 A Regularized Limited Memory Subspace Minimization Conjugate Gradient Algorithm
+In the section, combining the idea of subspace minimization and regularization quasi-Newton method,
+we present a regularized limited memory subspace minimization conjugate gradient algorithm. Firstly,
+we give the choices of search direction under different iterations. Subsequently, we develop a very effec-
+tive acceleration technique, a modified initial step selection strategy and generalized nonmonotonic line
+search technology to optimize the performance of the proposed algorithm. Finally, the details of algorithm
+RL SMCG are described.
+2.1 Direction Selection of SMCG Iteration and Regularized Quasi-Newton Iteration
+The regularized limited memory subspace minimization conjugate gradient method mainly contains two
+kinds of iterations which are SMCG iteration and regularized quasi-Newton(RQN) iteration, respectively.
+Furthermore, the search direction derivation of the two iterations is also different.
+2.1.1 SMCG iteration
+The search direction selection of SMCG iteration is closely related to the properties of the objective function
+f(x) at the iteration point xk. By reference [3,38], defined
+tk =
+���2
+�
+fk−1 − fk + gT
+k sk−1
+�
+/
+�
+sT
+k−1yk−1
+�
+− 1
+��� ,
+(9)
+to describe how f(x) approaches a quadratic function on a line segment between xk−1 and xk. Literature
+[24] indicates that if the condition
+tk ≤ ¯ξ4 or
+�
+tk ≤ ¯ξ5 and tk−1 ≤ ¯ξ5
+�
+,
+(10)
+
+Title Suppressed Due to Excessive Length
+5
+is satisfied, where ¯ξ4 and ¯ξ5 are the smaller positive constants and ¯ξ4 < ¯ξ5, f(x) may be near to a quadratic
+function on a line between xk−1 and xk. Moreover, According to [32], we know that if the following condition
+¯ξ1 ≤ sT
+k−1yk−1
+∥sk−1∥2 ≤ ∥yk−1∥2
+sT
+k−1yk−1
+≤ ¯ξ2,
+(11)
+is satisfied, then the condition number of the Hessian matrix of the normal function may be not very large,
+here ¯ξ1 and ¯ξ2 are positive constants.
+Similar to [42], based on some certain properties of the function f(x) at the current point xk, we derive
+different search direction by dividing it into the following four cases.
+(i) If the condition (11) is satisfied while the condition (10) are not, this implies that the quadratic
+model may not be able to approach the objective function f(x) well at the present iteration point xk. Then,
+search direction dk will be obtained by minimizing the following cubic regular subproblem, i.e.
+min
+dk∈Ωk mk (dk) = dT
+k gk + 1
+2dT
+k Bkdk + 1
+3σk ∥dk∥3
+Bk ,
+(12)
+where Ωk is a subspace spanned by the vectors gk and sk−1, Bk ∈ Rn×n is an approximation of Hessian
+matrix, which is positive definite and symmetric and satisfying the secant condition Bksk−1 = yk−1, σk ≥ 0
+is an adaptive regularization parameter obtained from interpolation condition and dk is determined by
+dk = ukgk + vksk−1,
+(13)
+where vk and uk are parameters to be established. Obviously, we could obtain (12) by giving (4) a weighted
+regularization term 1
+3σk ∥dk∥3
+Bk. Substituting (13) to (12), it is easy to obtain that (12) is equivalent to
+min
+uk,vk∈R
+
+ ∥gk∥2
+gT
+k sk−1
+
+
+T 
+ uk
+vk
+
+ + 1
+2
+
+ uk
+vk
+
+
+T
+¯Bk
+
+ uk
+vk
+
+ + σk
+3
+������
+
+ uk
+vk
+
+
+������
+3
+¯
+Bk
+.
+(14)
+where ¯Bk =
+
+
+ρk
+gT
+k yk−1
+gT
+k yk−1 sk−1yk−1
+
+ is a positive definite and symmetric matrix, ρk is an estimate of
+gT
+k Bkgk. Similar to BBCG3 [6], we also use
+3
+2
+∥yk−1∥2
+sT
+k−1yk−1 I to estimate Bk in the term ρk, which means
+ρk = 3
+2
+∥yk−1∥2
+sT
+k−1yk−1 ∥gk∥2. Then, by solving problem (14) we obtain the following solutions about uk and vk:
+
+ uk
+vk
+
+ =
+
+
+1
+(1+σk(̟∗))∆k
+�
+gT
+k yk−1gT
+k sk−1 − sT
+k−1yk−1∥gk∥2�
+1
+(1+σk(̟∗))∆k
+�
+gT
+k yk−1∥gk∥2 − ρkgT
+k sk−1
+�
+
+ ,
+(15)
+among them,
+∆k =
+������
+ρk
+gT
+k yk−1
+gT
+k yk−1 sk−1yk−1
+������
+= ρksk−1yk−1 − (gT
+k yk−1)2 > 0,
+(16)
+σk and ̟∗ are the same as those in literature [42], which will not be repeated here.
+(ii) If both conditions (11) and (10) hold, this indicates that the objective function f(x) may approach
+the quadratic model at the current iteration point xk. Since that is the case, let σk = 0, i.e. we consider deriv-
+ing the search direction by solving the minimization problem (6). Like (i), we choose ρk = 3
+2
+∥yk−1∥2
+sT
+k−1yk−1 ∥gk∥2
+and ∆k is determined by (16), then we obtain the following unique solution of quadratic approximate
+
+6
+Wumei Sun1 et al.
+problem (6):
+
+ ¯uk
+¯vk
+
+ =
+
+
+1
+∆k (gT
+k yk−1gT
+k sk−1 − sT
+k−1yk−1∥gk∥2)
+1
+∆k (gT
+k yk−1∥gk∥2 − ρkgT
+k sk−1)
+
+ ,
+(17)
+here the search direction is calculated by dk = ¯ukgk + ¯vksk−1, where ¯uk and ¯vk are determined by (17).
+(iii) If condition (11) is not satisfied and the conditions
+���gT
+k yk−1gT
+k sk−1
+��� ≤ ¯ξ3sT
+k−1yk−1∥gk∥2 and sT
+k−1yk−1 ≥ ¯ξ1∥sk−1∥2,
+(18)
+are satisfied, where 0 ≤ ¯ξ3 ≤ 1, the condition number of the Hessian matrix may be lager, hence the search
+direction obtained in cases (i) and (ii) may not be better. However, the condition (18) can ensure sufficient
+descent and linear growth in HS conjugate gradient method. Moreover, because of the finite termination
+nature of the HS conjugate gradient method for solving exact convex quadratic minimization problems, this
+choice of direction allows for faster convergence of the algorithm. Then, in this case, the search direction is
+determined by (3) and βk = βHS
+k
+.
+(iv) If neither condition (11) nor (18) holds, then we pick the following direction, i.e. :
+dk = −gk.
+(19)
+In summary, the search direction in the SMCG iteration can be described as in the following:
+dk =
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ukgk + vksk−1,
+if (11) holds and (10) does not hold,
+¯ukgk + ¯vksk−1,
+if (11) holds and (10) holds,
+−gk + βHS
+k
+dk−1,
+if (11) does not hold and (18) holds,
+−gk,
+if neither (11) nor (18) holds,
+(20)
+where uk and vk are determined by (15); ¯uk and ¯vk are determined by (17).
+If the successive gradients have orthogonality or the lost orthogonality is restored, the algorithm performs
+SMCG iteration. On the contrary, if the orthogonality is lost, the iteration will turn to the following
+regularized quasi-Newton iteration to improve the orthogonality.
+2.1.2 Regularized Quasi-Newton(RQN) iteration
+When the successive gradients lose their orthogonality, the iteration switches from SMCG iteration to RQN
+iteration. In other words, a modified regularized BFGS algorithm in subspace Sk is proposed to restore the
+orthogonality, where Sk is a subspace generated by the following limited memory m search directions
+Sk = span {dk−1, dk−2, · · · , dk−m} ,
+where m > 0 and m is the number of limited memory. In this article, the limited memory m selected in our
+algorithm does not exceed 11. Then, as soon as orthogonality is corrected, the RQN iteration is terminated
+and the SMCG iteration is triggered immediately.
+First, we introduce some preparations for turning to RQN iteration. Let Sk ∈ Rn×m be a matrix which
+has columns consisting of dk−1, dk−2, · · · , dk−m. In similar fashion to limited memory CG method [15], we
+also assume that columns of Sk are line-independent. Let the QR factorization of Sk be Sk = Zk ¯Rk, where
+
+Title Suppressed Due to Excessive Length
+7
+the columns of Zk ∈ Rn×m form the normal orthogonal bases for subspace Sk and ¯Rk ∈ Rm×m is the
+upper triangular matrix with positive diagonal terms.
+If gk is included almost in subspace Sk, then we think that the orthogonality property of the algorithm
+may be lost. In this case, we interrupt the SMCG iteration and move to minimize the objective function in
+the subspace Sk:
+min
+z∈Sk f(xk + z).
+(21)
+The solution to the subspace problem (21) will improve the orthogonality and guide us to a suitable search
+direction that will lead us out of the subspace Sk. Similar to [15], we utilize the distance from gk to subspace
+Sk to judge whether orthogonality is lost. If the condition
+dist {gk, Sk} ≤ ˜η0∥gk∥
+(22)
+is satisfied, where 0 < ˜η0 < 1 and ˜η0 is small, we think gk is almost contained in Sk, it means that the
+orthogonality of the successive gradients has lost. Then, we switch to RQN iteration to solve the subspace
+problem (21) until the gradient is nearly orthogonal enough to the subspace to meet the condition
+dist {gk, Sk} ≥ ˜η1∥gk∥,
+(23)
+where 0 < ˜η0 < ˜η1 < 1. At this time, the algorithm iteration will go away subspace Sk and turn to the
+SMCG iteration. Because the column of Zk is the orthonormal basis of Sk, it’s not hard to know from the
+definition of dist {gk, Sk} that (22) and (23) can be expressed as
+�
+1 − ˜η2
+0
+�
+∥gk∥2 ≤
+���ZT
+k gk
+���
+2
+,
+(24)
+and
+�
+1 − ˜η2
+1
+�
+∥gk∥2 ≥
+���ZT
+k gk
+���
+2
+.
+(25)
+In [15], Hager and Zhang utilized the limited memory BFGS (L-BFGS) [22,28] method to solve the subspace
+problem (21) for restoring the orthogonality, and achieved better numerical results. However, it should
+be noted that the convergence analysis of the limited memory CG method [15] requires imposing strict
+assumptions (8) on the preprocessors (7). Because the dimension m of the chosen subspace Sk is usually
+small and when orthogonality is lost, the properties of the function at the iteration point maybe not
+very good. Based on these, we consider a regularized L-BFGS method in the subspace Sk for solving the
+subproblem (21).
+The search direction of general quasi-Newton method [40] for unconstrained optimization (1) is the
+form of dk = −B−1
+k gk, where Bk is a positive definite and symmetric approximation to the Hessian matrix.
+As one of the most popular methods of quasi-Newton method, L-BFGS method stores the approximate
+Hessian matrix of the objective function using small memory and computes the search direction dk using
+the nearest m vector pairs of (sk−i, yk−i), i = 0, 1, . . . , m − 1.
+
+8
+Wumei Sun1 et al.
+Ueda and Yamashita [35] presented a regularized Newton method for nonconvex unconstrained opti-
+mization, whose search direction dk is obtained by solving the following linear equations:
+�
+∇2f(xk) + µI
+�
+dk = −∇f(xk),
+(26)
+where µ > 0 is referred to as the regularized parameter. The regularized Newton method [35] generally
+defaults to a step size of 1, and global convergence is guaranteed by controlling the parameter µk. However,
+as a type of Newton method, the regularized Newton method in [35] must solve the Hessian matrix of
+f which is particularly computationally complex. To address this drawback, some scholars proposed the
+regularized limited memory BFGS-type method [33,23] for solving unconstrained optimization problems,
+i.e. the search direction dk is the solution of the following equations
+(Bk + µI) dk = −∇f(xk),
+(27)
+where matrix Bk is an approximate Hessian determined by a particular quasi-Newton method. Regular-
+ization technology can effectively improve the efficiency of quasi-Newton method in solving ill-conditioned
+problems. Nevertheless, when computing Bk by the L-BFGS method, it is very hard to calculate (Bk + µI)−1.
+Hence, motivated by [34], we present a regularized quasi-Newton method which combines the BFGS method
+with the regularized technique to improve orthogonality in the m-dimensional subspace Sk. In this paper,
+we consider Bk + µI as an approximation of ∇2f(xk) + µI. Because the matrix Bk is the approximate
+Hessian of f(xk) and Bk + µI can be used as an approximate Hessian of f(xk) + µ
+2 ∥x∥2. At this point, we
+utilize (sk, yk(µ)) instead of (sk, yk), where
+yk(µ) = (∇f(xk+1) + µxk+1) − (∇f(xk) + µxk) = yk + µsk.
+Note that the regularized BFGS method stores as many vector pairs as the traditional BFGS method and
+hence it does not require additional memory.
+In [19], a effective BFGS quasi-Newton method for solving nonconvex unconstrained minimization was
+proposed by Li and Fukushima [19], in which the matrix Bk+1 is updated by
+Bk+1 =
+
+
+
+Bk − BksksT
+k Bk
+sT
+k Bksk
++ ykyT
+k
+sT
+k yk ,
+if
+sT
+k yk
+∥sk∥2 > υ∥gk∥α,
+Bk,
+otherwise ,
+where υ > 0 and α > 0. Some recent advances about modified BFGS method can be found in [18,11,34].
+Inspired by the quasi-Newton methods described above, we propose an improved regularized BFGS
+method to solve the subproblem (21) in subspace Sk.
+Remark 1. In what follows, the variables with hats belong to subspace Sk , distinguished from the ones
+found in the full space Rn.
+Let ˆx = (ˆx1, ˆx2, · · · , ˆxm, )T ∈ Rm. The subproblem (21) can be expressed as
+min
+ˆx∈Rm f(xk + ˆx1dk−1 + ˆx2dk−2 + · · · + ˆxmdk−m).
+(28)
+
+Title Suppressed Due to Excessive Length
+9
+Similar to [27], because the regularized quasi-Newton directions in the subspace Sk always transform to
+the full space Rn and QR decomposition of matrix Sk, we can obtain dk = Zk ˆdk, ˆgk = ZT
+k gk, ˆyk = ZT
+k yk,
+ˆsT
+k ˆyk = sT
+k yk, ∥ˆsk∥2 = ∥sk∥2 and ˆfk = fk.
+Let Bk(µ) = Bk + µI, then inspired by Li and Fukushima [19], we develop an improved regularized
+BFGS method to solve the above subproblem (28) with a search direction of the form
+ˆdk+1 = − ˆB−1
+k+1(µ)ˆgk+1,
+(29)
+where ˆBk+1(µ) is given by
+ˆBk+1(µ) =
+
+
+
+ˆBk(µ) −
+ˆ
+Bk(µ)ˆskˆsT
+k ˆ
+Bk(µ)
+ˆsT
+k ˆ
+Bk(µ)ˆsk
++ ˆyk(µ)ˆyT
+k (µ)
+ˆsT
+k ˆyk(µ) ,
+if mod(k, l) ̸= 0 and ˆsT
+k ˆyk(µ)
+ˆsT
+k ˆsk
+≥ υ,
+ˆI,
+otherwise ,
+(30)
+where υ > 0, mod(k, l) ̸= 0 represents the remainder for k modulo l, ˆyk(µ) = ˆyk + µˆsk and µ > 0 is an
+important regularized parameter. The condition mod(k, l) ̸= 0 means the matrix ˆBk(µ) will be reset to
+the identity matrix ˆI after updating l times, which ensures the good convergence of the algorithm. In the
+paper, we set l = max(m2, 20). Obviously, ˆsT
+k ˆyk(µ) > 0, and as soon as the matrix ˆBk(µ) is symmetric and
+positive definitive, it is not hard to prove that the matrix ˆBk+1(µ) is symmetric and positive definitive.
+As a very important regularization parameter, µ is closely related to the convergence analysis of the
+regularized BFGS method. In this paper, the idea of the trust-region radius is used to find the suitable
+search direction by controlling µ, in other words, The ratio of objective function value reduction to model
+function value reduction is utilized. Then, give the definition of a ratio function rk( ˆdk, µ) as follows
+rk( ˆdk, µ) =
+ˆf(xk) − ˆf(xk + αk ˆdk)
+ˆf(xk) − ˆqk( ˆdk, µ)
+,
+(31)
+where ˆqk : Rm × R → R is a function of the form
+ˆqk( ˆdk, µ) = ˆf(xk) + αkˆgT
+k ˆdk + 1
+2α2
+k ˆdT
+k ˆBk(µ) ˆdk.
+(32)
+Then, if the ratio function rk( ˆdk, µ) is relatively large, this means that compared with the reduction of the
+model function, the reduction of the objective function is large enough, we choose to reduce the parameter
+µ. On the flip side, if the ratio function rk( ˆdk, µ) is relatively small, i.e., ˆf(xk) − ˆf(xk + αk ˆdk) is small,
+we will increase µ. In addition, to ensure that the algorithms converge well, we limit µ to an interval, i.e.
+0 < µmin < µ < µmax. In general, if the next iteration point is closer to the current iteration point, the
+reduction of the function value may not be obvious. At this time, we hope to get a new iteration point by
+modifying the search direction, then the search direction improved by regular parameter µ may be a good
+choice. Therefore, if ∥ˆsk∥2 ≤ ˆτ (ˆτ > 0), our choice and update of µ are as follows:
+µk+1 =
+
+
+
+max {µmin, σ1µk} ,
+if rk( ˆdk, µ) ≥ σ3,
+min {µmax, σ2µk} ,
+otherwise,
+(33)
+where 0 < σ1 ≤ 1, σ2 > 1 and 0 < σ3 ≤ 1. Otherwise, we choose µ = 0, i.e., the regularized BFGS method
+is reduced to a general BFGS method.
+
+10
+Wumei Sun1 et al.
+Remark 2. In order to simplify the symbol and facilitate writing, we still record the updated symbol
+µk+1 as µ.
+In the process of algorithm implementation, the search direction (29) in subspace Sk always converts
+to the full space Rn at each RQN iteration, i.e.,
+dk+1 = −Pkgk+1,
+(34)
+where
+Pk = Zk ˆB−1
+k+1(µ)ZT
+k
+(35)
+and ˆBk+1(µ) is given by (30).
+In Section 3, we will show that matrices ˆBk+1(µ) and Pk have some good properties in the RQN
+iteration, which is critical for the convergence analysis.
+2.2 An Effective Acceleration Technique
+In order to optimize the performance of the algorithm, Sun et al. [32] proposed an acceleration technique,
+which replaces (2) with the following new iterative form
+xk+1 = xk + ¯ηkαkdk,
+(36)
+where ¯ηk ≥ 0 is an acceleration parameter obtained from an interpolation function. In view of the numerical
+effect of the acceleration technique, our algorithm also takes it into account. Similar to reference [32], we
+minimize the following interpolation function to get the acceleration parameter ¯ηk:
+¯ηk = arg min q(ϕk(¯η)),
+(37)
+where ¯η ≥ 0, ϕk(¯η) = f(xk + ¯ηαkdk), and q(ϕk(¯η)) represents the interpolation function defined by ϕk(¯η).
+In the paper, we consider minimizing the quadratic interpolation function [29] q(ϕk(0), ϕ′
+k(0),ϕ′
+k(1)), then,
+¯ηk = arg min q(ϕk(0), ϕ′
+k(0), ϕ′
+k(1)),
+(38)
+By minimizing (38) we have
+¯ηk = −¯ak
+¯bk
+, ¯bk ≥ ¯ǫ,
+(39)
+where ¯ak = αkgT
+k dk, ¯bk = αk(g¯z − gk)Tdk, g¯z = ∇f(¯z), ¯z = xk + αkdk and ¯ǫ > 0 is a small constant.
+We propose the following acceleration criterion, which is simpler than the rule in reference [32], that is
+¯bk ≥ ¯ǫ, ∥s¯z∥2 ≤ ¯τ, ∥gk∥2 ≤ ˆτ, |¯tk+1| < ¯c, and |sT
+k g¯z| ≥ Max(ς, ¯ς · ¯bk)
+(40)
+where ¯ǫ, ¯τ, ˆτ, ¯c, ς and ¯ς are all small positive constants, ¯bk = αk(g¯z − gk)T dk, s¯z = ¯z − xk, ¯z = xk + αkdk,
+|¯tk+1| = | 2(fk−f¯z+gT
+¯z s¯z)
+sT
+¯z g¯z
+− 1|, f¯z = f(¯z) and g¯z = ∇f(¯z). When the condition (40) holds, we accelerate
+the algorithm and update the relevant variables. In addition, one of the necessary conditions for successful
+acceleration is that the trial iteration point must satisfy the line search condition. Therefore, if the algorithm
+
+Title Suppressed Due to Excessive Length
+11
+accelerates successfully, update the iteration point xk+1 by using (36). Otherwise the algorithm acceleration
+fails and returns to the original algorithm, at which point ¯ηk = 1, update the iteration point xk+1 with (2).
+In reference [32], the acceleration criterion is divided into three cases, which seems to be more complex,
+while our acceleration criterion has only one case and the form is simpler.
+2.3 Choices of the Initial Stepsize and the Generalized Nonmonotone Wolfe Line Search
+It is well known that the design of the search direction and the conditions of the line search are two
+critical factors which affect the efficiency of the line search algorithm. In this subsection, we will develop
+an improved nonmonotone Wolfe line search which can be regarded as an extension of the Zhang-Hager’s
+[41] nonmonotone line search. In addition, an improved initial step selection strategy is designed.
+For the sake of convenience, we express the one-dimensional line search function as
+φk(α) = f(xk + αdk), α ≥ 0.
+The choice of the initial stepsize α0
+k is of great importance for a line search in an optimization method. For
+the Newton-like methods, choosing the initial step α0
+k = 1 is important to speed up convergence. For the
+conjugate gradient methods, it is essential to use information from the current iteration of the problem to
+make initial guesses [29]. In the conjugate gradient method, there have been various ways to choose the
+initial stepsize, for example, see [5,12,15,29]. However, it did not have an agreement on which is the best.
+In particular, Hager and Zhang [15] select the initial step in CG DESCENT as below:
+α0
+k =
+
+
+
+arg min ¯q
+�
+φk (0) , φ′
+k (0) , φk (¯τ1αk−1)
+�
+, if φk (¯τ1αk−1) ≤ φk (0) ,
+¯τ2αk−1,
+otherwise,
+(41)
+where ¯q
+�
+φk (0) , φ′
+k (0) , φk (τ1αk−1)
+�
+represents the interpolation function given by the three values φk (0) ,
+φ′
+k (0) and φk (τ1αk−1) , ¯τ1 and ¯τ2 are positive parameters. In CGOPT, Dai and Kou [5] determined the
+initial stepsize in the following way:
+α0
+k =
+
+
+
+α
+if |φk (α) − φk (0)| / (τ3 + φk (0)) > τ4,
+arg min ¯q
+�
+φk (0) , φ′
+k (0) , φk (α)
+�
+, otherwise,
+(42)
+where α = max
+�
+τ5αk−1, −2 |fk − fk−1| /gT
+k dk
+�
+, τ3 > 0, τ4 > 0 and τ5 > 0. Most recently, Liu and Liu
+[26] discussed the development a very effective initial stepsize selection strategy for SMCG method by
+combining the BB methods and the interpolation technique.
+Based on the above research, we devise an improved strategy to obtain the initial stepsize. We first
+consider the initial stepsize for the search direction in the RQN iteration.
+(i) Initial stepsize of the search direction (34) with Bk+1(µ) ̸= I.
+Since the search direction ˆd is a quasi-Newton direction in the subspace Sk, then the initial stepsize
+α0
+k = 1 may be a good choice. Therefore, the trial initial stepsize can be stated as
+α0
+k =
+
+
+
+ˆαk,
+if
+((10) or ̟ ≤ τ2) holds and ¯αk > 0,
+1,
+otherwise,
+(43)
+
+12
+Wumei Sun1 et al.
+where
+ˆαk = min{max{¯αk, αmin}, αmax},
+¯αk = min ¯q(φk(0), φk
+′(0),φk(1)),
+̟ = |φk (1) − φk (0)| / (τ1 + φk (0)) , τ1 > 0, τ2 > 0 and αmax > αmin > 0.
+Here, ¯q
+�
+φk (0) , φ′
+k (0) , φk (1)
+�
+is a quadratic interpolation function for φk (0) , φ′
+k (0) , and φk (1) , and
+αmax and αmin represent two positive constants.
+(ii) Initial stepsize of the search direction (34) with Bk+1(µ) = I.
+α0
+k =
+
+
+
+ˆαk,
+if
+((10) or ̟ ≤ τ2) holds and ¯αk > 0,
+¯¯αk,
+otherwise,
+(44)
+where
+¯¯αk =
+
+
+
+max{min{αBB2
+k
+, αmax}, αmin}, if gT
+k sk−1 > 0,
+max{min{αBB1
+k
+, αmax}, αmin}, if gT
+k sk−1 ≤ 0,
+(45)
+For the initial stepsize of the search direction in the SMCG iteration. If the search direction dk is
+calculated by (20) with dk ̸= −gk, the initial stepsize is chosen in the same way as the RQN iteration,
+which is determined by (43). If the search direction dk is given by (19), the initial stepsize is determined
+by
+α0
+k =
+
+
+
+min{max{˜˜αk, αmin}, αmax}, if (10) holds, ∥gk∥2 ≤ 1, dk−1 ̸= −gk−1 and ˜˜αk > 0,
+¯¯αk,
+otherwise,
+(46)
+where ¯¯αk is determined by (45) and ˜˜αk = min q(φk(0), φk′(0),φk(¯¯αk)).
+Next, we introduce a generalized line search condition, which can be regarded as a development of the
+Zhang-Hager’s nonmonotone line search. We recall the nonmonotone line search introduced by Zhang and
+Hager [41]
+f(xk + αkdk) ≤ Ck + δαkgT
+k dk,
+(47)
+where
+Ck+1 = ηkQkCk + f k+1
+Qk+1
+, Qk+1 = ηkQk + 1,
+(48)
+0 < δ < 1, and ηk ∈ [0, 1]. From (48), it is easy to see that Ck+1 is a convex combination of fk+1 and
+Ck. If C0 = f(x0), it is thus clear that Ck can be regard as a convex combination of the function values
+f(x0), f(x1), · · · , f(xk). It means that Ck can employ information about the known function values from
+the previous iteration. The Zhang-Hager’s nonmonotone line search (47) is reduced to the standard Armijo
+line search condition when ηk = 0 for each k.
+As it was reported in [41], the nonmonotone line search proposed by Zhang and Hager plays a crucial
+role in generating an appropriate stepsize compared to the monotone line search method. Based on (47)
+and (48), Huang et al. [17] presented a very effective nonmonotone line search technique, which can be
+
+Title Suppressed Due to Excessive Length
+13
+regard as an extension of Zhang-Hager’s nonmonotone line search, that is
+Ck+1 = ηkQkCk + fk+1
+Qk+1
+≤ Ck + δkαkgT
+k dk,
+(49)
+where ηk ∈ [ηmin, ηmax], δmax < 1, 0 < δmin < (1 − ηmax)δmax, δmin ≤ δk ≤
+δmax
+Qk+1 and Qk+1 is computed
+by (48).
+Inspired by the previous discussion, we will study a generalized nonmonotone Wolfe line search technique
+based on (48) and (49). Considering the acceleration technique, the generalized nonmonotone Wolfe line
+search conditions are as follows:
+Ck+1 ≤ Ck + δk¯ηkαkgT
+k dk,
+(50)
+gT
+k+1dk ≥ σgT
+k dk,
+(51)
+where 0 < δmin < δk < δmax < 1, σ ∈ (0, 1), Q0 = 1, C0 = f0, ¯ηk is an acceleration parameter determined
+by (39), Ck and Qk are updated as follows
+Ck+1 = ηkQkCk + f(xk+1)
+Qk+1
+, Qk+1 = ηkQk + 1, f(xk+1) = f(xk + ¯ηkαkdk),
+(52)
+where ηk ∈ [0, 1]. Specially,
+Q1 = 2.0, C1 = min{C0, f1 + 1.0},
+(53)
+when k ≥ 1, Ck+1 and Qk+1 are updated by (52), and ηk is given as
+ηk =
+
+
+
+1,
+if Ck − fk+1 > 0.95|Ck| and k > 100,
+0.9, otherwise.
+(54)
+Here ηk is a parameter that controls the degree of non-monotonicity, referred to [25].
+Furthermore, we demonstrate that the generalized nonmonotone Wolfe line search is an extension of
+the Zhang-Hager’s nonmonotone Wolfe line search method. It follows from (50) that we get
+f(xk + ¯ηkαkdk) ≤ (Qk+1 − ηkQk)Ck + Qk+1δk¯ηkαkgT
+k dk.
+(55)
+Since Qk+1 − ηkQk = 1, (50) is equivalent to
+f(xk + ¯ηkαkdk) ≤ Ck + Qk+1δk¯ηkαkgT
+k dk,
+(56)
+It is easy to see that if δk =
+δ
+Qk+1 , nonmonotone line search condition (56) reduces to the Zhang-Hager’s
+nonmonotone Wolfe line search condition (47). This means that the Zhang-Hager’s nonmonotone Wolfe
+line search condition in [41] can be considered as a particular version of (50).
+2.4 A Regularized Limited Memory Subspace Minimization Conjugate Gradient Algorithm(RL SMCG)
+In this subsection, we describe the regularized limited memory subspace minimization conjugate gradient
+algorithm in detail. As mentioned above, the regularized limited memory subspace minimization conjugate
+gradient algorithm is made of two kinds of iterations. The “state” in Algorithm 1 represents for the type of
+
+14
+Wumei Sun1 et al.
+iteration, i.e., state= “SMCG” means that SMCG iteration will be carried out, and state= “RQN” means
+that RQN iteration will be performed.
+Algorithm 1 RL SMCG
+Step 0. Chosen x0 ∈ Rn, ε > 0, ˜η0, ˜η1, υ, m, ξ1, ξ2, ξ3, ξ4, ξ5, σ1, σ2, σ3, µmin, µmax, τ, ¯τ, ¯c, ς, ¯ς, ¯ǫ, τ1,
+τ2, δk, σ, IterRestart := 0, IterQuad := 0 and MinQuad. Set state = “SMCG” and k := 0.
+Step 1. If ∥gk∥∞ ≤ ε, stop.
+Step 2. Compute the search direction.
+If (state = “SMCG”), then
+If k = 0, then d0 = −g0.
+elseif (IterQuad = MinQuad and IterQuad ̸= IterRestart), set
+dk = −gk, IterQuad = 0, and IterRestart = 0.
+else
+Determine the search direction dk by (20).
+end
+elseif (state = “RQN”), then
+Compute Pk by (35), and compute the search direction dk by (34).
+end
+Step 3. Determine the corresponding initial step size α0
+k from (43), (44) and (46) according to the different
+iteration directions in the Step 2.
+Step 4. Determine a stepsize αk satisfying the generalized nonmonotone Wolfe line search (50) and (51)
+with initial stepsize α0
+k.
+Step 5.Compute the trial iteration ¯z = xk + αkdk and g¯z = ∇f(¯z). If ∥g¯z∥∞ ≤ ε, then stop; otherwise, go
+to Step 6.
+Step 6. Acceleration procedure.
+If the condition (40) holds, then go to 6.1.
+6.1. Compute ¯ak = αkgT
+k dk, ¯bk = αk(g¯z − gk)T dk and ¯ηk by (39).
+6.2. Update the iteration point as xk+1 = xk + ¯ηkαkdk and compute fk+1 and gk+1.
+6.3. If fk+1 satisfies (50) and gk+1 satisfies (51), go to Steps 8. Otherwise, go to Steps 7.
+else
+go to Steps 7.
+end
+Step 7. Update the variable as xk+1 = xk + αkdk. Compute fk+1 and gk+1.
+Step 8. Update restart conditions.
+Step 9. Update Qk+1 and Ck+1 with (52).
+Step 10. Update iteration type.
+If (state = “SMCG”), then
+If (24) holds, then state = “RQN”.
+elseif (state = “RQN”), then
+If (25) holds, then state = “SMCG”.
+end
+Step 11. Set k := k + 1 and go to Step 1.
+Remark 3. Notably, when the lost orthogonality is corrected, our algorithm terminates the RQN
+iteration and immediately calls the SMCG iteration. However, the limited memory CG method [15] first
+
+Title Suppressed Due to Excessive Length
+15
+carries out the complex preprocessing CG iteration after the orthogonality is improved. This means that
+algorithm RL SMCG is more simple compared to the limited memory CG method [15].
+3 Convergence Analysis
+In the section, we establish the global convergence of the algorithm RL SMCG under the following assump-
+tions and properties.
+Define N to be an open neighborhood of the level set L (x0) = {x ∈ Rn : f (x) ≤ f (x0)} , where x0 is
+an initial point.
+Assumption 1 (i) The objective function f is continuously differentiable in N and the level set is bounded
+from below. (ii) The gradient g of the objective function is Lipschitz continuous in N, i.e., there exists a
+constant L > 0 such that ∥g(x) − g(y)∥ ≤ L ∥x − y∥ , ∀x, y ∈ N.
+Under these assumptions, we have the following several properties.
+Lemma 1 Suppose that Assumption 1 holds. Then, for ˆBk+1(µ) in (30), there exist three constants ˆξ1 >
+0, ˆξ2 > 0 and ˆξ3 > 0 such that
+λmax
+�
+ˆBk+1(µ)
+�
+≤ ˆξ1, λmax
+�
+ˆB−1
+k+1(µ)
+�
+≤ ˆξ2,
+��� ˆB−1
+k+1(µ)
+��� ≤ ˆξ3.
+Proof We know that Zk is a normal orthogonal basis of Sk and the dimension m < +∞, hence we have
+ξ0 > 0 such that ∥Zk∥ ≤ ξ0. According to (30) and the property of the matrix norm in finite dimensional
+spaces, we can get that λmax
+�
+ˆBk(µ)
+�
+= 1 or
+λmax
+�
+ˆBk+1(µ)
+�
+≤ λmax
+�
+ˆBk(µ)
+�
++ λmax
+�
+−
+ˆBk(µ)ˆskˆsT
+k ˆBk(µ)
+ˆsT
+k ˆBk(µ)ˆsk
+�
++ λmax
+� ˆyk(µ)ˆyT
+k (µ)
+ˆsT
+k ˆyk(µ)
+�
+(57)
+≤ λmax
+�
+ˆBk(µ)
+�
++ ˆyT
+k (µ)ˆyk(µ)
+ˆsT
+k ˆyk(µ)
+.
+Further, by ˆyk(µ) = ˆyk + µˆsk, µ > 0, we get
+ˆyT
+k (µ)ˆyk(µ)
+ˆsT
+k ˆyk(µ)
+= ∥ˆyk∥2 + µ2
+k∥ˆsk∥2 + 2µˆsT
+k ˆyk
+ˆsT
+k ˆyk + µ∥ˆsk∥2
+= ∥ˆyk∥2 + µˆsT
+k ˆyk
+ˆsT
+k ˆyk + µ∥ˆsk∥2 + µˆsT
+k ˆyk + µ2
+k∥ˆsk∥2
+ˆsT
+k ˆyk + µ∥ˆsk∥2
+≤ ∥ˆyk∥2 + µˆsT
+k ˆyk
+ˆsT
+k ˆyk
++ µ
+≤ L2ξ2
+0∥ˆsk∥2
+ˆsT
+k ˆyk
++ 2µ
+≤ L2ξ2
+0
+υ
++ 2µmax.
+The fourth inequality above is obtained from ˆyk = ZT
+k yk, ∥Zk∥ ≤ ξ0 and Assumption 1 (ii). Because
+ˆBk(µ) will be set to ˆI after a maximum of l updates, combining with (57) easy to get λmax
+�
+ˆBk+1(µ)
+�
+≤
+1 + lL2ξ2
+0
+υ
++ 2lµmax ≜ ˆξ1.
+
+16
+Wumei Sun1 et al.
+Let ˆPk(µ) = ˆB−1
+k+1(µ). According to (30) and some simple matrix operations, we have that ˆPk(µ) = ˆI
+or
+ˆPk(µ) =
+�
+ˆI − ˆyk(µ)ˆsT
+k
+ˆsT
+k ˆyk(µ)
+�T
+ˆPk−1(µ)
+�
+ˆI − ˆyk(µ)ˆsT
+k
+ˆsT
+k ˆyk(µ)
+�
++
+ˆskˆsT
+k
+ˆsT
+k ˆyk(µ).
+(58)
+It is not difficult to that λmax
+��
+ˆI − ˆyk(µ)ˆsT
+k
+ˆsT
+k ˆyk(µ)
+�T �
+ˆI − ˆyk(µ)ˆsT
+k
+ˆsT
+k ˆyk(µ)
+��
+= ∥ˆyk(µ)∥2∥ˆsk∥2
+(ˆsT
+k ˆyk(µ))
+2
+. For any ˆz ̸= 0 ∈ Rm and
+ˆPk(µ) in (58), we have
+ˆzT ˆPk(µ)ˆz = ˆzT
+�
+ˆI − ˆyk(µ)ˆsT
+k
+ˆsT
+k ˆyk(µ)
+�T
+ˆPk−1(µ)
+�
+ˆI − ˆyk(µ)ˆsT
+k
+ˆsT
+k ˆyk(µ)
+�
+ˆz +
+�
+ˆsT
+k ˆz
+�2
+ˆsT
+k ˆyk(µ)
+≤ λmax
+�
+ˆPk−1(µ)
+�
+ˆzT
+�
+ˆI − ˆyk(µ)ˆsT
+k
+ˆsT
+k ˆyk(µ)
+�T �
+ˆI − ˆyk(µ)ˆsT
+k
+ˆsT
+k ˆyk(µ)
+�
+ˆz +
+�
+ˆsT
+k ˆz
+�2
+ˆsT
+k ˆyk(µ)
+≤ λmax
+�
+ˆPk−1(µ)
+�
+λmax
+��
+ˆI − ˆyk(µ)ˆsT
+k
+ˆsT
+k ˆyk(µ)
+�T �
+ˆI − ˆyk(µ)ˆsT
+k
+ˆsT
+k ˆyk(µ)
+��
+∥ˆz∥2 +
+�
+ˆsT
+k ˆz
+�2
+ˆsT
+k ˆyk(µ)
+≤ λmax
+�
+ˆPk−1(µ)
+� ∥ˆyk(µ)∥2∥ˆsk∥2
+�
+ˆsT
+k ˆyk(µ)
+�2
+∥ˆz∥2 +
+∥ˆsk∥2
+ˆsT
+k ˆyk(µ)∥ˆz∥2.
+The above inequality is divided by ∥ˆz∥2, and the resulting inequality is maximized, then we have
+λmax
+�
+ˆPk(µ)
+�
+≤ λmax
+�
+ˆPk−1(µ)
+� ∥ˆyk(µ)∥2∥ˆsk∥2
+�
+ˆsT
+k ˆyk(µ)
+�2
++
+∥ˆsk∥2
+ˆsT
+k ˆyk(µ)
+≤ λmax
+�
+ˆPk−1(µ)
+�
+
+
+∥ˆyk(µ)∥2
+ˆsT
+k ˆyk(µ)
+∥ˆsk∥2
+ˆsT
+k ˆyk(µ)
+
+ + ∥ˆsk∥2
+ˆsT
+k ˆyk
+≤ λmax
+�
+ˆPk−1(µ)
+� �L2ξ2
+0
+υ
++ 2µmax
+� ∥ˆsk∥2
+ˆsT
+k ˆyk
++ ∥ˆsk∥2
+ˆsT
+k ˆyk
+≤
+�L2ξ2
+0
+υ2
++ 2µmax
+υ
+�
+λmax
+�
+ˆPk−1(µ)
+�
++ 1
+υ .
+The third inequality above is obtained from ˆyk = ZT
+k yk, ∥Zk∥ ≤ ξ0 and Assumption 1 (ii). Because ˆPk(µ)
+will be set to ˆI after a maximum of l updates, it is easy to know that there exists a constant ˆξ2 > 0 such
+that λmax
+�
+ˆB−1
+k+1(µ)
+�
+= λmax
+�
+ˆPk(µ)
+�
+≤ ˆξ2.
+Since ˆB−1
+k+1(µ) is a positive definite and symmetric matrix, we have
+��� ˆB−1
+k+1(µ)
+���
+2 = λmax
+�
+ˆB−1
+k+1(µ)
+�
+≤
+ˆξ2. As a result, using the equivalence property of matrix norm in a finite dimensional space, it follows that
+there exists a constant ˆξ3 > 0 such that
+��� ˆB−1
+k+1(µ)
+��� ≤ ˆξ3. The proof is completed.
+⊓⊔
+Lemma 2 Suppose that Assumption 1 holds. Then, for Pk in (35), there exist three constants γ0 > 0, γ1 > 0
+and γ2 > 0 such that
+∥Pk∥ ≤ γ0, gT
+k+1Pkgk+1 ≥ γ1 ∥gk+1∥2 , dT
+k P −1
+k
+dk ≥ γ2 ∥dk∥2 ,
+(59)
+where P −1
+k
+denotes the pseudoinverse of Pk.
+Proof By (25), (35) and Lemma 1, we obtain that
+∥Pk∥ =
+���Zk ˆB−1
+k+1(µ)ZT
+k
+��� =
+��� ˆB−1
+k+1(µ)
+��� ≤ ˆξ3 ≜ γ0,
+gT
+k+1Pkgk+1 = gT
+k+1Zk ˆB−1
+k+1(µ)ZT
+k gk+1
+
+Title Suppressed Due to Excessive Length
+17
+= ˆgT
+k+1 ˆB−1
+k+1(µ)ˆgk+1
+≥ λmin
+�
+ˆB−1
+k+1(µ)
+�
+∥ˆgk+1∥2
+≥ 1
+ˆξ1
+�
+1 − ˜η2
+1
+�
+∥gk+1∥2 ≜ γ1 ∥gk+1∥2 ,
+dT
+k P −1
+k
+dk = dT
+k Zk ˆB−1
+k+1(µ)ZT
+k dk = ˆdT
+k ˆB−1
+k+1(µ) ˆdk ≥ 1
+ˆξ2
+��� ˆdk
+���
+2
+= 1
+ˆξ2
+∥dk∥2 ≜ γ2 ∥dk∥2 .
+Therefore, we can get the conclusions. The proof is completed.
+⊓⊔
+Subsequently, we provide some properties of the search directions produced by the algorithm RL SMCG,
+which are crucial for the following convergence analysis.
+Lemma 3 Suppose that Assumption 1 holds. Then, there exists a constant c1 > 0 such that the search
+directions (20) and (34) are calculated by algorithm RL SMCG satisfy the sufficient descent condition:
+gT
+k dk ≤ −¯c1∥gk∥2.
+(60)
+Proof We divide the proof into the following two cases.
+(i) SMCG iteration. Similar to the proof of Lemma 4.1 of [42], it is easy to have
+gT
+k dk ≤ −c1∥gk∥2,
+where c1 = min
+�
+1
+2, 1 − ¯ξ3,
+2
+3¯ξ2 ,
+1
+3¯ξ2 ,
+2
+5¯ξ2
+�
+.
+(ii) RQN iteration. According to Lemma 2, we have
+gT
+k dk = −gT
+k Pk−1gk ≤ −γ1 ∥gk∥2 .
+By setting ¯c1 = min {c1, γ1}, we can obtain (60). The proof is completed.
+⊓⊔
+Lemma 4 Suppose that Assumption 1 holds. Then, there exists a constant c1 > 0 such that the search
+directions (20) and (34) are calculated by algorithm RL SMCG satisfy
+∥dk∥ ≤ ¯c2∥gk∥.
+(61)
+Proof We divide the proof into the following two cases.
+(i) SMCG iteration. Referring to the proof procedure of Lemma 4.2 of [42], it is easy to get
+∥dk∥ ≤ c2∥gk∥,
+where c2 = max
+�
+1, 1 + L
+¯ξ1 , 20
+¯ξ1
+�
+.
+(ii) RQN iteration. According to Lemma 2, we obtain ∥dk∥ = ∥−Pk−1gk∥ ≤ γ0 ∥gk���.
+By setting ¯c2 = min {c2, γ0}, we can obtain (61). The proof is completed.
+⊓⊔
+The following lemmas are very critical for the convergence analysis of algorithm RL SMCG.
+
+18
+Wumei Sun1 et al.
+Lemma 5 Suppose that Assumption 1 holds, and the sequence {xk} is generated by the algorithm RL SMCG.
+Then,
+If acceleration succeeds:
+¯ηkαk ≥
+�1 − σ
+L
+� ��gT
+k dk
+��
+∥dk∥2 .
+(62)
+If acceleration fails:
+αk ≥
+�1 − σ
+L
+� ��gT
+k dk
+��
+∥dk∥2 .
+(63)
+Where σ are given by (51).
+Proof We divide the proof into the following two cases.
+(i) If acceleration succeeds:
+From (51) and Assumptions 1 (ii), we obtain that
+(σ − 1)gT
+k dk ≤ g(xk + ¯ηkαkdk)T dk − gT
+k dk = (g(xk + ¯ηkαkdk) − gk)T dk ≤ L¯ηkαk∥dk∥2,
+which yields
+¯ηkαk ≥
+�σ − 1
+L
+� gT
+k dk
+∥dk∥2 .
+This means that (62) holds.
+(ii) If acceleration fails:
+Let ¯ηk = 1, and the rest of the proof procedure is the same as before.
+⊓⊔
+Lemma 6 Suppose that Assumption 1 holds, and the sequence {xk} is generated by the algorithm RL SMCG.
+Then, there holds that fk ≤ Ck for each k.
+Proof We divide the proof into the following two cases.
+(i) If acceleration succeeds:
+The new iterative update format is xk+1 = xk + ¯ηkαkdk, where ¯ηk = − ¯ak
+¯bk . Through (56), we have
+fk+1 = f(xk + ¯ηkαkdk) ≤ Ck + Qk+1δk¯ηkαkgT
+k dk. Combining (52), δk > 0, lemma 5 and the sufficiently
+descent property of the direction dk+1, we have fk+1 < Ck. The remaining proof process refers to Lemma
+5.1 in [42], we can obtain fk+1 ≤ Ck+1, hence fk ≤ Ck is established for each k.
+(ii) If acceleration fails:
+Let ¯ηk = 1, and the rest of the proof procedure is the same as before.
+⊓⊔
+Theorem 1 Suppose that Assumption 1 holds, the sequence {xk} is generated by the algorithm RL SMCG.
+Then,
+lim
+k→∞ ∥gk∥ = 0.
+(64)
+Proof We divide the proof into the following two cases.
+(i) If acceleration succeeds:
+By Assumptions 1, lemmas 3 - 5 and the generalized nonmonotone Wolfe line search conditions (50)
+and (51), we get that
+Ck+1 ≤ Ck + δk¯ηkαkgT
+k dk
+(65)
+
+Title Suppressed Due to Excessive Length
+19
+≤ Ck + δmin¯ηkαkgT
+k dk
+≤ Ck + δmin 1 − σ
+L
+(gT
+k dk)2
+∥dk∥2
+≤ Ck + δmin(1 − σ)¯c2
+1
+L¯c2
+2
+∥gk∥2
+= Ck + β∥gk∥2.
+Where β =
+δmin(1−σ)¯c2
+1
+L¯c2
+2
+. Combined with (53), we have C1 ≤ C0 that means that Ck is monotonically
+decreasing. According to lemma 6 and Assumption 1 (i), we know Ck is bounded from below. Then
+∞
+�
+k=0
+β∥gk∥2 < ∞,
+therefore,
+lim
+k→∞ ∥g(xk)∥ = 0.
+(ii) If acceleration fails:
+Let ¯ηk = 1, and the rest of the proof procedure is the same as before.
+⊓⊔
+4 Numerical Experiments
+In this section, we compare the numerical performance of RL SMCG with ASMCG PR [32], CG DESCENT(6.8)
+[15] and CGOPT(2.0) [27] for the 145 test problems from CUTEr library [10]. The codes of CG DESCENT(6.8)
+[15] and CGOPT(2.0) [27] can be downloaded from http://users.clas.ufl.edu/hager/papers/Software and
+https://web.xidian.edu.cn/xdliuhongwei/en/paper.html or http://lsec.cc.ac.cn/ dyh/software.html, respec-
+tively.
+In the numerical experiments, we set the parameters of RL SMCG as: ¯ξ1 = 10−10, ¯ξ2 = 1.2 × 104,
+¯ξ3 = 5 × 10−5, ¯ξ4 = 10−4, ¯ξ5 = 0.08, ˜η0 = 10−9, ˜η1 = 0.5, υ = 5 × 10−7, m = min{n, 11}, σ1 = 0.1,
+σ2 = 5, σ3 = 0.85, ˆτ = 1, ¯τ = 0.225, ¯c = 0.1, ς = 5 × 10−5(n ≤ 11), ς = 5 × 10−6(n > 11), ¯ς = 5 × 10−3,
+τ1 = 0.1, τ2 = 135, δk = 0.0005 and σ = 0.9999. CG DESCENT(6.8) and CGOPT(2.0) take the default
+parameters in their codes but the stopping conditions. Note that the number of memory m for RL SMCG is
+min{n, 11} while the number of memory for CG DESCENT(6.8) is 11. All test methods in the experiment
+are terminated if ∥gk∥∞ ≤ 10−6 is satisfied, and we set the number of iterations for all test algorithms to
+be no more than 200,000. In addition, all algorithms are running in Ubuntu 10.04 LTS.
+We will show the performances of the test methods using the performance profiles introduced by Dolan
+and Mor´e [8]. In the following Figs. 1-12, “Niter”,“Nf”,“Ng” and “Tcpu” represent the number of iterations,
+the number of function evaluations, the number of gradient evaluations and CPU time(s), respectively.
+We divided the numerical experiments in three teams.
+In the first set of numerical experiments, figures 1-4 illustrate the performance profiles of RL SMCG
+and ASMCG PR [32]. From Figs. 1, 2, 3 and 4, we can observe that RL SMCG has a quite significant
+improvement over ASMCG PR in terms of the number of iterations, the number of function evaluations,
+
+20
+Wumei Sun1 et al.
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+τ
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+P(τ)
+ARL_SMCG
+ASMCG_PR
+Fig. 1: Niter
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+τ
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+P(τ)
+ARL_SMCG
+ASMCG_PR
+Fig. 2: Nf
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+τ
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+P(τ)
+ARL_SMCG
+ASMCG_PR
+Fig. 3: Ng
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+τ
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+P(τ)
+ARL_SMCG
+ASMCG_PR
+Fig. 4: Tcpu
+the number of gradient evaluations and CPU time. It indicates that the limited memory technique equipped
+in RL SMCG indeed brings quite significant numerical improvements.
+In the second set of numerical experiments, we give a comparison of the performance profiles of
+RL SMCG with CG DESCENT(6.8) [15]. Regarding the number of iterations and the number of func-
+tion evaluations in Fig. 5 and Fig. 6 respectively, we observe that RL SMCG is a little better than
+CG DESCENT(6.8) for the number of iterations and the number of function evaluations. As shown in
+Fig. 7, we can see that RL SMCG is much better than CG DESCENT(6.8) in terms of the number of
+gradient evaluations, because RL SMCG outperforms for about 71.5% of the CUTEr test problems, while
+the percentage of software CG DESCENT(6.8) is below 40%. It can be observe from Fig. 8 that RL SMCG
+is faster than CG DESCENT(6.8) in terms of CPU time. By Theorem 1, RL SMCG is globally conver-
+gent with the generalized nonmonotone Wolfe line search, while CG DESCENT (6.8) does not guarantee
+global convergence when using the rather efficient approximate Wolfe (AWolfe) line search. This means that
+RL SMCG is superior to CG DESCENT(6.8) for CUTEr library in theory and numerical performance.
+In the third set of the numerical experiments, comparing the performance of RL SMCG with CGOPT(2.0)
+[27]. As shown in Figs. 9 and 10, we can take a look at RL SMCG performs almost always better than
+CGOPT(2.0) in terms of the number of iterations and the number of function evaluations. Figures. 11 and
+12 indicates that RL SMCG outperforms CGOPT(2.0) in terms of the number of gradient evaluations and
+CPU time for the CUTEr library.
+
+Title Suppressed Due to Excessive Length
+21
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+τ
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+P(τ)
+ARL_SMCG
+CG_DESCENT(6.8)
+Fig. 5: Niter
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+τ
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+P(τ)
+ARL_SMCG
+CG_DESCENT(6.8)
+Fig. 6: Nf
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+τ
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+P(τ)
+ARL_SMCG
+CG_DESCENT(6.8)
+Fig. 7: Ng
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+τ
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+P(τ)
+ARL_SMCG
+CG_DESCENT(6.8)
+Fig. 8: Tcpu
+From the results of the above three numerical experiments, it is clear that the proposed algorithm
+RL SMCG is quite effective.
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+τ
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+P(τ)
+ARL_SMCG
+CGOPT(2.0)
+Fig. 9: Niter
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+τ
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+P(τ)
+ARL_SMCG
+CGOPT(2.0)
+Fig. 10: Nf
+
+22
+Wumei Sun1 et al.
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+τ
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+P(τ)
+ARL_SMCG
+CGOPT(2.0)
+Fig. 11: Ng
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+τ
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+P(τ)
+ARL_SMCG
+CGOPT(2.0)
+Fig. 12: Tcpu
+5 Conclusions
+In this paper, combined subspace minimization conjugate gradient method with limited memory technique,
+we presented a regularized limited memory subspace minimization conjugate gradient method, which con-
+tains two types of iteration. In the proposed algorithm, a modified regularized quasi-Newton method is
+given in small dimensional subspace to correct the orthogonality, and an improved initial step size selection
+strategy and some simple acceleration criteria are designed. Moreover, we establish the global convergence
+of the proposed algorithm by utilizing generalized nonmonotone Wolfe line search under some mild as-
+sumptions. Some numerical results suggest that our algorithm yields a tremendous improvement over the
+ASMCG PR and outperforms the most up-to-date limited memory CG software packages CG DESCENT
+(6.8) and CGOPT(2.0).
+6 Declarations
+6.1 Ethical Approval
+Not Applicable
+6.2 Availability of supporting data
+Data sharing not applicable to this article as no datasets were generated or analyzed during the current
+study.
+6.3 Competing interests
+The authors declare no competing interests.
+
+Title Suppressed Due to Excessive Length
+23
+6.4 Funding
+This research was supported by the National Natural Science Foundation of China (No. 11901561), the
+Natural Science Foundation of Guizhou (No. ZK[2022]084) and the Natural Science Basic Research Program
+of Shaanxi (No. 2021JM-396).
+6.5 Authors’ contributions
+Wumei Sun wrote the main manuscript text. Hongwei Liu and Zexian Liu reviewed and revised the
+manuscript.
+6.6 Acknowledgments
+The authors would like to thank the editor and the anonymous referees for their valuable suggestions and
+comments which have greatly improved the presentation of this paper.
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+

9tE4T4oBgHgl3EQfdwyk/content/tmp_files/2301.05094v1.pdf.txt

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+NV center magnetometry up to 130 GPa as if at ambient pressure
+Antoine Hilberer,1 Lo¨ıc Toraille,2, 3 Cassandra Dailledouze,1 Marie-Pierre Adam,1 Liam
+Hanlon,1 Gunnar Weck,2, 3 Martin Schmidt,1 Paul Loubeyre,2, 3 and Jean-Fran¸cois Roch1, ∗
+1Universit´e Paris-Saclay, CNRS, ENS Paris-Saclay,
+CentraleSupelec, LuMIn, F-91190 Gif-sur-Yvette, France
+2CEA DAM DIF, F-91297 Arpajon, France
+3Universit´e Paris-Saclay, CEA, Laboratoire Mati`ere en Conditions Extrˆemes, 91680 Bruy`eres-le-Chˆatel, France
+(Dated: January 13, 2023)
+Engineering a layer of nitrogen-vacancy (NV) centers on the tip of a diamond anvil creates a
+multipurpose quantum sensors array for high pressure measurements, especially for probing magnetic
+and superconducting properties of materials. Expanding this concept above 100 GPa appears to be
+a substantial challenge. We observe that deviatoric stress on the anvil tip sets a limit at 40-50 GPa
+for practical magnetic measurements based on optically detected magnetic resonance (ODMR) of
+NV centers under pressure. We show that this limit can be circumvented up to at least 130 GPa
+by machining a micropillar on the anvil tip to create a quasi-hydrostatic stress environment for the
+NV centers. This is quantified using the pressure dependence of the diamond Raman shift, the NV
+ODMR dependence on applied magnetic field, and NV photoluminescence spectral shift. This paves
+the way for direct and reliable detection of the Meissner effect in superconductors above 100 GPa,
+such as super-hydrides.
+Introduction. The diamond anvil cell (DAC) is rou-
+tinely used to synthesize compounds under megabar
+(100 GPa) pressures, exhibiting novel phenomena and
+remarkable properties. Recent examples such as the ob-
+servation of metal hydrogen [1], superconductivity close
+to ambient temperature in superhydrides [2–4], or su-
+perionic water ice [5] are lacking detailed magnetic or
+transport measurements for their definite proof and clear
+understanding. In particular, magnetic measurements re-
+main challenging at megabar pressures because they are
+mainly based on flux detection by inductive coils and
+must thus extract the signal of the few-micrometers sam-
+ples from the much larger magnetic background signal of
+the bulky DAC apparatus. This constraint can be cir-
+cumvented by implementing in the DAC sensing methods
+that exploit the magnetic sensitivity of nitrogen-vacancy
+(NV) centers in diamond [6–9].
+This method offers a
+tabletop optical microscopy instrumentation, the map-
+ping of the magnetic field in the sample chamber with
+micrometer spatial resolution and the absence of any sen-
+sitivity decrease with the sample size down to the mi-
+crometer scale. Another key feature is the easy combi-
+nation with synchrotron X-ray characterizations to cor-
+relate the magnetic or superconducting properties with a
+well-defined crystallographic structure [10]. Yet, the ex-
+tension of this technique to extreme pressures remains a
+challenge [11]. We investigate here how the existence of a
+deviatoric stress in the diamond anvil sets effective limits
+to the magnetic response of NV centers localized at the
+anvil tip to maximize sample proximity [6, 7]. We then
+propose and implement a method that overcomes that
+limit and keeps the full NV quantum sensing capabilities
+at pressures above 100 GPa.
+Experimental configuration.
+The negatively charged
+NV center is a point defect of diamond that emits visible
+photoluminescence (PL) by absorbing green photons
+and re-emitting red photons (at ambient pressure), with
+an electronic spin s = 1 in the ground and excited
+states. In the absence of external magnetic and stress
+fields, the ms = ±1 spin sublevels of the ground state
+are degenerate and separated by D = 2.87 GHz from
+the ms = 0 sublevel (Fig. 1a). Spin-dependent PL arises
+from a spin-selective difference in the non-radiative
+coupling to metastable singlet states, which also induces
+optical pumping into the ms = 0 state under green
+illumination [12].
+The energy difference between the
+sublevels of the ground state can then be read out
+from the change of the NV luminescence intensity upon
+scanning the frequency of an additional microwave
+excitation.
+Dips in the PL intensity indicate that
+the excitation microwave frequency is resonant with a
+transition between two sublevels, leading to optically
+detected magnetic resonance (ODMR) that can be easily
+implemented by optically addressing the NV centers
+through the diamond anvil [6].
+Here we use the same experimental configuration as in
+Ref. [6], keeping two crucial characteristics: 1) the NV
+centers are integrated in the DAC device by mounting a
+IIas ultra-pure Almax-Boehler design [100]-cut diamond
+anvil with a dense ensemble of NV centers (typically
+104
+NV/µm2) implanted at about 10 nm beneath
+the anvil surface using a nitrogen Focused Ion Beam
+(FIB) [13] (Fig. 1b); 2) the microwave excitation is
+applied using an external single-turn coil above the
+rhenium gasket of the DAC. The metallic gasket is
+machined with a slit, filled with an epoxy-glue mixture
+ensuring
+sample
+confinement
+and
+DAC
+mechanical
+stability, that re-distributes the induced currents in the
+metal, leading to a focusing and amplification of the
+microwave flux in the sample chamber similarly to a
+arXiv:2301.05094v1  [quant-ph]  12 Jan 2023
+
+2
+(a)
+(b)
+(d)
+(c)
+NV layer
+Anvil 1
+Anvil 2
+Gasket
+Pressure
+ms=0
+ms=±1
+D
+D+훿
+FIG. 1.
+(a) Energy diagram of the NV center ground state
+and evolution under stress. (b) Schematic cross-section of the
+location of NV centers implanted as a layer below the anvil
+culet surface. (c) Design of the machined gasket compatible
+with the MW excitation of the NV centers. Red arrows show
+initial MW excitation current in the wire loop, blue arrows
+are currents induced into the gasket. The areas shaded in red
+indicate the intensity of the MW field. (d) ODMR spectra
+of NV centers implanted in the tip of a standard diamond
+anvil at different pressures, as a function of a magnetic field
+applied along the [100] diamond axis. Green dashed lines are
+fits of the eigenfrequencies computed with the NV ground
+state Hamiltonian given by eq. 1.
+Lenz lens [14] (Fig. 1c).
+Upon pressure increase, the
+PL excitation wavelength was decreased to match the
+blueshift of the NV absorption spectrum [11] by using
+continuous-wave (cw) lasers at successive wavelengths
+532, 488, 457 and 405 nm. A customized confocal optical
+microscope was used to collect the PL. A static vector
+magnetic field was applied on the DAC using three
+Helmholtz coil pairs with an amplitude ranging between
+0 and 10 mT. The magnetic field was aligned along
+the DAC axis with accuracy ±0.5◦.
+This orientation
+corresponds to the diamond [100] crystal axis for which
+all NV centers have equivalent responses to stress and
+magnetic field. Pressure in the DAC was measured using
+the calibrated diamond Raman phonon mode at the
+anvil tip [15].
+Stress effect on the NV magnetic response. We per-
+formed cw-ODMR experiments on the NV centers un-
+der pressures ranging from 10 GPa to 70 GPa. At each
+pressure point, we collected the ODMR spectrum for the
+ensemble of NV centers under varying amplitude of the
+applied magnetic field.
+1300
+1400
+1500
+1600
+1700
+Raman shift ν (cm−1)
+Intensity (a.u.)
+P = 92 GPa
+Culet
+Micropillar
+40 µm
+(a)
+(b)
+(c)
+(d)
+*
+25
+50
+75
+Pressure (GPa)
+3.0
+3.2
+3.4
+Vmol (cm3.mol−1)
+1.95
+2.00
+2.05
+2.10
+2.15
+2.20
+2.25
+2.30
+NV− ZPL energy (eV)
+Linear fits
+Micropillar
+Standard anvil
+600
+670
+λ (nm)
+PL (a.u.)
+Anvil 1
+Anvil 2
+Gasket
+PTM
+NV layer
+−0.20 −0.15 −0.10 −0.05 0.00
+ln(V/V0)
+0.00
+0.05
+0.10
+0.15
+0.20
+ln(ν/ν0)
+Occelli et al. [16]
+Micropillar
+FIG. 2.
+(a) Scanning electron microscope image of a FIB-
+machined micropillar on a diamond anvil culet of 100 µm
+diameter. The bottom panel shows a schematic cross-section
+with the distortion under pressure of the culet. (b) Energy of
+the NV center zero-phonon line (ZPL) as a function of pres-
+sure and diamond volume, recorded for NV centers implanted
+in and out of the micropillar. Inset: typical PL spectra of the
+NV centers recorded at 0, 37 and 78 GPa (bottom to top).
+The arrows indicate the ZPL position. (c) Diamond Raman
+spectra recorded on a pressurized microstructured diamond
+anvil at 92 GPa, on and outside the micropillar. In the spec-
+trum taken on the micropillar, the peak indicated by the star
+reveals hydrostatic compression. (d) Raman frequency shift
+measured on the micropillar as a function of relative diamond
+volume. Data from [16] is a reference of the Raman shift of
+diamond under hydrostatic pressure.
+The data are shown in Fig. 1d. Four effects of stress
+on the ODMR signals are observed. First, the zero-field
+center frequency D = 2.87 GHz increases almost linearly
+with a slope of 9.6 MHz/GPa to a value D + δ, where δ
+is the pressure induced variation. Second, a splitting ∆σ
+appears between the transition lines in the absence of an
+external magnetic field. This splitting increases almost
+linearly with pressure with a slope of 3.9 MHz/GPa and
+originates in deviatoric stress at the anvil culet. Conse-
+quently, at a given pressure, the quasi-linear evolution of
+the Zeeman splitting due to the applied magnetic field
+can only be recovered above a compensating amplitude
+of the magnetic field that increases with pressure. This
+detrimental influence of stress hence weakens the NV
+
+3.8
+51 GPa
+1.025
+α =0.56
+41 GPa
+3.6
+1.000
+MW frequency (GHz)
+α =0.57
+30 GPa
+PL intensity (a.u.)
+α =0.56
+0.975
+20 GPa
+3.4
+α =0.56
+0.950
+10 GPa
+3.2
+α =0.67
+0.925
+0.900
+3.0
+0.875
+2.8
+0.850
+0
+5
+10
+5
+10
+5
+10
+5
+10
+5
+10
+α model fit
+B applied in [100] (mT)3
+sensing magnetic sensitivity. Furthermore, the required
+larger applied bias magnetic field isn’t aligned with a
+given NV axis here, to overlap responses from all NV
+orientations, and thus mixes the sublevels of the ground
+state. This mixing perturbs the optically induced spin
+polarization and quenches the PL [17]. Third, the shape
+of the ODMR spectra differs from the conventional
+symmetrical pair of peaks.
+The contrast of the low
+frequency branch becomes gradually smaller than the
+high frequency branch.
+After vanishing at a pressure
+around 40 GPa, a slightly positive contrast reappears
+(increase of PL at resonance) above 50 GPa under high
+enough magnetic field.
+Finally, the overall observed
+ODMR contrast decreases severely under pressure.
+In the diamond lattice under mechanical stress (or
+equivalently strain), the Hamiltonian describing the NV
+center ground state is modified by a spin-mechanical in-
+teraction [18, 19] related to the stress tensor
+↔σ.
+The
+stress tensor must exhibit the cylindrical symmetry of
+the anvil. At the anvil tip, the stress components parallel
+(σ∥) and perpendicular (σ⊥) to the surface differ. Due to
+continuity of the normal stress component, σ⊥ is equal to
+the experimental pressure P in the DAC chamber. The
+tangential component, σ∥, is reduced by a factor α com-
+pared to σ⊥. Using a simplified model of a semi-infinite
+anvil with a flat face and a circularly symmetric distribu-
+tion of pressure applied to this face, the α parameter was
+estimated about 0.6 [20]. Neglecting off-diagonal shear
+stress components, the stress tensor then reads as:
+↔σ=
+�
+�
+αP
+0
+0
+0
+αP
+0
+0
+0
+P
+�
+� .
+(1)
+Using this stress tensor, the diagonalization of the NV
+ground state Hamiltonian yields modified spin resonance
+frequencies which can be approximated to first order as:
+ν± = D + δ ± ∆/2
+(2)
+where δ is the spectral shift due to compression, and
+∆ =
+�
+∆2σ + ∆2
+B is the quadratic sum of the splittings
+respectively induced by the stress and by the magnetic
+field (see Supplementary Material for the full expression).
+Since eq. (2) is exact only for low off-axis magnetic field,
+a full numerical diagonalization was used to accurately
+fit the measured resonance frequencies, as shown by the
+green dashed lines in fig. 1d. Only two parameters, α
+and P, are hence needed to predict the magnetic field re-
+sponse under stress. We obtained a value α = 0.56 that is
+essentially constant with pressure, quantifying deviatoric
+stress close to the 0.6 value given in Ref. [20].
+Deviatoric stress thus introduces major modifications
+to the NV behavior as the anisotropic compression of
+the diamond host lattice distorts the C3v symmetry of
+the NV center. Here we quantified changes within the
+NV ground triplet states, but the stress dependence of
+the singlet states and the excited triplet states remains
+unexplored and is difficult to assess. As a hypothesis, we
+attribute the observed modification and ultimate loss of
+ODMR contrast to the effect of deviatoric stress on these
+levels involved in the contrast mechanism [21].
+This
+hypothesis is corroborated by recent results obtained
+on
+microdiamonds
+compressed
+quasi-hydrostatically
+inside the sample chamber of a DAC, for which the
+ODMR signal could be conserved up to 140 GPa [22].
+These results converge toward a possible circumventing
+strategy by ensuring hydrostatic compression of the NV
+centers.
+Restoring hydrostaticity with diamond microstructura-
+tion. A strategy to try to mitigate deviatoric stress can
+be implemented by microstructuring the diamond anvil
+culet. A successful geometry is presented in Fig. 2a. A
+pillar, 7 µm in diameter and with a 2 µm deep trench
+around it was FIB-machined on an NV-implanted dia-
+mond anvil culet. The pillar surface is thus disconnected
+from the anvil surface submitted to deviatoric stress in-
+duced by anvil cupping tension [23, 24].
+This also al-
+lows the pressure-transmitting medium (PTM) to fill the
+trench to immerse the pillar in a stress field close to hy-
+drostatic conditions. The pillar is then equivalent to a di-
+amond microdisk that would be integrated in the sample
+chamber of the DAC but ensures perfect reproducibility
+and removes any interface with the diamond culet to op-
+timize PL measurements. As seen below, this design is
+also very robust and can withstand extreme pressures.
+The hydrostaticity of the stress exerted on diamond
+under pressure can be tested by measuring the Raman
+frequency of the diamond optical phonon.
+Under hy-
+drostatic conditions, the dependence of the frequency of
+the Raman scattering with diamond volume follows a
+Gruneisen relation of parameter γ = 0.97(1) whereas the
+frequency shift is smaller under deviatoric stress [16]. As
+seen in Fig. 2c, the Raman spectra measured at the dia-
+mond anvil culet on the micropillar and away from it dif-
+fer. In both cases, the broad asymmetric peak is associ-
+ated to the stress distribution within the thickness of the
+anvil that is optically probed and the high frequency edge
+is used to estimate the pressure [15]. At the micropil-
+lar, a well separated peak appears with higher frequency
+shift. The pressure evolution of its center wavenumber
+perfectly matches the value obtained for diamond under
+hydrostatic pressure [16] as shown in Fig. 2d. This indi-
+cates that the tip of the micropillar hosting part of the
+NV center layer is then close to hydrostatic pressure.
+Accordingly the PL spectrum of the NV layer in
+the micropillar shows a pressure induced blue shift
+(Fig. 2b) that can be quantified with the zero-phonon line
+(ZPL) [11]. While the NV ZPL dependence with pressure
+is not linear, its evolution becomes linear when plotted
+versus the compressed diamond volume estimated using
+
+4
+0
+5
+10
+3.0
+3.5
+4.0
+4.5
+5.0
+MW frequency (GHz)
+20 GPa
+α =0.95
+α model fit
+5
+10
+50 GPa
+α =0.95
+5
+10
+B applied in [100] (mT)
+74 GPa
+α =0.95
+5
+10
+103 GPa
+α =0.95
+5
+10
+131 GPa
+α =0.97
+0.93
+0.94
+0.95
+0.96
+0.97
+0.98
+0.99
+1.00
+PL intensity (a.u.)
+(a)
+4.0
+4.5
+Microwave frequency (GHz)
+PL intensity (a.u.)
+|B| = 6 mT
+P = 73 GPa
+P = 103 GPa
+P = 131 GPa
+(b)
+FIG. 3.
+(a) ODMR spectra obtained from NV centers implanted in a micropillar at varying pressures, as a function of magnetic
+field applied along the diamond [100] axis. Fitted values of the stress anisotropy parameter α ≃ 0.95 indicate quasi-hydrostatic
+conditions. (b) ODMR spectra recorded for NV centers in the micropillar for a magnetic field of 6 mT amplitude. The signals
+at 73 GPa, 103 GPa, and 131 GPa are normalized for clarity, with contrast values of 5%, 3% and 1.5% respectively.
+the diamond equation of state [25].
+Linear fit gives a
+slope of −769±4 meV/(cm3·mol−1). A similar measure-
+ment performed on a non modified diamond anvil yields
+a weaker slope of −434 ± 2 meV/(cm3·mol−1). This sig-
+nificant difference in the pressure dependence of the ZPL
+is another indication of the deviatoric stress reduction
+caused by the microstructuration.
+ODMR measurements were also performed for the NV
+centers hosted in the micropillar.
+As shown in Fig. 3
+corresponding to the pressure evolution up to 130 GPa,
+most of the detrimental effects previously observed and
+attributed to deviatoric stress are now suppressed. The
+spectra consistently show a negative contrast remaining
+almost constant up to at least 100 GPa. Increasing fur-
+ther the pressure up to 130 GPa (where the experiment
+was stopped by one of the anvils breaking), a slight
+decrease of the contrast was observed and is attributed
+to a degraded efficiency of the microwave excitation
+for frequencies higher than 4 GHz. The magnetic field
+response remains also unchanged across the whole tested
+pressure range.
+The ODMR spectra exhibit a very
+low zero-field splitting ∆σ of 0.29 ± 0.03 MHz/GPa
+with increasing pressure, and a shift of the zero-field
+center frequency D + δ of 13.42 ± 0.14 MHz/GPa. As
+shown in Fig. 4 these values differ significantly from
+those measured for NV centers in standard anvils, and
+were consistent across four experimental runs performed
+on different anvils, with pillars machined either using
+a FIB or a femtosecond laser.
+Applying the model
+described above for the spin-mechanical interaction,
+the evolution of the ODMR eigenfrequencies versus
+the applied magnetic field were well-fitted using an
+anisotropy parameter α ≃ 0.95 that stays constant
+within the pressure range tested (Fig. 3a). Since α ≃ 1
+(a)
+(b)
+0
+25
+50
+75
+100
+125
+150
+Pressure (GPa)
+3.0
+3.5
+4.0
+4.5
+D + δ (GHz)
+Standard anvil
+Micropillar run 1, 2, 3, 4
+Doherty et al. [11]
+Dai et al. [22]
+0
+25
+50
+75
+100
+125
+150
+Pressure (GPa)
+0
+50
+100
+150
+200
+250
+∆σ (MHz)
+Standard anvil
+Micropillar run 1, 2, 3, 4
+FIG. 4.
+(a) Pressure dependence of ODMR center frequency
+D + δ, showing a quasi-linear shift of 13.42 ± 0.14 MHz/GPa
+on the micropillar compared to 9.68 ± 0.8 MHz/GPa on the
+standard anvil. The extrapolation of the values measured up
+to 60 GPa in [11] and the fit up to 140 GPa from [22] are
+given for comparison.
+(b) Pressure dependence of ODMR
+frequency splitting ∆σ at zero magnetic field.
+At the mi-
+cropillar, ∆σ increases by 0.29 ± 0.03 MHz/GPa instead of
+3.89±0.06 MHz/GPa with the standard geometry of the anvil.
+
+5
+would indicate perfect hydrostaticity, this result gives
+an independent confirmation of the almost hydrostatic
+pressure applied on the NV centers in the micropillar.
+Consequently, the microstructuration strategy enables
+efficient magnetic field sensing at pressures higher
+than 100 GPa with a sensitivity improved by orders of
+magnitude compared to the use of a standard anvil with
+a flat tip (see Supplementary Material).
+Conclusion.
+Microstructuration of diamond anvils,
+implemented here by machining a micropillar on the
+culet,
+provides quasi-hydrostatic conditions for NV
+centers implanted in the anvil up to 100 GPa and
+above.
+With this design NV magnetic sensing can be
+implemented under such extreme pressures as if at
+ambient pressure. This work opens the way to sensitive
+and spatially resolved magnetic measurements in the
+constrained environment of the DAC which should now
+be used for a convincing observation of the Meissner
+effect in super-hydrides.
+We are grateful to Olivier Marie and Gr´egoire Le
+Caruyer for machining of the diamond culets, to Flo-
+rent Occelli for assistance in DACs preparation and to
+Doroth´ee Colson and Anne Forget for annealing the dia-
+mond anvils after nitrogen implantation. This work has
+received funding from the EMPIR program co-financed
+by the Participating States and the European Union’s
+Horizon 2020 research and innovation program (20IND05
+QADeT), from the Agence Nationale de la Recherche
+under the project SADAHPT and the ESR/EquipEx+
+program (grant number ANR-21-ESRE-0031), and from
+the Paris ˆIle-de-France R´egion in the framework of DIM
+SIRTEQ. JFR acknowledges support from Institut Uni-
+versitaire de France.
+∗ [email protected]
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+Coherent driving of direct and indirect excitons in a quantum dot molecule
+Frederik Bopp,1, ∗ Johannes Schall,2 Nikolai Bart,3 Florian Vogl,1 Charlotte Cullip,1 Friedrich
+Sbresny,4 Katarina Boos,4 Christopher Thalacker,1 Michelle Lienhart,1 Sven Rodt,2 Dirk Reuter,5
+Arne Ludwig,3 Andreas Wieck,3 Stephan Reitzenstein,2 Kai M¨uller,4 and Jonathan J. Finley1, †
+1Walter Schottky Institut, School of Natural Sciences, and MCQST,
+Technische Universit¨at M¨unchen, Am Coulombwall 4, 85748 Garching, Germany
+2Technische Universit¨at Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
+3Faculty of Physics and Astronomy, Ruhr-Universit¨at Bochum,
+Universit¨atsstraße 150, 44801 Bochum, Germany
+4Walter Schottky Institut, School of Computation, Information and Technology, and MCQST,
+Technische Universit¨at M¨unchen, Am Coulombwall 4, 85748 Garching, Germany
+5Paderborn University, Department of Physics, Warburger Straße 100, 33098 Paderborn, Germany
+(Dated: February 1, 2023)
+Quantum dot molecules (QDMs) are one of the few quantum light sources that promise deter-
+ministic generation of one- and two-dimensional photonic graph states.
+The proposed protocols
+rely on coherent excitation of the tunnel-coupled and spatially indirect exciton states. Here, we
+demonstrate power-dependent Rabi oscillations of direct excitons, spatially indirect excitons, and
+excitons with a hybridized electron wave function. An off-resonant detection technique based on
+phonon-mediated state transfer allows for spectrally filtered detection under resonant excitation.
+Applying a gate voltage to the QDM-device enables a continuous transition between direct and
+indirect excitons and, thereby, control of the overlap of the electron and hole wave function. This
+does not only vary the Rabi frequency of the investigated transition by a factor of ≈ 3, but also
+allows to optimize graph state generation in terms of optical pulse power and reduction of radiative
+lifetimes.
+I.
+INTRODUCTION
+The use of single photons as flying qubits facilitates
+transmission of quantum information at the speed of
+light. However, transfer over large distances unavoidably
+comes with losses and decoherence. Encoding quantum
+information on an ensemble of entangled photons, a so-
+called graph state [1], instead of a single photon, provides
+a possibility to mitigate the losses is transmission chan-
+nels [2, 3].
+Furthermore, other specific forms of graph
+states such as photonic cluster states promise realization
+of measurement-based quantum computing [4] as well as
+quantum error correction [5, 6].
+Following
+the
+Lindner-Rudolph
+protocol [7],
+one-
+dimensional photonic cluster states can be deterministi-
+cally generated by utilizing single spins in semiconductor
+quantum dots (QDs). The polarization entanglement of
+up to five photons has been achieved in a one-dimensional
+cluster state has been achieved [8] and most recent ex-
+periments demonstrate localizable entanglement over ten
+photons [9]. While the nanophotonic environment of QDs
+provides high photon emission rates, the cluster state cre-
+ation fidelity is limited by spin dephasing and modified
+selection rules in the presence of a transverse magnetic
+field [9]. These challenges can be overcome by using a
+pair of tunnel coupled and vertically stacked QDs, so
+called quantum dot molecules (QDMs) [10]. Besides pro-
+longing the spin coherence compared to single quantum
+∗ [email protected]
+† fi[email protected]
+dots [11], QDMs possess an unique level structure [12].
+This level structure enables, for example, spin rotations
+and spin readout transitions without application of a
+magnetic field. The ability to create spatially indirect
+excitons, with one charge carrier occupying the upper
+and one the lower QD [13], provides a cycling transi-
+tion which can be used for generating time-bin entangled
+photons [10]. Moreover, QDMs are proposed to generate
+two-dimensional photonic cluster states by harnessing the
+tunnel coupling between the two QDs and inter-dot con-
+trol gates [14].
+The foundation for creating one- and two-dimensional
+photonic cluster states is the occurrence of excitons in
+spatially direct, spatially indirect, and hybridized config-
+urations [15]. In these different configurations, the charge
+carriers of an electron-hole pair are located in the same
+QD, in different QDs, or one of the charge wave func-
+tions is hybridized over both quantum dots, respectively.
+In each configuration, the overlap of the electron and
+hole wave functions and, therefore, the transition dipole
+moment (TDM) of the corresponding optical transition
+differs. This results in a change of both the lifetime of the
+excited state and the pulse area needed for maximal pop-
+ulation inversion [16]. While the lifetime influences the
+cluster state creation efficiency and rate, the π-pulse area
+sets the intensity of the required optical control pulses.
+Hence, the TDM of the addressed transitions influences
+the generation process of photonic cluster states. Fur-
+thermore, the proposed protocols require coherent exci-
+tation of electron-hole pairs in various exciton configura-
+tions to control and readout the exciton spin state.
+In this work, we demonstrate coherent Rabi oscillations
+arXiv:2301.13628v1  [cond-mat.mes-hall]  31 Jan 2023
+
+2
+of direct, spatially indirect, and hybridized excitons in a
+single QDM. An off-resonant detection technique is in-
+troduced and applied, relying on phonon-mediated state
+transfers. We examine the dependence of the Rabi fre-
+quency on the excitonic configuration, as the overlap of
+the electron and hole wave functions changes. Tuning the
+electric field via a gate voltage allows electrical control of
+this wave function overlap and, therefore, of the pulse
+area needed for population inversion.
+In this way, we
+demonstrate and quantify electric control of the TDM.
+Finally, a simple one-dimensional model of a double-well
+potential allows us to model the voltage-dependence of
+the TDM.
+II.
+RESULTS
+By vertically stacking two QDs with a separation in
+the nm regime, charge wave functions can hybridize
+across both QDss.
+In addition, both direct and spa-
+tially indirect excitons can form. Figure 1 (a) illustrates
+a schematic band-diagram of a QDM. The two QDs
+are depicted by a double-well potential, in which elec-
+trons (filled circle) and holes (empty circle) are trapped.
+The design of the investigated sample, described in Ap-
+pendix A, energetically favours the location of a hole
+in the top QD. Consequently, a direct/indirect exciton
+(red/blue ellipse) forms, when an electron is trapped in
+the top/bottom QD. The QDM is embedded in a p-i-n
+diode structure; applying a gate voltage V facilitates tun-
+ing of the energy levels of both QDs relative to each other.
+In this way, the direct and indirect exciton energies can
+be brought into resonance. At the resonance condition,
+the electron wave function hybridizes across both dots,
+molecular bonding and anti-bonding states form, and an
+avoided crossing between the orbital states occurs. Since
+we can control the tunnel coupling between the two QDs
+by varying the gate voltage, we use this dependency to
+investigate coherent driving of different exciton configu-
+rations.
+The most elemental charge state exhibiting the hy-
+bridization of wave functions is the neutral exciton (X0).
+Figure 1 (b) shows a voltage-dependent photolumines-
+cence measurement of the X0. We make use of a two-
+phase electrical and optical sequence to deterministi-
+cally prepare the QDM in a zero-charge ground-state
+and individually adjust the tunnel coupling [17]. Excit-
+ing the energetically higher p-shell orbital of the upper
+dot at 1353.6 meV enables the unimpeded detection of
+the X0 s-shell emission for multiple coupling conditions.
+At 0.16 V, the electron wave function hybridizes and an
+avoided crossing forms.
+The resulting electron eigen-
+states are described by symmetric and antisymmetric
+wave functions [13]. The corresponding lower and higher
+energy transitions of the avoided crossing are denoted
+LOW and UP in Figure 1 (b). The red and blue dashed
+lines depict the energies of a direct and indirect exciton,
+respectively. By increasing the gate voltage, the exciton
+0
+50
+100
+150
+0
+5
+10
+0.05
+0.1
+0.15
+0.2
+1336
+1338
+1340
+1342
+Energy (meV)
+Gate Voltage (V)
+Power1/2 (nW1/2)
+Emission (cts/3s)
+kCounts (/s)
+103
+102
+101
+V
+z
+E
+(a)
+(b)
+AlGaAs
+Excita�on
+Emission
+Emission
+UP
+LOW
+cgs
+UP
+LOW
+(c)
+(d)
+0.1 V
+𝛾P
+FIG. 1. Rabi oscillations of the neutral exciton in a QDM.
+(a) Schematic band structure of a QDM represented by a
+double-well potential. An AlGaAs barrier below the molecule
+prolongs tunneling times for electrons while not affecting tun-
+neling for holes. One hole (empty circle) is located in the up-
+per QD, while electrons (filled circles) occur in both dots. As
+a consequence, direct (red ellipse) and indirect (blue ellipse)
+excitons arise. A gate voltage V applied to the sample facil-
+itates tuning of the direct and indirect exciton energies rela-
+tive to each other. (b) Voltage-dependent photoluminescence
+of the neutral exciton. The red and blue dashed lines indicate
+the energies of the direct and indirect excitons. tunnel cou-
+pling between the two QDs leads to an avoided crossing with
+a symmetric (pink) and an anti-symmetric (green) electron
+eigenstate. The upper (lower) energy transition is called UP
+(LOW). Triangles indicate the excitation energy and voltage
+applied in Figure 2. (c) Neutral exciton state diagram illus-
+trating the excitation and detection scheme for monitoring
+Rabi oscillations. While a resonant light field (green) is driv-
+ing UP, a phonon-mediated state transfer with rate γP (black
+arrow) is enabling emission from both UP and the energet-
+ically detuned LOW. (d) Power-dependent Rabi oscillations
+when exciting UP and detecting UP (green) or LOW (pink)
+at 0.1 V.
+
+ge1336
+0050.150.210e
+1338
+uS
+C1340n
+01342SMSGaateVolta3
+character changes from direct to hybridized to indirect
+for the upper energy branch, and vice versa for the lower
+energy branch. As a result, the overlap of the electron
+and hole wave functions changes.
+The change of the wave function overlap is quanti-
+fied by coherently driving Rabi oscillations on the ex-
+citon transition.
+The Rabi frequency of a resonantly
+excited two-level system ΩR =
+�� E0D
+ℏ
+�� is linearly depen-
+dent on the TDM D, which in return is proportional to
+the overlap of the electron and hole wave function [18].
+In addition, ΩR depends linearly on the electric driving
+field amplitude E0. The E0-dependence allows the ob-
+servation of power-dependent Rabi oscillations [19]. For
+this purpose, a 5 ps laser pulse is applied to resonantly
+drive the crystal ground state (cgs)-to-X0 transition in
+the QDM. The occupation of the excited state is mon-
+itored by detecting the photons emitted by the driven
+two-level system. Commonly, emission from resonantly
+excited states is detected in a cross-polarized setup con-
+figuration to suppress the excitation laser [20]. At high
+excitation power, however, laser light can leak into the
+detection channel and reduce the signal-to-noise ratio.
+We propose and demonstrate a readout technique utiliz-
+ing a phonon-mediated state transfer [21], which detunes
+the emitted photons energetically from the two-level sys-
+tem.
+Thereby, the limitation of an insufficiently sup-
+pressed excitation laser is eliminated via spectral filter-
+ing, and the visibility of the Rabi oscillations is increased.
+Figure 1 (c) visualizes the state diagram of the X0. The
+two excited states UP and LOW can both radiatively de-
+cay into the cgs. A phonon emission process with rate
+γP can transfer the electron from the UP to the LOW
+configuration [21].
+Since the excitation pulse length is
+short compared to the decay rates, the cgs-UP system
+is well approximated by a two-level system. It is coher-
+ently driven by a 5 ps laser pulse (green arrow). Fig-
+ure 1 (d) shows the power-dependent resonance fluores-
+cence emission of the UP transition as green data points.
+The measurement is performed at 0.1 V, such that the
+driven transition exhibits a direct exciton character, as
+shown in Figure 1 (b). Rabi oscillations are observed up
+to a pulse area of slightly above 2π and 602 nW. However,
+a decreasing signal-to-noise ratio prevents the detection
+of oscillations above 602 nW due to nsufficient suppres-
+sion of the excitation laser.
+To improve the signal-to-noise ratio, which decreases
+with increasing power,
+we make use of a phonon-
+mediated state transfer.
+The emission of a phonon
+transfers the electron from the UP into the LOW
+configuration.
+This process can only occur as long as
+the system is in the excited state. Thus, the ensemble
+occupation of LOW is proportional to the ensemble
+occupation of UP, and so is the number of emitted
+photons of both transitions.
+In addition, due to the
+avoided crossing, the emission of LOW is at least
+2.1 meV detuned from the driving energy for any gate
+voltage, which allows the spectral filtering of the emis-
+sion from the excitation laser pulse. Thus, the resonant
+kCounts (/s)
+Power1/2 (nW1/2)
+UP
+LOW
+0.1 V
+0.22 V
+0
+50
+100
+0
+1
+2
+3
+0
+50
+100
+0
+5
+10
+0
+50
+100
+0
+2
+4
+0
+50
+100
+0
+0.1
+0.2
+FIG. 2.
+Rabi oscillations of the UP and LOW branch at 0.1 V
+(left) and 0.22 V (right) by phonon-mediated state transfers.
+The red data points correspond to a direct, the blue to an
+indirect driven transition.
+excitation of the two-level system and the off-resonant
+monitoring of its excited state occupation are achieved
+simultaneously.
+The power-dependent emission of the
+LOW transition when exciting UP is shown by the
+pink data points in Figure 1 (d).
+Below 602 nW, both
+readout techniques show the same Rabi frequency as
+expected, confirming the proportionality of occupancy
+between UP and LOW. However, in contrast to the
+resonant detection (green), Rabi oscillations are well
+resolvable up to a pulse area of 7π.
+The reduction of
+the oscillation amplitude arises from interactions with
+phonons [22], while the increase of the mean is attributed
+to a slightly chirped excitation laser pulse [23]. From the
+relative intensities of both transitions, we can conclude
+that the phonon induced relaxation rate is compara-
+ble to the radiative decay rate of the direct UP transition.
+Electric control of the tunnel coupling between the two
+QDs allows coherent excitation of electron-hole pairs in
+different occupation configurations. Figure 2 shows the
+power-dependent emission of the QDM while resonantly
+exciting UP and detecting LOW (green dashed box, UP).
+The measurements are performed at 0.1 V (left) and
+0.22 V (right), on either side of the avoided crossing. The
+red and blue data points indicate a direct and indirect
+character of the excited transition, respectively. We ob-
+serve Rabi oscillations for both the direct and indirect
+transitions, which confirms that coherent excitation of a
+spatially indirect exciton is possible. However, the Rabi
+
+4
+0
+0.1
+0.2
+0.3
+Gate Voltage (V)
+0
+0.05
+0.1
+0.15
+Rabi Frequency (1/nW1/2)
+0
+0.2
+0.4
+0.6
+0.8
+1
+FIG. 3.
+Measured voltage dependent Rabi frequency of
+UP (green) and LOW (pink), plotted on the left axes. The
+right axes visualizes the calculated overlap of the electron
+and hole wave functions as a function of the voltage, where
+the pink/green dashed line corresponds to the lowest/second-
+lowest electron eigenenergy. The red/blue shaded background
+indicates the direct/indirect character of the transition.
+frequency of the indirect configuration is reduced com-
+pared to the direct configuration. This is caused by the
+reduced overlap of the electron and hole wave functions
+and the accompanying decrease of the TDM for the in-
+direct exciton.
+A verification of these results is found by performing
+the same experiments on the lower branch (Figure 2,
+pink box, LOW). Similar to the previous case, a phonon
+absorption process facilitates a state transfer from LOW
+to UP and, therefore, the off-resonant detection of UP
+while exciting LOW. This allows us to off-resonantly
+monitor Rabi oscillations of the LOW branch.
+The
+investigated gate voltages of both excitation cases are
+chosen to be ±0.06 V away from the avoided crossing.
+Therefore, the electron configuration of the upper branch
+at 0.1 V (0.22 V) resembles the electron configuration of
+the lower branch at 0.22 V (0.1 V). This leads to a com-
+parable Rabi frequency of the direct (red) and indirect
+(blue) exciton of the upper and lower energy transition
+at the two voltages. The difference in absolute counts
+between the excitation of UP and LOW is attributed
+to the underlying phonon process.
+When exciting UP
+(LOW), we rely on the emission (absorption) of a
+phonon to detect the signal.
+Since the measurements
+are performed at 10 K, the probability of absorbing a
+phonon is strongly reduced compared to the emission.
+Since our QDM device allows continuous tuning the
+gate voltage while maintaining the prepared charge state,
+arbitrary exciton configurations can be set.
+Thereby,
+the overlap of the electron and hole wave functions is
+analyzable for any coupling condition. Figure 3 shows
+the power-dependent Rabi frequencies as a function of
+the gate voltage for the upper (green) and lower branch
+(pink). The Rabi frequencies are extracted by fitting a
+sin2(laser power) function to the data, with an exponen-
+tial decay to take phonon dephasing into account, and a
+linear increase with intensity to compensate for a chirped
+excitation pulse. We observe a continuous increase (de-
+crease) of the frequency when transitioning from an indi-
+rect (direct) to a direct (indirect) exciton. By raising the
+gate voltage and following the UP transition, the elec-
+tron occupation shifts from the top to the bottom dot.
+The opposite holds for the LOW transition. This leads
+to a continuous variation of the overlap of the electron
+and hole wave functions and, consequently, to a change
+in the Rabi frequency. Within the investigated range be-
+tween 0.1 V and 0.22 V, we are able to electrically tune
+the Rabi frequency by a factor of ≈ 3.
+The wave function overlap is modeled by calculating
+the eigenenergies and -values of a tilted, one-dimensional
+double-squarewell potential representing the QDM. By
+fitting the energy difference between the two lowest eigen-
+states to the voltage-dependent separation of UP and
+LOW, the depth of the squarewell potential and the effec-
+tive electron mass are determined. A detailed description
+of the model is provided in Appendix B. The right axes
+of Figure 3 shows the overlap of the electron and hole
+wave functions |⟨ψe|ψh⟩| for the lowest (pink) and sec-
+ond lowest (green) electron eigenenergy by dashed lines.
+The electron eigenenergies correspond to the LOW and
+UP transition, respectively. The one-dimensional model
+provides a remarkably good description of the measured
+voltage-dependent Rabi frequencies. Thereby, the Rabi
+frequency can be related to the TDM of direct, indirect,
+and hybridized excitons, which allows determination the
+π-pulse area as well as the difference in radiative lifetime
+of the corresponding transition.
+III.
+DISCUSSION AND SUMMARY
+Adressing direct, indirect, and hybridized excitons is
+fundamental for using QDMs as spin-photon interfaces.
+In addition, the electrical tuneability of the TDM of the
+adressed transitions at and around the tunnel coupling
+regime is one key parameter of a QDM. It not only de-
+termines the π-pulse power of the addressed transition,
+as shown in this work, but is also directly related to the
+lifetime of the excited state. Therefore, it is one of the
+parameters setting the creation rate for generating one-
+and two- dimensional photonic cluster states as well as
+for performing quantum-repeater protocols.
+We have demonstrated the coherent excitation of di-
+rect, indirect, and hybridized excitons – one of the el-
+ementary building blocks for creating photonic cluster
+states from QDMs. We use non-resonant readout, which
+is facilitated by phonon-mediated charge relaxation and
+excitation between the two lowest energy eigenstates of
+the electron.. Voltage-dependent Rabi oscillations show
+a continuous increase of the Rabi frequency when tran-
+sitioning from an indirect to a direct exciton.
+This is
+
+5
+attributed to an electrically controlled increase of the
+TDM of a direct compared to an indirect transition. Fur-
+thermore, we apply a one-dimensional model to calculate
+the overlap of the X0 electron and hole wave functions.
+Within the voltage range presented, we are able to tune
+the Rabi frequency and consequently the TDM by a fac-
+tor of ≈ 3. This corresponds to a variation of the radia-
+tive lifetime between a direct and an indirect exciton by
+a factor of ≈ 9, as it scales quadratically with the TDM.
+The coherent excitation and the electrical tunability
+between various exciton configurations in QDMs not only
+paves the way towards the generation of entangled multi-
+photon states. It might also enable protocols which uti-
+lize fast electrical switching between the exciton config-
+urations. This can reduce their lifetime and the π-pulse
+power and highly improve the cluster state generation
+rate.
+ACKNOWLEDGMENTS
+The authors gratefully acknowledge financial sup-
+port from the German Federal Ministry of Educa-
+tion and Research (BMBF) via Q.Link.X (16KIS0874,
+16KIS086), QR.X (16KISQ027, 16KISQ014, 16KISQ012
+and 16KISQ009), the European Union’s Horizon 2020 re-
+search and innovation program under grant agreement
+862035 (QLUSTER) and the Deutsche Forschungsge-
+meinschaft (DFG, German Research Foundation) via
+SQAM (FI947-5-1), DIP (FI947-6-1), and the Excellence
+Cluster MCQST (EXC-2111, 390814868). F.B. gratefully
+acknowledges the Exploring Quantum Matter (ExQM)
+programme funded by the State of Bavaria. F.S., K.B.,
+and K.M. gratefully acknowledge the BMBF for financial
+support via project MOQUA (13N14846).
+Appendix A: Sample
+The investigated InAs QDM is enclosed in a GaAs
+matrix and was grown by molecular beam epitaxy.
+It
+consists of two vertically stacked QDs.
+The inter-dot
+coupling strength is determined by a wetting layer-
+to-wetting layer separation of 10 nm.
+In addition, an
+AlxGa(x−1)As barrier (x = 0.33) with a thickness of
+2.5 nm is placed between the dots to reduce the coupling
+strength. The height of the top (bottom) QD was fixed
+to 2.9 nm (2.7 nm) via the indium-flush technique during
+growth.
+This height configuration facilitates electric
+field-induced tunnel coupling of orbital states in the
+conduction band. A 50 nm thick AlxGa(x−1)As tunnel
+barrier (x = 0.33) was grown 5 nm below the QDM
+to prolong electron tunneling times.
+The molecule is
+embedded in a p-i-n diode, with the doped regions used
+as contacts to gate the sample. The diode contacts are
+placed more than 150 nm away from the molecule to
+prevent uncontrolled charge tunneling into the QDM.
+Furthermore, a distributed Bragg reflector was grown
+below the diode and a circular Bragg grating was
+positioned deterministically via in-situ electron beam
+lithography above an individual and pre-selected QDM
+to improve photon in- and outcoupling efficiencies [24].
+All measurements are performed at 10 K.
+Appendix B: Double-well potential model
+To calculate the overlap of the electron and hole wave
+functions, we set up a model consisting of a double-
+squarewell potential representing the conduction band of
+the QDM. We assume that the variation of the in-plane
+wave functions is small compared to the wave functions
+along the growth direction z, when changing the gate
+voltage. This assumption is reasonable, since the confine-
+ment of charges along the growth axes and the translation
+introduced by the gate voltage along the growth axes ex-
+ceeds the in-plane variation. We can, therefore, approach
+the problem with a one-dimensional model and expect
+an acceptable degree of accuracy. The potential V (z) is
+designed to match the dimensions of the QDM with re-
+spect to the tunnel barrier width and dot heights (see
+Section A). z is here the growth direction of the sample.
+In addition, we tilt the potential to imitate the presence
+of an applied gate voltage. Solving the time-independent
+Schr¨odinger equation for a given gate voltage allows us to
+obtain the envelope functions Ψ and their eigenenergies
+E of an electron with mass me trapped in the double-well
+potential. The envelope functions are then used to rep-
+resent the wave function of the electron since the Bloch
+part of the wave functions are only weakly sensitive to
+electric fields of the magnitude applied here.
+To define the free parameters of the double-well model,
+we determine the effective electron mass me and the po-
+tential depth by fitting the difference between the cal-
+culated two lowest eigenenergies to the measured energy
+difference ∆E between UP and LOW. ∆E between the
+two X0 branches is purely determined by the energy dif-
+ference between the electron eigenstates. For calculating
+the hole wave function, the potential is inverted to repre-
+sent the valance band. Its depth is set to match half the
+depth of the electron potential whereas the heavy hole
+mass is set to match mh = 10 me [25]. Since we are inter-
+ested in calculating the overlap of wave functions rather
+than absolute transition energies, it is sufficient to work
+with relative values in the model.
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+

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+page_content='Coherent driving of direct and indirect excitons in a quantum dot molecule  Frederik Bopp,1, ∗ Johannes Schall,2 Nikolai Bart,3 Florian Vogl,1 Charlotte Cullip,1 Friedrich  Sbresny,4 Katarina Boos,4 Christopher Thalacker,1 Michelle Lienhart,1 Sven Rodt,2 Dirk Reuter,5  Arne Ludwig,3 Andreas Wieck,3 Stephan Reitzenstein,2 Kai M¨uller,4 and Jonathan J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Finley1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' †  1Walter Schottky Institut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' School of Natural Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' and MCQST,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  Technische Universit¨at M¨unchen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Am Coulombwall 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' 85748 Garching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Germany  2Technische Universit¨at Berlin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Hardenbergstraße 36,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' 10623 Berlin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Germany  3Faculty of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Ruhr-Universit¨at Bochum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  Universit¨atsstraße 150,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' 44801 Bochum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Germany  4Walter Schottky Institut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' School of Computation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Information and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' and MCQST,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  Technische Universit¨at M¨unchen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Am Coulombwall 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' 85748 Garching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Germany  5Paderborn University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Warburger Straße 100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' 33098 Paderborn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Germany  (Dated: February 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' 2023)  Quantum dot molecules (QDMs) are one of the few quantum light sources that promise deter-  ministic generation of one- and two-dimensional photonic graph states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  The proposed protocols  rely on coherent excitation of the tunnel-coupled and spatially indirect exciton states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Here, we  demonstrate power-dependent Rabi oscillations of direct excitons, spatially indirect excitons, and  excitons with a hybridized electron wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' An off-resonant detection technique based on  phonon-mediated state transfer allows for spectrally filtered detection under resonant excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  Applying a gate voltage to the QDM-device enables a continuous transition between direct and  indirect excitons and, thereby, control of the overlap of the electron and hole wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' This  does not only vary the Rabi frequency of the investigated transition by a factor of ≈ 3, but also  allows to optimize graph state generation in terms of optical pulse power and reduction of radiative  lifetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  INTRODUCTION  The use of single photons as flying qubits facilitates  transmission of quantum information at the speed of  light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' However, transfer over large distances unavoidably  comes with losses and decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Encoding quantum  information on an ensemble of entangled photons, a so-  called graph state [1], instead of a single photon, provides  a possibility to mitigate the losses is transmission chan-  nels [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  Furthermore, other specific forms of graph  states such as photonic cluster states promise realization  of measurement-based quantum computing [4] as well as  quantum error correction [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  Following  the  Lindner-Rudolph  protocol [7],  one-  dimensional photonic cluster states can be deterministi-  cally generated by utilizing single spins in semiconductor  quantum dots (QDs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' The polarization entanglement of  up to five photons has been achieved in a one-dimensional  cluster state has been achieved [8] and most recent ex-  periments demonstrate localizable entanglement over ten  photons [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' While the nanophotonic environment of QDs  provides high photon emission rates, the cluster state cre-  ation fidelity is limited by spin dephasing and modified  selection rules in the presence of a transverse magnetic  field [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' These challenges can be overcome by using a  pair of tunnel coupled and vertically stacked QDs, so  called quantum dot molecules (QDMs) [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Besides pro-  longing the spin coherence compared to single quantum  ∗ frederik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='bopp@wsi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='de  † finley@wsi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='de  dots [11], QDMs possess an unique level structure [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  This level structure enables, for example, spin rotations  and spin readout transitions without application of a  magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' The ability to create spatially indirect  excitons, with one charge carrier occupying the upper  and one the lower QD [13], provides a cycling transi-  tion which can be used for generating time-bin entangled  photons [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Moreover, QDMs are proposed to generate  two-dimensional photonic cluster states by harnessing the  tunnel coupling between the two QDs and inter-dot con-  trol gates [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  The foundation for creating one- and two-dimensional  photonic cluster states is the occurrence of excitons in  spatially direct, spatially indirect, and hybridized config-  urations [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' In these different configurations, the charge  carriers of an electron-hole pair are located in the same  QD, in different QDs, or one of the charge wave func-  tions is hybridized over both quantum dots, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  In each configuration, the overlap of the electron and  hole wave functions and, therefore, the transition dipole  moment (TDM) of the corresponding optical transition  differs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' This results in a change of both the lifetime of the  excited state and the pulse area needed for maximal pop-  ulation inversion [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' While the lifetime influences the  cluster state creation efficiency and rate, the π-pulse area  sets the intensity of the required optical control pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  Hence, the TDM of the addressed transitions influences  the generation process of photonic cluster states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Fur-  thermore, the proposed protocols require coherent exci-  tation of electron-hole pairs in various exciton configura-  tions to control and readout the exciton spin state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  In this work, we demonstrate coherent Rabi oscillations  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='13628v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='mes-hall]  31 Jan 2023  2  of direct, spatially indirect, and hybridized excitons in a  single QDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' An off-resonant detection technique is in-  troduced and applied, relying on phonon-mediated state  transfers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' We examine the dependence of the Rabi fre-  quency on the excitonic configuration, as the overlap of  the electron and hole wave functions changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Tuning the  electric field via a gate voltage allows electrical control of  this wave function overlap and, therefore, of the pulse  area needed for population inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  In this way, we  demonstrate and quantify electric control of the TDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  Finally, a simple one-dimensional model of a double-well  potential allows us to model the voltage-dependence of  the TDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  RESULTS  By vertically stacking two QDs with a separation in  the nm regime, charge wave functions can hybridize  across both QDss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  In addition, both direct and spa-  tially indirect excitons can form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Figure 1 (a) illustrates  a schematic band-diagram of a QDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' The two QDs  are depicted by a double-well potential, in which elec-  trons (filled circle) and holes (empty circle) are trapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  The design of the investigated sample, described in Ap-  pendix A, energetically favours the location of a hole  in the top QD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Consequently, a direct/indirect exciton  (red/blue ellipse) forms, when an electron is trapped in  the top/bottom QD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' The QDM is embedded in a p-i-n  diode structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' applying a gate voltage V facilitates tun-  ing of the energy levels of both QDs relative to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  In this way, the direct and indirect exciton energies can  be brought into resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' At the resonance condition,  the electron wave function hybridizes across both dots,  molecular bonding and anti-bonding states form, and an  avoided crossing between the orbital states occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Since  we can control the tunnel coupling between the two QDs  by varying the gate voltage, we use this dependency to  investigate coherent driving of different exciton configu-  rations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  The most elemental charge state exhibiting the hy-  bridization of wave functions is the neutral exciton (X0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  Figure 1 (b) shows a voltage-dependent photolumines-  cence measurement of the X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' We make use of a two-  phase electrical and optical sequence to deterministi-  cally prepare the QDM in a zero-charge ground-state  and individually adjust the tunnel coupling [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Excit-  ing the energetically higher p-shell orbital of the upper  dot at 1353.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='6 meV enables the unimpeded detection of  the X0 s-shell emission for multiple coupling conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  At 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='16 V, the electron wave function hybridizes and an  avoided crossing forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  The resulting electron eigen-  states are described by symmetric and antisymmetric  wave functions [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' The corresponding lower and higher  energy transitions of the avoided crossing are denoted  LOW and UP in Figure 1 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' The red and blue dashed  lines depict the energies of a direct and indirect exciton,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' By increasing the gate voltage, the exciton  0  50  100  150  0  5  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='2  1336  1338  1340  1342  Energy (meV)  Gate Voltage (V)  Power1/2 (nW1/2)  Emission (cts/3s)  kCounts (/s)  103  102  101  V  z  E  (a)  (b)  AlGaAs  Excita�on  Emission  Emission  UP  LOW  cgs  UP  LOW  (c)  (d)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='1 V  𝛾P  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Rabi oscillations of the neutral exciton in a QDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  (a) Schematic band structure of a QDM represented by a  double-well potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' An AlGaAs barrier below the molecule  prolongs tunneling times for electrons while not affecting tun-  neling for holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' One hole (empty circle) is located in the up-  per QD, while electrons (filled circles) occur in both dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' As  a consequence, direct (red ellipse) and indirect (blue ellipse)  excitons arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' A gate voltage V applied to the sample facil-  itates tuning of the direct and indirect exciton energies rela-  tive to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' (b) Voltage-dependent photoluminescence  of the neutral exciton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' The red and blue dashed lines indicate  the energies of the direct and indirect excitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' tunnel cou-  pling between the two QDs leads to an avoided crossing with  a symmetric (pink) and an anti-symmetric (green) electron  eigenstate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' The upper (lower) energy transition is called UP  (LOW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Triangles indicate the excitation energy and voltage  applied in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' (c) Neutral exciton state diagram illus-  trating the excitation and detection scheme for monitoring  Rabi oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' While a resonant light field (green) is driv-  ing UP, a phonon-mediated state transfer with rate γP (black  arrow) is enabling emission from both UP and the energet-  ically detuned LOW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' (d) Power-dependent Rabi oscillations  when exciting UP and detecting UP (green) or LOW (pink)  at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='1 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  ge1336  0050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='210e  1338  uS  C1340n  01342SMSGaateVolta3  character changes from direct to hybridized to indirect  for the upper energy branch, and vice versa for the lower  energy branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' As a result, the overlap of the electron  and hole wave functions changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  The change of the wave function overlap is quanti-  fied by coherently driving Rabi oscillations on the ex-  citon transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  The Rabi frequency of a resonantly  excited two-level system ΩR =  �� E0D  ℏ  �� is linearly depen-  dent on the TDM D, which in return is proportional to  the overlap of the electron and hole wave function [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  In addition, ΩR depends linearly on the electric driving  field amplitude E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' The E0-dependence allows the ob-  servation of power-dependent Rabi oscillations [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' For  this purpose, a 5 ps laser pulse is applied to resonantly  drive the crystal ground state (cgs)-to-X0 transition in  the QDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' The occupation of the excited state is mon-  itored by detecting the photons emitted by the driven  two-level system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Commonly, emission from resonantly  excited states is detected in a cross-polarized setup con-  figuration to suppress the excitation laser [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' At high  excitation power, however, laser light can leak into the  detection channel and reduce the signal-to-noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  We propose and demonstrate a readout technique utiliz-  ing a phonon-mediated state transfer [21], which detunes  the emitted photons energetically from the two-level sys-  tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  Thereby, the limitation of an insufficiently sup-  pressed excitation laser is eliminated via spectral filter-  ing, and the visibility of the Rabi oscillations is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  Figure 1 (c) visualizes the state diagram of the X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' The  two excited states UP and LOW can both radiatively de-  cay into the cgs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' A phonon emission process with rate  γP can transfer the electron from the UP to the LOW  configuration [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  Since the excitation pulse length is  short compared to the decay rates, the cgs-UP system  is well approximated by a two-level system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' It is coher-  ently driven by a 5 ps laser pulse (green arrow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Fig-  ure 1 (d) shows the power-dependent resonance fluores-  cence emission of the UP transition as green data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  The measurement is performed at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='1 V, such that the  driven transition exhibits a direct exciton character, as  shown in Figure 1 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Rabi oscillations are observed up  to a pulse area of slightly above 2π and 602 nW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' However,  a decreasing signal-to-noise ratio prevents the detection  of oscillations above 602 nW due to nsufficient suppres-  sion of the excitation laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  To improve the signal-to-noise ratio, which decreases  with increasing power,  we make use of a phonon-  mediated state transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  The emission of a phonon  transfers the electron from the UP into the LOW  configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  This process can only occur as long as  the system is in the excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Thus, the ensemble  occupation of LOW is proportional to the ensemble  occupation of UP, and so is the number of emitted  photons of both transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  In addition, due to the  avoided crossing, the emission of LOW is at least  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='1 meV detuned from the driving energy for any gate  voltage, which allows the spectral filtering of the emis-  sion from the excitation laser pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Thus, the resonant  kCounts (/s)  Power1/2 (nW1/2)  UP  LOW  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='1 V  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='22 V  0  50  100  0  1  2  3  0  50  100  0  5  10  0  50  100  0  2  4  0  50  100  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  Rabi oscillations of the UP and LOW branch at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='1 V  (left) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='22 V (right) by phonon-mediated state transfers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  The red data points correspond to a direct, the blue to an  indirect driven transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  excitation of the two-level system and the off-resonant  monitoring of its excited state occupation are achieved  simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  The power-dependent emission of the  LOW transition when exciting UP is shown by the  pink data points in Figure 1 (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  Below 602 nW, both  readout techniques show the same Rabi frequency as  expected, confirming the proportionality of occupancy  between UP and LOW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' However, in contrast to the  resonant detection (green), Rabi oscillations are well  resolvable up to a pulse area of 7π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  The reduction of  the oscillation amplitude arises from interactions with  phonons [22], while the increase of the mean is attributed  to a slightly chirped excitation laser pulse [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' From the  relative intensities of both transitions, we can conclude  that the phonon induced relaxation rate is compara-  ble to the radiative decay rate of the direct UP transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  Electric control of the tunnel coupling between the two  QDs allows coherent excitation of electron-hole pairs in  different occupation configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Figure 2 shows the  power-dependent emission of the QDM while resonantly  exciting UP and detecting LOW (green dashed box, UP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  The measurements are performed at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='1 V (left) and  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='22 V (right), on either side of the avoided crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' The  red and blue data points indicate a direct and indirect  character of the excited transition, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' We ob-  serve Rabi oscillations for both the direct and indirect  transitions, which confirms that coherent excitation of a  spatially indirect exciton is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' However, the Rabi  4  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='3  Gate Voltage (V)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='15  Rabi Frequency (1/nW1/2)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='8  1  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  Measured voltage dependent Rabi frequency of  UP (green) and LOW (pink), plotted on the left axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' The  right axes visualizes the calculated overlap of the electron  and hole wave functions as a function of the voltage, where  the pink/green dashed line corresponds to the lowest/second-  lowest electron eigenenergy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' The red/blue shaded background  indicates the direct/indirect character of the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  frequency of the indirect configuration is reduced com-  pared to the direct configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' This is caused by the  reduced overlap of the electron and hole wave functions  and the accompanying decrease of the TDM for the in-  direct exciton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  A verification of these results is found by performing  the same experiments on the lower branch (Figure 2,  pink box, LOW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Similar to the previous case, a phonon  absorption process facilitates a state transfer from LOW  to UP and, therefore, the off-resonant detection of UP  while exciting LOW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' This allows us to off-resonantly  monitor Rabi oscillations of the LOW branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  The  investigated gate voltages of both excitation cases are  chosen to be ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='06 V away from the avoided crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  Therefore, the electron configuration of the upper branch  at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='1 V (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='22 V) resembles the electron configuration of  the lower branch at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='22 V (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='1 V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' This leads to a com-  parable Rabi frequency of the direct (red) and indirect  (blue) exciton of the upper and lower energy transition  at the two voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' The difference in absolute counts  between the excitation of UP and LOW is attributed  to the underlying phonon process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  When exciting UP  (LOW), we rely on the emission (absorption) of a  phonon to detect the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  Since the measurements  are performed at 10 K, the probability of absorbing a  phonon is strongly reduced compared to the emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  Since our QDM device allows continuous tuning the  gate voltage while maintaining the prepared charge state,  arbitrary exciton configurations can be set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  Thereby,  the overlap of the electron and hole wave functions is  analyzable for any coupling condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Figure 3 shows  the power-dependent Rabi frequencies as a function of  the gate voltage for the upper (green) and lower branch  (pink).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' The Rabi frequencies are extracted by fitting a  sin2(laser power) function to the data, with an exponen-  tial decay to take phonon dephasing into account, and a  linear increase with intensity to compensate for a chirped  excitation pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' We observe a continuous increase (de-  crease) of the frequency when transitioning from an indi-  rect (direct) to a direct (indirect) exciton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' By raising the  gate voltage and following the UP transition, the elec-  tron occupation shifts from the top to the bottom dot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  The opposite holds for the LOW transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' This leads  to a continuous variation of the overlap of the electron  and hole wave functions and, consequently, to a change  in the Rabi frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Within the investigated range be-  tween 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='1 V and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='22 V, we are able to electrically tune  the Rabi frequency by a factor of ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  The wave function overlap is modeled by calculating  the eigenenergies and -values of a tilted, one-dimensional  double-squarewell potential representing the QDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' By  fitting the energy difference between the two lowest eigen-  states to the voltage-dependent separation of UP and  LOW, the depth of the squarewell potential and the effec-  tive electron mass are determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' A detailed description  of the model is provided in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' The right axes  of Figure 3 shows the overlap of the electron and hole  wave functions |⟨ψe|ψh⟩| for the lowest (pink) and sec-  ond lowest (green) electron eigenenergy by dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  The electron eigenenergies correspond to the LOW and  UP transition, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' The one-dimensional model  provides a remarkably good description of the measured  voltage-dependent Rabi frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Thereby, the Rabi  frequency can be related to the TDM of direct, indirect,  and hybridized excitons, which allows determination the  π-pulse area as well as the difference in radiative lifetime  of the corresponding transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  DISCUSSION AND SUMMARY  Adressing direct, indirect, and hybridized excitons is  fundamental for using QDMs as spin-photon interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  In addition, the electrical tuneability of the TDM of the  adressed transitions at and around the tunnel coupling  regime is one key parameter of a QDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' It not only de-  termines the π-pulse power of the addressed transition,  as shown in this work, but is also directly related to the  lifetime of the excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Therefore, it is one of the  parameters setting the creation rate for generating one-  and two- dimensional photonic cluster states as well as  for performing quantum-repeater protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  We have demonstrated the coherent excitation of di-  rect, indirect, and hybridized excitons – one of the el-  ementary building blocks for creating photonic cluster  states from QDMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' We use non-resonant readout, which  is facilitated by phonon-mediated charge relaxation and  excitation between the two lowest energy eigenstates of  the electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='. Voltage-dependent Rabi oscillations show  a continuous increase of the Rabi frequency when tran-  sitioning from an indirect to a direct exciton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  This is  5  attributed to an electrically controlled increase of the  TDM of a direct compared to an indirect transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Fur-  thermore, we apply a one-dimensional model to calculate  the overlap of the X0 electron and hole wave functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  Within the voltage range presented, we are able to tune  the Rabi frequency and consequently the TDM by a fac-  tor of ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' This corresponds to a variation of the radia-  tive lifetime between a direct and an indirect exciton by  a factor of ≈ 9, as it scales quadratically with the TDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  The coherent excitation and the electrical tunability  between various exciton configurations in QDMs not only  paves the way towards the generation of entangled multi-  photon states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' It might also enable protocols which uti-  lize fast electrical switching between the exciton config-  urations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' This can reduce their lifetime and the π-pulse  power and highly improve the cluster state generation  rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  The authors gratefully acknowledge financial sup-  port from the German Federal Ministry of Educa-  tion and Research (BMBF) via Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='Link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='X (16KIS0874,  16KIS086), QR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='X (16KISQ027, 16KISQ014, 16KISQ012  and 16KISQ009), the European Union’s Horizon 2020 re-  search and innovation program under grant agreement  862035 (QLUSTER) and the Deutsche Forschungsge-  meinschaft (DFG, German Research Foundation) via  SQAM (FI947-5-1), DIP (FI947-6-1), and the Excellence  Cluster MCQST (EXC-2111, 390814868).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' gratefully  acknowledges the Exploring Quantum Matter (ExQM)  programme funded by the State of Bavaria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=', K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=',  and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' gratefully acknowledge the BMBF for financial  support via project MOQUA (13N14846).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  Appendix A: Sample  The investigated InAs QDM is enclosed in a GaAs  matrix and was grown by molecular beam epitaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  It  consists of two vertically stacked QDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  The inter-dot  coupling strength is determined by a wetting layer-  to-wetting layer separation of 10 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  In addition, an  AlxGa(x−1)As barrier (x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='33) with a thickness of  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='5 nm is placed between the dots to reduce the coupling  strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' The height of the top (bottom) QD was fixed  to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='9 nm (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='7 nm) via the indium-flush technique during  growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  This height configuration facilitates electric  field-induced tunnel coupling of orbital states in the  conduction band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' A 50 nm thick AlxGa(x−1)As tunnel  barrier (x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='33) was grown 5 nm below the QDM  to prolong electron tunneling times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  The molecule is  embedded in a p-i-n diode, with the doped regions used  as contacts to gate the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' The diode contacts are  placed more than 150 nm away from the molecule to  prevent uncontrolled charge tunneling into the QDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  Furthermore, a distributed Bragg reflector was grown  below the diode and a circular Bragg grating was  positioned deterministically via in-situ electron beam  lithography above an individual and pre-selected QDM  to improve photon in- and outcoupling efficiencies [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  All measurements are performed at 10 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  Appendix B: Double-well potential model  To calculate the overlap of the electron and hole wave  functions, we set up a model consisting of a double-  squarewell potential representing the conduction band of  the QDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' We assume that the variation of the in-plane  wave functions is small compared to the wave functions  along the growth direction z, when changing the gate  voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' This assumption is reasonable, since the confine-  ment of charges along the growth axes and the translation  introduced by the gate voltage along the growth axes ex-  ceeds the in-plane variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' We can, therefore, approach  the problem with a one-dimensional model and expect  an acceptable degree of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' The potential V (z) is  designed to match the dimensions of the QDM with re-  spect to the tunnel barrier width and dot heights (see  Section A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' z is here the growth direction of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  In addition, we tilt the potential to imitate the presence  of an applied gate voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Solving the time-independent  Schr¨odinger equation for a given gate voltage allows us to  obtain the envelope functions Ψ and their eigenenergies  E of an electron with mass me trapped in the double-well  potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' The envelope functions are then used to rep-  resent the wave function of the electron since the Bloch  part of the wave functions are only weakly sensitive to  electric fields of the magnitude applied here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content='  To define the free parameters of the double-well model,  we determine the effective electron mass me and the po-  tential depth by fitting the difference between the cal-  culated two lowest eigenenergies to the measured energy  difference ∆E between UP and LOW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' ∆E between the  two X0 branches is purely determined by the energy dif-  ference between the electron eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' For calculating  the hole wave function, the potential is inverted to repre-  sent the valance band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Its depth is set to match half the  depth of the electron potential whereas the heavy hole  mass is set to match mh = 10 me [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Since we are inter-  ested in calculating the overlap of wave functions rather  than absolute transition energies, it is sufficient to work  with relative values in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
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+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
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+page_content=' Wieck, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Reiter,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' Kuhn, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFRT4oBgHgl3EQftjjG/content/2301.13628v1.pdf'}
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+Anomalously high supercurrent density in a two-dimensional topological material 
+ 
+Qi Zhang1*†, Md Shafayat Hossain1*†, Brian Casas2, Wenkai Zheng2, Zi-Jia Cheng1, Zhuangchai Lai3,4, Yi-Hsin 
+Tu5, Guoqing Chang6, Yao Yao3, Siyuan Li3, Yu-Xiao Jiang1, Sougata Mardanya5, Tay-Rong Chang5, Jing-Yang 
+You7, Yuan-Ping Feng7,8, Guangming Cheng9, Jia-Xin Yin1, Nana Shumiya1, Tyler A. Cochran1, Xian P. Yang1, 
+Maksim Litskevich1, Nan Yao9, Kenji Watanabe10, Takashi Taniguchi11, Hua Zhang3,12,13†, Luis Balicas2, M. 
+Zahid Hasan1,14† 
+1Laboratory for Topological Quantum Matter and Advanced Spectroscopy (B7), Department of Physics, Princeton 
+University, Princeton, New Jersey, USA. 
+2National High Magnetic Field Laboratory, Tallahassee, Florida 32310, USA. 
+3Department of Chemistry, City University of Hong Kong, Hong Kong, China.  
+4Department of Applied Physics, The Hong Kong Polytechnic University, Hong Kong, China. 
+5Department of Physics, National Cheng Kung University, 701 Tainan, Taiwan 
+6Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological 
+University, Singapore 637371, Singapore. 
+7Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117551, Singapore 
+8Centre for Advanced 2D Materials, National University of Singapore, 6 Science Drive 2, Singapore 117546, Singapore 
+9Princeton Institute for Science and Technology of Materials, Princeton University, Princeton, NJ, USA. 
+10Research Center for Functional Materials, National Institute for Materials Science, Tsukuba, Japan.  
+11International Center for Materials Nanoarchitectonics, National Institute for Materials Science, Tsukuba, Japan. 
+12Hong Kong Branch of National Precious Metals Material Engineering Research Center (NPMM), City University of 
+Hong Kong, Hong Kong, China. 
+13Shenzhen Research Institute, City University of Hong Kong, Shenzhen 518057, China. 
+14Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA. 
+ 
+* These authors contributed equally to this work 
+†Corresponding 
+to: 
+[email protected]; 
+[email protected]; 
+[email protected]; 
+[email protected]  
+Ongoing advances in superconductors continue to revolutionize technology thanks to the increasingly 
+versatile and robust availability of lossless supercurrent. In particular high supercurrent density can lead to 
+more efficient and compact power transmission lines, high-field magnets, as well as high-performance 
+nanoscale radiation detectors and superconducting spintronics. Here, we report the discovery of an 
+unprecedentedly high superconducting critical current density (17 MA/cm2 at 0 T and 7 MA/cm2 at 8 T) in 
+1T′-WS2, exceeding those of all reported two-dimensional superconductors to date. 1T′-WS2 features a 
+strongly anisotropic (both in- and out-of-plane) superconducting state that violates the Pauli paramagnetic 
+limit signaling the presence of unconventional superconductivity. Spectroscopic imaging of the vortices 
+further substantiates the anisotropic nature of the superconducting state. More intriguingly, the normal state 
+of 1T′-WS2 carries topological properties. The band structure obtained via angle-resolved photoemission 
+spectroscopy and first-principles calculations points to a Z2 topological invariant. The concomitance of 
+topology and superconductivity in 1T′-WS2 establishes it as a topological superconductor candidate, which 
+is promising for the development of quantum computing technology. 
+ 
+Since the discovery of superconductivity by Heike 
+Kamerlingh Onnes back in 1911 [1], superconductors have 
+revolutionized science and technology through numerous 
+applications ranging from superconducting qubits to high-
+field magnets [2-4]. High-field magnets fabricated from 
+superconductors with high critical current density, have 
+enabled scientific discoveries across physical, chemical, 
+and biological sciences [5-7]. On the other hand, 
+superconducting materials exhibiting topological properties 
+offer possibilities beyond this classical application 
+paradigm, opening a new frontier to implement fault-
+tolerant quantum information technologies. Recently, two-
+dimensional 
+(2D) 
+transition 
+metal 
+dichalcogenides 
+(TMDCs) attracted considerable interests thanks to their 
+abundant crystal structures and novel physical properties 
+[8-11]. Specifically, hole-doped TMDCs have been 
+considered as candidates for topological superconductivity 
+based on momentum-space-split spinless fermions [8]. For 
+
+example, the coexistence of superconductivity with a 
+topologically non-trivial electronic state makes 2M-WS2 a 
+good candidate for topological superconductivity [12]. 
+Here, we access both avenues and demonstrate an 
+unprecedentedly high superconducting critical current 
+density and topological features in the 2D superconductor 
+1T′-WS2. 
+1T′-WS2 is composed of a distorted [WS6] octahedral and 
+crystallizes in a monoclinic layered structure [13], as shown 
+in Fig. 1a. High purity 1T′-WS2 crystals were synthesized 
+via a previously reported method [13]. The single-phase 
+nature can be observed in the cross-sectional scanning 
+transmission electron microscope (STEM) image (Fig. 1b). 
+STEM image unveils the atomic stacking pertaining to a 
+monoclinic and distorted structure. This atomic-resolution 
+characterization confirms the high crystallinity and phase 
+purity of the as-synthesized 1T′-WS2 crystals, consistent 
+with the previous report [13]. After characterizing the bulk 
+material, we fabricated devices based on few-layer 1T′-WS2 
+for transport measurements. Thin flakes of 1T′-WS2 
+obtained via mechanical exfoliation were transferred onto a 
+SiO2 (285 nm)/Si substrate (inset of Fig. S1a). The Raman 
+spectrum of the as-prepared 1T′-WS2 flake shows a series 
+of peaks at ~112, ~178, ~270, ~316, and ~407 cm-1 (Fig. 
+S1a), consistent with single-phase 1T′-WS2 [13]. The 
+thickness of the flakes used in our measurements is ~6.1 
+nm as measured by the atomic force microscope (Fig. S1b). 
+The 
+device 
+was 
+fabricated 
+following 
+a 
+Hall-bar 
+configuration and measured from T = 300 K to 2.0 K in a 
+Physical Property Measurement System. Figure 1c depicts 
+the four-probe resistance as a function of temperature and 
+captures the electrical transport behavior of the sample. At 
+high temperatures, it exhibits metallic behavior (dR/dT > 0), 
+indicating phonon-scattering-dominated transport [14]. The 
+superconducting transition occurs at 7.7 K, which is slightly 
+lower than the bulk critical temperature (Tc) of 1T′-WS2 
+(8.6 K) [13].  We also measured the Hall effect of 1T′-WS2 
+above the critical temperature. Strikingly, the carrier 
+concentration in 1T′-WS2 approaches 1015~1016 cm-2 at T = 
+10 K (Fig. S2a and S2b). This value is much higher than the 
+typical 
+carrier 
+concentration 
+(~1014 
+cm-2) 
+of 
+2D 
+superconductors with electrostatic gating [15]. 
+To investigate the superconducting state of 1T′-WS2, we 
+performed magneto-transport measurements (Fig. 1d). We 
+start with the angular dependence of the upper critical 
+magnetic field (Hc2), defined as the magnetic field at which 
+the resistance drops to 50% of its normal state value. The 
+details of the angular dependent measurement are described 
+in Fig. S3. For a clear visualization, we normalized the 
+resistance by Rn, i.e., the normal state resistance for all the 
+samples. Figure 1e summarizes the magnetic field 
+dependence of the resistance at different angles (θ) at T = 
+0.33 K, where θ is the angle between the z-axis and the 
+magnetic field direction (Fig. 1d). As the sample is rotated 
+from the perpendicular (θ = 0º) to a parallel field (θ = 90º) 
+configuration, the transition towards superconductivity 
+progressively shifts to higher fields, manifesting a clear 
+superconducting anisotropy (Fig. 1e). In Fig. 1f, we present 
+a plot of Hc2 as a function of θ, showing that the highest Hc2 
+occurs when the magnetic field is applied parallel to the 
+sample plane. To understand the anisotropic nature of Hc2, 
+we fitted our data to the Tinkham formula, which describes 
+the angular dependence of Hc2 for a 2D superconductor 
+[16]: 
+ 
+where Hc2,⊥ and Hc2,// are the upper critical field for fields 
+perpendicular and parallel to the plane of the sample, 
+respectively. As shown in Fig. 1f, the blue fitting curve 
+matches the data quite well and thus confirms the 2D nature 
+of the superconductivity in 1T′-WS2. The fitting of the 
+angle dependent critical field for smaller angular regimes to 
+the 2D Tinkham formula is shown in Fig. S4. 
+ After exploring the anisotropy of Hc2 along the out-of-
+plane directions, we then examined how Hc2 evolved along 
+the in-plane directions. As the device is rotated from the x-
+axis (φ = 0°; φ is the angle between the magnetic field and 
+the x-axis as shown in Fig. 1d) to the y-axis (φ = 90°), the 
+superconducting transition progressively shifts from higher 
+fields to lower fields (Fig. 1g). Careful measurements were 
+performed to rule out the possibility of an accidental out-of-
+plane component (Fig. S5). Such a planar anisotropy is 
+likely to result from the reduced crystal symmetry due to 
+the distorted structure of 1T′-WS2, as clearly seen in Fig. 
+1a. Figure 1h, which shows Hc2 as a function of φ, reveals 
+an emergent two-fold symmetry. Furthermore, we observed 
+that the largest value of the in-plane Hc2 (28 T). To obtain a 
+quantitative understanding of such a large value, we 
+compared it to the expected Pauli paramagnetic limiting 
+field. In conventional superconductors, a sufficiently high 
+external magnetic field can suppress superconductivity 
+through the orbital [17] and spin Zeeman effect [18,19]. For 
+a few-layer sample, the suppression from the orbital effect 
+is nearly absent when the magnetic field is parallel to the 
+sample plane. Consequently, the Zeeman effect imposes an 
+upper bound on Hc2, known as the Pauli limit (Hp = 1.84×Tc 
+T/K) [20]. We find that the in-plane Hc2 (28 T) in 1T���-WS2 
+clearly violates the Pauli limit (14 T for Tc = 7.7 K). Such a 
+violation combined with the emergence of two-fold 
+symmetry for the in-plane Hc2 suggests unconventional 
+superconductivity in 1T′-WS2. 
+We further explored the superconducting transition via 
+systematic temperature dependent measurements. Figures 
+1i and 1j show such data taken when the magnetic field was 
+perpendicular and parallel to the sample plane, respectively. 
+In both cases, the superconducting transition shifts 
+gradually to lower magnetic fields as the temperature 
+increases. The temperature dependence of the out-of-plane 
+upper critical field (Hc2,⊥) and in-plane upper critical field 
+
+COS 0 
+FIG.1 Crystal structure and Superconductivity of 1T′-WS2. a, Schematic illustration of the structure of 1T′-WS2. Top 
+panel: side view of the crystallographic structure; bottom panel: top view of a typical monolayer. b, Cross-sectional STEM 
+image of 1T′-WS2. Inset: high-magnification STEM image of layered structure with atomic resolution. c, Temperature-
+dependent electrical resistance of the mechanically exfoliated 1T′-WS2 without magnetic field. Insets: optical image of the 
+1T′-WS2 device covered by h-BN with Hall-bar configuration (top) and small range Rxx-T plot of 1T′-WS2 around Tc shown in 
+the area within the red rectangle (bottom). d, Schematic illustration of a 1T′-WS2 device and the rotation experiment setup, 
+where the x-axis is parallel to c-axis of the crystal and z-axis is perpendicular to crystalline plane. θ is the angle between the 
+out-of-plane magnetic field and the z-axis; φ is the angle between the in-plane magnetic field and the x-axis. e, Magnetic field 
+dependence of the normalized resistance of the 1T′-WS2 device at T = 0.33 K with different out-of-plane rotation angles θ. f, 
+The θ-dependence of the upper critical field. The blue curve denotes a fit to the data following the Tinkham formula for a 2D 
+superconductor. g, Magnetic field dependence of the normalized resistance of the 1T′-WS2 device at T = 0.33 K with 
+different in-plane rotation angles φ. h, The φ-dependence of the upper critical field. The green dashed line indicates the Pauli 
+limit. i, j, Superconducting transition of the 1T′-WS2 device under a perpendicular magnetic field (i) and under a parallel 
+magnetic field (j) at different temperature. k, Temperature dependence of the upper critical field with magnetic field 
+directions parallel and perpendicular to the crystal plane. The red curve represents the linear relationship between Hc2,⊥ and T 
+according to the 2D GL theory. 
+
+c
+0.3
+10 um
+BN
+0.2
+IT'WS
+0.05
+C
+0.1
+0.0
+12
+T (K)
+0
+100
+200
+300
+T (K)
+2D-Tinkham (Hc2,//) are summarized in Fig. 1k. Hc2,⊥ displays a linear 
+dependence on temperature, that is well fitted by the 
+standard 
+Ginzburg-Landau 
+(GL) 
+theory 
+for 
+2D 
+superconductors [16]: 
+ 
+where 
+(0) is the zero-temperature GL in-plane 
+coherence length, Φ0 is the magnetic flux quantum, and Tc 
+is the critical temperature at which the resistance drops to 
+50% of its value in the normal state. From the fit we can 
+estimate the coherence length 
+(0) ≈ 9.6 nm. The 
+temperature dependence of Hc2,//, on the other hand, follows 
+the GL formula expected for 2D superconductors [16]: 
+ 
+where dSC is the superconducting thickness. From the 
+fitting of Hc2,//, the superconducting thickness is around 3.2 
+nm, which is smaller than 
+(0) and consistent with 2D 
+superconductivity.  
+ 
+ 
+FIG.2 Scanning tunneling microscopy measurements on 
+1T′-WS2. a, Topographic image of the bc plane of 1T′-
+WS2. Top inset: a zoom-in view of the topographic image 
+showing the atomic arrangements. Bottom inset: Fast 
+Fourier transform of the topographic image. b, A zero-bias 
+conductance map of vortices at 1 T. Inset: Fourier 
+transform of the dI/dV map. c, d, The conductance map of a 
+single vortex at 1 T with zero bias (c) and 1 mV bias (d). e, 
+Tunneling spectroscopy spectrum taken at 4.2 K, revealing 
+a superconducting gap. Light blue curves are the 
+differential spectra taken at different positions on the 
+surface; the dark blue curve denotes the average spectra. f, 
+Field dependence of tunneling spectroscopy taken at 0 T, 1 
+T and 5 T.  
+   To further characterize the anisotropic superconductivity 
+in 1T′-WS2, we performed scanning tunneling microscopy 
+(STM) measurements and directly imaged the vortices 
+under magnetic field. A single crystal was cleaved in-situ at 
+T = 77 K and measured at T = 4.2 K. Figure 2a shows the 
+topography of 1T′-WS2 over a large area. The atomically 
+resolved STM topographic image reveals a clean surface 
+featuring zigzag chains along the b-axis of the crystal (top 
+inset of Fig. 2a). In addition, the corresponding fast Fourier 
+transform pattern also exhibits the distorted octahedral 
+coordination feature (bottom inset of Fig. 2a). A zero-
+energy conductance map under 1 T applied perpendicularly 
+to the bc plane is shown in Fig. 2b. The Fourier transform 
+of the dI/dV map is two-fold symmetric (inset of Fig. 2b). 
+The conductance maps of a single vortex at 0.1 T taken at V 
+= 0 mV (Fig. 2c) and 1 mV (Fig. 2d) further highlight the 
+anisotropic nature of the superconductivity. Consistent with 
+the anisotropy observed in our transport data, the vortices 
+are anisotropic and elongated along the b direction, 
+reflecting the anisotropy of the Ginzburg-Landau coherence 
+length between both directions. Tunneling differential 
+conductance collected from an atomically resolved lattice 
+illustrates a superconducting gap with sharp coherence 
+peaks (Fig. 2e). This superconducting gap disappears 
+gradually as the magnetic field is increased (Fig. 2f). 
+     Subsequently, we performed critical current density (Jc) 
+measurements. As alluded in the introduction, an important 
+aspect of a superconductor is its Jc, which dictates several 
+practical 
+applications. 
+The 
+higher 
+the 
+Jc 
+of 
+a 
+superconductor, the smaller and more efficient the 
+superconducting devices that can be fabricated from it or 
+the larger the magnetic fields that can be generated. We 
+measured differential resistance of the 1T′-WS2 device with 
+thickness of 6 nm as a function of direct current (DC) bias 
+current at different temperatures (Fig. 3a). Note that, Jc is 
+defined as the current density at which the differential 
+resistance (dV/dI) reaches its maximum, as reported in 
+previous works [21,22]. Remarkably, as seen in Fig. 3b, 
+1T′-WS2 exhibits ultrahigh critical current densities 
+reaching 17 MA/cm2 at T = 0.33 K. Figure 3b highlights the 
+temperature dependence of the critical current density, 
+featuring an enormous Jc = 13 MA/cm2 at liquid He 
+temperature (4.2 K). In addition, we systematically 
+measured the critical currents of samples with different 
+layer thicknesses, as shown in Fig. S6. The thickness 
+dependence of the critical current density is summarized in 
+Fig. 3c. There is no obvious difference among the samples 
+with thicknesses exceeding 20 nm. The critical current 
+densities increase as the devices become thinner, which is 
+also observed in atomically thin TaS2 [23]. Furthermore, we 
+evaluated the field dependence of Jc (Fig. S7). The critical 
+current density falls rapidly as the perpendicular magnetic 
+field increases (Fig. 3d). In contrast, the critical current 
+density is rarely influenced by a parallel magnetic field 
+since 1T′-WS2 shows extremely high in-plane upper critical 
+
+GJH
+100nm
+0
+0.2nS
+100nm
+0mV
+e
+T=4K
+T=4K
+0.2
+0.2
+(su)
+(su) Λp/Ip
+my
+ΛP/Ip
+0.1
+0
+5T
+1T
+-OT
+0
+-10
+-5
+0
+5
+10
+-10
+-5
+0
+5
+10
+20nm
+Bias (mV)
+Bias (mV)fields. Even under an 8 T in-plane magnetic field, Jc is 
+substantially large (7 MA/cm2).  
+Experimentally, numerous 2D superconducting transition 
+metal dichalcogenides have been studied [20-33]. In-plane 
+anisotropic upper critical fields were observed in 2H-NbSe2 
+[24] and Td-MoTe2 [25]. 2H-NbSe2 [20] and ionic-gated 
+2H-MoS2 [26] also exhibited high in-plane upper critical 
+fields. However, we emphasize that 1T′-WS2 is the only 2D 
+material to our knowledge that shows the suitable critical 
+temperature and high critical current under high in-plane 
+magnetic field, which are crucial for building high-field 
+magnets. Even for a thick sample, the in-plane critical field 
+surpasses 8 T at 4 K (Fig. S8). We summarize the 
+parameters of 2D superconductors in Fig. 3e. As for 1T-
+MoS2 [27], 2H-TaS2 [23], 3R-TaSe2 [28], Td-MoTe2 [25], 
+2H-NbS2 [29], their critical temperatures are below the 
+temperature of liquid helium (4.2 K), rendering the 
+construction of high-field magnets impractical. Gated MoS2 
+displays a relatively high critical temperature and also very 
+high critical fields, but superconductors under ionic gating 
+are not suitable for applications [30]. Lastly, 2H-NbSe2 is 
+comparable to 1T′-WS2 in critical fields and critical 
+temperatures. However, its critical current density is two 
+orders of magnitude lower than that of 1T′-WS2 [31]. The 
+significance of our work is that we 
+report 
+an 
+unprecedentedly high superconducting critical current 
+density (17 MA/cm2 at 0 T) in 1T′-WS2, which exceeds 
+those of all the known 2D superconductors to date [21-33]. 
+Notably, it even exceeds the Jc of MgB2 films [34], a well-
+known superconductor for high-critical-current applications 
+(Fig. 3e). Even under an 8 T in-plane magnetic field, the Jc 
+of 1T′-WS2 is substantially large (7 MA/cm2). As a 
+reference, the critical currents of commercial magnet 
+building materials are listed here, such as Nb-Ti alloy (0.1 
+MA/cm2 at 10 T) and Nb3Sn (0.5 MA/cm2 at 10 T) [35]. 
+The large Jc at zero and finite magnetic fields makes 1T′-
+WS2 a potential candidate for future study on building next-
+generation superconducting magnets. 
+Having explored the superconductivity of 1T′-WS2, we 
+turn to the topological features pertinent to its electronic 
+band structure using a series of theoretical calculations and 
+angle-resolved 
+photoemission 
+spectroscopy 
+(ARPES) 
+experiments. The calculated bulk band structure is shown in 
+Fig. S9. Besides the continuous energy gap between 
+conduction band and valence band around the Fermi level, 
+we observe a band inversion at the -point between W d 
+and S p orbitals, which leads to a strong topological 
+insulating phase. Furthermore, the surface-projected 
+calculation shows the topological Dirac surface state 
+emerging from the valence band and merging into 
+conduction bands (Fig. 4a). The corresponding ARPES data 
+(Fig. 4b), taken at T = 10 K (above Tc) matches the first 
+principles calculations below EF. In particular we identify 
+the linear-dispersed hole pocket at 
+to be the lower cone 
+of the topological surface state, as it shows no photon- 
+ 
+FIG.3 Ultrahigh critical current density of 1T′-WS2. a, 
+Differential resistance of a 6-nm-thick 1T′-WS2 sample as a 
+function of the direct current (DC) bias at different 
+temperatures. b, Critical current density for a 1T′-WS2 
+device as a function of the temperature. c, Critical current 
+density of the 1T′-WS2 device plotted as a function of 
+sample thickness. d, Critical current densities of the 1T′-
+WS2 device plotted as a function of perpendicular and 
+parallel magnetic fields. e, Comparison of critical current 
+densities among 1T′-WS2 and other representative 2D 
+superconductors, such as twisted bilayer graphene (TBG) 
+and transition metal dichalcogenides. Commercial magnet 
+building materials are also included for reference. Here, the 
+superconducting critical temperatures Tc of the different 
+materials were determined under zero magnetic field. 
+energy dependence and agrees well with the calculated 
+dispersion of the Dirac state (More details of ARPES data 
+analysis are shown in Figs. S10-12). ARPES Fermi surface 
+map also visualizes the highly anisotropic Fermi surface 
+(Fig. 4c), which possibly contributes to the extremely 
+anisotropic 
+Hc2 
+in 
+1T′-WS2. 
+The 
+calculated 
+superconducting gap of 1T′-WS2 on the Fermi surface is 
+presented in Fig. 4d. These results lend crucial credence to 
+the in-plane anisotropy of superconductivity. It is noted that 
+so far there is no clear relationship between the topological 
+nature and high critical current.  
+
+0.4K -
+ 1.8K—2.5K—3.5K
+4.5K—5.5K—
+6.5K—7.5K
+8.4nm 1T-MoS2
+2nm 1T-WS2
+7nm 3R-TaSe2
+6nm 1T'-WS2★
+6.5nm T.-MoTe2
+1T" WS, @8T
+1.6nm TBG
+6nm 2H-NbS2
+7.6nm a-Mo2C
+MgB- filn
+4.2nm 2H-TaS2
+Nb,Sn @10T
+3nm 2H-NbSe2
+1.6nm gated
+Nb-Ti @10T
+MoS2 
+FIG.4 Topological features of 1T′-WS2. a, Calculated surface band structure of 1T′-WS2 at ky = 0, featuring a topological 
+Dirac surface state near the Fermi level. b, Energy-momentum cut acquired through ARPES. c, Fermi surface of 1T′-WS2. d, 
+Calculated superconducting gap which all kz are projected in the surface Brillouin zone at 2.5 K on the Fermi surface. The 
+unit of the color bar is meV. 
+In summary, combining a series of experimental and 
+numerical techniques, we comprehensively studied 1T′-
+WS2 and find a unique blend of ultrahigh critical 
+supercurrent density, large superconducting anisotropy (in-
+plane versus out-of-plane) along with topological features. 
+Our findings not only provide a promising material 
+platform for high magnetic field technologies but also 
+unveil a promising platform for future exploration of 
+topological superconductivity, which may be used to 
+fabricate topologically protected qubits for future quantum 
+computing schemes. 
+ 
+ACKNOWLEDGEMENTS. Experimental and theoretical 
+work at Princeton University was supported by the Gordon 
+and 286 Betty Moore Foundation (GBMF4547; M.Z.H.). 
+The material characterization is supported by the United 
+States 287 Department of Energy (US DOE) under the 
+Basic Energy Sciences program (grant number DOE/BES 
+DE-FG-288 02-05ER46200). L.B. is supported by DOE-
+BES through award DE-SC0002613. The National High  
+ 
+ 
+Magnetic Field Laboratory acknowledges support from the 
+US-NSF Cooperative agreement Grant number DMR-
+1644779 and the state of Florida. The authors acknowledge 
+the sample characterization of Imaging and Analysis Center 
+(IAC) at Princeton University, partially supported by the 
+Princeton Center for Complex Materials (PCCM) and the 
+NSF-MRSEC program (MRSEC; DMR-2011750). G.C. 
+acknowledges the support of the National Research 
+Foundation, Singapore under its Fellowship Award (NRF-
+NRFF13-2021-0010) 
+and 
+the 
+Nanyang 
+Assistant 
+Professorship 
+grant 
+from 
+Nanyang 
+Technological 
+University. J.Y.Y. and Y.P.F. is supported by the Ministry 
+of Education, Singapore, under its MOE AcRF Tier 3 
+Award MOE2018-T3-1-002. H.Z. thanks the support from 
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+Metals Material Engineering Research Center (NPMM), the  
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+Research Grants Council of Hong Kong (AoE/P-701/20), 
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+of 
+Shenzhen 
+Municipality 
+(grant 
+no. 
+JCYJ20200109143412311). K.W. and T.T. acknowledge 
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+

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+MNRAS 000, 1–?? (2023)
+Preprint 27 January 2023
+Compiled using MNRAS LATEX style file v3.0
+Divergence of the local large-scale structure velocity field
+and its implications for Tilted Cosmology
+Erick Past´en1⋆ Sebasti´an G´alvez2† V´ıctor H. C´ardenas1‡
+1Instituto de F´ısica y Astronom´ıa, Universidad de Valpara´ıso, Gran Breta˜na 1111, Valpara´ıso, Chile
+2Centro cient´ıfico tecnol´ogico de Valpara´ıso, Universidad Federico Santa Mar´ıa, Av. Espa˜na 1680, Valpara´ıso Chile
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+We characterize the peculiar velocity field of the local large-scale structure reconstructed from the 2M + + survey,
+by treating it as a fluid, extracting the gradient and the divergence via different approximations. This reconstructed
+field is important for cosmology, since it was used to correct the peculiar redshifts of the last SNIA compilation
+Pantheon+. We conclude that the local velocity field can be described on average as a slightly contracting fluid,
+with intriguing implications for the “Tilted Cosmology” model. We compute representative values of the apparent
+deceleration parameter (˜q) measured by observers inside the contracting region, in order to compare our results with
+the theoretical predictions of the tilted-universe scenario. As predicted, the computed values are found to be negative
+on a range of averaged scales, allowing for a possible explanation of dark energy as an effect induced by our peculiar
+motion relative to the universal expansion.
+Key words: dark energy – large-scale structure of Universe
+1 INTRODUCTION
+Dark Energy (DE) is the usual explanation for the apparent
+universal acceleration implied by the SNIA data (Riess et al.
+1998; Perlmutter et al. 1999). However, the suggestion for
+the existence of dark energy is ultimately based on the cos-
+mological principle, that is on the assumption of a globally
+homogeneous and isotropic Friedmann universe. The require-
+ment of an extra parameter ΩΛ is then necessary to explain
+the dimming of the supernovae magnitudes at large redshifts.
+Nevertheless, new interesting ideas have emerged in recent
+years, putting in doubt the cosmological principle, the Fried-
+mann models and the existence of dark energy.
+On sufficiently large scales the universe appears homo-
+geneous and isotropic, according to the Cosmic Microwave
+Background (CMB) observations. On small scales, however,
+our cosmos is far from that, due to complex structures that
+produce overdensities/underdensities (Keenan et al. 2013),
+fractal-like structures (Labini et al. 1998; Labini 2011) and
+bulk peculiar motions that are not at rest with respect to
+the Hubble flow (Hudson et al. 1999; Feindt 2013; Magoulas
+et al. 2014). There have been many works claiming that some
+of these effects can mimic an apparent acceleration. Possibly
+the combination of some (perhaps of all) of these contribu-
+tions may have an effect stronger than we have previously
+⋆ E-mail: [email protected]
+† E-mail: [email protected]
+‡ E-mail: [email protected]
+thought (Celerier 2006; Enqvist 2007; Tsagas 2011; Cosmai
+et al. 2019; Asvesta et al. 2022).
+One of the proposed scenarios is the “tilted cosmological
+model” (Tsagas 2011). The latter offers a natural environ-
+ment for the theoretical study of the observed large-scale
+peculiar motions, by allowing for two groups of relatively
+moving observers. One group is aligned with the reference
+frame of the cosmos, which is identified with the coordinate
+system of the CMB, where the associated dipole vanishes by
+construction. The second group are the real observers, living
+in typical galaxies like our Milky Way and moving relative
+to the CMB frame (e.g. see (Tsagas et al. 2008; Ellis et al.
+2012)). Adopting a tilted almost-Friedmann universe and us-
+ing linear relativistic cosmological perturbation theory, it was
+shown that relative-motion effects can lead to an apparent
+change in the sign of the deceleration parameter inside lo-
+cally contracting bulk flows. Although the effect is a local
+artefact of the observers’ peculiar motion, the affected scales
+can be large enough to have cosmological relevance. Then,
+observers inside (slightly) contracting bulk peculiar flows can
+be misled to believe that their universe recently entered a
+phase of accelerated expansion. Put another way, the unsus-
+pecting observers may misinterpret the local contraction of
+the bulk flow they live in, as global acceleration of the sur-
+rounding universe (see (Tsagas & Kadiltzoglou 2015; Tsagas
+2021, 2022) for further discussion and details). Our aim is to
+investigate this possibility by comparing theory to observa-
+tions.
+We use the velocity field reconstruction from the 2M++
+galaxy survey (Carrick et al. 2015). This reconstruction pro-
+© 2023 The Authors
+arXiv:2301.11246v1  [astro-ph.CO]  26 Jan 2023
+
+2
+E. Past´en et al.
+vides a pair of data-cubes containing the density contrast and
+the velocity vectors in galactic coordinate. One can use these
+data to apply basic calculus and also to perform corrections
+due to peculiar velocities in cosmological data, as it was done
+in the last Pantheon+ SNIA compilation (Scolnic et al. 2022;
+Carr et al. 2022). In the present paper we estimate the av-
+erage volume scalar of this local velocity-field reconstruction
+by different methods and on different scales. In all cases, the
+local bulk flow is found to contract on average, leading to neg-
+ative values for the local deceleration parameter on a range
+of scales. These results seem to support the tilted cosmolog-
+ical scenario as an alternative natural explanation of the DE
+problem.
+In section 2 we provide a brief but concise description of
+tilted cosmological scenario, referring the reader to the re-
+lated literature for more details. In section 3 we discuss how
+to relate the parameters obtained from the velocity-field re-
+construction with the theory and in section 4 we present the
+data used. Finally, we summarize the method and the results
+in sections 5 and 6 and discuss their implications for cosmol-
+ogy at the end of this paper. In addition to cosmology, our
+analysis has potential applications to astrophysics and to the
+local structure dynamics.
+2 TILTED COSMOLOGY MODEL
+Consider a perturbed Friedmann-Robertson-Walker (FRW)
+universe with two groups of relatively moving observers. As-
+suming that ua and ˜ua are the 4-velocities of these observers
+and va is the (non-relativistic) peculiar velocity of the latter
+group with respect to the former, we have
+˜ua = ua + va ,
+(1)
+to first approximation (with uava = 0 always). Introducing
+two sets of observers means that (strictly speaking) there are
+two temporal directions (along ua and ˜ua) and two associated
+3-spaces (orthogonal ua to and ˜ua). Then, the corresponding
+(covariant) differential operators are ˙ = ua∇a and ′ = ˜ua∇a
+for the time derivatives, with Da = ha
+b∇b and ˜Da = ˜ha
+b∇b for
+the spatial gradients. Also, the tensors hab = gab + uaub and
+˜hab = gab + ˜ua˜ub project orthogonal to ua and ˜ua respectively.
+The kinematic information of the observers’ motion is de-
+coded by decomposing the gradient of their 4-velocity field
+as follows
+∇bua = 1
+3 Θhab + σab + ωab − Aaub .
+(2)
+In the above, Θ is the volume expansion/contraction scalar
+(when positive/negative respectively), σab is the shear, ωab
+is the vorticity and Aa is the 4-acceleration (e.g. see Tsagas
+et al. (2008); Ellis et al. (2012)). In an exactly analogous way,
+the ˜ua-field splits as ∇b˜ua = (˜Θ/3)˜hab + ˜σab + ˜ωab − ˜Aa˜ub, with
+the tildas denoting variables evaluated in the tilted frame of
+the bulk flow. Relative to the same coordinate system, the
+peculiar-velocity field splits as
+˜Db˜va = 1
+3
+˜θ˜hab + ˜ςab + ˜ϖab ,
+(3)
+where ˜θ, ˜ςab and ˜ϖab are the volume scalar, the shear and the
+vorticity of the bulk peculiar motion (Tsagas & Kadiltzoglou
+2015). Of the last three variables, the most important for our
+purposes is the peculiar volume scalar (˜θ), which takes pos-
+itive/negative values in locally expanding/contracting bulk
+flows respectively.
+The three kinematic sets defined above are related by
+lengthy nonlinear expressions (e.g. see Maartens (1998) for
+the full list). Assuming non-relativistic peculiar motions on
+an FRW background, we obtain the linear relations
+˜Θ = Θ + ˜θ
+and
+˜Θ
+′ = ˙Θ + ˜θ
+′ ,
+(4)
+between the volume scalars and between their time deriva-
+tives evaluated in the two frames. At this point, we note
+that Θ and ˜Θ monitor the expansion rate of the universe,
+namely the Hubble parameters, as measured in their corre-
+sponding frames (that is Θ = 3H and ˜Θ = 3 ˜H). Then, equa-
+tions (4a) and (4b) imply that the expansion and the accel-
+eration/deceleration rates measured in the tilted coordinate
+system differ from those measured in its CMB counterpart
+solely due to relative-motion effects. In particular, recalling
+that
+q = −1 − 3 ˙Θ
+Θ2
+and
+˜q = −1 − 3˜Θ′
+˜Θ2 ,
+(5)
+define the deceleration parameters in the CMB and the bulk-
+flow frames respectively, the following useful relation between
+˜q and q can be obtained (Tsagas & Kadiltzoglou 2015; Tsagas
+2021):
+˜q = q +
+˜θ′
+2 ˙H
+,
+(6)
+to first approximation. Recall that ˜Θ = Θ = 3H in the Fried-
+mann background. Also note that, whereas ˜θ/H ≪ 1 at the
+linear level, the ratio ˜θ′/ ˙H of their time derivatives is not
+always small. Finally, using relativistic linear cosmological
+perturbation theory, we arrive at:
+˜q = q + 1
+9 (λH
+λ )
+2 ˜θ
+H .
+(7)
+with λH = 1/H and λ representing the Hubble horizon and
+the scale of the bulk flow in question. Note that we have fo-
+cused on bulk peculiar flows with sizes considerably smaller
+than the Hubble length (i.e. λ ≪ λH – see (Tsagas & Kadilt-
+zoglou 2015; Tsagas 2021) for the full details of the deriva-
+tion).1
+Following (7), the deceleration parameter measured locally
+by the bulk flow observers (˜q) differs from that of the global
+universe, which by definition coincides with the deceleration
+parameter measured in the idealised CMB frame (q). The
+difference is entirely due to the peculiar motion of the tilted
+observer, since ˜q = q when ˜θ = 0. Also, the “correction” term
+in (7) is scale-dependent and it gets stronger on progressively
+smaller scales (i.e. for λ ≪ λH), despite the fact that ˜θ/H ≪ 1
+throughout the linear regime. Moreover, in accord with (7),
+the overall impact of relative motion on ˜q is also determined
+by the sign of the peculiar volume scalar (˜θ). The latter is
+positive in locally expanding bulk flows, which means that
+1 Expression (7) has been obtained on an Einstein-de Sitter back-
+ground, primarily for reasons of mathematical simplicity. It is fairly
+straightforward to show that the linear result (7) holds on essen-
+tially all FRW backgrounds, irrespective of their equation of state
+and spatial curvature (Tsagas 2022).
+MNRAS 000, 1–?? (2023)
+
+Divergence of the local large-scale structure velocity field and its implications for Tilted Cosmology
+3
+the deceleration parameter measured by observers residing in
+them will be larger than that of the actual universe (i.e. ˜q > q
+when ˜θ > 0). In the opposite case, that is inside locally con-
+tracting bulk flows, the local deceleration parameter becomes
+smaller (i.e. ˜q < q for θ < 0). The latter case is clearly the most
+intriguing, since it allows for the sign of the deceleration pa-
+rameter to change, from positive to negative, when measured
+by observers inside locally contracting bulk flows. Although
+the sign-change of ˜q is simply an illusion and a local artefact
+of the observer’s relative motion, the affected scales can be
+large enough to make it look as a recent global event. If so,
+an unsuspecting observer may be misled to believe that their
+universe has recently entered a phase of accelerated expan-
+sion. According to (7), the “transition scale”, where the local
+deceleration parameter crosses the ˜q = 0 threshold is (Tsagas
+2021)
+λT = 1
+3
+√
+˜∣θ∣
+qH λH ,
+(8)
+where q > 0 always (with q = 1/2 in the case of the Einstein-de
+Sitter background).
+Although theoretically the model outlined above is well
+developed, it is not obvious yet how one should relate the
+tilted cosmological scenario to the observations. Parametriz-
+ing the deceleration function as ˜q = ˜q(z) and then using it
+in (7), has led to a good fit with the Pantheon SNIA sam-
+ple (Asvesta et al. 2022). Also, an apparent (Doppler-like)
+dipole anisotropy is expected to appear in the observed dis-
+tribution of the local deceleration parameter (˜q), due to the
+bulk-flow motion relative to the CMB frame (Tsagas 2011).
+However, which observational frame (heliocentric, geocen-
+tric, galactic, or cosmological) should be employed and what
+peculiar-velocity corrections should be applied to the data, in
+order to observe the aforementioned dipolar anisotropy, are
+the subjects of ongoing debate (Colin et al. 2019a,b; Rubin &
+Heitlauf 2020; ?). In this paper, our aim is to study the dy-
+namical structure of the peculiar velocity field directly from
+data reconstruction. We choose the velocity field reconstruc-
+tion of the 2M++ survey (Carrick et al. 2015), which has been
+previously used in cosmology to correct the peculiar velocities
+of the SNIA data in the last Pantheon+ compilation (Scolnic
+et al. 2022; Carr et al. 2022). The methods used to character-
+ize this peculiar velocity field and the procedures employed
+to relate our results with those of the tilted cosmologies are
+discussed in the next sections.
+3 CLASSICAL FLUID APPROXIMATION
+The kinematic analysis outlined in the previous section, is
+straightforwardly adapted to the Newtonian framework as
+well (e.g. see Ellis (1971, 1990) for further discussion and de-
+tails). In so doing, one replaces the projector (hab), which
+also acts as the metric tensor of the 3-space, with the Kro-
+necker delta (δij). Also, time derivatives and 3-dimensional
+covariant gradients are replaced by convective derivatives and
+by ordinary partial derivatives respectively. Note that, given
+the near spatial flatness of the observed universe, any cur-
+vature corrections due to a nonzero connection (Γa
+bc) will
+be of the second perturbative order. Then, focusing on the
+Figure 1. Peculiar velocities of the Pantheon+ SNIA compilation
+peculiar-velocity field (v = vi), we have ˜θij = ∇v = ∂jvi and
+˜θij = 1
+3
+˜θδij + ˜ςij + ˜ϖij .
+(9)
+Here, the (local) volume expansion/contraction scalar, the
+shear tensor and the vorticity tensor of the bulk peculiar flow
+are respectively defined as
+˜θ
+=
+∂ivi = δ
+ij∂jvi ,
+(10)
+˜ςij
+=
+1
+2 (∂jvi + ∂ivj) − 1
+3
+˜θ δij,
+(11)
+˜ϖij
+=
+1
+2 (∂jvi − ∂ivj) .
+(12)
+It ts possible to evaluate the gradient tensor of the local
+peculiar-velocity field using this approach. In particular, the
+gradient tensor can be reduced to the 3 × 3-matrix of the
+partial derivatives of the ˜vi-field as:
+˜θij = ∂jvi =
+⎛
+⎜⎜⎜⎜⎜⎜
+⎝
+∂vx
+∂x
+∂vx
+∂y
+∂vx
+∂y
+∂vy
+∂x
+∂vy
+∂y
+∂vy
+∂y
+∂vz
+∂x
+∂vz
+∂y
+∂vz
+∂y
+⎞
+⎟⎟⎟⎟⎟⎟
+⎠
+,
+(13)
+directly relating ˜θij to the Jacobian tensor of the field.
+4 THE 2M++ VELOCITY FIELD
+RECONSTRUCTION
+The
+last
+Supernovae
+IA
+compilation,
+namely
+Pan-
+theon+ (Scolnic et al. 2022), was released in 2022 showing
+a great improvement in the utility of data at low redshifts
+for cosmological uses. Part of this improvement is due to
+better corrections of the peculiar velocities of the SNIA
+data (Carr et al. 2022) (see Figure 1). This was done by
+using a velocity field reconstruction based on the 2M++
+galaxy survey (Carrick et al. 2015) (see Figure 2). The
+reconstruction procedure can be summarized as follows. If
+MNRAS 000, 1–?? (2023)
+
+400
+75
+50
+200
+25
+0
+-25
+-200
+-50
+75
+-400
+0
+50
+100
+150
+200
+250
+300
+350
+Ideg4
+E. Past´en et al.
+Figure 2. Peculiar Velocity field reconstruction from the 2M++ density field in galactic coordinates. Visualizations in 3D, with (left)
+and without (right) external dipole component.
+δ(r) is the density contrast, then the peculiar velocity field
+can be approximated as proportional to the gravitational
+acceleration when the fluctuations are small:
+v(r) = f(Ωm)
+4π
+∫ d
+3r
+′δ(r
+′) r′ − r
+∣r′ − r∣3 .
+(14)
+Here, f is the growth rate of cosmic structures defined as
+f = Ωγ
+m, where γ = 0.5 for ΛCDM cosmology. Also, r = HR
+is measured in km/s where R, with R being the comoving
+distance in Mpc and H the Hubble parameter.
+Since the total density perturbation (δ) cannot be directly
+observed, a bias parameter (b) has been introduced to relate
+the observed density contrast (δg) with the real one:
+δ =
+δg
+b ,
+(15)
+at the linear level. Therefore, the important parameter in
+evaluating the velocity field is the ratio β = f/b, since we can
+write:
+v(r) = β
+4π ∫ d
+3r
+′δg(r
+′) r′ − r
+∣r′ − r∣3 ,
+(16)
+to relate directly the peculiar velocity with the observed
+galaxy density. Also, as the observations extend only up
+to a maximum scale (Rmax), the contribution beyond this
+length can be added as a constant external velocity parame-
+ter (Vext), so that finally:
+v(r) = β
+4π ∫
+Rmax
+d
+3r
+′δg(r
+′) r′ − r
+∣r′ − r∣3 + Vext ,
+(17)
+where β and Vext are determined empirically from the recon-
+struction of the density field.
+We use the density contrast and the velocity field (see Fig-
+ure 3)) given by (Carrick et al. 2015), which can be easily
+downloaded from https://cosmicflows.iap.fr/. There, the au-
+thors provide two useful data-cubes containing the density
+contrast δ and the velocity field v using the best-fit param-
+eters β = 0.431 ± 0.021 and Vext = (89 ± 21, −131 ± 23, 17 ±
+26)km/s (with ∣Vext∣= 159±23km/s) in galactic Cartesian co-
+ordinates. It is also important to note that a different value
+of β, namely β = 0.341+0.031
+−0.047, was used to correct the Pan-
+theon+ data (Said et al. 2020; Carr et al. 2022), claiming
+that it gives a better fit when comparing the SDSS Funda-
+mental Plane peculiar velocities to the predicted peculiar ve-
+locity field. Overall, we can write v = βvrec, where vrec gives
+the directions and relative magnitudes of the velocity field.
+Then, it is easy to use both values and compare the results.
+In order to apply the same corrections to Pantheon+, the
+whole velocity field was approximated by a radially decaying
+function along the direction of the bulk flow. The latter is a
+200 Mpc sphere, composed by the sum of an external Vext
+and a small average internal velocity v200. Interestingly, the
+external dipole component does not contribute to the gradi-
+ent as it is a constant.2 Therefore:
+∇v = ∇(βvrec + Vext) = β∇vrec .
+(18)
+5 METHODS
+5.1 Finite differences
+We use the central finite difference method to compute
+derivatives in each pixel of the data-cube as:
+∇v = ∂jvi ≈
+vi(xj + s) − vi(xj − s)
+2s
+,
+(19)
+where xi is the central point of a pixel (I, J, K) of the array.
+Also, s is the physical size of a pixel, so we can use directly the
+2 Even a radially decaying Vext function, with a fixed direction,
+does not affect the average divergence of the velocity field.
+MNRAS 000, 1–?? (2023)
+
+150
+100
+Mpc)
+50
+T-)
+0
+-100
+-150
+-200
+200
+150
+100
+50
+-200
+-150
+-100
+-50
+0
+-100
+50
+-150
+100
+150
+200
+200150
+100
+Mpc)
+50
+T-)
+0
+-100
+-150
+-200
+200
+150
+100
+50
+-200
+-150
+-100
+-50
+-100
+50
+-150
+100
+150
+200
+200Divergence of the local large-scale structure velocity field and its implications for Tilted Cosmology
+5
+Figure 3. The density contrast and the vector velocity field projected over the GZ = 0 and GZ = 50h−1Mpc galactic planes.
+cube data to ensure the right conversion of a pixel to the phys-
+ical value, which fortunately is the same for each coordinate.
+For the case of the velocity field used here, with 2573 pixels
+in a datacube of length 400/h (where h/100 km/secMpc is
+the dimensionless normalised Hubble parameter), we have:
+s = 400Mpc
+257h
+,
+(20)
+Neglecting the borders of the sample, leads to a 2553 array
+containing each pixel in a 3 × 3 matrix that corresponds to
+the gradient tensor of the peculiar velocity field. For a central
+finite difference approximation of a function f, one may write:
+f
+′(x) = f(x + s) − f(x − s)
+2s
+− s2
+6 f
+′′′(ξ)
+= f
+′
+1(x) − ϵ ,
+(21)
+for some ξ ∈ [x − s, x + s]. In the above, f ′
+1(x) is the cen-
+tral approximation for the derivative and ϵ ∝ s2f ′′′(ξ) is the
+truncation error.
+MNRAS 000, 1–?? (2023)
+
+GZ= 0 (h-1 Mpc)
+200
+150 -
+10
+100
+GY (h-1 Mpc)
+50
+0
+OS-
+100
+150
+200
+200
+150
+100
+-50
+100
+150
+200
+GX (h-1 Mpc)GZ= 50 (h-1 Mpc)
+200
+150 -
+5
+100
+GY (h-1 Mpc)
+50
+0
+-50
+-100
+150
+200
+200
+150
+100
+-50
+50
+100
+150
+200
+Gx (h-1 Mpc)GZ= 0 (h-1 Mpc)
+200 -
+150
+1000
+100 -
+ 800
+50 
+GY (h-1 Mpc)
+600
+-50
+400
+-100 -
+200
+-150
+-200 -
+-200
+-150
+-100
+-50
+50 
+100
+150
+200
+GX (h-1 Mpc)GZ= 50 (h-1 Mpc)
+200 -
+150
+800
+100 -
+50 
+ 600
+GY (h-1 Mpc)
+ 400
+-50
+-100 
+200
+-150
+-200 -
+-200
+-150
+-100
+-50
+50 
+100
+150
+200
+GX (h-1 Mpc)6
+E. Past´en et al.
+Figure 4. Graphic representation of the upper face of a box with side 2s enclosing the pixel [I, J, K]. The finite difference method
+approximate the flux across this surface as simply the contribution of v[I, J, K + 1] over it (left). Meanwhile, the integral approximation
+technique considers the contributions of the side and of the diagonal pixels as well (right).
+5.2 Integral Approximations
+A direct approximation of the divergence could be computed
+recalling the definition of the operator:
+∇ ⋅ v = lim
+V →0
+1
+V ∯
+∂V v ⋅ dA .
+(22)
+In so doing, we choose a box-like volume of size 2s around
+the central point of each pixel and compute the flux of the
+velocity field through the box. Then, by dividing the flux
+over the volume, we can extract an approximate value for
+the divergence. Note the difference with the central finite dif-
+ference method in equation (19), as this approach ignores the
+contribution over diagonal pixels (see Figure 4) .
+5.3 Theoretical estimation
+Following equation (17) the divergence of the velocity field
+and the density contrast are also related by the linear ex-
+pression:
+∇ ⋅ v(r) = β
+4π ∫
+Rmax
+d
+3r
+′δg(r
+′)∇ ⋅
+r′ − r
+∣r′ − r∣3 = −βδg(r) .
+(23)
+Note that, as the r coordinate is measured in km/s, to ex-
+press the divergence in
+km/s
+Mpc/h units, we need to multiply this
+quantity by a factor of 100h2.
+6 RESULTS
+We have computed the gradient matrix using the Numpy pack-
+age from Python to manipulate matrix and arrays. The fol-
+lowing three methods of estimating the volume scalar have
+been used:
+(i) A decomposition of the full gradient tensor of the ve-
+locity field employing finite differences.
+(ii) A integration approximation for the divergence using
+a box volume around the pixels.
+(iii) A theoretical estimation by means of relation (23).
+The results obtained via these different methods are plot-
+ted in Figure 5. To relate the latter with the tilted cosmology
+scenario, we need to estimate an average value for ˜θ and thus
+put the predictions of the tilted model to the test. In our
+analysis, this corresponds to an average divergence of the
+entire fluid, which is then averaged over a spherical volume
+V = 4πλ3/3 as:
+˜θ = 1
+V ∫
+V (∇ ⋅ v)dV ≈ s3
+V ∑
+i
+(∇ ⋅ v)i ,
+(24)
+where the sum is over the pixels that reside inside a sphere
+of radius λ.
+Assuming that λ = 200/h Mpc for example, which is the
+radial scale of the survey, and setting β ≃ 0.43, as provided
+by (Carrick et al. 2015), the finite difference method leads to:
+˜θ ≈ −0.24 km/s
+Mpc/h .
+(25)
+Surprisingly, we have a locally contracting peculiar veloc-
+ity field, as the average divergence is negative. Alternatively,
+if we use the value β ≃ 0.34 that was used to correct the Pan-
+theon+ redshifts, we have:
+˜θ ≈ −0.19 km/s
+Mpc/h .
+(26)
+The corresponding values of the local divergence field
+through the integral approximation method are:
+˜θ ≈ −0.21 km/s
+Mpc/h,
+(27)
+MNRAS 000, 1–?? (2023)
+
+v[I-1,J+1,K+1]
+v[],J+1,K+1]
+v[I+1,J+1,K+1]
+v[I-1,J+1,K+1]
+v[lJ+1,K+1]
+v[I+1,J+1,K+1]
+v[1-1,J,K+1]
+v[I+1,J,K+1]
+v[I-1,J,K+1]
+v[l,J+1,K+1]
+v[],J,K+1]
+v[l,J,K+1]
+v[I-1,J-1,K+1]
+v[l,J-1,K+1]
+v[I+1,J-1,K+1]
+v[I-1,J-1,K+1]
+v[I,J-1,K+1]
+v[I+1,J-1,K+1]
+zDivergence of the local large-scale structure velocity field and its implications for Tilted Cosmology
+7
+Figure 5. Divergence of the velocity field in
+km/s
+Mpc/h using the finite difference, integral approximation and theoretical computation,
+projected over GZ = 0 and GZ = 50h−1Mpc galactic planes.
+MNRAS 000, 1–?? (2023)
+
+GZ= 50 (h-1 Mpc)
+150
+100
+OS-
+GY (h-1 Mpc)
+50 
+100
+50
+150
+-100
+200
+150
+-250
+150
+100
+-50
+50
+100
+150
+Gx (h-1 Mpc)GZ= 0 (h-1 Mpc)
+150
+100
+-100
+GY (h-1 Mpc)
+50 
+0
+200
+-50
+O0E-
+100
+150
+400
+150
+-100
+50
+100
+150
+Gx (h-1 Mpc)GZ= 50 (h-1 Mpc)
+150
+100
+50
+GY (h-1 Mpc)
+50
+0
+100
+-50
+150
+-100
+150
+200
+150
+-100
+50
+50
+100
+150
+Gx (h-1 Mpc)GZ= 0 (h-1 Mpc)
+200
+150 -
+100
+100
+GY (h-1 Mpc)
+50
+200
+0
+0E-
+-50
+400
+100
+500
+150
+200
+600
+-200
+150
+-100
+-50
+50
+100
+150
+200
+Gx (h-1 Mpc)GZ= 50 (h-1 Mpc)
+200
+150
+100
+50
+GY (h-1 Mpc)
+50
+100
+0
+150
+-50
+200
+-100
+250
+150
+00E-
+200
+-200
+150
+-100
+-50
+0
+50
+100
+150
+200
+Gx (h-1 Mpc)GZ= 0 (h-1 Mpc)
+150
+100
+100
+GY (h-1 Mpc)
+50 
+-200
+0
+50
+00E-
+-100
+400
+150
+500
+150
+100
+-50
+100
+150
+Gx (h-1 Mpc)8
+E. Past´en et al.
+Figure 6. Residual curl vector field in
+km/s
+Mpc/h units from finite difference method, projected over GZ = 0 and GZ = 50h−1Mpc
+when β ≃ 0.43 and
+˜θ ≈ −0.17 km/s
+Mpc/h.
+(28)
+for β ≃ 0.34.
+Finally, employing the theoretical estimation method, set-
+ting h ≃ 0.7 and integrating the density contrast over the
+same scale gives
+˜θ ≈ −0.29 km/s
+Mpc/h
+(29)
+and:
+˜θ ≈ −0.23 km/s
+Mpc/h.
+(30)
+respectively. Therefore, the theoretical estimation provides
+the higher values for ˜θ, while the integral approximation gives
+the lowest. What is most important, however, is that all three
+methods are consistent both in the sign and in the magnitude
+of ˜θ.
+Substituting ˜θ into the right-hand side of equation (7), we
+can compute representative estimates of the local decelera-
+tion parameter (˜q) measured by the bulk-flow observers on
+MNRAS 000, 1–?? (2023)
+
+GZ=  (h-1 Mpc)
+200 -
+150 
+20
+100 
+15
+50 -
+GY (h-1 Mpc)
+0
+:::::
+：：
+:::::::::
+10
+-50 -
+.
+-100
+5
+-150 -
+-200 
+-200
+-150
+-100
+-50
+0
+50 
+100
+150
+200
+GX (h-1 Mpc)GZ= 50 (h-1 Mpc)
+200 -
+150 
+35
+100 -
+ 50 -
+- 25 
+GY (h-1 Mpc)
+0
+20
+-50 -
+15 
+-100
+10
+-150 -
+-200 
+-200
+-150
+-100
+-50 
+0
+ 50
+100
+150
+200
+GX (h-1 Mpc)Divergence of the local large-scale structure velocity field and its implications for Tilted Cosmology
+9
+Figure 7. Projections of the shear tensor in
+km/s
+Mpc/h units estimated with finite difference method over cartesian planes that pass on the
+origin.
+different scales (λ). The results, which assign negative values
+to ˜q on scales up to 200 Mpc through all three estimation
+methods, are summarized in Table 1. Note that we have set
+h ≃ 0.7 in all cases. Also, although (7) holds in essentially
+all tilted FRW models, here we have assumed an Einstein-de
+Sitter background (with q = 0.5) for mathematical simplicity.
+6.1 Uncertainties
+We have identified both controlled and uncontrolled uncer-
+tainties in our estimations. In the former group we have the fit
+uncertainties for the reconstruction parameters β and Vext.
+Of those two, we are mainly interested in β, given that Vext
+does not enter the gradient calculation. Then, if we define the
+divergence of the relative velocity field vrec as ˜θrec, we have:
+˜θ = β˜θrec ,
+(31)
+while the uncertainty in ˜θ due to the β parameter can be
+written as:
+∆˜θβ = ˜θrec∆β .
+(32)
+This is the uncertainty recorded in Table 1.
+Turning to the uncontrolled uncertainties, we can group
+different possible biases and systematic effects coming from
+the reconstruction process, as well as errors between approx-
+imations and real values. A detailed summary of the first
+MNRAS 000, 1–?? (2023)
+
+GX= 0 (h-1 Mpc)
+200 
+150 -
+70
+100 
+- 60 
+50 -
+(h-1 Mpc)
+ 50
+0
+:::::
+ 40
+-50 -
+30
+-100
+20 
+-150
+10
+-200 
+-200
+-150
+-100
+-50
+0
+50 
+100
+150
+200
+GY (h-1 Mpc)GY= 0 (h-1 Mpc)
+200 -
+100
+150 -
+80
+100 
+50 -
+GX (h-1 Mpc)
+ 60
+0
+:::::
+：：
+-50 -
+ 40
+-100
+-20 
+-150 -
+-200 
+-200
+-150
+-100
+-50 
+0
+50 
+100
+150
+200
+GZ (h-1 Mpc)GZ= 0 (h-1 Mpc)
+200 
+150
+70
+100 
+- 60 
+50 -
+GY (h-1 Mpc)
+ 50
+0
+:::::
+：
+ 40
+-50 -
+30
+-100
+20 
+-150
+10
+-200 
+-200
+-150
+-100
+-50
+0
+50 
+100
+150
+200
+GX (h-1 Mpc)10
+E. Past´en et al.
+type can be found in (Carrick et al. 2015). With regard to
+the approximation errors, we can estimate the precision of
+the estimation by comparing to the theoretical result. In this
+respect, the finite difference method seems more precise than
+the volume integration method, as it is closer to the theoret-
+ically predicted values. Moreover, according to relation (14),
+the velocity field should be irrotational as the field is propor-
+tional to a Newtonian gravity potential in the linear regime.
+However, when the anti-symmetric part of the gradient tensor
+is computed we got a non-zero value, leading to a residual low
+vorticity term that could be related with a deviation of the fi-
+nite difference method with respect to theoretical estimation.
+(potential velocity) A symmetric trace-less part of the gradi-
+ent can also be computed via finite difference method. Resid-
+ual Curl and projections of the estimated Shear are plotted
+in Figures 6 and 7.
+7 DISCUSSION
+We have estimated the average volume scalar of the recon-
+structed peculiar velocity of the local universe via different
+methods. The volume scalar is related to the divergence of the
+velocity field. This is so because the velocity divergence mea-
+sures the change in the local volume of the associated bulk
+flow and therefore its tendency to locally expand or contract.
+Then, a positive divergence implies that the fluid tends to
+expand locally, whereas a negative one indicates a contract-
+ing region. We have plotted the divergence scalar for different
+galactic planes in Figure 5. There, one can see that the pecu-
+liar velocity divergence is highly negative in regions where the
+density contrast is high, while it is positive in regions where
+matter content is low. This is to be expected, of course, given
+the attractive nature of gravity. At this point, it also helps
+to recall the familiar divergence theorem:
+∰
+V (∇ ⋅ v)dV = ∯
+∂V v ⋅ dA .
+(33)
+Integrating the divergence over the region V reveals whether
+the latter contracts or expands, as the right-hand side of the
+equation represents the fluid fraction that ”enters” or ”goes
+out” of the volume surface ∂V over time.
+Surprisingly, the values of the local volume scalar (˜θ) as-
+sociated with the reconstructed peculiar velocity field, were
+found negative over a range of scales and by means of different
+estimation methods. This result has direct implications for
+the tilted cosmological scenario (Tsagas 2011; Asvesta et al.
+2022),. The latter predicts that observers living in contracting
+bulk peculiar flows could measure a negative deceleration pa-
+rameter locally, even when the universe is decelerating glob-
+ally (Tsagas & Kadiltzoglou 2015; Tsagas 2021, 2022). Also,
+as predicted, we found that the impact of the observer’s pecu-
+liar motion becomes stronger on progressively smaller scales,
+namely closer to the observer, while it decays away from them
+(see Table 1). The transition length (λT ), that is the max-
+imum scale where the local deceleration parameter appears
+to cross the ˜q = 0 mark and turn negative, also depends on
+the observer’s position inside the bulk flow. Following (8),
+for observes residing within 70/h Mpc from the centre of
+the bulk flow, we find λT ≳ 360 Mpc, λT ≳ 310 Mpc and
+λT ≳ 390 Mpc, when adopting the Finite Difference method,
+the Integral Approximation method and the Discrete Density
+Integration method respectively. Overall, the closer the ob-
+server is to the bulk-flow centre, the more negative the local
+value of ˜ϑ and the larger the associated transition length.
+As appealing these results may be, it is important to re-
+main vigilant. It is possible, for example, that the values of
+the average divergence could change, as more refined surveys
+and models are developed. It is also still unknown whether
+matter residing outside the survey range could impact the
+mean divergence of the peculiar velocity field. Recall that in
+the reconstruction used here this contribution was approxi-
+mated by a constant velocity term. In addition, there have
+been recent claims that we live in a large void extending up
+to ∼ 300 Mpc. However, a negative expansion scalar is not
+compatible with the idea of a large void, where one expects to
+find an expanding bulk flow rather than a contracting one.
+In this respect, our analysis does not seem to support the
+presence of a large underdensity.
+Finally, peculiar velocities seem unlikely to change the lo-
+cal value of the Hubble parameter appreciably and therefore
+to solve the H0 tension. One can immediately realise this by
+looking at the linear relation (4a). Indeed, keeping in mind
+that ∣˜θ∣/Θ = ∣˜θ∣/3H ≪ 1 on sufficiently large scales, the im-
+pact of the observer’s relative motion on the Hubble param-
+eter should be minimal.3 Instead, there might be other ex-
+planations, such as systematics, the evolution of cosmological
+parameters with redshift, etc (e.g. see Krishnan et al. (2020);
+Colgain et al. (2022)).
+8 CONCLUSIONS
+We have computed the average divergence (˜θ) of the peculiar
+velocity field reconstructed from the 2M++ survey, which was
+used to correct cosmological redshifts in the last SNIA com-
+pilation Pantheon+. In so doing, we employed three differ-
+ent approximation methods, coming from standard numerical
+analysis, the divergence theorem and from a linear theoretical
+derivation of the peculiar velocity formulae. In all cases, the
+resulting values of the velocity divergence were found neg-
+ative over a range of scales, suggesting that we live inside
+a contracting bulk flow. According to the tilted cosmologi-
+cal scenario, the deceleration parameter measured locally by
+observers residing in contracting bulk flows can be negative,
+although the surrounding universe is globally decelerating.
+Our numerical results support this scenario, thus allowing
+for the recent accelerated expansion to be just an illusion
+produced by our peculiar motion relative to the CMB rest
+frame. Nevertheless, this possibility should be treated with
+care, as the computed values are still representative of the
+measurements a typical bulk-flow observer will make. There-
+fore, better surveys with refined precision and broader range
+are needed to improve the values computed here. In any case,
+however, our results support the need for a deeper study and
+for the proper understanding of the implications the observed
+large-scale peculiar motions may have for our interpretation
+of the cosmological parameters,
+3 Recall that, although ∣˜θ∣/H ≪ 1 always during the linear regime,
+this is not necessarily the case for the ratio ∣˜θ′∣/ ˙H.
+MNRAS 000, 1–?? (2023)
+
+Divergence of the local large-scale structure velocity field and its implications for Tilted Cosmology
+11
+λ( Mpc
+h
+)
+˜θ( km/s
+Mpc/h )
+˜q
+Finite Difference
+70
+−3.36+0.07
+−0.07 (−2.65+0.08
+−0.12)
+-6.36 (-4.90)
+100
+−2.77+0.06
+−0.06 (−2.19+0.10
+−0.07)
+-2.27 (-1.70)
+125
+−1.48+0.03
+−0.03 (−1.17+0.06
+−0.04)
+-0.45 (-0.25)
+150
+−0.65+0.01
+−0.01 (−0.51+0.02
+−0.02)
++0.21 (+0.27)
+200
+−0.24+0.005
+−0.005 (−0.19+0.008
+−0.005)
++0.44 (+0.45)
+Integral Approximation
+70
+−2.45+0.05
+−0.05 (−1.94+0.09
+−0.06)
+-4.5 (-3.45)
+100
+−1.99+0.04
+−0.04 (−1.57+0.07
+−0.05)
+-1.49 (-1.07)
+125
+−1.13+0.02
+−0.02 (−0.90+0.04
+−0.03)
+-0.22 (-0.07)
+150
+−0.47+0.01
+−0.01 (−0.37+0.007
+−0.01 )
++0.29 (+0.33)
+200
+−0.21+0.004
+−0.004 (−0.17+0.008
+−0.005)
++0.45 (+0.46)
+Discrete Density Integration
+70
+−3.94+0.08
+−0.08 (−3.11+0.15
+−0.10)
+-7.54 (-5.86)
+100
+−3.17+0.07
+−0.07 (−2.50+0.12
+−0.08)
+-2.67 (-2.00)
+125
+−1.66+0.03
+−0.03 (−1.31+0.06
+−0.04)
+-0.56 (-0.34)
+150
+−0.76+0.02
+−0.02 (−0.59+0.03
+−0.02)
++0.16 (+0.23)
+200
+−0.29+0.006
+−0.006 (−0.23 +0.01
+−0.007)
++0.42 (+0.44)
+Table 1. Representative values for ˜q on different scales (λ), using β as it is in the datacube with the finite difference approximation,
+integral approximation and theoretical estimation. In parenthesis are the values of ˜q obtained after using β from Pantheon+. Note that,
+for numerical simplicity and demonstration purposes, we have set q = 0.5 in the CMB frame and h ≃ 0.7. Note that according to equation
+7, the error propagation for ˜q estimations are negligible
+ACKNOWLEDGMENTS
+EP
+acknowledges
+support
+from
+the
+graduate
+scholar-
+ship ANID-Subdirecci´on de Capital Humano/Doctorado
+Nacional/2021-21210824. We also wish to thank Christos
+Tsagas for his comments, which helped us understand fur-
+ther the tilted cosmological scenario.
+DATA AVAILABILITY
+The data underlying this article, including the programs and
+the results of gradient estimations, will be shared on reason-
+able request to the corresponding author.
+REFERENCES
+Asvesta K., Kazantzidis L., Perivolaropoulos L., Tsagas C. G.,
+2022, Mon. Not. R. Astron. Soc., 513, 2394
+Carr A., Davis T. M., Scolnic D., Said K., Brout D., Peterson E. R.,
+Kessler R., 2022, Publications of the Astronomical Society of
+Australia, 39
+Carrick J., Turnbull S. J., Lavaux G., Hudson M. J., 2015, Monthly
+Notices of the Royal Astronomical Society, 450, 317
+Celerier M.-N., 2006, arXiv
+Colgain E., Sheikh-Jabbari M. M., Solomon R., Dainotti M. G.,
+Stojkovic D., 2022, doi:10.48550/ARXIV.2206.11447, https:
+//arxiv.org/abs/2206.11447
+Colin
+J.,
+Mohayaee
+R.,
+Rameez
+M.,
+Sarkar
+S.,
+2019a,
+A
+response
+to
+Rubin
+&;
+Heitlauf:
+”Is
+the
+expansion
+of
+the
+universe
+accelerating?
+All
+signs
+still
+point
+to
+yes”, doi:10.48550/ARXIV.1912.04257, https://arxiv.org/
+abs/1912.04257
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+& Astrophysics, 631, L13
+Cosmai L., Fanizza G., Labini F. S., Pietronero L., Tedesco L.,
+2019, Classical and Quantum Gravity, 36, 045007
+Ellis G. F. R., 1971, in R. K. Sachs ed., General Relativity and
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+Feindt U. e. a., 2013, A&A, 560, A90
+Hudson M. J., Smith R. J., Lucey J. R., Schlegel D. J., Davies
+R. L., 1999, The Astrophysical Journal, 512, L79
+Keenan R. C., Barger A. J., Cowie L. L., 2013, The Astrophysical
+Journal, 775, 62
+Krishnan C., Colg´ain E., Ruchika Sen A., Sheikh-Jabbari M., Yang
+T., 2020, Physical Review D, 102
+Labini F. S., 2011, Classical and Quantum Gravity, 28, 164003
+Labini F., Montuori M., Pietronero L., 1998, Physics Reports, 293,
+61
+Maartens R., 1998, Phys. Rev. D, 58, 124006
+Magoulas C., Springob C., Colless M., Mould J., Lucey J., Erdo˘gdu
+P., Jones D. H., 2014, Proceedings of the International Astro-
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+Rubin D., Heitlauf J., 2020, The Astrophysical Journal, 894, 68
+Said K., Colless M., Magoulas C., Lucey J. R., Hudson M. J., 2020,
+Monthly Notices of the Royal Astronomical Society, 497, 1275
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+MNRAS 000, 1–?? (2023)
+

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+DATASET OF FLUORESCENCE SPECTRA AND CHEMICAL
+PARAMETERS OF OLIVE OILS
+AN OPEN SOURCE DATASET
+Francesca Venturini ∗
+Institute of Applied Mathematics and Physics
+Zurich University of Applied Sciences
+Winterthur, Switzerland, [email protected]
+Artificial Intelligence Research and Development
+TOELT LLC, Switzerland
+Michela Sperti
+PolitoBIOMed Lab
+Department of Mechanical and
+Aerospace Engineering
+Politecnico di Torino, Turin, Italy
+Umberto Michelucci
+Artificial Intelligence Research and Development
+TOELT LLC, Switzerland
+[email protected]
+Computer Science Department
+Lucerne University of Applied Sciences and Arts
+Lucerne, Switzerland
+Arnaud Gucciardi
+Artificial Intelligence Research and Development
+TOELT LLC, Switzerland
+[email protected]
+Artificial Intelligence Laboratory
+University of Ljubljana, Ljubljana, Slovenia
+Vanessa M. Martos
+Department of Plant Physiology
+Faculty of Sciences
+Biotechnology Institute
+University of Granada, Spain
+Marco A. Deriu
+PolitoBIOMed Lab
+Department of Mechanical and
+Aerospace Engineering
+Politecnico di Torino, Turin, Italy
+January 12, 2023
+ABSTRACT
+This dataset encompasses fluorescence spectra and chemical parameters of 24 olive oil samples from
+the 2019–2020 harvest provided by the producer Conde de Benalúa, Granada, Spain. The oils are
+characterized by different qualities: 10 extra virgin olive oil (EVOO), 8 virgin olive oil (VOO), and
+6 lampante olive oil (LOO) samples. For each sample, the dataset includes fluorescence spectra
+obtained with two excitation wavelengths, oil quality, and five chemical parameters necessary for
+the quality assessment of olive oil. The fluorescence spectra were obtained by exciting the samples
+at 365 nm and 395 nm under identical conditions. The dataset includes the values of the following
+chemical parameters for each olive oil sample: acidity, peroxide value, K270, K232, ethyl esters,
+and the quality of the samples (EVOO, VOO, or LOO). The dataset offers a unique possibility for
+researchers in food technology to develop machine learning models based on fluorescence data for
+the quality assessment of olive oil due to the availability of both spectroscopic and chemical data.
+The dataset can be used, for example, to predict one or multiple chemical parameters or to classify
+samples based on their quality from fluorescence spectra.
+Keywords Fluorescence · Olive Oil · Chemical Parameters · Quality control
+∗Contact email: [email protected]
+arXiv:2301.04471v1  [q-bio.QM]  10 Jan 2023
+
+Dataset of Fluorescence Spectra and Chemical Parameters of Olive Oils
+DATASET
+1
+Summary
+The dataset presented is a compilation of measurements of analytical chemistry and fluorescence spectroscopy. The
+dataset includes fluorescence spectra and chemical parameters of 24 Spanish olive oils from the 2019–2020 harvest. The
+24 samples were collected at SCA San Sebastián Puente del Ventorro, Benalua de las Villas, Spain. The data were later
+measured at the Institute of Applied Mathematics and Physics, Zurich University of Applied Sciences, Technikumstrasse
+9, 8401 Winterthur, Switzerland. The fluorescence spectroscopy data was acquired by a miniature spectrometer with a
+1024 element CCD array that acquires the entire spectrum in one single measurement. The dataset includes a total of
+960 spectra (24 oil samples × 2 excitation wavelengths x 20 repeated measurements). Each of the 960 spectra is an
+array of 1024 values whose elements are the intensity at the different pixel positions. The chemical parameters were
+determined by accredited laboratories using the procedures described in the European Commission regulation and its
+amendment Commission [2013, 1991]. These regulations control the methods for the quality assessment of olive oils
+and provide a decision tree to verify whether an olive oil class is consistent with the declared quality.
+The value of the dataset for research purposes is summarized in the points below.
+• The data are useful for studying the link between optical properties (fluorescence and absorption spectroscopy),
+chemical characteristics (such as oil acidity, peroxide value, and fatty acid content), and olive oil quality (extra
+virgin, virgin, and lampante olive oil).
+• This dataset is the first available that contains fluorescence spectra and chemical analysis obtained by accredited
+laboratories on samples coming from a single producer.
+• Many researchers can benefit from the data: computer scientists can use the data to develop machine learning
+models that link optical to chemical properties; researchers in food technology that are interested in studying
+chemical properties of olive oil samples of different qualities; engineers that want to develop new optical
+analysis techniques alternative to the current expensive and time-consuming analytical chemistry methods.
+• This dataset can be used to perform explainability analysis to identify spectral characteristics that are related
+to different chemical properties (e.g., the acidity of the oil). An example is given in the paper Venturini et al.
+[2023]. This will further advance the understanding of the complex chemical composition of olive oil and its
+link to its quality and health benefits.
+• This dataset can be used to develop instruments based on fluorescence spectroscopy for the rapid and cost-
+effective quality assessment of olive oil.
+2
+Data Description
+The dataset consists of one CSV file that contains the columns described in Table 1.
+A background file2 is also provided. The file contains 1024 values that correspond to the intensity measured by the
+spectrometer without any light (dark counts). This spectrum can be subtracted from the raw fluorescence spectra to
+remove the effect of the dark counts. The same file can be used for the spectra taken at both 365 nm and 395 nm.
+The raw fluorescence spectra of selected oils obtained with excitation at 365 nm and 395 nm are shown in Fig. 1.
+3
+Materials and methods
+3.1
+Olive Oil Samples
+The dataset contains the fluorescence spectra and the chemical parameters of 24 oils. The oils are characterized by
+different quality categories: 10 extra virgin olive oil (EVOO), 8 virgin olive oil (VOO), and 6 lampante olive oil (LOO)
+samples. All samples were provided by Conde de Benalúa, Granada, southern Spain, and were prepared from the
+2019–2020 harvest. The properties and values of the chemical parameters of the oil samples are listed in Table 2.
+For data acquisition, the samples were placed in commercial 4 ml clear glass vials, taking care that no headspace was
+present to reduce oxidation. All oils were stored in the dark and at 20 °C during the entire time of the measurements.
+3.2
+Fluorescence Data Acquisition
+The fluorescence spectroscopy data were acquired using the portable sensor described in Venturini et al. [2021]. Since
+already published, only the most relevant characteristics are reported here. The reader is referred to this publication
+2Fluorescence_olive_oil_dataset_background.csv
+2
+
+Dataset of Fluorescence Spectra and Chemical Parameters of Olive Oils
+DATASET
+Feature
+Datatype
+Description
+Sample
+String
+Oil sample name: the values are ’D03’,’D04’,’D05’, ’D06’, ’D07’ ,’D08’, ’D09’,
+’D10’, ’D 19’, ’D20’, ’D35’, ’D38’, ’D45’, ’D46’, ’D47’, ’D49’, ’D51’, ’D52’,
+’D53’, ’D64’, ’D77’, ’D81’, ’D92’,’D73’
+Repetition
+Integer
+Repetition number. There are 20 repetition for each oil and led: the iteration
+number goes from 0 to 19)
+Led
+Integer
+Excitation LED identifier: 1 (395 nm), 2 (365 nm)
+Data
+Float
+The fluorescence spectra. The feature is a string composed of 1024 values given
+between square brackets and seprated by a comma, as for example [1491.0,
+1508.0, ..., 1545.0]. Each value is the raw intensity of the fluorescence signal at
+the given pixel of the detector of the spectrometer.
+Quality
+String
+Quality of the oil. Possible values are ‘EXTRA’, ‘VIRGIN’, ‘LAMPANTE’
+FAEES
+Float
+Fatty acid ethyl esters in mg/Kg: content of waxes, fatty acid methyl esters and
+fatty acid ethyl esters
+K232
+Float
+UV Absorbance at 232 nm (K270)
+K270
+Float
+UV Absorbance at 270 nm (K232)
+Acidity
+Float
+Acidity: expressed as percentage (%) of oleic acid
+Peroxide Index
+Float
+Quantity of those substances in the sample, expressed in terms of milliequivalents
+of active oxygen per kilogram (mEqO2/Kg), which oxidize potassium iodide.
+Table 1: Information on each feature available in the dataset.
+for more details. The schematic design of the spectrometer is sown in Fig. 2. The excitation light was provided by
+two UV LEDs with emission at 365 nm and 395 nm driven by a current driver (MIC4801, Micrel Inc., San Jose, CA,
+USA) to adjust the excitation intensity. The fluorescence signal was collected by a miniature spectrometer (STS-Vis,
+Ocean Optics, Dunedin, FL, USA) with a 1024-element CCD array which acquires the entire spectrum in one single
+measurement with a resolution of 16 nm. The spectrometer was placed at 90° with respect to the LEDs to avoid the
+excitation light transmitted by the sample to reach the spectrometer. The sensor has a recess where standard 4 ml clear
+glass vials with the sample can be inserted.
+All spectra of the dataset were acquired on undiluted samples at room temperature under identical conditions (illumina-
+tion intensity, integration time, and geometry) for a quantitative comparison. The integration time was 1 s. During the
+measurements, the setup was kept in complete darkness to minimize the effect of stray light.
+Each spectrum consists of an array of 1024 values (one for each pixel). The value corresponds to the intensity in counts
+at the different positions of the pixels. To obtain the wavelength (in nanometers) corresponding to each pixel, the
+following formula can be used:
+i = a + b · i + c · i2 + d · i3
+(1)
+where i indicates the pixel (i = 0, ..., 1023) and
+a = 337.92288208 nm
+b = 0.4470772743 nm
+c = 3.55128 · 10−5 nm
+d = −8.38601 · 10−9 nm
+(2)
+Calibration parameters were provided by the spectrometer manufacturer. All spectra correspond to the raw data without
+any data processing (smoothing, background subtraction, or normalization). Since all the measurements were done
+under identical conditions the intensities are directly comparable.
+3.3
+Chemical Analysis
+For each olive oil sample, the dataset includes the values of the following chemical parameters: acidity, peroxide value,
+K270, K232, ethyl esters concentration and the samples quality class (EVOO, VOO, or LOO) (see Tab. 2).
+The chemical parameters were determined by accredited laboratories using the procedures described in the European
+Commission regulation and its amendment (Commission [2013, 1991]).
+3
+
+Dataset of Fluorescence Spectra and Chemical Parameters of Olive Oils
+DATASET
+Excitation 365 nm
+EVOO
+Excitation 395 nm
+VOO
+LOO
+0
+4'000
+8'000
+Intensity (a.u.)
+0
+4'000
+8'000
+Intensity (a.u.)
+12'000
+0
+4'000
+8'000
+Intensity (a.u.)
+678 nm
+722 nm
+678 nm
+722 nm
+500
+550
+600
+650
+700
+750
+500
+Wavelength (nm)
+550
+600
+650
+700
+750
+800
+Wavelength (nm)
+Figure 1: Fluorescence emission spectra of selected olive oils divided in the quality classes EVOO, VOO and LOO. On
+the left: spectra obtained with excitation at 365 nm; on the right: spectra obtained with excitation at 395 nm. Each
+curve shows a single spectrum without averaging or smoothing after the background subtraction. Reproduced from
+Venturini et al. [2023].
+4
+Funding
+This research was supported by the projects: “VIRTUOUS” funded by the European Union’s Horizon 2020 Project
+H2020-MSCA-RISE-2019 Grant No. 872181; “SUSTAINABLE” funded by the European Union’s Horizon 2020
+Project H2020-MSCA-RISE-2020 Grant No. 101007702; “Project of Excellence” from Junta de Andalucia-FEDER-
+Fondo de Desarrollo Europeo 2018. Ref. P18–H0-4700.
+5
+Author Contributions
+Conceptualization: Francesca Venturini and Umberto Michelucci; methodology: Francesca Venturini and Umberto
+Michelucci; software, Michela Sperti and Arnaud Gucciardi; validation, Francesca Venturini and Umberto Michelucci;
+formal analysis, Francesca Venturini and Umberto Michelucci; investigation, Francesca Venturini and Umberto
+Michelucci; resources, Vanessa M. Martos; data curation, Michela Sperti and Arnaud Gucciardi; writing, original draft
+preparation, Francesca Venturini and Umberto Michelucci; writing, review and editing, Francesca Venturini, Umberto
+Michelucci, Arnaud Gucciardi and Marco A. Deriu; funding acquisition, Vanessa M. Martos and Marco A. Deriu. All
+authors have read and agreed to the published version of the manuscript.
+6
+Data Availability
+The data presented in this study are openly available in Dataset of Fluorescence Spectra and Chemical Parameters of
+Olive Oils at https://data.mendeley.com/datasets/thkcz3h6n6/6, DOI: 10.17632/thkcz3h6n6.6.
+4
+
+Dataset of Fluorescence Spectra and Chemical Parameters of Olive Oils
+DATASET
+Label
+Acidity
+Peroxide value
+K270
+K232
+FAEES
+Quality
+(%)
+(mEq O2/kg)
+(mg/Kg)
+D03
+0.35
+8.4
+0.123
+1.435
+26
+VOO
+D04
+0.34
+8.6
+0.108
+1.403
+40
+VOO
+D05
+0.36
+10.3
+0.112
+1.44
+18
+VOO
+D06
+0.31
+9.2
+0.151
+1.484
+18
+VOO
+D07
+0.50
+8.9
+0.150
+1.537
+47
+VOO
+D08
+0.40
+8.5
+0.158
+1.546
+25
+VOO
+D09
+-
+-
+-
+-
+-
+LOO
+D10
+-
+-
+-
+-
+-
+LOO
+D19
+0.25
+4.9
+0.13
+1.540
+10
+EVOO
+D20
+0.26
+4.6
+0.14
+1.540
+10
+EVOO
+D35
+0.17
+6.4
+0.12
+1.63
+8
+EVOO
+D38
+0.16
+6.4
+0.12
+1.63
+9
+EVOO
+D45
+0.17
+4.9
+0.12
+1.63
+7
+EVOO
+D46
+0.18
+5.0
+0.13
+1.63
+8
+EVOO
+D47
+0.18
+5.2
+0.13
+1.64
+16
+EVOO
+D49
+0.9
+9.9
+-
+-
+-
+LOO
+D51
+2.16
+-
+-
+-
+-
+LOO
+D52
+1.78
+22
+-
+-
+-
+LOO
+D53
+0.7
+8.7
+-
+-
+-
+LOO
+D64
+0.2
+7.1
+0.13
+1.63
+29
+VOO
+D73
+0.2
+8.9
+0.14
+1.66
+15
+EVOO
+D77
+0.24
+10.4
+0.13
+1.74
+26
+VOO
+D81
+0.16
+4.9
+0.12
+1.63
+9
+EVOO
+D92
+0.18
+5
+0.17
+1.91
+15
+EVOO
+Table 2: List of olive oil samples and their physicochemical characteristics. FAEES: fatty acid ethyl esters, EVOO:
+extra virgin olive oil, VOO: virgin olive oil, LOO: lampante olive oil.
+7
+Ackowledgments
+The authors would like to thank Michael Baumgartner and Ivo Herzig (Institute of Applied Mathematics and Physics,
+Zurich University of Applied Sciences, Winterthur, Switzerland) for help for the realization of the sensor, and Josep
+Palau Caballero and Arturo Jimenez (SCA San Sebastián Puente del Ventorro, s/n, 18566 Benalua de las Villas, Spain)
+for providing the oil samples.
+8
+Conflicts of Interest
+The authors declare no conflicts of interest and no known competing financial interests or personal relationships that
+could have appeared to influence the work reported in this paper.
+9
+Abbreviations
+The following abbreviations are used in this manuscript:
+LOO
+Lampante Olive Oil
+EVOO
+Extra Vigrin Olive Oil
+VOO
+Virgin Olive Oil
+CCD
+Charge-Coupled Device
+LED
+Light Emitting Diode
+UV
+Ultraviolet
+FAEES
+Fatty Acid Ethyl Ester
+5
+
+Dataset of Fluorescence Spectra and Chemical Parameters of Olive Oils
+DATASET
+LED
+Driver
+Spectrometer
+Excitation
+LED
+Sample
+Fluorescence
+Raspberry Pi
+Figure 2: Schematics of the portable fluorescence sensor. Blue: excitation light, red: fluorescence light. From Venturini
+et al. [2021].
+References
+European Commission. Commission implementing regulation no 1348/2013 of december 17 2013. Official Journal of
+the European Union, 338:31–67, 2013.
+European Commission. Commission regulation (eec) no. 2568/91 of 11 july 1991 on the characteristics of olive oil and
+olive-residue oil and on the relevant methods of analysis official journal l 248, 5 september 1991. Offic. JL, 248:1–83,
+1991.
+Francesca Venturini, Michela Sperti, Umberto Michelucci, Arnaud Gucciardi, Vanessa M Martos, and Marco A Deriu.
+Extraction of physicochemical properties from the fluorescence spectrum with 1d convolutional neural networks:
+Application to olive oil. Journal of Food Engineering, 336:111198, 2023.
+Francesca Venturini, Michela Sperti, Umberto Michelucci, Ivo Herzig, Michael Baumgartner, Josep Palau Caballero,
+Arturo Jimenez, and Marco Agostino Deriu. Exploration of spanish olive oil quality with a miniaturized low-cost
+fluorescence sensor and machine learning techniques. Foods, 10(5):1010, 2021.
+6
+

HtE3T4oBgHgl3EQfWwou/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf,len=241
+page_content='DATASET OF FLUORESCENCE SPECTRA AND CHEMICAL  PARAMETERS OF OLIVE OILS  AN OPEN SOURCE DATASET  Francesca Venturini ∗  Institute of Applied Mathematics and Physics  Zurich University of Applied Sciences  Winterthur, Switzerland, vent@zhaw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='ch  Artificial Intelligence Research and Development  TOELT LLC, Switzerland  Michela Sperti  PolitoBIOMed Lab  Department of Mechanical and  Aerospace Engineering  Politecnico di Torino, Turin, Italy  Umberto Michelucci  Artificial Intelligence Research and Development  TOELT LLC, Switzerland  umberto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='michelucci@toelt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='ai  Computer Science Department  Lucerne University of Applied Sciences and Arts  Lucerne, Switzerland  Arnaud Gucciardi  Artificial Intelligence Research and Development  TOELT LLC, Switzerland  arnaud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='gucciardi@toelt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='ai  Artificial Intelligence Laboratory  University of Ljubljana, Ljubljana, Slovenia  Vanessa M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' Martos  Department of Plant Physiology  Faculty of Sciences  Biotechnology Institute  University of Granada, Spain  Marco A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' Deriu  PolitoBIOMed Lab  Department of Mechanical and  Aerospace Engineering  Politecnico di Torino, Turin, Italy  January 12, 2023  ABSTRACT  This dataset encompasses fluorescence spectra and chemical parameters of 24 olive oil samples from  the 2019–2020 harvest provided by the producer Conde de Benalúa, Granada, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' The oils are  characterized by different qualities: 10 extra virgin olive oil (EVOO), 8 virgin olive oil (VOO), and  6 lampante olive oil (LOO) samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' For each sample, the dataset includes fluorescence spectra  obtained with two excitation wavelengths, oil quality, and five chemical parameters necessary for  the quality assessment of olive oil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' The fluorescence spectra were obtained by exciting the samples  at 365 nm and 395 nm under identical conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' The dataset includes the values of the following  chemical parameters for each olive oil sample: acidity, peroxide value, K270, K232, ethyl esters,  and the quality of the samples (EVOO, VOO, or LOO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' The dataset offers a unique possibility for  researchers in food technology to develop machine learning models based on fluorescence data for  the quality assessment of olive oil due to the availability of both spectroscopic and chemical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='  The dataset can be used, for example, to predict one or multiple chemical parameters or to classify  samples based on their quality from fluorescence spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='  Keywords Fluorescence · Olive Oil · Chemical Parameters · Quality control  ∗Contact email: vent@zhaw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='ch  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='04471v1  [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='QM]  10 Jan 2023  Dataset of Fluorescence Spectra and Chemical Parameters of Olive Oils  DATASET  1  Summary  The dataset presented is a compilation of measurements of analytical chemistry and fluorescence spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' The  dataset includes fluorescence spectra and chemical parameters of 24 Spanish olive oils from the 2019–2020 harvest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' The  24 samples were collected at SCA San Sebastián Puente del Ventorro, Benalua de las Villas, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' The data were later  measured at the Institute of Applied Mathematics and Physics, Zurich University of Applied Sciences, Technikumstrasse  9, 8401 Winterthur, Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' The fluorescence spectroscopy data was acquired by a miniature spectrometer with a  1024 element CCD array that acquires the entire spectrum in one single measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' The dataset includes a total of  960 spectra (24 oil samples × 2 excitation wavelengths x 20 repeated measurements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' Each of the 960 spectra is an  array of 1024 values whose elements are the intensity at the different pixel positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' The chemical parameters were  determined by accredited laboratories using the procedures described in the European Commission regulation and its  amendment Commission [2013, 1991].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' These regulations control the methods for the quality assessment of olive oils  and provide a decision tree to verify whether an olive oil class is consistent with the declared quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='  The value of the dataset for research purposes is summarized in the points below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='  The data are useful for studying the link between optical properties (fluorescence and absorption spectroscopy),  chemical characteristics (such as oil acidity, peroxide value, and fatty acid content), and olive oil quality (extra  virgin, virgin, and lampante olive oil).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='  This dataset is the first available that contains fluorescence spectra and chemical analysis obtained by accredited  laboratories on samples coming from a single producer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='  Many researchers can benefit from the data: computer scientists can use the data to develop machine learning  models that link optical to chemical properties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' researchers in food technology that are interested in studying  chemical properties of olive oil samples of different qualities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' engineers that want to develop new optical  analysis techniques alternative to the current expensive and time-consuming analytical chemistry methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='  This dataset can be used to perform explainability analysis to identify spectral characteristics that are related  to different chemical properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=', the acidity of the oil).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' An example is given in the paper Venturini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='  [2023].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' This will further advance the understanding of the complex chemical composition of olive oil and its  link to its quality and health benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='  This dataset can be used to develop instruments based on fluorescence spectroscopy for the rapid and cost-  effective quality assessment of olive oil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='  2  Data Description  The dataset consists of one CSV file that contains the columns described in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='  A background file2 is also provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' The file contains 1024 values that correspond to the intensity measured by the  spectrometer without any light (dark counts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' This spectrum can be subtracted from the raw fluorescence spectra to  remove the effect of the dark counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' The same file can be used for the spectra taken at both 365 nm and 395 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='  The raw fluorescence spectra of selected oils obtained with excitation at 365 nm and 395 nm are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='  3  Materials and methods  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='1  Olive Oil Samples  The dataset contains the fluorescence spectra and the chemical parameters of 24 oils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' The oils are characterized by  different quality categories: 10 extra virgin olive oil (EVOO), 8 virgin olive oil (VOO), and 6 lampante olive oil (LOO)  samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' All samples were provided by Conde de Benalúa, Granada, southern Spain, and were prepared from the  2019–2020 harvest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' The properties and values of the chemical parameters of the oil samples are listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='  For data acquisition, the samples were placed in commercial 4 ml clear glass vials, taking care that no headspace was  present to reduce oxidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' All oils were stored in the dark and at 20 °C during the entire time of the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='2  Fluorescence Data Acquisition  The fluorescence spectroscopy data were acquired using the portable sensor described in Venturini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' [2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' Since  already published, only the most relevant characteristics are reported here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' The reader is referred to this publication  2Fluorescence_olive_oil_dataset_background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='csv  2  Dataset of Fluorescence Spectra and Chemical Parameters of Olive Oils  DATASET  Feature  Datatype  Description  Sample  String  Oil sample name: the values are ’D03’,’D04’,’D05’, ’D06’, ’D07’ ,’D08’, ’D09’,  ’D10’, ’D 19’, ’D20’, ’D35’, ’D38’, ’D45’, ’D46’, ’D47’, ’D49��, ’D51’, ’D52’,  ’D53’, ’D64’, ’D77’, ’D81’, ’D92’,’D73’  Repetition  Integer  Repetition number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' There are 20 repetition for each oil and led: the iteration  number goes from 0 to 19)  Led  Integer  Excitation LED identifier: 1 (395 nm), 2 (365 nm)  Data  Float  The fluorescence spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' The feature is a string composed of 1024 values given  between square brackets and seprated by a comma, as for example [1491.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='0,  1508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=', 1545.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' Each value is the raw intensity of the fluorescence signal at  the given pixel of the detector of the spectrometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='  Quality  String  Quality of the oil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' Possible values are ‘EXTRA’,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' ‘VIRGIN’,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' ‘LAMPANTE’  FAEES  Float  Fatty acid ethyl esters in mg/Kg: content of waxes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' fatty acid methyl esters and  fatty acid ethyl esters  K232  Float  UV Absorbance at 232 nm (K270)  K270  Float  UV Absorbance at 270 nm (K232)  Acidity  Float  Acidity: expressed as percentage (%) of oleic acid  Peroxide Index  Float  Quantity of those substances in the sample,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' expressed in terms of milliequivalents  of active oxygen per kilogram (mEqO2/Kg),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' which oxidize potassium iodide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='  Table 1: Information on each feature available in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='  for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' The schematic design of the spectrometer is sown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' The excitation light was provided by  two UV LEDs with emission at 365 nm and 395 nm driven by a current driver (MIC4801, Micrel Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=', San Jose, CA,  USA) to adjust the excitation intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' The fluorescence signal was collected by a miniature spectrometer (STS-Vis,  Ocean Optics, Dunedin, FL, USA) with a 1024-element CCD array which acquires the entire spectrum in one single  measurement with a resolution of 16 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' The spectrometer was placed at 90° with respect to the LEDs to avoid the  excitation light transmitted by the sample to reach the spectrometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' The sensor has a recess where standard 4 ml clear  glass vials with the sample can be inserted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='  All spectra of the dataset were acquired on undiluted samples at room temperature under identical conditions (illumina-  tion intensity, integration time, and geometry) for a quantitative comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' The integration time was 1 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' During the  measurements, the setup was kept in complete darkness to minimize the effect of stray light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='  Each spectrum consists of an array of 1024 values (one for each pixel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' The value corresponds to the intensity in counts  at the different positions of the pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' To obtain the wavelength (in nanometers) corresponding to each pixel, the  following formula can be used:  i = a + b · i + c · i2 + d · i3  (1)  where i indicates the pixel (i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=', 1023) and  a = 337.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='92288208 nm  b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='4470772743 nm  c = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='55128 · 10−5 nm  d = −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='38601 · 10−9 nm  (2)  Calibration parameters were provided by the spectrometer manufacturer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' All spectra correspond to the raw data without  any data processing (smoothing, background subtraction, or normalization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' Since all the measurements were done  under identical conditions the intensities are directly comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='3  Chemical Analysis  For each olive oil sample, the dataset includes the values of the following chemical parameters: acidity, peroxide value,  K270, K232, ethyl esters concentration and the samples quality class (EVOO, VOO, or LOO) (see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='  The chemical parameters were determined by accredited laboratories using the procedures described in the European  Commission regulation and its amendment (Commission [2013, 1991]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content="  3  Dataset of Fluorescence Spectra and Chemical Parameters of Olive Oils  DATASET  Excitation 365 nm  EVOO  Excitation 395 nm  VOO  LOO  0  4'000  8'000  Intensity (a." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=")  0  4'000  8'000  Intensity (a." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=")  12'000  0  4'000  8'000  Intensity (a." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=')  678 nm  722 nm  678 nm  722 nm  500  550  600  650  700  750  500  Wavelength (nm)  550  600  650  700  750  800  Wavelength (nm)  Figure 1: Fluorescence emission spectra of selected olive oils divided in the quality classes EVOO, VOO and LOO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' On  the left: spectra obtained with excitation at 365 nm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' on the right: spectra obtained with excitation at 395 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' Each  curve shows a single spectrum without averaging or smoothing after the background subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' Reproduced from  Venturini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' [2023].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='  4  Funding  This research was supported by the projects: “VIRTUOUS” funded by the European Union’s Horizon 2020 Project  H2020-MSCA-RISE-2019 Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' 872181;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' “SUSTAINABLE” funded by the European Union’s Horizon 2020  Project H2020-MSCA-RISE-2020 Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' 101007702;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' “Project of Excellence” from Junta de Andalucia-FEDER-  Fondo de Desarrollo Europeo 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' P18–H0-4700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='  5  Author Contributions  Conceptualization: Francesca Venturini and Umberto Michelucci;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' methodology: Francesca Venturini and Umberto  Michelucci;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' software, Michela Sperti and Arnaud Gucciardi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' validation, Francesca Venturini and Umberto Michelucci;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='  formal analysis, Francesca Venturini and Umberto Michelucci;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' investigation, Francesca Venturini and Umberto  Michelucci;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' resources, Vanessa M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' Martos;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' data curation, Michela Sperti and Arnaud Gucciardi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' writing, original draft  preparation, Francesca Venturini and Umberto Michelucci;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' writing, review and editing, Francesca Venturini, Umberto  Michelucci, Arnaud Gucciardi and Marco A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' Deriu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' funding acquisition, Vanessa M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' Martos and Marco A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' Deriu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' All  authors have read and agreed to the published version of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='  6  Data Availability  The data presented in this study are openly available in Dataset of Fluorescence Spectra and Chemical Parameters of  Olive Oils at https://data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='mendeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='com/datasets/thkcz3h6n6/6, DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='17632/thkcz3h6n6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='  4  Dataset of Fluorescence Spectra and Chemical Parameters of Olive Oils  DATASET  Label  Acidity  Peroxide value  K270  K232  FAEES  Quality  (%)  (mEq O2/kg)  (mg/Kg)  D03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='35  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='123  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='435  26  VOO  D04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='34  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='108  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='403  40  VOO  D05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='36  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='112  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='44  18  VOO  D06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='31  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='151  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='484  18  VOO  D07  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='50  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='150  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='537  47  VOO  D08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='40  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='158  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='546  25  VOO  D09 LOO  D10 LOO  D19  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='25  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='13  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='540  10  EVOO  D20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='26  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='14  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='540  10  EVOO  D35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='17  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='12  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='63  8  EVOO  D38  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='16  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='12  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='63  9  EVOO  D45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='17  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='12  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='63  7  EVOO  D46  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='18  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='13  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='63  8  EVOO  D47  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='18  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='13  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='64  16  EVOO  D49  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='9  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='9 LOO  D51  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='16 LOO  D52  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='78  22 LOO  D53  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='7  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='7 LOO  D64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='2  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='13  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='63  29  VOO  D73  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='2  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='14  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='66  15  EVOO  D77  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='24  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='13  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='74  26  VOO  D81  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='16  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='12  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='63  9  EVOO  D92  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='18  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='17  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='91  15  EVOO  Table 2: List of olive oil samples and their physicochemical characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' FAEES: fatty acid ethyl esters, EVOO:  extra virgin olive oil, VOO: virgin olive oil, LOO: lampante olive oil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='  7  Ackowledgments  The authors would like to thank Michael Baumgartner and Ivo Herzig (Institute of Applied Mathematics and Physics,  Zurich University of Applied Sciences, Winterthur, Switzerland) for help for the realization of the sensor, and Josep  Palau Caballero and Arturo Jimenez (SCA San Sebastián Puente del Ventorro, s/n, 18566 Benalua de las Villas, Spain)  for providing the oil samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='  8  Conflicts of Interest  The authors declare no conflicts of interest and no known competing financial interests or personal relationships that  could have appeared to influence the work reported in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='  9  Abbreviations  The following abbreviations are used in this manuscript:  LOO  Lampante Olive Oil  EVOO  Extra Vigrin Olive Oil  VOO  Virgin Olive Oil  CCD  Charge-Coupled Device  LED  Light Emitting Diode  UV  Ultraviolet  FAEES  Fatty Acid Ethyl Ester  5  Dataset of Fluorescence Spectra and Chemical Parameters of Olive Oils  DATASET  LED  Driver  Spectrometer  Excitation  LED  Sample  Fluorescence  Raspberry Pi  Figure 2: Schematics of the portable fluorescence sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' Blue: excitation light, red: fluorescence light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' From Venturini  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' [2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='  References  European Commission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' Commission implementing regulation no 1348/2013 of december 17 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' Official Journal of  the European Union, 338:31–67, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='  European Commission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' Commission regulation (eec) no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' 2568/91 of 11 july 1991 on the characteristics of olive oil and  olive-residue oil and on the relevant methods of analysis official journal l 248, 5 september 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' Offic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' JL, 248:1–83,  1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='  Francesca Venturini, Michela Sperti, Umberto Michelucci, Arnaud Gucciardi, Vanessa M Martos, and Marco A Deriu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='  Extraction of physicochemical properties from the fluorescence spectrum with 1d convolutional neural networks:  Application to olive oil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' Journal of Food Engineering, 336:111198, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='  Francesca Venturini, Michela Sperti, Umberto Michelucci, Ivo Herzig, Michael Baumgartner, Josep Palau Caballero,  Arturo Jimenez, and Marco Agostino Deriu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' Exploration of spanish olive oil quality with a miniaturized low-cost  fluorescence sensor and machine learning techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content=' Foods, 10(5):1010, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}
+page_content='  6' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE3T4oBgHgl3EQfWwou/content/2301.04471v1.pdf'}

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+Prepared for submission to JINST
+The EXTRA-BL4S experiment for the measurement of the
+energy and angular distributions of transition radiation
+X-rays
+M. N. Mazziotta,𝑎,1 F. Loparco, 𝑎,𝑏,1 A. Anelli,𝑐 M. M. Belviso,𝑐 A. Buquicchio,𝑐
+E. V. Cassano,𝑐 M. De Cosmo,𝑐 P. Ginefra,𝑐 M. L. Martulli,𝑐 C. Picci,𝑐 D. Picicci,𝑐
+R. D. Soriano,𝑐 A. P. Tatulli,𝑐 G. Tripaldella,𝑐 V. M. Zupo,𝑐 M. F. Muscarella,𝑐 S. Turbacci,𝑐
+M. Boselli,𝑑 C. B. da Cruz E Silva,𝑑,2 M. Joos𝑑 and P. Schütze𝑒
+𝑎Istituto Nazionale di Fisica Nucleare, Sezione di Bari,
+via Orabona 4, I-70126 Bari, Italy
+𝑏Dipartimento di Fisica dell’Università e del Politecnico di Bari,
+via Amendola 173, I-70126 Bari, Italy
+𝑐The EXTRA Team Liceo Scientifico Statale "A. Scacchi",
+Corso Cavour 241, I-70121 Bari, Italy
+𝑑CERN, the European Organization for Nuclear Research,
+Esplanade des Particules 1, 1211 Geneva, Switzerland
+𝑒DESY, Notkestrasse 85, D-22607 Hamburg
+E-mail: [email protected], [email protected]
+Abstract: We have designed and implemented an experiment to measure the angular distributions
+and the energy spectra of the transition radiation X-rays emitted by fast electrons and positrons
+crossing different radiators. Our experiment was selected among the proposals of the 2021 Beamline
+for Schools contest, a competition for high-school students organized every year by CERN, and
+was performed at the DESY II Test Beam facility area TB21, using a high-purity beam of electrons
+or positrons with momenta in the range from 1 to 6 GeV/c. The measurements were performed
+using a 100 𝜇m thick silicon pixel detector, with a pitch of 55 𝜇m. Our results are consistent with
+the expectations from the theoretical models describing the production of transition radiation in
+multilayer regular radiators.
+Keywords: Transition radiation detectors; Particle identification methods
+1Corresponding authors.
+2Now at LIP - Laboratório de Instrumentação e Física Experimental de Partículas Avenida Prof. Gama Pinto 2,
+Complexo Interdisciplinar (3is), 1649-003 Lisboa, Portugal
+arXiv:2301.11247v1  [hep-ex]  26 Jan 2023
+
+Contents
+1
+Introduction
+1
+2
+The BL4S competition
+1
+3
+The EXTRA experiment
+3
+4
+Data analysis
+7
+4.1
+Conversion & Clustering
+7
+4.2
+Detector alignment procedure
+7
+4.3
+Data selection and analysis
+8
+5
+Results
+10
+6
+Conclusions
+15
+1
+Introduction
+In recent years, high-school physics curricula increasingly include topics related to modern high-
+energy physics and particle detectors. Universities and research centers promote several programs
+to bring high-school students in touch with modern physics and the scientific research. The Liceo
+Scientifico “A. Scacchi” in Bari has taken part in such projects for years, and in 2021 the school
+promoted the participation of a team of students of the 12𝑡ℎ and 13𝑡ℎ grade in the Beamline for
+Schools (BL4S) competition.
+BL4S is organized by CERN in collaboration with DESY, and offers to groups of high-school
+students the unique opportunity to propose a scientific experiment at a particle accelerator facility
+and to win a trip to perform it. Because of the maintenance of CERN accelerators, the experiment
+was performed at the DESY II Beam Test facility in Hamburg. The students, coordinated by their
+physics teachers and under the supervision of experienced researchers from the Physics Department
+of the Bari University and from the INFN Unit in Bari, won the competition.
+The goal of the experiment conceived by the team was to study the transition radiation emitted
+by fast electrons and positrons crossing different kinds of radiators. This paper provides a short
+presentation of the BL4S competition and presents the experiment and the result obtained by the
+team during their beam time in Hamburg in September 2021.
+2
+The BL4S competition
+Beamline for Schools (BL4S) [1] is a physics competition organised by CERN and DESY, which
+invites high-school students from all over the world to propose an experiment to be performed at a
+– 1 –
+
+particle accelerator. Each team has to write an original scientific proposal, explaining the theoretical
+background of the selected topic, and describing both the procedure to carry it out at a test beam
+facility and the results that they expect to find. A jury of experts, including scientists of CERN
+and DESY, review the proposal and select two teams (three from 2022 on) that win a trip to a fully
+equipped beam line of a particle accelerator.
+From 2014 to 2018 the winning experiments took place at the test beam area of the CERN
+Proton Synchrotron (PS) accelerator. In 2019 the competition moved to the DESY II Test Beam
+Facility (Hamburg) [2]. The partnership between CERN and the German laboratory allowed BL4S
+to continue during the three-year long shutdown of the CERN accelerator complex for upgrade and
+maintenance.
+The competition is structured in several preparatory phases, which include conferences and
+meetings with the organisers. Once the competition is announced, usually in Autumn, interested
+teams start preparing their proposals. Teams can include students either from the same school or
+from different schools. Having teams representing two schools or more is not unusual. During
+the proposal preparation, students are involved in an intense research project. After the conception
+and design of their experiment, the participants must write a well structured proposal and submit
+it on time. The students are not alone in this process, but they are guided by their coaches, who
+provide them with details on particle physics and teach them the necessary technical skills. Team
+coaches can be teachers, parents or scientists of local universities. It is important that students are
+well aware of each scientific detail of the proposed experiment, so that the theoretical background is
+clear and solid. The students are required to write down in detail how they intend to use the particle
+beam for their measurements and which equipment and detectors they need. Moreover, participants
+often complement their theoretical hypothesis with computer simulations. In fact, it is fundamental
+that students acquire the rudimentary programming skills that will be required in case of victory.
+Lastly, conclusions must contain the team’s expectations and motivation, which play a significant
+role in the jury’s decision. The BL4S organisers are always available to answer questions that the
+teams might have during the preparation of their proposals. Many teams contact them to discuss
+the feasibility of their experiments or practical problems that they encounter.
+In the final phase of the competition, a jury consisting of more than 50 volunteers selects
+the teams that are invited to a research institute to perform their experiment together with support
+scientists. Prior to the visit, the winning teams work remotely with the BL4S scientists to refine
+their experiments and perform a detailed planning.
+The beam time of the winning teams usually happens just after the summer, and the students
+have 12 full days of access to the experimental area to perform their measurements, supervised by
+the support scientists. During their stay, they work as a team of professional scientist would do and
+they complement their scientific experience with visits and lectures.
+After taking the data at the beam line, the teams are encouraged to analyse their data to answer
+the scientific question of the initial proposal, and to write a paper. During this phase, the team
+members stay in close contact with the BL4S support scientists and the team coaches.
+– 2 –
+
+Figure 1. Schematic view of the experimental setup.
+3
+The EXTRA experiment
+The EXTRA (Electron X-ray Transition RAdiation) experiment is designed to study the transition
+radiation (TR) [3] emitted by fast electrons and positrons crossing different radiators.
+Highly relativistic particles crossing the boundary between materials with different dielectric
+constants can produce TR in the X-ray region. However, since the yield of TR photons emitted at
+a single interface is considerably small (it is of the order of the fine structure constant 𝛼 ≈ 1/137),
+multiple boundaries are needed to enhance the X-ray production. Periodic radiators, consisting
+of stacks of thin foils of dielectric material separated by thicker air gaps, are commonly used in
+transition radiation detectors (TRDs) [4].
+The main features of the TR emitted by a periodic radiator depend on the kinematic properties
+of the radiating particles and on the radiator properties. They can be summarized as follows [5]:
+1. The effective TR photon emission starts at a threshold Lorentz factor, which is given by
+𝛾𝑡ℎ𝑟 = 𝑑1𝜔1/𝑐, where 𝑑1 is the thickness of the foils, while 𝜔1 is the plasma frequency of
+the foil material.
+2. The TR emission increases with the Lorentz factor 𝛾 until it reaches saturation at 𝛾𝑠𝑎𝑡 =
+𝛾𝑡ℎ𝑟
+√︁
+𝑑2/𝑑1, where 𝑑2 is the thickness of the air gaps.
+3. Most of the TR energy is emitted near the energy ℏ𝜔𝑚𝑎𝑥 = ℏ𝜔2
+1𝑑1/2𝜋𝑐.
+4. The angular distribution of TR photons exhibits a few maxima and extends up to 𝜃𝑚𝑎𝑥 =
+√︃
+1/𝛾2 + 𝜔2
+1/𝜔2.
+We have designed an experimental setup to measure the energy spectra and the angular
+distributions of the TR X-rays emitted by fast electrons and positrons crossing different radiators.
+Similar measurements were performed in the past at the CERN SPS with beams of 20 GeV/c
+electrons and of 120, 180 and 290 GeV/c muons, using silicon strip detectors [6], silicon pixel
+– 3 –
+
+Beam
+Radiator
+Timepix3
+BeamTelescopeScintillatorsFigure 2. Pictures of the experimental setup.
+– 4 –
+
+beamling
+forschools
+cern.ch/bl4sRadiator
+Foil/gap material
+d1 (𝜇m)
+d2 (𝜇m)
+N 𝑓
+EXTRA
+polyethylene/air
+23
+500
+150
+INFN
+polyethylene/air
+25
+300
+155
+CERN
+polyethylene/air
+25
+240
+190
+Table 1. Parameters of radiators used in the beam test: 𝑑1 and 𝑑2 are the thickness of the foils and the gap
+respectively; 𝑁 𝑓 is the number of foils.
+Radiator
+distance ( cm)
+Beam particle
+Beam momenta ( GeV/c)
+EXTRA
+40.5
+𝑒−
+1, 2, 3, 4, 5, 6
+88.0
+𝑒−
+1, 2, 3, 4, 5, 6
+132.0
+𝑒−
+1, 2, 3, 4, 5, 6
+INFN
+88.9
+𝑒−
+1, 2, 3, 4, 5, 6
+CERN
+88.4
+𝑒−/𝑒+
+1, 2, 3, 4, 5
+Table 2. Summary of the data taking configurations. For each radiator the beam particle, their momenta and
+the distance between the radiator and the X-ray detector are reported.
+detectors [7, 8] and GaAs pixel detectors [8, 9]. Parallel to the measurements, an effort to develop
+accurate Monte Carlo simulations of the TR process is being carried out [10]. One of the goals of
+these activities is that of exploiting TR for the identification of charged hadrons in the TeV energy
+region [11]. In this region all hadrons have Lorentz factor exceeding the typical threshold values
+for TR production (usually 𝛾𝑡ℎ𝑟 ∼ 500 ÷ 1000), and the simultaneous measurement of the energies
+and of the emission angles of TR X-rays can help to discriminate among different hadron species.
+Our measurements were performed at the DESY II Test Beam Facility [2] area TB21, using
+a beam of either electrons or positrons with momenta in the range from 1 to 6 GeV/c. A scheme
+of the setup is shown in Fig. 1, while some pictures are shown in Fig. 2. The radiator is followed
+by a Timepix3 assembly containing a thin silicon pixel sensor, which is used to detect the TR
+X-rays. A downstream beam telescope, composed by an array of six silicon pixel detectors, is
+used to reconstruct the tracks of the beam particles [12]. A set of two trigger scintillators, located
+downstream of the last plane of the beam telescope, is used for triggering the data acquisition.
+In our experiment we used three different radiators, which in the following will be labelled
+as "EXTRA", "INFN" and "CERN" respectively. Their features are summarized in Tab. 1. In
+particular, the EXTRA radiator was assembled for this measurement by the students at the Liceo
+Scientifico "A. Scacchi" in Bari. Fig. 3 shows some picture taken during the assembly of this
+radiator. The INFN and CERN radiators were borrowed from the Bari INFN Group and were used
+in a beam test campaign performed in 2006 [13].
+With these radiators, several measurements were performed, changing the beam composition
+and momentum, and the distance between the radiator and the X-ray detector. The different data
+taking configurations are summarized in Tab. 2.
+The TR X-rays were detected by a 100 𝜇m thick silicon sensor, bump-bonded to a Timepix3
+readout chip [14], consisting of a pixel matrix of 256×256 pixels with a pitch of 55 𝜇m. This silicon
+– 5 –
+
+Figure 3. Assembly of the EXTRA radiator at the Liceo Scientifico "A. Scacchi".
+detector assembly was placed such that the sensor faces the radiator to mitigate prior absorption in
+the readout chip. The sensor of the assembly with the ID W5_E2 was operated at a bias voltage of
+−21 V to ensure full depletion [15].
+The pixel pitch of the silicon sensor and its distance from the radiator determine the minimum
+detectable angular separation 𝜃𝑚𝑖𝑛 of TR X-rays from the direction of the radiating particles, as
+they should be separated of at least one pixel. Its value is in fact given by 𝜃𝑚𝑖𝑛 ≳ 𝑤/𝑑, where
+𝑤 = 55 𝜇m is the pixel pitch and 𝑑 is the distance of the silicon detector from the radiator. The
+configurations with larger distances allow to detect smaller angular separations; however, due to the
+X-ray absorption in the radiator and in the air gap between the radiator and the sensor, the number
+of detected TR X-rays will decrease with the distance from the radiator, and the angular resolution
+will deteriorate due to multiple Coulomb scattering of the primary particles in air.
+While the TR X-rays are likely absorbed by the front sensor, the radiating charged particles
+traverse the detector and leave an ionization track in the detectors of the beam telescope, which
+consists of an array of six regularly spaced silicon pixel detectors. In this configuration, scattering
+in air is limited to a minimum, enabling a track resolution of a few 𝜇m extrapolated to the Timepix3
+detector [12], which is more than sufficient for an identification of the charged particle among two
+or more clusters in the Timepix3 detector with cluster distances larger than a pixel pitch.
+Finally, the two scintillators, approximately shadowing the size of the telescope sensor planes
+and located at the end of the beam line, are used for triggering the data acquisition.
+The data acquisition was performed using the software framework EUDAQ2 [16], which
+– 6 –
+
+语integrates the control and readout of the Timepix3 assembly and the beam telescope, and features a
+graphical user interface for the configuration of connected devices, starting and stopping runs and
+data storage. An AIDA TLU [17] was used to form a trigger signal as a coincidence of the signals
+from the two scintillators while enabling a busy-handshake with the detectors.
+4
+Data analysis
+4.1
+Conversion & Clustering
+The raw data contains a collection of hit pixels per detector plane per trigger, which defines a
+so-called "event", including the corresponding pixel addresses; for the data from the Timepix3
+assembly, the corresponding information on the energy deposit, in form of a digitised signal, is
+also stored, while for the beam telescope no charge information is recorded. The collected data
+are converted to ROOT TTree format [18] using the data analysis framework Corryvreckan [19].
+In addition, this software performs a clustering procedure, which identifies adjacent hit pixels and
+connects them to form a so-called "cluster" under the hypothesis that pixel hits in one cluster are
+caused by a single incident particle. The cluster center, as an estimation on the incidence position
+of the particle, is calculated either as the center-of-gravity using the charge information, or as the
+arithmetic mean of the pixel hit positions in case of binary hit information.
+The energy calibration of the silicon pixel detector is performed assuming that the most probable
+energy loss of 5 GeV/c electrons crossing a 100 𝜇m thick silicon layer is 25.41 keV. This value
+has been calculated using a dedicated Monte Carlo simulation developed by H. Bichsel for the
+calculation of the energy losses of charged particles in thin silicon absorbers [20].
+4.2
+Detector alignment procedure
+The positions of the clusters in each silicon detector are evaluated in the local detector reference
+frame, with the 𝑧-axis oriented along the beam direction and the 𝑥 − 𝑦 plane corresponding to
+the detector plane, with the origin in the center of the detector. In the global reference frame the
+𝑧-axis is also directed along the beam direction, and the detectors are disposed on planes parallel
+to the 𝑥 − 𝑦 plane, with their centers at the coordinates (𝑥𝑖
+0, 𝑦𝑖
+0, 𝑧𝑖
+0). Due to mechanical tolerances
+in the assembly of the detectors, the coordinates (𝑥𝑖
+0, 𝑦𝑖
+0) are slightly misaligned with respect to the
+reference values (0, 0).
+A dedicated alignment run has been therefore performed to evaluate the coordinates (𝑥𝑖
+0, 𝑦𝑖
+0) of
+the centers of the silicon detectors (the index 𝑖 = 0 refers to the Timepix3 sensor, while the indices
+𝑖 = 1 . . . 6 refer to the detectors of the beam telescope). The alignment run has been performed
+removing the radiator from the beam line and using 5 GeV/c electrons.
+We have implemented an iterative alignment procedure selecting a sample of events with only
+one cluster in each silicon detector. This choice is aimed to select events with only one electron
+track across all the detectors. In the first iteration we assume 𝑥𝑖
+0 = 0 and 𝑦𝑖
+0 = 0 for all detectors. We
+fit all the tracks with a straight line and, for each track, we evaluate the residuals in each detector as
+𝑟𝑖
+𝑥 = 𝑥𝑖 − 𝑥𝑖
+𝑓 𝑖𝑡 and 𝑟𝑖
+𝑦 = 𝑦𝑖 − 𝑦𝑖
+𝑓 𝑖𝑡, where (𝑥𝑖, 𝑦𝑖) and (𝑥𝑖
+𝑓 𝑖𝑡, 𝑦𝑖
+𝑓 𝑖𝑡) are respectively the true and fitted
+positions of the cluster in the 𝑖-th detector. We then build the distributions of the residuals 𝑟𝑖
+𝑥 and 𝑟𝑖
+𝑦
+and, in the next iteration, we set 𝑥𝑖
+0 = −𝜇𝑖
+𝑥 and 𝑦𝑖
+0 = −𝜇𝑖
+𝑦, where 𝜇𝑖
+𝑥 and 𝜇𝑖
+𝑦 are the average values
+– 7 –
+
+Figure 4. Distributions of the residuals in the silicon detector equipped with the TimePix3 chip after the
+alignment procedure.
+of these distributions. The iterative procedure is terminated when |𝜇𝑖
+𝑥| < 1 𝜇m and |𝜇𝑖
+𝑦| < 1 𝜇m
+for all detectors. Convergence is reached after the second iteration.
+Fig. 4 shows the distributions of the residuals in the silicon detector equipped with the Timepix3
+chip after the alignment procedure. The RMS of the residual distributions in both the 𝑥 and 𝑦 views
+are of about 10 𝜇m.
+Fig. 5 shows the distributions of the direction cosines of the electron tracks in the alignment
+run. We see that the average values of the direction cosines 𝑐𝑥 and 𝑐𝑦 are slightly different from
+zero. This result implies that the 𝑧-axis of our reference frame is not perfectly aligned with the
+direction of the beam. The tilt angle can be estimated from the average value of 𝑐𝑧, and is of about
+5 mrad. Finally, from the values of the RMS of the distributions of 𝑐𝑥 and 𝑐𝑦 we can deduce that
+the beam divergence is of about 1 mrad in both the 𝑥 and 𝑦 directions.
+4.3
+Data selection and analysis
+As discussed in Sec. 3, several runs in different configurations have been taken, by changing the
+beam composition and momentum, the radiator and its distance from the silicon pixel detector.
+In each of these runs we have selected events with at least one cluster in the silicon pixel sensor
+and at least 3 clusters in different detectors of the beam telescope. This choice is motivated by the
+need of identifying, among the clusters in the silicon sensor, the one originated by the ionization
+energy deposit of the beam particle and those eventually originated by the absorption of TR X-rays
+produced in the upstream radiator.
+Fig. 6 shows the distribution of the total number of clusters in the detectors of the beam
+telescope for all the runs performed with electrons crossing the EXTRA radiator, which was placed
+at a distance of 88.9 cm from the silicon pixel sensor. As expected, the distribution is peaked at 6
+clusters, corresponding to clean electron tracks, yielding one cluster in each detector. Events with
+less than 6 clusters can be originated from inefficiencies of some detectors in the beam telescope
+or from beam particles which do not cross all the telescope planes. Events with more than 6
+clusters can be originated from delta rays accompanying the primary electron track or from TR
+X-rays passing through the upstream silicon sensor and being absorbed in any detector of the silicon
+– 8 –
+
+X103
+Entries
+542808
+1.2
+Mean
+4.227e-04
+RMS
+1.064e-02
+x? / ndf
+1.047e+04/2389
+Constant
+1.126e+03±1.961e+00
+Mean
+4.303e-04±1.287e-05
+Sigma
+9.346e-03±9.969e-06
+0.8
+ents
+0.4
+0.2
+0
+/x10~3
+60
+40
+20
+0
+20
+40
+60
+Residualsinthex-view(mmX103
+Entries
+542808
+1.2
+Mean
+3.268e-04
+RMS
+1.050e-02
+x? / ndf
+1.044e+04/2379
+Constant
+1.140e+03±1.981e+00
+Mean
+3.287e-04± 1.271e-05
+Sigma
+0.8
+9.229e-03±9.775e-06
+ents
+0.4
+0.2
+/x10~3
+0
+60
+40
+20
+0
+20
+40
+60
+Residuals in the y-view (mm)Figure 5. Distributions of the direction cosines of the electron tracks in the silicon detector and in the beam
+telescope in the alignment run.
+telescope. We also see two peaks, at 12 and 18 clusters respectively, which include less than 1% of
+the total number of events, and which likely correspond to double and triple electron tracks.
+The clusters in the detectors of the beam telescope are used to reconstruct the tracks of the
+beam particles in the telescope. To select events with single electron (positron) tracks, we require
+less than 10 clusters in the beam telescope. Candidate tracks are built by selecting all the possible
+cluster combinations with only one cluster per plane of the telescope. The clusters of each candidate
+track are then fitted with a straight line and the 𝜒2 of the fit is evaluated. The track with the best 𝜒2
+is then selected.
+Once the track of the radiating particle in the beam telescope is reconstructed, we evaluate the
+coordinates (𝑥𝑡𝑟𝑎𝑐𝑘, 𝑦𝑡𝑟𝑎𝑐𝑘) of its intersection with the upstream silicon pixel sensor. Then, if more
+clusters are found in the sensor, the cluster nearest to the track is associated to the particle ("particle
+cluster"), while other clusters are associated to possible TR X-rays ("X-ray clusters"). Clearly, if
+only one cluster is found in the silicon pixel sensor, it is associated to the particle and no X-rays are
+detected.
+– 9 –
+
+X103
+Entries
+542808
+Mean 4.254e-03
+25
+RMS
+9.283e-04
+20
+Events
+15
+10
+L0
+×10~3
+-10
+-8
+-6
+4
+-2
+0
+2
+4
+6
+80
+10
+CxX103
+Entries
+542808
+30
+Mean -1.903e-03
+RMS
+8.606e-04
+25
+20
+Events
+15
+10
+LO
+0
+/×10~3
+-10
+-8
+-6
+-4
+2
+0
+2
+4
+9
+8
+10
+CyX103
+Entries
+542808
+Mean 1.160e-05
+30
+RMS4.175e-06
+25
+20
+Events
+15
+10
+L0
+0
+×10~6
+0
+5
+10
+15
+20
+25
+30
+35
+40
+45
+50
+1-CzFigure 6. Distribution of the total number of clusters in the beam telescope for all the runs performed with
+electrons crossing the EXTRA radiator, placed at a distance of 88.9 cm from the silicon pixel sensor.
+5
+Results
+In Figs. 8, 9 and 10 the results obtained in the runs with the EXTRA radiator are summarized. The
+plots in each figure correspond to the configurations with the silicon detector placed at the distances
+of 40.5 cm, 88 cm and 132 cm from the radiator respectively. The plots are built selecting events
+with the particle cluster inside a square of a 3 × 3 mm2 area, in the centre of the TimePix3 detector.
+All the distributions shown in the above plots are normalized to the total number of selected events.
+The top panels of each figure show the distributions of the relative positions of the TR X-
+rays (evaluated from the X-ray clusters) with respect to the radiating electron (evaluated from the
+particle cluster). As expected, TR photons tend to accumulate in rings centered on the position of
+the radiating particle and the number of photons per electron increases with the beam momentum
+(and consequently with the Lorentz factor of the radiating particles).
+The central panels show the distributions of the TR X-ray energies as a function of their
+angular separation from the radiating particle. Most X-rays are emitted at angles 𝜃 ≲ 2 mrad from
+the radiating particle, with energies peaked at energies < 10 keV. A second peak of X-rays emitted
+at angles ∼ 3.5 mrad and with the same energies as the first peak can also be seen, and it becomes
+more evident as the beam momentum increases.
+Finally, the bottom panels show the energy distributions of the absorbed TR X-rays compared
+with the distributions of the energies deposited by the parent electrons in the TimePix3 detector.
+As discussed in Sec. 4.1, the energy losses of the electrons follow Landau distributions with a most
+probable value of 25.4 keV, while X-ray energies are peaked at less than 10 keV. We see that the
+area of the X-ray energy spectra increases with increasing electron momentum. This behaviour is
+– 10 –
+
+Entries7541667
+Mean
+6.177
+RMS
+1.049
+10
+10~3
+10-4
+0
+20
+Numberofclusters inthebeamtelescopeFigure 7. Distribution of the distances of the "particle clusters" from the track in the silicon sensor for the
+runs performed with electrons crossing the EXTRA radiator, placed at a distance of 88.9 cm from the silicon
+pixel sensor.
+expected since the spectra are normalized to the total number of electrons and the TR yield increases
+with the Lorentz factor of the radiating particle.
+A summary of the results obtained in all the configurations explored is shown in Fig. 11.
+The average number of detected TR X-rays per electron is shown as a function of the beam
+momentum. We see that for all configurations the number of detected photons increases with the
+beam momentum and saturates above 4 GeV/c. This behavior is expected, since the threshold
+Lorentz factor for all radiators is 𝛾𝑡ℎ𝑟 ≃ 103 and the saturation Lorentz factors are in the range
+4÷5×103. Comparing the results obtained with the EXTRA radiator in the different configurations
+we see that the average number of detected TR X-rays decreases when the radiator-detector distance
+is increased. The increase of the distance causes an increase of the X-ray absorption in the air gap
+between the radiator and the detector, which is not compensated by the lower minimum detectable
+angle between the photons and the radiating particles. We remark here that the results shown in
+Fig. 11 referred to the CERN radiator have been obtained from a joint analysis of the data samples
+collected with both the electron and positron beams (see Tab. 2). This choice is motivated by the
+fact that the separate analyses of the electron and positron data samples yield the same results. This
+feature was expected, since the properties of TR are independent of the sign of the charge of the
+radiating particle.
+The experimental results shown in Fig. 11 are compared with the predictions obtained by
+folding the TR yield, evaluated with the theoretical formulae for regular radiators [4, 21] with the
+X-ray absorption probabilities in the air gap between the radiator and the TimePix3 detector and in
+– 11 –
+
+X10~3
+Entries7541668
+25
+Mean
+0.0408
+RMS
+0.0645
+20
+Fractionofevents
+15
+10
+5
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+Distance betweenthe track and theparticle cluster (mm)Figure 8.
+Summary of the results obtained in the runs with the EXTRA radiator at 40.5 cm from the
+TimpePix3 detector. Top panel: distribution of the relative positions of the TR photons (X-ray clusters) with
+respect to the electrons (particle clusters); middle panel: distribution of X-ray energies as a function of their
+angular separation from the electrons; bottom panel: electron and X-ray energy distributions.
+– 12 –
+
+EXTRA radiator, d = 40.5 cm
+10
+Ypart(mm)
+>
+6Gevc
+2
+0
+X- Xpart (mm)EXTRA radiator,d = 40.5 cm
+50
+Gevic
+2Gevic
+=3GeV/c
+40
+30
+/Electron
+(keV)
+20
+Energy
+Photons/
+10
+0
+50
+Photon
+5Gev
+=6Gev
+40
+30
+20
+10
+0
+2
+4
+6
+8
+0
+2
+4
+6
+8
+0
+2
+4
+6
+8
+Electron-Photonangularseparation(mradEXTRA radiator, d = 40.5 cm
+0.12
+p = 1 GeV/c
+p = 2 GeV/c
+p = 3 GeV/c
+Electron
+Electron
+Electron
+0.10
+Photons
+Photons
+Photons
+0.08
+0.06
+Entries
+0.04
+0.02
+of
+Fraction 
+0.00
+0.12
+p = 4 GeV/c
+p = 5 GeV/c
+p = 6 GeV/c
+Electron
+Electron
+Electron
+0.10
+Photons
+Photons
+Photons
+0.08
+0.06
+0.04
+0.02
+0.00
+0
+20
+40
+60
+80
+1000
+20
+40
+60
+80
+100 0
+20
+40
+60
+80
+100
+Energy (keV)Figure 9. Summary of the results obtained in the runs with the EXTRA radiator at 88 cm from the TimpePix3
+detector. Top panel: distribution of the relative positions of the TR photons (X-ray clusters) with respect to
+the electrons (particle clusters); middle panel: distribution of X-ray energies as a function of their angular
+separation from the electrons; bottom panel: electron and X-ray energy distributions.
+– 13 –
+
+EXTRA radiator, d = 88 cm
+=
+Gevlo
+=
+2GeVIc
+3GeVIc
+10-3
+Photons/Electron
+2
+=4 GeV/c
+=
+5GeVic
+6GeV/c
+-
+10-5
+0
+2
+2
+0
+1
+2
+-2
+2
+1
+2
+X - Xpart (mm)EXTRA radiator, d = 88 cm
+50
+1 GeV/o
+2Gev/o
+=3GeV/o
+40
+30
+10
+lectror
+20
+10
+-
+E
+Energy
+10
+itons/
+0
+Phot
+50
+Photon
+p=4GeV/c
+of
+40
+10-5
+er
+lumbe
+30
+10-6之
+20
+10
+0
+2
+2
+6
+8
+0
+4
+6
+8EXTRA radiator, d = 88 cm
+0.12
+p = 1 GeV/c
+p = 2 GeV/c
+p = 3 GeV/c
+Electron
+Electron
+Electron
+0.10
+Photons
+Photons
+Photons
+0.08
+0.06
+Entries
+0.04
+0.02
+of
+Fraction 
+0.00
+0.12
+p = 4 GeV/c
+p= 5 GeV/c
+p = 6 GeV/c
+Electron
+Electron
+Electron
+0.10
+Photons
+Photons
+Photons
+0.08
+0.06
+0.04
+0.02
+0.00
+0
+20
+40
+60
+80
+1000
+20
+40
+60
+80
+100 0
+20
+40
+60
+80
+100
+Energy (keV)Figure 10. Summary of the results obtained in the runs with the EXTRA radiator at 132 cm from the
+TimpePix3 detector. Top panel: distribution of the relative positions of the TR photons (X-ray clusters) with
+respect to the electrons (particle clusters); middle panel: distribution of X-ray energies as a function of their
+angular separation from the electrons; bottom panel: electron and X-ray energy distributions.
+– 14 –
+
+EXTRA radiator, d = 132 cm
+Gew
+p = 2 GeV/c
+=3GeV/C
+10
+()d
+>
+=4GeV/C
+p=5GeV/c
+-
+0
+0
+1
+0
+1
+2
+-2
+-1
+0
+1
+2
+X - Xpart (mm)EXTRA radiator, d = 132 cm
+50
+=2GeV/o
+3Gev/c
+40
+30
+10
+lectron
+20
+Energy
+10
+Photons
+S
+50
+Photon
+40
+Jumber
+30
+10~5之
+20
+10
+1
+0
+2
+4
+6
+8
+0
+2
+4
+6
+0
+2
+4
+8
+Electron-Photonangularseparation(mradEXTRA radiator, d = 132 cm
+0.12
+p = 1 GeV/c
+p = 2 GeV/c
+p = 3 GeV/c
+Electron
+Electron
+Electron
+0.10
+Photons
+Photons
+Photons
+0.08
+0.06
+Entries
+0.04
+0.02
+of
+Fraction 
+0.00
+0.12
+p = 4 GeV/c
+p= 5 GeV/c
+Electron
+Electron
+0.10
+Photons
+Photons
+0.08
+0.06
+0.04
+0.02
+0.00
+0
+20
+40
+60
+80
+1000
+20
+40
+60
+80
+100 0
+20
+40
+60
+80
+100
+Energy (keV)Figure 11. Average number of TR photons as a function of electron beam momentum for the three radiator
+types and for the different distances from the TimpePix3 detector. The dashed lines show the predictions
+for the different configurations. The results obtained in a run without radiator and in two runs with dummy
+radiators are also shown.
+the silicon layer of the detector. The theoretical curves seem to be in a reasonable agreement with
+the experimental results.
+Finally, we have performed some control runs to check our results.
+A run with 5 GeV/c
+electrons without any radiator was performed to evaluate the possible contamination to the detected
+TR signal from bremsstrahlung photons produced in the upstream materials and accompanying
+the beam particles and the possible contamination from noisy pixels. In this run we found about
+0.03 X-rays per electron; in addition, since all X-ray clusters are found very close to the particle
+cluster, the contamination from noisy pixels can be considered negligible. We also performed two
+additional runs with 6 GeV/c electrons, in which we replaced the radiator with some "dummy"
+radiators: in particular, we used a set of paper towels, which were arranged in a stack simulating
+a regular radiator, and a piece of sponge, which simulates an irregular radiator1. In both cases we
+observed a TR signal of about 0.17 X-rays per electron.
+6
+Conclusions
+In the framework of the BL4S competition we have designed and implemented an experiment to
+measure the TR emitted by fast electrons and positrons crossing different kind of radiators. The
+1Irregular radiators made of foams or fiber mats are sometimes used in TRDs.
+– 15 –
+
+1.2
+Photons
+EXTRA 40.5 cm
+CERN 88.4 cm
+EXTRA 88.0 cm
+No Radiator
+1.0
+EXTRA132cm
+Towels 88.4cm
+INFN88.9cm
+Sponge 88.4 cm
+TR
+0.8
+0.6
+0.4
+Average
+0.2
+0.0
+0
+1
+2
+3
+4
+5
+6
+Electronmomentum(GeY/c)measurement has been performed at the DESY II Test Beam Facility area TB21, using electrons and
+positron beams with momenta up to 6 GeV/c. We have measured the energy spectra and the angular
+distribution of the TR X-rays using a 100 𝜇m thick pixel silicon detector, with a pitch of 55 𝜇m.
+The experimental results are well reproduced by the theoretical curves obtained from standard TR
+models.
+BL4S has offered the students the chance to be actively involved in all the aspects of an
+experimental research: during the preparation of the proposal, they have learned how to design
+an experimental setup, optimizing the detectors available for the measurement; after their proposal
+was selected, they have been involved in the design and in the assembly of their own radiator; then,
+at DESY, they had the chance to run a real beam test; finally, they have taken part to the analysis of
+the data collected in the test. However, the most important educational result of this experience is
+that the students learned how to apply the scientific approach not only in the field of research, but
+also to the solution of everyday life challenges.
+Acknowledgments
+The members of the EXTRA team thank the CERN and DESY support scientists, the beamline
+scientists, the volunteers and the BL4S organisers who helped them during the preparation and the
+implementation of their experiments. All the scientists involved in the competition dedicated a lot
+of their time to answer all the questions the students had, giving them precious advises for their
+future career. The team was really pleased to find such wonderful people, who showed them what
+unconditional love for science really means.
+A big thank to the Teomizli team from Mexico, the other winning team of the BL4S 2021.
+Meeting peers from the other side of the world and work with them as a unique team of scientists
+has been an enriching opportunity.
+Beamline for Schools is an education and outreach project funded by the CERN & Society
+Foundation and supported by individual donors, foundations and companies. In 2021, the project
+was funded by the Wilhelm and Else Heraeus Foundation. Additional contributions have been
+received from the Arconic Foundation, Amgen Switzerland AG, and the Ernest Solvay Fund
+managed by the King Baudouin Foundation.
+The EXTRA team also acknowledges financial support from CERN and DESY for their
+participation to the beam test campaign.
+The EXTRA team thanks B. Fanti, I. Iusco, D. Ricchiuti and all the personnel of the Liceo
+Scientifico “A. Scacchi” for their support to the project activities.
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