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+Numerous studies are carried out and published as stand-alone experiments +that often vary only slightly with respect to problem setups and solution +approaches. The programmatic core of these experiments is typically very +similar. Despite this fact, no standardized and resilient framework for ex- +perimentation on PS problems with DRL algorithms could be established so +far. In this paper, we introduce schlably, a Python-based framework that +provides researchers a comprehensive toolset to facilitate the development +of PS solution strategies based on DRL. schlably eliminates the redundant +overhead work that the creation of a sturdy and flexible backbone requires +and increases the comparability and reusability of conducted research work. +Keywords: +Scheduling, Reinforcement Learning, JSSP, Framework +Preprint submitted to SoftwareX +January 12, 2023 +arXiv:2301.04182v1 [cs.LG] 10 Jan 2023 + +Required Metadata +Current code version +Nr. +Code metadata description +Please fill in this column +C1 +Current code version +v0.1.0 +C2 +Permanent link to code/repository +used for this code version +https://github.com/tmdt- +buw/schlably +C4 +Legal Code License +Apache License, 2.0 (Apache-2.0) +C5 +Code versioning system used +git +C6 +Software code languages, tools, and +services used +Python, +OpenAI +Gym, +RLlib, +Weights and Biases +C7 +Compilation requirements, operat- +ing environments & dependencies +Python 3.10 +C8 +If available Link to developer docu- +mentation/manual +https://github.com/tmdt- +buw/schlably/tree/main/docs +C9 +Support email for questions +schlably@uni-wuppertal.de +Table 1: Code metadata (mandatory) +1. Motivation and Significance +Production scheduling (PS) is a challenging and highly researched prob- +lem in operations research (OR) and optimization. It deals with allocating +resources to production jobs over time to minimize criteria such as time, +effort, and cost [1]. PS problems are considered NP-hard and therefore re- +quire much computation effort to be solved sufficiently well with increasing +problem sizes. In recent years, with increasingly powerful algorithms and +computational hardware, deep reinforcement learning (DRL) has emerged as +a promising tool to address PS problems [2, 3, 4, 5, 6, 7, 8]. DRL is a ma- +chine learning paradigm, in which deep learning models are trained through +interaction with an environment to autonomously derive solution strategies +for sequential tasks [9]. +The young research field of DRL-based PS lies at the intersection of op- +erations research and artificial intelligence and as such is characterized by +a heterogeneous community with varying problem-solving approaches and +technical skill sets. Yet, all empirical studies apply a very similar experi- +mental setup consisting of the same main software components: An envi- +ronment representing the production facility layout and logic, a scheduling +problem generator, a DRL agent algorithm, and logging as well as evalua- +tion tools. The difference between different experimental setups usually lies +2 + +within one or more of these components, for example by incorporating a new +DRL algorithm [10, 11, 12], interaction logic between agent and environ- +ment [6, 3], training procedure [13], learning objective [14, 12] or additional +problem constraint [14, 15]. Regardless of large overlaps, all researchers im- +plement their own individual experimentation framework with the following +two consequences: Large initial ramp-up efforts when experimenting with +new methodologies or custom problem settings, and scarcity of empirical +comparisons to other works. In this paper, we address these shortcomings +and present the software framework schlably for developing and evaluating +DRL-based PS solutions. schlably provides the following contributions: +• It is modular, so individual changes may be adapted without much +overhead. +• It works out of the box with functioning environments, data-generation +scripts, agents, logging functions, training, and testing scripts. +• It provides benchmark datasets for different scheduling problem classes +and sizes. +• It includes widely recognized benchmark algorithms and results. +• It facilitates the application of algorithms designed for one problem +class and size to other problems. +schlably will accelerate the research area of DRL-based PS under real- +world conditions by lowering the barrier of entry for researchers from different +domains with different perspectives, levels of expertise, and objectives. +2. Background and Related Work +schlably started as a code base for experiments on a real-world inspired +scheduling problem in the context of a university research project with in- +dustrial partners. As such, several requirements became apparent early on +that can be summarized in four general design goals. First, from the ap- +plied industrial perspective, schlably has to offer the integration of DRL +methods and heuristics which work out of the box. Second, it also has to +cover different scheduling scenarios, e.g. including such bounded by resource +constraints. Third, from the scientific research perspective, schlably should +support detailed comparable evaluations of methods. Lastly, it has to be +easy to interact with at the code level, to enable students with limited expe- +rience to quickly understand the topic, concepts, and implementation. The +implications of these design goals are discussed in this section. +3 + +In the following, we present an overview of related published experi- +mental frameworks and compare them to our design goals in schlably. In +the comparison, we include frameworks dedicated to being used by others +[16, 17, 18, 19, 20] and frameworks published in a supplementary fashion +along with research papers and projects [21, 22, 23, 24, 25, 26], because in +practice both may serve as starting points for additional experiment designs. +The frameworks were found via references in scientific publications and a +search of "Scheduling Reinforcement Learning" on GitHub. We do not claim +the list to be exhaustive but are not aware of any other popular frameworks +at the time of writing this paper. Commercial or proprietary scheduling soft- +ware was excluded because license fees and other accompanying challenges +introduce a major barrier. However, we are not aware of any commercial +software that provides the tight integration of DRL and PS. Table 2 provides +an overview of the frameworks. +In addition, we assessed them regarding +their fulfillment of our design goals which are formally categorized into the +following four groups. +Pre-Implemented Benchmarks. Several frameworks either provide pre- +implemented DRL agents and scripts for training or the easy integration of +agents from popular DRL libraries, e.g. from StableBaselines [27] or RLlib +[28]. Like [20], our goal is to enable and facilitate both, the manual extensions +of basic DRL algorithms, like DQN and PPO, as well as the usage of powerful +third-party libraries. Both options are important to empower users to choose +the appropriate approach for their respective research interests. Additionally, +for the sake of comparability, it is crucial to provide common benchmarks +in the form of popular priority dispatching rules (PDRs), such as Shortest- +Processing-Time-First, and a flexible optimal solver that can handle several +scheduling problem types. Most frameworks only cover a few PDRs, often +missing competitive ones, such as Most-Tasks-Remaining and even random +baselines, where agent actions are sampled from a normal distribution. +Scheduling Instance Generation. Generating new data with varying prob- +lem cases is necessary to enable comprehensive training and testing of a DRL +agent. Accordingly, a suitable framework must implement a flexible problem +instance generator. This generator enables the user to create scheduling prob- +lems of different popular categories (e.g. the Job Shop Scheduling Problem +(JSSP) or the Flexible Job Shop Scheduling Problem (FJSSP) [1]) instances +with any combination of instance variables, like the number of jobs, number +of tasks, runtimes, and more. Moreover, its design simplifies the integration +4 + +[16] [17] [18] [21] [19] [20] [22] [23] [24] [25] [26] schlably +B +Implemented RL-agents +� +� +� +� +� +� +� +� +� +� +� +� +Implemented PDRs +� +� +� +� +� +� +� +� +� +� +� +� +Implemented opt. solver +� +� +� +� +� +� +� +� +� +� +� +� +Interface for RLLib +� +� +� +� +� +� +� +� +� +� +� +� +S +Flexible Data Generation +� +� +� +� +� +� +� +� +� +� +� +� +JSSP +� +� +� +� +� +� +� +� +� +� +� +� +FJSSP +� +� +� +� +� +� +� +� +� +� +� +� +Different problem types +� +� +� +� +� +� +� +� +� +� +� +� +Resource constraint tool +� +� +� +� +� +� +� +� +� +� +� +� +L +Log achieved results +� +� +� +� +� +� +� +� +� +� +� +� +Evaluate achieved results +� +� +� +� +� +� +� +�� +� +�� +� +� +Visualize Gantt-Chart +� +� +� +� +� +� +� +� +� +� +� +� +Comparison to solver +� +� +� +� +� +� +� +� +� +� +� +� +Comparison to PDRs +� +� +� +� +� +� +� +� +� +� +� +� +C +Paper +� +� +� +� +� +� +� +� +� +� +� +� +README +� +� +� +� +� +� +� +� +� +� +� +� +Code documentation +� +� +� +� +� +� +� +� +� +� +� +� +Easily personalizable +� +� +� +� +� +� +� +� +� +� +� +� +Works out-of-the-box +� +� +� +� +� +� +� +� +� +� +� +� +User manual in Readme +� +� +� +� +� +� +�� +� +�� +� +� +� +OpenAI Gym Env +� +� +� +� +� +� +� +� +� +� +� +� +Legend: +�not fulfilled +�half-fulfilled +�fulfilled +Table 2: Overview of related frameworks and their fulfillment regarding pre- +implemented benchmarks (B), scheduling instances (S), logging and evalua- +tion (L), and code usability (C) +5 + +of additional scheduling problem types to encourage the implementation of +individual, more complex, or more specific use cases. Finally, to the best +of our knowledge, this is the first framework providing an optional resource +constraint. Concretely, the user is able to specify a required tool per opera- +tion. +Logging and Evaluation. Logging results and evaluation metrics in a struc- +tured manner is key for quick feedback during training runs, but also for +identifying patterns in and drawing conclusions from large-scale experiments. +Our objective with schlably is to provide extensive logging options that may +be turned on and off, and where results and models may be shared to promote +collaboration on projects. This design goal is met to a large degree by [20] +from which we took much inspiration in this regard. All other frameworks +do not address this design goal. For the evaluation of solutions generated +by DRL agents, a comprehensive framework should apply the benchmarks +mentioned above and provide an overview of the overall performance. Most +of the reviewed frameworks lack functionality in this aspect. Moreover, to +get a graphical overview and to visually support the tracking of very specific +actions in a production schedule, a Gantt chart plotter is useful for human +inspection. The Gantt chart should display all metadata of operations, e.g. +the runtime or required tool, and has been found to be helpful for debugging +and evaluating DRL agents. Many other frameworks, but not all, include a +Gantt chart plotter. +Code Usability. As usability is of utmost importance, a framework has to +offer easy access through full documentation and must include a README, +user application programming interface (API) manual, and formal functional- +ity description. Within the reviewed frameworks, only [20] covers all criteria. +To be usable with different skill sets our explicit goal is to enable users to +start experimenting with small design decisions by only using the configura- +tion files but at the same time facilitate substantial logical changes to routines +and components by means of their own implementations. For that reason, we +would favor comprehensibility over efficiency wherever a trade-off is unavoid- +able. This requires a careful balance. In our opinion, all other frameworks +overemphasize one side: [20] offers many functional changes through configu- +ration files but at the cost of a comparatively complex software architecture. +On the other hand, all other frameworks are much smaller and easier to get +an overview of, at the expense of limited functionality. Lastly, a framework +with a claim to widespread use should stick to conventional APIs. In the +6 + +context of DRL, the most commonly used API is the OpenAI Gym [29] API. +Only half of the reviewed frameworks adhere to it. +3. Software Architecture +This chapter describes our framework with focus on implementation- +specific details. We are providing a general overview of the code itself, fo- +cusing on currently existing exemplary implementations while also pointing +out open interfaces. Furthermore, we describe details regarding the main +components of schlably to demonstrate the realization of the design goals, as +introduced in Chapter 2, and to enable users to fit schlably to their needs. +The overall structure is illustrated in Figure 1. We divided the code base +into six main components, which are described below in detail. Following +this component-oriented approach, and in combination with comprehensive +code documentation, schlably adheres to the objective of the fourth design +goal, which requires easy interaction and usability at the code level. +agents +code_tests +environments +utils +visuals_generator +data_generator +GanttChartPlotter ++ VISUALS_DIRECTORY ++ get_gantt_chart_image ++ get_gantt_chart_gif_and_save +EvaluationHandler ++ rewards ++ makespan ++ record_environment_episode ++ evaluate_test +Logger ++ WANDB_PROJECT ++ LOG_MODE_DEFAULT ++ record ++ dump ++ dump_wandb +ui_tools +file_handler +ConfigHandler +DataHandler +FileChooser +ProgressBar +MessageBox +agent_tests +data_generator_tests +visuals_generator_tests +Runner +Env ++ num_jobs ++ num_machines ++ reset ++ step +EnvIndirectAction ++ get_action_mask +EnvironmentLoader + + ENVIRONMENT_MAPPER_DICT ++ load ++ check_environment_agent_compatibility +InstanceFactory +SPFactory ++ SP(Enum) ++ generate_instances +Task ++ job_index ++ task_index +heuristic +reinforcement_learning +solver +dqn +ppo +ppo_masked +intermediate_test +test +train +Figure 1: Overview of schlably project and code structure. +7 + +Data generator. The general data structure of scheduling problems, as +used in schlably, is represented by so-called instances. A user can generate +infinite instances of a scheduling problem, however, each instance is a spe- +cific configuration and entity. The specific configuration, contained within +an instance, is given by a number of jobs, with a job simply being an encom- +passing logical container consisting of individual tasks. The data_generator +component incorporates the necessary classes to generate such an instance +and the individual tasks. From the scheduling problem point-of-view, it is +the centerpiece of the problem formulation and representation. The Task +class is a specifically designed data class and its entities are the atomic +units of a scheduling problem instance. +Such an instance can be created +via the SPFactory, which allows the generation of different types of schedul- +ing problems that are given via the included Enum. If users would like to +introduce a new type of scheduling problem into schlably, they would have +to include their function in this class and add it to the Enum. Finally, the +InstanceFactory enables high-level access to the problem factory class and +manages the configuration-based creation of batches of instances. Thus, the +data_generator component realizes the foundation of the second design +goal, which requires the implementation and handling of different scheduling +scenarios. +Environment. An environment defines the observation space, action space, +and reward strategy. +Thus, it represents a simulation of the agent’s en- +vironment and interaction dynamics and is the central piece of any DRL +approach. All schlably environments are included in the environments com- +ponent. Exemplary, we provide a simple scheduling Env as well as a derived +version named EnvIndirectAction to showcase the expandability. All en- +vironments adhere to the Gym API and are explicitly derived from a base +Gym environment. The EnvironmentLoader class enables high-level access +and management of the different environment types and appropriate algo- +rithms, as not all algorithms are feasible for every environment. New envi- +ronments have to be included in this component and added to the managing +EnvironmentLoader. This encapsulated approach, in conjunction with the +data_generator component, represents the implementation of the second +design goal. +Agent. The agents component combines the heuristic functionalities, the +solver, and implementations of DRL algorithms as well as the train and test +functions for the DRL approach. Users can to integrate functionalities from +8 + +other DRL frameworks, like more extensive training procedures, model types, +and learning algorithms via pre-defined interfaces. As such, the agents com- +ponent realizes the first design goal, to support and simplify the integration +of out-of-the-box-methods as well as pre-implemented benchmarks. +Visual generator. The component visuals_generator incorporates all +classes and scripts which are used to create visualizations of the problem +instances and generated solutions. +These functionalities are intentionally +isolated as different scheduling problem environments and still share the same +visualization approach. schlably, for example, introduces a GanttChartPlotter +that enables a user to generate individual Gantt chart images (see Figure 2b) +or create a GIF of the scheduling progress. Thus, it is part of the implemen- +tation of the third design goal. +Utils. The utils component aggregates classes and functions which have a +supporting character for the main functionalities of schlably. Specifically, it +includes user interface components (ui_tools), data interface components +(file_handler), e.g. to load and save data, and the high-level Logger class. +Accordingly, the utils component realizes the third design goal, facilitating +logging and evaluations for comparisons. +Code tests. All code tests that ensure the crucial functionality of the de- +scribed components are collected in the code_tests component. Up to this +point, we included multiple unit tests with a central Runner. These are also +intended as an example for users that plan to extend the code base. +4. Illustrative Example +To illustrate a typical use case, we consider a scenario in which an ML +engineer wants to compare the learning behavior of two PPO agents. It is also +part of our tutorial in the documentation. One agent is trained on 6x6 JSSP +instances and receives a reward based on the change in the time to complete +all tasks (i.e. the makespan) per step, as proposed in [3]. This setup is also the +default setting delivered in the framework. The other one is trained on a 3x4 +tool-constrained JSSP instance and receives a zero reward per step with the +exception of the last step, where the reward is equal to the overall achieved +makespan. The remaining training parameters are kept constant. The second +training requires only minimal manual changes to the base model. These +include setting different configuration parameters, generating new data, and +9 + +Figure 2: Comparing agent runs in Weights&Biases (screenshot from the web interface +shown on the left-hand side). a) Visualized training curves for interpreting the learning +performance of the agent. b) Gantt chart depicting the solution of the trained agent on a +selected test instance. c) Table providing evaluation results and comparison of the trained +agents and benchmark methods on the test instances. +changing the reward function in the base environment. Details may be found +in the documentation. The integrated interface to Weights&Biases [30] makes +it easy to compare the training curves and achieved results, as depicted in +Figure 2. +The described short example reflects several of our design goals. Figure +2c) demonstrates that the agents’ performance is automatically compared +to many other benchmarks and with respect to different dimensions such +as the reward or the gap to the optimal solver. +The continuous logging +and graphical depiction are visible in Figure 2a) and b). The example also +showcases our understanding of high code usability. The experiments could +be defined by changing training parameters (only a few lines in configuration +files) and minimal intended changes to the source code. Examples of the +most common changes which are intended to be coded are explained in more +detailed follow-along tutorials in the provided documentation. +10 + +wandb web interface +agent_training/entropy_loss +agent_training/policy_gradient_loss +agent_training/value_loss +a + still-river-17 +swept-blaze-15 + still-river-17 +- swept-blaze-15 + still-river-17 +swept-blaze-15 +400 +0.1 +0.08 +300 +agent_.training/loss +1.42 +0.06 +200 +0.04 +1.44 +100 +1.46 +runs.summary["Final Evaluation Table"] +Agent +Mean Rewal +Mean Tardin +Tardiness M +Mean Makesp +MakespanS +Tardiness ST +Gap To Solv +1 agent +-81.333 +35.167 +20.667 +81.333 +13.237 +53.108 +17.167 +Ganttchart +2 rand +-94.667 +10.949 +19.117 +30.5 +b +3EDD +-180.5 +15.575 +68.966 +116.333 +4 SPT +155.16 +24.423 +56.782 +91 +5 MTR +-76.333 +23.5 +17.833 +76.333 +11.146 +31.261 +12.167 +C +6 LTR +-187.333 +255.5 +93.5 +187.333 +18.436 +52.677 +123.167 +- 6 of 14 +Export as CSV Columns... +Reset Table +lidden Panels 5. Impact +schlably is useful for the entire community around PS with DRL. Com- +pared to other frameworks, it is particularly useful to reduce the entry barrier +for researchers from the OR or other related domains, who want to empir- +ically explore a new methodology for scheduling problems, and for DRL +researchers who want to test a new algorithm on a challenging and impactful +problem domain. We believe that the seamless interchangeability of prob- +lem settings offered by schlably will also encourage researchers in the domain +of PS with DRL to try out methodologies applied to one particular prob- +lem setting (e.g. 6x6 JSSP) on different problem settings (e.g. 11x11 tool- +constrained JSSP). This has the potential to greatly speed up the transfer of +research from academic problems to real-world problems. +In several projects where our test partners and we have used schlably, it +has significantly increased the throughput of experiments. This is achieved +because new methodological ideas can be integrated more quickly and the +results of experiments can be compared more easily. schlably facilitates the +generation of new problem instances and the training and evaluation of cus- +tom DRL agents. Due to the various pre-implementations in the framework, +such as training and testing routines, well-known scheduling benchmarks, +and visualization of logged results, it is much easier to conduct experimental +research in DRL for PS. In addition, collaboration has become more effec- +tive because design changes can be compared easily and the results of peers +can be viewed online through Weights&Biases. We have further experienced +a substantial increase in productivity in research projects, where new re- +searchers and university students, who had no prior domain knowledge and +little coding skills, had to conduct experiments on the PS domain. This, we +mainly attribute to the code documentation and modular structure, but also +to the fact that schlably is 100% written in Python and therefore runs on all +relevant operation systems. +6. Discussion and limitations +In its current state, schlably serves as a useful framework for empiri- +cal DRL-based PS research. It has reached a maturity level, at which it +works out-of-the-box and, to the best of our knowledge, offers the broadest +range of different easy-to-implement design choices compared to any pub- +lished framework. schlably, on the one hand, is intended to be abstract and +modular enough to offer different instance generation, training, and testing +configurations without many lines of code. On the other hand, it is designed +to not be too interwoven in its code structure to hinder the extension with +11 + +fundamentally different features experts might find desirable. As such, the +development required a balancing act and certain compromises, which some +may see as limitations. +For example, one deliberate choice was made in +favor of a class-based problem description as opposed to a vector representa- +tion. The class-based description simplifies the search and usage of certain +information about the current state of jobs and increases code readability +compared to a vector problem representation. Hence, the choice was made +between readability and computational efficiency in favor of the former. +7. Conclusions +In this paper, we introduced schlably, a software framework for research +on DRL-based PS. With the release of the framework, we strive towards +two main goals: the first is to lower the entry barrier for researchers, who +have little experience with production scheduling, deep reinforcement learn- +ing (DRL) and/or coding. The second goal is to encourage researchers al- +ready active in the field to apply and test their methods on other problem +settings, which is largely facilitated by schlably. Both goals aim at promoting +the transfer of DRL methods to real-world scheduling applications. In the +future, we plan to include more problem settings, such as the dynamic JSSP +and stochastic properties of environments like machine breakdowns to get +even closer to real-world scenarios. +8. Conflict of Interest +We wish to confirm that there are no known conflicts of interest associated +with this publication and there has been no significant financial support for +this work that could have influenced its outcome. +Acknowledgements +This research work was undertaken within the research project AlphaMES +funded by the German Federal Ministry for Economic Affairs and Climate +Action (BMWK). +References +[1] M. Pinedo, Scheduling: +Theory, algorithms, and systems, fifth edi- +tion Edition, Springer International Publishing, 2016. doi:10.1007/ +978-3-319-26580-3. +12 + +[2] I. Bello, H. Pham, Q. Le V, M. Norouzi, S. Bengio, Neural combinatorial +optimization with reinforcement learning (2016). doi:10.48550/ARXIV. +1611.09940. +URL http://arxiv.org/pdf/1611.09940v3 +[3] C. Zhang, W. Song, Z. Cao, J. Zhang, P. S. Tan, X. Chi, Learning +to dispatch for job shop scheduling via deep reinforcement learning, +Advances in Neural Information Processing Systems 33 (2020) 1621– +1632. +[4] A. Kuhnle, J.-P. Kaiser, F. Theiß, N. Stricker, G. Lanza, Designing +an adaptive production control system using reinforcement learning, +Journal of Intelligent Manufacturing 32 (44) (2021) 855–876. +doi: +10.1007/s10845-020-01612-y. +[5] T. van Ekeris, R. Meyes, T. Meisen, Discovering heuristics and meta- +heuristics for job shop scheduling from scratch via deep reinforcement +learning, Proceedings of the Conference on Production Systems and Lo- +gistics : CPSL 2021 1 (2021) 709–718. doi:10.15488/11231. +[6] V. Samsonov, K. B. Hicham, T. Meisen, Reinforcement learning in man- +ufacturing control: Baselines, challenges and ways forward, Engineering +Applications of Artificial Intelligence vol. C (112) (2022). +[7] C. W. de Puiseau, R. Meyes, T. Meisen, On reliability of reinforcement +learning based production scheduling systems: a comparative survey, +Journal of Intelligent Manufacturing 33 (4) (2022) 911–927. doi:10. +1007/s10845-022-01915-2. +[8] Sebastian Pol, Schirin Baer, Danielle Turner, Vladimir Samsonov, To- +bias Meisen, Global reward design for cooperative agents to achieve +flexible production control under real-time constraints, in: Proceedings +of the 23rd International Conference on Enterprise Information Sys- +tems - Volume 1: ICEIS„ INSTICC, SciTePress, 2021, pp. 515–526. +doi:10.5220/0010455805150526. +[9] R. S. Sutton, A. Barto, Reinforcement learning: An introduction, second +edition Edition, Adaptive computation and machine learning, The MIT +Press, Cambridge, Massachusetts, London, England, 2018. +[10] A. Rinciog, C. Mieth, P. M. Scheikl, A. Meyer, Sheet-metal produc- +tion scheduling using alphago zero, Proceedings of the Conference on +Production Systems and Logistics : +CPSL 2020 1 (2020) 342–352. +doi:10.15488/9676. +13 + +[11] M. Monaci, V. Agasucci, G. Grani, An actor-critic algorithm with deep +double recurrent agents to solve the job shop scheduling problem (2021). +doi:10.48550/ARXIV.2110.09076. +URL https://arxiv.org/pdf/2110.09076 +[12] S. Luo, Dynamic scheduling for flexible job shop with new job insertions +by deep reinforcement learning, Applied Soft Computing 91 (2020). doi: +10.1016/j.asoc.2020.106208. +[13] Z. Iklassov, D. Medvedev, R. Solozabal, M. Takac, Learning to gener- +alize dispatching rules on the job shop scheduling, Advances in Neural +Information Processing Systems 33 (2020) 1621–1632. doi:10.48550/ +ARXIV.2206.04423. +URL https://arxiv.org/pdf/2206.04423 +[14] A. H. Sakr, A. Aboelhassan, S. Yacout, S. Bassetto, Simulation and +deep reinforcement learning for adaptive dispatching in semiconductor +manufacturing systems, Journal of Intelligent Manufacturing 1 (2021) +1–14. doi:10.1007/s10845-021-01851-7. +URL +https://link.springer.com/article/10.1007/ +s10845-021-01851-7 +[15] P. C. Luo, H. Q. Xiong, B. W. Zhang, J. Y. Peng, Z. F. Xiong, Multi- +resource constrained dynamic workshop scheduling based on proximal +policy optimisation, International journal of production research 60 (19) +(2022) 5937–5955. doi:10.1080/00207543.2021.1975057. +[16] P. +Tassel, +M. +Gebser, +K. +Schekotihin, +Job_shop_scheduling_problem_with_reinforcement_learning, +GitHub (2021). +URL +https://github.com/dmksjfl/Job_Shop_Scheduling_ +Problem_with_Reinforcement_Learning +[17] L. Zheng, L. Zijun, Y. Dai, X. Li, B. Yuan, Gymjsp, GitHub (2022). +URL https://github.com/yunhui1998/gymjsp +[18] Dr-ilyassPHx, Auto-rl-competition: Dynamic job shop scheduling prob- +lem challenge, GitHub (2022). +URL https://github.com/Dr-ilyassPHx/Auto-RL-Competition +[19] P. Tassel, P. Willms, Jssenv: An openai gym environment for the job +shop scheduling problem., GitHub (2022). +URL https://github.com/prosysscience/JSSEnv +14 + +[20] V. Samsonov, optimization-with-rl-in-manufacturing-control, GitHub +(2021). +URL +https://github.com/v-samsonov/ +optimization-with-rl-in-manufacturing-control +[21] tejaswini medi, Rl_scheduling_system, GitHub (2022). +URL https://github.com/tejaswini-medi/RL_scheduling_system +[22] D. Venturelli, D. Marchand, G. Rojo, job-shop-scheduling: Determine a +schedule for running a set of jobs., GitHub (2015). +URL https://github.com/dwave-examples/job-shop-scheduling +[23] samy barrech, Flexible-job-shop-scheduling-problem, GitHub (2018). +URL +https://github.com/samy-barrech/ +Flexible-Job-Shop-Scheduling-Problem +[24] C. Zhang, W. Song, Z. Cao, J. Zhan, P. Tan, X. Chi, L2d: Official +implementation of paper "learning to dispatch for job shop scheduling +via deep reinforcement learning", GitHub (2020). +URL https://github.com/zcaicaros/L2D +[25] V. Kumar, Jobschedulingrlenv: +Reinforcement learning environment +for job scheduling written in python., GitHub (2019). +URL +https://github.com/TimeTraveller-San/ +JobSchedulingRLenv +[26] T. van Ekeris, jobshop: Deep reinforcement learning (drl) for jobshop +scheduling problems (jsp) - an evaluation framework, GitLab (2020). +URL https://gitlab.com/tvanekeris/jobshop +[27] A. Raffin, A. Hill, A. Gleave, A. Kanervisto, M. Ernestus, N. Dor- +mann, Stable-baselines3: Reliable reinforcement learning implementa- +tions, Journal of Machine Learning Research 22 (268) (2021) 1–8. +URL http://jmlr.org/papers/v22/20-1364.html +[28] Eric Liang, Richard Liaw, Robert Nishihara, Philipp Moritz, Roy Fox, +Ken Goldberg, Joseph Gonzalez, Michael Jordan, Ion Stoica, Rllib: Ab- +stractions for distributed reinforcement learning, International Confer- +ence on Machine Learning (2018) 3053–3062. +URL https://proceedings.mlr.press/v80/liang18b.html +[29] G. Brockman, V. Cheung, L. Pettersson, J. Schneider, J. Schulman, +J. Tang, W. Zaremba, Openai gym (2016). arXiv:arXiv:1606.01540. +15 + +[30] L. Biewald, Experiment tracking with weights and biases (2020). +URL https://www.wandb.com/ +Current executable software version +Ancillary data table required for sub version of the executable software: +(x.1, x.2 etc.) +kindly replace examples in right column with the correct +information about your executables, and leave the left column as it is. +Nr. +(Executable) +software +meta- +data description +Please fill in this column +S1 +Current software version +v0.1.0 +S2 +Permanent link to executables of +this version +https://github.com/tmdt- +buw/schlably +S3 +Legal Software License +Apache License, 2.0 (Apache-2.0) +S4 +Computing +platforms/Operating +Systems +Python, +OpenAI +Gym, +DRL- +lib,Weights and Biases +S5 +Installation requirements & depen- +dencies +Python 3.10 +S6 +If available, link to user manual - if +formally published include a refer- +ence to the publication in the refer- +ence list +https://github.com/tmdt- +buw/schlably/docs +S7 +Support email for questions +schlably@uni-wuppertal.de +Table 3: Software metadata (optional) +16 + diff --git a/0NE2T4oBgHgl3EQf4gjD/content/tmp_files/load_file.txt b/0NE2T4oBgHgl3EQf4gjD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e4fa02c34b8683b7b31b8fad09bc49dea4947bdd --- /dev/null +++ b/0NE2T4oBgHgl3EQf4gjD/content/tmp_files/load_file.txt @@ -0,0 +1,772 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf,len=771 +page_content='schlably: A Python Framework for Deep Reinforcement Learning Based Scheduling Experiments Constantin Waubert de Puiseau, Jannik Peters, Christian Dörpelkus, Tobias Meisen Institute for Technologies and Management of the Digital Transformation University of Wuppertal Rainer-Gruenter-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' 21, Wuppertal, 42119, NRW, Germany Abstract Research on deep reinforcement learning (DRL) based production schedul- ing (PS) has gained a lot of attention in recent years, primarily due to the high demand for optimizing scheduling problems in diverse industry settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Numerous studies are carried out and published as stand-alone experiments that often vary only slightly with respect to problem setups and solution approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' The programmatic core of these experiments is typically very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Despite this fact, no standardized and resilient framework for ex- perimentation on PS problems with DRL algorithms could be established so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' In this paper, we introduce schlably, a Python-based framework that provides researchers a comprehensive toolset to facilitate the development of PS solution strategies based on DRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' schlably eliminates the redundant overhead work that the creation of a sturdy and flexible backbone requires and increases the comparability and reusability of conducted research work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Keywords: Scheduling, Reinforcement Learning, JSSP, Framework Preprint submitted to SoftwareX January 12, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='04182v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='LG] 10 Jan 2023 Required Metadata Current code version Nr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Code metadata description Please fill in this column C1 Current code version v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='0 C2 Permanent link to code/repository used for this code version https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='com/tmdt- buw/schlably C4 Legal Code License Apache License, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='0 (Apache-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='0) C5 Code versioning system used git C6 Software code languages, tools, and services used Python, OpenAI Gym, RLlib, Weights and Biases C7 Compilation requirements, operat- ing environments & dependencies Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='10 C8 If available Link to developer docu- mentation/manual https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='com/tmdt- buw/schlably/tree/main/docs C9 Support email for questions schlably@uni-wuppertal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='de Table 1: Code metadata (mandatory) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Motivation and Significance Production scheduling (PS) is a challenging and highly researched prob- lem in operations research (OR) and optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' It deals with allocating resources to production jobs over time to minimize criteria such as time, effort, and cost [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' PS problems are considered NP-hard and therefore re- quire much computation effort to be solved sufficiently well with increasing problem sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' In recent years, with increasingly powerful algorithms and computational hardware, deep reinforcement learning (DRL) has emerged as a promising tool to address PS problems [2, 3, 4, 5, 6, 7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' DRL is a ma- chine learning paradigm, in which deep learning models are trained through interaction with an environment to autonomously derive solution strategies for sequential tasks [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' The young research field of DRL-based PS lies at the intersection of op- erations research and artificial intelligence and as such is characterized by a heterogeneous community with varying problem-solving approaches and technical skill sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Yet, all empirical studies apply a very similar experi- mental setup consisting of the same main software components: An envi- ronment representing the production facility layout and logic, a scheduling problem generator, a DRL agent algorithm, and logging as well as evalua- tion tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' The difference between different experimental setups usually lies 2 within one or more of these components, for example by incorporating a new DRL algorithm [10, 11, 12], interaction logic between agent and environ- ment [6, 3], training procedure [13], learning objective [14, 12] or additional problem constraint [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Regardless of large overlaps, all researchers im- plement their own individual experimentation framework with the following two consequences: Large initial ramp-up efforts when experimenting with new methodologies or custom problem settings, and scarcity of empirical comparisons to other works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' In this paper, we address these shortcomings and present the software framework schlably for developing and evaluating DRL-based PS solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' schlably provides the following contributions: It is modular, so individual changes may be adapted without much overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' It works out of the box with functioning environments, data-generation scripts, agents, logging functions, training, and testing scripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' It provides benchmark datasets for different scheduling problem classes and sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' It includes widely recognized benchmark algorithms and results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' It facilitates the application of algorithms designed for one problem class and size to other problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' schlably will accelerate the research area of DRL-based PS under real- world conditions by lowering the barrier of entry for researchers from different domains with different perspectives, levels of expertise, and objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Background and Related Work schlably started as a code base for experiments on a real-world inspired scheduling problem in the context of a university research project with in- dustrial partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' As such, several requirements became apparent early on that can be summarized in four general design goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' First, from the ap- plied industrial perspective, schlably has to offer the integration of DRL methods and heuristics which work out of the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Second, it also has to cover different scheduling scenarios, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' including such bounded by resource constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Third, from the scientific research perspective, schlably should support detailed comparable evaluations of methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Lastly, it has to be easy to interact with at the code level, to enable students with limited expe- rience to quickly understand the topic, concepts, and implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' The implications of these design goals are discussed in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' 3 In the following, we present an overview of related published experi- mental frameworks and compare them to our design goals in schlably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' In the comparison, we include frameworks dedicated to being used by others [16, 17, 18, 19, 20] and frameworks published in a supplementary fashion along with research papers and projects [21, 22, 23, 24, 25, 26], because in practice both may serve as starting points for additional experiment designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' The frameworks were found via references in scientific publications and a search of "Scheduling Reinforcement Learning" on GitHub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' We do not claim the list to be exhaustive but are not aware of any other popular frameworks at the time of writing this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Commercial or proprietary scheduling soft- ware was excluded because license fees and other accompanying challenges introduce a major barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' However, we are not aware of any commercial software that provides the tight integration of DRL and PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Table 2 provides an overview of the frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' In addition, we assessed them regarding their fulfillment of our design goals which are formally categorized into the following four groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Pre-Implemented Benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Several frameworks either provide pre- implemented DRL agents and scripts for training or the easy integration of agents from popular DRL libraries, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' from StableBaselines [27] or RLlib [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Like [20], our goal is to enable and facilitate both, the manual extensions of basic DRL algorithms, like DQN and PPO, as well as the usage of powerful third-party libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Both options are important to empower users to choose the appropriate approach for their respective research interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Additionally, for the sake of comparability, it is crucial to provide common benchmarks in the form of popular priority dispatching rules (PDRs), such as Shortest- Processing-Time-First, and a flexible optimal solver that can handle several scheduling problem types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Most frameworks only cover a few PDRs, often missing competitive ones, such as Most-Tasks-Remaining and even random baselines, where agent actions are sampled from a normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Scheduling Instance Generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Generating new data with varying prob- lem cases is necessary to enable comprehensive training and testing of a DRL agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Accordingly, a suitable framework must implement a flexible problem instance generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' This generator enables the user to create scheduling prob- lems of different popular categories (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' the Job Shop Scheduling Problem (JSSP) or the Flexible Job Shop Scheduling Problem (FJSSP) [1]) instances with any combination of instance variables, like the number of jobs, number of tasks, runtimes, and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Moreover, its design simplifies the integration 4 [16] [17] [18] [21] [19] [20] [22] [23] [24] [25] [26] schlably B Implemented RL-agents � � � � � � � � � � � � Implemented PDRs � � � � � � � � � � � � Implemented opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' solver ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='User manual in Readme ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='OpenAI Gym Env ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='Legend: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='�not fulfilled ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='�half-fulfilled ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='�fulfilled ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='Table 2: Overview of related frameworks and their fulfillment regarding pre- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='implemented benchmarks (B),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' scheduling instances (S),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' logging and evalua- tion (L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' and code usability (C) 5 of additional scheduling problem types to encourage the implementation of individual,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' more complex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' or more specific use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Finally, to the best of our knowledge, this is the first framework providing an optional resource constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Concretely, the user is able to specify a required tool per opera- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Logging and Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Logging results and evaluation metrics in a struc- tured manner is key for quick feedback during training runs, but also for identifying patterns in and drawing conclusions from large-scale experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Our objective with schlably is to provide extensive logging options that may be turned on and off, and where results and models may be shared to promote collaboration on projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' This design goal is met to a large degree by [20] from which we took much inspiration in this regard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' All other frameworks do not address this design goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' For the evaluation of solutions generated by DRL agents, a comprehensive framework should apply the benchmarks mentioned above and provide an overview of the overall performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Most of the reviewed frameworks lack functionality in this aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Moreover, to get a graphical overview and to visually support the tracking of very specific actions in a production schedule, a Gantt chart plotter is useful for human inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' The Gantt chart should display all metadata of operations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' the runtime or required tool, and has been found to be helpful for debugging and evaluating DRL agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Many other frameworks, but not all, include a Gantt chart plotter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Code Usability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' As usability is of utmost importance, a framework has to offer easy access through full documentation and must include a README, user application programming interface (API) manual, and formal functional- ity description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Within the reviewed frameworks, only [20] covers all criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' To be usable with different skill sets our explicit goal is to enable users to start experimenting with small design decisions by only using the configura- tion files but at the same time facilitate substantial logical changes to routines and components by means of their own implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' For that reason, we would favor comprehensibility over efficiency wherever a trade-off is unavoid- able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' This requires a careful balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' In our opinion, all other frameworks overemphasize one side: [20] offers many functional changes through configu- ration files but at the cost of a comparatively complex software architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' On the other hand, all other frameworks are much smaller and easier to get an overview of, at the expense of limited functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Lastly, a framework with a claim to widespread use should stick to conventional APIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' In the 6 context of DRL, the most commonly used API is the OpenAI Gym [29] API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Only half of the reviewed frameworks adhere to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Software Architecture This chapter describes our framework with focus on implementation- specific details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' We are providing a general overview of the code itself, fo- cusing on currently existing exemplary implementations while also pointing out open interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Furthermore, we describe details regarding the main components of schlably to demonstrate the realization of the design goals, as introduced in Chapter 2, and to enable users to fit schlably to their needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' The overall structure is illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' We divided the code base into six main components, which are described below in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Following this component-oriented approach, and in combination with comprehensive code documentation, schlably adheres to the objective of the fourth design goal, which requires easy interaction and usability at the code level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='agents ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='code_tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='environments ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='utils ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='visuals_generator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='data_generator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='GanttChartPlotter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='+ VISUALS_DIRECTORY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='+ get_gantt_chart_image ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='+ get_gantt_chart_gif_and_save ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='EvaluationHandler ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='+ rewards ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='+ makespan ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='+ record_environment_episode ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='+ evaluate_test ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='Logger ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='+ WANDB_PROJECT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='+ LOG_MODE_DEFAULT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='+ record ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='+ dump ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='+ dump_wandb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='ui_tools ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='file_handler ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='ConfigHandler ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='DataHandler ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='FileChooser ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='ProgressBar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='MessageBox ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='agent_tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='data_generator_tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='visuals_generator_tests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='Runner ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='Env ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='+ num_jobs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='+ num_machines ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='+ reset ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='+ step ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='EnvIndirectAction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='+ get_action_mask ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='EnvironmentLoader ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='+ ENVIRONMENT_MAPPER_DICT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='+ load ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='+ check_environment_agent_compatibility ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='InstanceFactory ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='SPFactory ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='+ SP(Enum) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='+ generate_instances ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='Task ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='+ job_index ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='+ task_index ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='heuristic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='reinforcement_learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='solver ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='dqn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='ppo ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='ppo_masked ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='intermediate_test ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='test ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='train ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='Figure 1: Overview of schlably project and code structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' 7 Data generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' The general data structure of scheduling problems, as used in schlably, is represented by so-called instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' A user can generate infinite instances of a scheduling problem, however, each instance is a spe- cific configuration and entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' The specific configuration, contained within an instance, is given by a number of jobs, with a job simply being an encom- passing logical container consisting of individual tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' The data_generator component incorporates the necessary classes to generate such an instance and the individual tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' From the scheduling problem point-of-view, it is the centerpiece of the problem formulation and representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' The Task class is a specifically designed data class and its entities are the atomic units of a scheduling problem instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Such an instance can be created via the SPFactory, which allows the generation of different types of schedul- ing problems that are given via the included Enum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' If users would like to introduce a new type of scheduling problem into schlably, they would have to include their function in this class and add it to the Enum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Finally, the InstanceFactory enables high-level access to the problem factory class and manages the configuration-based creation of batches of instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Thus, the data_generator component realizes the foundation of the second design goal, which requires the implementation and handling of different scheduling scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' An environment defines the observation space, action space, and reward strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Thus, it represents a simulation of the agent’s en- vironment and interaction dynamics and is the central piece of any DRL approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' All schlably environments are included in the environments com- ponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Exemplary, we provide a simple scheduling Env as well as a derived version named EnvIndirectAction to showcase the expandability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' All en- vironments adhere to the Gym API and are explicitly derived from a base Gym environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' The EnvironmentLoader class enables high-level access and management of the different environment types and appropriate algo- rithms, as not all algorithms are feasible for every environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' New envi- ronments have to be included in this component and added to the managing EnvironmentLoader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' This encapsulated approach, in conjunction with the data_generator component, represents the implementation of the second design goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' The agents component combines the heuristic functionalities, the solver, and implementations of DRL algorithms as well as the train and test functions for the DRL approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Users can to integrate functionalities from 8 other DRL frameworks, like more extensive training procedures, model types, and learning algorithms via pre-defined interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' As such, the agents com- ponent realizes the first design goal, to support and simplify the integration of out-of-the-box-methods as well as pre-implemented benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Visual generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' The component visuals_generator incorporates all classes and scripts which are used to create visualizations of the problem instances and generated solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' These functionalities are intentionally isolated as different scheduling problem environments and still share the same visualization approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' schlably, for example, introduces a GanttChartPlotter that enables a user to generate individual Gantt chart images (see Figure 2b) or create a GIF of the scheduling progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Thus, it is part of the implemen- tation of the third design goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Utils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' The utils component aggregates classes and functions which have a supporting character for the main functionalities of schlably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Specifically, it includes user interface components (ui_tools), data interface components (file_handler), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' to load and save data, and the high-level Logger class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Accordingly, the utils component realizes the third design goal, facilitating logging and evaluations for comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Code tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' All code tests that ensure the crucial functionality of the de- scribed components are collected in the code_tests component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Up to this point, we included multiple unit tests with a central Runner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' These are also intended as an example for users that plan to extend the code base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Illustrative Example To illustrate a typical use case, we consider a scenario in which an ML engineer wants to compare the learning behavior of two PPO agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' It is also part of our tutorial in the documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' One agent is trained on 6x6 JSSP instances and receives a reward based on the change in the time to complete all tasks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' the makespan) per step, as proposed in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' This setup is also the default setting delivered in the framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' The other one is trained on a 3x4 tool-constrained JSSP instance and receives a zero reward per step with the exception of the last step, where the reward is equal to the overall achieved makespan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' The remaining training parameters are kept constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' The second training requires only minimal manual changes to the base model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' These include setting different configuration parameters, generating new data, and 9 Figure 2: Comparing agent runs in Weights&Biases (screenshot from the web interface shown on the left-hand side).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' a) Visualized training curves for interpreting the learning performance of the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' b) Gantt chart depicting the solution of the trained agent on a selected test instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' c) Table providing evaluation results and comparison of the trained agents and benchmark methods on the test instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' changing the reward function in the base environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Details may be found in the documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' The integrated interface to Weights&Biases [30] makes it easy to compare the training curves and achieved results, as depicted in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' The described short example reflects several of our design goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Figure 2c) demonstrates that the agents’ performance is automatically compared to many other benchmarks and with respect to different dimensions such as the reward or the gap to the optimal solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' The continuous logging and graphical depiction are visible in Figure 2a) and b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' The example also showcases our understanding of high code usability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' The experiments could be defined by changing training parameters (only a few lines in configuration files) and minimal intended changes to the source code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Examples of the most common changes which are intended to be coded are explained in more detailed follow-along tutorials in the provided documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' 10 wandb web interface agent_training/entropy_loss agent_training/policy_gradient_loss agent_training/value_loss a still-river-17 swept-blaze-15 still-river-17 swept-blaze-15 still-river-17 swept-blaze-15 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='08 300 agent_.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='training/loss 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='06 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='44 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='46 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='summary["Final Evaluation Table"] Agent Mean Rewal Mean Tardin Tardiness M Mean Makesp MakespanS Tardiness ST Gap To Solv 1 agent 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='333 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='167 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='667 81.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='167 6 of 14 Export as CSV Columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Reset Table lidden Panels 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Impact schlably is useful for the entire community around PS with DRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Com- pared to other frameworks, it is particularly useful to reduce the entry barrier for researchers from the OR or other related domains, who want to empir- ically explore a new methodology for scheduling problems, and for DRL researchers who want to test a new algorithm on a challenging and impactful problem domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' We believe that the seamless interchangeability of prob- lem settings offered by schlably will also encourage researchers in the domain of PS with DRL to try out methodologies applied to one particular prob- lem setting (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' 6x6 JSSP) on different problem settings (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' 11x11 tool- constrained JSSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' This has the potential to greatly speed up the transfer of research from academic problems to real-world problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' In several projects where our test partners and we have used schlably, it has significantly increased the throughput of experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' This is achieved because new methodological ideas can be integrated more quickly and the results of experiments can be compared more easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' schlably facilitates the generation of new problem instances and the training and evaluation of cus- tom DRL agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Due to the various pre-implementations in the framework, such as training and testing routines, well-known scheduling benchmarks, and visualization of logged results, it is much easier to conduct experimental research in DRL for PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' In addition, collaboration has become more effec- tive because design changes can be compared easily and the results of peers can be viewed online through Weights&Biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' We have further experienced a substantial increase in productivity in research projects, where new re- searchers and university students, who had no prior domain knowledge and little coding skills, had to conduct experiments on the PS domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' This, we mainly attribute to the code documentation and modular structure, but also to the fact that schlably is 100% written in Python and therefore runs on all relevant operation systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Discussion and limitations In its current state, schlably serves as a useful framework for empiri- cal DRL-based PS research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' It has reached a maturity level, at which it works out-of-the-box and, to the best of our knowledge, offers the broadest range of different easy-to-implement design choices compared to any pub- lished framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' schlably, on the one hand, is intended to be abstract and modular enough to offer different instance generation, training, and testing configurations without many lines of code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' On the other hand, it is designed to not be too interwoven in its code structure to hinder the extension with 11 fundamentally different features experts might find desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' As such, the development required a balancing act and certain compromises, which some may see as limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' For example, one deliberate choice was made in favor of a class-based problem description as opposed to a vector representa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' The class-based description simplifies the search and usage of certain information about the current state of jobs and increases code readability compared to a vector problem representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Hence, the choice was made between readability and computational efficiency in favor of the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Conclusions In this paper, we introduced schlably, a software framework for research on DRL-based PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' With the release of the framework, we strive towards two main goals: the first is to lower the entry barrier for researchers, who have little experience with production scheduling, deep reinforcement learn- ing (DRL) and/or coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' The second goal is to encourage researchers al- ready active in the field to apply and test their methods on other problem settings, which is largely facilitated by schlably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Both goals aim at promoting the transfer of DRL methods to real-world scheduling applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' In the future, we plan to include more problem settings, such as the dynamic JSSP and stochastic properties of environments like machine breakdowns to get even closer to real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Conflict of Interest We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Acknowledgements This research work was undertaken within the research project AlphaMES funded by the German Federal Ministry for Economic Affairs and Climate Action (BMWK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' References [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Pinedo, Scheduling: Theory, algorithms, and systems, fifth edi- tion Edition, Springer International Publishing, 2016.' metadata={'source': 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+page_content=' Brockman, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Cheung, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Pettersson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Schneider, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Schulman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Tang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Zaremba, Openai gym (2016).' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='com/ Current executable software version Ancillary data table required for sub version of the executable software: (x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='1, x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content='2 etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=') kindly replace examples in right column with the correct information about your executables, and leave the left column as it is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} +page_content=' Nr.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE2T4oBgHgl3EQf4gjD/content/2301.04182v1.pdf'} diff --git a/0dAzT4oBgHgl3EQfRPuW/content/tmp_files/2301.01213v1.pdf.txt b/0dAzT4oBgHgl3EQfRPuW/content/tmp_files/2301.01213v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9e58413168372237b357172e43267f3260b771e3 --- /dev/null +++ b/0dAzT4oBgHgl3EQfRPuW/content/tmp_files/2301.01213v1.pdf.txt @@ -0,0 +1,1369 @@ +arXiv:2301.01213v1 [physics.optics] 3 Jan 2023 +Numerical study of magneto-optical binding between two dipolar particles under +illumination by two counter-propagating waves +Ricardo Mart´ın Abraham-Ekeroth +Instituto de F´ısica Arroyo Seco, IFAS (UNCPBA), Tandil, Argentina and +CIFICEN (UNCPBA-CICPBA-CONICET), Grupo de Plasmas Densos, Pinto 399, 7000 Tandil, Argentina∗ +The formation of a stable magneto plasmonic dimer with THz resonances is theoretically studied +for the principal directions of the system. Unlike a recent report, our work provides a complete +description of the full photonic coupling for arbitrary magnetic fields as, for instance, unbalanced +particle spins. As an illustration, we consider two small, n-doped InSb nanoparticles under illumina- +tion by two counter-propagating plane waves. Remarkably, when an external magnetic field exists, +the symmetry in the system is broken, and a resonant radiation pressure for the dimer appears. +Similarly, tunable inter-particle forces and spins are exerted on the non-reciprocal dimer. The sys- +tem is also characterized when the magnetic field is absent. Moreover, we show how the mechanical +observables truly characterize the dimer since their resonance dependency contains detailed informa- +tion about the system. Unlike far-field observables like absorption, mechanical magnitudes depend +on the system’s near-field. In addition, the nature of the particle spins is originally explained by +the energy flow’s behavior around the dimer. This work constitutes a generalization of any previ- +ous approach to optical binding between small nanoparticles. It paves the way for fully controlling +optical matter and nano factory designs based on surface plasmon polaritons. +Keywords: +Magneto plasmonics, +Spin torques, +Dimers, Optical Binding, Photonics, Poynting field, +Radiation Pressure, Optical Matter +I. +INTRODUCTION +Optical matter (OM) consists of arrays of micro or +nanoparticles that are somehow bound and controlled +by light [10]. +An OM able to self-assemble at will +to develop solid technology is a long-standing goal in +photonics [38]. The background for OM is the particle +manipulation by optical forces; the first results were +applied to microparticles due to the lack of technology +and the presence of thermal noise for smaller systems +[4, 5]. +Generally, OM comprises multiple particles +subjected to electromagnetic forces that come from +their mutually scattered light. However, this multiple +scattering phenomena can be very complex and bring +about unusual effects such as “non-reciprocal” forces, +torque opposite to the illumination angular momen- +tum, and non-conservative forces [2, 38]. Appropri- +ate control of OM by forces and torques could lead to +programmable materials for optomechanical, rheologi- +cal, and biological applications. In this respect, many +works studied the optical binding between nanopar- +ticles as a primary tool to develop OM. Some ap- +proached specific combinations of optical beams like +those with programmable phase [24, 31]. For example, +light-induced rotation of objects holds potential for +various applications such as sensing, cargo transporta- +tion, drug delivery, and micro/nanosurgery [3, 7, 42]. +Optical traps use the combination of beams as a po- +tent characterization tool for material science and bio- +physics, as in Ref. [23], which uses electrostatic focus- +ing to obtain the mass spectrum of SARS-CoV-2 and +∗ mabraham@ifas.exa.unicen.edu.ar +BoHV-1 virions. However, the high intensity at the +focal spot may introduce laser heating, which is an +issue for bio applications [31]. +On the other hand, recent advances in THz tech- +nology call for new devices and materials that exhibit +a non-reciprocal behavior for photonic networks and +optical information processing [41]. +Non-reciprocal +devices are a crucial component of modern commu- +nication technology. They are nowadays required for +miniaturized electronic and photonic devices [32]. One +way to create optical non-reciprocity at the THz range +is using magneto-optical (MO) systems like graphene, +hexaferrites, and semiconductors [27]. For instance, +Ref. [15] presented MO measurement of several sam- +ples of InSb with different carriers and carrier con- +centrations for low external magnetic field and room +temperatures. +With these advantages, dimers and +trimers of InSb particles have been studied to enhance +THz spectroscopy by forming electric and magnetic +hotspots in the gap between them [6, 40]. Anisotropic +materials like InSb in OM would make it strongly +dependent on the beam combinations, allowing for +countless possibilities [39]. +Recently, several efforts +have focused on MO nanoparticle systems to shape +OM and optical traps with a reasonable degree of con- +trol and accuracy, besides other relevant applications +[1, 18, 20, 22, 26]. on a broader sense, magnetoplas- +monics relates the plasmonic behavior of nanoparticles +with the presence of external magnetic fields. +The +modulation and tunability of plasmonic resonances +offered by magnetoplasmonics results auspicious for +ultra-sensitive sensors and active plasmonic devices +[28]. +In particular, the formation of stable optical +binding between two small magnetoplasmonic parti- +cles has been lately studied [19]. Equilibrium binding +distances were predicted and found tunable by the +incoming wave’s polarization state and the magnetic +field’s magnitude. However, the model developed in +that report is valid only for relatively small magnetic + +2 +field values. Moreover, it predicts stable dimers only +using alternating magnetic static fields and polariza- +tion angles to remove azimuthal, unbalanced forces. +More importantly, this work needs to discuss possible +rotations of the particles due to angular momentum +transfer in the multiple scattering scheme. +In this paper, we study the formation of stable MO +dimers for small nanoparticles in a complete frame- +work involving all the possible optomechanical induc- +tions. The dimer’s isotropic and anisotropic responses +are assessed as a base for OM designs, i.e., under the +presence/absence of an external magnetic field. This +field can be of arbitrary magnitude in our model. The +illumination consists of two counter-propagating plane +waves with circular polarization, which simulates a +simple optical trap in the vacuum. +We found sev- +eral possibilities to create stable dimers even when +the magnetic field is off. The beams do not exert net +forces for reciprocal dimers but may exert torques on +them. On the contrary, there is a net radiation pres- +sure and spin for the whole system when the static +field is on, allowing complete control of the system’s +movement. The results will enable one to infer that +the mechanical variables can be used as near-field ob- +servables to explore the content of unknown samples. +Conversely, they can be used to accurately control the +dimer’s creation/destruction and its mobility. Finally, +the spins predicted are explained in terms of the en- +ergy flows around the dimer, which constitutes a novel +scattering-force effect for interacting particle arrays. +II. +MODEL +In the following, we assume two equal particles of +the same non-reciprocal material immersed in the vac- +uum. Then the method of discrete dipoles (MDD) [17] +simplifies considerably to +p1 = ǫ0ˆαE0,1 + k2 +0 ˆα ˆGp2, +(1) +p2 = ǫ0ˆαE0,2 + k2 +0 ˆα ˆGp1, +(2) +where ˆα is the polarizability tensor representing the +particles. The following definition automatically in- +cludes the radiative corrections necessary to fulfill the +optical theorem [21] +ˆα = +� +ˆα−1 +0 +− ik3 +0 ˆI +6π +�−1 +(3) +where ˆα0 is the so-called quasistatic polarizability, +which can be given by +ˆα−1 +0 += 1 +V +� +ˆL + [ˆǫr − ˆI]−1� +(4) +being V the particles’ volume, ˆL = ˆI/3 is the elec- +trostatic depolarization tensor specified for spheres or +cubes, and ˆǫr is the relative dielectric tensor. +The +system of equations 1 can be solved straightforwardly, +leading to +p1 = ǫ0 ˆF +� +E0,1 + k2 +0 ˆα ˆGˆαE0,2 +� +, +(5) +p2 = ǫ0 ˆF +� +E0,2 + k2 +0 ˆα ˆGˆαE0,1 +� +, +(6) +where we define ˆF = +� +ˆα − k4 +0 ˆGˆα ˆG +�−1 +. In this work, +a counter-propagating configuration is assumed as a +superposition of two left-handed circularly polarized +(LCP) plane waves with the same intensity I0 [11, 18], +see Fig. 1, namely, +E0 = E0 +√ +2 +� +(ˇx + iˇy) eik0z + (ˇx − iˇy) e−ik0z� +. +(7) +This field is used in Eq. 5 to calculate the incident +field at the particles’ positions E0,1 and E0,2. +The absorption cross section of the system can be +calculated once the dipole moments are known by +σabs = +k0 +ǫ0wtot +E +Im +� +p1 · +� +ˆα−1 +0 p1 +�∗ + p2 · +� +ˆα−1 +0 p2 +�∗� +(8) +where wtot +E += ǫ0|E0|2. The i-component of the forces +exerted on each particle can be obtained from the +time-averaged force within the Rayleigh approxima- +tion [13]. This is +F1,i = 1 +2Re{pt +1[∂iE∗(r, ω)|r=r1} +(9) +F2,i = 1 +2Re{pt +2[∂iE∗(r, ω)|r=r2} +(10) +where the derivatives of the total field ∂iE(r, ω)|r=rn +at the dipoles’ positions rn, n = {1, 2} can be obtained +from [12]: +∂iE(r, ω)|r=r1 = ∂iE0(r, ω)|r=r1+ ++ k2 +0 +ǫ0 +(∂iG(r, r2))r=r1p2]} +(11) +∂iE(r, ω)|r=r2 = ∂iE0(r, ω)|r=r2+ ++ k2 +0 +ǫ0 +(∂iG(r1, r))r=r2p1]} +(12) +The total force exerted on the dimer results from +adding the force components for each particle, namely, +Ftot,i = F1,i + F2,i. In particular, the net radiation +pressure for the dimer under the illumination given +by Eq. 7 is defined by taking i = 3, or the z compo- +nents, as +Ftot,z = F1,z + F2,z +(13) +Another useful mechanical variable is the binding +force, which in the present case is defined as +∆ = (F1 − F2) · ˇn +(14) +where ˇn = +r2−r1 +|r2−r1| is the dimer’s versor. The optical +torques can also be calculated, as given in Ref. [14]: +Nspin,1 = +1 +2ǫ0 +Re +� +p1 × +�� +ˆα−1 +0 +�∗ p∗ +1 +�� +(15) +Norb,1 = r1 × F1 +(16) +N1 = Nspin,1 + Norb,1 +(17) + +3 +FIG. 1. +(Color online) Dimer configurations and in- +cident +waves +treated +in +this +work. +Two +counter- +propagating waves with left circular polarization illumi- +nate the magneto-optical dimer. The leading example con- +sists of two n-doped InSb particles separated by a gap of +a particle’s diameter, g = 2R. (a) Parallel [(b) perperdic- +ular] configuration. The static magnetic field B is parallel +to +z direction (green arrow). In (b), φ is the azimuthal +angle of the dimer’s position. +Nspin,2 = +1 +2ǫ0 +Re +� +p2 × +�� +ˆα−1 +0 +�∗ p∗ +2 +�� +(18) +Norb,2 = r2 × F2 +(19) +N2 = Nspin,2 + Norb,2 +(20) +The definitions of the orbital and spin torques were +discussed previously in Refs. [14, 33, 34], among oth- +ers. The spin torques are always defined with respect +to the centers of the particles. Otherwise, the refer- +ence system is located at the dimer’s center of mass, +and orbital torques are set. Thus, the total torque +exerted on the dimer is +Norb = Norb,1 + Norb,2 +(21) +Nspin = Nspin,1 + Nspin,2 +(22) +Ntot = N1 + N2 = Norb + Nspin +(23) +In particular, this study simulates nanoparticles made +of n-doped Indium antimonide (n-InSb) [37]. +In- +dium antimonide (InSb) is one example of the most +widely studied polar semiconductors for magnetoplas- +monic applications because it can be easily doped +for sizable magnetic-induced effects [15, 30]. As re- +viewed in Refs. [30, 37], n-InSb is an exciting mate- +rial that has two kinds of surface resonances in the +absence of static field, namely, the phonon polari- +ton (SPhP, higher-energy) and the plasmon polariton +(SPP, lower-energy). +Its model properties were de- +scribed on Refs. [1, 18, 30], among others. Since we +are interested in the near-field interactions between +the particles, the study focuses on an example for +which the interparticle’s gap equals one particle di- +ameter (g = 2R); see Fig. 1. We add complementary +examples for other values of the gap in the Supple- +mentary Material (SM). +III. +RESULTS +In this section, all the optical variables were scaled +by the proper factors to make them adimensional. +The following characteristic magnitudes, namely, wtot +E , +Ap = πR2, Vp = +4 +3πR3, and Vint = +4 +3π (|r2 − r1|)3 +redefine the variables as Qabs = σabs +2Ap for the absorp- +tion efficiency, Frad = +Ftot,z +wtot +E Ap and ∆′ = +∆ +wtot +E Ap for +the radiation pressure and the binding forces, and +N′ +spin = +Nspin +wtot +E Vp for the spin torque. +The variable +N′ +orb = +Norb +wtot +E Vint for the orbital torque is only shown +by an example in the SM since it gave negligible re- +sults unless the gap is minimal, see Figs. S1 and S2 +for details. We calculate the scaled Poynting vector +as S = +1 +2I0 Re{E × H∗} where the magnetic field H +comes from an MDD equation similar to that for E +[36]. The curl of S is calculated using an appropriate +tridimensional mesh around the system’s near-field. +A. +Parallel Illumination. +Fig. 2 shows the spectral results for parallel illumi- +nation when the magnetic field is off (B = 0, black +line) and on (B = 1 T, red line with squares). The +absorption efficiencies, Fig. 2a, result independent of +the direction of the dimer so that the same spectra +remain for any other illumination configuration. The +low-energy resonance (around 73µm) corresponds to +an SPP, while the high-energy resonance (48.7µm) +corresponds to an SPhP [30, 37]. Making use of the +Plasmon Hybridization Model (PHM) for two dipo- +lar particles, both kinds of surface modes show as +bright antibonding modes for transverse electric fields +according to the configuration shown in Fig. 1a, see +Ref. [35] for details. +When B is on, each isotropic surface mode splits +into two modes due to degeneracy removal. The ab- +sorption is the only far-field observable shown in this +work since negligible scattering occurs for small sys- +tems [8]. As it is illumination-independent, the spec- +tra obtained remain invariant for all illumination di- +rections concerning the dimer’s axis. Thus, the only +available far-field observable is neither adequate to +study the interactions occurring in the dimer nor valu- +able to predict the dimer’s dynamics. In Fig. 2b, there +is no net radiation pressure when B is off due to the +high symmetry of both the system and incident field. +On the other hand, there is a resonant pressure for the +MO dimer when B is on, revealing the magnetoplas- +monic resonances and directing the dimer upwards or +downwards according to their energy. As a result of + +IKA +LCP +B +KB +B +Z +X +X +KA +1 +- +2R +g +LCP +kB +(a) +(b)4 +the interaction, the radiation pressure identifies the +modes by the sign of the force. Fig. 2c shows that +the binding force leads to repulsion between the par- +ticles for both cases, B = 0 and B = 1 T. In other +words, there cannot be a stable dimer under this par- +allel configuration. The response is still resonant but +less sensitive than the radiation pressure. +Remark- +ably, the results agree with the interpretation given +by the PHM. +Notably, although we are dealing with a dimer sys- +tem, our results agree with those reported in [18] +for a single particle under the same illumination. In +general, the absorption efficiency Qabs behaves like +Re{α11}, both the radiation pressure Frad and the +spin N ′ +spin,z on z behave like Im{α12}, and the bind- +ing force ∆′ resembles −Re{α11}, being αij the carte- +sian components of the polarizability tensor ˆα. +In +the case of the spins in Fig. 2d, these behave like +±Im{α33} for each particle respectively (polarizabil- +ities not shown here). These functional dependencies +are due exclusively to the type of illumination; oth- +erwise, other αij-terms would appear in the spectral +variables [18]. +Following +angular +momentum’s +conservation, +Fig. 2d shows that the net spin for the system is zero +when B is off (black line). +To put it another way, +the spins for each particle are equal and opposite, +showing the resonant modes for the isotropic case +(see red and blue lines with symbols). +When B is +on, however (Fig. 2e), there is a net spin for the +system, black line, which is twice the spin for each +particle (red line with squares). The spin resonates +sensitively with the dimer’s modes, quite like the +radiation pressure. Consequently, the spins become +much stronger than those for B off, compare the +scales of Fig. 2d and e. Thus, the radiation pressure +and spins constitute the most sensitive observables +in the near field, giving a common spectral shape +(compare spectra in Figs. 2b and e). +B. +Perpendicular Illumination. +Now we vary the dimer’s azimuthal angle since a de- +pendency on the net polarization is expected. Figs. 3 +and 4 show maps as a function of the incident wave- +length and azimuthal angle for B = 0 and B = 1 T, +respectively. Similarly to that found in the previous +subsection, there is no net radiation pressure when B +is off (not shown). Yet this time, a different behav- +ior is found for the binding force, Fig. 3a. The sys- +tem offers a resonant spectral response but depends on +the angle φ. Maxima [minima] of binding are found +around φ = 90, 270 [φ = 0, 180] deg, meaning inter- +particle attraction [repulsion]. This fact also defines +stable positions for the dimer around the strongest +optical resonance, namely the SPhP at 48.7µm, and +around φ = 34, 145.6, 214.5, 325.7 deg for all wave- +lengths when B is off (follow the black lines). A sim- +ilar situation is found for the second resonant wave- +length ≈ 72.6µm (SPP), where the variations are less +pronounced. Regarding the spin, Fig. 3b shows a re- +maining behavior for the whole system, which is res- +onant with the surface modes and coordinated with +the binding phenomenon. Remarkably, the spin gets +its extremals (maxima or minima) when the dimer +reaches its stable positions; namely, neither attraction +nor repulsion, compare Figs. 3a and b. As mentioned +above, the most sensitive resonance corresponds to the +excitation of the SPhP. +Noteworthy, our results are consistent with the in- +terpretation of the PHM for isotropic, dipolar par- +ticles [35]. +In particular, each value φ = nπ [φ = +(n + 1/2)π] rad with n ∈ Z, the binding force shows +repulsion [attraction] for both types of resonances, +namely, the SPhP and SPP, see Fig. 3a. This outcome +is due to the net polarization; the electric field is along +ˇy, see the map for φ = 0 at the SPhP wavelength in +Fig. 3c. Thus, φ = 0 corresponds to a transverse elec- +tric field compared with the dimer’s direction, mean- +ing an antibonding bright mode in the context of the +PHM. Differently, φ = 90 deg corresponds to a parallel +electric field compared with the dimer’s axis, meaning +a bright bonding mode in the PHM (map not shown). +In Fig. 4a, there is a remaining radiation pressure for +the whole system due to the symmetry breaking that +appears only at the resonances’ locations. This spec- +trum results invariant with φ and follows the same +resonances as in absorption in Fig. 2a when B is on +(red line with squares). Thus the presence of a static +magnetic field induces the dimer to move forward or +backward in the illumination’s direction when the in- +cident energy is that of a surface mode. +Likewise, +the system shows a resonant binding (Fig. 4b). As in +Fig. 3a, the black lines follow the values of zero force. +Note that the map strongly distorts by the presence +of the resonances when B is on, making the dynam- +ics more complex and even reducing the extremals of +the binding force. However, the possibility to obtain +stable binding enhances around the SPhP due to the +overlapping of MO modes, which means more degree +of control in the dimer’s creation and stability. +In Fig. 4c, the system’s spin follows a trend simi- +lar to that for the radiation pressure in Fig. 4a. This +behavior is quite different from that found for B = 0. +Note that spin is enhanced when B is on; the color- +bar limits show values ≈ 6.2 − 7.5 times higher than +those for B off, compare Figs. 3b and 4c. As a re- +sult, the MO system could be readily identified in an +experiment by observing the dimer’s dynamics at the +resonance wavelengths. +Fig. 4d shows the electric field around the dimer’s +plane z = 0 for the SPhP found at 49.85µm. This +wavelength corresponds to the most robust resonance +when B is on. The rest of the configuration is equal to +that given in Fig. 3c. The field hot spots are leaned on +the right ≈ 65 deg from the x axis by the MO effect. +Up to this point, we have explored a few examples +of MO dimers to approach the idea of controlling the +particle dynamics and ”photonic molecule” stability +[25] in the presence/absence of a static magnetic field +B. + +5 +FIG. 2. (Color online) Optical properties for parallel configuration. (a) Absorption efficiency, which is independent of +the dimer’s orientation. (b) Radiation pressure (total force along z). (c) Binding force. Black line [red line with squares] +for magnetic field B = 0 [B = 1] T. (d) [(e)] Spin torques for B = 0 [B = 1] T. In (d), the net spin torque is zero for all +wavelengths. +Below, we discuss the behavior of the dynamic ob- +servables in terms of the information contained in +the Poynting field. The reader is reminded that the +particles’ photonic interaction matches a multiple- +scattering framework [16, 29]. The near fields involve +the evanescent waves, which play a crucial role in the +particles’ interaction for surface modes and close par- +ticles. This phenomenon can be seen through the en- +ergy flows because they may have all the information +of the near fields E and H. +Generally, the magni- +tudes obtained from far-field calculations lose some of +the information about the system [36]. +C. +Nature of the spins through an examination +of the Poynting fields +We explore the spins exerted on the system by show- +ing a few calculations of the Poynting field around +the dimer for perpendicular configuration. The paral- +lel configuration is less interesting since it would only +lead to repulsion states without dimer formation for +any gap under both cases B = 0 and 1 T, see Fig. 2c +for the example g = 2R. More clarifications can be +found in the SM. +Figs. 5a-d [e-f] consist of maps related to the en- +ergy flow when B is off [on] upon different azimuthal +angles. The wavelengths coincide with that for the +strongest SPhP in each case. The left column (a-c-e) +shows the Poynting field S when z = 0. +Similarly, +the right column (b-d-f) shows the z-component of +∇ × S. The white arrows are rescaled to visualize the +maps easily. Interestingly, Figs. 5a-b show an exam- +ple of a repulsion state with zero spins when φ = 0, +see Figs. 3a and b. Even though S aligns in a single +direction, a resonant magnitude and a non-negligible +curl appear near the surface of the particles. This res- +onance is due to the excitation of the SPhP. However, +the contributions to the spin cancel out due to high +symmetry evidenced by these maps and zero net spin +results for the system. +Figs. 5c-d show the attrac- +tion state with maximum positive spin when φ = 135 +deg, see Figs. 3a and b. This time, two hot spots of +maximum magnitude face each other, and a kind of +saddle point emerges in the gap region between the +particles, Fig. 5c. As a result, the values of the curl +clearly show a rotational state for light as the field +spots have ”turbine-blade” shapes, Fig. 5d, explain- +ing the net positive spin calculated for the system in +Fig. 3b for φ = 135 deg. It is also evident from Fig. 5d +that the two particles have the same spin, visually +showing that the net spin is two times the spin of one +particle. Finally, we show the example when B is on +and φ = 67.2 deg, which coincides with the hot spot +of maximum attraction at the resonance 49.85µm of +the SPhP, see Fig. 4b. +Notice in passing that this +value for φ is close to the angle of the electric spots in +Fig. 4d, namely, ≈ 65 deg. Remarkably, Fig. 5e illus- +trates that the energy flow would make the particles +spin counterclockwise. Moreover, it is also notewor- +thy the vortex that appears in the field region between +the particles with clockwise orientation, resembling a +”gear” mechanism which coordinates field and parti- +cles [9, 21, 38]. Consistently, the curl’s map shows a + +2.0 +0 +4. +B=O T +B=O T +1.5 +11 +1.0 +×10-1 +3 +-2 +0.5 +1 0.0 +2 +Qabs +-0.5 +-1.0 +B=O T +(a) +(b) +(c) +-1.5 +1T +5 +-2.0 +20 +60 +80 +100 +20 +60 +80 +40 +40 +100 +20 +40 +60 +80 +100 +120 +120 +120 +Wavelength (μm) +Wavelength (μm) +Wavelength (μm) +5 +321 +之 +23 +system + system +-3 +45 +np. 1 = np. 2 +(d) +(e) +np. 2 +6- +20 +40 +60 +100 +120 +20 +40 +60 +80 +100 +80 +120 +Wavelength (μm) +Wavelength (μum)6 +FIG. 3. +(Color online) Near-field observables for per- +pendicular configuration in the absence of magnetic field, +B = 0 T. (a-b) Maps of the mechanical variables as a +function of wavelength and dimer’s azimuthal angle. (a) +Binding force. The black lines correspond to zero force. +(b) Spin torque for the system. In this case, the net radi- +ation pressure is zero for all wavelengths (not shown). (c) +Distribution of electric field around the dimer for z = 0 at +the resonance wavelength 48.85µm for φ = 0. +structure similar to that in Fig. 5d but this time en- +hanced and possessing a structure in the gap region +that contains the vortex indicated in Fig. 5e, see in- +set in Fig. 5f. The inset zooms this region so that a +connection between the curl hot spots is appreciated. +FIG. 4. (Color online) Near-field observables for perpen- +dicular configuration and B = 1 T. (a-c) Maps of the me- +chanical variables as a function of wavelength and dimer’s +azimuthal angle. (a) Radiation pressure for the system. +(b) Binding force. The black lines correspond to zero force. +(c) Spin torque for the system. (d) Distribution of electric +field around the dimer for z = 0 at the resonant wave- +length 49.85µm for φ = 0. + +360 +10.2 +(a) +315 +0.14 +270 +0.09 +225 +0.03 +180 +-0.02 +135 +-0.08 +90 +45 +-0.14 +0.18 +20 +30 +40 +50 +60 +70 +80 +90 +100 +120 +Wavelength (μm) +360 +0.79 +315 +(b) +270 +0.5 +225 +0.25 +135 +0 +90 +45 +0.15 +0.32 +20 +30 +40 +50 +60 +70 +80 +90 +100110 +120 +Wavelength(μum) +N' +spin,z +360 +(C) +6 +315 +4 +270 +225 +2 +0 +135 +90 +45 +-4 +-5 +0 +20 +30 +40 +50 +60 +7080 +90 +100 +110120 +Wavelength (μum) +[E +Eo +一 +1 +0.75 +6 +(d) +0.5 +5 +0.25 +(wn) +4 +0 +y +-0.25 +3 +-0.5 +2 +-0.75 +1 +-1-0.75-0.5-0.2500.25 0.5 0.75 +x (μum)△(-) +360 +1.2 +315 +1 +270 +0.7 +225 +0.5 +leg) +0. 3 +0180 +135 +106 +-0.2 +45 +-0.4 +0 +-0.6 +20 +30 +40 +50 +60 +70 +80 +90 +100 +110 +120 +Wavelength (μm) +360 +0.8 +315 +(b) +270 +0.4 +0 +135 +90 +-0.4 +45 +-0.8 +0 +20 +30 +40 +50 +60 +70 +80 +90 +100 +110 +120 +Wavelength (μm) +[E +Eo +(c) +0.75 +6 +0.5 +5 +0.25 +(un) +4 +0 +y +3 +-0.25 +-0.5 +2 +-0.75 +-1-0.75-0.5-0.25 00.25 0.50.75 +x(μm)7 +FIG. 5. Scaled energy flows around the dimer for z = 0 under perpendicular configuration and at the strongest SPhP. +The color scale corresponds to the magnitude; the white arrows show the Poynting flow (2× their original size). Left +[right] column for the [z-component of the curl of the] Poynting field. (a-d) [(e-f)] Examples for B = 0 [B = 1] T; the +wavelength is 48.85µm [49.85µm]. (a-b) For φ = 0 deg, (c-d) φ = 135 deg, and (e-f) 67.2 deg. The inset in (f) zooms the +gap region up to a maximum of 0.05 in the colorbar. + +[Sxy] +2RI(V × S)zl +1o +o +0.8 +0.8 +0.4 +6 +(b) +(a) +0.6 +0.6 +5 +0.4 +0.4 +0.3 +00 +4 +0.2 +0.2 +(wn +0 +0 +0.2 +-0.2 +2 +-0.4 +0.1 +-0.4 +-0.6 +-0.6 +-0.8 +-0.8 +0 +-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 +-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 +x (μm) +x (μm) +0.8 +0.8 +0.4 +(d) +0.6 +0.6 +6 +0.4 +0.4 +0.3 +5 +0.2 +0.2 +(wn) +(wn) +0 +0.2 +0 +3 +y +-0.2 +2 +-0.4 +0.1 +-0.4 +-0.6 +-0.6 +-0.8 +0 +-0.8 +0 +-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 +-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 +x (μm) +x (um) +0.8 +0.8 +4 +(f) +e) +0.6 +0.6 +0.8 +0.4 +0.4 +3 +0.2 +0.2 +0.6 +(wn) +(wn) +0 +0 +0.4 +-0.2 +-0.4 +-0.4 +0.2 +-0.6 +-0.6 +-0.8 +-0.8 +-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 +-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 +x (μum) +x (μum)8 +IV. +CONCLUSIONS +By a simple dipolar model, this work explores the +behavior of small nanoparticle dimers when magneto- +optical materials like n-doped InSb and moderate +magnetic fields are used. +Two counter-propagating +waves with equal circular polarization are used as il- +lumination to simulate a simple optical trap with nei- +ther net gradient nor scattering forces. Our results +show that the system can be thoroughly character- +ized by observing its mechanical inductions, provided +these latter depend on the near field. +Besides, we +found no possibility of forming stable dimers when +the dimer is aligned with the illumination since the +inter-particle force only leads to repulsion. +On the +contrary, under ”perpendicular” alignment, there are +several ways to obtain stable dimers or inter-particle +attraction, at least under this ”static” model for which +the particles’ velocities and accelerations are not con- +sidered. As the results strongly depend on the mag- +netic field’s presence, this study constitutes a novel +background to build OM or photonic molecule nano +factories and control their movements, or conversely, +to study samples containing this class of multimers. +Finally, we give an original explanation for the ap- +pearance of particle spins based on the energy flows. +This interpretation offers satisfactory results from the +”scattering” forces produced by the interaction be- +tween the particles. Gradient forces were also inves- +tigated but showed no appreciable influence on the +spins’ appearances (results not shown in this work). +ACKNOWLEDGMENTS +The author would like to thank A. Garc´ıa-Mart´ın +from IMN-CSIC and M.I. Marqu´es from Universidad +Aut´onoma de Madrid for their valuable discussions +during his postdoc in Spain. +CONFLICTS OF INTEREST +The authors declare that the research was con- +ducted in the absence of any commercial or financial +relationships that could be construed as a potential +conflict of interest. +[1] Abraham Ekeroth, R. M., Garc´ıa-Mart´ın, A., and +Cuevas, J. C. (2017). +Thermal discrete dipole ap- +proximation for the description of thermal emis- +sion and radiative heat transfer of magneto-optical +systems. +Phys. +Rev. +B +95, +235428. +doi: +10.1103/PhysRevB.95.235428 +[2] Albaladejo, +S., +Marqu´es, +M. +I., +Laroche, +M., +and S´aenz, J. J. (2009). +Scattering Forces from +the Curl of the Spin Angular Momentum of a +Light Field. +Phys. Rev. Lett. 102, 113602. +doi: +10.1103/PhysRevLett.102.113602 +[3] Arita, Y., Simpson, S. H., Zem´anek, P., and Dholakia, +K. (2020). 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Optical Forces: From Fundamental +to Biological Applications. +Advanced Materials 32, +2001994. doi:10.1002/adma.202001994 + diff --git a/0dAzT4oBgHgl3EQfRPuW/content/tmp_files/load_file.txt b/0dAzT4oBgHgl3EQfRPuW/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7284742cfa1db1ff4d8af7131b6bfada65e30d3e --- /dev/null +++ b/0dAzT4oBgHgl3EQfRPuW/content/tmp_files/load_file.txt @@ -0,0 +1,958 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf,len=957 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='01213v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='optics] 3 Jan 2023 Numerical study of magneto-optical binding between two dipolar particles under illumination by two counter-propagating waves Ricardo Mart´ın Abraham-Ekeroth Instituto de F´ısica Arroyo Seco, IFAS (UNCPBA), Tandil, Argentina and CIFICEN (UNCPBA-CICPBA-CONICET), Grupo de Plasmas Densos, Pinto 399, 7000 Tandil, Argentina∗ The formation of a stable magneto plasmonic dimer with THz resonances is theoretically studied for the principal directions of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Unlike a recent report, our work provides a complete description of the full photonic coupling for arbitrary magnetic fields as, for instance, unbalanced particle spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' As an illustration, we consider two small, n-doped InSb nanoparticles under illumina- tion by two counter-propagating plane waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Remarkably, when an external magnetic field exists, the symmetry in the system is broken, and a resonant radiation pressure for the dimer appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Similarly, tunable inter-particle forces and spins are exerted on the non-reciprocal dimer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' The sys- tem is also characterized when the magnetic field is absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Moreover, we show how the mechanical observables truly characterize the dimer since their resonance dependency contains detailed informa- tion about the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Unlike far-field observables like absorption, mechanical magnitudes depend on the system’s near-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' In addition, the nature of the particle spins is originally explained by the energy flow’s behavior around the dimer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' This work constitutes a generalization of any previ- ous approach to optical binding between small nanoparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' It paves the way for fully controlling optical matter and nano factory designs based on surface plasmon polaritons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Keywords: Magneto plasmonics, Spin torques, Dimers, Optical Binding, Photonics, Poynting field, Radiation Pressure, Optical Matter I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' INTRODUCTION Optical matter (OM) consists of arrays of micro or nanoparticles that are somehow bound and controlled by light [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' An OM able to self-assemble at will to develop solid technology is a long-standing goal in photonics [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' The background for OM is the particle manipulation by optical forces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' the first results were applied to microparticles due to the lack of technology and the presence of thermal noise for smaller systems [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Generally, OM comprises multiple particles subjected to electromagnetic forces that come from their mutually scattered light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' However, this multiple scattering phenomena can be very complex and bring about unusual effects such as “non-reciprocal” forces, torque opposite to the illumination angular momen- tum, and non-conservative forces [2, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Appropri- ate control of OM by forces and torques could lead to programmable materials for optomechanical, rheologi- cal, and biological applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' In this respect, many works studied the optical binding between nanopar- ticles as a primary tool to develop OM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Some ap- proached specific combinations of optical beams like those with programmable phase [24, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' For example, light-induced rotation of objects holds potential for various applications such as sensing, cargo transporta- tion, drug delivery, and micro/nanosurgery [3, 7, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Optical traps use the combination of beams as a po- tent characterization tool for material science and bio- physics, as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' [23], which uses electrostatic focus- ing to obtain the mass spectrum of SARS-CoV-2 and ∗ mabraham@ifas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='exa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='unicen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='ar BoHV-1 virions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' However, the high intensity at the focal spot may introduce laser heating, which is an issue for bio applications [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' On the other hand, recent advances in THz tech- nology call for new devices and materials that exhibit a non-reciprocal behavior for photonic networks and optical information processing [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Non-reciprocal devices are a crucial component of modern commu- nication technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' They are nowadays required for miniaturized electronic and photonic devices [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' One way to create optical non-reciprocity at the THz range is using magneto-optical (MO) systems like graphene, hexaferrites, and semiconductors [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' For instance, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' [15] presented MO measurement of several sam- ples of InSb with different carriers and carrier con- centrations for low external magnetic field and room temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' With these advantages, dimers and trimers of InSb particles have been studied to enhance THz spectroscopy by forming electric and magnetic hotspots in the gap between them [6, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Anisotropic materials like InSb in OM would make it strongly dependent on the beam combinations, allowing for countless possibilities [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Recently, several efforts have focused on MO nanoparticle systems to shape OM and optical traps with a reasonable degree of con- trol and accuracy, besides other relevant applications [1, 18, 20, 22, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' on a broader sense, magnetoplas- monics relates the plasmonic behavior of nanoparticles with the presence of external magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' The modulation and tunability of plasmonic resonances offered by magnetoplasmonics results auspicious for ultra-sensitive sensors and active plasmonic devices [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' In particular, the formation of stable optical binding between two small magnetoplasmonic parti- cles has been lately studied [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Equilibrium binding distances were predicted and found tunable by the incoming wave’s polarization state and the magnetic field’s magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' However, the model developed in that report is valid only for relatively small magnetic 2 field values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Moreover, it predicts stable dimers only using alternating magnetic static fields and polariza- tion angles to remove azimuthal, unbalanced forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' More importantly, this work needs to discuss possible rotations of the particles due to angular momentum transfer in the multiple scattering scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' In this paper, we study the formation of stable MO dimers for small nanoparticles in a complete frame- work involving all the possible optomechanical induc- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' The dimer’s isotropic and anisotropic responses are assessed as a base for OM designs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=', under the presence/absence of an external magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' This field can be of arbitrary magnitude in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' The illumination consists of two counter-propagating plane waves with circular polarization, which simulates a simple optical trap in the vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' We found sev- eral possibilities to create stable dimers even when the magnetic field is off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' The beams do not exert net forces for reciprocal dimers but may exert torques on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' On the contrary, there is a net radiation pres- sure and spin for the whole system when the static field is on, allowing complete control of the system’s movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' The results will enable one to infer that the mechanical variables can be used as near-field ob- servables to explore the content of unknown samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Conversely, they can be used to accurately control the dimer’s creation/destruction and its mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Finally, the spins predicted are explained in terms of the en- ergy flows around the dimer, which constitutes a novel scattering-force effect for interacting particle arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' MODEL In the following, we assume two equal particles of the same non-reciprocal material immersed in the vac- uum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Then the method of discrete dipoles (MDD) [17] simplifies considerably to p1 = ǫ0ˆαE0,1 + k2 0 ˆα ˆGp2, (1) p2 = ǫ0ˆαE0,2 + k2 0 ˆα ˆGp1, (2) where ˆα is the polarizability tensor representing the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' The following definition automatically in- cludes the radiative corrections necessary to fulfill the optical theorem [21] ˆα = � ˆα−1 0 − ik3 0 ˆI 6π �−1 (3) where ˆα0 is the so-called quasistatic polarizability, which can be given by ˆα−1 0 = 1 V � ˆL + [ˆǫr − ˆI]−1� (4) being V the particles’ volume, ˆL = ˆI/3 is the elec- trostatic depolarization tensor specified for spheres or cubes, and ˆǫr is the relative dielectric tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' The system of equations 1 can be solved straightforwardly, leading to p1 = ǫ0 ˆF � E0,1 + k2 0 ˆα ˆGˆαE0,2 � , (5) p2 = ǫ0 ˆF � E0,2 + k2 0 ˆα ˆGˆαE0,1 � , (6) where we define ˆF = � ˆα − k4 0 ˆGˆα ˆG �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' In this work, a counter-propagating configuration is assumed as a superposition of two left-handed circularly polarized (LCP) plane waves with the same intensity I0 [11, 18], see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 1, namely, E0 = E0 √ 2 � (ˇx + iˇy) eik0z + (ˇx − iˇy) e−ik0z� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' (7) This field is used in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 5 to calculate the incident field at the particles’ positions E0,1 and E0,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' The absorption cross section of the system can be calculated once the dipole moments are known by σabs = k0 ǫ0wtot E Im � p1 · � ˆα−1 0 p1 �∗ + p2 · � ˆα−1 0 p2 �∗� (8) where wtot E = ǫ0|E0|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' The i-component of the forces exerted on each particle can be obtained from the time-averaged force within the Rayleigh approxima- tion [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' This is F1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='i = 1 2Re{pt 1[∂iE∗(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' ω)|r=r1} (9) F2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='i = 1 2Re{pt 2[∂iE∗(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' ω)|r=r2} (10) where the derivatives of the total field ∂iE(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' ω)|r=rn at the dipoles’ positions rn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' n = {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 2} can be obtained from [12]: ∂iE(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' ω)|r=r1 = ∂iE0(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' ω)|r=r1+ + k2 0 ǫ0 (∂iG(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' r2))r=r1p2]} (11) ∂iE(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' ω)|r=r2 = ∂iE0(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' ω)|r=r2+ + k2 0 ǫ0 (∂iG(r1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' r))r=r2p1]} (12) The total force exerted on the dimer results from adding the force components for each particle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' namely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Ftot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='i = F1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='i + F2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' In particular, the net radiation pressure for the dimer under the illumination given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 7 is defined by taking i = 3, or the z compo- nents, as Ftot,z = F1,z + F2,z (13) Another useful mechanical variable is the binding force, which in the present case is defined as ∆ = (F1 − F2) · ˇn (14) where ˇn = r2−r1 |r2−r1| is the dimer’s versor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' The optical torques can also be calculated, as given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' [14]: Nspin,1 = 1 2ǫ0 Re � p1 × �� ˆα−1 0 �∗ p∗ 1 �� (15) Norb,1 = r1 × F1 (16) N1 = Nspin,1 + Norb,1 (17) 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' (Color online) Dimer configurations and in- cident waves treated in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Two counter- propagating waves with left circular polarization illumi- nate the magneto-optical dimer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' The leading example con- sists of two n-doped InSb particles separated by a gap of a particle’s diameter, g = 2R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' (a) Parallel [(b) perperdic- ular] configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' The static magnetic field B is parallel to +z direction (green arrow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' In (b), φ is the azimuthal angle of the dimer’s position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Nspin,2 = 1 2ǫ0 Re � p2 × �� ˆα−1 0 �∗ p∗ 2 �� (18) Norb,2 = r2 × F2 (19) N2 = Nspin,2 + Norb,2 (20) The definitions of the orbital and spin torques were discussed previously in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' [14, 33, 34], among oth- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' The spin torques are always defined with respect to the centers of the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Otherwise, the refer- ence system is located at the dimer’s center of mass, and orbital torques are set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Thus, the total torque exerted on the dimer is Norb = Norb,1 + Norb,2 (21) Nspin = Nspin,1 + Nspin,2 (22) Ntot = N1 + N2 = Norb + Nspin (23) In particular, this study simulates nanoparticles made of n-doped Indium antimonide (n-InSb) [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' In- dium antimonide (InSb) is one example of the most widely studied polar semiconductors for magnetoplas- monic applications because it can be easily doped for sizable magnetic-induced effects [15, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' As re- viewed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' [30, 37], n-InSb is an exciting mate- rial that has two kinds of surface resonances in the absence of static field, namely, the phonon polari- ton (SPhP, higher-energy) and the plasmon polariton (SPP, lower-energy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Its model properties were de- scribed on Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' [1, 18, 30], among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Since we are interested in the near-field interactions between the particles, the study focuses on an example for which the interparticle’s gap equals one particle di- ameter (g = 2R);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' We add complementary examples for other values of the gap in the Supple- mentary Material (SM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' RESULTS In this section, all the optical variables were scaled by the proper factors to make them adimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' The following characteristic magnitudes, namely, wtot E , Ap = πR2, Vp = 4 3πR3, and Vint = 4 3π (|r2 − r1|)3 redefine the variables as Qabs = σabs 2Ap for the absorp- tion efficiency, Frad = Ftot,z wtot E Ap and ∆′ = ∆ wtot E Ap for the radiation pressure and the binding forces, and N′ spin = Nspin wtot E Vp for the spin torque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' The variable N′ orb = Norb wtot E Vint for the orbital torque is only shown by an example in the SM since it gave negligible re- sults unless the gap is minimal, see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' S1 and S2 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' We calculate the scaled Poynting vector as S = 1 2I0 Re{E × H∗} where the magnetic field H comes from an MDD equation similar to that for E [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' The curl of S is calculated using an appropriate tridimensional mesh around the system’s near-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Parallel Illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 2 shows the spectral results for parallel illumi- nation when the magnetic field is off (B = 0, black line) and on (B = 1 T, red line with squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' The absorption efficiencies, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 2a, result independent of the direction of the dimer so that the same spectra remain for any other illumination configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' The low-energy resonance (around 73µm) corresponds to an SPP, while the high-energy resonance (48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='7µm) corresponds to an SPhP [30, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Making use of the Plasmon Hybridization Model (PHM) for two dipo- lar particles, both kinds of surface modes show as bright antibonding modes for transverse electric fields according to the configuration shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 1a, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' [35] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' When B is on, each isotropic surface mode splits into two modes due to degeneracy removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' The ab- sorption is the only far-field observable shown in this work since negligible scattering occurs for small sys- tems [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' As it is illumination-independent, the spec- tra obtained remain invariant for all illumination di- rections concerning the dimer’s axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Thus, the only available far-field observable is neither adequate to study the interactions occurring in the dimer nor valu- able to predict the dimer’s dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 2b, there is no net radiation pressure when B is off due to the high symmetry of both the system and incident field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' On the other hand, there is a resonant pressure for the MO dimer when B is on, revealing the magnetoplas- monic resonances and directing the dimer upwards or downwards according to their energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' As a result of IKA LCP B KB B Z X X KA 1 2R g LCP kB (a) (b)4 the interaction, the radiation pressure identifies the modes by the sign of the force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 2c shows that the binding force leads to repulsion between the par- ticles for both cases, B = 0 and B = 1 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' In other words, there cannot be a stable dimer under this par- allel configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' The response is still resonant but less sensitive than the radiation pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Remark- ably, the results agree with the interpretation given by the PHM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Notably, although we are dealing with a dimer sys- tem, our results agree with those reported in [18] for a single particle under the same illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' In general, the absorption efficiency Qabs behaves like Re{α11}, both the radiation pressure Frad and the spin N ′ spin,z on z behave like Im{α12}, and the bind- ing force ∆′ resembles −Re{α11}, being αij the carte- sian components of the polarizability tensor ˆα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' In the case of the spins in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 2d, these behave like ±Im{α33} for each particle respectively (polarizabil- ities not shown here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' These functional dependencies are due exclusively to the type of illumination;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' oth- erwise, other αij-terms would appear in the spectral variables [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Following angular momentum’s conservation, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 2d shows that the net spin for the system is zero when B is off (black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' To put it another way, the spins for each particle are equal and opposite, showing the resonant modes for the isotropic case (see red and blue lines with symbols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' When B is on, however (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 2e), there is a net spin for the system, black line, which is twice the spin for each particle (red line with squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' The spin resonates sensitively with the dimer’s modes, quite like the radiation pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Consequently, the spins become much stronger than those for B off, compare the scales of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 2d and e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Thus, the radiation pressure and spins constitute the most sensitive observables in the near field, giving a common spectral shape (compare spectra in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 2b and e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Perpendicular Illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Now we vary the dimer’s azimuthal angle since a de- pendency on the net polarization is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 3 and 4 show maps as a function of the incident wave- length and azimuthal angle for B = 0 and B = 1 T, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Similarly to that found in the previous subsection, there is no net radiation pressure when B is off (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Yet this time, a different behav- ior is found for the binding force, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' The sys- tem offers a resonant spectral response but depends on the angle φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Maxima [minima] of binding are found around φ = 90, 270 [φ = 0, 180] deg, meaning inter- particle attraction [repulsion].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' This fact also defines stable positions for the dimer around the strongest optical resonance, namely the SPhP at 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='7µm, and around φ = 34, 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='6, 214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='5, 325.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='7 deg for all wave- lengths when B is off (follow the black lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' A sim- ilar situation is found for the second resonant wave- length ≈ 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='6µm (SPP), where the variations are less pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Regarding the spin, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 3b shows a re- maining behavior for the whole system, which is res- onant with the surface modes and coordinated with the binding phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Remarkably, the spin gets its extremals (maxima or minima) when the dimer reaches its stable positions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' namely, neither attraction nor repulsion, compare Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 3a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' As mentioned above, the most sensitive resonance corresponds to the excitation of the SPhP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Noteworthy, our results are consistent with the in- terpretation of the PHM for isotropic, dipolar par- ticles [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' In particular, each value φ = nπ [φ = (n + 1/2)π] rad with n ∈ Z, the binding force shows repulsion [attraction] for both types of resonances, namely, the SPhP and SPP, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' This outcome is due to the net polarization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' the electric field is along ˇy, see the map for φ = 0 at the SPhP wavelength in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Thus, φ = 0 corresponds to a transverse elec- tric field compared with the dimer’s direction, mean- ing an antibonding bright mode in the context of the PHM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Differently, φ = 90 deg corresponds to a parallel electric field compared with the dimer’s axis, meaning a bright bonding mode in the PHM (map not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 4a, there is a remaining radiation pressure for the whole system due to the symmetry breaking that appears only at the resonances’ locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' This spec- trum results invariant with φ and follows the same resonances as in absorption in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 2a when B is on (red line with squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Thus the presence of a static magnetic field induces the dimer to move forward or backward in the illumination’s direction when the in- cident energy is that of a surface mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Likewise, the system shows a resonant binding (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' As in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 3a, the black lines follow the values of zero force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Note that the map strongly distorts by the presence of the resonances when B is on, making the dynam- ics more complex and even reducing the extremals of the binding force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' However, the possibility to obtain stable binding enhances around the SPhP due to the overlapping of MO modes, which means more degree of control in the dimer’s creation and stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 4c, the system’s spin follows a trend simi- lar to that for the radiation pressure in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' This behavior is quite different from that found for B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Note that spin is enhanced when B is on;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' the color- bar limits show values ≈ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='2 − 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='5 times higher than those for B off, compare Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 3b and 4c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' As a re- sult, the MO system could be readily identified in an experiment by observing the dimer’s dynamics at the resonance wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 4d shows the electric field around the dimer’s plane z = 0 for the SPhP found at 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='85µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' This wavelength corresponds to the most robust resonance when B is on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' The rest of the configuration is equal to that given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' The field hot spots are leaned on the right ≈ 65 deg from the x axis by the MO effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Up to this point, we have explored a few examples of MO dimers to approach the idea of controlling the particle dynamics and ”photonic molecule” stability [25] in the presence/absence of a static magnetic field B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' (Color online) Optical properties for parallel configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' (a) Absorption efficiency, which is independent of the dimer’s orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' (b) Radiation pressure (total force along z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' (c) Binding force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Black line [red line with squares] for magnetic field B = 0 [B = 1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' (d) [(e)] Spin torques for B = 0 [B = 1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' In (d), the net spin torque is zero for all wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Below, we discuss the behavior of the dynamic ob- servables in terms of the information contained in the Poynting field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' The reader is reminded that the particles’ photonic interaction matches a multiple- scattering framework [16, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' The near fields involve the evanescent waves, which play a crucial role in the particles’ interaction for surface modes and close par- ticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' This phenomenon can be seen through the en- ergy flows because they may have all the information of the near fields E and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Generally, the magni- tudes obtained from far-field calculations lose some of the information about the system [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Nature of the spins through an examination of the Poynting fields We explore the spins exerted on the system by show- ing a few calculations of the Poynting field around the dimer for perpendicular configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' The paral- lel configuration is less interesting since it would only lead to repulsion states without dimer formation for any gap under both cases B = 0 and 1 T, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 2c for the example g = 2R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' More clarifications can be found in the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 5a-d [e-f] consist of maps related to the en- ergy flow when B is off [on] upon different azimuthal angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' The wavelengths coincide with that for the strongest SPhP in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' The left column (a-c-e) shows the Poynting field S when z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Similarly, the right column (b-d-f) shows the z-component of ∇ × S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' The white arrows are rescaled to visualize the maps easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Interestingly, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 5a-b show an exam- ple of a repulsion state with zero spins when φ = 0, see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 3a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Even though S aligns in a single direction, a resonant magnitude and a non-negligible curl appear near the surface of the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' This res- onance is due to the excitation of the SPhP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' However, the contributions to the spin cancel out due to high symmetry evidenced by these maps and zero net spin results for the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 5c-d show the attrac- tion state with maximum positive spin when φ = 135 deg, see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 3a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' This time, two hot spots of maximum magnitude face each other, and a kind of saddle point emerges in the gap region between the particles, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 5c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' As a result, the values of the curl clearly show a rotational state for light as the field spots have ”turbine-blade” shapes, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 5d, explain- ing the net positive spin calculated for the system in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 3b for φ = 135 deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' It is also evident from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 5d that the two particles have the same spin, visually showing that the net spin is two times the spin of one particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Finally, we show the example when B is on and φ = 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='2 deg, which coincides with the hot spot of maximum attraction at the resonance 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='85µm of the SPhP, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Notice in passing that this value for φ is close to the angle of the electric spots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 4d, namely, ≈ 65 deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Remarkably, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 5e illus- trates that the energy flow would make the particles spin counterclockwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Moreover, it is also notewor- thy the vortex that appears in the field region between the particles with clockwise orientation, resembling a ”gear” mechanism which coordinates field and parti- cles [9, 21, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Consistently, the curl’s map shows a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='0 0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' B=O T B=O T 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='5 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='0 ×10-1 3 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='0 2 Qabs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='0 B=O T (a) (b) (c) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='5 1T 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='0 20 60 80 100 20 60 80 40 40 100 20 40 60 80 100 120 120 120 Wavelength (μm) Wavelength (μm) Wavelength (μm) 5 321 之 23 system system 3 45 np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 1 = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 2 (d) (e) np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 2 6- 20 40 60 100 120 20 40 60 80 100 80 120 Wavelength (μm) Wavelength (μum)6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' (Color online) Near-field observables for per- pendicular configuration in the absence of magnetic field, B = 0 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' (a-b) Maps of the mechanical variables as a function of wavelength and dimer’s azimuthal angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' (a) Binding force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' The black lines correspond to zero force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' (b) Spin torque for the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' In this case, the net radi- ation pressure is zero for all wavelengths (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' (c) Distribution of electric field around the dimer for z = 0 at the resonance wavelength 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='85µm for φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' structure similar to that in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 5d but this time en- hanced and possessing a structure in the gap region that contains the vortex indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 5e, see in- set in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 5f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' The inset zooms this region so that a connection between the curl hot spots is appreciated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' (Color online) Near-field observables for perpen- dicular configuration and B = 1 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' (a-c) Maps of the me- chanical variables as a function of wavelength and dimer’s azimuthal angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' (a) Radiation pressure for the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' (b) Binding force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' The black lines correspond to zero force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' (c) Spin torque for the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' (d) Distribution of electric field around the dimer for z = 0 at the resonant wave- length 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='85µm for φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 360 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='2 (a) 315 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='14 270 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='09 225 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='03 180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='02 135 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='08 90 45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='18 20 30 40 50 60 70 80 90 100 120 Wavelength (μm) 360 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='79 315 (b) 270 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='5 225 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='25 135 0 90 45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content="32 20 30 40 50 60 70 80 90 100110 120 Wavelength(μum) N' spin,z 360 (C) 6 315 4 270 225 2 0 135 90 45 4 5 0 20 30 40 50 60 7080 90 100 110120 Wavelength (μum) [E Eo 一 1 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='75 6 (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='5 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='25 (wn) 4 0 y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='25 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='5 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='75 1 1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='75-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='5-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='2500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='75 x (μum)△(-) 360 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='2 315 1 270 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='8 315 (b) 270 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='4 0 135 90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='4 45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='8 0 20 30 40 50 60 70 80 90 100 110 120 Wavelength (μm) [E Eo (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='75 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='5 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='25 (un) 4 0 y 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='5 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='75 1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='75-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='5-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='25 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='75 x(μm)7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Scaled energy flows around the dimer for z = 0 under perpendicular configuration and at the strongest SPhP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' The color scale corresponds to the magnitude;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' the white arrows show the Poynting flow (2× their original size).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Left [right] column for the [z-component of the curl of the] Poynting field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' (a-d) [(e-f)] Examples for B = 0 [B = 1] T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' the wavelength is 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='85µm [49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='85µm].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' (a-b) For φ = 0 deg, (c-d) φ = 135 deg, and (e-f) 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='2 deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' The inset in (f) zooms the gap region up to a maximum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='05 in the colorbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' [Sxy] 2RI(V × S)zl 1o o 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='4 6 (b) (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='6 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='3 00 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='2 (wn 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='4 0.' metadata={'source': 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this work explores the behavior of small nanoparticle dimers when magneto- optical materials like n-doped InSb and moderate magnetic fields are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Two counter-propagating waves with equal circular polarization are used as il- lumination to simulate a simple optical trap with nei- ther net gradient nor scattering forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Our results show that the system can be thoroughly character- ized by observing its mechanical inductions, provided these latter depend on the near field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Besides, we found no possibility of forming stable dimers when the dimer is aligned with the illumination since the inter-particle force only leads to repulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' On the contrary, under ”perpendicular” alignment, there are several ways to obtain stable dimers or inter-particle attraction, at least under this ”static” model for which the particles’ velocities and accelerations are not con- sidered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' As the results strongly depend on the mag- netic field’s presence, this study constitutes a novel background to build OM or photonic molecule nano factories and control their movements, or conversely, to study samples containing this class of multimers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Finally, we give an original explanation for the ap- pearance of particle spins based on the energy flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' This interpretation offers satisfactory results from the ”scattering” forces produced by the interaction be- tween the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Gradient forces were also inves- tigated but showed no appreciable influence on the spins’ appearances (results not shown in this work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' ACKNOWLEDGMENTS The author would like to thank A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Garc´ıa-Mart´ın from IMN-CSIC and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Marqu´es from Universidad Aut´onoma de Madrid for their valuable discussions during his postdoc in Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' CONFLICTS OF INTEREST The authors declare that the research was con- ducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' [1] Abraham Ekeroth, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=', Garc´ıa-Mart´ın, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=', and Cuevas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Thermal discrete dipole ap- proximation for the description of thermal emis- sion and radiative heat transfer of magneto-optical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' B 95, 235428.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='1103/PhysRevB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='235428 [2] Albaladejo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=', Marqu´es, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=', Laroche, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=', and S´aenz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Scattering Forces from the Curl of the Spin Angular Momentum of a Light Field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' 102, 113602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} +page_content='102.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfRPuW/content/2301.01213v1.pdf'} diff --git a/1dFKT4oBgHgl3EQfOy1w/content/tmp_files/2301.11760v1.pdf.txt b/1dFKT4oBgHgl3EQfOy1w/content/tmp_files/2301.11760v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..721a4ce8bf6f8c3c9281143e82b572fd1c055c69 --- /dev/null +++ b/1dFKT4oBgHgl3EQfOy1w/content/tmp_files/2301.11760v1.pdf.txt @@ -0,0 +1,4592 @@ +Renaissance of Analogue Optical Computing +Nikita Stroev1 and Natalia G. Berloff2∗ +1Department of Physics of Complex Systems, +Weizmann Institute of Science, Rehovot 76100, Israel and +2Department of Applied Mathematics and Theoretical Physics, +University of Cambridge, Cambridge CB3 0WA, United Kingdom +Abstract +This review paper examines the physics and mathematics of optical computing, which utilizes +photons and optics-related technologies for effective and efficient computational purposes. We dis- +cuss the history and development of optical computing, as well as modern analogue computing +platforms and architectures, focusing on neural network implementations. Furthermore, we cover +special-purpose optimisers and mathematical descriptions of optical optimisers, as well as their +various applications and interconnections. We also explore the main directions of technological +development in optical computing and estimates of its efficiency. Finally, we discuss future per- +spectives and the domain of optical quantum computing. This review provides a comprehensive +overview of the current state-of-the-art in optical computing and its potential applications. +∗ correspondence address: N.G.Berloff@damtp.cam.ac.uk +1 +arXiv:2301.11760v1 [physics.optics] 27 Jan 2023 + +I. +INTRODUCTION +In 1965 Intel co-founder Gordon Moore formulated an empirical observation that the +number of transistors in a microprocessor will double nearly every two years, the statement +which is known as Moore’s law [1, 2]. This prediction was followed by the forecast of reaching +a saturation point by 2015. The progress of conventional computer architectures was very +close to Moore’s vision. However, reaching the saturation point was just a matter of time. +The miniaturization of silicon transistors recently managed to break the 7-nanometre barrier, +which was believed to be the limit. Also, Moore’s law usually comes with several essential +indicators, such as the processor’s thread performance and clock frequency, which reached +the point of saturation much faster than the density of the transistors. All of these factors +limit the scaling performance of modern computers. However, there are other reasons for the +saturation of conventional computing power growth, which are the consequences of Moore’s +law. For example, increasing the number of transistors allows one to obtain more powerful +systems. Still, the processing speed will inevitably decrease with the concomitant increase +in heat production, while increased energy consumption is connected with the growth of +the performance. Another critical issue is the so-called von Neumann bottleneck [3], arising +from the architecture design. It refers to the computer system throughput limitation due +to the characteristic of bandwidth for incoming and outcoming data [4, 5]. All these issues +pose severe problems to the future of conventional computer development. As a result, the +alternatives to von Neumann systems started to emerge [6, 7]. +One turns to alternative hardware architectures and purpose-built devices to keep up +with the scaling performance. As such, universal quantum computing promises to decrease +the algorithmic complexity of solving challenging tasks by exploiting the entangled states. +However, in contrast to this high-risk and high-reward strategy (also discussed in the current +review in the optical setting), there is an option to replace electrons with photons but remain +in the scope of classical or classical with a transient quantum coherence regime of optical +computing. The motivation for such transition is clear since photons move at the speed of +light, have low heat production, have high density and can be efficiently coupled to matter +to exploit nonlinear behaviours. Moreover, optical technologies have matured and entered +our everyday lives, such as fibre optic channels that carry the global traffic of information +or optical readers of compact disks. However, the conversion of photons into electrons is +2 + +required for compatibility with CMOS architectures. Such conversion takes a significant +portion of energy, slows down the overall process of information processing, and presents a +severe technological bottleneck in this type of hybrid technology. +Despite these difficulties, optical hardware is exploited in computing devices. For ex- +ample, different application-specific photonic hardware can operate on a reasonable scale +in data centres for heavy machine learning (ML) applications and large-scale optimization. +Moreover, neural network (NN) architectures are nearly ideally suited for optical hardware +with the potential to achieve high efficiency, fast computing times, and low energy consump- +tion due to the desired physical properties of the photonic systems. Nevertheless, at this +point, optical computing can not be associated with mainstream technology. It is unlikely +that optics will ever replace electronics as the universal platform in the foreseeable future. +The additional reason is the technological inertia accumulated through the years by signif- +icant investments in CMOS technologies. Partially, the rapid development of what we call +conventional computers in the early years led to an ever-increasing gap with computing using +photonics, which will occupy its own place in the domain of application-specific hardware. +There are many excellent reviews on the topic of optical computing. The challenges of +modern computing and new opportunities for optics are discussed in [8]. This work presents +the latest research progress of analogue optical computing, focusing on three main direc- +tions: vector/matrix manipulation, reservoir computing (RC) and photonic Ising machine. +Moreover, it covers the topic of computing efficiencies, such as the ratio of performance +and power dissipation and the error/precision interplay of such hardware. Another excel- +lent review considers analogue optical computing in the context of artificial intelligence (AI) +applications [9]. +This work provides an overview of the latest accomplishments of opti- +cal computing, considering the realization of different AI models and NN paradigms. One +can find additional information in other reviews [10–14], which appeared due to the recent +interest in deep learning methods and their success in many domains. +What differentiates our review from those listed above is that we treat analogue optical +computing using the concept of universality of the underlying dynamical systems description. +The advantage of optical computing comes from ultrafast emulation of the dynamics [15]. +We focus on physical optimisers that exploit bifurcation dynamics and threshold operation +and aim at solving nonlinear problems, therefore, going beyond the speed-up of performing +the linear operations that optics is so efficient at. +3 + +We organised our review as follows. Section II provides a short history of optical comput- +ing together with the modern analogue computing platforms focusing on NN implementation +and other neuromorphic systems. Section III discusses the special-purpose optimisers and +several examples of such devices. This section connects the operational regimes of such +machines with the complexity classes and addresses the scalability of this approach. Sec- +tion IV focuses on the physics of optical computing devices based on laser networks, optical +parametric oscillators (OPOs) in fibre, photon or polariton networks, as well as their math- +ematical models. The second part of this review investigates the mathematical structures +of different assignments and their emulation by the physical systems. The following Section +V lists a wide range of possible applications across different applied domains. The final +part consists of our subjective perspective on the future technological development of op- +tical computing field in Section VI and passing remarks about quantum optical devices in +Section VII. Finally, Section VIII summarises the results. +II. +ANALOG OPTICAL COMPUTING +Modern technologies demand vast data flows, creating various challenges for the develop- +ment of the semiconductor industry and pushing classic electrical circuits to their physical +limits. The developments range from more mainstream such as optical components that can +be integrated into traditional computers or play the role of specific hardware, dealing with +computationally heavy tasks or supplementing such calculations, to ambitious ones, such as +all-optical digital computer architecture. +II.1. +Brief prehistory of the optical computing +Although optical computing is an emerging technology that has gained more momentum +over time (especially considering the popularity and efficiency of the latest data-driven ap- +proaches), many significant advances have been made in previous decades. Therefore, before +describing the particular systems, their advantages and their applications, we briefly discuss +the progress that enabled. More information and additional details can be found in [16]. +The generic optical processor architecture comprises three plane parts: the input, the +processing and the output planes. Early on, the input plane was a fixed image slide with +4 + +its later change to a spatial light modulator (SLM), introduced to perform the input signal +conversion. +The processing plane can be composed of lenses, nonlinear components, or +holograms, while the final output part is composed of photodetectors or a camera. +The first promising applications for optical processors were pattern recognition tasks, +which influenced the prototypes of optical correlators. The simple architecture called 4-f +was based on the work on spatial filtering, see [17]. The Fourier transform property of a +lens is the standard function of many optical computing schemes, taking advantage of the +speed and parallelism of light. The second type of correlator architecture was presented in +1966 by Weaver and Goodman [18], which is called the joint transform correlator (JTC), +see Fig. 1 +FIG. 1. The optical setup of the joint transform correlator (JTC). The figure is taken from +[16]. +Before 1950 there were significant steps in development of optical technologies such as +the theory of image formation in the microscope [19], developed by Abbe, the development +of phase contrast filter by Zernike [20] and the appearance of the information optics after +Elias Snitzer in 1952 [21, 22]. +Other major inventions of that period were holography +5 + +Reference r(x, y) +Joint spectrum +xoT +TSLM +fi +CCD +Scene s(x, y) +fi +Optional processing +Crosscorrelations +2x0 +SLM ! +2x0 +f2 +f2 +Autocorrelations(by Gabor,1948) [23] and the development of the laser in 1960 [24, 25]. The consequent +introduction of the off-axis hologram allowed the separation of the different terms of the +reconstruction giving remarkable 3D reconstructions [26, 27] in 1962 by Leith and Upatnieks, +which basically led to practical holography and was further enhanced by Lohmann, creating +the first computer-generated hologram [28, 29] in 1966. +Early SLMs were based on the +Pockels effect with few prospective devices [30–32]. Liquid crystal technology is the most +commonly used technology for SLMs today. Another significant step was the invention of +the first optical transistor [33], the hope for small integrated circuits. +The period from 1980 to 2004 was vibrant and productive. Active progress was going in +the field of holography, particularly new encoding methods and the point-oriented methods +were developed to achieve high quality and high diffraction efficiency optical reconstructions +of the CGHs [34]. More than 50 types of SLM were introduced in the eighties and nineties +[35]. Optical transistors presented another active area of research with the appearance of +the micro-electromechanical systems (MEMS) technology [36]. In the 1990s, vertical-cavity +surface-emitting lasers (VCSELs) and the self-electrooptic effect (SEED) devices became +available [37]. In general, many aspects of modern optical interconnections and their com- +ponents were introduced and studied during this period. +The optical technologies development provided the necessary experience in the capabilities +of the optical devices and led to the maturation of the experimental element base. Optical +computing received a second chance after the success of so-called deep NNs, which share +many similarities with the previous neural-like optical architectures. +II.2. +Modern optical computing +Today, numerous research topics benefit from the progress in optical computing; therefore, +the field is no longer so well defined. For example, some of the algorithms initially devel- +oped for pattern recognition using optical processors are now used successfully in digital +computers. Other fields, such as biophotonics, largely benefit from past optical processing +research. +The fundamental building block of modern electronic computers is a transistor. There- +fore, one must find an equivalent optical transistor to replace electronic components with +optical ones. To assemble such transistors into the higher-level components to create an +6 + +all-optical computer’s central processing unit (CPU), one has to design the optical processor +and optical storage and organise the optical data transfer. However, such an approach faces +many challenges, while the potential of optics in large architecture consisting of higher- +level components can be seen as somewhat speculative [38]. Among persisting problems +are the scalability of the optical logic devices due to the bad logic-level restoration, cascad- +ability, fan-out and input-output isolation, energy consumption issues and non-miniature +device footprint. Moreover, coupling these potential devices with the electrical components +will require data format conversion from photons to electrons, which is relatively slow and +energy-consuming. +However, the development of integrated photonics continued [39]. It led to attempts to +create linear logic elements, such as all-optical logic gates [40, 41], improve the existing +optical transistors and develop new ones in the context of the all-optical processing [42, 43]. +One can use SLM, micro-lens array and holographic elements in free space to realize optical +linear interconnection. +Such linear elements are essential components in various optical +computing devices. +Nonlinearity is another essential component in optical schemes; however, its realisation +meets specific difficulties as light beams pass through each other unperturbed in a pure +vacuum. To force these beams to interact, one has to set up a high-energy experiment, +which is challenging to realise in practice. There are two other ways to realise the nonlinear- +ity: introduce the digital readout mechanism, implemented by the charged-coupled device +(CCD), send it to a computer with further nonlinear processing before feeding it back to +SLM, or develop fully optical nonlinear activation materials with high enough intensity of +the beams (utilising absorption, refraction or scattering processes). Nonlinearities can be +divided into local (as needed in neural architectures) and global systems (such as reservoir +computing systems, see below). Combining the linear and nonlinear elements led to the +developing of specialised isolated devices. As a result, optical computing research has seen a +resurgence in activity, centring around new developments in photonic hardware accelerators +and neuromorphic computing. +Neuromorphic computing usually denotes the use of integrated systems to mimic neuro- +biological architectures. Although it is very close to the domain of AI, with the stress on +the word “artificial”, which deals with the intelligent designed machines or agents, we will +use neuromorphic computing in the general sense to describe any neural systems, be it +7 + +brain or nature-inspired or artificially designed. Modern key focus areas are concerned with +emulating the neural structure and operation of the human brain, including probabilistic +computing, which creates algorithmic approaches to dealing with uncertainty, ambiguity, +and contradiction in the natural world. Optics has required ingredients to emulate NNs +[13, 44]. +II.3. +Optical neural networks +Optics has long been a promising medium for matrix multiplication and interconnects. +Artificial neural networks (ANN) have been widely used for industrial and fundamental +applications, and this new technological demand created a renewed case for photonic NNs. +Although most ANN hardware systems are electronic-based, their optical implementation +is particularly attractive because of their intrinsic parallelism and low energy consumption. +Disparate ANNs vary by types of constituent elements, mathematical operations and the +architecture used. In photonic approaches to ANN, the mathematical operations are mapped +to the dynamics of optical propagation set by programmable linear optics and nonlinearity. +A scalar synaptic weight describes pairwise connections between artificial neurons. At the +same time, the layout of interconnections can be represented as a matrix-vector operation, +where the input to each neuron is the dot product of the output from connected neurons +with assigned weights. +Photonic realizations of ANNs fall into three categories. First, free-space systems rely +on diffraction, Fourier transforms, etc. [45, 46]. They have high scalability and can simul- +taneously process large numbers of neurons but suffer limited connectivity. One example +is a reconfigurable, scalable two-layer NN for the classifying phases of a statistical Ising +model [47]. Second, SLMs program linear operations and Fourier lenses implement the sum- +mation by collecting the light power encoded signal. However, in the case of free optics, +the nonlinear optical activation functions are realized in a complicated manner, e.g. with +the laser-cooled atoms with electromagnetically induced transparency [47]. Finally, on-chip +approaches based on wavelength multiplexing [48] or beamsplitter meshes [49] can achieve +programmable all-to-all coupling but need to scale better. One on-chip design was proposed +in [50], where the optical platform takes advantage of encoding information in both phase +and magnitude, thus making it possible to execute complex arithmetic by optical interfer- +8 + +ence, which suits performing handwriting recognition tasks. Mach–Zehnder Interferometers +(MZIs) can perform many functions, such as dividing and modulating the light signals, sep- +arating the reference and the main light beams, and implementing a complex-valued weight +matrix. +FIG. 2. (a) One layer of an optical NN with k layers consists of matrix-vector multiplica- +tion (grey) and non-linearities (red). (b) One-level implementation. Matrix multiplication +is performed by combining the input and weight signals and performing balanced homodyne +detection. The final signals are sent through a non-linear function (red), serialized, and sent +to the following layer’s input. Figure from [51] +. +Tunable waveguides can multiply optical signals, while wavelength-division multiplexing +can add signals. Wavelength-division multiplexing can be achieved by the accumulation of +carriers in semiconductors [52, 53], electronic currents [54, 55], or photon-induced changes +of the material [56]. To achieve the full potential in on-chip architectures, one must require +long-range connections between neurons, assisted with photonic waveguides that outper- +form metal wires connections of conventional electronics but fall behind free-optics solutions. +In particular, silicon photonic platforms demonstrated efficient neuromorphic architectures +[48, 49, 54]. An array of beam splitters and phase shifters can implement unitary matrix +9 + +(0) +文(1) +(2) +文(3) +(a) +(+ +口 +口 +口 +f +4 (0) +A (K-1) +口 +口 +Copies of x(k) +Combiner +Multiplier, +Nonlinear +Readout, +Conversion +(b) +(beam splitter) +integrator +function +serialization +to optical +f +(k) +(k+ 1) +f +Source +F +Output +Input +? +f +Xi +Weights +A +(k) +(k) +.N +Optical signal +Electronic signal +Optical signaltransformations using interference between different paths of coherent input light, where +inputs are assigned to different waveguides and power modulated [57]. +Modulating the +effective refractive index of signal-carrying waveguides is another optical mode-based ap- +proach to weight configuration. Non-volatile synapse implementations have been referred to +as all-optical because they do not need electrical inputs for tuning. These may use optically +induced changes in chalcogenide materials to control the light propagation in waveguides +[58]. Weight configurations based on non-volatile optical materials could lead to improved +heat dissipation. +FIG. 3. In fully functioned 2-layer all optical NN the first layer comprises a linear operation +by the first SLM (SLM1) which encodes a certain pattern and a nonlinear activation function +based on the electromagnetic induced transparency at magneto-optical trap (MOT). The +second layer contains the second SLM (SLM2), converting four beams into two output beams +at camera C3. The collimated coupling laser beam passing lens L1 is incident on the SLM1, +which generates four beams at the focal plane of L3, which is monitored by a flip mirror (FM) +and camera C1. Four beams are imaged on the MOT through a 4-f system comprising L4 +and L5. A probe laser is going opposite the coupling beam, which is imaged on camera C2 +through L5 and L6. L7 and L8 achieve further amplification. Four beams are incident on +SLM2, generating two beams and then focusing on camera C3. Figure from [47]. +A scheme based on homodyne detection has several scaling advantages over on-chip ap- +10 + +L2 +MOT +L5 +L4 +FM +L3 +SMF +2/4 +SLM1 +2/4 PBS +Probelaser +L6 +L1 +C1 +FM +SMF +L7 +Coupling laser +L8 +L9 +8 +SLM2proaches, including linear (rather than quadratic) chip-area scaling and constant circuit +depth [51]. The input vector in this implementation is encoded onto a pulse train, which +is fanned out to an array of homodyne detectors where each detector computes a product +between the input and a row of a matrix encoded into optical pulse trains. The accumulated +charge on the homodyne detector performs matrix-vector multiplication. The output is sent +through an electrical nonlinearity and converted back to optical signal using a modulator. +The advantage of the homodyne detection scheme is that the matrix elements (weights) are +encoded optically and can be dynamically reconfigured. This procedure requires a reduced +number of photonic components: the number of modulators, detectors, and beamsplitter +grows linearly with the number of neurons. The homodyne detection architecture can be +parallelized to implement general matrix-matrix multiplication by routing the light out of +plane [51]. This is useful in practical NNs that reuse weights (either natively in convolutional +layers or through batching). +Nonlinearity in ANN is required to implement the thresholding effect of the neuron. Some +photonic devices exhibit nonlinear neuron-like (gate-like) transfer functions. However, the +challenge is to achieve cascadability. Photonic neurons must be capable of reacting to multi- +ple optical inputs, applying a nonlinearity and producing an optical output suitable to drive +other photonic neurons. Integrated photonic solutions use either optical/electrical/optical +(O/E/O) or all-optical design to achieve such cascadability. In the O/E/O approach, nonlin- +earities may be introduced during the E/O conversion stage by employing lasers, saturated +modulators or photodetector–modulators [59] or in the electronic domain only (e.g. the non- +linear dynamics of spiking photonic neurons could be implemented with a superconducting +electronic signal pathway [60]). +NN architectures can take different forms: feed-forward and back-forward, layered and +recurrent, spiking or continuous etc. Each neural model has a different signal representation, +training method and network topology. Weight configurations can differ depending on the +training type: supervised training, unsupervised or programmatic ‘compilation’. Topology +describes the graph structure of neuron connectivity, and often it is advantageous to ANN +operation to constrain the topology to guide weight configurations. Therefore, hardware im- +plementation details may differ between different ANN, while the key technologies necessary +for practical realization include active on-chip electronics and light sources. +Many pho- +tonic architectures have already been demonstrated: recurrent ANN, continuous-time and +11 + +FIG. 4. Complex-valued coherent optical NN. (a) Scheme with an input layer, multiple hidden +layers, and an output layer. (b) The schematic of the optical neural chip in implementing +complex-valued networks. The single chip performs all stages, such as the input preparation, +weight multiplication and coherent detection. The division and modulation of the light signals +(i1 − i6) are realized by the MZIs (red). Green MZI separates the reference light. Blue MZIs +are used to implement the 6×6 complex-valued weight matrix. Grey MZIs are used for on-chip +coherent detection. (c) The workflow of the ONC system. Figure from [50]. +programmed by compiler [48]; feed-forward, single-valued and externally trained ANN [49]; +spiking, feed-forward ANN with both external and local training [56]; feed-forward multilayer +ANN with semiconducting few-photon light-emitting diodes and superconducting-nanowire +single-photon detectors [54]; diffractive networks with a nonlinearity [47]. The computa- +tional tasks solved by these platforms cover the main functions attributed to ML and AI: +image and audio recognition and classification, simulation of dynamical systems, combina- +12 + +(a) +Thearchitectureofopticalneuralnetworks +A single hidden layer +P1,01 +P2,02 +P3,03 +W1 +Wn +Pn,ON +input +hidden +output +(b) +Complex-valuedneural network chip +Electricalcontrol +nnnnn +Lase +Detector +Network input preparation (Win) +Weightmultiplication&accumulation(W) +Coherentdetection +(c) +External ML +model +Feedback signal +signal light +0 +Amplitude & +Optical +Coherent +Coherent +Classical +phase +neural +detection +output +laser +PC +processing +modulator +network +Reference lighttorial optimization and many other applications, which we will discuss in Section V. Some +of the architectures and their different experimental realizations are shown in Fig. 2, 3 and +4. +The key merit of NN hardware is the level of energy consumption, which can be evalu- +ated as petaMAC (multiply-accumulate operations) per second per mm2 processing speeds +[61] and attojoule per MAC energy efficiencies [62]. In general, current optoelectronic hard- +ware offers great advantages for implementing ANN, but eliminating the electrical contri- +bution will inevitably be beneficial. For practical applications of neuromorphic photonic +systems, one needs to reduce heat dissipation during information transfer between electrons +and photons. Such reduction can be achieved by improving optical sources, high-efficiency +modulators, and photonic analogue-to-digital interfaces. Current photonic platforms lack +the functionality of electronic processors such as logic gates, high-level compilers and as- +semblers, analogue–digital–analogue conversion and memory. Although photonics provides +advantages in connectivity and linear operations over electronics, on-chip memory is chal- +lenging. ‘In-memory’ computing, where processing is performed in situ with an array of +memory devices called memristors, has been established [63, 64]; however, reading and writ- +ing at high frequencies is still challenging. The recent trends in the development of the ANN +show the increasing demand to lower the power consumption of the devices. At the same +time, the requirements for parallelism and scalability remain the same through the years +[65]. Thus, the optical domain offers a promising solution to future hardware requirements. +II.4. +Reservoir and other neuromorphic computing systems +Reservoir computing (RC) is a recurrent NN-based framework for computation that maps +input signals into a specific computational space of the fixed nonlinear system dynamics. +This system is usually called a ”reservoir”, and its state is passed to a simple readout +mechanism, specifically trained to get the final output [66]. The original concepts of RC +can be traced to the liquid-state machines [67] and echo-state networks [68]. Many physical +systems can reproduce this computational framework, and the optical/photonics domain +is no exception. The extension of RC to deep hierarchical RC allows one to create more +efficient models and simultaneously investigate the inherent role of layered composition in +recurrent structures. Another promising research direction is to combine RC with quantum +13 + +physical systems to access larger computational space. +The idea of RC is to exploit the rich nonlinear dynamics of controllable nonlinear sys- +tems and simultaneously overcome the disadvantages of recurrent architectures with their +challenging and time-consuming training for both hardware and software systems. The RC +training is performed only at the readout stage, as the reservoir dynamics are fixed. This +readout framework enjoys the benefits of a particular photonic physical system, such as speed +or energy consumption, reducing learning costs. Another RC benefit is learning temporal +(dynamic) dependencies compared to the feed-forward architectures used for static (non- +temporal) data processing. The simplicity of the training procedure in RC is attractive. +However, accessing complex dynamics without rigorous understanding can lead to many +problems. Operating within the RC framework usually needs extensive experiments and +experimental verification due to the need for a unified theory of RC. Another disadvantage +is the performance instability due to the noise present, typical for nearly chaotic dynamical +systems. +Nevertheless, many successful cases of RC are being applied to practical problems, such as +temporal pattern classification, time series forecasting, pattern generation, adaptive filtering +and control, and system approximations. +Moreover, RC can be used conventionally for +static data processing. The first all-optical implementation of RC was demonstrated within +a simple optical delayed feedback loop combined with the nonlinearity of an optical amplifier +[70]. Concerning the free-space optics principles, an image processing task was successfully +solved using a predesigned configuration with a diffraction grating and Fourier imaging with +randomly interconnected microring resonators [71]. The reservoir consisting of a diffractive +optical element was described based on an 8 × 8 laser array (of VCSELs) and an SLM. +It showed rich dynamics with the potential for scaling up [72]. Further modifications of +this setup with a laser illumination field and digital micro-mirror device allowed one to +realise the large-scale RC scheme with 2025 diffractively coupled photonic nodes applied +to a time series prediction task [73]. The recurrent 24-node silicon photonic NN, in which +microring weight banks configure connections, was programmed using a “neural compiler” to +solve a differential system emulation task with a 294-fold acceleration against a conventional +benchmark [48]. +Some hybrid architectures, such as opto-electronic devices, similarly benefit from the +RC concept. For example, excellent performance has been obtained for speech recognition +14 + +FIG. 5. (a) Schematics of proposed reservoir computing (RC) architecture. The electric input +signal u(t) is coded on optical pulse, δ(t) is coded by optical modulator. +The sinusoidal +nonlinearity is achieved by the electro-optic conversion (F in). RC system comprises the L- +array of optical cavities with N temporal nodes with N × L virtual nodes. Photodetectors +(PDs) get the output signals from optical circuit and generate nonlinear conversion (F out). +The digital processing unit detects signals |xl|2 and weights and sums them to obtain the +final output y(t). (b)Schematic of the waveguide layout for input mask circuit with the three +types of installations for optical connection, and (c) - input mask reservoir circuit. The input +weights hl and reservoir parameters can be tuned by the phase shifter, MZI and variable +optical attenuator setup. Figure from [69]. +[74–76], chaotic time series prediction [75, 77, 78], and radar signal forecasting [79], with +the operating speed in the megahertz range and the potential to increase it to gigahertz +speed, at the same time preserving the state-of-the-art numerical accuracy. Additional cases +of successful RC have been reported in literature [66, 74, 80]. We will consider the NN +and RC architectures cases involving quantum effects in Section VII.1. Another example is +15 + +a +Photonic input mask +Photonicreservoir +Digital readout +101 +N temporal nodes +a +s;(t) +h,u(t) +s(t) +Fout(x) +x;(t) +y(t) +u'(t) +[Fin(u) +L spatial nodes +N ×L reservoir nodes +Opticalpulse +1:N +Delay lines +N xL opticali +L-arrayed coherent cavities +generator +splitter +connection +(t) +(t)n +NO +Digital +(t) +(0),n +: (N-1)e +processing +MM +Optical +20 +M +modulator +0 +VOA and PS +PDs +b +c +Setinputmask +Setinputmask +parameters +parameters +MZI unit cell +heaters +1:N splitter +Coherentcavities +Delay lines +withVOAandPS +Opticalconnection +axi(n-1) +Type I: Sparse +Type Il: Dense +Type lll:Fullyprogrammable +(PI +a +si(n)- +(1-α)xi(n) +IPS +MZI +I VOAillustrated by Fig. 5. +III. +NONLINEAR OPTIMIZATION SPECIFIC OPTICAL MACHINES +A large class of problems that can be solved by optical hardware includes nonlinear +programming problems. They seek to minimize a nonlinear objective function E(x) of real +or complex variables in x subject to a series of constraints in the form of equalities or +inequalities, i.e. g(x) ≤ 0 and h(x) = 0. Such a general framework can include many +applications across social sciences, finance, telecommunications, aerospace, biological and +chemical industries [81]. +Many nonlinear optimisation problems present a significant challenge as the number of +operations to solve them usually grows exponentially fast with the number of variables. +This algorithmic complexity is the reason for using specialised techniques such as genetic +algorithms, particle swarm optimisation, simulation and population annealing. Quadratic +programming (QP) for minimising quadratic functions of variables subject to linear con- +straints is a usual simplification of such problems because nonlinear optimisation problems +are quadratic to second order around the vicinity of the optimal solution. Such approxima- +tion can be successfully performed even outside the feasible solutions space. QPs can be +met in the least squares regression or as a part of a bigger problem, such as support vector +machine (SVM) training. The apparent correspondence between the QP objective function +and 2-local spin Hamiltonians of various physical systems allows one to map the problem +into the physical setup. Here, the degrees of freedom x are associated with “spins” and +the cost function E(x) is associated with a “Hamiltonian” that specifies the interactions +patterns and strengths between spins. +There are several possible ways such a system can find the optimal solution or the ground +state of the corresponding spin Hamiltonian. Depending on the nature of the system, it can +use either quasi-equilibrium or non-equilibrium regimes. +The system in thermodynamic +equilibrium may find the optimal solution by quantum annealing, which is executed with +the time-dependent Hamiltonian +H(t) = +� +1 − t +τ +� +H0 + t +τ Hobjective, +(1) +16 + +where H0 is the initial trivial Hamiltonian with known ground state, and Hobjective is the +final Hamiltonian at t = τ which encodes an original objective function E. One can keep +the system at the ground state during adiabatic evolution from H0 to Hobjective. For the +adiabatic transformation, the time scaling τ for the system to remain in the ground state +must be much larger than that defined by the inverse of the spectral gap [82]. However, the +process becomes inefficient for larger systems and sophisticated (glassy) Hobjective because +the spectral gap typically shrinks exponentially fast with the system size. The excited states +lead to large errors while simultaneously slowing down the annealing procedure. +Many open non-equilibrium gain-dissipative systems, such as lasers and photonic or po- +laritonic condensates are non-Hermitian systems and, therefore, do not have a ground state. +Instead, they tend to minimise losses on the route to coherence. One can use geometric +analogies to describe their operational principle as the approach of the surface of the op- +timisation cost function (loss landscape) from below. There are two main processes that +lead to loss minimisation: bosonic stimulation below the threshold and the coherence of +operations at the threshold responsible for the quality of the solution. After increasing the +gain to the point where it overcomes the linear losses and is stabilised by the nonlinearity +of the gain saturation, the emergent coherent state minimises the losses (equivalent to max- +imisation of the total number of particles). It hence achieves the loss minimisation mapped +into the objective spin Hamiltonian. The system elements’ resulting evolution closely resem- +bles the Hopfield Networks’ dynamics, proposed to be used to solve quadratic optimisation +problems forty years ago [83]. Despite the successes and a lot of excitement generated back +then, the optimisers based on Hopfield networks were almost forgotten primarily due to +the high connectivity required between neurons and the concomitant evolution time of the +networks used by classical architecture. The recent interest in Hopfield networks reemerged +as it became possible to emulate them with physical systems such as electronic circuits or +photonic NNs. Photonic systems have an advantage over their electronic counterparts due +to the picosecond to the femtosecond time scale of their operation. At the same time, many +signals can flow through a single optical waveguide. As a result, a photonic implementation +of Hopfield networks as optimisers can have large dimensionality, dense connectivity, and a +fast convergence time to the optimum solution. +17 + +III.1. +Spin Hamiltonians +The most real-life decision or optimisation problems present a severe challenge to con- +ventional classical computers, with classic examples of a so-called “hard optimisation task” +being the travelling salesman problem, the dynamic analysis of financial markets, the pre- +diction of new chemical materials, and ML. Mathematically, many of these optimisation +problems admit a reformulation into the problem of finding the ground state of a particular +spin Hamiltonian, which can be emulated with a given simulator (e.g. solid-state or optical +system). The overhead in the number of additional variables needed during this mapping is, +at most, polynomial [84]. However, better still, such a system needs to easily map the vari- +ables of the desired objective function into spin Hamiltonian elements (spins, currents etc.). +Additionally, one wants to independently tune short and long-range interactions between +the elements and perform measurements to obtain the answer with the required precision. +Such spin model Hamiltonians are experimentally challenging to implement and control. +Still, the possible advantages in dealing with large problem sizes lead to an intensive search +for a superior simulator. Such simulators have been realised to various extents in different +physical systems and are covered in this review. Through all these systems, two classes of +spin Hamiltonians are generally considered: Ising and XY. The Ising model attracts the +most attention since an extensive range of challenging discrete combinatorial optimisation +problems, e.g. travelling salesman, graph colouring, graph partitioning, etc., can be mapped +into it [84]. This model is formulated for N classical “spins” sj that take discrete values +{−1, 1} to minimise the quadratic unconstrained binary optimisation problem (QUBO): +min +si +− +N +� +i=1 +N +� +j=1,j 2 and would allow for a direct mapping +of higher-order binary optimization problems (HOBO) including Max-SAT [85] or number +18 + +factorization [86] +min +si +− +N +� +i1,i2,...ik +Qi1,i2,...il,...,iksi1si2...sil...sik +subject to +sil ∈ {−1, 1}. +(3) +In the XY model “spins” are continuous vectors sj = (cos θj, sin θj) and the corresponding +quadratic continuous optimization problem (QCO) can be formulated as +min +si − +� +i,j 2 the model (4) recovers the n−state Potts model +with applications in protein folding [91]. +The appearance of continuous spins is a common feature in many optical systems because +short photonic impulses can be characterized through amplitude and phase variables. Some +of the optical hardware for ML take advantage of this feature. For example, complex-valued +NNs [50], or more unusual concept of analogue transformations using a nonlinear set of +functions were proposed [92]. +III.2. +P, NP, NP-complete problems +The computational complexity of a problem is determined through the dependence of the +problem’s size on time or the number of operations required to solve it. A problem belongs to +a P class when a polynomial algorithm exists for solving it (e.g. searching for the maximum +element in an array with no prior information). Suppose there exists a polynomial algorithm +for verifying a solution. In that case, the problem belongs to a non-deterministic polynomial- +time (P ∋ NP), which does not always have an efficient (polynomial time) method of finding +a solution. Whether P = NP true or not is a major unsolved problem in computer science, +although it is widely believed to be untrue. Most difficult problems in NP are called NP- +complete. They are equivalent to each other in that either all of them or none admit a +polynomial-time algorithm (e.g. the travelling salesman problem, spin glass models and +integer linear programming are in general NP-complete problems). A problem is called NP- +hard (informally, the hardest problems in NP) if the existence of an efficient algorithm for +19 + +its solution implies the existence of such an algorithm for all the NP-complete problems. +In general, if a decision problem with a yes or no answer (e.g. does a particular Ising +Hamiltonian have a ground state energy less than some value?), is NP-complete, then its +corresponding optimization problem (e.g. +what is the ground state energy of this Ising +Hamiltonian?), is said to be NP-hard. It means that NP-hard problems are not any easier to +solve than the corresponding NP-complete decision problems. The computational complexity +of the Ising model has been studied before [93] where the Ising model with a magnetic field +(2) and equal antiferromagnetic couplings was shown to be NP-hard for planar graphs. NP- +hardness was demonstrated for the three-dimensional Ising model with nearest neighbour +interactions and coupling strengths randomly drawn from {−1, 0, 1}. +The NP-hardness +implies the hardness of the hardest instances for the considered problems, while the average +problem can be polynomially easy. +The existence of universal spin Hamiltonians has been established [94]. +Universality +means that all classical spin models can be reproduced within such a model, and certain +simple Hamiltonians, such as the 2D Ising model on a square lattice with transverse fields +and nearest neighbour interactions of infinite precision are universal [94]. +Thus, due to +NP-hardness of the Ising model, there should exist a polynomial time mapping of many +practically relevant NP-complete problems to the Ising Hamiltonian, whose decision version +solves the NP-the complete problem of interest. +The mapping of various NP problems, +including Karp’s 21 NP-complete problems, to Ising models with a polynomial overhead +were explicitely formulated [84]. +A problem belongs to P class only if all its instances can be solved in polynomial time, +while for a problem to be NP-complete is enough to have some instances that are hard to +solve. Such instances are said to represent worst-case scenario behaviour. How to distinguish +hard instances from simple ones is a cornerstone question of analogue physical optimisers. +Such understanding is necessary to evaluate their scalability and efficiency [95]. +It is believed that the procedure for creating “hard” instances for spin Hamiltonians may +be found at the intersection of computational complexity and statistical physics, e.g. the +hardness of problems can be connected to the existence of a first-order phase transition in +a system (see [96–99] and references therein). Indeed, even a medium size hard instance is +difficult to solve on a classical computer due to the exponential growth of operations with +size. Thus, the time required to find the ground state energy depends on the coupling matrix +20 + +structure J. For instance, finding the global minimum of the XY model for positive definite +matrices remains NP-hard due to the non-convex constraints. Still, it can be effectively ap- +proximated using a semidefinite programming relaxation with some performance guarantee +[100, 101]. Sparsity also plays an important role, and for sufficiently sparse coupling matri- +ces, fast methods exist [102]. Having a unified set of optimization problems with tunable +hardness and known solutions is an ongoing research direction. It will allow for an objective +benchmark of classical and/or quantum simulators and algorithms. Otherwise, it would be +hard to evaluate the performance of state-of-the-art platforms and methods. +Current research made a good starting point in developing a standardised procedure for +performance evaluation. For example, the “optimisation simplicity criterion” was recently +proposed to identify computationally simple instances [95]. Optical machines with their +mode selection operation often follow the dominant eigenvalue of the coupling matrix and +find minimisers that correspond to the signs of the principal eigenvector components. If +the minimisers of a given problem have this property, the solution will be found easily in +polynomial (at most quadratic) time. One such popular example is the Ising model on the +M¨obius ladder graph [95]. By rewiring the M¨obius ladder graph to random 3-regular graphs, +one can probe an intermediate computational complexity between P and NP-hard classes +with several numerical methods. Another way to construct instances for testing involves +planted ensemble technique [99, 103]. +IV. +DESCRIPTION OF PHYSICAL OPTICAL PLATFORMS FOR OPTIMIZA- +TION +Rather than trying to model nature, one can consider a reverse idea of exploiting physical +phenomena for solving NP-complete problems. The concept of using simulators or analogue +processing devices is quite old; see, for example, [104]. +However, in the last years, one +can observe a competition of different physical platforms in solving classical optimization +problems faster than it can be achieved on conventional hardware. This competition resulted +in the rapid emergence of a new field of Hamiltonian simulators at the intersection of laser +and condensed matter physics, engineering and complexity theories. Here we discuss various +physical systems that appeared as promising platforms for solving computational problems. +21 + +IV.1. +Complex laser networks +A new generation of complex lasers such as degenerate cavity lasers, multimode fibre +amplifiers, large-aperture VCSEL, and random lasers have many advantages compared with +the relatively simple traditional laser resonators of their computing properties [105]. They +have many spatial degrees of freedom, their nonlinear interactions within the gain material +can be controlled by adjusting the spatial structures of lasing modes, the spatial coherence +of emission can be tuned over a wide range, and the output beams may have arbitrary +profiles. These properties allow the complex lasers to be used for RC [106] or mapped to +hard computational problems. +In laser networks, the coupling can be engineered by mutual light injection from one laser +to another. This introduces losses that depend on the relative phases between the lasers. +Such dissipative coupling drives the system to a phase-locking that minimises losses. If the +amplitudes of lasers are about the same, a steady-state minimum of the XY Hamiltonian +is found [107]. +Degenerate cavity lasers are beneficial as solvers as all their transverse +modes have nearly identical Q. This implies that a large number of transverse modes lase +simultaneously since they all have similar lasing thresholds [105]. +The evolution of the N single transverse and longitudinal modes class–B lasers can be +described by the rate equations [108, 109] on the amplitude Ai, phase θi, and gain Gi of the +i-th laser +dAi +dt = (Gi − αi)Ai +τp ++ +� +j +Jij +Aj +τp +cos(θi − θj), +(5) +dθi +dt = Ωi − +� +j +Jij +Aj +τpAi +sin(θi − θj), +(6) +dGi +dt = 1 +τc +[Pi − Gi(1 + |Ai|2)], +(7) +where Pi, αi, Ωi represent the pump strength, loss, frequency detuning of laser i, respectively, +whereas τp and τc denote the cavity round trip time and the carrier lifetime, respectively. +The coupling strengths between i-th and j-th lasers are represented by Jij. If the amplitudes +of all lasers are equal, Eq. (6) reduces to the Kuramoto equation of coupled phase oscillators +dθi +dt = Ωi − 1 +τp +� +j +Jij sin(θi − θj). +(8) +22 + +Equation (8) is a celebrated Kuramoto model of identical oscillators, which is widely used +to describe the emergence of coherent behaviour in complex systems [110, 111]. By LaSalle +invariance principle [112], every trajectory of the Kuramoto model converges to a minimum +of the XY Hamiltonian. +It was shown that the probability of finding the global minimum of the XY Hamilto- +nian agrees between the experimental realization of the laser array and with the numerical +simulation of Eqs. (5-7). However, simulating the Kuramoto model Eq. (8) on the same +matrix of coupling strength gives a much lower probability of finding the global minimum. +This result implies that the amplitude dynamics described by Eq. (5) provide a mechanism +to reach lower energy states by pumping from below [109]. Consequently, the cavity lasers +can be used as an efficient physical simulator for finding the global minimum of the XY +Hamiltonian and, therefore, for solving phase retrieval problems. A particularly successful +in these tasks was a digital degenerate cavity laser [90]. It is an all-optical system that uses +a nonlinear lasing process to find a solution that best satisfies the constraint on the Fourier +magnitudes of the light scattered from an object. To ensure that the solution to the phase +retrieval problem is found, the compact support aperture is introduced inside the cavity, +ensuring that different configurations of laser phases compete to find the one with minimal +losses. The system combines the advantages of short round-trip times of the order of 20ns +and high parallelism in selecting the winning mode. +IV.2. +Coherent Ising machine +A network of coupled optical parametric oscillators (OPOs) is an alternative physical +system for solving the Ising problem ([113–119] and references therein). Each OPO is a non- +linear oscillator with two possible phase states above the threshold that can be interpreted +as the Ising spins. These artificial Ising spins are encoded by the optical phase of short laser +pulses generated by a nonlinear optical process, i.e. optical parametric amplification. The +OPO-based simulator, coherent Ising machine (CIM), is a gain-dissipative system in which +the ground state of the Ising Hamiltonian corresponds to the lowest loss configuration. The +optimal solution is found by driving the system close to the near-threshold regime, where +other local energy minima are still unstable. +Currently, most successful implementations of CIMs use a fibre-based degenerate OPOs +23 + +(DOPOs) and a measurement-based feedback coupling, in which a matrix-vector multipli- +cation is performed on the FPGA embedded in the feedback loop, see the scheme depicted +in Fig. 6. The computational performance of such a scalable optical processor, bounded +by the electronic feedback, was demonstrated for various large-scale Ising problems [113– +115]. The comparison of a possible CIM’s speedup over classical algorithms is an ongoing +study [116, 120]. Furthermore, the ability to implement arbitrary coupling connections [113] +between any two spins implies better scalability than the solid-state based annealer, i.e. +D-Wave machine [114]. +FIG. 6. Schematics of the coherent Ising machine (CIM) with the feedback mechanism. The +time-multiplexed pulse degenerate parametric oscillator is formed by a non-linear crystal (pe- +riodically polarized lithium niobate (PPLN)) in a fibre optic ring cavity containing 160 pulses. +The feedback signal couples the independent pulses in the cavity and is computed from the +measurements from different pulse fractions.IM - intensity modulator; PM - phase modulator; +LO - local oscillator; SHG - second-harmonic generation; FPGA - field-programmable gate +array. The figure is taken from [113]. +In CIM, each Ising spin corresponds to a DOPO that is described by a stochastic equation +for the complex amplitude of the signal field ai: +dai +dt = pa∗ +i − ai − |ai|2ai + +� +j +Jijaj, +(9) +24 + +Pump +PPLN Waveguide +SHG +OPO160 +OPO1 +Pulsed Laser +OPOs4to157 +OPO 2 +1560 nm +OPO159 +OPO158 +OPO3 +Injection +Measurement +IM +FPGA +PM +LO +Feedback Calculations +LO +Fiberbeamsplitterwhere the dynamics is defined by a linear pump term p, normalised linear and nonlinear +losses, and mutual couplings Jij. To experimentally realise these couplings, a portion of +the light is extracted from the cavity after each round trip. That light is then homodyned +against a reference pulse to produce ai that is supplied to the FPGA, where a feedback +signal is computed for each pulse. Lastly, an optical modulator is applied to convert the +signal back to light for the next round trip. The equations (9) are often reformulated in +terms of the in-phase and quadrature components ai = ci + isi giving the equations in real +terms: +dci +dt = +� +p − 1 − (c2 +i + s2 +i ) +� +ci + +� +j +Jijcj +(10) +dsi +dt = +� +− p − 1 − (c2 +i + s2 +i ) +� +si + +� +j +Jijsj. +(11) +The computational effectiveness of these equations has been demonstrated by tackling small +size Ising type problems of order up to 20 [118]. In part devoted to polariton condensates, +we will show that for achieving the global minimum, the realisation of an individual pump +variation pi for equalising all signal amplitudes |ai| is crucial. +Phase stability for the cavity’s whole length is required, making the DOPOs system +highly susceptible to external perturbations that can affect performance [114]. Furthermore, +the nonlinear DOPO generation process demands powerful laser systems and temperature- +controlled nonlinear materials, which results in large and complex optical setups. These +issues have led to recent proposals of other physical platforms for implementing a CIM-like +machine. A CIM based on optoelectronic oscillators with self-feedback was suggested to +be more stable and cheaper based on solving Ising optimisation problems on regular and +frustrated graphs with up to 100 spins, and similar or better performance compared to the +original DOPO-based CIM [117]. An analogue all-optical implementation of a CIM based on +a network of injection-locked multicore fibre lasers demonstrated the possibility of solving +Ising Hamiltonians for up to thirteen nodes [121]. The dynamics of a network of injection- +locked lasers were based on nonlinear coupled photon rate equations, and the couplings +were implemented using SLMs. The couplings were reported to be dependent on the photon +numbers that have yet to be discovered beforehand, which can be a significant obstacle in +solving a given Ising Hamiltonian with the proposed photonic CIM. To resolve this issue +25 + +approaches similar to gain feedback [122, 123] may be considered in future. Another large- +scale optical Ising machine based on the use of an SLM was experimentally demonstrated by +using the binary phases in separated spatial points of the optical wavefront of an amplitude- +modulated laser beam and realising configurations with thousands of spins with tunable +all-to-all pairwise interactions [124]. +CIM’s essential elements are DOPOs with an unconventional operating mechanism called +mode selection or gain-dissipative principle. Here we briefly describe this operational regime: +Each neuron is prepared in a linear superposition state of different excitations to implement +a quantum parallel search. The cost function is mapped to the effective loss, photon decay +rate, of the given network by setting the coupling coefficient proportional to the Jij, which +encodes the information about the given task. The ground state of the Ising Hamiltonian +corresponds to an oscillation mode with the minimum network loss. The system reaches +the ground state with a minimum loss at the threshold pump rate. It starts oscillating as a +single stable mode, which triggers photons’ stimulated emission and affects the saturation +for all the other modes. Detecting this single oscillation mode will give us the solution to +the desired problem. +IV.3. +Photon and polariton networks +Photons have both attractive and not properties concerning computational assignments. +However, despite the commonly known optical platforms, such as free optical setups or sys- +tems of lasers, it is possible to bind the photons with the matter wave excitations. This +gives rise to unique designs, combining the photons with matter, such as exciton-polaritons. +Microcavity exciton-polaritons, or simply polaritons, are quasi-particles that result from the +hybridisation of light confined inside semiconductor microcavities and bound electron-hole +pairs (excitons). +The steady states in these nonequilibrium systems are set by the bal- +ance between the pumping intensity, coming from the interconversion rate of the exciton’s +reservoir into polaritons, and losses, happening due to the leakage of photons. Polaritons +are bosons and obey Bose-Einstein statistics. Therefore, they can form a condensed (co- +herent) state above a critical density [125]. Thus, polaritons offer a unique playground to +explore nonequilibrium condensation and related effects in solids. The advantage for such +explorations comes from the polariton’s small effective mass of 4-5 orders of magnitude +26 + +smaller than the electron’s mass. The design and choice of material allow one to control +the polariton mass and realise such solid-state nonequilibrium condensates not only at cryo- +genic temperatures but also at room temperature in organic structures. The weak coupling +at high temperatures and high pumping intensities transitions continuously to strong cou- +pling at lower temperatures and lower pumping intensities. In the limit of a small gain, +i.e. +small losses, solid-state condensates resemble equilibrium Bose-Einstein condensates +(BECs). They approach the lasers in the regime of high gain, i.e. high losses. This transi- +tion from the equilibrium BECs to normal lasers was described with a unified approach via +polariton condensates [126]. +In another system, closely resembling the physics of polariton condensates, macroscopic +occupation of the lowest mode for gas of photons confined in a dye-filled optical microcavity +was recently shown [127–130]. The rapid thermalization of rovibrational modes of the dye +molecules by their collisions with the solvent and phonon dressing of the absorption and +emission by the dye molecules leads to the thermal equilibrium distribution of photons and +concomitant accumulation of low-energy photons. Such systems resemble microlasers [131], +but unlike microlasers, they exhibit a sharp threshold that occurs far below the inversion. +Many techniques have been proposed and realised in experiments to construct the lattices +of polariton or photon condensates. Polariton lattices can be optically engineered by inject- +ing polaritons in specific areas of the sample using the SLM [132–136]. Various potential +landscapes to confine polariton or photons have also been engineered [137–139]. The rate +equations describing the evolution of gain-dissipative condensates in a lattice were derived +using the tight-binding approximation of the space and time-resolved mean-field equations +[123, 140] and take the form of the Stuart-Landau equations +˙Ψi = −iU|Ψi|2Ψi + (γi − |Ψi|2)Ψi + +� +j̸=i +CijΨj, +(12) +where Ψi = √ρi exp[iθi] is the complex amplitude of the i−th condensate, U is the strength +of self-interactions between the quasi-particles, γi is the effective injection rate (the difference +between the pumping of the quasi-particles into the system and linear losses). The coupling +strength Cij = Jij + iGij is generally a complex number and consists of the Heisenberg +coupling Jij mediated by the injection reservoir and the Josephson part Gij that comes from +exchange interactions between the condensates. The system described by Eq. (12) reaches +27 + +the fixed point when Jij ≫ Gij and the pumping feedback is introduced in the system [123]. +The feedback on the pumping intensity ensures that all the occupations are the same at +the fixed point by adjusting the pumping if the occupation exceeds the set threshold value +|Ψi|2 = ρth. The total injection of the particles in the system of N condensates at the fixed +point is given by +N +� +i=1 +γi = Nρth − +N +� +i=1 +N +� +j maxi +� +j |Jij| at the +threshold. At the fixed point, Eq. (13) is replaced with +N +� +i=1 +γi = Nρth − +N +� +i=1 +N +� +j 2, the n-state Potts +Hamiltonian is minimized. The minimization of HOBO may be achieved when the system +operates much above the threshold, and higher-order terms must be addressed [142]. +FIG. 7. Top: Schematic of the condensate density map for a five-vertex polariton graph. The +sign of the coupling depends on the separation distance between the sites and is either ferro- +magnetic (solid-blue lines) or anti-ferromagnetic (dashed-red lines). Each vertex of the graph +polaritons represents a local phase mapped to a classical vector spin. Bottom: schematics of +different types of annealing for finding the global minimum of the energy landscape of the +simulated XY Hamiltonian [136]. +If the time evolution of the reservoir of noncondensed particles is slow, the system of N +interacting coherent centres is better described by the following equations [140]: +˙Ψi = −iU|Ψi|2Ψi + (Ri − γc)Ψi + +� +j̸=i +JijΨj, +(17) +˙Ri = Γi − γRRi − Ri|Ψi|2, +(18) +where Ri is the occupation of the i−th reservoir, Γi, γR and γc characterize the rate of +particle injection into the reservoir and the linear losses of the reservoir and condensate, +29 + +Energylandscape +(a) +Energylandscape +(b) +abovethreshold +Classical +Quantum +Annealing +tunnelling +Bosonic +Bosonic +stimulation +stimulationrespectively. If one replaces Ψi by the electric field and Ri by the population inversion of +the i−th laser, the result is a form of the Lang-Kobayashi equations normally derived to +describe the dynamical behaviour of coupled lasers from Lamb’s semiclassical laser theory +[143, 144]. The total injection of the particles in the system of N condensates at the fixed +point is given by +N +� +i=1 +Γi = (γR + ρth)[Nγc − +N +� +i=1 +N +� +j 0, +−1, +otherwise, +(21) +with the same notation used in 20. The continuous version has the form: +dxi +dt = −xi +τ + +� +j +Jijg (xj) + hi, +(22) +where xi denotes the mean state of the i-th neuron that can get continuous values in the +initially defined range, hi is a direct input or bias coefficient in case the Lyapunov function +(20) has non-zero field, g is a monotone function that bounds the continuous states and con- +verts them into the discrete in the final state of convergence, i.e. makes the correspondence +between the variables σi = g(xj), and τ is the characteristic time (22) of the convergence to +an optimal or suboptimal solution. +The analogue computation with the NN can be described as an evolution of the vector- +state variables in the high-dimensional continuous space. One can precisely trace it using +Eq. (22). +The vital aspect of such a differential equation structure is an existence of a +Lyapunov function. This Lyapunov function H behind the Hopfield NN can lead to the un- +37 + +derstanding of possible final states, which appear to be attractors of the system’s dynamical +behaviour. For both models, one can realise the dynamical state update using a particular +hardware system described previously. However, one should differentiate between different +regimes that can be realised on the hardware level: the task of finding the ground state (the +global minimum) of the model and pattern restoration (descending on the surface of the +Lyapunov function towards its nearest minimum). +The explicit formula for the Lyapunov function in the discrete variant of the model with +the non-zero field is: +H = −1 +2 +N +� +i,j=1 +σiJijσj − +N +� +i=1 +hiσi. +(23) +In the case of continuous variables Eq. (22), the same function has a slightly different forms: +H = −1 +2 +N +� +i,j=1 +σiJijσj − +N +� +i=1 +hiσi + 1 +τ +N +� +i=1 +� σi +g−1(Z)dZ, +(24) +where the last term appears due to the correspondence between the discrete and continuous +state σi = g(xi). For g(x), one usually picks the g(x) = tanh(x/β) function, where the β +parameter tends to zero value during the evolution of the Hopfield NN forcing the last term +of the Eq. (24) to disappear, see [175] with the additional emphasis on the optimization +problems. The essential property of the dynamical update rules is that the energy decreases +through the system evolution, which leads to the final stable patterns in the phase space. +The classical Hopfield NN has many modifications for the Lyapunov function, variables +update rules and other features. +One version is known as modern Hopfield NNs [157]. +Modern Hopfield networks with continuous states can be integrated into deep learning ar- +chitectures because they are continuous and differentiable with respect to their parameters. +Moreover, they retrieve patterns with just one update, conforming to deep learning layers. +For these reasons, modern Hopfield networks can serve as specialised layers in deep networks +to equip them with memories. Possible applications of Hopfield layers in deep network ar- +chitectures find their way in multiple instance learning, defence against adversarial attacks +[176], processing of and learning with point sets, sequence analysis and time series prediction, +storing and retrieving reference data, e.g. the training data, outliers, high error data points, +prototypes and many other purposes [157]. Even more importantly, the functionality of the +modern Hopfield networks can be compared with various methods from the ML domain, +38 + +such as SVMs, random forest, boosting, decision trees, Bayesian methods and many others +[177, 178]. +As we mentioned above, many optical systems can perform optimization tasks. Since +there are intrinsic similarities between this task and the associative memory model, one can +exploit this relation to realize Hopfield NN on the optical setup. Such realizations include +previously discussed laser networks, Ising machines, photon [179] and polariton systems +[136], and confocal cavity QED NN [180], see Fig. 9. The connection between the optical +networks and the Hopfield model is important since it allows one to incorporate such layers +into more complex optical architectures without complicated adjustments. +FIG. 9. (a) Four nodes with the all-to-all coupling and sign-changing connectivity between +spin ensembles. Blue and red show ferromagnetic versus antiferromagnetic Jij links. One can +find the physical details in [181]. (b) The realization of the Hopfield NN by the spin ensemble. +Binary neurons si of a single-layer network are recurrently fed back and subjected to a linear +transform J with the consequent element-wise threshold operation. (c) The Hopfield model +exhibits an energy landscape with many metastable states. +Energy-minimizing dynamics +drive similar spin configurations to the stored local minimum, characterized by the basin of +attraction. Too many memories make the basins of attraction vanish. (d) Schematic of the +associative memory problem - recalling multiple stored patterns by completing distorted input +images. Figure from [181]. +IV.6. +Higher-order systems +One significant extension of the Hopfield model is incorporating the tensor terms, which +depend on the σi variables polynomially in n [182]. +The such extension allows one to +increase the number of stored patterns to Kmax = αnN n−1, where αn is a numerical constant. +Moreover, it is possible to observe the so-called ”feature to prototype transition” when +increasing n in the NN training. The prototype theory provides an alternative approach to +39 + +(a) +(b) +(c) +(d) +EnergyLandscape +Distorted +Corrected +Inputs +Outputs +Cs +Rb +Js +0E3 +Rb +S3 +Stored Patterns +Dy +Rb +Dylearning in which objects are recognized as a whole. Although tensor terms are assumed not +to be biologically plausible [158], they can be reproduced on some artificial physical setups +[142]. From this perspective, artificial tensor platforms can significantly benefit from such +technological opportunities. The higher order Hopfield NNs [183] can be written as +dxl +dt = −xl +τ + +� +¯Ω +Ak +l,i1,···,iksi1· · ·sik; +sl = g +�xl(t) +β +� +, +(25) +where xl are real continuous variables, g(x) is the threshold function and β is the scaling +parameter that can depend on time. Such systems can solve HOBO, see Eq. (3), because of +the k-local coupling. +It was shown [142] that polariton systems above the threshold are described by +dΨl +dt = Ψl(γl(t) − |Ψl|2) + +� +¯Ω +Ak +i1,···ikΨi1...Ψ∗ +ik, +(26) +dγl +dt = ϵ(ρth − |Ψl|2), +(27) +where ¯Ω is the set of indices that excluded index l. Eq. (27) describes the feedback mechanism +that drives all ρi to a priori set values ρth, ϵ characterizes how fast γi adjusts to changes +in ρi. Next, we proceed with the different ways of connecting the practical computational +tasks with the actual physical behaviour of the presented systems. +V. +MATHEMATICAL FORMULATION OF APPLICATIONS +This section considers a range of generic applications that follow from the network’s +ability to solve optimization problems or/and act as Hopfield networks. We start with the +simple problems from classical computer science with the corresponding mapping to the +QUBO problem. We then move to modern tasks that differ in information capacity and are +considered to suffer from the so-called ”curse of dimensionality”, where it is more suitable +to work with the probability distributions instead of the individual variables. +However, +in both cases, we do not pay attention to whether the presented mapping is efficient (like +in the following subsection) or not (when one needs multiple sequential operations with +a considerable amount of pre and post-processing in between). +Some of the inefficient +embeddings can still possess mathematical challenges and can be improved either in the +40 + +general formulation or with task-specific information. At the end of this chapter, we discuss +the NN architectures and their capabilities. +V.1. +Direct encoding/decoding +This subsection describes the connections/correspondences between different computa- +tional tasks established during the last 50 years [174, 184, 185]. +The propositional satisfiability problem (SAT) lies at the heart of such correspondence. +It is a fundamental problem determining whether a set of sentences in propositional logic +is satisfactory. A clause is built as the disjunction, the logical OR (denoted by ∨) of some +Boolean variables or their negations. +A set of several clauses, which must be satisfied +simultaneously, is the conjunction, logical AND (denoted by ∧) of the clauses. One can +write a satisfiability problem in the general form: +(x1 ∨ x2 ∨ ...) ∧ (y1 ∨ y2 ∨ ...) ∧ ...(...), +(28) +where the xi, yi are ”literals”, any of the original variables or their negations. The form (28) +is called a conjunctive normal form (CNF), and one can easily see that any logical statement +between Boolean variables can be written as a CNF. +SAT is the first problem that was proven to be NP-complete [174, 184]. Currently, no +known algorithm efficiently solves each SAT instance. The question of its existence is equiv- +alent to the famous P vs NP problem. Nevertheless, many heuristics SAT algorithms can +solve problem instances involving a significant number of variables, sufficient for many ap- +plications. +Additionally, many versions of the SAT problems exist, like 3-SAT and the +generalization k-SAT, HORN-SAT, and XOR-SAT, which can better suit particular uncon- +ventional tasks. +One specific SAT version - weighted MAX-2-SAT allows one to easily reformulate the task +as QUBO, often appearing in this review. A simple 2-SAT has m clauses of 2 literals each. A +MAX-2-SAT is the problem of assigning values that maximize the number of satisfied clauses. +Weighted MAX-SAT gives each clause a positive weight so that the measure of violating +the cost appears in the problem. +To reformulate a weighted MAX-2-SAT problem as a +QUBO, one has to use the fact that maximizing the weight of satisfied clauses is equivalent +41 + +to minimizing the weight of unsatisfied clauses, and using the logic xi ∨ xj = xi ∧ xj. The +final form looks then: +max +xi +� +i,j 0) guarantees that only the space of valid routes +is explored. Reshaping this two-dimensional spin matrix with elements xv,i to a spin vector +of size N 2 allows one to recover the coupling matrix J and magnetic field h to formulate the +corresponding Ising Hamiltonian. One can reduce the size of the Ising problem to (N − 1)2 +by fixing a particular city to be the first in the route. Note that the Hamiltonian HTSP +can represent both directed and undirected graphs, and the generalisation for the cycles +optimisation problem is straightforward. It has also been used for finding transportation +routes that minimise costs. +V.3. +Portfolio optimization +Optimizing the portfolio selection means finding the most optimal combination of invest- +ments for an institution or individual. One of the modern portfolio optimization problem +formulations has the following form [186]: +min +0≤xi≤1 λ +� +N +� +i=1 +N +� +j=1 +Jijxixj +� +− (1 − λ) +� +N +� +i=1 +µixi +� +, +N +� +i=1 +xi = 1, +(31) +where N is the number of different assets, and xi is the decision variable representing the +proportion of capital invested in asset i. Here coupling coefficient Jij represents the covari- +ance between returns of assets i and j, µi is the mean return of asset i, and λ ∈ [0, 1] is the +risk aversion parameter. When λ = 0, the model maximizes the portfolio’s mean return, +and the optimal solution will be formed only by the assets with the greatest mean return. +When λ = 1, only the total risk associated with the portfolio is minimized. +There are different modifications to the portfolio optimization problem. For instance, +one can introduce bounding and cardinality constraints that specify that there should be K +different assets in the portfolio or/and the portion of some assets should be within certain +bounds. This is achieved by +N +� +i=1 +zi = K, +ϵizi ≤ xi ≤ δizi, +zi ∈ {0, 1}. +(32) +43 + +The cardinality-constrained mean-variance model is a mixed quadratic and integer program- +ming problem in the NP-hard class of problems. Although the problem is not a combinatorial +optimisation, we take advantage of the fact that the objective function has the same form +as the energy function in Hopfield networks. Consequently, it will be minimised if we follow +the Hopfield dynamics. Hopfield NNs have efficiently solved this problem [187, 188]. The +discrete dynamics becomes +xi(t + 1) = Gi[−2λ +� +j +Jijxj(t) + (1 − λ)µi)], +(33) +where Gi is a sigmoid with values in [ϵi, δi]. When solving any optimization problem, con- +straints usually appear in the energy function. However, in many cases of Hopfield networks, +this is not necessary. Constraints on xi are satisfied using a sigmoid’s activation function +since its outputs already lie inside the desired interval. To fulfil the cardinality constraints, +we begin with 3K/2 neurons. After getting a minimum for the objective function, we remove +the asset with the smallest output and repeat this process until the network has precisely +K assets. These remaining assets solve the original portfolio selection problem. To satisfy +the constraint � xi = 1, one can use various adjustments, for instance, to evaluate the +feasibility of every portfolio and change the proportions of capital xi to be invested in each +selected asset [187]. +V.4. +Phase retrieval +The minimisation of the XY model (solving QCO) is directly related to the notoriously +hard-to-solve phase retrieval problem. The problem’s objective is to recover a general signal +(or image) from the magnitude of its Fourier transform [87–89]. This problem arises because +the signal detectors can usually record only the modulus of the diffraction pattern, therefore, +losing the information about the phase of the optical wave. Mathematically, one needs to +recover a signal x ∈ Cm from the amplitude b = |Ax|, where A ∈ Cn×m, b ∈ Rn. Then the +phase recovery problem [189] can be formulated as: +min +xj,ui +� +i +� � +j +Aijxj − biui +�2 +(34) +44 + +where u ∈ Cn is a phase vector that satisfies Ax = diag(b)u, |ui| = 1 for i = 1, n. This +optimization problem can be further rewritten as +min +� +ij +Mijuiuj +subject to +|ui| = 1, i = 1, n, +(35) +where M = diag(b)(I − AA†)diag(b) is the Hermitian matrix, I is the identity matrix, and +A† is the Moore-Penrose inverse of a matrix A (see [189] for details). +V.5. +Machine learning +The data growth now surpasses our capabilities to process it concerning human and com- +putational resources. The development of data-driven methods also marks the transition +from the classical computer science paradigm to the modern ML setting. The related ques- +tions concerning the abilities of the precisely crafted algorithms and worst-case scenarios are +changed by the most probable cases and the design of the NN architectures. +One of the main ML field’s goals is to predict specific outcomes from the given data. The +richness of the data dramatically influences the methods’ performance, making it easier to +find patterns and expect accurate results. Considering complicated methods and deep NN +architectures, there are three crucial components in ML: data, features, and algorithms. +Practically speaking, one can meet the data in many ways: e-mails, stock prices time- +series, users databases and collection of the experimental measurements. Moreover, one can +collect in different ways, either manually, usually quite long and costly, with few errors or +automatically feeding everything to some sorting algorithms. Depending on the context, the +collected data (or datasets) can be of great value, determining the demand for suitable rare +datasets. +Features represent the properties of the considered objects. Therefore, a small amount of +essential and sorted features in most cases can guarantee the success of the ML approach to +the problem. However, it is very time-consuming to determine the feature in the so-called +’raw’ big datasets and select the right ones. Moreover, sometimes one has to avoid human- +based decisions to prevent introducing subjectivity and opinion-based bias to optimize the +model performance. Therefore, the latest deep learning success is partially tied to automatic +feature engineering compared to the previous partially empiric ML models. +45 + +The last part of the considered scheme is the algorithm. Choosing the method of solving +a particular task depends on the context and influences such parameters as the final model’s +accuracy, speed, and computational complexity. In general, one can solve a problem in many +different ways. +The components were presented according to their significance in the ML pipeline. Simply +saying, one can only extract useful information from a noisy and meaningful dataset. The +following subsection starts the discussion with the simple classical algorithms, which are the +basis of many existing applications. Then, we outline the central ideas behind the main +ML methods that will be the centre of attention for transferring into the special-purpose +hardware. At the end of this chapter, we cover the wide range of capabilities of the NNs. +V.5.1. +Regression +Regression analysis is one of the earliest methods in statistical modelling that allows +estimating the relationships between a dependent variable and independent variables. The +most common form of regression analysis is linear regression. This model assumes that the +dependent variables denoted by yi have a linear relationship depending on the m-vector of +points {xi1, . . . , xim}n +i=1 with an addition of the disturbance terms ϵi in each case. This +relationship can be written in the following form: +yi = β0 + β1xi1 + · · · + βmxim + ϵi = +m +� +j=0 +βjxij + ϵi. +(36) +To shorten notation we use the matrix form y = Xβ + ϵ where: y = {yi}, X = {xij}, β = +{βj}, ϵ = {ϵi}, (i = 1, . . . , n), (j = 0, . . . , m), with xi0 = 1. The linear regression task is +the estimation of the values of the regression coefficients βj given the data points xij and +observables yi, so that the error term ϵ = y − Xβ is minimized. One can use different +metrics for that purpose, such as the sum of squared errors of ϵi or others. +The most common parameter estimation technique is called the least-squares estimation. +Here, the optimum parameter is defined through the minimization of the sum of the mean +squared loss +min +βj +n +� +i=1 +� m +� +j=0 +βjxij − yi +�2 +, +(37) +46 + +which can be connected with the conventional QP. The optimal solution can be obtained +by differentiating Eq. (37) and equating it to zero with respect to parameters βj. In matrix +notation, the solution can be written as +β = (XTX)−1XTy. +(38) +There exist different modifications of the proposed procedure: generalized least squares, +where one introduces a certain degree of correlation between the residuals ϵi (37), or the +weighted least squares, where the knowledge of the variance of observations is incorporated +as the coefficients wk before each of the residual. Moreover, intrinsically different techniques +can be based on maximum likelihood estimation, Bayesian methods, or regularization. +A natural extension of linear regression is in replacing linear dependence with a polyno- +mial. In the case of one argument, it is possible to rewrite Eq. (36) as +yi = β0 + β1xi + β2x2 +i + · · · + βmxm +i + ϵi = +m +� +j=0 +βjxj +i + ϵi. +(39) +Given the data points xj +i, the task is the same as Eq. (37) except for the change in variables +xij → xj +i. Similarly, it is possible to replace the polynomial basis with a set of some nonlinear +functions f(xi)j, so that x2 +i → f(xi)j. +Multiple linear regression is a generalization of linear regression with more than one +independent variable. The basic model for multiple linear regression can be written in a +similar form: +yi = β0 + β1Xi1 + β2Xi2 + · · · + βmXim + ei = +m +� +j=0 +βjXij + ei, +(40) +where instead of variables xij one has a set of matrix elements Xij of size k × k. Depending +on the chosen norm for the matrix, it is possible to formulate the task of finding the regres- +sion coefficients. Taking the square Frobenius norm of the matrix, the search for optimal +coefficients βi is equivalent to solving Eq.(37), except for the additional sum over the k2 +matrix elements: +min +βj +k2 +� +l=1 +n +� +i=1 +� m +� +j=0 +βjxl +ij − yl +i +�2 +. +(41) +47 + +This can be extended further for multivariate linear regression or combined with the nonlin- +ear basis with minor consequences concerning the parameters search and hardware opera- +tions, except for the much more complicated procedure for preprocessing the coefficients for +any modification. Regression can be considered the simplest form of supervised learning. +V.5.2. +Classification +Classification is one of the popular tasks for ML. The purpose of classification is to sort +the objects among the initially defined classes. The earliest algorithms include naive Bayes +and decision trees. Here, we only consider Markov random field (MRF) encoding, which is +the general case for such models. +The k-nearest neighbours algorithm is a non-parametric classification method used in +statistics [190, 191]. It aims to classify the objects by considering their k nearest neigh- +bours with the defined class. The consequent attaching objects to a particular group is +repeated until the convergence. We omit the explicit corresponding formulas because of +their similarity with the k-means, the unsupervised clusterization algorithm, presented be- +low. Both methods are usually based on Euclidean distances and can easily be transferred +to special-purpose optimization hardware. +Another classification method is called a support vector machine (SVM). SVM is a su- +pervised learning model that analyses data for classification purposes. It aims to construct +a hyperplane between the classes of training data points in a high-dimensional space, em- +phasising a good separation achieved by maximising its margin. SVM was introduced in +[192] and standardised in [193]. +Linear SVM deals with the n points x in the m-dimensional space, where each point has +been assigned a binary class yi = ±1. The task is to construct a hyperplane that divides +these two groups with the maximum distance between them. The so-called ”hard margin” +scenario assumes that the initial data is linearly separable. One can start by constructing +two parallel hyperplanes, separating groups of different classes with the largest distance +between these two surfaces. +The target surface between these hyperplanes is called the +maximum margin hyperplane. To mathematically describe these surfaces, one can write: +wTxi − b = +� +j +wjxi +j − b = ±1, +(42) +48 + +where wj are the components of the normal vector for both of the hyperplanes, xi +j are m- +dimensional coordinates of the vector with the serial number i, b defines the surface shift +concerning the zero coordinates and ±1 defines the class. Everything above y = 1 belongs to +one class, and everything below y = −1 belongs to another. The offset of the hyperplane is +determined by b/ ∥w∥, while the marginal distance equals 2/ ∥w∥. To maximize the marginal +distance, one has to minimize the norm of ∥w∥ and hence its square ∥w∥2. This task can be +reformulated as the optimization problem, adding the constraints that prevent data points +from being positioned into the margin +min ∥w∥ +s.t. yi +� +wTxi − b +� +≥ 1, for i = 1, ..., n +(43) +The natural extension of SVM is in considering a so-called ”soft margin” case. +It is +assumed that the given data points are not linearly separable. In this case, one has to +introduce a new kind of variable ξi = max(0, 1 − yi +� +wTxi − b +� +) for each point i, which is +usually referred to as the hinge loss function, playing a regularizer role. Thus, it is possible +to rewrite Eq. (43) as +min 1 +n +n +� +i=1 +ξi + C∥w∥2 +s.t.yi +� +wTxi − b +� +≥ 1 − ξi and ξi ≥ 0, for all i, +(44) +where the constant C regulates the interplay between the pure hard margin classifier and +the soft margin one. We can reformulate the problem using the Lagrangian duality: +max +ai +n +� +i=1 +ai − 1 +2 +n +� +i=1 +n +� +j=1 +yiai +� +xT +i xj +� +yjaj +s.t. +n +� +i=1 +aiyi = 0, and 0 ≤ ai ≤ +1 +2nC for all i, +(45) +where the norm vector w is expressed through the new variables ai, so that w = �n +i=1 aiyixi, +and the initial task of determining the offset of the surface is expressed via b = wTxi − yi. +Thus, it is possible to obtain the problem, which has an exact QP formulation. This problem +can be solved with the standard quadratic algorithms, thus, can be solved using special- +49 + +purpose hardware. +It is helpful to mention the nonlinear extension of the SVM, which solves nonlinear +classification task and can exploit the different functional forms of kernels. One can modify +the scalar dot product in the quadratic form Eq. (45) by a different kernel function k(xi, xj) +depending on the properties of the analogue hardware. +V.5.3. +Finding the principal eigenvector +Finding the principal (dominant) eigenvector of a given matrix J belongs to the P-class +of problems. However, finding such a dominant eigenvector on an ever-growing large matrix +becomes a computationally intensive task incompatible with Moore’s law. At the same time, +a range of real-life problems would benefit from fast calculation of the principal eigenvector. +For instance, the PageRank algorithm [194, 195] evaluates the relative importance of pages +by exploiting the web link structure. The web network is represented as a directed graph, +where each page is a node of the graph, and each hyperlink is an edge connecting one page +to another. +For the entire database of web pages, the PageRank algorithm computes a +single score vector, the PageRank. The algorithm’s key underlying assumption is that pages +transfer the importance to other pages via links; hence, PageRank components determine +the importance of pages. +Mathematically, finding the PageRank vector is equivalent to +calculating the principal eigenvector of the link-structure matrix, Google matrix. Besides, +calculating the principle eigenvector is required in social network analysis, recommendation +systems, bibliometrics, bioinformatics, DNA sequencing, and distributed computing systems +[196–198]. +There are numerous applications of PageRank to chemistry and engineering sciences net- +works to investigate and analyse complex systems. As engineered systems grow in size, they +become increasingly complex, with networks and submodules interacting in unpredictable, +nonlinear ways. Network analysis methods like PageRank help to organise and study these +complexities [197]. For instance, MonitorRank diagnoses root causes of issues in a modern +distributed system: error logs and tracing debugging information [199]. PageRank has been +used for road and urban space networks, which help predict traffic flow and human move- +ment. It was shown that PageRank is the best network measure in predicting traffic on +individual roads [200]. +50 + +The advantage of using optical systems for calculating the principal eigenvector has been +recently shown [198]. For a certain choice of control parameters of these optical systems, the +steady state of optical networks can solve an eigenvalue maximization problem [201], which +results in finding the energy state dictated by signs of the eigenvector corresponding to the +largest eigenvalue of the interaction matrix, i.e. principal eigenvector. In particular, the +estimates presented [198] show that special-purpose optical machines for PageRank calcula- +tions may provide dramatic improvements in power consumption over classical computing +architectures. +V.5.4. +Dimensionality reduction +Dimensionality reduction involves the transformation of data from the space with many +dimensions into a low-dimensional space, usually preserving meaningful and valuable prop- +erties from the original data. It isn’t easy to handle high-dimensional data in practice due +to the growth of the space volume. Dimensionality reduction is standard in data-intensive +fields. It can be used in signal processing, neuroinformatics, and bioinformatics [202, 203]. +One can find its applications in recommender systems [204], semantic search [205] or as a +primary tool in many domains involving numerical analysis. +One of the well-known methods for dimensionality reduction is the principal component +analysis (PCA) [206]. The idea behind PCA is to approximate particular data with linear +manifolds of lower dimensions. PCA can be alternatively interpreted as finding subspaces +of lower dimension in the orthogonal projection on which the data variation is maximum. +The initial task behind the PCA is to find the best approximation of the data points +using lines and surfaces. Given the set of vectors x1, x2, . . . , xm ∈ Rn, the aim is at finding +the sequence of k k-dimensional affine spaces Lk ⊂ Rn that find +min +Lk +m +� +i=1 +d2 (xi, Lk) = min +ajl +m +� +i=1 +n +� +l=1 +� +xil − a0l − +k +� +j=1 +ajl +n +� +q=1 +ajq (xiq − a0q) +�2 +, +(46) +for each k, where d (xi, Lk) is the Euclidean distance from the point xi to the Lk. Affine +spaces Lk are defined as the sets of linear combinations Lk = {a0 + α1a1 + · · · + αkak} with +coefficients αi ∈ R, while the vectors {a1, a2, . . . , ak} ⊂ Rn form orthonormal basis in Rn. +Eq. (46) is an optimization problem. The initial vector a0 is simply defined as the solution +51 + +to +min +a0 +m +� +i=1 +d2 (xi, L0) = 1 +m +m +� +i=1 +xi. +(47) +The next component is found iteratively by subtracting the projection xi = xi − a0 +� +aT +0 xi +� +(with the scalar product aT +0 xi) for the vectors corresponding to Lj: +aj = argmin +∥aj∥=1 +� m +� +i=1 +� +xi − aj +� +aT +j xi +��2 +� +. +(48) +The iterations continue until the number of the affine space k reaches the n−1 of the initial +problem space dimension. Using the identity ||xi − aj +� +aT +j xi +� +||2 = ||xi||2 − +� +aT +j xi +�2, one can +easily map this task into the QP in ai variables with the normalization constraints and the +coupling matrix Jij = −xixj. To shorten the presented notation, the iterative procedure +can be written similarly to maximization tasks +ˆXk = X − +k−1 +� +s=1 +Xw(s)wT +(s), +(49) +w(k) = arg max +∥w∥=1 +���� ˆXkw +��� +2� +, +(50) +where k is the number of principal component, X is the data matrix of size n × m, ws = +(w1, . . . , wm)(s) are the weight coefficients. +If the sequential operation is limited on the +specific hardware system, one can still use the first iteration of the PCA method to obtain the +largest eigenvalues of a matrix. One can find many alternative formulations of the PCA task, +such as cancelling correlations between coordinates, i.e. covariance matrix diagonalization +or singular value decomposition. +Singular value decomposition (SVD) is a special form of a rectangular matrix decompo- +sition in the form +X = UΣV⊤, +(51) +where U is the unitary matrix (representing the rotation as the linear transformation of +the space in the geometrical interpretation), Σ is the rectangular diagonal matrix with non- +negative real numbers on the diagonal (which are called the singular values, the action of +the matrix has the interpretation of the corresponding scaling by diagonal elements) and +52 + +V⊤ is another unitary matrix (with the same additional rotation interpretation). +SVD is essentially vital in the standard techniques of the latent semantic analysis (LSA) +[207, 208], which purpose is to process documents and detect the relationship between li- +braries and terms. +There is a direct correspondence between PCA and SVD decomposition. To perform +the PCA, one has to find the eigenvectors of the covariance matrix XX⊤ (without the +appropriate scaling factor +1 +n−1). +The covariance matrix is diagonalizable, and with the +normalized eigenvectors, one can write +XX⊤ = WDW⊤. +(52) +Applying SVD to the same data matrix X gives +XX⊤ = +� +UΣV⊤� � +UΣV⊤�⊤ = +� +UΣV⊤� � +VΣU⊤� +, +(53) +which gives +WDW⊤ = UΣ2.U⊤, +(54) +Using this correspondence, one can perform the SVD decomposition as PCA on the special- +purpose hardware. +V.5.5. +Clusterization +The most detailed description of clusterization is the separation of the objects on a spe- +cific basis. The goal can be defined as a classification without any prior information about +the classes. The machine can set the number of clusters in advance or define them automat- +ically. The algorithm determines objects’ similarity by their marked features and puts the +objects with many similar features in the same class. There are successful applications of +clusterization in market analysis (consumer analytics), image compressing, data analytics, +and anomaly detection. +K-means clustering is a clustering method that aims to partition n observations into k +clusters. Each of these observations is located in the cluster with the nearest mean, also +called a centroid [209–211]. There are heuristic algorithms that deal with such an assignment; +53 + +however, the problem is NP-hard. +Given a set of observations {x1, ..., xn} in a d-dimensional space k-means algorithm aims +to partition these observations into k sets {S1, S2, ..., Sk} to minimise the within-cluster sum +of squares (or variance): +arg min +Si +k +� +i=1 +� +x∈Si +��x − µSi +��2 , +(55) +where µSi is the mean of points in the set Si. +One usually uses an iterative technique +consisting of two steps to perform such an optimisation task. +Given an initial set of k +means m1 +1, ..., m1 +k, the first step is to assign each observation to the cluster with the nearest +mean, according to the Euclidean distance. The next step is to recalculate the centroids: +mt+1 +i += � +xj∈Si,(t) xj. Finally, the loop is run until the convergence. The algorithm uses the +assigning of objects to the nearest cluster by Euclidean distance, and it is a suitable method +for transferring its sequential operations to the specific hardware. +Mean shift is a high-dimensional-space analysis method for locating the maximum density +function given a discrete number of data sampled from this arbitrary density function. It is +helpful in complex hierarchical algorithms and is used in different computer vision or image +processing domains. +Given data points xi in n-dimensional space, one can use the kernel function k(r), acting +on the norm value r, to determine the mean shift’s value. The kernel function has to be +non-negative, non-increasing and continuous. One can use the flat kernel so that k(r) = 1 +if r < r0 and 0 outside. Each iteration consists of calculating the function +F(x) = +� +i +k +�(x − xi)2 +α2 +� +, +(56) +where there are α states. The maximum of F(x) is computed using the square norm. +V.6. +Neural networks +ANNs are often associated with ML. We considered the associative memory model, a +simple recurrent shallow NN in the subsection IV.5. This model can be extended to higher- +order systems, simultaneously gaining many useful properties. However, optical systems are +not tied only to this type of architecture [10]. +54 + +Any NN can be defined as a set of neurons and connections between them. An artificial +neuron’s task is to take input numbers, process them in a certain way (executing a special +function), and output the results. The standard mathematical transformation of one NN +layer can be written as ϕ(�N +i=0 wixi + b), where wi denote the weights for the input data +points xi (or independent variables), and the constant b is the shift called bias. Here, the ϕ +is a nonlinear activation function. A single-layer NN that performs a similar transformation +and produces a single output number is called a perceptron. The perceptrons, assembled +into multilayered structures, are called multilayer perceptrons. The introduction to the NNs +is presented in [212–214] with more modern work [215] and the latest results after the deep +learning breakthrough [216]. +The activation function ϕ plays an essential role in the NN design because the output +signal would be simply a linear function in its absence. There are many functional activation +functions, such as binary step function, sigmoid (or logistic function), hyperbolic tangent, +etc. They allow a NN to map an input to the output appropriately. Thus, NN is considered +a universal function approximator [217]. To choose the NN weights, one usually uses the +backpropagation procedure [218, 219], although there are many alternatives. Backpropaga- +tion consists of tuning the NN weights according to the difference between the actual output +value of the network and the predicted one, with the final goal of minimizing this discrep- +ancy or the cost function. The tuning procedure involves computing the total discrepancy +gradients on each layer, starting from the final one and updating the corresponding weight +values. Through the extensive number of such iterations, there is a chance that the weights +will be tuned in the desired way. +Many deep NN (NN with many layers) can be mapped into a shallow one with a signifi- +cant overhead on the number of neurons in the standard layer. That means that any deep +NN functionality can, in principle, be performed on a physical device suitable to a shallow ar- +chitecture. With an appropriate mapping, both networks will have the same approximation +qualities [220–223]. +A well-trained NN can approximate many complicated algorithms, some of which are pre- +sented in this review. However, one has to provide enough input conditions and good output +answers, especially when the problem is of high computational complexity. In addition, how- +ever, the resulting correlations need to be better understood. The valuable properties of the +NNs go far beyond the optimization domain. We will consider some of them below. +55 + +V.6.1. +Neural networks and dynamical systems +Using ML models in the domain of physical sciences, i.e. incorporating physical laws +and domain knowledge into neural architectures, is called physics-informed machine learn- +ing (PIML). It provides a powerful approach to modelling different physical phenomena. +This rapidly growing field can pursue many other goals. +Among them are constructing +better predictive models with high accuracy and reliable generalization abilities, increasing +data processing rate, accelerating the dynamical processes through optimized architecture, +and solving inverse problems with interpretable models. One should expect that emulating +complex nonlinear dynamics should benefit from the PIML. This can be seen in the appli- +cations in weather forecasting [224], modelling of turbulence [225, 226], nonlinear dynamics +[227, 228], applications of the ML to the Koopman operator theory [229]. Optical hardware +can be potentially used to speed up these applications. +The correspondence between NN architectures and dynamical systems is straightforward. +Some of the NN can be viewed as discretizations of dynamical systems, which is true in re- +verse order - one can design NNs to have specific properties, such as invertibility [230]. +This correspondence can broaden the applicability of the potential optical hardware. Their +connection with dynamical systems and deep learning can be found in [231]. The gener- +alization of the optimization algorithms inspired by different optical systems has canonical +universality property [232]. +V.7. +Probabilistic graphical models +Graphical models provide a natural tool for dealing with uncertainty and complexity +throughout applied mathematics and engineering. The graph theoretic side of graphical +models offers an appealing interface where one can model a data structure that lends itself +naturally to the design of efficient general-purpose algorithms. Many models in statistics, +systems engineering, information theory, and pattern recognition are special cases of the +general graphical model formalism. It is vital for representing joint probability distribution +and inference based on the given observations [233–235]. +Probabilistic graphical models (PGMs) are graphs with the nodes represented by random +variables, while edges connecting them represent conditional independence assumptions. +56 + +PGM can be thought of as a compact representation of joint probability distributions. There +are two main kinds of graphical models: undirected, which are known as Markov random +fields (MRFs), widely used in the physics and vision communities, and directed, also known +as Bayesian networks (BNs), belief networks or causal models that are more popular with +the AI and ML communities [233]. +The spin Hamiltonians are particularly useful for PGMs. We recall the Ising spin model +of Eq. (2) with zero-field coefficients (hi = 0). Each spin variable si can be treated as a +random binary variable so that their coupling strengths serve as the connections between +random variables. Certain configurations of spin variables X = (x1, . . . , xN) ∈ {−1, +1}N +is called an assignment. The probability of an assignment in the PGM is given by +P(si = xi) = 1 +Z exp +� +− +N +� +i=1 +N +� +j=1,j