Optimal control of driven quantum many-body systems

驱动系统提供了一个令人兴奋的分析工具来研究固态模型的清洁实现，具有高温超导性，分数量子霍尔效应，自旋霍尔效应和具有拓扑顺序的状态。最有希望的实现之一是基于光学晶格，其中原子被困在一个站立的激光场中。光学晶格允许在不同几何形状的晶格中捕获原子，并且原子的动力学可以通过周期性调制（震动）晶格来控制。当前实验的一个主要限制是震动会使样品升温，最终会破坏预期的效果。实际上，加热仅仅是根据系统震动频率的选择来控制的。如果这个频率离系统的共振频率很远，那么加热就不会太大。然而，如果相消干涉阻止原子占据高激发态，人们会期望加热可以大大减少。为此目的，可以多色驱动系统，并确定驱动配置文件，以最大限度地减少不希望发生的过程。为了达到这一目标，可以采用不同的技术。一方面，各种摄动技术已被用于处理周期驱动系统。它提供了一种分析方法来找到系统的最佳调制，以便在确保抑制有害副作用的同时设计有效的动力学。当许多身体相互作用开始发挥作用时，这种方法受到复杂性指数增长的限制。另一种方法是使用机器学习（ML）技术。机器学习已经成功地用于寻找经典系统的最优控制方案，但直到最近才成功地应用于量子系统。研究的结果表明，解决任何量子控制问题的优化应该是容易的。在量子系统中使用机器学习的例子包括高保真目标量子态的制备、超冷注视的冷却过程、超快光脉冲的形成以及物质波的相干操纵。在众多可能的技术中，（物理）无模型方法（例如强化学习，贝叶斯推理，进化算法）为更常用的梯度下降方法提供了有趣的替代方案。这个项目是与剑桥的一个实验小组合作建立调制光学晶格，我可以与他们讨论现实的实验限制和开发的控制方案的可行性。为了准备这个项目，我在微软研究院实习，以熟悉ML（强化学习）技术。驱动系统提供了一个令人兴奋的分析工具来研究固态模型的清洁实现，具有高温超导性，分数量子霍尔效应，自旋霍尔效应和具有拓扑顺序的状态。最有希望的实现之一是基于光学晶格，其中原子被困在一个站立的激光场中。光学晶格允许在不同几何形状的晶格中捕获原子，并且原子的动力学可以通过周期性调制（震动）晶格来控制。当前实验的一个主要限制是震动会使样品升温，最终会破坏预期的效果。实际上，加热仅仅是根据系统震动频率的选择来控制的。如果这个频率离系统的共振频率很远，那么加热就不会太大。然而，如果相消干涉阻止原子占据高激发态，人们会期望加热可以大大减少。

Driven systems offer an exciting analytic tool to investigate clean realizations of solid state models, featuring high temperature super-conductivity, fractional quantum Hall effect, spin Hall effect and states with topological order. One of the most promising implementations is based on optical lattices in which atoms are trapped in a standing laser field. Optical lattices permit to trap atoms in lattices of different geometries, and the dynamics of the atoms can be controlled by periodically modulating (shaking) the lattice. A central limitation on current experiments is that the shaking heats up the sample, what ultimately destroys the desired effects. In practice, heating is controlled merely in terms of the choice of frequency with which the system is shaken. If this frequency is far off the resonance frequencies of the system, then the heating is not too large. One would, however, expect that heating can be substantially reduced if destructive interference prevents atoms from occupying highly excited states. For this purpose, one can drive the system poly-chromatically and identify driving profiles that minimize the occurrence of undesired processes. For reaching such a goal different techniques can be employed. On the one hand a variety of perturbative techniques have been employed to deal with periodically driven systems. It provides a path to analytically find optimal modulation of the system in order to engineer effective dynamics of interest while ensuring the suppression of detrimental side-effects. This approach is limited by an exponential increase in complexity when many body interactions come into play. Another way would be through the use of Machine Learning (ML) techniques. ML has been successfully employed in finding optimal control schemes for classical systems, however only more recently it has been successfully applied to quantum system. Research has been driven by results showing in that solving any optimization of quantum control problem should be easy. Examples of the use of ML for quantum systems include the preparation of target quantum states with high fidelities, cooling procedures of ultra-cold gazes, shaping of ultrafast light pulses, and coherent manipulation of matter waves. Among the plethora of possible techniques, (physical) model-free approaches (e.g. reinforcement learning, Bayesian inferences, evolutionary algorithms) provide interesting alternatives to the more commonly used gradient descent methods. This project is in collaboration with an experimental group at Cambridge setting up modulated optical lattices, with whom I can discuss realistic experimental constraints and the viability of the control schemes developed. In order to prepare for the project I did a placement at Microsoft Research to familiarize myself with ML (reinforcement learning) techniques.Driven systems offer an exciting analytic tool to investigate clean realizations of solid state models, featuring high temperature super-conductivity, fractional quantum Hall effect, spin Hall effect and states with topological order. One of the most promising implementations is based on optical lattices in which atoms are trapped in a standing laser field. Optical lattices permit to trap atoms in lattices of different geometries, and the dynamics of the atoms can be controlled by periodically modulating (shaking) the lattice. A central limitation on current experiments is that the shaking heats up the sample, what ultimately destroys the desired effects. In practice, heating is controlled merely in terms of the choice of frequency with which the system is shaken. If this frequency is far off the resonance frequencies of the system, then the heating is not too large. One would, however, expect that heating can be substantially reduced if destructive interference prevents atoms from occupying highly excited states.