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，自旋霍尔效应和拓扑有序状态。其中最有前途的实现是基于光学晶格中的原子被困在一个常设的激光场。光学晶格允许将原子捕获在不同几何形状的晶格中，并且可以通过周期性地调制（摇动）晶格来控制原子的动力学。目前实验的一个主要限制是摇动会使样品升温，最终破坏了预期的效果。在实践中，加热仅根据振动系统的频率的选择来控制。如果该频率远离系统的谐振频率，则加热不会太大。然而，如果相消干涉阻止原子占据高激发态，人们预计加热可以大大减少。为此，可以多色地驱动系统，并识别使不期望的过程的发生最小化的驱动曲线。为了达到这样的目标，可以采用不同的技术。一方面，各种微扰技术已被用来处理周期驱动系统。它提供了一条途径来分析找到系统的最佳调制，以便设计有效的感兴趣的动态，同时确保抑制有害的副作用。当许多身体相互作用发挥作用时，这种方法受到复杂性指数增加的限制。另一种方法是使用机器学习（ML）技术。ML已经成功地用于寻找经典系统的最优控制方案，然而直到最近它才成功地应用于量子系统。研究的结果表明，解决量子控制问题的任何优化都应该很容易。ML用于量子系统的例子包括制备具有高结晶度的目标量子态，超冷凝视的冷却程序，超快光脉冲的成形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约束和控制方案的可行性。为了准备这个项目，我在微软研究院做了一个实习，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，用于研究固态模型的干净实现，具有高温超导性，分数量子霍尔效应，自旋霍尔效应和拓扑有序状态。其中最有前途的实现是基于光学晶格中的原子被困在一个常设的激光场。光学晶格允许将原子捕获在不同几何形状的晶格中，并且可以通过周期性地调制（摇动）晶格来控制原子的动力学。目前实验的一个主要限制是摇动会使样品升温，最终破坏了预期的效果。在实践中，加热仅根据振动系统的频率的选择来控制。如果该频率远离系统的谐振频率，则加热不会太大。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.