Implementing optimal Control with principla Component Analysis

最优控制，特别是量子最优控制[1]，是量化“控制”应用的实践，通常是电磁场，对量子系统进行控制，使其从初始量子态进入所需的最终态。如果控制在执行期望任务的同时描述了某些参数空间的最小值，则称为最优控制，通常是花费的时间或消耗的能量。这些问题（除了大规模简化近似，如两能级系统的旋转波近似[2]）往往不允许解析解，因此必须通过数值推导，然后在实际系统中进行测试。一个系统可能有几种控制策略，这些策略可以减少集群，例如，基于密度的集群[5]。控制可能倾向于在其集群中具有相似的属性，例如，鲁棒性可以在集群规模上进行分析。控制器也可以使用PCA分解成有效的特征基，从而大幅降低优化问题的必要维数。这种降维对于物理系统中的闭环控制是至关重要的，在物理系统中，控制的不忠可能被测量到。对从地面驱动到第二激发态的BEC系统[3]进行了理论研究。我们的目标是将BEC建模扩展到一个更复杂的系统（带噪声），我们可以物理访问，比如Alex Clark的系统。这是建立在之前工作的基础上的，但应该为其他寻找控制器和聚类/降低一般系统的维数提供一个配方。这将导致控制策略的识别，其鲁棒性思想可以可行地进行闭环优化。这将在系统上实施，希望能显示性能的提高，并描述不同路线的改进。这将进一步提供不同过渡之间的控制比较，其中应该有相当多的重叠。这个项目的第二个方面涉及一个振荡晶格干涉仪，这是一个基于物质的干涉仪，原子被困在一个相位调制的光学晶格[4]中。相对于喷泉方法的关键优势是物理足迹，晶格伪动量提供状态分裂。目标是将Carrie（和Meagan）之前在抖动晶格干涉仪上的工作扩展到更高的自由度。该模型将包含多达六轴传感器，包括旋转传感，然而，这样的设备实质上比纯粹的可分离轴更复杂。当使用必要的传播和控制进行建模时，这可以转化为在博尔德或布里斯托尔的实际系统中的执行。

Optimal control, and in particular quantum optimal control [1], is the practice of quantifying the application of 'controls', generally EM fields, to a quantum system to drive it from an initial quantum state into a desired final state. The control is termed optimal if it performs the desired task while describing a minimum in some parameter space, generally either time taken, or energy expended. These problems (beyond massively simplifying approximations like the rotating wave approximation for two level systems [2]) tend to not admit analytical solutions so must be numerically derived, and subsequently tested on real systems. There may be several control strategies for a system, and these can be cluster educing, for example, density-based clustering [5]. Controls may tend to have similar properties in their cluster, robustness for example can be analysed on a cluster scale. The controllers can also be decomposed into an effective eigen basis using PCA, giving a sharp reduction in the necessary dimensionality of the optimisation problem. This dimensionality reduction is vital for closed loop control in a physical system where the infidelity of a control may be measured [3]. This was studied theoretically for a BEC system [3], driving from the ground to second excited state. The goal would be to extend the BEC modelling to a more complex system (with noise) which we have physical access to, such as Alex Clark's. This builds from the previous work but should provide a recipe for other finding controllers for and clustering/reducing dimensionality of general systems. This should result in identification of control strategies with an idea of their robustness which can feasibly be close loop optimised. This would then be implemented on the system to hopefully show performance gains and characterise the improvements for different routes. This should then further provide for a comparison of controls between different transitions where naively there should be considerable overlap. The second aspect of this project involves a shaken lattice interferometer, this is a matter-based interferometer with atoms trapped in a phase-modulated optical lattice [4]. The key advantage over a fountain approach is the physical footprint, with lattice pseudo momentum providing the state splitting. The goal would be to extend Carrie's (and Meagan's) previous work in a shaken lattice interferometer to higher degrees of freedom. This modelling would incorporate up to a six-axis sensor including rotation sensing, however such a device is substantially more involved than purely separable axis. When modelled with the necessary propagation and controls this can translate to implementation in a real system either in Boulder or Bristol.