Implementing optimal Control with principla Component Analysis

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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.