Topics on filtering and coherent control of nonlinear quantum systems

Abstract

This thesis covers several related topics concerning nonlinear filtering and control for quantum systems. There are two main approaches to the feedback control of a quantum system: measurement-based feedback control and coherent feedback control. Measurement-based feedback control involves quantum measurement, quantum filtering to estimate the state of the system, and a feedback control law based on this state estimate to generate an appropriate control signal. Coherent feedback control involves cascading the output of the quantum system into another coherent quantum system, which modulates the output field and then coherently cascades it back into the system input as the control signal. Chapter 2 of this thesis studies necessary and sufficient conditions for the existence of a quantum filter in the case where the system has multiple output channels. It is shown that when measurements are taken at each output channel, there is an algebraic condition for which there exists a corresponding joint probability space, which is required in establishing conditional expectations as required for the existence of the filter. Chapter 3 studies an approximation technique for quantum filters arising in nonlinear quantum systems. A first order Taylor expansion of the nonlinear quantum Markovian generator of the dynamics is used to compute the filter. This approach is referred to as the quantum extended Kalman filter. Chapter 4 provides a foundation for nonlinear coherent control design by analyzing the physical realizability condition of a class of nonlinear quantum stochastic differential equations. It is shown that both the quantum Markovian generator and the diffusion drift coefficient have to be conservative vectors of potential operators under suitable choice of variables. Chapter 5 studies a stability condition for the system’s density operator. It is shown how to analyze the stability of this set via a candidate Lyapunov operator. The analysis of the set of invariant density operators is concluded by introducing an analog of the Barbashin-Krasovskii-LaSalle Theorem for quantum systems. Finally, Chapter 6 studies the feasibility of developing a quantum coherent nonlinear controller for disturbance attenuation. The weak asymptotic stability of the closed-loop system is shown via the stability results derived in Chapter 5.