Feedback stabilization of discrete-time quantum systems subject to non-demolition measurements with imperfections and delays

Details Feedback stabilization of discrete-time quantum systems subject to non-demolition measurements with imperfections and delays Feedback stabilization of discrete-time quantum systems subject to non-demolition measurements with imperfections and delays

2012-01-06

arxiv.org/abs/1201.1387v1

Mathematics

Optimization and Control

submitted (29 pages, 4 figures)

Scientific paper

The mathematical methods underlying a recent quantum feedback experiment stabilizing photon-number states is developed. It considers a controlled system whose quantum state, a finite dimensional density operator, is governed by a discrete-time nonlinear Markov process. In open-loop, the measurements are assumed to be quantum non-demolition (QND) measurements. This Markov process admits a set of stationary pure states associated to an orthonormal basis. These stationary states provide martingales crucial to prove the open-loop stability: under simple assumptions, almost all trajectories converge to one of these stationary states; the probability to converge to a stationary state is given by its overlap with the initial quantum state. From these open-loop martingales, we construct a supermartingale whose parameters are given by inverting a Metzler matrix characterizing the impact of the control input on the Kraus operators defining the Markov process. This supermartingale measures the "distance" between the current quantum state and the goal state chosen from one of the open-loop stationary pure states. At each step, the control input minimizes the conditional expectation of this distance. It is proven that the resulting feedback scheme stabilizes almost surely towards the goal state whatever the initial quantum state. This state feedback takes into account a known constant delay of arbitrary length in the control loop. This control strategy is proved to remain also convergent when the state is replaced by its estimate based on a quantum filter. It relies on measurements that can be corrupted by random errors with conditional probabilities described by a known left stochastic matrix. Closed-loop simulations corroborated by experimental data illustrate the interest of such nonlinear feedback scheme for the photon box.

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