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

• Published in: Automatica (Oxford) Vol. 49; no. 9; pp. 2683 - 2692
• Main Authors: Amini, Hadis, Somaraju, Ram A., Dotsenko, Igor, Sayrin, Clément, Mirrahimi, Mazyar, Rouchon, Pierre
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Ltd 01.09.2013; Elsevier
• Subjects: Applied sciences; Automation; Classical and quantum physics: mechanics and fields; Computer science; control theory; systems; Control system analysis; Control system synthesis; Control theory. Systems; Convergence; Delay; Exact sciences and technology; Feedback; Feedback stabilization; Markov chain; Markov processes; Mathematical analysis; Mathematics; Measurement-based feedback; Modelling and identification; Optimization and Control; Photon-number states (Fock states); Photons; Physics; Quantum computation; Quantum filter; Quantum information; Quantum non-demolition measurements; Quantum Physics; Quantum theory; Strict control Lyapunov function; Markov chain; Strict control Lyapunov function; Photon-number states (Fock states); Measurement-based feedback; Quantum filter; Feedback stabilization; Quantum non-demolition measurements; Markov process; State feedback; Quantum system; Fock space; Error probability; Feedback regulation; Conditional probability; Discrete time processes; Modeling; Delay; Quantum nondemolition measurement; Defect; System identification; Measurement error; Closed loop; Role playing; Almost sure convergence; Stochastic matrix; Stationary state; Finite state machine; Cavity electrodynamics; Observable; Kraus operator; Filter; Discrete time; Open loop; Output feedback; Lyapunov function; Signal to noise ratio
• DOI: 10.1016/j.automatica.2013.06.012
• ISSN: 0005-1098

We consider a controlled quantum system whose finite dimensional state is governed by a discrete-time nonlinear Markov process. In open-loop, the measurements are assumed to be quantum non-demolition (QND). The eigenstates of the measured observable are thus the open-loop stationary states: they are used to construct a closed-loop supermartingale playing the role of a strict control Lyapunov function. The parameters of this supermartingale are calculated by inverting a Metzler matrix that characterizes the impact of the control input on the Kraus operators defining the Markov process. The resulting state feedback scheme, taking into account a known constant delay, provides the almost sure convergence to the target state. This convergence is ensured even in the case where the filter equation results from imperfect measurements corrupted by random errors with conditional probabilities given as a left stochastic matrix. Closed-loop simulations corroborated by experimental data illustrate the interest of such nonlinear feedback scheme for the photon box, a cavity quantum electrodynamics system.