Approximate stabilization of an infinite dimensional quantum stochastic system

Ram Abhinav Somaraju (1) , Mazyar Mirrahimi (1) , Pierre Rouchon (2)

1 SISYPHE - SIgnals and SYstems in PHysiology & Engineering
2 CAS - Centre Automatique et Systèmes

We study the state feedback stabilization of a quantum harmonic oscillator near a pre-specified Fock state (photon number state). Such a state feedback controller has been recently implemented on a quantized electromagnetic field in an almost lossless cavity. Such open quantum systems are governed by a controlled discrete-time Markov chain in the unit ball of an infinite dimensional Hilbert space. The control design is based on an unbounded Lyapunov function that is minimized at each time-step by feedback. This ensures (weak-*) convergence of probability measures to a final measure concentrated on the target Fock state with a pre-specified probability. This probability may be made arbitrarily close to 1 by choosing the feedback parameters and the Lyapunov function. They are chosen so that the stochastic flow that describes the Markov process may be shown to be tight (concentrated on a compact set with probability arbitrarily close to 1). Convergence proof uses Prohorov's theorem and specific properties of this Lyapunov function.

Mots clés: Quantum feedback control; stochastic stabilization; infinite dimensional stochastic systems

Domaines: Sciences de l'ingénieur [physics]; Automatique / Robotique

Ram Abhinav Somaraju, Mazyar Mirrahimi, Pierre Rouchon. Approximate stabilization of an infinite dimensional quantum stochastic system. Reviews in Mathematical Physics, 2013, 25 (1), pp.1350001. ⟨10.1142/S0129055X13500013⟩. ⟨hal-00829864⟩