# Approximate stabilization of an infinite dimensional quantum stochastic system

Authors: A. Somaraju, M. Mirrahimi, P. Rouchon, Reviews in Mathematical Physics, Vol 25, No 1, 1350001, DOI: 10.1142/S0129055X13500013

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.

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BibTeX:

@Article{,

author = {M. Mirrahimi A. Somaraju, P. Rouchon},

title = {Approximate stabilization of an infinite dimensional quantum stochastic system},

journal = {Reviews in Mathematical Physics},

volume = {25},

number = {1},

pages = {1350001},

year = {2013},

}

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.

Download PDF

BibTeX:

@Article{,

author = {M. Mirrahimi A. Somaraju, P. Rouchon},

title = {Approximate stabilization of an infinite dimensional quantum stochastic system},

journal = {Reviews in Mathematical Physics},

volume = {25},

number = {1},

pages = {1350001},

year = {2013},

}