# Approximate stabilization of an infinite dimensional quantum stochastic system

**Authors**: Ram Somaraju, Mazyar Mirrahimi, Pierre Rouchon, 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC) 2011, pp. 6248 - 6253, 12-15 Dec. 2011, Orlando, USA. DOI: 10.1109/CDC.2011.6160560

We propose a feedback scheme for preparation of photon number states in a microwave cavity. Quantum Non-Demolition (QND) measurements of the cavity field and a control signal consisting of a microwave pulse injected into the cavity are used to drive the system towards a desired target photon number state. Unlike previous work, we do not use the Galerkin approximation of truncating the infinite-dimensional system Hilbert space into a finite-dimensional subspace. We use an (unbounded) strict Lyapunov function and prove that a feedback scheme that minimizes the expectation value of the Lyapunov function at each time step stabilizes the system at the desired photon number state with (a pre-specified) arbitrarily high probability. Simulations of this scheme demonstrate that we improve the performance of the controller by reducing “leakage” to high photon numbers.

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**BibTeX**:

@Proceedings{,

author = {Ram Somaraju, Mazyar Mirrahimi, Pierre Rouchon},

editor = {},

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

booktitle = {50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC) 2011},

volume = {},

publisher = {},

address = {},

pages = {6248 - 6253},

year = {2011},

abstract = {We propose a feedback scheme for preparation of photon number states in a microwave cavity. Quantum Non-Demolition (QND) measurements of the cavity field and a control signal consisting of a microwave pulse injected into the cavity are used to drive the system towards a desired target photon number state. Unlike previous work, we do not use the Galerkin approximation of truncating the infinite-dimensional system Hilbert space into a finite-dimensional subspace. We use an (unbounded) strict Lyapunov function and prove that a feedback scheme that minimizes the expectation value of the Lyapunov function at each time step stabilizes the system at the desired photon number state with (a pre-specified) arbitrarily high probability. Simulations of this scheme demonstrate that we improve the performance of the controller by reducing “leakage” to high photon numbers.},

keywords = {}}