Approximate stabilization of an infinite dimensional quantum stochastic system

TL;DR

This paper introduces a feedback control method for preparing specific photon number states in a microwave cavity, avoiding finite-dimensional approximations and using Lyapunov functions to ensure high-probability stabilization.

Contribution

It presents a novel feedback scheme that stabilizes infinite-dimensional quantum systems at a target state without truncating the Hilbert space, improving control performance.

Findings

The scheme stabilizes the system at the desired photon number state with high probability.

Simulations show reduced leakage to high photon numbers.

The method outperforms previous approaches using Galerkin approximations.

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.