Stabilizing Open Quantum Systems by Markovian Reservoir Engineering

TL;DR

This paper characterizes the steady states of open quantum systems governed by Lindblad equations, providing conditions for stability and methods for reservoir engineering to achieve desired quantum states.

Contribution

It offers a comprehensive characterization of steady states, stability conditions, and applications to physical models for quantum reservoir engineering.

Findings

Unique steady states imply attractivity and global stability.

Derived necessary and sufficient conditions for steady state existence.

Applicable to various physical quantum systems.

Abstract

We study open quantum systems whose evolution is governed by a master equation of Kossakowski-Gorini-Sudarshan-Lindblad type and give a characterization of the convex set of steady states of such systems based on the generalized Bloch representation. It is shown that an isolated steady state of the Bloch equation cannot be a center, i.e., that the existence of a unique steady state implies attractivity and global asymptotic stability. Necessary and sufficient conditions for the existence of a unique steady state are derived and applied to different physical models including two- and four-level atoms, (truncated) harmonic oscillators, composite and decomposable systems. It is shown how these criteria could be exploited in principle for quantum reservoir engineeing via coherent control and direct feedback to stabilize the system to a desired steady state. We also discuss the question of limit points of the dynamics. Despite the non-existence of isolated centers, open quantum systems can have nontrivial invariant sets. These invariant sets are center manifolds that arise when the Bloch superoperator has purely imaginary eigenvalues and are closely related to decoherence-free subspaces.