Coherent feedback $H^\infty$ control of quantum linear systems

TL;DR

This paper presents a simplified and efficient method for designing coherent feedback $H^ Infty$ controllers for linear quantum systems, ensuring stability and disturbance attenuation with reduced computational complexity.

Contribution

It introduces a new design approach that simplifies controller synthesis by solving Lyapunov equations instead of Riccati equations, applicable to general and passive quantum systems.

Findings

Controller design reduces to solving at most four Lyapunov equations.

Provides necessary and sufficient conditions for passive systems using uncoupled Lyapunov pairs.

Demonstrates effectiveness on quantum optical devices like cavities and parametric amplifiers.

Abstract

The purpose of this paper is to investigate the coherent feedback $H^\infty$ control problem for linear quantum systems. A key contribution is a simplified design methodology that guarantees closed-loop stability and a prescribed level of disturbance attenuation. It is shown that for general linear quantum systems, a physically realizable quantum controller can be obtained by solving at most four Lyapunov equations. In the passive case, a necessary and sufficient condition is provided in terms of two uncoupled pairs of Lyapunov equations. These results represent a significant simplification over the standard approach, which requires solving two coupled algebraic Riccati equations. The effectiveness of the proposed method is demonstrated through two typical quantum optical devices: an empty optical cavity and a degenerate parametric amplifier. These results provide a computationally efficient procedure for the robust and optimal control of quantum optical and optomechanical systems.