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

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.