Stochastic Quantum Dynamics Stabilization: A Lyapunov-Based Control Approach with Homodyne-Mediated Filtering

Efficient control of stochastic dynamics in quantum systems is pivotal for various applications, including quantum information processing and metrology. This paper introduces a Lyapunov-based control approach with homodyne-mediated filtering. We employ a modified extended Kalman filtering method to directly estimate the evolution of the quantum density operator ρ, considering sequential homodyne current measurements. Our method explicitly addresses the dynamics of a stochastic master equation with correlated noise, ensuring by construction the quantum properties of the estimated state variable ρ̂. Moreover, our proposed switching based Lyapunov control scheme that is fed with ρ̂, guarantees noise-to-state practically stable in probability of the desired quantum stationary target set with respect to the estimation error variance. We demonstrate our approach's efficacy in stabilizing a qubit coupled to a leaky cavity under homodyne detection.