A Lyapunov-based Shaking Function for a Class of Non-bilinear Quantum Systems

This paper introduces a Lyapunov-based feedback law for quantum systems that are controlled by shifting the phase of an optical beam. The objective is to drive the system to a target eigenstate of the nominal Hamiltonian (i.e. phase equal zero) by designing a shaking function such that the Hilbert-Schmidt distance is monotonically decreasing. After identifying the control law that maximises the descent rate, two possible tuning strategies are proposed to address bounded input requirements. Convergence of the proposed controller is shown using the Krasovskii-LaSalle principle under the same conditions commonly found for bilinear quantum systems. Numerical simulations showcase the effectiveness of the proposed shaking function.