An Adaptive Observer Design for State and Parameter Estimation of Quantum Systems via Averaging Theory

This paper presents an adaptive asymptotic observer-based approach for estimating the states and the atom-laser coupling constant (parameter) of a two-level, closed quantum system driven by a constant control input. Using the interaction frame and averaging theory, the proposed observer ensures asymptotic estimation of both the system states and the unknown parameter. In addition to the asymptotic observer, we also propose an adaptive finite-time observer for the estimation of states in finite time, considering the parameter is unknown. The proposed finite-time observer is designed to provide the minimum output horizon required for accurate state and parameter estimation. To ensure the estimation of the unknown parameter, the extended Lyapunov method is employed. Through simulations on a two-level quantum dynamical system, the proposed observers are demonstrated to exhibit robust performance in the presence of noise in the control and output measurement.