In this thesis, we study measurement-based feedbacks stabilizing discrete-time quantum systems subject to quantum non-demolition (QND) measurements and stability of continuous-time quantum filters. This thesis contains two parts. In the first part, we generalize the mathematical methods underlying a discrete-time quantum feedback experimentally tested in Laboratoire Kastler Brossel (LKB) at Ecole Normale Supérieure (ENS) de Paris. In fact, we contribute to a control algorithm which has been used in this recent quantum feedback experiment. This experiment prepares and stabilizes on demand photon-number states (Fock states) of a microwave field in a superconducting cavity. We design real-time filters allowing estimation of the state despite measurement imperfections and delays, and we propose a feedback law which ensures the stabilization of a predetermined target state. This stabilizing feedback is obtained by stochastic Lyapunov techniques and depends on a filter estimating the quantum state. We prove that such control strategy extends to other discrete-time quantum systems under QND measurements. The second part considers an extension, to continuous-time, of a stability result for discrete-time quantum filters. Indeed, we prove the stability of a quantum filter associated to usual stochastic master equation driven by a Wiener process. This stability means that a “distance” between the physical state and its associated quantum filter decreases in average. Another subject that we study in this part is related to the design of a continuous-time optimal filter, in the presence of measurement imperfections. To this aim, we extend a construction method for discrete-time quantum filters with measurement imperfections. Finally, we obtain heuristically generalized continuous-time optimal filters whose dynamics are given by stochastic master equations driven by both Poisson and Wiener processes. We conjecture the stability of such optimal filters.
