Section: New Results

System theory approach of some quantum systems

Design of strict control-Lyapunov functions for quantum systems with QND measurements

Participants : Hadis Amini, Mazyar Mirrahimi, Pierre Rouchon.

We have proposed a feedback scheme for quantum systems undergoing discrete-in-time non-destructive measurements. Under some observability assumptions, the proposed feedback law ensures the stabilization of any desired equilibrium state of the measurement process. This theoretical contribution has given rise to a primary conference paper [40] and we are currently working on a more complete journal paper that proves the convergence in presence of various measurement noises and uncertainties. The proposed feedback scheme has been applied in a recent experiment realized by Serge Haroche and Jean-Michel Raimond's group at Ecole Normale Supérieure and has given rise to a journal paper [29] .

Approximate stabilization of an infinite dimensional quantum stochastic system

Participants : Ram Somaraju, Mazyar Mirrahimi, Pierre Rouchon.

Extending our previous feedback schemes dealing only with finite dimensional quantum systems, we have proven the approximate stabilization of any desired Fock state in the microwave cavity setup of Ecole Normale Supérieure. By appropriately choosing the Lyapunov function, we avoid the mass-loss phenomena through high energy levels and ensure the pre-compactness (in probability) of the trajectories. This work has given rise to a conference paper [48] and a submitted journal paper [56] .

On stability of continuous-time quantum filters

Participants : Hadis Amini, Mazyar Mirrahimi, Pierre Rouchon.

We have studied the stability of quantum filters for continuous-in-time measurement processes. Indeed, we have proven that the filter between the quantum state governed by a continuous time stochastic master equation driven by a Wiener process and its associated quantum-filter state is a sub-martingale. This result has given rise to a conference paper [41] .