International audienceFeedback stabilization of an ensemble of non interacting half spins described by the Bloch equations is considered. This system may be seen as an interesting example for infinite dimensional systems with continuous spectra. We propose an explicit feedback law that stabilizes asymptotically the system around a uniform state of spin 1/2 or -1/2. The proof of the convergence is done locally around the equilibrium in the H1 topology. This local convergence is shown to be a weak asymptotic convergence for the H1 topology and thus a strong convergence for the C0 topology. The proof relies on an adaptation of the LaSalle invariance principle to infinite dimensional systems. Numerical simulations illustrate the efficiency of these feedback laws, even for initial conditions far from the equilibrium

For a right-invariant and controllable driftless system on SU(n), we consider a time-periodic reference trajectory along which the linearized control system generates su(n): such trajectories always exist and constitute the basic ingredient of Coron's Return Method. The open-loop controls that we propose, which rely on a left-invariant tracking error dynamics and on a fidelity-like Lyapunov function, are determined from a finite number of left-translations of the tracking error and they assure global asymptotic convergence towards the periodic reference trajectory. The role of these translations is to avoid being trapped in the critical region of this Lyapunov-like function. The convergence proof relies on a periodic version of LaSalle's invariance principle and the control values are determined by numerical integration of the dynamics of the system. Simulations illustrate the obtained controls for n = 4 and the generation of the C-NOT quantum gate.

6 pages, 2 figuresInternational audienceWe consider the feedback stabilization of a variable profile for an ensemble of non interacting half spins described by the Bloch equations. We propose an explicit feedback law that stabilizes asymptotically the system around a given arbitrary target profile. The convergence proof is done when the target profile is entirely in the south hemisphere or in the north hemisphere of the Bloch sphere. The convergence holds for initial conditions in a H^1 neighborhood of this target profile. This convergence is shown for the weak H^1 topology. The proof relies on an adaptation of the LaSalle invariance principle to infinite dimensional systems. Numerical simulations illustrate the efficiency of these feedback laws, even for initial conditions far from the target profile

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This paper considers right-invariant and controllable driftless quantum systems with state X(t) evolving on the unitary group U(n) and m inputs u = (u 1 , . . . , u m ). The T -sampling stabilisation problem is introduced and solved: given any initial condition X 0 and any goal state X goal , find a control law u = u(X, t) such that lim j →∞ X(jT ) = X goal for the closed-loop system. The purpose is to generate arbitrary quantum gates corresponding to X goal . This is achieved by the tracking of T -periodic reference trajectories (X a (t), u a (t)) of the quantum system that pass by X goal using the framework of Coron's return method. The T -periodic reference trajectories X a (t) are generated by applying controls u a (t) that are a sum of a finite number M of harmonics of sin(2πt/T ), whose amplitudes are parameterised by a vector a. The main result establishes that, for M big enough, X(jT ) exponentially converges towards X goal for almost all fixed a, with explicit and completely constructive control laws. This paper also establishes a stochastic version of this deterministic control law. The key idea is to randomly choose a different parameter vector of control amplitudes a = a j at each t = jT , and keeping it fixed for t ∈ [jT , (j + 1)T ). It is shown in the paper that X(jT ) exponentially converges towards X goal almost surely. Simulation results have indicated that the convergence speed of X(jT ) may be significantly improved with such stochastic technique. This is illustrated in the generation of the C-NOT quantum logic gate on U(4).