Models and Feedback Stabilization of Open Quantum Systems

At the quantum level, feedback-loops have to take into account measurement back-action. We present here the structure of the Markovian models including such back-action and sketch two stabilization methods: measurement-based feedback where an open quantum system is stabilized by a classical controller; coherent or autonomous feedback where a quantum system is stabilized by a quantum controller with decoherence (reservoir engineering). We begin to explain these models and methods for the photon box experiments realized in the group of Serge Haroche (Nobel Prize 2012). We present then these models and methods for general open quantum systems.Comment: Extended version of the paper attached to an invited conference for the International Congress of Mathematicians in Seoul, August 13 - 21, 201

Singular perturbations and Lindblad-Kossakowski differential equations

We consider an ensemble of quantum systems whose average evolution is described by a density matrix, solution of a Lindblad-Kossakowski differential equation. We focus on the special case where the decoherence is only due to a highly unstable excited state and where the spontaneously emitted photons are measured by a photo-detector. We propose a systematic method to eliminate the fast and asymptotically stable dynamics associated to the excited state in order to obtain another differential equation for the slow part. We show that this slow differential equation is still of Lindblad-Kossakowski type, that the decoherence terms and the measured output depend explicitly on the amplitudes of quasi-resonant applied field, i.e., the control. Beside a rigorous proof of the slow/fast (adiabatic) reduction based on singular perturbation theory, we also provide a physical interpretation of the result in the context of coherence population trapping via dark states and decoherence-free subspaces. Numerical simulations illustrate the accuracy of the proposed approximation for a 5-level systems.Comment: 6 pages, 2 figure

Fidelity is a sub-martingale for discrete-time quantum filters

Fidelity is known to increase through any Kraus map: the fidelity between two density matrices is less than the fidelity between their images via a Kraus map. We prove here that, in average, fidelity is also increasing for any discrete-time quantum filter: fidelity between the density matrix of the underlying Markov chain and the density matrix of its associated quantum filter is a sub-martingale. This result is not restricted to pure states. It also holds true for mixed states

Symmetry-preserving Observers

This paper presents three non-linear observers on three examples of engineering interest: a chemical reactor, a non-holonomic car, and an inertial navigation system. For each example, the design is based on physical symmetries. This motivates the theoretical development of invariant observers, i.e, symmetry-preserving observers. We consider an observer to consist in a copy of the system equation and a correction term, and we give a constructive method (based on the Cartan moving-frame method) to find all the symmetry-preserving correction terms. They rely on an invariant frame (a classical notion) and on an invariant output-error, a less standard notion precisely defined here. For each example, the convergence analysis relies also on symmetries consideration with a key use of invariant state-errors. For the non-holonomic car and the inertial navigation system, the invariant state-errors are shown to obey an autonomous differential equation independent of the system trajectory. This allows us to prove convergence, with almost global stability for the non-holonomic car and with semi-global stability for the inertial navigation system. Simulations including noise and bias show the practical interest of such invariant asymptotic observers for the inertial navigation system.Comment: To be published in IEEE Automatic Contro

On stability of continuous-time quantum-filters

We prove that the fidelity 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 is a generalization to non-pure quantum states where fidelity does not coincide in general with a simple Frobenius inner product. This result implies the stability of such filtering process but does not necessarily ensure the asymptotic convergence of such quantum-filters

Design of Strict Control-Lyapunov Functions for Quantum Systems with QND Measurements

We consider discrete-time quantum systems subject to Quantum Non-Demolition (QND) measurements and controlled by an adjustable unitary evolution between two successive QND measures. In open-loop, such QND measurements provide a non-deterministic preparation tool exploiting the back-action of the measurement on the quantum state. We propose here a systematic method based on elementary graph theory and inversion of Laplacian matrices to construct strict control-Lyapunov functions. This yields an appropriate feedback law that stabilizes globally the system towards a chosen target state among the open-loop stable ones, and that makes in closed-loop this preparation deterministic. We illustrate such feedback laws through simulations corresponding to an experimental setup with QND photon counting

Approximate stabilization of an infinite dimensional quantum stochastic system

We propose a feedback scheme for preparation of photon number states in a microwave cavity. Quantum Non-Demolition (QND) measurements of the cavity field and a control signal consisting of a microwave pulse injected into the cavity are used to drive the system towards a desired target photon number state. Unlike previous work, we do not use the Galerkin approximation of truncating the infinite-dimensional system Hilbert space into a finite-dimensional subspace. We use an (unbounded) strict Lyapunov function and prove that a feedback scheme that minimizes the expectation value of the Lyapunov function at each time step stabilizes the system at the desired photon number state with (a pre-specified) arbitrarily high probability. Simulations of this scheme demonstrate that we improve the performance of the controller by reducing "leakage" to high photon numbers.Comment: Submitted to CDC 201

Euler-Lagrange models with complex currents of three-phase electrical machines and observability issues

A new Lagrangian formulation with complex currents is developed and yields a direct and simple method for modeling three-phase permanent-magnet and induction machines. The Lagrangian is the sum a mechanical one and of a magnetic one. This magnetic Lagrangian is expressed in terms of rotor angle, complex stator and rotor currents. A complexification procedure widely used in quantum electrodynamic is applied here in order to derive the Euler-Lagrange equations with complex stator and rotor currents. Such complexification process avoids the usual separation into real and imaginary parts and simplifies notably the calculations. Via simple modifications of such magnetic Lagrangians we derive new dynamical models describing permanent-magnet machines with both saturation and saliency, and induction machines with both magnetic saturation and space harmonics. For each model we also provide its Hamiltonian thus its magnetic energy. This energy is also expressed with complex currents and can be directly used in Lyapunov and/or passivity based control. Further, we briefly investigate the observability of this class of Euler-Lagrange models, in the so-called sensorless case when the measured output is the stator current and the load torque is constant but unknown. For all the dynamical models obtained via such variational principles, we prove that their linear tangent systems are unobservable around a one-dimensional family of steady-states attached to the same constant stator voltage and current. This negative result explains why sensorless control of three-phase electrical machines around zero stator frequency remains yet a difficult control problem.Comment: Revised version. Submitted for publicatio