arXiv subjects

Pierre Rouchon

Publications and source records attributed to Pierre Rouchon.

At least 73 records · Page 4Linked to original sources

arXiv2015

Convergence and adiabatic elimination for a driven dissipative quantum harmonic oscillator

We prove that a harmonic oscillator driven by Lindblad dynamics where the typical drive and loss channels are two-photon processes instead of single-photon ones, converges to a protected subspace spanned by two coherent states of opposite amplitude. We then characterize the slow dynamics induced by a perturbative single-photon loss on this protected subspace, by performing adiabatic elimination in the Lindbladian dynamics.

quant-ph↗

arXiv2015

Parameter estimation from measurements along quantum trajectories

The dynamics of many open quantum systems are described by stochastic master equations. In the discrete-time case, we recall the structure of the derived quantum filter governing the evolution of the density operator conditioned to the measurement outcomes. We then describe the structure of the corresponding particle quantum filters for estimating constant parameter and we prove their stability. In the continuous-time (diffusive) case, we propose a new formulation of these particle quantum filters. The interest of this new formulation is first to prove stability, and also to provide an efficient algorithm preserving, for any discretization step-size, positivity of the quantum states and parameter classical probabilities. This algorithm is tested on experimental data to estimate the detection efficiency for a superconducting qubit whose fluorescence field is measured using a heterodyne detector.

math.OC↗

arXiv2015

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.

math.OC↗

arXiv2015

Efficient Quantum Filtering for Quantum Feedback Control

We discuss an efficient numerical scheme for the recursive filtering of diffusive quantum stochastic master equations. We show that the resultant quantum trajectory is robust and may be used for feedback based on inefficient measurements. The proposed numerical scheme is amenable to approximation, which can be used to further reduce the computational burden associated with calculating quantum trajectories and may allow real-time quantum filtering. We provide a two-qubit example where feedback control of entanglement may be within the scope of current experimental systems.

quant-ph↗

arXiv2014

A Flat Triangular Form for Nonlinear Systems with Two Inputs: Necessary and Sufficient Conditions

The present work establishes necessary and sufficient conditions for a nonlinear system with two inputs to be described by a specific triangular form. Except for some regularity conditions, such triangular form is flat. This may lead to the discovery of new flat systems. The proof relies on well-known results for driftless systems with two controls (the chained form) and on geometric tools from exterior differential systems. The paper also illustrates the application of its results on an academic example and on a reduced order model of an induction motor.

math.OC↗

arXiv2014

Controllability of the 1D Schrodinger equation by the flatness approach

We derive in a straightforward way the exact controllability of the 1-D Schrodinger equation with a Dirichlet boundary control. We use the so-called flatness approach, which consists in parameterizing the solution and the control by the derivatives of a "flat output". This provides an explicit control input achieving the exact controllability in the energy space. As an application, we derive an explicit pair of control inputs achieving the exact steering to zero for a simply-supported beam.

math.OC↗

arXiv2014

Energy-based modeling of electric motors

We propose a new approach to model electrical machines based on energy considerations and construction symmetries of the motor. We detail the approach on the Permanent-Magnet Synchronous Motor and show that it can be extended to Synchronous Reluctance Motor and Induction Motor. Thanks to this approach we recover the usual models without any tedious computation. We also consider effects due to non-sinusoidal windings or saturation and provide experimental data.

math.OC↗

arXiv2013

Stability of continuous-time quantum filters with measurement imperfections

The fidelity between the state of a continuously observed quantum system and the state of its associated quantum filter, is shown to be always a submartingale. The observed system is assumed to be governed by a continuous-time Stochastic Master Equation (SME), driven simultaneously by Wiener and Poisson processes and that takes into account incompleteness and errors in measurements. This stability result is the continuous-time counterpart of a similar stability result already established for discrete-time quantum systems and where the measurement imperfections are modeled by a left stochastic matrix.

quant-ph↗

arXiv2013

Trajectory control of a bipedal walking robot with inertial disc

In this paper we exploit some interesting properties of a class of bipedal robots which have an inertial disc. One of this properties is the ability to control every position and speed except for the disc position. The proposed control is designed in two hierarchic levels. The first will drive the robot geometry, while the second will control the speed and also the angular momentum. The exponential stability of this approach is proved around some neighborhood of the nominal trajectory defining the geometry of the step. This control will not spend energy to adjust the disc position and neither to synchronize the trajectory with the time. The proposed control only takes action to correct the essential aspects of the walking gait. Computational simulations are presented for different conditions, serving as a empirical test for the neighborhood of attraction.

math.OC↗

arXiv2013

Null controllability of the heat equation using flatness

We derive in a direct and rather straightforward way the null controllability of the N-dimensional heat equation in a bounded cylinder with boundary control at one end of the cylinder. We use the so-called flatness approach, which consists in parameterizing the solution and the control by the derivatives of a "flat output". This yields an explicit control law achieving the exact steering to zero. Replacing the involved series by partial sums we obtain a simple numerical scheme for which we give explicit error bounds. Numerical experiments demonstrate the relevance of the approach.

math.OC↗

arXiv2013

Inertial-sensor bias estimation from brightness/depth images and based on SO(3)-invariant integro/partial-differential equations on the unit sphere

Constant biases associated to measured linear and angular velocities of a moving object can be estimated from measurements of a static scene by embedded brightness and depth sensors. We propose here a Lyapunov-based observer taking advantage of the SO(3)-invariance of the partial differential equations satisfied by the measured brightness and depth fields. The resulting asymptotic observer is governed by a non-linear integro/partial differential system where the two independent scalar variables indexing the pixels live on the unit sphere of the 3D Euclidian space. The observer design and analysis are strongly simplified by coordinate-free differential calculus on the unit sphere equipped with its natural Riemannian structure. The observer convergence is investigated under C^1 regularity assumptions on the object motion and its scene. It relies on Ascoli-Arzela theorem and pre-compactness of the observer trajectories. It is proved that the estimated biases converge towards the true ones, if and only if, the scene admits no cylindrical symmetry. The observer design can be adapted to realistic sensors where brightness and depth data are only available on a subset of the unit sphere. Preliminary simulations with synthetic brightness and depth images (corrupted by noise around 10%) indicate that such Lyapunov-based observers should be robust and convergent for much weaker regularity assumptions.

math.OC↗

arXiv2013

Quantum Gate Generation by T-Sampling Stabilization

This paper considers right-invariant and controllable driftless quantum systems with state X(t) evolving on the unitary group U(n) and m inputs. The T-sampling stabilization problem is introduced and solved: given any initial condition X(0) any goal state Xg find a feedback law such that X(jT) converges to Xg as the integer j tends to infinity. The purpose is to generate arbitrary quantum gates corresponding to Xg. This is achieved by the tracking of T-periodic reference trajectories that pass by Xg using the framework of Coron's Return Method. The T-periodic reference trajectories are generated by applying open-loop controls that are a sum of a finite number M of odd T-periodic function whose amplitudes are parameterized by a vector. The main result establishes that, for M big enough, X(jT) exponentially converges towards Xg for almost all fixed amplitude vectors, 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 amplitude vector at each sampling time jT, and keeping it fixed until the next sampling time (j+1)T. It is also shown that X(jT) exponentially converges towards Xg almost surely. Simulation results have indicated that the convergence speed may be significantly improved with such stochastic technique. This is illustrated in the generation of the C--NOT quantum logic gate on U(4).

math.OC↗

arXiv2013

Null Controllability of the 2D Heat Equation Using Flatness

We derive in a direct and rather straightforward way the null controllability of a 2-D heat equation with boundary control. We use the so-called flatness approach, which consists in parameterizing the solution and the control by the derivatives of a "flat output". This provides an explicit control law achieving the exact steering to zero. Numerical experiments demonstrate the relevance of the approach.

math.OC↗

arXiv2013

Null controllability of the 1D heat equation using flatness

We derive in a straightforward way the null controllability of a 1-D heat equation with boundary control. We use the so-called {\em flatness approach}, which consists in parameterizing the solution and the control by the derivatives of a "flat output". This provides an explicit control law achieving the exact steering to zero. We also give accurate error estimates when the various series involved are replaced by their partial sums, which is paramount for an actual numerical scheme. Numerical experiments demonstrate the relevance of the approach.

math.OC↗

arXiv2013

Feedback stabilization of discrete-time quantum systems subject to non-demolition measurements with imperfections and delays

We consider a controlled quantum system whose finite dimensional state is governed by a discrete-time nonlinear Markov process. In open-loop, the measurements are assumed to be quantum non-demolition (QND). The eigenstates of the measured observable are thus the open-loop stationary states: they are used to construct a closed-loop supermartingale playing the role of a strict control Lyapunov function. The parameters of this supermartingale are calculated by inverting a Metzler matrix that characterizes the impact of the control input on the Kraus operators defining the Markov process. The resulting state feedback scheme, taking into account a known constant delay, provides the almost sure convergence of the controlled system to the target state. This convergence is ensured even in the case where the filter equation results from imperfect measurements corrupted by random errors with conditional probabilities given as a left stochastic matrix. Closed-loop simulations corroborated by experimental data illustrate the interest of such nonlinear feedback scheme for the photon box, a cavity quantum electrodynamics system.

math.OC↗

arXiv2013

Contraction and stability analysis of steady-states for open quantum systems described by Lindblad differential equations

For discrete-time systems, governed by Kraus maps, the work of D. Petz has characterized the set of universal contraction metrics. In the present paper, we use this characterization to derive a set of quadratic Lyapunov functions for continuous-time systems, governed by Lindblad differential equations, that have a steady-state with full rank. An extremity of this set is given by the Bures metric, for which the quadratic Lyapunov function is obtained by inverting a Sylvester equation. We illustrate the method by providing a strict Lyapunov function for a Lindblad equation designed to stabilize a quantum electrodynamic "cat" state by reservoir engineering. In fact we prove that any Lindblad equation on the Hilbert space of the (truncated) harmonic oscillator, which has a full-rank equilibrium and which has, among its decoherence channels, a channel corresponding to the photon loss operator, globally converges to that equilibrium.

math.OC↗

arXiv2012

Low rank approximation for the numerical simulation of high dimensional Lindblad and Riccati equations

A systematic numerical approach to approximate high dimensional Lindblad equations is described. It is based on a deterministic rank m approximation of the density operator, the rank m being the only parameter to adjust. From a known initial value, this rank m approximation gives at each time-step an estimate of the largest m eigen-values with their eigen-vectors of the density operator. A numerical scheme is proposed. Its numerical efficiency in the case of a rank 12 approximation is demonstrated for oscillation revivals of 50 atoms interacting resonantly with a slightly damped coherent quantized field of 200 photons. The approach may be employed for other similar equations. We in particularly show how to adapt such low-rank approximation for Riccati differential equations appearing in Kalman filtering.

physics.comp-ph↗

arXiv2012

Robust open-loop stabilization of Fock states by time-varying quantum interactions

A quantum harmonic oscillator (spring subsystem) is stabilized towards a target Fock state by reservoir engineering. This passive and open-loop stabilization works by consecutive and identical Hamiltonian interactions with auxiliary systems, here three-level atoms (the auxiliary ladder subsystem), followed by a partial trace over these auxiliary atoms. A scalar control input governs the interaction, defining which atomic transition in the ladder subsystem is in resonance with the spring subsystem. We use it to build a time-varying interaction with individual atoms, that combines three non-commuting steps. We show that the resulting reservoir robustly stabilizes any initial spring state distributed between 0 and 4n+3 quanta of vibrations towards a pure target Fock state of vibration number n. The convergence proof relies on the construction of a strict Lyapunov function for the Kraus map induced by this reservoir setting on the spring subsystem. Simulations with realistic parameters corresponding to the quantum electrodynamics setup at Ecole Normale Superieure further illustrate the robustness of the method.

quant-ph↗