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ABSTRACT: We propose to apply the Back and Forth Nudging (BFN) method used for geophysical data assimilations [1] to estimate the initial state of a quantum system. We consider a cloud of atoms interacting with a magnetic field while a single observable is being continuously measured over time using homodyne detection. The BFN method relies on designing an observer forward and backwards in time. The state of the BFN observer is continuously updated by the measured data and tends to converge to the system's state. The proposed estimator seems to be globally asymptotically convergent when the system is observable. A detailed convergence proof and simulations are given in the 2-level case. An extension of the algorithm to the multilevel case is also presented.
American Control Conference (ACC), 2011; 08/2011
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ABSTRACT: This paper proposes simple feedback loops, inspired from extremum-seeking, that use the photon emission times of a single quantum system following quantum Monte-Carlo trajectories in order to lock in real time a probe frequency to the system's transition frequency. Two specific settings are addressed: a 3-level system coupling one ground to two excited states (one highly unstable and one metastable) and a 3-level system coupling one excited to two ground states (both metastable). Analytical proofs and simulations show the accurate and robust convergence of probe frequency to system-transition frequency in the two cases.
Decision and Control (CDC), 2010 49th IEEE Conference on; 01/2011
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ABSTRACT: We propose an estimation method allowing to identify the initial state of a quantum system based on the continuous weak measurement of a certain physical observable over a fixed interval of time. The algorithm is based on the back-and-forth nudging method consisting in iterative application of Luenberger observers for the time-forward and time-backward dynamics. A clever change of variables unveils the needed symmetry in the observer design leading to the decrease of a certain distance (in an appropriate metric) between the estimator and the main system, both in forward and backward directions.
Communications, Control and Signal Processing (ISCCSP), 2010 4th International Symposium on; 04/2010
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ABSTRACT: A feedback scheme for preparation of photon number states in a microwave cavity is proposed. Quantum Non Demolition (QND) measurement of the cavity field provides information on its actual state. The control consists in injecting into the cavity mode a microwave pulse adjusted to increase the population of the desired target photon number. In the ideal case (perfect cavity and measures), we present the feedback scheme and its detailed convergence proof through stochastic Lyapunov techniques based on super-martingales and other probabilistic arguments. Quantum Monte-Carlo simulations performed with experimental parameters illustrate convergence and robustness of such feedback scheme.
Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference. CDC/CCC 2009. Proceedings of the 48th IEEE Conference on; 01/2010
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ABSTRACT: An observer-based Hamiltonian identification algorithm for quantum systems has been recently proposed by two of the authors to estimate the dipole moment matrix of a quantum system requiring the measurement of the populations on all states. This could be experimentally difficult to achieve. We propose here an extension to a 3-level quantum system, having access to the population of the ground state only. By a suitable choice of the control field, we show that a continuous measurement of this population, alone, is enough to identify the field coupling parameters (dipole moment). Simulations with 20% of noise on the measured population illustrate the robustness of the proposed estimation algorithm and confirm the convergence analysis.
Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference. CDC/CCC 2009. Proceedings of the 48th IEEE Conference on; 01/2010
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ABSTRACT: We explore experimentally the fundamental projective properties of a quantum measurement and their application in the control of a system's evolution. We perform quantum non-demolition (QND) photon counting on a microwave field trapped in a very-high-Q superconducting cavity, employing circular Rydberg atoms as non-absorbing probes of light. By repeated measurement of the cavity field we demonstrated the freeze of its initially coherent evolution, illustrating the back action of the photon number measurement on the field's phase. On the contrary, by utilizing a weak QND measurement in combination with the control injection of coherent pulses, we plan to force the field to deterministically evolve towards any target photon-number state. This quantum feedback procedure will enable us to prepare and protect photon-number states against decoherence.
Phys. Scr. T140. 01/2010;
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ABSTRACT: We propose a quantum feedback scheme for the preparation and protection of photon-number states of light trapped in a high-Q microwave cavity. A quantum nondemolition measurement of the cavity field provides information on the photon-number distribution. The feedback loop is closed by injecting into the cavity a coherent pulse adjusted to increase the probability of the target photon number. The efficiency and reliability of the closed-loop state stabilization is assessed by quantum Monte Carlo simulations. We show that, in realistic experimental conditions, the Fock states are efficiently produced and protected against decoherence.
Phys. Rev. A. 07/2009; 80(1).
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ABSTRACT: We consider an ensemble of quantum systems 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 with 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.
IEEE Transactions on Automatic Control 07/2009; · 2.11 Impact Factor
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ABSTRACT: A symmetry-preserving observer-based parameter identification algorithm for quantum systems is proposed. Starting with a 2-level quantum system (qubit), where the unknown parameters consist of the atom-laser frequency detuning and coupling constant, we prove an exponential convergence result. The analysis is inspired by Lyapunov and adaptive control techniques and is based on averaging theory. The observer is then extended to the multi-level case where all the atom-laser coupling constants are unknown. The extension of the convergence analysis is discussed through some heuristic arguments. The relevance and the robustness with respect to various noises are tested through numerical simulations.
Automatica. 01/2009;
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ABSTRACT: We consider an ensemble of quantum systems whose average evolution is described by a density matrix solution of a Lindbladian differential equation. We will suppose that the decoherence is only due to a highly unstable excited state. We measure then the spontaneously emitted photons. Whenever we consider resonant laser fields, we can remove the fast 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 type. The decoherence terms are then of second order and the measurement structure depends explicitly on the resonant laser field. The later can be adjusted to give information on a specific linear combination of the density matrix entries. The case of a 3-level system is treated in details and before the general case. On this 3-level system, we show how a simple PI regulator allows us to robustly control the 2-level slow subsystem
Decision and Control, 2006 45th IEEE Conference on; 01/2007
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ABSTRACT: A result (see, e.g., [3][complement B III , page 217]) of quantum electrodynamics (based on Glauber theorem) says that classical currents and sources only generate classical light (quasi-classical states of the field). This paper provides a control theoretic interpretation of this result when the classical currents and sources are considered as control inputs: the dynamics of the quantified electrodynamics field is not controllable; the controllable part is contained into the classical dynamics. This result can be seen as the infinite dimensional analogue of the following fact (see [8]): the controllable part of the quantum harmonic oscillator corresponds to the classical dynamics of the average <q̈>=- +u (u is the control input). Thus controllability can only be achieved when the field dynamics is coupled to some localized quantum dynamics. We describe here two typical models of such controlled systems: the Jaynes-Cummings model [6] (one field mode coupled to a two-level atom) and trapped ions vibrational model. Both systems correspond to experiments conducted by physicists. We recall their main approximations and validity domains. We also provide their associated formulation in terms of PDE with control.
Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC '05. 44th IEEE Conference on; 01/2006
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ABSTRACT: It is proven in a previous paper that any modal approximation of the one-dimensional quantum harmonic oscillator is controllable. We prove here that, contrary to such finite-dimensional approximations, the original infinite-dimensional system is not controllable: Its controllable part is of dimension 2 and corresponds to the dynamics of the average position. More generally, we prove that, for the quantum harmonic oscillator of any dimension, similar lacks of controllability occur whatever the number of control is: the controllable part still corresponds to the average position dynamics. We show, with the quantum particle in a moving quadratic potential, that some physically interesting motion planning questions can be however solved.
IEEE Transactions on Automatic Control 06/2004; · 2.11 Impact Factor
