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Quantum Electronics and Laser Science Conference

Feedback control of linear quantum systems

Andrew C. Doherty, Howard M. Wiseman

Published Dec 10, 2005• 3 citations
Publisher

Abstract

We compare quantum and classical feedback control of linear systems, investigating the effect of back-action noise on control performance. One consequence is the existence of an optimal measurement for fixed system-environment coupling.
DOI
10.1109/qels.2005.1548668

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Wecompare quantum andclassical feedback control oflinear systems, investigating the effect ofback-action noise oncontrol performance. Oneconsequence istheexistence ofan optimal measurement forfixed system-environment coupling. ©2005 Optical Society ofAmerica OCIScodes: (270.5570) Quantum detectors; (270.250) Fluctuations, relaxations andnoise. 1.Introduction andBackground Experiments inoptical andatomic physics areincreasingly able tomakemeasurements onindividual quantum systems with signal tonoise atornearthequantum limit. Since quantum measurements irreducibly disturb the system, feedback control loops containing such quantum-limited sensors musttake this back-action noise into account incontroller design. Asaconsequence thestudy ofquantum feedback control hasbeen developing rapidly inrecent years, both intheory andnotably inquantum optics experiments (1-3). Linear quantum systems arethose forwhich theHeisenberg equations ofmotion for thecanonical positions and momenta (and hence also theequations fortheir expectation values) arelinear. WithGaussian initial conditions the equations ofmotion forthemeanvalues andcovariance matrix form aclosed system andpropagate sufficient statistics forthefull quantum state. This immensely simplifies theanalysis andmakes itpossible toutilize and compare toresults inclassical control theory forlinear systems with Gaussian noises, theso-called Linear- Quadratic-Gaussian
In many optical and atomic systems it is now possible to monitor an individual quantum system with high signal to noise and feed back by altering the system dynamics in real time. This motivates further development of quantum mechanical descriptions of feedback control. We discuss recent work on closed loop control of open quantum systems, focusing on general linear systems for which the statistics of the problem are Gaussian. Such systems may be realized with linear optics, parametric amplifiers and homodyne detection. This problem allows a direct comparison with the classical linear, quadratic cost, Gaussian noise (LQG) optimal control problem. This highlights the key distinction between the quantum and classical theories for linear systems: that increased measurement sensitivity may run counter to the control objectives since it increases the backaction noise. While in an idealized classical control problem it is always preferable to obtain a better sensor, quantum mechanical problems generically have an optimal sensitivity since quantum mechanical measurements irreducibly disturb the system. The general theory will be illustrated by reference to specific simple examples.
For quantum systems with linear dynamics in phase space much of classical feedback control theory applies. However, there are some questions that are sensible only for the quantum case: Given a fixed interaction between the system and the environment what is the optimal measurement on the environment for a particular control problem? We show that for a broad class of optimal (state-based) control problems (the stationary linear-quadratic-Gaussian class), this question is a semidefinite program. Moreover, the answer also applies to Markovian (current-based) feedback.
We investigate feedback control of linear quantum systems subject to feedback-loop time delays. In particular, we examine the relation between the potentially achievable control performance and the time delays and provide theoretical guidelines for the future experimental setup in two physical systems, which are typical in this research field. The evaluation criterion for the analysis is given by the optimal control performance formula, the derivation of which is from the classical control theoretic results about the input-output delay systems.
Quantum feedback control is the control of the dynamics of a quantum system by feeding back (in real time) the results of monitoring that system. For systems with linear dynamics, the control problem is amenable to exact analysis. In these cases, the quantum system is equivalent to a stochastic system of classical phase-space variables with linear drift and constant diffusion, and with a measured current (e.g. a homodyne photocurrent) linear in the system variables. However, the classical evolution is constrained in order to represent valid quantum evolution. We quantify this in terms of a linear matrix inequality (LMI) relating the drift and diffusion (a sort of zero temperature fluctuation-dissipation theorem), and another LMI for the covariance matrix of the possible conditioned states (i.e. under all possible monitoring schemes consistent with the master equation). For manipulable systems (i.e. where the experimenter has arbitrary control over the parameters in a Hamiltonian linear in the system variables) the covariance of the conditioned state is all that is needed to calculate the effectiveness of the feedback. In this case the double optimization problem reduces to a semidefinite program, which can be solved efficiently in general. We illustrate this with an example drawn from quantum optics.