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Feedback Control ofLinear QuantumSystems

Andrew C. Doherty

Published Jan 1, 2005• 0 citations
OpenAlex

Abstract

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

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Optimal Unravellings for Feedback Control in Linear Quantum Systems

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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.

arXiv:quant-ph/0408099•Physical Review Letters•Feb 25, 2005•151 citations
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.
Introduction In the preceding chapter we introduced quantum trajectories: the evolution of the state of a quantum system conditioned on monitoring its outputs. As discussed in the preface, one of the chief motivations for modelling such evolution is for quantum feedback control. Quantum feedback control can be broadly defined as follows. Consider a detector continuously producing an output, which we will call a current. Feedback is any process whereby a physical mechanism makes the statistics of the present current at a later time depend upon the current at earlier times. Feedback control is feedback that has been engineered for a particular purpose, typically to improve the operation of some device. Quantum feedback control is feedback control that requires some knowledge of quantum mechanics to model. That is, there is some part of the feedback loop that must be treated (at some level of sophistication) as a quantum system. There is no implication that the whole apparatus must be treated quantum mechanically. The structure of this chapter is as follows. The first quantum feedback experiments (or at least the first experiments specifically identified as such) were done in the mid 1980s by two groups [WJ85a, MY86]. They showed that the photon statistics of a beam of light could be altered by feedback. In Section 5.2 we review such phenomena and give a theoretical description using linearized operator equations. Section 5.3 considers the changes that arise when one allows the measurement to involve nonlinear optical processes.
Quantum engineering has seen rapid growth in the past two decades. Physicists, mathematicians and engineers have been working in unison to control a number of diverse systems in the quantum regime. Quantum control involving feedback has become particularly topical, as using information gained from a system can lead to more stable operation of a control protocol. Quantum feedback can be split into two paradigms: measurement based feedback and coherent feedback. Although it is increasingly evident that retaining the coherence of the feedback signal provides an intrinsic advantage over measurement-based feedback, coherent feedback is still a new paradigm. In particular, there are a limited number of options for coherently estimating a state within a feedback loop. As a step towards better understanding and implementation of quantum feedback, this thesis reports on modelling, estimation and control of quantum systems. The first topic is physical realisability of a variety of quantum systems, especially finite level systems and their outgrowths. Bilinear quantum stochastic differential equations (QSDEs) have to be employed to characterise these systems, which is a significant complement to the previous work on linear QSDEs. In light of the fact that direct and indirect couplings play a vital role in quantum network and control, we also provide state-space models for different classes of coupled open quantum systems incorporating both bidirectional and directional interactions. The second topic is coherent observers and optimal filtering. It is well established that classically estimation using the Kalman filter can provide improved performance over direct feedback schemes, and similar demonstrations have been performed for measurement-based quantum feedback. In this thesis I present coherent observers, including least mean squares estimators, which are driven by the coherent output of a specified quantum plant and designed such that some subset of the observer and plant's expectation values converge in the asymptotic limit. Not only can estimators "observe" mean quantities of quantum plant, but also they can "estimate" quantum correlations such as entanglement. This is of much importance to coherent feedback design in the absence of measurement steps without loss of fidelity. Last but not least, several topics regarding stabilisation and control design of quantum systems are discussed in my thesis. To be specific, tools for quantum stability analysis (e.g. Lyapunov conditions) are provided. Both measurement-based feedback and coherent feedback control design are taken into account. A measurement-based optimal controller for opto-mechanical systems aimed at synchronising different mechanical modes is presented, which is application-oriented. Moreover, A general observer-based coherent feedback control framework is studied, and I demonstrate a pole-placement approach via coherent observers that can be applied to various scenarios.
The purpose of this tutorial is to give a brief introduction to linear quantum control systems. The mathematical model of linear quantum control systems is presented first, then some fundamental control-theoretic notions such as stability, controllability and observability are given, which are closely related to several important concepts in quantum information science such as decoherence-free subsystems, quantum non-demolition variables, and back-action evasion measurements. After that, quantum Gaussian states are introduced, in particular, an information-theoretic uncertainty relation is presented which often gives a better bound for mixed Gaussian states than the well-known Heisenberg uncertainty relation. The quantum Kalman filter is presented for quantum linear systems, which is the quantum analogy of the Kalman filter for classical (namely, non-quantum-mechanical) linear systems. The quantum Kalman canonical decomposition for quantum linear systems is recorded, and its application is illustrated by means of a recent experiment. As single- and multi-photon states are useful resources in quantum information technology, the response of quantum linear systems to these types of input is presented. Finally, coherent feedback control of quantum linear systems is briefly introduced, and a recent experiment is used to demonstrate the effectiveness of quantum linear systems and networks theory.
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.