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IEEE Transactions on Automatic Control

Transfer function approach to quantum control- part I: dynamics of quantum feedback systems

M. Yanagisawa, Hidenori Kimura

Published Dec 1, 2003• 217 citations
Publisher

Abstract

This paper gives a unified approach to feedback control theory of quantum mechanical systems of bosonic modes described by noncommutative operators. A quantum optical closed-loop, including a plant and controller, is developed and its fundamental structural properties are analyzed extensively from a purely quantum mechanical point of view, in order to facilitate the use of control theory in microscopic world described by quantum theory. In particular, an input-output description of quantum mechanical systems which is essential in describing the behavior of the feedback systems is fully formulated and developed. This would then provide a powerful tool for quantum control and pave an avenue that connects control theory to quantum dynamics. This paper is divided into two parts. The first part is devoted to the basic formulation of quantum feedback control via quantum communication and local operations on an optical device, cavity, that can be regarded as a unit of quantum dynamics of bosonic modes. The formulation introduced in this paper presents the feature intrinsic in quantum feedback systems based on quantum stochastic differential equations. The input-output description provides a basis for developing quantum feedback control through the transfer function representation of quantum feedback systems. In the follow-up paper, the quantum mechanical representation of feedback is further elaborated to yield the control theoretical representation of fundamental notions of quantum theory, uncertainty principle, e.g., and some applications are presented.

DOI
10.1109/tac.2003.820063

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This paper gives a unified approach to feedback control theory of quantum mechanical systems of bosonic modes described by noncommutative operators. A quantum optical closed-loop, including a plant and controller, is developed and its fundamental structural properties are analyzed extensively from a purely quantum mechanical point of view, in order to facilitate the use of control theory in microscopic world described by quantum theory. In particular, an input-output description of quantum mechanical systems which is essential in describing the behavior of the feedback systems is fully formulated and developed. This would then provide a powerful tool for quantum control and pave an avenue that connects control theory to quantum dynamics. This paper is divided into two parts. The first part is devoted to the basic formulation of quantum feedback control via quantum communication and local operations on an optical device, cavity, that can be regarded as a unit of quantum dynamics of bosonic modes. The formulation introduced in this paper presents the feature intrinsic in quantum feedback systems based on quantum stochastic differential equations. The input-output description provides a basis for developing quantum feedback control through the transfer function representation of quantum feedback systems. In the follow-up paper, the quantum mechanical representation of feedback is further elaborated to yield the control theoretical representation of fundamental notions of quantum theory, uncertainty principle, e.g., and some applications are presented.
The paper gives an approach to feedback control for quantum systems described by noncummutative operators. We investigate the dynamics of a quantum system interacting with an optical field through which the output of the system is fed back. This process corresponds to quantum feedback via local operations and quantum channels. Based on an input/output description, the dynamics of the feedback system is translated in terms of control theory. This establishes a link between quantum optics and control theory, and the concept and tools of control theory contribute not only to understand quantum control, but enable to solve some of the fundamental problems in quantum mechanics.
Based on the stochastic differential equation of quantum mechanical feedback obtained in the first part of this paper, detailed control concepts and applications are discussed for quantum systems interacting with a noncommutative noise source. A feedback system in our framework is purely nonclassical in the sense that feedback control is performed via local operation and quantum communication through a quantum channel. The role of the controller is to alter the quantum dynamic characteristics of the plant through entanglement, shared between the plant and controller by sending quantum states, that is modulated by the Hamiltonian on the controller. The input-output relation of quantum systems provides a natural extension of control theory to the quantum domain. This enables one to present a control theoretical interpretation of some fundamental quantum mechanical notions such as the uncertainty principle, but also to find applications of ideas and tools of control theory. One of the most important applications is the production of squeezed states, which has been an important issue of quantum theory in relation to quantum computation and quantum communication. The method proposed here reduces the application to a conventional noise reduction problem with feedback. The H/sub /spl infin// control then leads to complete squeezing.
The paper gives an approach to feedback control for quantum systems described by noncummutative operators. We investigate the dynamics of a quantum system interacting with an optical field through which the output of the system is fed back. This process corresponds to quantum feedback via local operations and quantum channels. Based on an input/output description, the dynamics of the feedback system is translated in terms of control theory. This establishes a link between quantum optics and control theory, and the concept and tools of control theory contribute not only to understand quantum control, but enable to solve some of the fundamental problems in quantum mechanics.
In this review paper, we survey the main concepts and some of the recent developments in quantum feedback control. For consistency and clarity, essential ideas and notations in the theory of open quantum systems and quantum stochastic calculus, as well as continuous measurement theory are developed. We give a general description of quantum feedback control, set up a coherent model and compare it to open-loop designs. Objectives which can be achieved by feedback, such as rapid state preparation and purification or entanglement generation are formulated and analyzed, based on the relevant literature. The connection between quantum feedback and quantum chaos is also described and unravelled which, apart from its theoretical curiosity, can shed more light on some of the intrinsic properties of this control paradigm.
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.
A simple method of analyzing quantum feedback circuits is presented. The classical analysis of feedback circuits can be generalized to apply to quantum systems by mapping the field operators of various outputs to other inputs via the standard input-output formalism. Unfortunately, this has led to unphysical results such as the violation of the Heisenberg uncertainty principle for in-loop fields. This paper shows that this general approach can be redeemed by ensuring a self-consistently Hermitian Hamiltonian. The calculations are based on a noncommutative calculus of operator derivatives. A full description of several examples of quantum linear and nonlinear feedback for optical systems is presented.