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IEEE Conference on Decision and Control (CDC)

Transfer function approach to quantum control systems

M. Yanagisawa, H. Kimura

Published Nov 13, 2002• 0 citations
Publisher

Abstract

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.

DOI
10.1109/.2001.981127

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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.
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.
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.
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.
Abstract This chapter discusses a novel aspect of quantum control, namely quantum feedback. It explains the crucial distinction between measurement-based feedback and autonomous feedback. In the former, a detector performs a measurement of the variable to be controlled, and information passes from the quantum system to a classical system, which processes it to produce a correction signal sent to an actuator acting back on the quantum system. In the latter, the detector-processor-actuator part of the controller is another quantum system coupled to the controlled system, and no information need pass through classical channels. The chapter treats the useful formalism of the quantum stochastic master equation, a basic element of knowledge essential to the detailed understanding of measurement-based feedback, and the design of the filters for the corresponding controller. The autonomous aspect of quantum feedback is illustrated by a discussion of analog, continuous error correction of a qubit by quantum optics elements.
The concept of a controlled flow of a dynamical system, especially when the controlling process feeds information back about the system, is of central importance in control engineering. In this paper, we build on the ideas presented by Bouten and van Handel [Quantum Stochastics and Information: Statistics, Filtering and Control (World Scientific, 2008)] and develop a general theory of quantum feedback. We elucidate the relationship between the controlling processes, Z, and the measured processes, Y, and to this end we make a distinction between what we call the input picture and the output picture. We should note that the input-output relations for the noise fields have additional terms not present in the standard theory but that the relationship between the control processes and measured processes themselves is internally consistent—we do this for the two main cases of quadrature measurement and photon-counting measurement. The theory is general enough to include a modulating filter which post-processes the measurement readout Y before returning to the system. This opens up the prospect of applying very general engineering feedback control techniques to open quantum systems in a systematic manner, and we consider a number of specific modulating filter problems. Finally, we give a brief argument as to why most of the rules for making instantaneous feedback connections [J. Gough and M. R. James, Commun. Math. Phys. 287, 1109 (2009)] ought to apply for controlled dynamical networks as well.
Progress in quantum physics has made it possible to perform experiments in which individual quantum systems are monitored and manipulated in real time. The advent of such new technical capabilities provides strong motivation for the development of theoretical and experimental methodologies for quantum feedback control. The availability of such methods would enable radically new approaches to experimental physics in the quantum realm. Likewise, the investigation of quantum feedback control will introduce crucial new considerations to control theory, such as the uniquely quantum phenomena of entanglement and measurement back-action. The extension of established analysis techniques from control theory into the quantum domain may also provide new insight into the dynamics of complex quantum systems. We anticipate that the successful formulation of an input-output approach to the analysis and reduction of large quantum systems could have very general applications in nonequilibrium quantum statistical mechanics and in the nascent field of quantum information theory.