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arXiv:0806.4448 · SIAM Journal on Control and Optimization

Network Synthesis of Linear Dynamical Quantum Stochastic Systems

Hendra I. Nurdin, M. R. James, Andrew C. Doherty

Published Jan 1, 2009• 157 citations
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Abstract

The purpose of this paper is to develop a synthesis theory for linear dynamical quantum stochastic systems that are encountered in linear quantum optics and in phenomenological models of linear quantum circuits. In particular, such a theory will enable the systematic realization of coherent/fully quantum linear stochastic controllers for quantum control, amongst other potential applications. We show how general linear dynamical quantum stochastic systems can be constructed by assembling an appropriate interconnection of one degree of freedom open quantum harmonic oscillators and, in the quantum optics setting, discuss how such a network of oscillators can be approximately synthesized or implemented in a systematic way from some linear and nonlinear quantum optical elements. An example is also provided to illustrate the theory.

DOI
10.1137/080728652

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The purpose of this paper is to develop a synthesis theory for linear dynamical quantum stochastic systems that are encountered in linear quantum optics and in phenomenological models of linear quantum circuits. In particular, such a theory will enable the systematic realization of coherent/fully quantum linear stochastic controllers for quantum control, amongst other potential applications. We show how general linear dynamical quantum stochastic systems can be constructed by assembling an appropriate interconnection of one degree of freedom open quantum harmonic oscillators and, in the quantum optics setting, discuss how such a network of oscillators can be approximately synthesized or implemented in a systematic way from some linear and nonlinear quantum optical elements. An example is also provided to illustrate the theory.
In this paper we introduce a network synthesis theorem for linear dynamical quantum stochastic systems that are encountered in linear quantum optics and in phenomenological models of quantum linear circuits. In particular, such a theorem will be important in the physical realization of coherent/fully quantum linear stochastic controllers for quantum control. We show how general linear dynamical quantum stochastic systems can be systematically constructed by assembling an appropriate interconnection of one degree of freedom open quantum harmonic oscillators and, in the quantum optics setting, provide an explicit illustrative example of the systematic synthesis of a two degrees of freedom open quantum harmonic oscillator.
The aim of this paper is to contribute to a new synthesis algorithm for generalized linear quantum stochastic systems that are usually encountered in linear quantum optics and in phenomenological models of linear quantum circuits. In particular, such a theory will enable the systematic realization of generalized coherent quantum linear stochastic controllers for quantum control. We show how an any generalized linear quantum stochastic system can be synthesized in a systematic way from one degree of freedom generalized open quantum harmonic oscillators.
Recent theoretical and experimental investigations of coherent feedback quantum control, the feedback control of a quantum system with another quantum system, has raised the important problem of how to synthesize a class of quantum systems, called the class of linear quantum stochastic systems, from basic quantum optical components and devices in a systematic way. The synthesis theory sought in this case can be naturally viewed as a quantum analogue of linear electrical network synthesis theory and as such has potential for applications beyond the realization of coherent quantum feedback controllers. In earlier work, Nurdinet al.have established that an arbitrary linear quantum stochastic system can be realized as a cascade connection of simpler one degree of freedom quantum harmonic oscillators, together with a direct interaction Hamiltonian which is bilinear in the canonical operators of the oscillators. However, from an experimental perspective and based on current methods and technologies, direct interaction Hamiltonians are challenging to implement for systems with more than just a few degrees of freedom. In order to facilitate more tractable physical realizations of these systems, this technical note develops a new synthesis algorithm for linear quantum stochastic systems that relies solely on field-mediated interactions, including in implementation of the direct interaction Hamiltonian. Explicit synthesis examples are provided to illustrate the realization of two degrees of freedom linear quantum stochastic systems using the new algorithm.
In the last decade, there has been a growing interest in the integration of photonic components into VLSI circuits that in the long term may drive the need for systematic mixed electronic-photonic circuit design methods. On the other hand, theoretical and experimental investigations of coherent-feedback quantum control, the feedback control of a quantum system with another quantum system, have led to the development of a network synthesis theory for linear quantum stochastic models that are commonly employed in (linear) quantum optics, modelling such devices as optical cavities and optical parametric amplifiers. This paper makes theoretical contributions towards connecting these two independent developments, and anticipates potential future interest in systematic design methods for mixed linear electronic-photonic circuits, by establishing a network synthesis theory for linear stochastic systems with mixed quantum and classical degrees of freedom. It is shown how a physically realizable mixed quantum-classical linear stochastic system can be realized as a circuit composed of a feedback interconnection of a fully quantum linear subsystem, that can be implemented by quantum optical devices, and a classical linear sub-system, that can be implemented with standard electrical and electronic devices, together with appropriate interfaces that convert quantum signals to classical signals, and vice-versa. Two feedback architectures are proposed, and a decomposition lemma is derived that shows the structure of linear transformations of bosonic quantum signals into classical signals.
Recently, Nurdin, James and Doherty have demonstrated that an arbitrary linear quantum stochastic system can be realized as a cascade connection of simple one degree of freedom quantum harmonic oscillators together with a direct interaction Hamiltonian which is bilinear in the canonical operators of the oscillators. However, from an experimental point of view, direct interaction Hamiltonians are challenging to implement for systems with more than just a few degrees of freedom. In order to facilitate simpler experimental realizations of these systems, this paper develops a new synthesis algorithm for linear quantum stochastic systems that relies solely on field-mediated interactions, including in implementation of the direct interaction Hamiltonian. An explicit synthesis example is provided to illustrate the realization of a two degrees of freedom linear quantum stochastic system using the new algorithm.
This thesis makes some theoretical contributions towards mixed quantum feedback network synthesis, quantum optical realization of classical linear stochastic systems and quantum feedback control designs. A mixed quantum-classical feedback network is an interconnected system consisting of a quantum system and a classical system connected by interfaces that convert quantum signals to classical signal (using homodyne detectors), and vice versa (using electro-optic modulators). In the area of mixed quantum-classical feedback networks, we present a network synthesis theory, which provides a natural framework for analysis and design for mixed linear systems. Physical realizability conditions are derived for linear stochastic differential equations to ensure that mixed systems can correspond to physical systems. The mixed network synthesis theory developed based on physical realizability conditions shows that how a classical of mixed quantum-classical systems described by linear stochastic differential equations can be built as a interconnection of linear quantum systems and linear classical systems using quantum optical devices as well as electrical and electric devices. However, an important practical problem for the implementation of mixed quantum-classical systems is the relatively slow speed of classical parts implemented with standard electrical and electronic devices, since a mixed system will not work correctly unless the electronic processing of classical devices is fast enough. Therefore, another interesting work is to show how classical linear stochastic systems build using electrical and electric devices can be physically implemented using quantum optical components. A complete procedure is proposed for a stable quantum linear stochastic system realizing a given stable classical linear stochastic system. The thesis explains how it may be possible to realize certain measurement feedback loops fully at the quantum level. In the area of quantum feedback control design, two numerical procedures based on extended linear matrix inequality (LMI) approach are proposed to design a coherent quantum controller in this thesis. The extended synthesis linear matrix inequalities are, in addition to new analysis tools, less conservative in comparison to the conventional counterparts since the optimization variables related to the system parameters in extended LMIs are independent of the symmetric Lyapunov matrix. These features may be useful in the optimal design of quantum optical networks. Time delays are frequently encountered in linear quantum feedback control systems such as long transmission lines between quantum plants and linear controllers, which may have an effect on the performance of closed-loop plant controller systems. Therefore, this thesis investigates the problem of linear quantum measurement-based feedback control systems subject to feedback-loop time delay described by linear stochastic differential equations. Several numerical procedures are proposed to design classical controllers that make quantum measurement-based feedback control systems with time delay stable and also guarantee that their desired control performance specifications are satisfied.