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IEEE Conference on Decision and Control (CDC)

On synthesis of linear quantum stochastic systems by pure cascading

Hendra I. Nurdin

Published Dec 1, 2010• 2 citations
Publisher

Abstract

Recently, it has been demonstrated 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 point of view, realizations by pure cascading, without a direct interaction Hamiltonian, would be much simpler to implement and this raises the natural question of what class of linear quantum stochastic systems are realizable by cascading alone. This paper gives a precise characterization of this class of linear quantum stochastic systems and then it is proved that, in the weaker sense of transfer function realizability, all passive linear quantum stochastic systems belong to this class. A constructive example is given to show the transfer function realization of a two degrees of freedom passive linear quantum system by pure cascading.

DOI
10.1109/cdc.2010.5717695

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Recently, it has been demonstrated 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 point of view, realizations by pure cascading, without a direct interaction Hamiltonian, would be much simpler to implement and this raises the natural question of what class of linear quantum stochastic systems are realizable by cascading alone. This paper gives a precise characterization of this class of linear quantum stochastic systems and then it is proved that, in the weaker sense of transfer function realizability, all passive linear quantum stochastic systems belong to this class. A constructive example is given to show the transfer function realization of a two degrees of freedom passive linear quantum system by pure cascading.
Recently, it has been demonstrated 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 point of view, realizations by pure cascading, without a direct interaction Hamiltonian, would be much simpler to implement and this raises the natural question of what class of linear quantum stochastic systems are realizable by cascading alone. This technical note gives a precise characterization of this class of linear quantum stochastic systems and then it is proved that, in the weaker sense of transfer function realizability, all passive linear quantum stochastic systems belong to this class. A constructive example is given to show the transfer function realization of a two degrees of freedom passive linear quantum stochastic system by pure cascading.
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.
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 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 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.