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Australian Control Conference

How many quantum noises need to be added to make an LTI system physically realizable

Shanon L. Vuglar, Ian R. Petersen

Published Jan 1, 2011• 8 citations
OpenAlex

Abstract

In M. R. James, H. I. Nurdin, and I. R. Petersen, “H∞ control of linear quantum stochastic systems,” IEEE Transactions on Automatic Control, vol. 53, no. 8, pp. 1787–1803, 2008, the authors showed that an arbitrary LTI system can be made physically realizable if additional quantum noise components are permitted in the implementation. However, incorporating additional noise is undesirable, and this paper refines their method in order to provide a concrete expression for the number of additional quantum noise components needed to guarantee physical realizability.

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In M. R. James, H. I. Nurdin, and I. R. Petersen, “H∞ control of linear quantum stochastic systems,” IEEE Transactions on Automatic Control, vol. 53, no. 8, pp. 1787–1803, 2008, the authors showed that an arbitrary LTI system can be made physically realizable if additional quantum noise components are permitted in the implementation. However, incorporating additional noise is undesirable, and this paper refines their method in order to provide a concrete expression for the number of additional quantum noise components needed to guarantee physical realizability.
Physical Realizability addresses the question of whether it is possible to implement a given LTI system as a quantum system. It is in general not true that a given synthesized quantum controller described by a set of stochastic differential equations is equivalent to some physically meaningful quantum system. However, if additional quantum noises are permitted in the implementation it is always possible to implement an arbitrary LTI system as a quantum system. In this paper we give an expression for the exact number of noises required to implement a given LTI system as a quantum system. Furthermore, we focus our attention on proving that this is a minimum, that is, it is not possible to implement the system as a quantum system with a smaller number of additional quantum noises.
I present an experimental realization of a coherent-feedback control system that was recently proposed for testing basic principles of linear quantum stochastic control theory [M. R. James, H. I. Nurdin, and I. R. Petersen, e-print arXiv:quant-ph/0703150v2, IEEE Transactions on Automatic Control (to be published)]. For a dynamical plant consisting of an optical ring resonator, I demonstrate $\ensuremath{\sim}7\phantom{\rule{0.3em}{0ex}}\mathrm{dB}$ broadband disturbance rejection of injected laser signals via all-optical feedback with a tailored dynamic compensator. Comparison of the results with a transfer function model pinpoints critical parameters that determine the coherent-feedback control system's performance.
The purpose of this paper is to present a theoretic and numerical study of utilizing squeezing and phase shift in coherent feedback control of linear quantum optical systems. A quadrature representation with built-in phase shifters is proposed for such systems. Fundamental structural characterizations of linear quantum optical systems are derived in terms of the new quadrature representation. These results reveal considerable insights into the issue of the physical realizability of such quantum systems. The problem of coherent quantum linear quadratic Gaussian (LQG) feedback control studied in H. I. Nurdin, M. R. James, and I. R. Petersen, Automatica, IFAC, 45 (2009), pp. 1837--1846; G. Zhang and M. R. James, IEEE Trans. Automat. Control, 56 (2011), pp. 1535--1550 is reinvestigated in depth. First, the optimization methods in these papers are extended to a multistep optimization algorithm which utilizes ideal squeezers. Second, a two-stage optimization approach is proposed on the basis of controller parametrization. Numerical studies show that closed-loop systems designed via the second approach may offer LQG control performance even better than that when the closed-loop systems are in the vacuum state. When ideal squeezers in a closed-loop system are replaced by (more realistic) degenerate parametric amplifiers, a sufficient condition is derived for the asymptotic stability of the resultant new closed-loop system; the issue of performance convergence is also discussed in the LQG control setting.