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European Control Conference (ECC)

Cascade cavity realization for a class of complex transfer functions arising in coherent quantum feedback control

Ian R. Petersen

Published Aug 1, 2009• 14 citations
Publisher

Abstract

This paper presents a realization algorithm for a class of complex transfer functions corresponding to physically realizable complex linear quantum systems. The class of complex linear quantum systems under consideration includes interconnections of passive optical components such as cavities, beam-splitters, phase-shifters and interferometers. It is shown that for almost all quantum optical systems within this class, the corresponding transfer function can be realized as a cascade connection involving only cavities and phase-shifters.

DOI
10.23919/ecc.2009.7074402

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This paper presents a realization algorithm for a class of complex transfer functions corresponding to physically realizable complex linear quantum systems. The class of complex linear quantum systems under consideration includes interconnections of passive optical components such as cavities, beam-splitters, phase-shifters and interferometers. It is shown that for almost all quantum optical systems within this class, the corresponding transfer function can be realized as a cascade connection involving only cavities and phase-shifters.
This paper presents a method for approximating a class of complex transfer function matrices corresponding to physically realizable complex linear quantum systems. The class of complex linear quantum systems under consideration includes interconnections of passive optical components such as cavities, beam-splitters, phase-shifters and interferometers. This approximation method builds on a previous result for the cascade realization of these transfer function matrices and gives good approximations at low frequencies.
This paper considers the use of singular perturbation approximations for a class of linear complex quantum systems arising in the area of quantum optics. The class of linear complex quantum systems under consideration arises in systems consisting of interconnections of passive optical components such as cavities, beams splitters and phase shifters. The main result of the paper shows that for a class of singular perturbations of such systems, the approximate reduced dimension quantum system and guaranteed to be physically realizable.
The realization of transfer functions of Linear Quantum Stochastic Systems (LQSSs) is an issue of fundamental importance for the practical applications of such systems, especially as coherent controllers for other quantum systems. In this paper, we review two realization methods proposed by the authors in [1], [2], [3], [4]. The first one uses a cascade of a static linear quantum-optical network and single-mode optical cavities, while the second uses a feedback network of such cavities, along with static linear quantum-optical networks that pre- and post-process the cavity network inputs and outputs.
This paper surveys some recent results on the theory of quantum linear systems and presents them within a unified framework.Quantum linear systems are a class of systems whose dynamics, which are described by the laws of quantum mechanics, take the specific form of a set of linear quantum stochastic differential equations (QSDEs).Such systems commonly arise in the area of quantum optics and related disciplines.Systems whose dynamics can be described or approximated by linear QSDEs include interconnections of optical cavities, beam-splitters, phase-shifters, optical parametric amplifiers, optical squeezers, and cavity quantum electrodynamic systems.With advances in quantum technology, the feedback control of such quantum systems is generating new challenges in the field of control theory.Potential applications of such quantum feedback control systems include quantum computing, quantum error correction, quantum communications, gravity wave detection, metrology, atom lasers, and superconducting quantum circuits.A recently emerging approach to the feedback control of quantum linear systems involves the use of a controller which itself is a quantum linear system.This approach to quantum feedback control, referred to as coherent quantum feedback control, has the advantage that it does not destroy quantum information, is fast, and has the potential for efficient implementation.However, the design of coherent quantum feedback controllers remains a major challenge.This paper discusses recent results concerning the synthesis of H-infinity optimal controllers for linear quantum systems in the coherent control case.An important issue which arises both in the modelling of linear quantum systems and in the synthesis of linear coherent quantum controllers is the issue of physical realizability.This issue relates to the property of whether a given set of QSDEs corresponds to a physical quantum system satisfying the laws of quantum mechanics.The paper will cover recent results relating the question of physical realizability to notions occurring in linear systems theory such as lossless bounded real systems and dual J-J unitary systems.
We consider the transfer functions describing the input-output relation for a class of linear open quantum systems involving feedback with nonzero time delays. We show how such transfer functions can be factorized into a product of terms which are transfer functions of canonical physically realizable components. We prove under certain conditions that this product converges, and can be approximated on compact sets. Thus our factorization can be interpreted as a (possibly infinite) cascade. Our result extends past work where linear open quantum systems with a state-space realization have been shown to have a pure cascade realization [Nurdin, H. I., Grivopoulos, S., & Petersen, I. R. (2016). The transfer function of generic linear quantum stochastic systems has a pure cascade realization. Automatica, 69, 324-333.]. The functions we consider are inherently non-Markovian, which is why in our case the resulting product may have infinitely many terms.