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International Journal of Control

State space modelling of the quantum feedback control system in interacting fock space

Pradip Das, B. C. Roy

Published Mar 30, 2006• 4 citations
Publisher

Abstract

In this paper we introduce a formulation of the quantum mechanical feedback system in order to understand the quantum system in interacting Fock space which generalizes the transfer function approach to the quantum mechanical feedback system studied in bosonic mode. A mathematical model of the feedback control of the cavity quantum electrodynamic system in the interacting Fock space with the second cavity in the feedback path is described using a beam splitter.

DOI
10.1080/00207170600646501

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In this paper we introduce a formulation of the quantum mechanical feedback system in order to understand the quantum system in interacting Fock space which generalizes the transfer function approach to the quantum mechanical feedback system studied in bosonic mode. A mathematical model of the feedback control of the cavity quantum electrodynamic system in the interacting Fock space with the second cavity in the feedback path is described using a beam splitter.
The present article is concerned with the study of quantum electrodynamics (QED) in interacting Fock space and a generalised approach of modelling quantum feedback systems of the optical QED cavities. The dynamical behaviour and the state space representation in stochastic field of the optical QED system provide the basis for developing quantum feedback QED control system. This study is also motivated by the design of quantum networks of any number of optical QED baths connected in parallel. Nyquist stability of the feedback control system in the interacting Fock space using two-mode beam splitter device at the input and a single-mode fractional beam splitter device at the output of the cavity has been analysed. The degree of stability of the quantum feedback control system of the cavity QED is studied by measuring gain margin and phase margin of the system in terms of parameters of the beam splitter.
This paper is concerned with the state space modelling of closed optical QED feedback control system in interacting Fock space which is shown to be a generalization of Langevin stochastic model in boson Fock space. This work includes the design of quantum networks of the feedback systems of QED cavity using two-mode beam splitter device at the input to the cavity and a single mode fractional beam splitter device at the output of the cavity in interacting mode and their stability analysis using Nyquist criterion.
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.
The mathematical theory of quantum feedback networks has recently been developed [J. Gough and M. R. James, e-print arXiv:0804.3442v2] for general open quantum dynamical systems interacting with bosonic input fields. In this article we show, for the special case of linear dynamical Markovian systems with instantaneous feedback connections, that the transfer functions can be deduced and agree with the algebraic rules obtained in the nonlinear case. Using these rules, we derive the transfer functions for linear quantum systems in series, in cascade, and in feedback arrangements mediated by beam splitter devices.
In this proceeding we show how to formulate the quantum stochastic Schrodinger equation (QSSE) of the Jaynes-Cummings-model, where the cavity mode is subject to coherent feedback as control mechanism. We start from the Jaynes-Cummings Hamiltonian and the Hamiltonian for feedback on the cavity mode and derive the QSSE and show how to represent the system state as a matrix product state in order to derive a numerically tractable model for the case of the Jaynes-Cummings model subject to feedback. We compare the dynamics for stabilization of the Rabi oscillation with the solution of the Schrodinger equation in the single excitation regime. Furthermore, we compare this with the behavior for the case with two excitations in the system.
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.