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Journal of Modern Optics

Quantum Optics of Linear Systems

Stig Stenholm

Published Dec 1, 1994• 10 citations
Publisher

Abstract

The classical Jones calculus for optical networks is utilized to formulate a general theory of transfer of quantum fields through a linear quantum optics device. For input fields having Gaussian quasi-distribution functions, the output distributions can be calculated explicitly. The formalism is applied to the simple beam splitter as an illustration.

DOI
10.1080/09500349414552321

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The classical Jones calculus for optical networks is utilized to formulate a general theory of transfer of quantum fields through a linear quantum optics device. For input fields having Gaussian quasi-distribution functions, the output distributions can be calculated explicitly. The formalism is applied to the simple beam splitter as an illustration.
A simple method of analyzing quantum feedback circuits is presented. The classical analysis of feedback circuits can be generalized to apply to quantum systems by mapping the field operators of various outputs to other inputs via the standard input-output formalism. Unfortunately, this has led to unphysical results such as the violation of the Heisenberg uncertainty principle for in-loop fields. This paper shows that this general approach can be redeemed by ensuring a self-consistently Hermitian Hamiltonian. The calculations are based on a noncommutative calculus of operator derivatives. A full description of several examples of quantum linear and nonlinear feedback for optical systems is 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.
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A quantum optical formulation of the action of passive, lossless optical instruments on quantum light fields is developed. The quantum Maxwell equations are formally solved and the commutation relations for various combinations of field operators at different times are studied. General relationships between field correlation functions and correlation functions of source-quantity operators and free-field operators are derived. Formulas are presented for the case when the free field is the vacuum field. Furthermore, the mixing of source-field light with coherent free-field light is treated. The theory is applied to the calculation of the photocount distribution. The theory presented renders it possible to take into account the effects of light propagation through the optical system. It is shown that the effect of the optical instruments may be taken into account by introducing an apparatus function in a way which formally corresponds to that of classical optics. However, the calculation of the corresponding convolution integrals is governed by operator ordering rules, which are essential in the case of quantum light fields and which can give rise to substantial differences between classical and quantum optics.
The purpose of this paper is to extend linear systems and signals theory to include single photon quantum signals. We provide detailed results describing how quantum linear systems respond to multichannel single photon quantum signals. In particular, we characterize the class of states (which we call photon-Gaussian states) that result when multichannel photons are input to a quantum linear system. We show that this class of quantum states is preserved by quantum linear systems. Multichannel photon-Gaussian states are defined via the action of certain creation and annihilation operators on Gaussian states. Our results show how the output states are determined from the input states through a pair of transfer function relations. We also provide equations from which output signal intensities can be computed. Examples from quantum optics are provided to illustrate the results.
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