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Journal of the Optical Society of America B

Input and output in damped quantum systems II Methods in non-white-noise situations and application to inhibition of atomic phase decays

Crispin W. Gardiner, Matthew J. Collett, A. S. Parkins

Published Oct 1, 1987• 102 citations
Publisher

Abstract

A near-white-noise input–output formalism is developed that does not require white-noise assumptions. In particular, correlation formulas for outputs are developed, which enables account to be taken of reflections of near-white-noise inputs. The methods are applied to the inhibition of atomic phase decays by squeezed light from a degenerate parametric amplifier incident upon a two-level atom, and modifications from previous work carried out in the extreme-white-noise limit are computed.

DOI
10.1364/josab.4.001683

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A near-white-noise input–output formalism is developed that does not require white-noise assumptions. In particular, correlation formulas for outputs are developed, which enables account to be taken of reflections of near-white-noise inputs. The methods are applied to the inhibition of atomic phase decays by squeezed light from a degenerate parametric amplifier incident upon a two-level atom, and modifications from previous work carried out in the extreme-white-noise limit are computed.
The inhibition of atomic-phase decays by squeezed light, and consequent line narrowing, is reexamined for the case of finite-bandwidth squeezing, as produced in the output of a degenerate parametric amplifier. The extension of previous work to incorporate the effect of nonwhite, or colored, noise is accomplished via an adjoint equation, which gives the time evolution of a matrix functional of the incoming field. Simple equations for atomic variables are derived in which noise operators are defined so as to commute with each other and with all other quantities. This facilitates the reduction of a purely quantum-mechanical problem to a classical problem, solvable by classical stochastic-approximation methods. In particular, small-noise-approximation methods and simulations are carried out for a wide range of parameters describing the squeezed-field input. The method of simulation is extended to the computation of correlation functions and spectra for a realistically achievable set of parameters, with consideration given to reflections of the input in the emitted light. The fundamental effect is confirmed, and modifications to previous results are given.
We develop a formulation of quantum damping theory in which the explicit nature of inputs from a heat bath, and of outputs into it, is taken into account. Quantum Langevin equations are developed, in which the Langevin forces are the field operators corresponding to the input modes. Time-reversed equations exist in which the Langevin forces are the output modes, and the sign of damping is reversed. Causality and boundary conditions relating inputs to system variables are developed. The concept of ``quantum white noise'' is formulated, and the formal relationship between quantum Langevin equations and quantum stochastic differential equations (SDE's) is established. In analogy to the classical formulation, there are two kinds of SDE's: the Ito and the Stratonovich forms. Rules are developed for converting from one to the other. These rules depend on the nature of the quantum white noise, which may be squeezed. The SDE's developed are shown to be exactly equivalent to quantum master equations, and rules are developed for computing multitime-ordered correlation functions with use of the appropriate master equation. With use of the causality and boundary conditions, the relationship between correlation functions of the output and those of the system and the input is developed. It is possible to calculate what kind of output statistics result, provided that one knows the input statistics and provided that one can compute the system correlation functions.
The theory of a two-mode nondegenerate parametric amplifier in a cavity is reformulated in terms of quadrature-phase-amplitude variables. The corrrespondence with a genuine classical stochastic linear process is found (non-negative-definite diffusion matrices) for the case of a cavity device immersed in thermal or ordinary (nonsqueezed) vacuum sources. A special kind of squeezing, i.e., quadrature squeezing [B. L. Schumaker, Phys. Rep. 135, 318 (1986)], is found to be characteristic of the internal steady state for the case of a cavity model subjected to an additional phase-sensitive noise source coupled to one of the two internal modes. Finally, the usual squeezing spectrum of the output field is calculated in both cases by means of an input-output formalism based upon a symmetric ordering scheme of noncommuting operators, well adapted for the quadrature-phase description of the fields involved in the interaction.
The interaction between matter and squeezed light has mostly been treated within the approximation that the field-correlation time is small. Methods for treating squeezed light with more general correlations currently involve explicitly modeling the systems producing the light. We develop a general purpose input-output theory for a particular form of narrowband squeezed light, a squeezed wave-packet mode, that only concerns the statistics of the squeezed field and the shape of the wave packet. This formalism allows us to derive the input-output relations and the master equation. We also consider detecting the scattered field using photon counting and homodyne measurements, which necessitates the derivation of the stochastic master equation. The non-Markovian nature of the field manifests itself in the master equation as a coupled hierarchy of equations. We illustrate these with consequences for the decay and resonance fluorescence of two-level atoms in the presence of such fields.
A general input-output theory for quantum dissipative systems is developed in which it is possible to relate output to input via the internal dynamics of a system. This is applied to the problem of computing the squeezing produced by a degenerate parametric amplifier located inside a cavity. The results for the internal modes are identical with those of Milburn and Walls [Opt. Commun. 39, 401 (1981)]. The output field is also found to have only 50% of maximal squeezing. However, by taking the output for a degenerate parametric amplifier inside a single-ended cavity and feeding this into an empty single-ended cavity, one can produce a maximally squeezed state inside this second cavity.
We develop a semiclassical approximation method for calculating correlation functions of the outgoing field of dissipative quantum systems. The method is strictly valid for linear quantum systems, and can be extended consistently to the nonlinear case. It can be used even at bifurcation points where linearization is no longer possible. It is applied to the dc-pumped degenerated Josephson-junction parametric amplifier. We demonstrate that the linearized fluctuation analysis can severely overestimate the maximum obtainable degree of squeezing.