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arXiv:quant-ph/0309103

Quantum White Noises and The Master Equation for Gaussian Reference States

John Edward Gough

Published Sep 12, 2003• 10 citations
arXivHTMLPDF

Abstract

We show that a basic quantum white noise process formally reproduces quantum stochastic calculus when the appropriate normal / chronological orderings are prescribed. By normal ordering techniques for integral equations and a generalization of the Araki-Woods representation, we derive the master and random Heisenberg equations for an arbitrary Gaussian state: this includes thermal and squeezed states.

DOI
10.48550/arxiv.quant-ph/0309103

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We show that a basic quantum white noise process formally reproduces quantum stochastic calculus when the appropriate normal / chronological orderings are prescribed. By normal ordering techniques for integral equations and a generalization of the Araki-Woods representation, we derive the master and random Heisenberg equations for an arbitrary Gaussian state: this includes thermal and squeezed states.
The master equation of the quantum noise theory is derived by means of Feynman's path integral method. We propose an equivalent s ochastic process where the wavefunction satisfies a nonlinear SchrOdinger equation except for random moments at which it shows orthogonal jumps. Since the early works [ 1-3] on the theory of quantum noise, great interest has been shown in quantum-me-chanical systems affected by random external forces and also new aspects have appeared [4-6]. Usually, the quan-tum state of such systems i described by a density operator obeying a deterministic differential equation (master equation). If we attribute a state vector (instead of the density operator) to the given system, then the state vector should be considered as a stochastic variable governed by a certain stochastic process. In the first part of our work we derive the master equation for the density operator by a seemingly new applica-tion of the path integral technique. Then, we construct a stochastic process for the wavefunction of the system, while retaining its equivalence with the density operator formalism. In the final part we shortly discuss favoured features of the above stochastic process proposed for the wavefunction. This work is an anticipation of a more general and lengthy explanation of the stochastic properties of the wave-function in the presence of random forces. Now, we are going to introduce the notion of stochastic potential of white-noise type. Here we shall investigate the effect of gaussian white-noise potentials. For brevity, we restrict ourselves to the case of a single point.like par-ticle moving in one dimension. Let us assume that the potential V(x, t) acting on the particle is a stochastic variable (x, t stand for the coordi-nate and time). We define the probability distribution of V by the following generator functional G [h]:
White noise theory allows to formulate quantum white noises explicitly as elemental quantum stochastic processes. A traditional quantum stochastic differential equation of Itô type is brought into a normal-ordered white noise differential equation driven by lower powers of quantum white noises. The class of normal-ordered white noise differential equations covers quantum stochastic differential equations with highly singular noises such as higher powers or higher order derivatives of quantum white noises, which are far beyond the traditional Itô theory. For a general normal-ordered white noise differential equation unique existence of a solution is proved in the sense of white noise distribution. Its regularity properties are investigated by means of weighted Fock spaces interpolating spaces of white noise distributions and associated characterization theorems for S-transform and for operator symbols.
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.
# Part 1: # White Noise Functional Approach to Polymer Entanglements (C C Bernido & M V Carpio-Bernido) # White Noise Analysis, Quantum Field Theory, and Topology (A Hahn) # A Topic on Noncanonical Representations of Gaussian Processes (Y Hibino) # Integral Representation of Hilbert-Schimdt Operators on Boson Fock Space (U C Ji) # The Dawn of White Noise Analysis (I Kubo) # White Noise Stochastic Integration (H-H Kuo) # Connes-Hida Calculus and Bismut-Quillen Superconnections (R Leandre & H Ouerdiane) # A Quantum Decomposition of Levy Processes (Y-J Lee & H-H Shih) # Generalized Entanglement and Its Classification (T Matsuoka) # A White Noise Approach to Fractional Brownian Motion (D Nualart) # Adaptive Dynamics in Quantum Information and Chaos (M Ohya) # Micro-Macro Duality in Quantum Physics (I Ojima) # White Noise Measures Associated to the Solutions of Stochastic Differential Equations (H Ouerdiane) # A Remark on Sets of Infinite Dimensional Spaces with Full or Zero Capacity (J Ren & M Rockner) # An Infinite Dimensional Laplacian in White Noise Theory (K Saito) # Invariance of Poisson Noise (Si Si et al.) # Nonequilibrium Steady States with Bose-Einstein Condensates (S Tasaki & T Matsui) # Multidimensional Skew Reflected Diffusions (G Trutnau) # On Quantum Mutual Type Entropies and Quantum Capacity (N Watanabe) # Part 2: # White Noise Calculus and Stochastic Calculus (L Accardi & A Boukas)