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Open Systems & Information Dynamics

Master Fields, Drift and Dispersion in the Stochastic Limit of Quantum Theory

Luigi Accardi, F. Gómez-Cubillo

Published Dec 1, 2007• 1 citation
Publisher

Abstract

This work is a detailed study of the convergence of the rescaled creation and annihilation densities, which lead to the master fields, and the form of the drift in the stochastic limit of quantum theory. The approach, based on the distributional theory of Fourier transforms, dispenses with the “analytical condition” and other restrictions usually considered and also establishes the dependence of the stochastic golden rules upon the dispersion function of the quantum field.

DOI
10.1007/s11080-007-9068-9

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This work is a detailed study of the convergence of the rescaled creation and annihilation densities, which lead to the master fields, and the form of the drift in the stochastic limit of quantum theory. The approach, based on the distributional theory of Fourier transforms, dispenses with the “analytical condition” and other restrictions usually considered and also establishes the dependence of the stochastic golden rules upon the dispersion function of the quantum field.
The effects of decoherence for quantum system coupled with a bosonic field are investigated. An application of the stochastic golden rule shows that in the stochastic limit the dynamics of such a system is described by a quantum stochastic differential equation. The corresponding master equation describes convergence of a system to equilibrium. In particular it predicts exponential damping for off-diagonal matrix elements of the system density matrix, moreover these elements for a generic system will decay at least as \exp(-tN{kT\over\hbar}), where N is a number of particles in the system. As an application of the described technique a derivation from first principles (i.e. starting from a Hamiltonian description) of a quantum extension of the Glauber dynamics for systems of spins is given.
A new derivation of the quantum stochastic differential equation for the evolution operator in the low density limit is presented. We use the distribution approach and derive a new algebra for quadratic master fields in the low density limit by using the energy representation. We formulate the stochastic golden rule in the low density limit case for a system coupling with a Bose field via quadratic interaction. In particular, the vacuum expectation value of the evolution operator is computed and its exponential decay is shown.
A rigorous derivation of quantum Langevin equation from microscopic dynamics in the low density limit is given. We consider a quantum model of a microscopic system (test particle) coupled with a reservoir (gas of light Bose particles) via interaction of scattering type. We formulate a mathematical procedure (the so-called stochastic golden rule) which allows us to determine the quantum Langevin equation in the limit of large time and small density of particles of the reservoir. The quantum Langevin equation describes not only dynamics of the system but also the reservoir. We show that the generator of the corresponding master equation has the Lindblad form of most general generators of completely positive semigroups.
We apply new techniques based on the distributional theory of Fourier transforms to study, in the stochastic limit of quantum theory, the convergence of the rescaled creation and annihilation densities, which lead to the master fields, and the form of the drift term of the stochastic Schrodinger equation obtained in such limit. This approach permits us to dispense with the "analytical condition" and other restrictions usually considered and also to establish the dependence of the stochastic golden rules on certain properties of the dispersion function of the quantum field.
For physical systems with Hamiltonians analytic in the position and momentum operators the diagonal matrix elements of the density operator in the coherent state representation may be considered as a phase-space probability measure in the stochastic formulation of quantum mechanics. The correspondence rule relating quantum-mechanical observables to ordinary functions in phase space is obtained. It is shown that the dynamics of the time-dependent phase-space probability densities is again representable in the form of a stochastic process.