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Physical Review A

Influence of damping on quantum interference: An exactly soluble model

A. O. Caldeira, A. J. Leggett

Published Feb 1, 1985• 561 citations
Publisher

Abstract

This paper reports the result of a calculation which shows the effect of damping on the quantum interference of two Gaussian wave packets in a harmonic potential. We use the influence-functional method, which seems to be the most appropriate one for this kind of calculation. It is shown that quantum interference effects are severely diminished by the presence of damping even when its influence on the system is only light. The corrections to the undamped formulas are always expressible in terms of the phenomenological damping constant \ensuremath{\gamma}, the temperature T (in the high-temperature limit), the cutoff frequency (\ensuremath{\Omega}) of the reservoir oscillators, and the mean number (N) of quanta of energy initially present in the system. Both weakly and strongly damped systems are analyzed in the regime of low and high temperatures.

DOI
10.1103/physreva.31.1059

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This paper reports the result of a calculation which shows the effect of damping on the quantum interference of two Gaussian wave packets in a harmonic potential. We use the influence-functional method, which seems to be the most appropriate one for this kind of calculation. It is shown that quantum interference effects are severely diminished by the presence of damping even when its influence on the system is only light. The corrections to the undamped formulas are always expressible in terms of the phenomenological damping constant \ensuremath{\gamma}, the temperature T (in the high-temperature limit), the cutoff frequency (\ensuremath{\Omega}) of the reservoir oscillators, and the mean number (N) of quanta of energy initially present in the system. Both weakly and strongly damped systems are analyzed in the regime of low and high temperatures.
We study the influence of damping on quantum interference of a particle in free space and in a harmonic potential. Using the influence-functional method, it is shown that the suppression of quantum interference is described by the ratio of the thermal diffusion induced by the environment to the quantum-mechanical diffusion. We describe the interference pattern as a function of the friction constant and the temperature, and discuss the difference between the strong damping limit and the classical limit.
We have studied the quantum dissipative problem of a Gaussian wave packet under the influence of a harmonic potential. A phenomenological approach to dissipation is adopted in the light of the well-known model in which the environment is composed of a bath of noninteracting harmonic oscillators. As one of the effects of the coupling to the bath is the evolution of an initially pure wave packet into a statistical mixture, we estimate the characteristic time elapsed for this to occur for different regimes of temperature, damping, and also different initial states.
The quantum theory of a damped harmonic oscillator is developed by considering an exactly solvable model, previously discussed by Unruh and Zurek [Phys. Rev. D 40, 1071 (1989)]. A one-dimensional harmonic oscillator is coupled to a scalar field. The field is assumed to be initially in thermal equilibrium. The Heisenberg equations of motion are solved without approximation, and the first and second moments, 〈q(t)〉, 〈p(t)〉, 〈q2{\mathit{q}}^{2}q2(t)〉, 〈p2{\mathit{p}}^{2}p2(t)〉, and 〈q(t)p(t)+p(t)q(t)〉 of the coordinate and momentum of the oscillator are calculated. The high-temperature, zero-temperature, free-particle, and weak-damping limits are discussed. A Wigner function of Gaussian form is constructed. The density-matrix equation derived by Louisell $[---Quantum Statistical Properties of Radiation (Wiley, New York, 1973)] is examined and shown to have a solution in agreement with our weak-damping limit.
We investigate the exact quantum dynamics of a free particle damped through its interaction with an harmonic bath, when the effective coupling strength behaves as ωδ at low frequency. We find that various regimes can occur depending on the value of δ and of the temperature. At large times, the mean square displacement is shown either to diverge as tv, the exponent v being never greater than 2 or to tend towards a finite value, indicating a confinement in the broad sense. In addition, for δ < 1, an oscillation (absent in the uncoupled system) is found : due to the friction on the dominant low-frequency modes and to the existence of retardation effects, the particle is forced to relax in the mean towards its initial location, the net dynamical effect of the bath being similar to that of an applied potential. In the limit δ → 0, the particle is frozen.
We consider a quantum system linearly coupled to a reservoir of harmonic oscillators. For finite coupling strengths, the stationary distribution of the damped system deviates from the predictions of standard thermodynamics. With the help of the quantum Hamiltonian of mean force, we quantify this deviation exactly for a harmonic oscillator and provide approximations in the limit of high and low temperatures and weak and strong couplings. Moreover, in the semiclassical regime, we use the quantum Smoluchowski equation to obtain results valid for any potential. We finally give a physical interpretation of the deviation in terms of the initial system-reservoir coupling.
Abstract This thesis has two parts, in the first, the Caldeira-Leggett model is introduced; its derivation and general consequences are explored following a paper by Caldeira and Leggett[1]. An operator-formalism shortcut through some of the more mathematically cumbersome parts of the derivation of the model is also developed. The correlation of the force resulting from reservoir-interaction is examined in the high- and low-temperature limits, and the Langevin equation is shown to emerge in the classical limit.Abstract The second part introduces decoherence through a thought experiment that demonstrates the destructive effect of random phase shifts on interference terms, and then follows another paper by Caldeira and Leggett[2] in applying their model further to study the phenomenon of dissipative decoherence. The time-evolution of the interference terms in a superposition of Gaussian wave packets in a harmonic oscillator potential is studied when interacting with a heat bath, and they are shown to vanish at a rate much faster than the relaxation of the system.