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Physical Review

Influence of Ergodic Behavior on the Scattering of Slow Neutrons by a Harmonic Oscillator

Theodore J. Krieger

Published Mar 1, 1961• 11 citations
Publisher

Abstract

Random quantum jumps of a harmonic oscillator in contact with a temperature bath are shown to give rise to a general broadening of the slow neutron scattering peaks. An exception occurs in the case of the elastic scattering peak, a major component of which remains unbroadened. In general, the broadening increases with the number of quanta transferred. The analysis is based on the "damped" oscillator model of Senitzky.

DOI
10.1103/physrev.121.1388

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APS March Meeting Abstracts•Mar 1, 2003•1 citation
Random quantum jumps of a harmonic oscillator in contact with a temperature bath are shown to give rise to a general broadening of the slow neutron scattering peaks. An exception occurs in the case of the elastic scattering peak, a major component of which remains unbroadened. In general, the broadening increases with the number of quanta transferred. The analysis is based on the "damped" oscillator model of Senitzky.
We investigate the exact dynamics of the damped quantum harmonic oscillator under the (un)correlated initial conditions. The master equation is generalized to the cases of the arbitrary factorized state and/or Gaussian state. We show that the variances of the factorized Gaussian state do not sensitively depend on the initial oscillator-bath correlation, which however can remarkably affect the mean values even at high temperature. We also illustrate that the correlations among the factorized states still give rise to the initial dips during the purity evolutions, which can be smoothed out by increasing the amount of correlation to some extent. We finally study the effects of repeated measurements on the time evolution of the damped oscillator analytically, which are compared with the weak coupling results to indicate that they give rather different transient behaviors even for an intermediate coupling.
The nonzero ground-state energy of the quantum mechanical harmonic oscillator implies quantum fluctuations around the minimum of the potential with the mean-square value proportional to Planck's constant. On the other hand, thermal fluctuations occur when the oscillator is coupled to a heat bath. Using quantum statistical mechanics, this paper describes the transition from pure quantum fluctuations at zero temperature to classical thermal fluctuations in the high-temperature limit. It was early pointed out by Peierls that the classical mean-square thermal fluctuations in a harmonic chain—a linear chain of coupled harmonic oscillators—increase linearly with the distance of the masses from the fixed end of the chain, destroying long-range crystalline order. As shown here, the corresponding quantum fluctuations lead to a much slower logarithmic increase with the distance from the end. It is also shown that this behavior implies, for example, the absence of sharp Bragg peaks in x-ray scattering in an infinite chain at zero temperature, which instead shows power-law behavior typical for one-dimensional quantum liquids (called Luttinger liquids).
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.
The behavior of systems far from equilibrium is often complex and unpredictable, challenging and sometimes overturning the physical intuition derived from equilibrium scenarios. One striking example of this is the Mpemba effect, which implies that non-equilibrium states can sometimes relax more rapidly when they are further from equilibrium. Despite a rich historical background, the precise conditions and mechanisms behind this phenomenon remain unclear. Recently, there has been growing interest in investigating accelerated relaxation and Mpemba-like effects within quantum systems. In this work, we explore a quantum manifestation of the Mpemba effect in a simple and paradigmatic model of open quantum systems: the damped quantum harmonic oscillator, which describes the relaxation of a bosonic mode in contact with a thermal bath at finite temperature T . By means of an exact analytical analysis of the relaxation dynamics based on the method of moments in both population and coherence subspaces, we demonstrate that any initial distribution of populations with the first r moments exactly matching those of the equilibrium distribution shows a super-accelerated relaxation to equilibrium at a rate linearly increasing with r , leading to a pronounced Mpemba effect. In particular, one can find a broad class of far-from-equilibrium distributions that relax to equilibrium faster than any other initial thermal state with a temperature T ′ arbitrarily close to T . The super-accelerated relaxation effect is shown to persist even for a broad class of initial states with non-vanishing coherences, and a general criterion for the observation of super-accelerated thermalization is presented.
We analyze a model system of fermions in a harmonic oscillator potential under the influence of a fluctuating force generated by a bath of harmonic oscillators. This represents an extension of the well-known Caldeira-Leggett model to the case of many fermions. Using the method of bosonization, we calculate Green's functions and discuss relaxation and dephasing of a single extra particle added above the Fermi sea. We also extend our analysis to a more generic coupling between system and bath, that results in complete thermalization of the system.