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Communications in Theoretical Physics

Instantaneously Dynamical Response and Correlation Functions in Independent-Oscillator Model

Jianzhong Gu, Xizhen Wu, Zhuo Yizhong, En-Guang Zhao

Published Sep 15, 1998• 0 citations
Publisher

Abstract

For the independent-oscillator model (IOM), the explicit expressions of the instantaneously dynamical response functions and correlation functions for both the system and environment are derived. It is shown that for external perturbation the environment of IOM is dynamically stable and the system is oscillatory. Although the external perturbation exists, there is still strongly temporal correlation for the system, for the environment, however, the temporal correlation quickly decays with the second time variable and is almost independent of the first time variable , nevertheless is significantly affected by dissipation. It is shown that the environment behaves as a heat bath from the dynamical point of view.

DOI
10.1088/0253-6102/30/2/193

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For the independent-oscillator model (IOM), the explicit expressions of the instantaneously dynamical response functions and correlation functions for both the system and environment are derived. It is shown that for external perturbation the environment of IOM is dynamically stable and the system is oscillatory. Although the external perturbation exists, there is still strongly temporal correlation for the system, for the environment, however, the temporal correlation quickly decays with the second time variable and is almost independent of the first time variable , nevertheless is significantly affected by dissipation. It is shown that the environment behaves as a heat bath from the dynamical point of view.
The dynamics of a two-level system (TLS) coupled to an external heat bath has been studied. Correlation functions are obtained by combining analytical and numerical renormalization-group methods. It is shown that perturbation theory can be applied to obtain the low-energy, long-time behavior of the system. Analytical expressions are given for 〈${\ensuremath{\sigma}}_{z}$(t)${\ensuremath{\sigma}}_{z}$(0)〉T=0{〉}_{T=0}〉T=0​ valid in the long- and short-time limits (as compared with the renormalized frequency of the TLS). It is shown that other correlation functions, like 〈${\ensuremath{\sigma}}_{x}$(t)${\ensuremath{\sigma}}_{x}$(0)〉, depend sensitively on the nature of the high-energy cutoff of the environment. Thermodynamic functions, like the susceptibility to a bias potential and the correction to the specific heat of the environment, are also calculated. Finally, it is discussed how, in these systems, the experimental setup may influence the quantity being measured.
In many physical situations the behavior of a quantum system is affected by interaction with a larger environment. We develop, using the method of an influence functional, how to deduce the density matrix of the quantum system incorporating the effect of environment. After introducing the characterization of the environment by spectral weight, we first devise schemes to approximate the spectral weight, and then a perturbation method in field theory models, in order to approximately describe the environment. All of these approximate models may be classified as extended Ohmic models of dissipation whose differences are in the high frequency part. The quantum system we deal with in the present work is a general class of harmonic oscillators with an arbitrary time-dependent frequency. The late time behavior of the system is well described by an approximation that employs a localized friction in the dissipative part of the correlation function appearing in the influence functional. The density matrix of the quantum system is then determined in terms of a single classical solution obtained with the time-dependent frequency. With this one can compute the entropy, the energy distribution function, and other physical quantities of the system in a closed form. A specific application is made to the case of a periodically varying frequency. This dynamical system has a remarkable property when the environmental interaction is switched off: The effect of the parametric resonance gives rise to an exponential growth of the populated number in higher excitation levels, or particle production in field theory models. The effect of the environment is investigated for this dynamical system and it is demonstrated that there exists a critical strength of the friction for the parametric effect. In the model of a periodically oscillating field coupled to a system quantum field, it is verified that the parametric effect occurs in medium, with a somewhat diminished rate, if the relaxation time scale of the system field towards thermalization given by the friction term is larger than the time scale of the coherent parametric amplification. The effect persists until the back reaction against the periodic oscillation stops particle production. The resulting energy distribution of produced particles described by a universal function deviates from the thermal one, having an average energy that exponentially increases with time. The dynamical system driven by the parametric oscillator thus maintains and does not lose its quantum nature even in thermal bath. In the present work analytic formulas of how physical quantities behave at late times both in the high and in the low temperature regions are given, along with the results of a numerical computation displaying time evolution.
Abstract This review provides a brief and quick introduction to the quantum Langevin equation for an oscillator, while focusing on the steady-state thermodynamic aspects. A derivation of the quantum Langevin equation is carefully outlined based on the microscopic model of the heat bath as a collection of a large number of independent quantum oscillators, the so-called independent-oscillator model. This is followed by a discussion on the relevant ‘weak-coupling’ limit. In the steady state, we analyze the quantum counterpart of energy equipartition theorem which has generated a considerable amount of interest in recent literature. The free energy, entropy, specific heat, and third law of thermodynamics are discussed for one-dimensional quantum Brownian motion in a harmonic well. Following this, we explore some aspects of dissipative diamagnetism in the context of quantum Brownian oscillators, emphasizing upon the role of confining potentials and also upon the environment-induced classical-quantum crossover. We discuss situations where the system-bath coupling is via the momentum variables by focusing on a gauge-invariant model of momentum-momentum coupling in the presence of a vector potential; for this problem, we derive the quantum Langevin equation and discuss quantum thermodynamic functions. Finally, the topic of fluctuation theorems is discussed (albeit, briefly) in the context of classical and quantum cyclotron motion of a particle coupled to a heat bath.
The effects of system-environment correlations on the dynamics of an open quantum system are investigated for the standard model of a set of quantum harmonic oscillators interacting with a heat bath of oscillators. By definition, a subdynamics is described by transformations of the open system observables. It is shown that such a construction can reproduce the observable properties of the exact dynamics only when the states of system and environment are uncorrelated, while for classical systems there is always a subdynamics. A quantum subdynamics cannot have the properties we associate with thermal fluctuations; the KMS relation at a finite temperature for the open system implies that the system must be closed. The conditions for having a subdynamics as a good approximation to the exact closed dynamics are investigated, and so are the similar but stronger conditions for a Markovian dynamics. It is also shown that a subdynamics defines the response of the open system to some types of time dependent external forces.
The time correlation functions for a Gaussian wave-packet preparation of the dissipative harmonic oscillator evolving from three initial conditions for the heat bath are calculated and compared with each other for Ohmic heat baths. The three initial distributions for the bath are the factorized, partially factorized, and unfactorized distributions. Explicit analytical formulas are derived and then used to study the effect of the three initial distributions on the subsequent dynamics. We find that the transient behavior does not depend sensitively on the initial condition as long as the initial Gaussian wave function of the system is centered at the equilibrium point. Differences become noticeable as the center of the wave packet is significantly shifted from the equilibrium point. These observations justify to some extent the prevalent use of factorized initial conditions for studying real time quantum dynamics in dissipative systems. The total energy in the system is also calculated for the three initial states and its relation to features in the decay is pointed out.
The macroscopic description of a quantum particle with passive dissipation and moving in an arbitrary external potential is formulated in terms of the generalized Langevin equation. The coupling with the heat bath corresponds to two terms: a mean force characterized by a memory function \ensuremath{\mu}(t) and an operator-valued random force. Explicit expressions are given for the correlation and commutator of the random force. The random force is never Markovian. It is shown that \ensuremath{\mu}\ifmmode \tilde{}\else \~{}\fi{}(z), the Fourier transform of the memory function, must be a positive real function, analytic in the upper half-plane and with Re[\ensuremath{\mu}\ifmmode \tilde{}\else \~{}\fi{}(\ensuremath{\omega}+i0+{0}^{+}0+)] a positive distribution on the real axis. This form is then derived for the independent-oscillator model of a heat bath. It is shown that the most general quantum Langevin equation can be realized by this simple model. A critical comparison is made with a number of other models that have appeared in the literature.