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Physical Review A

Long-time tails in quantum Brownian motion

Roland Jung, Gert‐Ludwig Ingold, Hermann Grabert

Published Oct 1, 1985• 26 citations
Publisher

Abstract

The authors address the problem of quantum Brownian motion at low temperatures and with arbitrarily strong damping. For a harmonically bound particle the zero-temperature correlation functions are shown to display long-time tails. At finite temperatures an power-law decay at intermediate times is followed by an exponential decay with time constant \ensuremath{\Elzxh}/2\ensuremath{\pi}${k}_{B}$T. The case of free Brownian motion is treated, and some general conclusions for nonlinear systems are drawn.

DOI
10.1103/physreva.32.2510

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The authors address the problem of quantum Brownian motion at low temperatures and with arbitrarily strong damping. For a harmonically bound particle the zero-temperature correlation functions are shown to display long-time tails. At finite temperatures an power-law decay at intermediate times is followed by an exponential decay with time constant \ensuremath{\Elzxh}/2\ensuremath{\pi}kB{k}_{B}kB​T. The case of free Brownian motion is treated, and some general conclusions for nonlinear systems are drawn.
The quantum thermodynamic behavior of small systems is investigated in presence of finite quantum dissipation. We consider the archetype cases of a damped harmonic oscillator and a free quantum Brownian particle. A main finding is that quantum dissipation helps to ensure the validity of the Third Law. For the quantum oscillator, finite damping replaces the zero-coupling result of an exponential suppression of the specific heat at low temperatures by a power-law behavior. Rather intriguing is the behavior of the free quantum Brownian particle. In this case, quantum dissipation is able to restore the Third Law: Instead of being constant down to zero temperature, the specific heat now vanishes proportional to temperature with an amplitude that is inversely proportional to the ohmic dissipation strength. A distinct subtlety of finite quantum dissipation is the result that the various thermodynamic functions of the sub-system do not only depend on the dissipation strength but depend as well on the prescription employed in their definition.
The quantum thermodynamic behavior of small systems is investigated in presence of finite quantum dissipation. We consider the archetype cases of a damped harmonic oscillator and a free quantum Brownian particle. A main finding is that quantum dissipation helps to ensure the validity of the Third Law. For the quantum oscillator, finite damping replaces the zero-coupling result of an exponential suppression of the specific heat at low temperatures by a power-law behavior. Rather intriguing is the behavior of the free quantum Brownian particle. In this case, quantum dissipation is able to restore the Third Law: Instead of being constant down to zero temperature, the specific heat now vanishes proportional to temperature with an amplitude that is inversely proportional to the ohmic dissipation strength. A distinct subtlety of finite quantum dissipation is the result that the various thermodynamic functions of the sub-system do not only depend on the dissipation strength but depend as well on the prescription employed in their definition.
Chandrasekhar's phenomenological theory of brownian motion is transcribed to the quantum regime. The statistics used is that of Boltzmann but the free particle was treated, in addition, with Fermi-Dirac statistics followed by some discussion relating to the measurements of low temperatures. Difficulties arising from the dissipative processes involved are by-passed with the aid of a new methodology enabling one to obtain a propagator in configuration space, which embraces single-particle dissipative forces. This sort of propagator is in turn used for the construction of a type of a nonequilibrium density matrix for the quantum brownian particle. Exact nonequilibrium density matrices for the free and the harmonically bound brownian particles under the influence of external time dependent forces are obtained. Finally, a quantum Fokker-Planck type equation for the general brownian particle is derived. The analysis proceeds via functional integration.
We investigate quantum persistence by analyzing amplitude and phase fluctuations of the wave function governed by the time-dependent free-particle Schrödinger equation. The quantum system is initialized with local random uncorrelated Gaussian amplitude and phase fluctuations. In analogy with classical diffusion, the persistence probability is defined as the probability that the local (amplitude or phase) fluctuations have not changed sign up to time t. Our results show that the persistence probability in quantum diffusion exhibits exponential-like tails. More specifically, in d=1 the persistence probability decays in a stretched exponential fashion, while in d=2 and d=3 as an exponential. We also provide some insights by analyzing the two-point spatial and temporal correlation functions in the limit of small fluctuations. In particular, in the long-time asymptotic limit, the temporal correlation functions for both local amplitude and phase fluctuations become time-homogeneous. Hence, the zero-crossing events correspond to those governed by a stationary Gaussian process, with an autocorrelation-function power-law tail decaying sufficiently fast to imply an exponential-like tail of the persistence probabilities.
The Born-Markov approximation is widely used to study the dynamics of open quantum systems coupled to external baths. Using Keldysh formalism, we show that the dynamics of a system of bosons (fermions) linearly coupled to a noninteracting bosonic (fermionic) bath falls outside this paradigm if the bath spectral function has nonanalyticities as a function of frequency. In this case, we show that the dissipative and noise kernels governing the dynamics have distinct power-law tails. The Green's functions show a short-time ``quasi''-Markovian exponential decay before crossing over to a power-law tail governed by the nonanalyticity of the spectral function. We study a system of bosons (fermions) hopping on a one-dimensional lattice, where each site is coupled linearly to an independent bath of noninteracting bosons (fermions). We obtain exact expressions for the Green's functions of this system, which show power-law decay $\ensuremath{\sim}|t\ensuremath{-}{t}^{\ensuremath{'}}{|}^{\ensuremath{-}3/2}$. We use these to calculate the density and current profile, as well as unequal-time current-current correlators. While the density and current profiles show interesting quantitative deviations from Markovian results, the current-current correlators show qualitatively distinct long-time power-law tails $|t\ensuremath{-}{t}^{\ensuremath{'}}{|}^{\ensuremath{-}3}$ characteristic of non-Markovian dynamics. We show that the power-law decays survive in the presence of interparticle interaction in the system, but the crossover time scale is shifted to larger values with increasing interaction strength.
The analysis of a strongly coupled harmonic quantum Brownian motion has been performed in [1] for a special class of spectral densities obtained as a generalization of the Drude model. In the present scenario, we extend the study of the strongly coupled harmonic quantum Brownian motion to regular spectral densities that are structured as sub-Ohmic at low frequencies and arbitrarily shaped at high frequencies. The bosonic environment is initially in the vacuum state unentangled from the coherent state of the main oscillator. As a generalization of the previous results, we obtain that the long time dynamics is determined uniquely by the initial condition and the low frequency structure of the spectral density. Also in the present framework, inverse power law regressions to the asymptotics appear. The position and the momentum tend to undamped oscillations. The number of excitations relaxes to its initial value with damped oscillations enveloped in inverse power law relaxations. For the momentum and the number of excitations the inverse power law decays become arbitrarily slow in critical configurations by approaching the upper bound of the sub-Ohmic regime. The critical frequencies of the main oscillator are determined by the first negative moment of the spectral densities. Differences with respect to the weak coupling regime arise.