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arXiv:cond-mat/9807252 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Coherence correlations in the dissipative two-state system

Gunther Lang, Elisabetta Paladino, Ulrich Weiß

Published Oct 1, 1998• 5 citations
arXivHTMLPDF

Abstract

We study the dynamical equilibrium correlation function of the polaron-dressed tunneling operator in the dissipative two-state system. Unlike the position operator, this coherence operator acts in the full system-plus-reservoir space. We calculate the relevant modified influence functional and present the exact formal expression for the coherence correlations in the form of a series in the number of tunneling events. For an Ohmic spectral density with the particular damping strength $K=\frac{1}{2},$ the series is summed in analytic form for all times and for arbitrary values of temperature and bias. Using a diagrammatic approach, we find the long-time dynamics in the regime $K<1.$ In general, the coherence correlations decay algebraically as ${t}^{\ensuremath{-}2K}$ at $T=0.$ This implies that the linear static susceptibility diverges for $K<~\frac{1}{2}$ as $\stackrel{\ensuremath{\rightarrow}}{T}0,$ whereas it stays finite for $K>\frac{1}{2}$ in this limit. The qualitative differences with respect to the asymptotic behavior of the position correlations are explained.

DOI
10.1103/physreve.58.4288

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We study the dynamical equilibrium correlation function of the polaron-dressed tunneling operator in the dissipative two-state system. Unlike the position operator, this coherence operator acts in the full system-plus-reservoir space. We calculate the relevant modified influence functional and present the exact formal expression for the coherence correlations in the form of a series in the number of tunneling events. For an Ohmic spectral density with the particular damping strength K=12,K=\frac{1}{2},K=21​, the series is summed in analytic form for all times and for arbitrary values of temperature and bias. Using a diagrammatic approach, we find the long-time dynamics in the regime K<1.K<1.K<1. In general, the coherence correlations decay algebraically as t\ensuremath−2K{t}^{\ensuremath{-}2K}t\ensuremath−2K at T=0.T=0.T=0. This implies that the linear static susceptibility diverges for K< 12K<~\frac{1}{2}K< 21​ as T\ensuremath→0,\stackrel{\ensuremath{\rightarrow}}{T}0,T\ensuremath→​0, whereas it stays finite for K>12K>\frac{1}{2}K>21​ in this limit. The qualitative differences with respect to the asymptotic behavior of the position correlations are explained.
We study the equilibrium correlation function of the polaron-dressed tunnelling operator in the dissipative two-state system and compare the asymptoptic dynamics with that of the position correlations. For an Ohmic spectral density with damping strength K = ½, the correlation functions are obtained in analytic form for all times at any T and any bias. For K < 1, the asymptotic dynamics is found by using a diagrammatic approach within a Coulomb gas representation. At T = 0, the tunnelling or coherence correlations drop as t −2 K , whereas the position correlations show universal decay ∝ t −2 . The former decay law is a signature of unscreened attractive charge-charge interactions, while the latter is due to unscreened dipole-dipole interactions.
This paper presents the results of a functional-integral approach to the dynamics of a two-state system coupled to a dissipative environment. It is primarily an extended account of results obtained over the last four years by the authors; while they try to provide some background for orientation, it is emphatically not intended as a comprehensive review of the literature on the subject. Its contents include (1) an exact and general prescription for the reduction, under appropriate circumstances, of the problem of a system tunneling between two wells in the presence of a dissipative environment to the "spin-boson" problem; (2) the derivation of an exact formula for the dynamics of the latter problem; (3) the demonstration that there exists a simple approximation to this exact formula which is controlled, in the sense that we can put explicit bounds on the errors incurred in it, and that for almost all regions of the parameter space these errors are either very small in the limit of interest to us (the "slow-tunneling" limit) or can themselves be evaluated with satisfactory accuracy; (4) use of these results to obtain quantitative expressions for the dynamics of the system as a function of the spectral density $J(\ensuremath{\omega})$ of its coupling to the environment. If $J(\ensuremath{\omega})$ behaves as ${\ensuremath{\omega}}^{s}$ for frequencies of the order of the tunneling frequency or smaller, the authors find for the "unbiased" case the following results: For s<1s<1s<1 the system is localized at zero temperature, and at finite TTT relaxes incoherently at a rate proportional to $\mathrm{exp}\ensuremath{-}{(\frac{{T}_{0}}{T})}^{1\ensuremath{-}s}$. For s>2s>2s>2 it undergoes underdamped coherent oscillations for all relevant temperatures, while for 1<s<21<s<21<s<2 there is a crossover from coherent oscillation to overdamped relaxation as TTT increases. Exact expressions for the oscillation and/or relaxation rates are presented in all these cases. For the "ohmic" case, s=1s=1s=1, the qualitative nature of the behavior depends critically on the dimensionless coupling strength $\ensuremath{\alpha}$ as well as the temperature TTT: over most of the ($\ensuremath{\alpha}$,TTT) plane (including the whole region $\ensuremath{\alpha}>1$) the behavior is an incoherent relaxation at a rate proportional to ${T}^{2\ensuremath{\alpha}\ensuremath{-}1}$, but for low TTT and $0<\ensuremath{\alpha}<\frac{1}{2}$ the authors predict a combination of damped coherent oscillation and incoherent background which appears to disagree with the results of all previous approximations. The case of finite bias is also discussed.
We study the influence of system-bath correlations in the initial state on the dynamics of biased two-state systems. The formal solution for the equilibrium correlation function in the form of a power series in the number of transitions between the system states is determined. For Ohmic damping and a special value of the coupling strength (K=1/2), the exact solution for all times and temperatures is presented. We find that at low temperatures the long-time tails of the decay into equilibrium are strongly affected by the initial preparation. Correspondingly, the low-frequency behavior of the neutron-scattering function is qualitatively changed by the presence or absence of correlations in the initial state.
We study the nonlinear response of a periodically driven dissipative two-state system. The exact formal solution in the form of a series in the number of system transitions is derived. The series is summed in analytic form in the parameter region where incoherent transitions prevail. We also present the exact solution for the time evolution of the system in the entire region of temperatures and driving strength for a special value of the Ohmic viscosity. The destruction of quantum coherence by incoherent processes which are induced by the fluctuating and the driving forces is discussed on the basis of these solutions.
The motion of a quantum particle in a tight-binding lattice coupled to an Ohmic thermal reservoir is analyzed in the path-integral formalism. We derive exact formal expressions for moments of the probability distribution both for noncorrelated and correlated thermal initial states of the particle-plus-reservoir complex. Based on these expressions we discuss the role of initial conditions on the subsequent motion and calculate the differences in the transient behavior. For a particular value of the damping strength we find the exact solution of the dynamics in analytic form at any time, temperature, and bias for the two kinds of initial states.
Abstract The quantum dissipative dynamics of a particle which tunnels through the barrier of a double well potential and is coupled to an Ohmic heat bath is studied. Exact formal expressions for correlation functions are presented and examined. Attention is focused on the region in which coherence effects are important, and on the algebraic long‐time tails at zero temperature. Coherence effects in the dissipative quantum transport of a particle in a tight‐binding lattice are also investigated.