Dynamics of the dissipative two-state system

A. J. Leggett, S. Chakravarty, A. T. Dorsey, Matthew P. A. Fisher, Anupam Garg, and W. Zwerger
Rev. Mod. Phys. 59, 1 – Published 1 January 1987; Erratum Rev. Mod. Phys. 67, 725 (1995)
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Abstract

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(ω) of its coupling to the environment. If J(ω) behaves as ωs for frequencies of the order of the tunneling frequency or smaller, the authors find for the "unbiased" case the following results: For s<1 the system is localized at zero temperature, and at finite T relaxes incoherently at a rate proportional to exp(T0T)1s. For s>2 it undergoes underdamped coherent oscillations for all relevant temperatures, while for 1<s<2 there is a crossover from coherent oscillation to overdamped relaxation as T increases. Exact expressions for the oscillation and/or relaxation rates are presented in all these cases. For the "ohmic" case, s=1, the qualitative nature of the behavior depends critically on the dimensionless coupling strength α as well as the temperature T: over most of the (α,T) plane (including the whole region α>1) the behavior is an incoherent relaxation at a rate proportional to T2α1, but for low T and 0<α<12 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.

    DOI:https://doi.org/10.1103/RevModPhys.59.1

    ©1987 American Physical Society

    Erratum

    Erratum: Dynamics of the dissipative two-state system

    Anthony J. Leggett, S. Chakravarty, A. T. Dorsey, Matthew P. A. Fisher, Anupam Garg, and W. Zwerger
    Rev. Mod. Phys. 67, 725 (1995)

    Authors & Affiliations

    A. J. Leggett

    • Department of Physics, University of Illinois, 1110 West Green Street, Urbana, Illinois 61801 and Institute of Theoretical Physics, University of California, Santa Barbara, California 93106

    S. Chakravarty

    • State University of New York, Stony Brook, New York and Institute of Theoretical Physics, University of California, Santa Barbara, California 93106

    A. T. Dorsey

    • Department of Physics, University of Illinois, 1110 West Green Street, Urbana, Illinois 61801

    Matthew P. A. Fisher

    • Department of Physics, University of Illinois, 1110 West Green Street, Urbana, Illinois 61801

    Anupam Garg

    • Department of Physics, University of Illinois, 1110 West Green Street, Urbana, Illinois 61801 and Institute of Theoretical Physics, University of California, Santa Barbara, California 93106

    W. Zwerger

    • Department of Physics, University of Illinois, 1110 West Green Street, Urbana, Illinois 61801 and Institute of Theoretical Physics, University of California, Santa Barbara, California 93106

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    Issue

    Vol. 59, Iss. 1 — January - March 1987

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