Dynamics of the dissipative two-state system

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 $J\left(\omega \right)$ of its coupling to the environment. If $J\left(\omega \right)$ behaves as ${\omega }^s$ for frequencies of the order of the tunneling frequency or smaller, the authors find for the "unbiased" case the following results: For $s<\mathrm{}1\mathrm{}$ the system is localized at zero temperature, and at finite $T$ relaxes incoherently at a rate proportional to $\exp -{\left(\frac{T_0}{T}\right)}^{1-s}$. For $s>\mathrm{}2\mathrm{}$ it undergoes underdamped coherent oscillations for all relevant temperatures, while for $1<s<\mathrm{}2\mathrm{}$ 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, $s=1\mathrm{}$, the qualitative nature of the behavior depends critically on the dimensionless coupling strength $\alpha$ as well as the temperature $T$: over most of the ($\alpha$,$T$) plane (including the whole region $\alpha >1\mathrm{}$) the behavior is an incoherent relaxation at a rate proportional to $T^{2\alpha -1\mathrm{}}$, but for low $T$ and $0<\mathrm{}\alpha <\frac{1}{2}$ 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.