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<1 the system is localized at zero temperature, and at finite
T relaxes incoherently at a rate proportional to $\mathrm{exp}\ensuremath{-}{(\frac{{T}_{0}}{T})}^{1\ensuremath{-}s}$. 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 $\ensuremath{\alpha}$ as well as the temperature
T: over most of the ($\ensuremath{\alpha}$,
T) 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
T 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 propose an approximation scheme to describe the dynamics of the spin-boson model when the spectral density of the environment shows a peak at a characteristic frequency
Ω which can be very close (or even equal) to the spin Zeeman frequency
Δ. Mapping the problem onto a two-state system (TSS) coupled to a harmonic oscillator (HO) with frequency
ω0 we show that the representation of displaced HO states provides an appropriate basis to truncate the Hilbert space of the TSS-HO system and therefore a better picture of the system dynamics. We derive an effective Hamiltonian for the TSS-HO system, and show it furnishes a very good approximation for the system dynamics even when its two subsystems are moderately coupled. Finally, assuming the regime of weak HO-bath coupling and low temperatures, we are able to analytically evaluate the dissipative TSS dynamics.
Abstract. We propose an approximation scheme to describe the dynamics of the spin-boson model when the spectral density of the environment shows a peak at a characteristic frequency Ω which can be very close (or even equal) to the spin Zeeman frequency ∆. Mapping the problem onto a two-state system (TSS) coupled to a harmonic oscillator (HO) with frequency ω0 we show that the representation of displaced HO states provides an appropriate basis to truncate the Hilbert space of the TSS-HO system and therefore a better picture of the system dynamics. We derive an effective Hamiltonian for the TSS-HO system, and show it furnishes a very good approximation for the system dynamics even when its two subsystems are moderately coupled. Finally, assuming the regime of weak HO-bath coupling and low temperatures, we are able to analytically evaluate the dissipative TSS dynamics.
A generalized approximation scheme is proposed to describe the dynamics of the spin-boson problem. Being nonperturbative in the coupling strength nor in the tunneling frequency, it gives reliable results over a wide regime of temperatures and coupling strength to the thermal environment for a large class of bath spectral densities. We use a path-integral approach and start from the exact solution for the two-level system population difference in the form of a generalized master equation (GME). Then, we approximate interblip and blip-sojourn interactions up to linear order, while retaining all intrablip correlations to find the kernels entering the GME in analytical form. Our approximation scheme, which we call weakly interacting blip approximation, fully agrees with conventional perturbative approximations in the tunneling matrix element (noninteracting-blip approximation) or in the system-bath coupling strength in the proper parameter regime.
The dynamical behavior of a two-level system coupled linearly to a dissipative bath of two-level systems whose tunneling splittings and coupling parameters are distributed according to an Ohmic spectrum is studied using numerical path integral methods as well as the noninteracting blip approximation for an equivalent harmonic environment characterized by a scaled spectral density. Comparisons with the behaviors of the conventional spin-boson model of similar coupling strength reveal a weaker quenching of coherence in the present case. Finally, the boundary in parameter space between coherent and incoherent dynamics is mapped.
The dynamics of the spin-boson Hamiltonian is considered in the stochastic approximation. The Hamiltonian describes a two-level system coupled to an environment, and is widely used in physics, chemistry, and the theory of quantum measurement. We demonstrate that the method of stochastic approximation, a general method for consideration of dynamics of an arbitrary system interacting with environment, is powerful enough to reproduce qualitatively the striking results by Leggett et al. [Rev. Mod. Phys. 59, 1 (1987)] found earlier for this model system. The results include an exact expression of the dynamics in terms of the spectral density, and show the appearance of two interesting regimes for the system, i.e., pure oscillating and pure damping ones. Correlators describing the environment are also computed.