Dynamics of a two-level system with Ohmic dissipation driven by a time-dependent cw field in the noninteracting-blip approximation are studied. It is found that at low temperatures when \ensuremath{\alpha}, a friction parameter, lies between 1/2 and 1, the localization predicted by Chakravarty [Phys. Rev. Lett. 49, 681 (1982)] and Bray and Moore [Phys. Rev. Lett. 49, 1546 (1982)] disappears, and the rate constant for tunneling of the particle increases with the intensity and frequency of the field. For 01/2 quantum coherence is observed, and the transition temperature from the coherent to the decay regime depends on the field intensity. When the intensity parameter approaches the zeros of the zeroth Bessel function coherence disappears. This transition from coherent to incoherent motion oscillates with the intensity of the field, giving rise to a cascade of transitions at different temperatures.
The time-dependent average population in a two-level system interacting with an Ohmic boson bath and driven by a dc-ac field is obtained. The study is performed within the noninteracting-blip approximation and for the high-frequency driving case. The integro-differential master equation and the exact formal solution for the average transition probability are obtained. The dissipative quantum decay and the transition temperature from the coherent to the incoherent motion are investigated. The effect of the external electric field on the the dissipative two-level system is found to destroy its coherence.
The quantum dynamics of a two-level system coupled to an Ohmic spin bath is studied by means of the perturbation approach based on a unitary transformation. A scattering function xi(k) is introduced in the transformation to take into account quantum fluctuations. By the master equation within the Born approximation, nonequilibrium dynamics quantities are calculated. The method works well for the coupling constant 0<alpha<alpha(c) and a finite bare tunneling Delta. It is found that (i) only at zero temperature with small coupling or moderate one does the spin-spin-bath model display identical behavior as the well-known spin-boson-bath model; (ii) in comparison with the known results of spin-boson-bath model, the coherence-incoherence transition point, which occurs at alpha(c)=12[1+etaDelta/omega(c)], is temperature independent; and (iii) the nonequilibrium correlation function P(t)=tau(z)(t), evolves without temperature dependence, while tau(x)(t) depends on temperature. Both P(t) and tau(x)(t) not only satisfy their initial conditions, respectively, and also have correct long time limits. Besides, the Shiba relation and sum rule are exactly satisfied in the coherent regime for this method. Our results show that increasing temperature does not help the system suppress decoherence in the coherent regime, i.e., finite temperature does not favor the coherent dynamics in this regime. Thus, the finite-temperature dynamics induced by two kinds of baths (spin-bath and boson-bath) exhibit distinctly different physics.
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.
The dissipative quantum dynamics of a two-level system interacting with a structured reservoir consisting of a damped harmonic mode is investigated by means of the numerically exact path integral Monte Carlo method. This approach provides benchmark results in a broad range of parameter space, in particular, in those domains which are not accessible by approximate methods and alternative numerical schemes, that is, strong coupling between system and harmonic mode and from low to high temperatures. For low temperatures the numerical data are quantitatively in agreement with the noninteracting blip approximation only in the regimes of very weak and very strong coupling. It turns out that the entangled dynamics of the two-level system and the harmonic mode is relatively robust so that its signatures are observable up to relatively strong friction and high temperatures. Nonequilibrium initial preparations of the reservoir with respect to the initial state of the system give rise, for strong interactions, to a stepwise decay of the population, thus displaying coherent dynamics of the bath. The impact of an additional Ohmic bath coupled directly to the two-level system is studied as well, including the case where both reservoirs carry different temperatures.
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 decoherence of a two-state system coupled with a sub-Ohmic bath is investigated theoretically by means of the perturbation approach based on a unitary transformation. It is shown that the decoherence depends strongly and sensitively on the structure of environment. Nonadiabatic effect is treated through the introduction of a function ${\ensuremath{\xi}}_{k}$ which depends on the boson frequency and renormalized tunneling. The results are as follows: (i) the nonequilibrium correlation function
P(t), the dynamical susceptibility ${\ensuremath{\chi}}^{\ensuremath{''}}(\ensuremath{\omega})$, and the equilibrium correlation function
C(t) are analytically obtained for $s\ensuremath{\leqslant}1$; (ii) the phase diagram of thermodynamic transition shows the delocalized-localized transition point ${\ensuremath{\alpha}}_{l}$ which agrees with exact results and numerical data from the numerical renormalization group; (iii) the dynamical transition point ${\ensuremath{\alpha}}_{c}$ between coherent and incoherent phase is explicitly given. A crossover from the coherent oscillation to incoherent relaxation appears with increasing coupling (for $\ensuremath{\alpha}>{\ensuremath{\alpha}}_{c}$, the coherent dynamics disappear); (iv) the Shiba's relation and sum rule are exactly satisfied when $\ensuremath{\alpha}\ensuremath{\leqslant}{\ensuremath{\alpha}}_{c}$; (v) an underdamping-overdamping transition point ${\ensuremath{\alpha}}_{c}^{*}$ exists in the function $S(\ensuremath{\omega})$. Consequently, the dynamical phase diagrams in both ohmic and sub-Ohmic case are mapped out. For $\ensuremath{\Delta}⪡{\ensuremath{\omega}}_{c}$, the critical couplings (${\ensuremath{\alpha}}_{l},{\ensuremath{\alpha}}_{c}$, and ${\ensuremath{\alpha}}_{c}^{*}$) are proportional to ${\ensuremath{\Delta}}^{1\ensuremath{-}s}$.