We study the dissipative quantum dynamics and the asymptotic behavior of a particle in a bistable potential interacting with a sub-Ohmic broadband environment. The reduced dynamics, in the intermediate to strong dissipation regime, is obtained beyond the two-level system approximation by using a real-time path integral approach. We find a crossover dynamic regime with damped intra-well oscillations and incoherent tunneling and a completely incoherent regime at strong damping. Moreover, a nonmonotonic behavior of the left/right well population difference is found as a function of the damping strength.
The dynamics of open quantum systems connected with several reservoirs attract great attention due to their importance in quantum optics, biology, quantum thermodynamics, transport phenomena, etc. In many problems, the Born approximation is applicable, which implies that the influence of the open quantum system on the reservoirs can be neglected. However, in the case of long-time dynamics or mesoscopic reservoirs, the reverse influence can be crucial. In this paper, we investigate the transient dynamics of several bosonic reservoirs connected through an open quantum system. We use an adiabatic approach to study the temporal dynamics of temperatures of the reservoirs during relaxation to thermodynamic equilibrium. We show that there are various types of temperature dynamics that strongly depend on the values of dissipative rates and initial temperatures. We demonstrate that temperatures of the reservoirs, including the hottest and coldest ones, can exhibit nonmonotonic behavior. Moreover, there are moments of time during which the reservoir with an initially intermediate temperature becomes the hottest or coldest reservoir. The obtained results pave the way for managing energy flows in mesoscale and nanoscale systems.
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 standard approach for path-integral Monte Carlo simulations of open quantum systems is extended as an efficient tool to monitor the time evolution of coherences (off-diagonal elements of the reduced density matrix) also for strong coupling to environments. Specific simulations are performed for two-level systems embedded in Ohmic and sub-Ohmic reservoirs in the domains of coherent and incoherent dynamics of the polarization. In the latter regime, the notorious difficulty to access the long-time regime is overcome by combining simulations on moderate time scales with iteratively calculated initial densities. This allows us to extract relaxation rates for sub-Ohmic environments at finite temperatures and over a broad range of couplings and to compare them to analytical predictions. The time evolution of the von Neumann entropy provides insight into the quantum phase transition at thermal equilibrium from a delocalized to a localized state at zero temperature.
Abstract Small (but still containing many, about 102–104, atoms) quantum systems (traditionally termed nano-systems) are dramatically different from their macroscopic or genuine microscopic (atomic) cousins. Microscopic molecular systems (with a few atoms) obey a regular quantum dynamics (described by time dependent Schrödinger equation), whereas in macroscopic systems with continuous energy spectra, one can expect, also regular, although typically relaxation, dynamic behavior. The topic of our paper is in between these limits. Nano-scale systems are characterized by small (but finite) mean interlevel spacing. In such a case with recurrence periods in picosecond range, Loschmidt echo and double resonance phenomena come into the game. System behavior becomes nontrivial and manifests a sort of transitions between regular and chaotic dynamics. We show that such dynamic transitions occur when the Loschmidt echo time of life exceeds the typical recurrence cycle period. We illustrate this behavior in the frame work of a few versions of the exactly solvable quantum problem, proposed long ago by Zwanzig. It is based on the study of time evolution of the initially prepared vibrational state coupled to a reservoir with dense spectrum of its vibrational states. In the simplest version of the Zwanzig model, the reservoir has an equidistant spectrum, and the system—reservoir coupling matrix elements are independent of the reservoir states. We generalize the model to include into consideration the coupling of the initially prepared single state to system phonon excitations. The coupling results to temperature dependent broadening and decay of the echo components. Another generalization is to replace a single level by two states coupled to the Zwanzig reservoir. We anticipate that the basic ideas inspiring our work can be applied to a large variety of interesting for the applications nanosystems (e.g., dissipative free propagation of excitations along molecular chains, or as a model for exchange reactions).