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.
In this PhD thesis we have been concerned with dissipative and decoherence processes which take place specifically for quantum systems in a dissipative bistable potential. This potential describes a Hamiltonian system whose spectrum is composed by a ladder of levels, grouped as doublets under the potential barrier and being almost equally spaced above it. The system interacts with a thermal environment, fully characterized by the so-called bath spectral density. The dynamical quantity we looked at is the so-called reduced density matrix (RDM), obtained by tracing out the full density matrix the environmental degrees of freedom. The matrix elements for the RDM are exactly evaluated by means of the Feynman-Vernon real-time path-integral technique. Depending on which regime of parameters we are interested in, i.e. temperature and coupling strength with the reservoir, a few or several energy levels contribute to the dynamics. In particular, a two-level system (TLS) is first analyzed, corresponding to the regime where only the lowest doublet is energetically accessible. We discussed in the beginning the case of a TLS interacting with Ohmic and super-Ohmic baths and then the situation in which a symmetric TLS is assumed to be coupled with an external detector, like a dc-SQUID, the latter being coupled to an Ohmic environment. This model provides a better description of a real qubit, as frequently discussed in various experiments. In the former case, we developed a theory called Weakly-Interacting Blip Approximation (WIBA), which is nonperturbative in the coupling strength as well as in the tunneling frequency and gives reliable results over a wide range of temperatures and coupling strength to the thermal environment. In the latter case, the dynamics was evaluated upon starting from the Non-Interacting Blip Approximation (NIBA) and approximating the NIBA kernels up to first order in the detector-bath coupling strength. Analytical results, valid also at finite detuning, were obtained and we showed that our theory correctly reproduces the onset of multiple oscillation frequencies in the dynamics due to the entanglement between TLS and detector. Finally, we discussed the more general case of an M-level system coupled with an Ohmic environment. This situation describes the case of a bistable potential where the temperature is high enough that higher-lying energy levels are being populated and the system cannot be reduced to a simple TLS. Many are the experimental situations which can be described by our theory. We developed a novel approximation called Weakly-Interacting VR-Blip Approximation (VR-WIBA) for the dynamics in a driven dissipative bistable potential. In the regimes of low to moderate temperatures and coupling strengths, we exploited a separation of time scales between tunneling and vibrational relaxation events and we could derive analytical kernels for the intra- and inter-well dynamics.
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 dynamics of a two-level system coupled to a spin bath is investigated via the numerically exact multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) theory. Consistent with the previous work on linear response approximation [N. Makri, J. Phys. Chem. B 103, 2823 (1999)], it is demonstrated numerically that this spin-spin-bath model can be mapped onto the well-known spin-boson model if the system-bath coupling strength obeys an appropriate scaling behavior. This linear response mapping, however, may require many bath spin degrees of freedom to represent the practical continuum limit. To clarify the discrepancies resulted from different approximate treatments of this model, the population dynamics of the central two-level system has been investigated near the transition boundary between the coherent and incoherent motions via the ML-MCTDH method. It is found that increasing temperature favors quantum coherence in the nonadiabatic limit of this model, which corroborates the prediction in the previous work [J. Shao and P. Hanggi, Phys. Rev. Lett. 81, 5710 (1998)] based on the non-interacting blip approximation (NIBA). However, the coherent-incoherent boundary obtained by the exact ML-MCTDH simulation is slightly different from the approximate NIBA results. Quantum dynamics in other physical regimes are also discussed.
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.
We present a novel approximation scheme to describe the influence of a harmonic bath on the dynamics of a two-level particle over almost the whole regime of temperatures and coupling to the environment, for a wide class of bath spectral densities. Starting from the exact path integral solution for the two-level system density matrix, effective intra-blip correlations are fully included, while inter-blip and blip-sojourn interactions are considered up to first order. In the proper regimes, an excellent agreement with conventional perturbative approaches and ab initio path-integral results is found.