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Physical review. B, Condensed matter

Dynamics of the dissipative multiwell system

Ulrich Weiß, Matthias Wollensak

Published Feb 15, 1988• 39 citations
Publisher

Abstract

The motion of a quantum particle with Ohmic damping in a tight-binding lattice is discussed. An exact series representation in powers of the tunnelmatrix element \ensuremath{\Delta} for moments of the probability distribution is given. The dynamics at zero and at finite temperatures in the presence of an external force is solved exactly to all orders of \ensuremath{\Delta} for a particular value of the friction coefficient, \ensuremath{\eta}=\ensuremath{\pi}\ensuremath{\Elzxh}/${d}^{2}$, where d is the lattice constant, and also for very weak damping.

DOI
10.1103/physrevb.37.2729

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Acta Physica Polonica B•May 1, 2006•46 citations
The motion of a quantum particle with Ohmic damping in a tight-binding lattice is discussed. An exact series representation in powers of the tunnelmatrix element \ensuremath{\Delta} for moments of the probability distribution is given. The dynamics at zero and at finite temperatures in the presence of an external force is solved exactly to all orders of \ensuremath{\Delta} for a particular value of the friction coefficient, \ensuremath{\eta}=\ensuremath{\pi}\ensuremath{\Elzxh}/d2{d}^{2}d2, where d is the lattice constant, and also for very weak damping.
The motion of a quantum particle in a tight-binding lattice coupled to an Ohmic thermal reservoir is analyzed in the path-integral formalism. We derive exact formal expressions for moments of the probability distribution both for noncorrelated and correlated thermal initial states of the particle-plus-reservoir complex. Based on these expressions we discuss the role of initial conditions on the subsequent motion and calculate the differences in the transient behavior. For a particular value of the damping strength we find the exact solution of the dynamics in analytic form at any time, temperature, and bias for the two kinds of initial states.
The coherence-to-decoherence transition is studied in a nanowire modeled as a one-dimensional tight-binding lattice in the presence of an external field and in linear interaction with a boson heat bath, characterized by Ohmic dissipation. The focus of attention is the probability propagator which quantifies the likelihood of a quantum particle, such as an electron to end up at an arbitrary site at a time ttt, given that it was at the origin initially---and from it---the particle-current and the mean-squared displacement. If the bath is absent, the probability operator exhibits quantum coherence as can be captured by say, Bloch oscillation and Wannier-Stark (dynamic) localization. The coupling with the bath---which can be weak or strong---leads to decoherence that will be quantified in the text. The Ohmic model of the spectral function of bath excitations contains a cutoff frequency, which if greater than the temperature (in energy units) defines the ``low-temperature'' regime, whereas the opposite limit implies ``high temperatures.'' Results will be presented in both these regimes separately.
We investigate the exact quantum dynamics of a free particle damped through its interaction with an harmonic bath, when the effective coupling strength behaves as ωδ at low frequency. We find that various regimes can occur depending on the value of δ and of the temperature. At large times, the mean square displacement is shown either to diverge as tv, the exponent v being never greater than 2 or to tend towards a finite value, indicating a confinement in the broad sense. In addition, for δ < 1, an oscillation (absent in the uncoupled system) is found : due to the friction on the dominant low-frequency modes and to the existence of retardation effects, the particle is forced to relax in the mean towards its initial location, the net dynamical effect of the bath being similar to that of an applied potential. In the limit δ → 0, the particle is frozen.
Abstract The quantum dissipative dynamics of a particle which tunnels through the barrier of a double well potential and is coupled to an Ohmic heat bath is studied. Exact formal expressions for correlation functions are presented and examined. Attention is focused on the region in which coherence effects are important, and on the algebraic long‐time tails at zero temperature. Coherence effects in the dissipative quantum transport of a particle in a tight‐binding lattice are also investigated.
Real-time quantum Monte Carlo (QMC) methods have been used to study diffusion on a one-dimensional tight-binding lattice in a dissipative environment with Ohmic friction. For this system, the inherent sign problem of real-time QMC methods can be substantially reduced by employing a partial summation scheme, allowing direct calculations of long-time transport properties. At very low temperatures, the system undergoes a transition from a coherent transport mechanism to an incoherent mechanism as the Kondo parameter K goes through 1. For 0K1, perturbation theory which predicts a ${\mathit{T}}^{2\mathit{K}\mathrm{\ensuremath{-}}1}$ power-law temperature dependence for the diffusion coefficient D is grossly incorrect for low temperatures. Instead, D(T\ensuremath{\rightarrow}0)=0 for 0K1/2, and with increasing temperature D goes through a pronounced maximum. This maximum disappears for K>1/2 and D increases monotonically with increasing temperature. For all K1, perturbation theory becomes exact at sufficiently high temperatures, indicating that the transport mechanism eventually becomes incoherent. For K>1, perturbation theory is exact for all temperatures, and transport is always incoherent.
The quantum thermodynamic behavior of small systems is investigated in presence of finite quantum dissipation. We consider the archetype cases of a damped harmonic oscillator and a free quantum Brownian particle. A main finding is that quantum dissipation helps to ensure the validity of the Third Law. For the quantum oscillator, finite damping replaces the zero-coupling result of an exponential suppression of the specific heat at low temperatures by a power-law behavior. Rather intriguing is the behavior of the free quantum Brownian particle. In this case, quantum dissipation is able to restore the Third Law: Instead of being constant down to zero temperature, the specific heat now vanishes proportional to temperature with an amplitude that is inversely proportional to the ohmic dissipation strength. A distinct subtlety of finite quantum dissipation is the result that the various thermodynamic functions of the sub-system do not only depend on the dissipation strength but depend as well on the prescription employed in their definition.