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arXiv:cond-mat/9503133 · Physical review. B, Condensed matter

Dynamics of a strongly damped two-level system

Tabish Qureshi

Published Feb 1, 1996• 5 citations
arXivHTMLPDF

Abstract

Dynamics of a dissipative two-level system is studied using quantum relaxation theory. This calculation goes beyond the commonly used dilute bounce gas approximation (DBGA), even for strong damping. The results obtained here deviate from the DBGA results at low temperatures, however, the DBGA form is recovered at high temperatures. The results in the parameter regime 1/21, where the model has connection with the Kondo Hamiltonian, are of particular significance. In this regime, the spin shows a crossover to a slower exponential relaxation at intermediate times, which is roughly half the relaxation rate at short times, as also observed in quantum Monte Carlo simulation of the model. The asymptotic behavior of the spin in the Kondo regime is in agreement with the exact conformal field theory results for the Kondo model. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.53.3183

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Dynamics of a dissipative two-level system is studied using quantum relaxation theory. This calculation goes beyond the commonly used dilute bounce gas approximation (DBGA), even for strong damping. The results obtained here deviate from the DBGA results at low temperatures, however, the DBGA form is recovered at high temperatures. The results in the parameter regime 1/21, where the model has connection with the Kondo Hamiltonian, are of particular significance. In this regime, the spin shows a crossover to a slower exponential relaxation at intermediate times, which is roughly half the relaxation rate at short times, as also observed in quantum Monte Carlo simulation of the model. The asymptotic behavior of the spin in the Kondo regime is in agreement with the exact conformal field theory results for the Kondo model. \textcopyright{} 1996 The American Physical Society.
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.
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.
Dissipative processes in nonequilibrium many-body systems are fundamentally different than their equilibrium counterparts. Such processes are of great importance for the understanding of relaxation in single-molecule devices. As a detailed case study, we investigate here a generic spin-fermion model, where a two-level system couples to two metallic leads with different chemical potentials. We present results for the spin relaxation rate in the nonadiabatic limit for an arbitrary coupling to the leads using both analytical and exact numerical methods. The nonequilibrium dynamics is reflected by an exponential relaxation at long times and via complex phase shifts, leading in some cases to an ``antiorthogonality'' effect. In the limit of strong system-lead coupling at zero temperature we demonstrate the onset of a Marcus-like Gaussian decay with voltage difference activation. This is analogous to the equilibrium spin-boson model, where at strong coupling and high temperatures, the spin excitation rate manifests temperature activated Gaussian behavior. We find that there is no simple linear relationship between the role of the temperature in the bosonic system and a voltage drop in a nonequilibrium electronic case. The two models also differ by the orthogonality-catastrophe factor existing in a fermionic system, which modifies the resulting line shapes. Implications for current characteristics are discussed. We demonstrate the violation of pairwise Coulomb gas behavior for strong coupling to the leads. The results presented in this paper form the basis of an exact, nonperturbative description of steady-state quantum dissipative systems.
We investigate the dynamical correlation function of a quantum-mechanical two-state system that is coupled to a bosonic heat bath, utilizing the equivalence between the spin-boson Hamiltonian and the 1/r2{1/r}^{2}1/r2 Ising model. The imaginary-time correlation function is obtained from Monte Carlo simulations on the Ising system and then continued to real time by a Pad\'e approximation. In the unbiased system, the transition from oscillatory to strongly damped behavior is found to occur at a coupling strength close to $\ensuremath{\alpha}=1/2$. The biased system favors coherent relaxation and displays a significantly larger crossover value ${\ensuremath{\alpha}}_{c}.$ We introduce the quasiparticle picture to describe the relevant behavior at intermediate time scales. Within this approximation, we map out phase diagrams for the unbiased and biased systems.
We summarize in this work the relations of the spin-boson Hamiltonian, which was recently studied in connection with the phenomenon of quantum coherence in the presence of dissipation, to three different fermionic Hamiltonians. These relations were obtained through well-known equivalences between Fermi and Bose operators in one dimension. The fermionic Hamiltonians correspond to (a) a two-level system coupled linearly with a fermionic bath, (b) the resonant-level model, and (c) the anisotropic Kondo model. For the first Hamiltonian we reobtain and discuss the relationship between the dimensionless dissipation coefficient \ensuremath{\alpha} and the phase shift of the fermions. The resonant-level model allows us to study the properties of the two-level system for values of \ensuremath{\alpha} around (1/2). At \ensuremath{\alpha}=(1/2) the model reduces to the Toulouse limit, where the Hamiltonian is exactly soluble. A pure exponential decay is obtained for the relaxation of P(t)=〈${\ensuremath{\sigma}}_{z}$(t)〉 for t>0, given that for t<0 the system is known to be localized in one of the two states. The comparison with the Kondo model gives the long-time-limit behavior of the system for (1/2)1 and provides a connection between universal numbers obtained previously for the spin-boson Hamiltonian and the Kondo model.