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The Journal of Chemical Physics

On the relaxation of a two-level system: Beyond the weak-coupling approximation

David R. Reichman, R. Silbey

Published Jan 22, 1996• 83 citations
Publisher

Abstract

The model of two nondegenerate quantum levels coupled linearly and off-diagonally to a bath of quantum mechanical harmonic oscillators studied previously by Laird, Budimir, and Skinner is re-examined. Interpretations are made for both the fourth order population relaxation and dephasing processes. Some of the methods used are applied to the standard spin-boson problem. The question of experimental detection of predicted phenomena is discussed. An approximate method, based on a canonical transformation of the original Hamiltonian is proposed to study the problem.

DOI
10.1063/1.470916

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The model of two nondegenerate quantum levels coupled linearly and off-diagonally to a bath of quantum mechanical harmonic oscillators studied previously by Laird, Budimir, and Skinner is re-examined. Interpretations are made for both the fourth order population relaxation and dephasing processes. Some of the methods used are applied to the standard spin-boson problem. The question of experimental detection of predicted phenomena is discussed. An approximate method, based on a canonical transformation of the original Hamiltonian is proposed to study the problem.
The relaxation of a nondegenerate two-level quantum system linearly and off-diagonally coupled to a thermal bath of quantum-mechanical harmonic oscillators is studied. The population and phase relaxation times, T1 and T2, are calculated to fourth order in the system/bath interaction. Focus is on a specific model of the bath spectral density that is both Ohmic (proportional to frequency at low frequency) and Lorentzian, and which has the property that, in the semiclassical or high-temperature limit, it reproduces the stochastic model studied previously by Budimir and Skinner [J. Stat. Phys. 49, 1029 (1987)]. For this fully quantum-mechanical model, it is found that under certain conditions the standard inequality, T2≤2T1, is violated, demonstrating that this unusual result, which was originally derived from the (infinite-temperature) stochastic model, is valid at finite temperature as well.
Two nondegenerate quantum levels coupled off-diagonally and linearly to a bath of quantum-mechanical harmonic oscillators are considered. In the weak-coupling limit one finds that the equations of motion for the reduced density-matrix elements separate naturally into two uncoupled pairs of linear equations for the diagonal and off-diagonal elements, which are known as the Bloch equations. The equations for the populations form the simplest two-component master equation, and the rate constant for the relaxation of nonequilibrium population distributions is 1/T1, defined as the sum of the ‘‘up’’ and ‘‘down’’ rate constants in the master equation. Detailed balance is satisfied for this master equation in that the ratio of these rate constants is equal to the ratio of the equilibrium populations. The relaxation rate constant for the off-diagonal density-matrix elements is known as 1/T2. One finds that this satisfies the well-known relation 1/T2=1/2T1. In this paper the weak-coupling limit is transcended by deriving the Bloch equations to fourth order in the coupling. The equations have the same form as in the weak-coupling limit, but the rate constants are calculated to fourth order. For the population-relaxation rate constants this results in an extension to fourth order of Fermi’s golden rule. We find that these higher-order rate constants do indeed satisfy detailed balance. Comparing the dephasing and population-relaxation rate constants, we find that in fourth order 1/T2≠1/2T1.
We study a new model for treating quantum dissipative systems, in which the bath is modeled as a collection of spins coupled to the system of interest. We develop a quasiclassical method to study this model, approximating the quantum Heisenberg equations by the classical ones, supplemented with stochastic initial conditions carefully chosen so that the results obtained from the classical equations are as close as possible to the quantum results. Using this method we compare the dynamics of such a spin–spin system with that of a spin–boson system, in which the bath is modeled as a collection of harmonic oscillators. We verify numerically that when the system-bath coupling is spread over many bath spins (the Brownian motion limit), the spin–spin model can be mapped on the spin–boson model (although with a temperature dependent spectral density). We also demonstrate that the two dissipative models are qualitatively very different in a non-Brownian motion regime.
We study the dissipative dynamics of a biased two-level system (TLS) coupled to a harmonic oscillator (HO), the latter interacting with an Ohmic environment. Using Van-Vleck perturbation theory and going to second order in the coupling between TLS and HO, we show how the Hamiltonian of the TLS-HO system can be diagonalized analytically. Our model represents an improvement to the usually used Jaynes-Cummings Hamiltonian as an initial rotating wave approximation is avoided. By assuming a weak coupling to the thermal bath, analytical expressions for the time evolution of the populations of the TLS are\nfound: the population is characterized by a multiplicity of damped oscillations together with a complex relaxation dynamics towards thermal equilibrium. The long time evolution is characterized by a single relaxation rate, which is largest at resonance and whose expression can be given in closed analytic form.
We study the dissipative dynamics of a biased two-level system (TLS) coupled to a harmonic oscillator (HO), the latter interacting with an Ohmic environment. Using the Van Vleck perturbation theory and going to second order in the coupling between the TLS and the HO, we show how the Hamiltonian of the TLS–HO system can be diagonalized analytically. Our model represents an improvement on the usually used Jaynes–Cummings Hamiltonian as an initial rotating wave approximation is avoided. By assuming a weak coupling to the thermal bath, analytical expressions for the time evolution of the populations of the TLS are found: the population is characterized by a multiplicity of damped oscillations together with a complex relaxation dynamics towards thermal equilibrium. The long time evolution is characterized by a single relaxation rate, which is largest at resonance and whose expression can be given in a closed analytic form.
We discuss the long-time relaxation of a qubit linearly coupled to a finite bath of [Formula: see text] spins (two-level systems, TLSs), with the interaction Hamiltonian in rotating wave approximation. We focus on the regime [Formula: see text], assuming that the qubit–bath coupling is weak, that the range of spin frequencies is sufficiently broad, and that all the spins are initialized in the ground state. Despite the model being perfectly integrable, we make two interesting observations about the effective system relaxation. First, as one would expect, the qubit relaxes exponentially towards its zero-temperature state at a well characterized rate. Second, the bath spins, even when mutually coupled, do not relax towards a thermal distribution, but rather form a Lorentzian distribution peaked at the frequency of the initially excited qubit. This behaviour is captured by an analytical approximation that makes use of the property [Formula: see text] to treat the TLS frequencies as a continuum and is confirmed by our numerical simulations.