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The Journal of Chemical Physics

T 2 can be greater than 2T1 even at finite temperature

Brian B. Laird, James L. Skinner

Published Mar 15, 1991• 58 citations
Publisher

Abstract

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.

DOI
10.1063/1.460627

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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.
We consider a quantum–mechanical two-level system under the influence of both diagonal and off-diagonal stochastic perturbations, and focus on the decay times T1 and T2, which refer to the relaxation to equilibrium of the populations and relative phase of the two levels, respectively. From both theoretical and experimental viewpoints one traditionally expects that T2≤2T1. On the other hand, from a fourth-order cumulant expansion calculation of the asymptotic time dependence of the density matrix elements, Budimir and Skinner [J. Stat. Phys. 49, 1029 (1987)] showed that, in fact, in some instances T2>2T1. In this paper we solve the stochastic model numerically, which leads to the exact time dependence of the density matrix at all times. We find that the analytic prediction that T2>2T1 is not only correct, but also meaningful, in the sense that the density matrix elements decay exponentially after only a short transient time.
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 influence of a finite bath correlation time and the bath temperature on the onset of a non-Markovian behaviour in the relaxation of a harmonic oscillator linearly coupled to a coloured thermal bath is analysed. A fully quantum mechanical treatment of the time autocorrelation function of the oscillator-bath interaction is made. Special attention is paid to the temperature dependence. In the low-temperature regime a non-exponential time decay of the mean number of quanta is found even though the correlation time associated with the oscillator-bath interaction is zero.
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.
Exact calculations for a quantum oscillator linearly coupled to a thermal reservoir are presented. Explicit results (variances and correlations) for the response and the spontaneous fluctuations are evaluated for two different models of the heat bath; a reservoir with a spectrum which is similar to that of acoustic phonons and another which is analogous to optic phonons. The low-temperature behavior is shown to depend sensitively on the spectral density of the thermal reservoir. The mass and the frequency of the oscillator become renormalized through the coupling to the bath. These renormalizations are intrinsically frequency dependent. Moreover, the commonly used form of Ehrenfest's equation for the dynamics of the linearly damped oscillator is shown to only be a reasonable approximation within certain frequency regimes.
Abstract We generalize the formalism of open quantum systems to introduce anyon baths. In particular, we work out a dissipative anyon bath composed of independent pairs of one-dimensional Grundberg–Hansson harmonically bound anyons, which are characterized by one statistical parameter. Using a mapping of these anyons to a bosonic bath with rescaled oscillator frequencies, we show that the original bilinear system-bath coupling assumes a particular non-polynomial form. To determine the relaxation properties, we use the imaginary-time path integral formalism together with a generalization of Wick’s theorem in the form of a smearing formula. The latter allows to approximately calculate the anyon bath spectral density, which acquires a nontrivial temperature dependence. The corresponding relaxation dynamics of the dissipative harmonic oscillator in an anyon bath is found. Well defined limits are revealed for both low and high temperatures. Anyonic features turn out to be most pronounced in the regime of intermediate temperatures.