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Physical Review A

Asymptotically exact evaluation of T1 and T2 for a two-level system coupled to a bath with dichotomic colored noise

P. Reineker, Benjamin Kaiser, A. M. Jayannavar

Published Feb 1, 1989• 21 citations
Publisher

Abstract

For a two-level system coupled stochastically to a bath with dichotomic colored noise the relaxation behavior is investigated. An exact numerical calculation shows that it can be described by two (longitudinal and transverse) relaxation rates only when correlations of the stochastic process decay rapidly enough as compared to the amplitude of the fluctuations. The numerical evaluation of the model also shows that the relation ${T}_{2}^{\mathrm{\ensuremath{-}}1}$\ensuremath{\ge}0.5${T}_{1}^{\mathrm{\ensuremath{-}}1}$, well known from textbooks, is generally not satisfied, in agreement with a recent fourth-order perturbative calculation. Finally, it is shown that this relaxation model for a \ensuremath{\delta}-correlated bath is mathematically equivalent to the Haken-Strobl-Reineker model for the coupled coherent and incoherent motion of a quantum particle in the special case of transport in a dimer.

DOI
10.1103/physreva.39.1469

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For a two-level system coupled stochastically to a bath with dichotomic colored noise the relaxation behavior is investigated. An exact numerical calculation shows that it can be described by two (longitudinal and transverse) relaxation rates only when correlations of the stochastic process decay rapidly enough as compared to the amplitude of the fluctuations. The numerical evaluation of the model also shows that the relation T2\ensuremath−1{T}_{2}^{\mathrm{\ensuremath{-}}1}T2\ensuremath−1​\ensuremath{\ge}0.5T1\ensuremath−1{T}_{1}^{\mathrm{\ensuremath{-}}1}T1\ensuremath−1​, well known from textbooks, is generally not satisfied, in agreement with a recent fourth-order perturbative calculation. Finally, it is shown that this relaxation model for a \ensuremath{\delta}-correlated bath is mathematically equivalent to the Haken-Strobl-Reineker model for the coupled coherent and incoherent motion of a quantum particle in the special case of transport in a dimer.
The transport of a quantum particle in a dimer is investigated for the case in which the influence of phonons from different branches is modeled by a dichotomic stochastic process with exponentially decaying correlations. For physical reasons, these processes might be correlated. An exact equation for the correlations between matrix elements of the density operator and both stochastic processes is constructed, and eigenvalues of the corresponding matrix are found numerically. Correlations between the noises reduce the range of incoherent transport. Various criteria for distinguishing between coherent and incoherent motion are also discussed.
We investigate the relaxation of multilevel systems under the influence of stochastic fields with finite correlation time ${\mathrm{\ensuremath{\tau}}}_{\mathit{c}}$. First, we study a system with spin S=1/2 in a static magnetic field in the presence of dichotomic perturbations. The spectra of the components of the averaged spin vector S\ifmmode\bar\else\textasciimacron\fi{} are evaluated exactly by resorting to the memory-function approach. The model is then extended to include the influence on the relaxation of S\ifmmode\bar\else\textasciimacron\fi{} of a different species with spin I=1/2. The relaxation rates T1{\mathit{T}}_{1}T1​ and T2{\mathit{T}}_{2}T2​ are given in terms of cumulant expansions and compared with the exact results. Fourth-order calculations recover the inequality T2{\mathit{T}}_{2}T2​/2T1{\mathit{T}}_{1}T1​\ensuremath{\le}1 for correlated noise fields. Furthermore, it is found that T1{\mathit{T}}_{1}T1​ can be defined also for long ${\mathrm{\ensuremath{\tau}}}_{\mathit{c}}$. T1{\mathit{T}}_{1}T1​ is longer than ${\mathrm{\ensuremath{\tau}}}_{\mathit{c}}$ provided that ${\mathit{r}}^{(\mathrm{est})}$1, where r(est){\mathit{r}}^{(\mathrm{est})}r(est) is the squared ratio between the total amplitude of the fluctuating fields and the largest one between the Larmor frequency and 1/${\mathrm{\ensuremath{\tau}}}_{\mathit{c}}$. It is also found that 2r(est){\mathit{r}}^{(\mathrm{est})}r(est) is an upper bound of the relative error between the exact and the second-order cumulant expansion of T1{\mathit{T}}_{1}T1​. Generalizations on systems with S=1/2, I>1/2, and arbitrary kinds of noise are discussed.
The relaxation of a two-level system under the influence of multiple stochastic fields with finite correlation time ${\mathrm{\ensuremath{\tau}}}_{\mathit{c}}$ is investigated. Both dichotomic and Gaussian noises are considered with strong or negligible mutual correlations. The fluctuation regimes of the noises are not assumed to be equal, as is customarily done. By extending our previous findings [Phys. Rev. A 46, 6222 (1992)] the dominance of a single longitudinal relaxation time T1{\mathit{T}}_{1}T1​ is proven for most of the fluctuation regimes, provided that the fluctuating fields have smaller amplitudes ${\mathrm{\ensuremath{\Delta}}}_{\mathit{i}}$ than the Larmor frequency ${\mathrm{\ensuremath{\omega}}}_{0}$. The key role of the ratio tsumi{\mathit{tsum}}_{\mathit{i}}tsumi​ ${\mathrm{\ensuremath{\Delta}}}_{\mathit{i}}^{2}$/${\mathrm{\ensuremath{\omega}}}_{0}^{2}$ to provide an upper bound of the relative error between the exact value of T1{\mathit{T}}_{1}T1​ and its second-order cumulant expansion is shown. The crossover between the heterogeneous and the homogeneous relaxation, depending on the observation time, is investigated. In the case of a continuous distribution of relaxation times the generalized Langevin equation yields the correct behavior in terms of a finite number of exponential decays. Finally, a counterexample to the usual belief concerning the independence of T1{\mathit{T}}_{1}T1​ from fluctuations occurring along the direction of the static magnetic field is presented.
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.
We obtain kinetic equations for the stochastically averaged reduced density matrix of a quantum system driven by a dichotomous Markovian process and weakly coupled to a quantum thermal bath. These kinetic equations are exact in arbitrary dichotomous perturbation and provide a unified description of both small and large correlation time limits, along with an intermediate case. As an example, the problem of a dissipative two-level system with a dichotomically modulated difference in eigenergies is considered.