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Physical Review A

Non-Markovian relaxation of a spin-1/2 particle in a fluctuating transverse field: Cumulant expansion and stochastic simulation results

M. Aihara, Hannah Sevian, James L. Skinner

Published Jun 1, 1990• 32 citations
Publisher

Abstract

We consider the relaxation dynamics of a spin-1/2 particle in a fluctuating transverse magnetic field. With a cumulant expansion technique (up to sixth order) we calculate the time dependence of the approach to equilibrium of the population difference of the two spin levels and of the phase coherence. We find that for intermediate strength coupling to the fluctuations, the cumulant expansion converges, and the relaxation to equilibrium can have a pronounced non-Markovian character. We compare these analytic results with numerically exact data from stochastic simulations.

DOI
10.1103/physreva.41.6596

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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.