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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Longitudinal relaxation induced by colored noise

Laura Andreozzi, Claudio Donati, M. Giordano, D. Leporini

Published Apr 1, 1994• 11 citations
Publisher

Abstract

Following the analytical results presented in our recent paper [Phys. Rev. A 46, 6222 (1992)], we clarify the conditions leading to a single exponential decay of the longitudinal magnetization in a multilevel system. A direct expression of the longitudinal relaxation time is provided in terms of the transition rates induced by colored noise.

DOI
10.1103/physreve.49.3488

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Relaxation induced by colored noise: Analytical results for multilevel systems

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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 ${\mathit{T}}_{1}$ and ${\mathit{T}}_{2}$ are given in terms of cumulant expansions and compared with the exact results. Fourth-order calculations recover the inequality ${\mathit{T}}_{2}$/2${\mathit{T}}_{1}$\ensuremath{\le}1 for correlated noise fields. Furthermore, it is found that ${\mathit{T}}_{1}$ can be defined also for long ${\mathrm{\ensuremath{\tau}}}_{\mathit{c}}$. ${\mathit{T}}_{1}$ is longer than ${\mathrm{\ensuremath{\tau}}}_{\mathit{c}}$ provided that ${\mathit{r}}^{(\mathrm{est})}$1, where ${\mathit{r}}^{(\mathrm{est})}$ 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 2${\mathit{r}}^{(\mathrm{est})}$ is an upper bound of the relative error between the exact and the second-order cumulant expansion of ${\mathit{T}}_{1}$. Generalizations on systems with S=1/2, I>1/2, and arbitrary kinds of noise are discussed.

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H.A.W. van Vianen, John A. Tjon

Physica•Sep 1, 1970•4 citations
Following the analytical results presented in our recent paper [Phys. Rev. A 46, 6222 (1992)], we clarify the conditions leading to a single exponential decay of the longitudinal magnetization in a multilevel system. A direct expression of the longitudinal relaxation time is provided in terms of the transition rates induced by colored noise.
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.
The nonlinear response to external disturbances of a system of interest scrA (the ``atom'') in contact with a thermal bath is studied. The central issue is the relationship between the nonlinear susceptibility and the equilibrium relaxation spectra of observables of scrA. Paralleling the results of the linear-response theory, the nth-order nonlinear susceptibility is factorized by quantum-mechanical arguments into two terms pertaining to the disturbances and the spin-bath system, respectively. The role of slow atom states, i.e., states relaxing on times T longer than the bath correlation time ${\mathrm{\ensuremath{\tau}}}_{\mathit{c}}$, is pointed out, by recurring to recent findings on their general features [Phys. Rev. A 46, 6222 (1992)]. In the frame of a stochastic picture of the bath and a multilevel scheme of the system scrA, the conditions under which the nonlinear susceptibility provides the spectrum of the slow atom states are clarified. Illustrations are drawn from magnetic resonance to demonstrate that techniques based on the nonlinear response of a spin system to multiple continuous waves compete favorably with pulsed techniques to provide information on longitudinal spin relaxation.