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Physical Review A

Relaxation induced by colored noise: Analytical results for multilevel systems

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

Published Nov 1, 1992• 17 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.46.6222

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Relaxation induced by colored noise. II. Homogeneous and heterogeneous correlation loss

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Relaxational dynamics of the Edwards-Anderson model and the mean-field theory of spin-glasses

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Langevin equations for the relaxation of spin fluctuations in a soft-spin version of the Edwards-Anderson model are used as a starting point for the study of the dynamic and static properties of spin-glasses. An exact uniform Lagrangian for the average dynamic correlation and response functions is derived for arbitrary range of random exchange, using a functional-integral method proposed by De Dominicis. The properties of the Lagrangian are studied in the mean-field limit which is realized by considering an infinite-ranged random exchange. In this limit, the dynamics are represented by a stochastic equation of motion of a single spin with self-consistent (bare) propagator and Gaussian noise. The low-frequency and the static properties of this equation are studied both above and below ${T}_{c}$. Approaching ${T}_{c}$ from above, spin fluctuations slow down with a relaxation time proportional to ${|T\ensuremath{-}{T}_{c}|}^{\ensuremath{-}1}$ whereas at ${T}_{c}$ the damping function vanishes as ${\ensuremath{\omega}}^{\frac{1}{2}}$. We derive a criterion for dynamic stability below ${T}_{c}$. It is shown that a stable solution necessarily violates the fluctuation-dissipation theorem below ${T}_{c}$. Consequently, the spin-glass order parameters are the time-persistent terms which appear in both the spin correlations and the local response. This is shown to invalidate the treatment of the spin-glass order parameters as purely static quantities. Instead, one has to specify the manner in which they relax in a finite system, along time scales which diverge in the thermodynamic limit. We show that the finite-time correlations decay algebraically with time as ${t}^{\ensuremath{-}\ensuremath{\nu}}$ at all temperatures below ${T}_{c}$, with a temperature-dependent exponent $\ensuremath{\nu}$. Near ${T}_{c}$, $\ensuremath{\nu}$ is given (in the Ising case) as $\ensuremath{\nu}(T)\ensuremath{\sim}\frac{1}{2}\ensuremath{-}{\ensuremath{\pi}}^{\ensuremath{-}1}(\frac{1\ensuremath{-}T}{{T}_{c}})+\ensuremath{\sigma}{(\frac{1\ensuremath{-}T}{{T}_{c}})}^{2}$. A tentative calculation of $\ensuremath{\nu}$ at $T=0$ K is presented. We briefly discuss the physical origin of the violation of the fluctuation-dissipation theorem.

World Scientific lecture notes in physics•Nov 1, 1986•1 citation
We investigate the relaxation of multilevel systems under the influence of stochastic fields with finite correlation time \ensuremathτc{\mathrm{\ensuremath{\tau}}}_{\mathit{c}}\ensuremathτc​. 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 \ensuremathτc{\mathrm{\ensuremath{\tau}}}_{\mathit{c}}\ensuremathτc​. T1{\mathit{T}}_{1}T1​ is longer than \ensuremathτc{\mathrm{\ensuremath{\tau}}}_{\mathit{c}}\ensuremathτc​ provided that r(est){\mathit{r}}^{(\mathrm{est})}r(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/\ensuremathτc{\mathrm{\ensuremath{\tau}}}_{\mathit{c}}\ensuremathτc​. 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.
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.
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.
We investigate the dynamics of a spin-1/2 particle in a constant magnetic field driven by two transverse magnetic noise fields. These noise fields are exponentially correlated in time. By introducing suitable marginal averages and suitable expansions of these averages it is shown that the problem reduces to the solution of two independent tridiagonally coupled vector recurrence relations with vectors of dimension 3. One of these relations describes the decay of the spin component parallel to the constant field, the other the decay of the spin component transverse to the constant field. The equations are solved by a direct time integration and by determining the eigenvalues. It is shown that the Green's functions are determined by ordinary continued fractions. These continued fractions agree with those obtained by Shibata and Sato and are thus rederived by a completely different method. The long-time dependence is governed by the eigenvalue with the lowest real part and by its weight. The eigenvalues are found by determining the poles of the Green's function, the weights of the eigenvalues by evaluating the Green's function near the poles. The relaxation times T1{\mathit{T}}_{1}T1​ and T2{\mathit{T}}_{2}T2​ as well as a frequency shift of the perpendicular component are thus obtained. For certain parameters a single decay constant is not sufficient.
The dynamics of a spin in the presence of a deterministic and a fluctuating magnetic field is solved for analytically to obtain the averaged value of the spin as a function of time for various kinds of fluctuations (noise). Specifically, analytic results are obtained for the time dependence of the expectation value of the spin, averaged over fluctuations, for Gaussian white noise, Gaussian colored noise, and non-Gaussian telegraph noise. Fluctuations cause the decay of the average spin vector (decoherence). For noise with a finite temporal correlation time, a deterministic component of the field can suppress the decoherence of the spin component along the field. Hence, decoherence can be manipulated by controlling the deterministic magnetic field. A simple universal physical picture emerges which explains the mechanism for the suppression of decay.
Langevin equations for the relaxation of spin fluctuations in a soft-spin version of the Edwards-Anderson model are used as a starting point for the study of the dynamic and static properties of spin-glasses. An exact uniform Lagrangian for the average dynamic correlation and response functions is derived for arbitrary range of random exchange, using a functional-integral method proposed by De Dominicis. The properties of the Lagrangian are studied in the mean-field limit which is realized by considering an infinite-ranged random exchange. In this limit, the dynamics are represented by a stochastic equation of motion of a single spin with self-consistent (bare) propagator and Gaussian noise. The low-frequency and the static properties of this equation are studied both above and below Tc{T}_{c}Tc​. Approaching Tc{T}_{c}Tc​ from above, spin fluctuations slow down with a relaxation time proportional to ${|T\ensuremath{-}{T}_{c}|}^{\ensuremath{-}1}$ whereas at Tc{T}_{c}Tc​ the damping function vanishes as ${\ensuremath{\omega}}^{\frac{1}{2}}$. We derive a criterion for dynamic stability below Tc{T}_{c}Tc​. It is shown that a stable solution necessarily violates the fluctuation-dissipation theorem below Tc{T}_{c}Tc​. Consequently, the spin-glass order parameters are the time-persistent terms which appear in both the spin correlations and the local response. This is shown to invalidate the treatment of the spin-glass order parameters as purely static quantities. Instead, one has to specify the manner in which they relax in a finite system, along time scales which diverge in the thermodynamic limit. We show that the finite-time correlations decay algebraically with time as ${t}^{\ensuremath{-}\ensuremath{\nu}}$ at all temperatures below Tc{T}_{c}Tc​, with a temperature-dependent exponent $\ensuremath{\nu}$. Near Tc{T}_{c}Tc​, $\ensuremath{\nu}$ is given (in the Ising case) as $\ensuremath{\nu}(T)\ensuremath{\sim}\frac{1}{2}\ensuremath{-}{\ensuremath{\pi}}^{\ensuremath{-}1}(\frac{1\ensuremath{-}T}{{T}_{c}})+\ensuremath{\sigma}{(\frac{1\ensuremath{-}T}{{T}_{c}})}^{2}$. A tentative calculation of $\ensuremath{\nu}$ at T=0T=0T=0 K is presented. We briefly discuss the physical origin of the violation of the fluctuation-dissipation theorem.