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 ${\mathit{T}}_{1}$ and ${\mathit{T}}_{2}$ as well as a frequency shift of the perpendicular component are thus obtained. For certain parameters a single decay constant is not sufficient.
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 and T2 as well as a frequency shift of the perpendicular component are thus obtained. For certain parameters a single decay constant is not sufficient.
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 and T2 are given in terms of cumulant expansions and compared with the exact results. Fourth-order calculations recover the inequality T2/2T1\ensuremath{\le}1 for correlated noise fields. Furthermore, it is found that T1 can be defined also for long ${\mathrm{\ensuremath{\tau}}}_{\mathit{c}}$. T1 is longer than ${\mathrm{\ensuremath{\tau}}}_{\mathit{c}}$ provided that ${\mathit{r}}^{(\mathrm{est})}$1, where r(est) is the squared ratio between the total amplitude of the fluctuating fields and the largest one between the Larmor frequency and 1/${\mathrm{\ensuremath{\tau}}}_{\mathit{c}}$. It is also found that 2r(est) is an upper bound of the relative error between the exact and the second-order cumulant expansion of T1. Generalizations on systems with S=1/2, I>1/2, and arbitrary kinds of noise are discussed.
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.
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 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 ${\mathrm{\ensuremath{\Delta}}}_{\mathit{i}}^{2}$/${\mathrm{\ensuremath{\omega}}}_{0}^{2}$ to provide an upper bound of the relative error between the exact value of T1 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 from fluctuations occurring along the direction of the static magnetic field is presented.
Multiple-spin coupling is a key tool for investigating magnetic quantum systems. In particular, models with four-spin interactions became of great interest since they have been applied to describe material properties such as superconductivity. In this framework, the dynamics of the spin-1/2 Ising two-leg ladder with four-spin plaquette interaction in a transverse field is investigated. By means of the recurrence relations method, the first four recurrants are determined exactly for the general model, while five exact recurrants are calculated for two particular models. In all cases, higher-order recurrants are computed through extrapolation to obtain time-dependent autocorrelation functions and spectral densities in the long-time regime and infinite-temperature limit. It is found that the relaxation functions decay slower than the expected behavior of the one-dimensional transverse Ising model. The spectral lines, characterized by a Lorentzian behavior in the central body and a Gaussian shape in the tails, display exchange narrowing as the coupling intensities increase.
Within the universality class of ferromagnetic vector models with O( n ) symmetry and purely dissipative dynamics, we study the non-equilibrium critical relaxation from a magnetized initial state. Transverse correlation and response functions are exactly computed for Gaussian fluctuations and in the limit of infinite number n of components of the order parameter. We find that the fluctuation–dissipation ratios (FDRs) for longitudinal and transverse modes differ already at the Gaussian level. In these two exactly solvable cases we completely describe the crossover from the short-time to the long-time behaviour, corresponding to a disordered and a magnetized initial condition, respectively. The effects of non-Gaussian fluctuations on longitudinal and transverse quantities are calculated in the first order in the -expansion ( = 4− d ) and reliable three-dimensional estimates of the two FDRs are obtained.