Relaxation of a spin-1/2 particle driven by transverse colored Gaussian noise: Time dependence and eigenvalues

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.