Phase-space formulation of the nonlinear longitudinal relaxation of the magnetization in quantum spin systems

Nonlinear longitudinal relaxation of a spin in a uniform external dc magnetic field is treated using a master equation for the quasiprobability distribution function of spin orientations in the configuration space of polar and azimuthal angles (analogous to the Wigner phase space distribution for translational motion). The solution of the corresponding classical problem of the rotational Brownian motion of a magnetic moment in an external magnetic field essentially carries over to the quantum regime yielding in closed form the dependence of the longitudinal spin relaxation on the spin size $S$ as well as an expression for the integral relaxation time, which in linear response reduces to that previously given by D. A. Garanin [Phys. Rev. E 55, 2569 (1997)] using the density matrix approach. The nonlinear relaxation is dominated by a single exponential having as time constant the integral relaxation time. Thus a simple description in terms of a Bloch equation holds even for the nonlinear response of a giant spin.

Figure 1

Schematic representation of the longitudinal nonlinear transient response.

Figure 2

(Color online) Normalized integral relaxation time ${\tau }_{\mathrm{int}}∕{\tau }_N$ from Eq. (25) as a function of $S$ (a) and $\xi$ (b) for various values of $\delta$ (symbols). Dashed line: Eq. (28).

Figure 3

(Color online) Normalized correlation time ${\tau }_{\mathrm{cor}}∕{\tau }_N$ from Eq. (32) as a function of $\xi$ for various values of $S$ (symbols). Dashed lines: Eq. (28).

Figure 4

The real parts of the normalized spectra ${\tilde{f}}_1\left(\omega \right)∕f_1\left(0\right)$ vs the normalized frequency $\omega {\tau }_N$ evaluated from the exact continued fraction solution [Eq. (17): solid lines] for $S=5$, $\delta =0.1$, and various values of $\xi$ compared with those calculated from the single Lorentzian approximation Eq. (37) (symbols).

Figure 5

The real parts of the normalized spectra ${\tilde{f}}_1\left(\omega \right)∕f_1\left(0\right)$ vs $\omega {\tau }_N$ evaluated from the exact continued fraction solution [Eq. (17): solid lines] for $\xi =3$, $\delta =0.1$, and various values of $S$ compared with those calculated from the single Lorentzian approximation Eq. (37) (symbols).