Nonlinear longitudinal relaxation of a quantum superparamagnet with arbitrary spin value $S$: Phase space and density matrix formulations

The nonlinear relaxation of quantum spins interacting with a thermal bath is treated via the respective evolution equations for the reduced density matrix and phase space distribution function in the high temperature and weak spin-bath coupling limits using the methods already available for classical spins. The solution of each evolution equation is written as a finite series of the polarization operators and spherical harmonics, respectively, where the coefficients of the series (statistical averages of the polarization operators and spherical harmonics) are found from entirely equivalent differential-recurrence relations. Each system matrix has an identical set of eigenvalues and eigenfunctions. For illustration, the time behavior of the longitudinal component of the magnetization and its characteristic relaxation times are evaluated for a uniaxial paramagnet of arbitrary spin $S$ in an external constant magnetic field applied along the axis of symmetry. In the large spin limit, the quantum solutions reduce to those of the Fokker-Planck equation for a classical uniaxial superparamagnet. For linear response, the results entirely agree with existing solutions.

Figure 1

(Color online) Nonlinear integral relaxation time ${\tau }_{\text{int}}/{\tau }_N$ for the rise transient response as a function of the barrier parameter $\sigma$ for the spin value $S=8$, ${\xi }_{\text{I}}=0$, and various values of ${\xi }_{\text{II}}=\kappa \to 0,2,4,6$. Solid lines: calculations from Eq. (58); circles: Eq. (60).

Figure 2

(Color online) Nonlinear relaxation time ${\tau }_{\text{int}}/{\tau }_N$ for the rise transient response as a function of the barrier parameter $\sigma$ for ${\xi }_{\text{I}}=0$ and ${\xi }_{\text{II}}=6$ and various values of spin $S$. Solid lines: Eq. (58); circles: classical equation (A5).

Figure 3

(Color online) Correlation time ${\tau }_{cor}/{\tau }_N$ (a) and inverse smallest nonvanishing eigenvalue, $1/{\lambda }_1$, of the system matrix $\mathbf{X}$ (b) vs the field parameter $\xi$ for the barrier parameter $\sigma =10$ and various values of $S$. Filled circles: the classical limit.

Figure 4

(Color online) Correlation time ${\tau }_{cor}/{\tau }_N$ (a) and inverse smallest nonvanishing eigenvalue, $1/{\lambda }_1$, (b) vs the barrier parameter $\sigma$ for the field parameter $\xi =5$ and various values of $S$. Filled circles: the classical limit.

Figure 5

(Color online) Correlation time ${\tau }_{cor}/{\tau }_N$ (a) and inverse smallest nonvanishing eigenvalue, $1/{\lambda }_1$, vs the spin value $S$ for barrier parameter $\sigma =10$ and different field parameter $\xi$. Dashed lines: the classical limit.

Figure 6

(Color online) Correlation time ${\tau }_{cor}/{\tau }_N$ (a) and inverse smallest nonvanishing eigenvalue, $1/{\lambda }_1$, vs the spin value $S$ for field parameter $\xi =5$ and various values of the barrier parameter $\sigma$. Dashed lines: the classical limit.

Figure 7

(Color online) ${\tilde{c}}_1\left(\omega \right)/c_1\left(0\right)$ vs the normalized frequency $\omega {\tau }_N$ for the rise transient response for $S=8$, $\sigma =10$, ${\xi }_{\text{I}}=0$, and various values of ${\xi }_{\text{II}}=\kappa \to 0,2,4,6$. Solid lines: calculations from Eq. (32); stars: Eq. (67).

Figure 8

(Color online) ${\tilde{c}}_1\left(\omega \right)/c_1\left(0\right)$ vs $\omega {\tau }_N$ for the rise transient response for ${\xi }_{\text{I}}=0$, ${\xi }_{\text{II}}=6$, $S=8$, and various values of the anisotropy parameter $\sigma$. Solid lines: calculations from Eq. (32); stars: Eq. (67).

Figure 9

(Color online) Real and imaginary parts of $\chi \left(\omega \right)$ vs $\omega {\tau }_N$ for field parameter $\xi =2$, the anisotropy parameter $\sigma =10$, and various values of spin $S$. Crosses are the single Lorentzian approximation, Eq. (68), while straight dashed lines are the high-frequency asymptote, Eq. (49).

Figure 10

(Color online) Real and imaginary parts of $\chi \left(\omega \right)$ vs $\omega {\tau }_N$ for the barrier parameter $\sigma =10$, $S=6$, and various values of field parameter $\xi$. Crosses are the single Lorentzian approximation, Eq. (68), while the straight dashed lines are the high-frequency asymptote, Eq. (49).

Figure 11

(Color online) Real and imaginary parts of $\chi \left(\omega \right)$ vs $\omega {\tau }_N$ for $S=6$, for field parameter $\xi =2$, and various values of the barrier parameter $\sigma$. Crosses are the single Lorentzian approximation, Eq. (68), while straight dashed lines are the high-frequency asymptote, Eq. (49).