Spin Dynamics in a Dissipative Environment: From Quantal to Classical

We address the problem of spin dynamics in the presence of a thermal bath, by solving exactly the appropriate quantum master equations with continued-fraction methods. The crossover region between the quantum and classical domains is studied by increasing the spin value $S$, and the asymptote for the classical absorption spectra is eventually recovered. Along with the recognized relevance of the coupling strength, we show the critical role played by the structure of the system-environment interaction in the emergence of classical phenomenology.

Figure 1

Absorption line shape ${\chi }^{\prime \prime }(\Omega )$ for a spin $S=10$ with $D=0.5$ at $B_z=0$. Thick line: $\sigma =D{S}^2/T=5$ and spin-phonon coupling $\lambda =3\times {10}^{-8}$. Thin lines: effects of halving the damping at the same $T$ (solid), and of halving $T$ keeping $\lambda$ (dashed). Vertical lines: loci of the transition frequencies ${\Delta }_{m,m+1}=D(2m+1)$. Inset: classical dampingless asymptote (8) for $\sigma =1$, 2, and 3 (lines), and exact Fokker-Planck results for finite Landau-Lifshitz damping (${\lambda }_{\mathrm{LL}}=0.003$; symbols).

Figure 2

Spectral line shape ${\chi }^{\prime \prime }(\Omega )$ for $\sigma =1$ at $B_z=0$ (see the text for the scaled quantities). The thick dashed line is the classical Eq. (8). Top: $S=5$, 25, 50, and 100 with “constant” coupling $\lambda /S={10}^{-2}$. Bottom: fixed $S=50$ with $\lambda /S={10}^{-2}$ (as in top), $3\times {10}^{-2}$, and ${10}^{-1}$.

Figure 3

Line shape for $S=10$ at $\sigma =1$ with (i) the phonon-coupling model and (ii) bilinear coupling, both with super-Ohmic (hybrid) and Ohmic (electron-hole) spectral densities.