Bifurcation Phenomenon in Spin Relaxation

Spin relaxation in a strong-coupling regime (with respect to the spin system) is investigated in detail based on the spin-boson model in a stochastic limit. We find a bifurcation phenomenon in temperature dependence of relaxation constants, which is never observed in the weak-coupling regime. We also discuss inequalities among the relaxation constants in our model and show the well-known relation $2{\Gamma }_T\ge {\Gamma }_L$, for example, for a wider parameter region than before.

Figure 1

(a) Longitudinal and (b) transverse relaxations.

Figure 2

Temperature dependences of the longitudinal-relaxation constant ${\Gamma }_L$ (solid lines) and the transverse one ${\Gamma }_T$ (dashed lines) for (a) $\epsilon /\hslash {\Omega }_0=0.0$, (b) $0.2$, (c) $0.4$, and (d) $0.6$.

Figure 3

Characteristic functions $f(\lambda )=\det (\lambda -A)$ in the temperature regions I–III in Fig. 2b.

Figure 4

Temperature dependence of the longitudinal direction $\mathit{l}$ corresponding to Fig. 2b. In region II, only two directions which connect continuously with that in region I or III are displayed.

Figure 5

Temperature dependences of ${\Gamma }_L/{\Gamma }_T$, corresponding to Fig. 2. Note that there are regions where no transverse-relaxation constant exists.