Stochastic decoupling approach to the spin-boson dynamics: Perturbative and nonperturbative treatments

We develop a hierarchical functional derivative (HFD) method to investigate the reduced dynamics of a spin-boson model within the framework of a stochastic decoupling description. Keeping only the lowest-order truncation of the hierarchical functional derivatives, one can recover the second-order Nakajima-Zwanzig quantum master equation. Taking into account the higher-order corrections, our method can be implemented as a highly efficient numerical scheme for a general spin-boson model beyond the usual Markovian, rotating-wave, and weak-coupling approximations.

Figure 1

The dynamics of the population difference $\langle {\widehat{\sigma }}_z\left(t\right)\rangle$ of the spin-boson model at zero temperature with different spectrum widths: $\lambda =0.2$ (non-Markovian QSD approach: solid blue line, our numerical results: blue rectangles), $\lambda =0.4$ (non-Markovian QSD approach: dot-dashed purple line, our numerical results: purple diamonds), and $\lambda =0.8$ (non-Markovian QSD approach: dashed red line, our numerical results: red circles). Other parameters are chosen as ${\omega }_0=1$ and $\mathrm{\Lambda }\gamma =0.2$.

Figure 2

The dynamics of the spin $z$ component $\langle {\widehat{J}}_z\left(t\right)\rangle$ of the spin-1 model at finite temperature $\beta {\omega }_c=0.5$. The solid purple line is the numerical result according to our HFD method, and the red dashed line denotes the approximate solution given by the Born-Markov quantum master equation. The initial state is $|J=1,J_{\mathrm{m}}=1\rangle$. Other parameters are chosen as $\varepsilon =0.25{\omega }_c$, $\mathrm{\Delta }=0.5{\omega }_c$, $\chi =0.05$, and ${\omega }_c=1$.