Realization of hierarchical equations of motion from stochastic perspectives

The hierarchical equations of motion (HEOM) for a generalized quantum dissipative system is rigorously constructed in the frameworks of two different stochastic dynamical descriptions, i.e., the non-Markovian quantum state diffusion approach as well as the stochastic decoupling scheme. We demonstrated that the HEOMs obtained by these two different stochastic dynamical methods are identical. Moreover, we present some numerical examples to verify the feasibility of our formalism.

Figure 1

The reduced dynamics of the pure dephasing model with initial state $|\mathrm{\Phi }\left(0\right)\rangle =\frac{1}{\sqrt{2}}\left(\right|e\rangle +|g\rangle )$, where $|g\rangle$ and $|e\rangle$ denote the ground and the excited states of ${\widehat{\sigma }}_z$, respectively. The blue solid line is the exactly analytical result, and the red circles are the numerical results obtained by HEOM. Other parameters are chosen as $\chi =0.002$, ${\omega }_c=5$, $\beta =0.015$, and ${\omega }_0=1$.

Figure 2

The reduced dynamics of the spontaneous decay model with initial state $\left|\mathrm{\Phi }\right(0)\rangle =|e\rangle$. The blue solid line is the exactly analytical result, and the red circles are the numerical results obtained by HEOM. Other parameters are chosen as $\lambda =0.2$, $\gamma =5$, and ${\omega }_0=1$.

Figure 3

The coherent dynamics of the population difference $\langle {\widehat{\sigma }}_z\left(t\right)\rangle$ of the spin-boson model with initial state $\left|\mathrm{\Phi }\right(0)\rangle =|e\rangle$ at zero temperature. The purple solid line is the numerical result from the HEOM method, the red rectangles denote the results obtained by the generalized Silbey-Harris transformation approach, and the blue dashed line represents the Wigner-Weisskopf result from Ref. [61]. Other parameters are chosen as $\mathrm{\Delta }=0.5$, $\lambda =0.5\gamma$, $\gamma =0.5$, and ${\omega }_0=\mathrm{\Delta }$.