Hierarchical equations of motion method applied to nonequilibrium heat transport in model molecular junctions: Transient heat current and high-order moments of the current operator

We present a theoretical approach to study nonequilibrium quantum heat transport in molecular junctions described by a spin-boson type model. Based on the Feynman-Vernon path integral influence functional formalism, expressions for the average value and high-order moments of the heat current operators are derived, which are further obtained directly from the auxiliary density operators (ADOs) in the hierarchical equations of motion (HEOM) method. Distribution of the heat current is then derived from the high-order moments. As the HEOM method is nonperturbative and capable of treating non-Markovian system-environment interactions, the method can be applied to various problems of nonequilibrium quantum heat transport beyond the weak coupling regime.

Figure 1

Transient heat current $J_{\nu ,x}$ and $J_{\nu ,p}$ calculated using Eqs. (17) and (19). The other parameters are ${\beta }_L=1.5, {\beta }_R=0.5, \epsilon =0, \mathrm{\Delta }=0.1, {\gamma }_1=0.5, {\gamma }_2=1.0$, and ${\gamma }_3=1.5$.

Figure 2

Transient dynamics of the left heat current and the system population. $\mathrm{\Delta }=0.1$ for the left panel and 0.5 for the right panel. The other parameters are the same as Fig. 1.

Figure 3

Steady state heat current as a function of system-bath coupling strength $\eta$. The other parameters are the same as Fig. 1.

Figure 4

Transient higher (second–fifth) -order moments of the heat current ${\widehat{j}}_p$. $\mathrm{\Delta }=0.5, \eta =0.5$, and the other parameters are the same as Fig. 1.

Figure 5

Distributions of the transient heat current operator ${\widehat{j}}_p$ at different times. The parameters are the same as Fig. 4.

Figure 6

Same as Fig. 5, except for the heat current operator ${\widehat{j}}_x$.