Energy transfer in the nonequilibrium spin-boson model: From weak to strong coupling

To explore energy transfer in the nonequilibrium spin-boson model (NESB) from weak to strong system-bath coupling regimes, we propose a polaron-transformed nonequilibrium Green's function (NEGF) method. By combining the polaron transformation, we are able to treat the system-bath coupling nonperturbatively, thus in direct contrast to conventionally used NEGF methods which take the system-bath coupling as a perturbation. The Majorana-fermion representation is further utilized to evaluate terms in the Dyson series. This method not only allows us to deal with weak as well as strong coupling regimes but also enables an investigation on the role of bias in the energy transfer. As an application of the method, we study an Ohmic NESB. For an unbiased spin system, our energy current result smoothly bridges predictions of two benchmarks, namely, the quantum master equation and the nonequilibrium noninteracting blip approximation, a considerable improvement over existing theories. In case of a biased spin system, we found a bias-induced nonmonotonic behavior of the energy conductance in the intermediate coupling regime, resulting from the resonant character of the energy transfer. This finding may offer a nontrivial quantum control knob over energy transfer at the nanoscale.

Figure 1

Comparison between theoretical results on the energy current. The pink dashed line denotes the result of the QME [Eq. (42)], the blue dash-dotted line denotes the result of the NE-NIBA [Eq. (43)], the red dash-dotted line denotes the result of a previous NEGF method [Eq. (41)], and the green solid line denotes the result of the present PT-NEGF method [Eq. (7) together with Eq. (36)]. Parameters are $T_L/\mathrm{\Delta }=1.4$, $T_R/\mathrm{\Delta }=1.2$, ${\alpha }_L={\alpha }_R=\alpha$, ${\omega }_c/\mathrm{\Delta }=30$ such that the nonadiabatic limit is fulfilled.

Figure 2

Dependence of the energy conductance on the energy spacing $\mathrm{\Delta }$ of the spin system for different temperatures. We choose $\alpha =0.1$, $T_L=T_R=T$.

Figure 3

Temperature dependence of the energy conductance with (a) $\alpha =0.01$ and (b) $\alpha =0.5$ (Toulouse point). We choose ${\omega }_c/\mathrm{\Delta }=20$, $T_L=T_R=T$.

Figure 4

Behaviors of the energy current with varying bias (blue denotes $\varepsilon =1$, red denotes $\varepsilon =2$) as a function of coupling strength. We compare PT-NEGF results (solid lines) to QME [Eq. (46)] (dash-dotted line) and NE-NIBA [Eq. (47)] (dashed lines). The inset shows the dependence of heat conductance on the bias for different coupling strengths predicted by the PT-NEGF. We choose $T_L/\mathrm{\Delta }=1.4$, $T_R/\mathrm{\Delta }=1.2$, ${\alpha }_L={\alpha }_R=\alpha$, ${\omega }_c/\mathrm{\Delta }=30$ such that the nonadiabatic limit is fulfilled.

Figure 5

The dependence of energy conductance on the bias for different coupling strengths predicted by the PT-NEGF. We choose $T_L/\mathrm{\Delta }=T_R/\mathrm{\Delta }=1.2$, ${\omega }_c/\mathrm{\Delta }=30$ such that the nonadiabatic limit is fulfilled.

Figure 6

Behaviors of the heat conductance with varying bias as a function of coupling strength for (a) $T/\mathrm{\Delta }=0.3$ and (b) $T/\mathrm{\Delta }=6$. The inset shows details of $\kappa$ in the strong coupling regime. We choose $T_L=T_R=T$ and ${\omega }_c/T=30$ such that the nonadiabatic limit is fulfilled.