Dynamic control of quantum geometric heat flux in a nonequilibrium spin-boson model

We study the quantum geometric heat flux in the nonequilibrium spin-boson model. By adopting the noninteracting-blip approximation that is able to accommodate the strong system-bath coupling, we show that there exists a nonzero geometric heat flux only when the two-level system is nondegenerate. Moreover, the pumping, no pumping, and dynamic control of geometric heat flux are discussed in detail, compared to the results with Redfield weak-coupling approximation. In particular, the geometric energy transfer induced by modulation of two system-bath couplings is identified, which is exclusive to quantum transport in the strong system-bath coupling regime.

Figure 1

Comparison of heat flux and its high-order fluctuations in the NIBA and Redfield methods. (a) The mean value of heat flux: $J=-{\partial }_{i\chi }{E}_0{|}_{\chi =0}$. (b) The shot noise of heat flux: $S=-{\partial }_{i\chi }^2{E}_0{|}_{\chi =0}$. $T_L=150$ K (5), $T_R=90$ K (3), $\Delta =5.22$ meV (2). The cutoff of the Ohmic spectral function is set as 26.1 meV (10). The numbers in the parentheses are the corresponding dimensionless parameters used in Ref. 17.

Figure 2

Integrated geometric heat flux per period $Q_{\mathrm{geom}}\equiv J_{\mathrm{geom}}{\mathcal{T}}_p$ as a linear function of the system-bath coupling under the two-bath-temperature modulation. We set the symmetric coupling ${\Gamma }_L={\Gamma }_R=\Gamma$, and tunneling energy is $\Delta =5.22$ meV. When ${\varepsilon }_0\to -{\varepsilon }_0$, we observe the same lines.

Figure 3

(a) $Q_{\mathrm{geom}}$ as a linear function of $\Gamma$ under modulations of one bath temperature and the gap ${\varepsilon }_0\left(t\right)$. We set $T_L\left(t\right)=150+90\cos \left({\Omega }_{p}t\right)$ (K), ${\varepsilon }_0\left(t\right)={\overline{\varepsilon }}_0+0.78\sin \left({\Omega }_{p}t\right)$ (meV). $T_R=150$ K, $\Delta =5.22$ meV. (b) Integrated dynamic heat flux per period $Q_{\mathrm{dyn}}\equiv J_{\mathrm{dyn}}{\mathcal{T}}_p$ under the same conditions for comparison.

Figure 4

Emergence of the geometric phase effect and geometric heat flux for modulating two system-bath couplings. Although $Q_{\mathrm{geom}}$ is absent at $T_L=T_R$, the nonzero geometric skewness (${\partial }_{i\chi }^3{\mathcal{G}}_{\mathrm{geom}}{|}_{\chi =0}$) shows the existence of the geometric phase effect, manifesting itself as the high-order heat transfer fluctuations. The control protocol is ${\Gamma }_L=130.5+104.4\cos \left({\Omega }_{p}t\right)$ (meV) and ${\Gamma }_R=130.5+104.4\sin \left({\Omega }_{p}t\right)$ (meV). Other parameters are $T_L=150$ K, ${\omega }_0=2.61$ meV, and $\Delta =5.22$ meV.