Coherent destruction of tunneling of quantum energy transport in a driven nonequilibrium spin-boson model

We investigate the steady-state energy flow of the periodically driven nonequilibrium spin-boson model by means of combining counter-rotating-hybridized rotating-wave (CHRW) method with full counting statistics. In the case of weak driving $(A<0.1\Delta )$, where $A$ is the driving amplitude and $\Delta$ is the energy gap of the two-level system, our results are consistent with those obtained by traditional rotating-wave approximation (RWA) approach; in the case of high-frequency driving, our results are in agreement with the ones of the secular Floquet-Redfield method. As the driving amplitude increases, the steady-state flow exhibits nonlinear behavior: it reaches a maximum at medium driving, and then decreases to zero, which means that the coherent destruction of tunneling (CDT) appears, and then continues to grow slowly. We also find that, when the steady-state energy flow varies with driving frequency ${\omega }_d$, comparing with RWA results, the counter-rotating (CR) terms increase the energy flow at low-frequency driving $({\omega }_d<\Delta )$, where ${\omega }_d$ is the driving frequency, and decreases it at high-frequency driving $({\omega }_d>\Delta )$. In the case of strong driving, when the zero-order Bessel function of the first kind satisfies $J_0\left(\frac{A}{{\omega }_d}\zeta \right)=0$, instead of $J_0\left(\frac{A}{{\omega }_d}\right)=0$, the steady-state energy flow is 0, which corresponds to the CDT of the energy flow. The modified parameter $\zeta$ mainly defines the contribution of the CR terms. At high-frequency driving, the modified parameter $\zeta$ tends to $1,$ our result is consistent with previous results $J_0\left(\frac{A}{{\omega }_d}\right)=0$. We find that the CDT of flow only occurs when both CR terms of the driving and the dissipation are considered simultaneously. Our results show the influence of the CR terms on the energy flow and the modification of CDT condition and give theoretical guidance for energy transport of small quantum devices.