Unifying quantum heat transfer in a nonequilibrium spin-boson model with full counting statistics

To study the full counting statistics of quantum heat transfer in a driven nonequilibrium spin-boson model, we develop a generalized nonequilibrium polaron-transformed Redfield equation with an auxiliary counting field. This enables us to study the impact of qubit-bath coupling ranging from weak to strong regimes. Without external modulations, we observe maximal values of both steady-state heat flux and noise power in moderate coupling regimes, below which we find that these two transport quantities are enhanced by the finite-qubit-energy bias. With external modulations, the geometric-phase-induced heat flux shows a monotonic decrease upon increasing the qubit-bath coupling at zero qubit energy bias (without bias). While under the finite-qubit-energy bias (with bias), the geometric-phase-induced heat flux exhibits an interesting reversal behavior in the strong coupling regime. Our results unify the seemingly contradictory results in weak and strong qubit-bath coupling regimes and provide detailed dissections for the quantum fluctuation of nonequilibrium heat transfer.

Figure 1

Schematic of the nonequilibrium spin-boson model, composed of a central two-level qubit (purple circle) coupled to two individual thermal baths (red and blue regions), with temperatures $T_L$ and $T_R$, respectively. Wavy red (blue) arrowed lines describe the interaction between the qubit and the $L\mathrm{th}$ ($R\mathrm{th}$) bath. For the driven nonequilibrium spin-boson model, the system parameters appear to be time dependent, e.g., $T_L\left(t\right)$ and $T_R\left(t\right)$.

Figure 2

Representative processes involving phonons in quantum heat transfer: (a), (b) single-phonon sequentially incoherent processes, $Q_L\left(\omega \right)$ and $Q_R(-\omega )$, respectively; (c), (d) two-phonon cotunneling processes, $Q_L\left(\omega \right)Q_R(-\omega )$ and $Q_R\left(\omega \right)Q_L(-\omega )$, respectively.

Figure 3

Behaviors of the steady-state heat flux and noise power: (a), (b) with varying system-bath coupling strengths and (c), (d) with tuning of the qubit energy bias, respectively. Other parameters are $\mathrm{\Delta }=5.22$ meV, ${\omega }_c=26.1$ meV, $T_L=150$ K, and $T_R=90$ K.

Figure 4

Adiabatic modulation by two bath temperatures without bias (${\epsilon }_0=0$): (a) geometric-phase-induced heat pump $Q_{\mathrm{geom}}=J_{\mathrm{geom}}*T_p$; (b) coherence ($P_{10}$) in the local basis. The two bath temperatures are specified as $T_L\left(\tau \right)=(150+90cos{\mathrm{\Omega }}_p\tau )$ K and $T_R\left(\tau \right)=(150+90sin{\mathrm{\Omega }}_p\tau )$ K, with the period $T_p=1$ ns. Other parameters are $\mathrm{\Delta }=5.22$ meV and ${\omega }_c=26.1$ meV.

Figure 5

(a) Geometric-phase-induced heat pump $Q_{\mathrm{geom}}=J_{\mathrm{geom}}\times T_p$ under finite energy bias of the qubit (${\epsilon }_0=2.61$ meV); (b) influence of the qubit energy bias on the geometric-phase-induced heat pump, with modulation of the two bath temperatures; (c) log-log relation between ${\epsilon }_0$ and $Q_{\mathrm{geom}}$ in the strong coupling regime ($\alpha =4$); (d) linear relation between $Q_{\mathrm{geom}}$ and $\alpha$ in the strong coupling regime. Parameters are the same as in Fig. 4.