Quantum thermodynamics of the resonant-level model with driven system-bath coupling

We study nonequilibrium thermodynamics in a fermionic resonant-level model with arbitrary coupling strength to a fermionic bath, taking the wide-band limit. In contrast to previous theories, we consider a system where both the level energy and the coupling strength depend explicitly on time. We find that, even in this generalized model, consistent thermodynamic laws can be obtained, up to the second order in the drive speed, by splitting the coupling energy symmetrically between system and bath. We define observables for the system energy, work, heat, and entropy, and calculate them using nonequilibrium Green's functions. We find that the observables fulfill the laws of thermodynamics, and connect smoothly to the known equilibrium results.

Figure 1

Resonant-level model of a driven single-electron quantum dot at energy $\epsilon \left(t\right)$, with time-dependent tunnel coupling $\gamma \left(t\right)$ to a single metallic lead at inverse temperature $\beta$, with chemical potential $\mu$. The coupling results in broadening of the dot electron level with profile $A(t,\omega )$, see Eq. (14).

Figure 2

Keldysh integration contour $C$ with times $\tau$ and ${\tau }^{\prime }$, running from $-\infty$ to $+\infty$ in the lower half plane, before returning to $-\infty$ in the upper half plane.

Figure 3

Feynman diagrams contributing to the exact dot Green's function from Eq. (5).

Figure 4

Dot particle number as a function of time, for ${\epsilon }_0=0.5,{\mathrm{\Delta }}_{\epsilon }=0.5,{\omega }_{\epsilon }=0.5,{\mathrm{\Gamma }}_0=1,{\mathrm{\Delta }}_{\mathrm{\Gamma }}=0.2,{\omega }_{\mathrm{\Gamma }}=0.5$. Blue: Adiabatic $N_0\left(t\right)$ obtained using the Lorentzian spectral density $A_0(t,\omega )$. Red: Exact $N\left(t\right)$ from Eq. (14).

Figure 5

Difference of entropy production rate ${\partial }_{t}S\left(t\right)$ and inverse temperature multiplied by heat, for ${\epsilon }_0=0.5,{\mathrm{\Delta }}_{\epsilon }=0.5,{\omega }_{\epsilon }=0.5,{\mathrm{\Gamma }}_0=1,{\mathrm{\Delta }}_{\mathrm{\Gamma }}=0.2,{\omega }_{\mathrm{\Gamma }}=0.5$. Blue: Second-order quasiadiabatic entropy production rate ${({\partial }_{t}S-\beta \dot{Q})}^{\left(2\right)}$. This rate is positive for all times, in accordance with the second law as in Eq. (44). Red: ${\partial }_{t}S-\beta \dot{Q}$ as calculated from Eqs. (23) and (36). This rate can become negative beyond the adiabatic limit, reflecting the influence of higher orders in the quasiadiabatic expansion. Moreover, the non-Markovianity of the system itself may lead to negative transients.