Dynamical energy transfer in ac-driven quantum systems

We analyze the time-dependent energy and heat flows in a resonant level coupled to a fermionic continuum. The level is periodically forced with an external power source that supplies energy into the system. Based on the tunneling Hamiltonian approach and scattering theory, we discuss the different contributions to the total energy flux. We then derive the appropriate expression for the dynamical dissipation, in accordance with the fundamental principles of thermodynamics. Remarkably, we find that the dissipated heat can be expressed as a Joule law with a universal resistance that is constant at all times.

Figure 1

Energy diagram of the system under consideration. A single electronic level (the impurity with charge $e$) is coupled to a Fermi sea (the reservoir with chemical potential $\mu$). Energy is supplied into the system by a power source (amplitude $V_{\text{ac}}$ and frequency $\Omega$) attached to the quantum level. Thus, energy rates are created not only at the impurity ($W_D$) but also at the reservoir ($W_C$) and in the contact region ($W_T$).

Figure 2

Heat fluxes $\dot{Q}\left(t\right)$ (stars and triangles) and $\dot{\stackrel{̃}{Q}}\left(t\right)$ (solid and open circles) as a function of the charge current $I_C{\left(t\right)}^2$ within the slow-driving regime for two different amplitudes $V_{\text{ac}}=10,12$, respectively. Clearly, only the heat $\dot{Q}\left(t\right)$ satisfies $\dot{Q}\left(t\right)/I_C{\left(t\right)}^2=R$, with $R$ a constant independent of time. Parameters: $\mu =0$, ${\varepsilon }_0=-1.2$, $T=0$, and $\hslash \Omega ={10}^{-3}$. Energies are expressed in units of $\Gamma$. Inset: $\dot{Q}\left(t\right)$ (dashed lines with a vertical offset) and $\dot{\stackrel{̃}{Q}}\left(t\right)$ (solid lines) as a function of time.