Out‐of‐Equilibrium Fluctuation‐Dissipation Relations Verified by the Electrical and Thermoelectrical AC‐Conductances in a Quantum Dot

The electrical and heat currents flowing through a quantum dot are calculated in the presence of a time‐modulated gate voltage with the help of the out‐of‐equilibrium Green function technique. From the first harmonics of the currents, we extract the electrical and thermoelectrical trans‐admittances and ac‐conductances. Next, by a careful comparison of the ac‐conductances with the finite‐frequency electrical and mixed electrical‐heat noises, we establish the fluctuation‐dissipation relations linking these quantities, which are thus generalized out‐of‐equilibrium for a quantum system. It is shown that the electrical ac‐conductance associated to the displacement current is directly linked to the electrical noise summed over reservoirs, whereas the relation between the thermoelectrical ac‐conductance and the mixed noise contains an additional term proportional to the energy step that the electrons must overcome when traveling through the junction. A numerical study reveals however that a fluctuation‐dissipation relation involving a single reservoir applies for both electrical and thermoelectrical ac‐conductances when the frequency dominates over the other characteristic energies. The fluctuation‐dissipation theorem linking dc‐noise with ac‐conductance at equilibrium is well known and widely used. How this relation generalizes out‐of‐equilibrium? By calculating the time‐dependent Keldysh Green functions, it is shown here that one has to sum over noises in both source and drain reservoirs to obtain such a relationship. The existence of a similar relation between thermoelectrical ac‐conductance and mixed noise is also questioned.