Finite-frequency noise, Fano factor, $\Delta T$-noise and cross-correlations in double quantum dots

A theoretical study on electrical current fluctuations in a double quantum dot connected to electronic reservoirs is presented, with the aim of deriving the finite-frequency noise, the Fano factor and the $\Delta T$-noise. We establish a general expression for the noise in terms of Green functions in the double quantum dot and self-energies in the reservoirs. This result is then applied to model double quantum dots in various situations. For a non-interacting double quantum dot, we highlight several interesting features in the physical properties of this system. In particular, we demonstrate the possibility of obtaining a significant reduction in zero-frequency noise and Fano factor either when the system is placed in a given operating regime, or when a temperature gradient is applied between the two reservoirs, resulting in a negative $\Delta T$-noise being generated. In addition, in the vicinity of honeycomb vertices, a sign change is observed in the finite-frequency cross-correlator between the two reservoirs, in contrast to what is obtained for the zero-frequency cross-correlator, which remains negative throughout the $(\varepsilon_1,\varepsilon_2)$-plane, $\varepsilon_{1,2}$ being the level energies in each of the two dots. By using an approximate first-level numerical approach, we finally study how the finite-frequency noise in a double quantum dot evolves under the influence of Coulomb interactions.