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We consider the out-of-equilibrium behavior of a general class of mesoscopicdevices composed of several superconducting or/and normal metal leads separatedby quantum dots. Starting from a microscopic Hamiltonian description, weprovide a non-perturbative approach to quantum electronic transport in thetunneling amplitudes between dots and leads: using the equivalent of a pathintegral formulation, the lead degrees of freedom are integrated out in orderto compute both the current and the current correlations (noise) in this classof systems, in terms of the dressed Green's function matrix of the quantumdots. In order to illustrate the efficiency of this formalism, we apply ourresults to the "all superconducting Cooper pair beam splitter", a devicecomposed of three superconducting leads connected via two quantum dots, wherecrossed Andreev reflection operates Cooper pair splitting. Commensurate voltagedifferences between the three leads allow to obtain expressions for the currentand noise as a function of the Keldysh Nambu Floquet dressed Green's functionof the dot system. This voltage configuration allows the occurrence ofnon-local processes involving multiple Cooper pairs which ultimately lead tothe presence of non-zero DC currents in an out-of-equilibrium situation. Weinvestigate in details the results for the noise obtained numerically in thespecific case of opposite voltages, where the transport properties aredominated by the so called "quartet processes", involving the coherent exchangeof two Cooper pairs among all three superconducting terminals. We show thatthese processes are noiseless in the non-resonant case, and that this propertyis also observed for other voltage configurations. When the dots are in aresonant regime, the noise characteristics change qualitatively, with theappearance of giant Fano factors.

Theory of non-equilibrium noise in general multi-terminal superconducting hydrid devices: application to multiple Cooper pair resonances