We propose that a device composed of two vertically stacked monolayer graphene Josephson
junctions can be used for Cooper pair splitting. The hybridization of the Andreev bound states of
the two Josephson junction can facilitate non-local transport in this normal-superconductor hybrid
structure, which we study by calculating the non-local differential conductance. Assuming that one
of the graphene layers is electron and the other is hole doped, we find that the non-local Andreev
reflection can dominate the differential conductance of the system. Our setup does not require the
precise control of junction length, doping, or superconducting phase difference, which could be an
important advantage for experimental realization.

We study the quantum interference (QI) effects in three-terminal Andreev interferometers based on polyaromatic hydrocarbons (PAHs) under non-equilibrium conditions. The Andreev interferometer consists of a PAH coupled to two superconducting and one normal conducting terminals. We calculate the current measured in the normal lead as well as the current between the superconducting terminals under non-equilibrium conditions. We show that both the QI arising in the PAH cores and the bias voltage applied to a normal contact have a fundamental effect on the charge distribution associated with the Andreev Bound States (ABSs). QI can lead to a peculiar dependence of the normal current on the superconducting phase difference that was not observed in earlier studies of mesoscopic Andreev interferometers. We explain our results by an induced asymmetry in the spatial distribution of the electron- and hole-like quasiparticles. The non-equilibrium charge occupation induced in the central PAH core can result in a π transition in the current-phase relation of the supercurrent for large enough applied bias voltage on the normal lead. The asymmetry in the spatial distribution of the electron- and hole-like quasiparticles might be used to split Cooper pairs and hence to produce entangled electrons in four terminal setups.

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