Large Tunable Thermophase in Superconductor -- Quantum Dot -- Superconductor Josephson Junctions

In spite of extended efforts, detecting thermoelectric effects in superconductors have proven to be a challenging task, due to the inherent superconducting particle-hole symmetry. Here we present a theoretical study of an experimentally attainable Superconductor -- Quantum Dot -- Superconductor (SC-QD-SC) Josephson Junction with highly tunable thermoelectric properties and a thermal response that is far larger than previous suggestions. The QD energy level between the SCs breaks particle-hole symmetry in a gradual manner, allowing, in the presence of a temperature gradient, for gate controlled appearance of a superconducting thermo-phase. This thermo-phase increases up to a maximal value of $\pi/2$ after which thermovoltage is expected to develop. Using time-independent Keldysh countour Green's functions we derive the exact thermo-phase and thermal response on the junction. These are shown to be orders of magnitude larger than usual SC tunnel junctions with a sharp dependence on gate voltage.