Phase-tunable thermoelectricity in a Josephson junction

Superconducting tunnel junctions constitute the units of superconducting quantum circuits and are massively used both for quantum sensing and quantum computation. In previous works, we predicted the existence of a nonlinear thermoelectric effect in a electron-hole symmetric system, namely, a thermally biased tunnel junction between two different superconductors, where the Josephson effect is suppressed. In this paper, we investigate the impact of the phase-coherent contributions on the thermoelectric effect by tuning the size of the Josephson coupling changing the flux of a direct-current superconducting quantum interference device (dc-SQUID). For a suppressed Josephson coupling, the system generates a finite average thermoelectric signal, combined with an oscillation due to the standard ac Josephson phenomenology. At large Josephson couplings, the thermoelectricity induces an oscillatory behavior with zero average value of the current and voltage, with an amplitude and a frequency associated to the Josephson coupling strength, and ultimately tuned by the dc-SQUID magnetic flux. In conclusion, we demonstrate the ability to control the dynamics of the spontaneous breaking of the electron-hole symmetry. Furthermore, we compute how the flux applied to the dc-SQUID and the lumped elements of the circuit determine the frequency of the thermoelectric signal across the structure, and we envision a frequency modulation application.