A simple Floquet-Wannier-Stark-Andreev viewpoint and emergence of low-energy scales in a voltage-biased three-terminal Josephson junction

A three-terminal Josephson junction consists of three superconductors coupled coherently to a small nonsuperconducting island, such as a diffusive metal, a single or double quantum dot. A specific resonant single quantum dot three-terminal Josephson junction (Sa, Sb, Sc) biased with voltages (V,−V, 0) is considered, but the conclusions hold more generally for resonant semi-conducting quantum wire set-ups. A phase-sensitive DC-current due to the quartet or multipair mechanism appears also if V 6= 0. Floquet theory combined with band theory generalizes the equilibrium Andreev bound states (for V = 0) to nonequilibrium Floquet-Wannier-Stark-Andreev (FWS-Andreev) ladders of resonances (for V 6= 0). If V = 0 and at zero temperature, the supercurrent is related to the phase-derivative of the spectrum of Andreev bound states at negative energy (neglecting the contribution of the continua). If the gap is sent to infinity for V 6= 0, then the Josephson relation holds between the current carried by the FWS-Andreev resonances and the phase derivative of the Floquet energy. However, this relation cannot be demonstrated in the presence of strong effect of tiny relaxation (electron-phonon scattering or coupling to the quasiparticle semi-infinite continua by multiple Andreev reflections). In addition, the notion of “states at negative energy” (with V = 0) is meaningless for FWS-Andreev resonances (with V 6= 0). In practise, if V 6= 0, the relation between spectrum and transport can still be used in a weaker form as a guideline to understand qualitatively the numerical results for the current in the presence of relaxation. Namely, the low-energy scales in the current correlate qualitatively with their counterparts in the FWS-Andreev spectrum of resonances, as far as the voltage dependence is concerned. Three important low-energy scales are identified, and a perspective is to relate those low-energy scales to a recent noise cross-correlation experiment [Y. Cohen et al., arXiv:1606.08436].