Berry phase in superconducting multiterminal quantum dots

We report on a study of the nontrivial Berry phase in superconducting multiterminal quantum dots biased at commensurate voltages. Starting with the time-periodic Bogoliubov–de Gennes equations, we obtain a tight-binding model in Floquet space, and we solve these equations in the semiclassical limit. We observe that the parameter space defined by the contact transparencies and quartet phase splits into two components with a nontrivial Berry phase. We use the Bohr-Sommerfeld quantization to calculate the Berry phase. We find that if the quantum dot level sits at zero energy, then the Berry phase takes the values φB = 0 or φB = π. We demonstrate that this nontrivial Berry phase can be observed by tunneling spectroscopy in the Floquet spectra. Consequently, the Floquet-Wannier-Stark ladder spectra of superconducting multiterminal quantum dots are shifted by half-a-period if φB = π. Our numerical calculations based on the Keldysh Green's functions show that this Berry phase spectral shift can be observed from the quantum dot tunneling density of states.