## Transport in hybrid systems with superconductors

*Benoit Doucot*

Recently, a lot of theoretical and experimental works have investigated physical properties of periodically driven condensed matter systems. The main challenge here is that, unlike the equilibrium case, the density matrix of periodic stationary states is not given by an explicit formula, and it has to be computed from first principles. An interesting class of such systems is obtained by coupling one or several quantum dots to a finite number of superconducting reservoirs. In equilibrium, the energy spectrum of these systems hosts Andreev bound-states localized on the quantum dots, at energies lying inside the superconducting gap of the reservoirs. In the presence of finite and commensurate voltage bias, the corresponding Bogoliubov-De Gennes Hamiltonian is periodic in time. We have shown that this periodic driving turns the equilibrium Andreev bound-states into narrow resonances forming a pattern of coupled Floquet-Wannier-Stark ladders, which can be probed experimentally by measuring finite frequency noise fluctuations, or more directly by measuring the tunneling density of states on the dot. This spectrum is sensitive to interesting quantum effects, such as the Berry phase experienced by a coherent superposition of a particle and hole state, in the presence of a time-dependent Hamiltonian. We have studied in some detail the strong variations of physical observables in the stationary state, when external parameters are varied in the vicinity of a point where the two Floquet-Wannier-Stark ladders nearly cross. In her PhD thesis, Andriani Keliri has extended this study to the case of two quantum dots coupled by an intermediate superconductor. She has found a striking effect, namely the onset of long range correlations between these two dots induced by the periodic driving. We are currently attempting to see how local Coulomb interactions on the dots may modify the properties of these periodic stationary states.