Phase-coherent transport and Higgs dynamics in superconductor-quantum dot hybrids

In the present work, we investigate the transport and dynamic nonequilibrium properties of superconductor-quantum dot hybrids. This consists of a single-level quantum dot coupled to two BCS superconductors, which is brought into nonequilibrium by a temperature gradient and a phase difference. We analyze the interplay of the superconducting correlations induced on the quantum dot, the proximity effect, the strong Coulomb interaction, and the nonequilibrium dynamics. To describe these properties, we use an extension of the real-time diagrammatics. This can be used to describe sequential tunneling and the proximity effect, which has its origin in Andreev processes. In this system, an explicit phase dependence of the charge and heat flux already occurs in the stationary non-equilibrium, the measurement of which allows an experimental access to the coherences on the quantum dot. Also, a thermoelectric effect is revealed which persists even directly at the point of particle-hole symmetry as it is broken by renormalization of energy levels. Due to the temperature dependence of the superconductor band gap, the system can additionally be realized as a highly effective heat diode. In non-equilibrium, a superconductor pair amplitude is induced on the quantum dot by the proximity effect. We can understand its nonstationary dynamics in analogy to the dynamics of the order parameter in the bulk superconductor. In the case of an instantaneous change of the system (quench), there is a short-lived build-up of the pair amplitude. However, the following exponential decay does not allow observation of coherent oscillations. However, if individual parameters are periodically modulated, oscillations of the magnitude and phase of the pair amplitude occur, which can be understood as an analogy to the Higgs mode or Nambu-Goldstone mode in the bulk superconductor.