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An epitaxial semiconductor-superconductor nanowire is a superconductingsystem with a complex level structure originating from hybridization: inaddition to a dense set of higher-energy states derived predominantly from themetallic superconducting shell above the bulk gap $\Delta$, there is a smallernumber of lower-energy proximitized states from the semiconducting core thatdefine the induced gap $\Delta^*$. Nanostructures built from such nanowires canfurthermore incorporate quantum dots in order to obtain localized spins forstoring and manipulating quantum information. We discuss the magnetic fielddependence in three devices with different combinations of embedded quantumdots and superconducting islands. For strong fields, they show pinning ofexcitation energies to a uniform spacing, even if for weak fields they havenon-universal properties with different behaviors for even and odd number ofconfined electrons. We propose a quantum impurity model for hybrid devices thatincorporates all relevant physical effects. We show that the model accounts forthe key observations and permits unambiguous interpretation in terms ofmany-particle states. In particular, we study the replicas of theYu-Shiba-Rusinov states in the hybrid gap, their collapse and oscillationaround zero bias with increasing field, and the strong smoothing effect of thespin-orbit coupling (SOC) on these oscillations. We propose that theSOC-induced mixing of many-body states is a generic mechanism and that magneticpinning is likely to be a ubiquitous feature in hybrid nanowires.

Richardson model with complex level structure and spin-orbit coupling  for hybrid superconducting islands: stepwise suppression of pairing and  magnetic pinning