Josephson junctions with charging energy

Superconducting (SC) islands coupled to semiconducting quantum dots are important building blocks of hybrid quantum devices, including superconducting qubits. To realistically model microscopic SC islands, their charging energy has to be included in the description. This is at odds with the standard BCS description where charge is not well defined. We developed a charge conserving model for quantum dots coupled to SC islands with charging energy and found that its competition with superconductivity qualitatively changes the nature of the subgap states. This understanding is important as modern devices are typically small enough that the effects of charging energy should not be neglected. The model is successful at describing spectra of InAs nanowires with Al SC islands.

Here we extend our model for a description of a Josephson junction with an embedded quantum dot, a model applicable to various realizations of superconducting qubits. Charging energy changes the superconducting state by collapsing the superposition with a well defined phase difference across the junction into a state with well defined charge difference. We investigate the nature of low energy states, particularly focusing on the effect of phase collapse with increasing charging energy.