Quantum dynamics of capacitively and inductively shunted quantum dot Josephson junctions: charging effects, boundary conditions, and all that

For a single-channel superconductor-semiconductor quantum dot-superconductor junction, starting from a many-body quantum formalism in a path integral formulation [1] we obtain an effective single-particle Hamiltonian of the phase difference which reflects the presence of two subgap Andreev bound states. The Josephson-like energy term recovers previous results [2], however with the phase being a quantum variable. The charging energy obtains a capacitance renormalization and new charge offsets that are functions of the superconducting (SC) gaps in the leads, the hopping rates from leads to the dot, and the dot’s gate voltage. These charging effects are maximized for equal SC gaps in the two leads, and increase considerably for large gate voltages approaching the gap, and for asymmetric hopping. For a gatemon we derived new boundary conditions generalizing the 2π periodicity of a transmon for each of the four parity sectors: the odd/even parity of the dot occupation and the odd/even parity of the junction charge ("transmon parity"). We also discuss the gatemon anharmonicity to leading order in the expansion parameters.

For a fluxonium-like circuit (i.e., shunting the gatemon with an inductor) we rederive from the many-body theory the relevant effective Hamiltonian. We show that the capacitance renormalization is not altered in the leading order of the slow phase expansion. Finally, we comment on the boundary conditions in this case in order to directly compare to the ordinary Josephson junction.

[1] U. Güngördü et al., arXiv:2402.10330 (2024)

[2] P. D. Kurilovich et al., Phys. Rev. B 104, 174517 (2021)