Quantum charge fluctuations of a proximitized nanowire

Motivated by recent experiment by Albrecht et al., Nature (2016), we consider charging of a nanowire which is proximitized by a superconductor and connected to a normal-state lead by a single-channel junction. The charge $Q$ of the nanowire is controlled by gate voltage $e N_g/C$. A finite conductance of the contact allows for quantum charge fluctuations, making the function $Q(N_g)$ continuous. It depends on the relation between the superconducting gap $\Delta$ and the effective charging energy $E^*_C$. The latter is determined by the junction conductance, in addition to the geometrical capacitance of the nanowire. We investigate $Q(N_g)$ at zero magnetic field $B$, and at fields exceeding the critical value $B_c$ corresponding to the topological phase transition. Unlike the case of $\Delta = 0$, the function $Q(N_g)$ is analytic even in the limit of negligible level spacing in the nanowire. At $B=0$ and $\Delta>E^*_C$, the maxima of $dQ/d N_g$ are smeared by $2e$-fluctuations described by a single-channel ``charge Kondo'' physics, while the $B=0$, $\Delta