Quantum charge fluctuations of a proximitized nanowire

Motivated by a recent experiment [Nature (London) 531, 206 (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 $e{\mathcal{N}}_g/C$. A finite conductance of the contact allows for quantum charge fluctuations, making the function $Q({\mathcal{N}}_g)$ continuous. It depends on the relation between the superconducting gap $\mathrm{\ensuremath{\Delta}}$ and the effective charging energy ${E}_C^*$. The latter is determined by the junction conductance in addition to the geometrical capacitance of the proximitized nanowire. We investigate $Q({\mathcal{N}}_g)$ at zero magnetic field $B$ and at fields exceeding the critical value ${B}_c$ corresponding to the topological phase transition [Phys. Rev. Lett. 105, 077001 (2010); Phys. Rev. Lett. 105, 177002 (2010)]. Unlike the case of $\mathrm{\ensuremath{\Delta}}=0$, the function $Q({\mathcal{N}}_g)$ is analytic even in the limit of negligible level spacing in the nanowire. At $B=0$ and $\mathrm{\ensuremath{\Delta}}>{E}_C^*$, the maxima of $dQ/d{\mathcal{N}}_g$ are smeared by $2e$ fluctuations described by a single-channel ``charge Kondo'' physics, whereas the $B=0,\phantom{\rule{0.28em}{0ex}}\mathrm{\ensuremath{\Delta}}<{E}_C^*$ case is described by a crossover between the Kondo and the mixed-valence regimes of the Anderson impurity model. In the topological phase, $Q({\mathcal{N}}_g)$ is an analytic function of the gate voltage with $e$-periodic steps. In the weak-tunneling limit, $dQ/d{\mathcal{N}}_g$ has peaks corresponding to Breit-Wigner resonances, whereas in the strong-tunneling limit (i.e., small reflection amplitude $r$) these resonances are broadened, and $dQ/d{\mathcal{N}}_g\ensuremath{-}e\ensuremath{\propto}r\phantom{\rule{0.16em}{0ex}}cos(2\ensuremath{\pi}{\mathcal{N}}_g)$.