Capacitance-based Fermion parity read-out and predicted Rabi oscillations in a Majorana nanowire

Recent experiments have measured flux dependent capacitance at radio frequencies leading to the potential for a fast parity read-out of a Majorana qubit. In this work we argue that the quantum dot used in the capacitance measurement can be reasonably approximated by a non-interacting weakly coupled orbital. We then predict the measured flux and parity dependent capacitance for several parameter regimes of the disordered Majorana nanowire model that are both topological and trivial. Following this we study how such a fast capacitance read-out can be used to characterize the quantum coherence of a Majorana nanowire-based qubit using Rabi oscillations. We additionally show that such measurements, if made possible by coherent inter-wire tunneling, would provide a valuable way of characterizing the low energy states in the frequency domain.