Charge sensitivity of a cavity-embedded Cooper pair transistor limited by single-photon shot noise

Using an operator scattering approach, we analyze the quantum dynamics of an ultrasensitive electrometer -- a Cooper pair transistor embedded in a quarter-wave microwave cavity (cCPT). While the cCPT is inherently a tunable, strongly nonlinear system affording a diverse range of functionalities, we restrict our present analysis to a necessary first investigation of its linear charge sensing capabilities, limiting to low pump powers corresponding to an average cavity photon number $\lesssim 1$. Assuming realizable cCPT parameters (B. L. Brock et al., Phys. Rev. Applied, 044009), and not including noise from the subsequent amplifier chain, we predict the fundamental, photon shot noise-limited charge sensitivity to be $0.12\, \mu{\mathrm{e}}/\sqrt{\mathrm{Hz}}$ when the pumped cavity has an average of one photon and the cCPT is operated close to charge degeneracy. This is to be compared with a first reported charge sensitivity value $14\, \mu{\mathrm{e}}/\sqrt{\mathrm{Hz}}$ in the single-photon regime (B. L. Brock et al., arXiv:2102.05362).