  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).
