A quantum theory for RF reflectometry measurement of a semiconductor qubit

Radio frequency (RF) gate reflectometry is a promising tool in semiconductor qubit readout when it comes to CMOS compatibility, scalability, and bandwidth. In this measurement approach, an RF pulse is sent to a gate electrode controlling the qubit, which senses the change in the impedance of the system and measured by either a shift in frequency or a change in phase of a reflected signal. Here we develop a quantum theory for RF reflectometry using quantum Langevin equation and input-output formalism. We calculate a reflected signal based on system parameters such as the resonator detuning, leakage rate, and qubit relaxation rate. We also explore how an additional resonator, either in series or parallel, and may also be coupled to the qubit, could influence a reflected signal.