Using a qubit to measure photon-number statistics of a driven thermal oscillator

We demonstrate theoretically how photon-number statistics of a driven damped oscillator at finite temperature can be extracted by measuring the dephasing spectrum of a two-level system dispersively coupled to the oscillator; we thus extend the work of M. I. Dykman and M. A. Krivoglaz Sov. Phys. Solid State 29 210 (1987) and J. Gambetta et al. Phys. Rev. A 74 042318 (2006). We carefully consider the fidelity of this scheme—to what extent does the measurement reflect the initial number statistics of the mode? We also derive analytic results for the dephasing of a qubit by a driven thermal mode, and compare results obtained at different levels of approximation. Our results have relevance both to experiments in circuit cavity QED, as well as to nanoelectromechanical systems.