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.