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; previous results only dealt with the purely thermal or zero-temperature driven cases [1][2]. We also consider the fidelity of this scheme-- to what extent does the measurement reflect the initial number statistics of the mode? Finally, we make a connection to the theory of full counting statistics in mesoscopic physics. Our results have relevance both to experiments in circuit cavity QED, as well as with quantum nano-electromechanical systems. \newline \newline [1] M.I. Dykman and M.A. Krivoglaz, Sov. Phys. Solid State 29, 210, (1987). \newline [2] J. Gambetta et al, Phys. Rev. A 74, 042318 (2006).