Measurement of small photon numbers in circuit QED resonators

Off-resonant interaction of fluctuating photons in a resonator with a qubit increases the qubit dephasing rate. We use this effect to measure a small average number of intracavity photons that are coherently or thermally driven. For spectral resolution, we do this by subjecting the qubit to a Carr-Purcell-Meiboom-Gill (CPMG) sequence and record the qubit dephasing rate for various periods between qubit $\pi$-pulses. The recorded data is then analyzed with formulas for the photon-induced dephasing rate that we have derived for the non-Gaussian noise regime with an arbitrary ratio $2\chi/\kappa$, where $2\chi$ is the qubit frequency shift due to a single photon and $\kappa$ is the resonator decay rate. We show that the presented CPMG dephasing rate formulas agree well with experimental results and demonstrate measurement of thermal and coherent photon populations at the level of a few $10^{-4}$.