We experimentally study the evolution of photon-number states of a microwave field stored in a very high finesse Fabry-Perot cavity. To probe the field, we send one by one circular Rydberg atoms which interact dispersively with the cavity mode. Using the Ramsey interferometry technique, we perform a quantum non-demolition measurement of the photon number n. The standard estimate of the field’s state at time t is based on the analysis of all measurements performed before this time.
In this manuscript, we present the implementation of the “past quantum state” analysis which uses the complete ensemble of the probes detection results obtained both before and after time t. The past state is a better estimate of the real field state than the density matrix. The almost noiseless reconstructed quantum trajectories make the resolution of quantum jumps of the field more efficient. Furthermore, the past state method allows us to remove the main restriction of our QND measurement related to its interferometric periodicity. We are now able to observe photon number states with n>8, which were inaccessible with the standard analysis, and measure their decay rates.