Photon echo using imperfect X-ray pulse with phase fluctuation

We study the impact of inter-pulse phase fluctuation in free-electron X-ray laser on the signal in the photon echo spectroscopy, which is one of the simplest non-linear spectroscopic methods. A two-pulse echo model is considered with two-level atoms as the sample. The effect of both fluctuation amplitude and correlation strength of the random phase fluctuation is studied both numerically and analytically. We show that the random phase effect only affects the amplitude of the photon echo, yet not change the recovering time. Such random phase induces the fluctuation of recovering amplitude in the photon echo signals among different measurements. We show the normal method of measuring coherence time retains by averaging across the signals in different repeats in current paper.