Indistinguishability of temporally separated pairwise two-photon state of thermal photons in Franson-type interferometry.

The phenomenon of Franson interference with time-energy entangled photon pairs beyond the single-photon coherence length observed upon nonlocal measurement at two space-like separated locations is of particular research interest. Herein, we determine the coherence length of temporally separated pairwise two-photon (TSPT) states of thermal photons emitted from a warm atomic ensemble in Franson-type interferometry, with the setup consisting of two spatially separated unbalanced Michelson interferometers beyond the coherence length of a thermal photon. Using a novel method of square-modulated thermal photons, we show that the sinusoidal Franson-type interference fringe of thermal photons is determined by the presence or absence of TSPT states (corresponding to the time delay between the long and short paths in Franson-type interferometry). We find that the indistinguishability of the TSPT state in the Franson-type interference is independent of the temporal separation of the thermal photons in the TSPT states.