The origin of Franson-type nonlocal correlation

Franson-type nonlocal correlation is for the second-order intensity fringes measured between two remotely separated photons via coincidence detection, whereas their locally measured first-order intensities are uniform. This nonlocal intensity-product fringe shows a joint-phase relation of independent local parameters. Here, the Franson nonlocal correlation is investigated using a coherence approach based on the wave nature of quantum mechanics to understand the mysterious quantum feature of nonlocal fringes. For this, a typical Franson scheme based on entangled photon pairs is coherently analyzed for both local and nonlocal correlations, where the local intensities are due to many-wave interference between measured photos. For the nonlocal fringe, however, coincidence detection results in selective measurements, resulting in second-order amplitude superposition between locally measured photon basis products. Due to the intrinsic property of a fixed sum-phase relation between entangled photons in each pair, the joint-phase relation of the nonlocal fringe is immune to the random spectral detuning of photon pairs. As in the first-order amplitude superposition of a single photon self-interference, the second-order amplitude superposition between nonlocal basis-products is the origin of the nonlocal fringe.