Title: Quantum and Classical Two-photon Interference of Single Photons with Ultralong Coherence Time

Abstract: Two-photon interference (TPI) is a fundamental phenomenon in quantum optics and plays a crucial role in quantum information science and technology. TPI is commonly considered as quantum interference with an upper bound of $100\%$ for both the TPI visibility and the beat visibility in contrast to its classical counterpart with a maximum visibility of $50\%$. However, this is not always the case. Here we report a simultaneous observation of quantum and classical TPI of single photons with ultralong coherence time which is longer than the photon correlation time by five orders of magnitude. We observe a TPI visibility of $94.3\%\pm 0.2\%$ but a beat visibility of $50\%$. Besides an anti-bunching central dip due to single-photon statistics, we observe two bunching side peaks in cross-correlation curves for indistinguishable photons. Using either classical wave superposition theory or quantum field approach, we derive the same expressions for the cross-correlation functions which reproduce and explain the experiments well. We conclude that quantum TPI with a stream of single photons is equivalent to classical TPI, both of which are the fourth-order interference arising from the second-order interference occurring on the time scale of photon coherence time.