Understanding of coincidence detection in Franson-type nonlocal correlations for second-order quantum superposition

Coincidence detection is a key technique used in nonlocal quantum-correlation measurements to test Bell’s inequality violation between remotely separated local detectors. With individual uniform intensity of local measurements, the nonlocal correlation fringe is a mysterious quantum feature that cannot be achieved classically. Here, the coincidence detection is coherently investigated to understand the fundamental physics of the nonlocal correlation fringe via second-order quantum superposition between selected nonlocal measurement events. Because of the coherence feature of paired photons, the coincidence technique modifies the measurement events for the rule of thumb of indistinguishability between selected measurement bases of paired photons. This indistinguishability is quantum superposition between nonlocally detected events resulting from a selected time slot of coincidence, where coherence between individually measured photons is an absolute condition. Introduction The essential components of quantum technologies are quantum superposition and quantum entanglement [1,2]. Quantum superposition is normally understood as self-interference of a single photon in an interferometric system [3-7]. Quantum entanglement is between two correlated entities via quantum superposition with inseparable intensity product combinations [8-20]. Bell inequality violation has been intensively studied to oppose the local realism based on a hidden variable theory [8-14]. Nonlocal correlation fringe in a Franson scheme has been applied to loophole-free quantum communications, where the nonlocal correlation fringe has been left as a mysterious quantum feature because of the uniform intensity in individual and local measurements [15-20]. Although nonlocal quantum correlation has been vastly applied for various quantum technologies, including quantum key distributions [21-25], understanding the fundamental physics of the nonlocal phenomenon has been limited, where the particle nature-based interpretation does not provide a clear view of the weirdness. In this study, a pure classical approach of coherence optics is used to investigate the mystery of nonlocal correlation fringe in a Franson scheme of Bell inequality violation. For the light source, typical entangled photon pairs from a spontaneous parametric down conversion (SPDC) process are used to analyze both local and nonlocal correlations [26,27]. For the Franson-type nonlocal correlation fringe, a general method of coincidence detection is investigated for coherence optics-caused path-basis superposition via a selected measurement process [25]. This nonlocally selected measurement process results in path-basis superposition, satisfying indistinguishability between locally measured individual event products in a specified time slot of coincidence. Thus, the coincidence detection technique plays as a major role in second-order quantum superposition between locally measured events, resulting in the nonlocal correlation fringe, otherwise satisfying a classical intensity product between local detections. Local randomness versus decoherence of entangled photon pairs In the typical Franson scheme of Fig. 1(a), an ensemble of entangled photon pairs is provided by the second-order nonlinear optics-based SPDC processes [27]. The symbol S indicates the entangled photon pair source that fires paired photons in opposite directions to satisfy the momentum conservation law. Each paired photon from S enters a corresponding noninterfering Mach-Zehnder interferometer (NMZI) comprising short (S) and long (L) paths that are simultaneously measured by both detectors in each party, Alice or Bob. For nonlocal correlation measurements, each pair of detectors in both parties is jointly measured via coincidence detection. The path-length difference ΔL (L − S) of the NMZI is set to ΔL ≫ cΔff−1, where ∆ff is the full-width-at-half-maximum of the