The origin of Franson-type nonlocal correlation

Quantum entanglement is known as a weird phenomenon that cannot be explained by classical physics or achieved by any classical means.¹ Ever since the well-known thought experiment by Einstein, Podolsky, and Rosen (EPR) in 1935,² EPR has been a key aspect of quantum information science and technologies in computing,^3–5 communications,^6–9 and sensing areas.^10–12 Although Bell had mathematically demonstrated the so-called EPR paradox in 1964, the definition of classical physics rules out coherence optics.¹³ Since then, mutual coherence between paired photons has not been carefully considered.^13–26 Although the energy-time uncertainty relation prohibits definite phase information for a single photon, a mutual phase between entangled photons is free from such uncertainty relation without violating quantum mechanics. Here, we focus on the mutual coherence between interacting photons in Franson-type nonlocal correlation to understand the mysterious quantum feature. The classical understanding stands for a definite and clear reasoning-based physical nature.

Franson-type nonlocal correlation¹⁹ has been studied since 1987.^20–26 Unlike Bell inequality violations,^13–18 Franson nonlocal correlation is based on a set of unbalanced Mach–Zehnder interferometers (U-MZIs).^19–26 The U-MZI is designed with respect to interacting photons' bandwidth for both local randomness in each detector and nonlocal correlation fringe in coincidence detection. The local randomness-based nonlocal fringe is a unique feature of the Franson correlation. Although the mathematical form of the nonlocal correlation is clear to view the fringe, the physical understanding of the nonlocal fringe has been severely limited. Thus, the fringe in Franson nonlocal correlation has been left as a mysterious quantum feature due to the U-MZI resulting local randomness. Here, the origin of Franson nonlocal correlation is investigated to clear out the weirdness of nonlocal fringe based on local randomness. For this, we analyze the relation between U-MZI and entangled photon pairs generated from the spontaneous parametric down conversion (SPDC) process.^27–31 Specifically, the coherence relation between each entangled photon pair and the U-MZI is focused on for coincidence detection. From this understanding, a coherence version of the Franson correlation is proposed for macroscopic quantum information compatible with current technologies based on coherence optics.

Figure 1 shows the original Franson scheme for nonlocal correlation using entangled photon pairs generated from SPDC processes.²⁰ As studied²⁹ and applied for quantum key distributions,^24–26 the coincidence measurements between two remotely separated output photons show a path-length difference-dependent fringe, even though their local measurements do not. This nonlocal fringe for the second-order intensity correlation looks exactly the same as the first-order intensity correlation in a typical double-slit case. Regarding the U-MZIs in Fig. 1(a), whose entangled photon source S is depicted in Figs. 1(b) and 1(c), the nonlocal correlation fringe due to the coincidence measurements implies that each U-MZI may act as a coherence system for individual photon pairs. Considering this, the coherence time $τ c$ of each entangled photon pair from the SPDC should be much longer than the path-length difference $δL L − S$ of each U-MZI. This coherence relation can be achieved by choosing a narrow bandwidth pump laser (p) in Fig. 1(c) according to $χ ( 2 )$ nonlinear optics of SPDC.^20,30,31 As explained below in Eq. (11), the coherence washout among the broadband photon pairs is suppressed by coincidence measurements. Due to the ultrawide bandwidth $(Δ)$ of the entangled photon ensemble in Fig. 1(b), however, each U-MZI in Fig. 1(a) acts as a noninterfering interferometer for local measurements, resulting in the path-length independent uniform intensity. Thus, both local and nonlocal measurements are easily understood by many-wave interference in coherence optics.

Figure 1(c) shows schematic of the type I SPDC for the same polarization of entangled photon pairs. Due to the phase matching condition of SPDC governed by both energy and momentum conservation laws,^30,31 the signal (s) and idler (i) photons are interchangeable, satisfying an entangled state, $| ψ= 1 2 | s 1 | i 2 + | i 1 | s 2$⁠, where the subscripts indicate concentric circles. Thus, the $f 0$ in Fig. 1(b) is the half of the pump photon frequency, whose spectral width is ultranarrow compared with the generated photon bandwidth $Δ$⁠. Here, having a narrow linewidth pump laser is essential for the higher efficiency of nonlocal fringe visibility via stable concentric ring patterns in Fig. 1(c) according to the momentum conservation law.^20,31,32 According to the phase matching in SPDC nonlinear optics,³¹ the paired (signal and idler) photons satisfy symmetric frequency detuning (⁠ $δ f j$⁠) across the center frequency $f 0$⁠, resulting in a $±δ f j$ relation with $2 f 0= f s+ f i$ according to the energy conservation law.³¹ 

Over the last several decades, Franson-type nonlocal correlation has been applied to quantum key distributions based on the energy-time bin method using single photon-correlated entangled pairs.^24–26 Because this nonlocal phenomenon is successfully explained by coherence optics for coincidence measurements in Eq. (12), a coherence version of the Franson-type nonlocal correlation can be considered as well. In quantum mechanics, the believed thumb rule is that nonlocal correlation cannot be achieved by any classical means. Thus, the present analysis provides some insight into the quantum nature.

Figure 2 shows a coherence version of Fig. 1, where the entangled photon pairs are replaced by coherent photons from a commercial laser with some modifications. Unlike the SPDC-generated entangled photon pairs, the spectral bandwidth $Δ l$ of each coherent photon is the same as the laser linewidth according to cavity optics.³³ The function of an optical cavity is for spectral filtering of wide-bandwidth distributed photons generated from a gain medium. To satisfy both temporal and spectral integrations for nonlocal and local measurements, respectively, the laser-generated photons are modified to be inhomogeneous for each U-MZI in Fig. 2. This inhomogeneity of coherent photons results in the same function of overall decoherence in ensemble measurements of SPDC. Such spectral widening of coherent photons can be accomplished by laser scanning, such as in frequency modulation of continuous waves³⁴ or dc Stark effects.³⁵ 

Franson-type nonlocal correlation was analyzed and discussed using the wave nature of photons for both local randomness and nonlocal correlation fringe. For this, the original unbalanced MZI (U-MZI) is investigated with respect to the characteristics of entangled photons generated from type I SPDC. From this, the locally measured uniform intensity in each detector was analyzed as a dephasing effect due to many-wave interference of spectrally broadened photons. On the contrary, the U-MZI path-length dependent intensity fringe for the nonlocal correlation was analyzed as a coherence feature of each photon pair in the U-MZI, where the U-MZI is designed to be coherent with respect to each photon pair. This coherence condition is satisfied with a narrow-bandwidth pump laser via coincidence detection. Unlike local measurements in time averaging, resulting in overall decoherence, the nonlocal measurements require coincidence detection between two remotely separated local measurements. This coincidence detection is key to understand the nonlocal fringe due to ruling out the time average effect for all spectral bandwidth entangled photons. Thus, the origin of both local randomness and nonlocal Franson correlation was found in coherence optics, where the nonlocal property is due to measurement filtering via coincidence detections. Based on this understanding, a coherence version of Franson nonlocal correlation was proposed using spectral modification of a commercially available laser light for the local randomness. The nonlocal correlation was achieved using a pair of acousto-optic modulators mimicking the SPDC-generated signal and idler photon pairs in symmetric frequency detuning.

This work was supported by the ICT R&D program of MSIT/IITP (No. 2021-0-01810) via Development of Elemental Technologies for ultra-secure Quantum Internet.

The authors have no conflicts to disclose.

B.S.H. solely wrote the manuscript.

Data sharing is not applicable to this article as no new data were created or analyzed in this study.