The zeroing of second order correlation functions between output fields after interferences in a 50/50 beam splitter has been accepted decades-long in the quantum optics community as an indicator of the quantum nature of lights. But, a recent work  presented some notable discussions and experiments that classical electromagnetic fields can still exhibit the zero correlation under specific conditions. Here, we examine analytically classical and quantum electromagnetic field interferences in a 50/50 beam splitter in the context of the second order correlation function for various input conditions. Adopting the Heisenberg picture in quantum electromagnetics, we examine components of four-term interference terms in the numerator of second order correlation functions and elucidate their physical significance. As such, we reveal the fundamental difference between the classical and quantum interference as illustrated by the Hong-Ou-Mandel (HOM) effect. The quantum HOM effect is strongly associated with: (1) the commutator relation that does not have a classical analogue; (2) the property of Fock states needed to stipulate the one-photon quantum state of the system; and (3) a destructive wave interference effect. Here, (1) and (2) imply the indivisibility of a photon. On the contrary, the classical HOM effect requires the presence of two destructive wave interferences without the need to stipulate a quantum state.
1. Sadana, S., D. Ghosh, K. Joarder, A. N. Lakshmi, B. C. Sanders, and U. Sinha, "Near-100% two-photon-like coincidence-visibility dip with classical light and the role of complementarity," Phys. Rev. A, Vol. 100, 013839, 2019. doi:10.1103/PhysRevA.100.013839
12. Kaltenbaek, R., B. Blauensteiner, M. Zukowski, M. Aspelmeyer, and A. Zeilinger, "Experimental interference of independent photons," Phys. Rev. Lett., Vol. 96, 240502, Jun. 2006. doi:10.1103/PhysRevLett.96.240502
13. Prasad, S., M. O. Scully, and W. Martienssen, "A quantum description of the beam splitter," Optics Communications, Vol. 62, No. 3, 139-145, 1987. doi:10.1016/0030-4018(87)90015-0
15. Ham, B. S., "The origin of anticorrelation for photon bunching on a beam splitter," Scientific Reports, Vol. 10, 7309, 2020. doi:10.1038/s41598-020-64441-2
16. Branczyk, A. M., "Hong-Ou-Mandel interference,", arXiv:1711.00080, 2017.
17. Di Martino, G., Y. Sonnefraud, M. S. Tame, S. Kena-Cohen, F. Dieleman, K. Ozdemir, M. S. Kim, and S. A. Maier, "Observation of quantum interference in the plasmonic Hong-Ou-Mandel effect," Phys. Rev. Applied, Vol. 1, 034004, 2014. doi:10.1103/PhysRevApplied.1.034004
18. Longo, P., J. H. Cole, and K. Busch, "The Hong-Ou-Mandel effect in the context of few-photon scattering," Opt. Express, Vol. 20, 12 326-12 340, 2012. doi:10.1364/OE.20.012326
19. Lang, C., C. Eichler, L. Steffen, J. M. Fink, M. J. Woolley, A. Blais, and A. Wallraff, "Correlations, indistinguishability and entanglement in Hong-Ou-Mandel experiments at microwave frequencies," Nature Physics, Vol. 9, 345-348, 2013. doi:10.1038/nphys2612
20. Lopes, R., A. Imanaliev, A. Aspect, M. Cheneau, D. Boiron, and C. I. Westbrook, "Atomic Hong-Ou-Mandel experiment," Nature, Vol. 520, 66-68, 2015. doi:10.1038/nature14331
21. Kobayashi, T., R. Ikuta, S. Yasui, S. Miki, T. Yamashita, H. Terai, T. Yamamoto, M. Koashi, and N. Imoto, "Frequency-domain Hong-Ou-Mandel interference," Nature Photonics, Vol. 10, 441-444, 2016. doi:10.1038/nphoton.2016.74
22. Imany, P., O. D. Odele, M. S. Alshaykh, H.-H. Lu, D. E. Leaird, and A. M. Weiner, "Frequency-domain Hong-Ou-Mandel interference with linear optics," Opt. Lett., Vol. 43, No. 12, 2760-2763, 2018. doi:10.1364/OL.43.002760
23. Rohde, P. P. and T. C. Ralph, "Frequency and temporal effects in linear optical quantum computing," Phys. Rev. A, Vol. 71, 032320, 2005. doi:10.1103/PhysRevA.71.032320
24. Rohde, P. P., T. C. Ralph, and M. A. Nielsen, "Optimal photons for quantum-information processing," Phys. Rev. A, Vol. 72, 052332, 2005. doi:10.1103/PhysRevA.72.052332
25. Mahrlein, S., S. Oppel, R. Wiegner, and J. von Zanthier, "Hong-Ou-Mandel interference without beam splitters," Journal of Modern Optics, Vol. 64, 921-929, 2017. doi:10.1080/09500340.2016.1242790
26. Kim, M. S., W. Son, V. Buzek, and P. L. Knight, "Entanglement by a beam splitter: Nonclassicality as a prerequisite for entanglement," Phys. Rev. A, Vol. 65, 032323, 2002. doi:10.1103/PhysRevA.65.032323
27. Walschaers, M., "Signatures of many-particle interference," Journal of Physics B: Atomic, Molecular and Optical Physics, Vol. 53, 043001, 2020. doi:10.1088/1361-6455/ab5c30
28. Deng, Y.-H., H. Wang, X. Ding, Z.-C. Duan, J. Qin, M.-C. Chen, Y. He, Y.-M. He, J.-P. Li, Y.- H. Li, L.-C. Peng, E. S. Matekole, T. Byrnes, C. Schneider, M. Kamp, D.-W. Wang, J. P. Dowling, S. Hofling, C.-Y. Lu, M. O. Scully, and J.-W. Pan, "Quantum interference between light sources separated by 150 million kilometers," Phys. Rev. Lett., Vol. 123, 080401, 2019. doi:10.1103/PhysRevLett.123.080401
29. Chew, W. C., Waves and Fields in Inhomogeneous Media, Van Nostrand, 1990.
30. Electromagnetic field theory, Lecture Notes for ECE 604 at Purdue U, 2020, https://engineering.purdue.edu/wcchew/ece604s20/EMFTAll.pdf.
31. Na, D.-Y., J. Zhu, F. L. Teixeira, and W. C. Chew, "Quantum information propagation preserving computational electromagnetics,", arXiv preprint arXiv:1911.00947, 2019.
32. Na, D.-Y., J. Zhu, W. C. Chew, and F. L. Teixeira, "Quantum information preserving computational electromagnetics," Phys. Rev. A, Vol. 102, No. 1, 013711, Jul. 2020. doi:10.1103/PhysRevA.102.013711
33. Chew, W. C., A. Y. Liu, C. Salazar-Lazaro, and W. E. I. Sha, "Quantum electromagnetics: A new look — Part I and Part II," J. Multiscale and Multiphys. Comput. Techn., Vol. 1, 73-97, 2016. doi:10.1109/JMMCT.2016.2617018
34. Kirk, D. E., Optimal Control Theory: An Introduction, Courier Corporation, 2004.
35. Schrodinger, E., "An undulatory theory of the mechanics of atoms and molecules," Phys. Rev., Vol. 28, No. 6, 1049, 1926. doi:10.1103/PhysRev.28.1049
36. Chew, W., A. Liu, C. Salazar-Lazaro, D.-Y. Na, and W. Sha, "Hamilton equations, commutator, and energy conservation," Quantum Reports, Vol. 1, 295-303, 2019. doi:10.3390/quantum1020027
37. Louisell, W. H. and W. H. Louisell, Quantum Statistical Properties of Radiation, Vol. 7, Wiley, New York, 1973.
38. Haken, H., Quantum Field Theory of Solids, an Introduction, North-Holland, 1976.
39. Cohen-Tannoudji, C., J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions: Basic Processes and Applications, Wiley-VCH, New York, NY, USA, 1988.
40. Vogel, W. and D.-G. Welsch, "Quantum Optics," John Wiley & Sons, 2006.
41. Walls, D. F. and G. J. Milburn, Quantum Optics, Springer Science & Business Media, 2007.
42. Scheel, S. and S. Y. Buhmann, "Macroscopic quantum electrodynamics," Acta Physica Slovaca, Vol. 58, 675-809, 2008.