本研究利用光子波動力學(PWM)的理論框架，模擬Hong-Ou-Mandel (HOM) 干涉現象。在此，不同於量子場論的創生、湮滅算符表述，光子態得以由光子波函數依E+icB的複數形式精確地描述，其所遵循的運動方程式正好是古典電磁學中的Maxwell方程，因而適合以FDTD進行數值計算。
　　光子波函數的機率由非定域積分的形式給出。此形式要求孿生模態的引入，才能和波包模態構成雙正交基底系統，從而具有Lorentz不變性。因此，其係數平方被詮釋為定域能量密度。
　　二維分光器設計為漸逝波耦合的機制，透過數值計算，√3的介質折射率確保了不同的入射模態能有同樣的分光效果。
　　經過模擬後，所得結果與HOM 實驗並不相同。有趣的是，其下部包絡符合HOM dip的形狀。看來PWM的理論旨趣並不僅止於理論的完備，或是模擬的需求，它更有助於我們更深地理解基本粒子的特性。

This work simulates the Hong-Ou-Mandel (HOM) interference phenomenon by using the theoretical framework of the photon wave mechanics (PWM). Here, comparing with the creation operator and annihilation operator formalism of the quantum field theory (QFT), photon wave function (PWF) describes the photon state explicitly in the E+icB form, and the equation of motion obeyed is just the Maxwell’s equations in the classical electromagnetics. Therefore, the finite-difference time domain method is suitable for numerical calculations. The probability of the photon wave function is given in the form of non-local integral. This form requires the introduction of dual modes to form the biorthogonal basis system with the wave-packet mode, thus possessing the Lorentz-invariance. Therefore, their modulus square is interpreted as the local energy density. The two-dimensional beam splitter is designed in the mechanism of evanescent wave coupling. Through the numerical calculation, the medium refractive index of √3 ensures that the splitting effects of different incident modes are the same. After the simulation, the result is different from HOM experiment. But interestingly, its lower envelope matches the shape of the HOM dip. This work shows that the theoretical interest of PWM is not only for the completeness of theories or the demand of simulations, but also helping us to understand more deeply the properties of fundamental particles.

[1] N. Bohr, “The Quantum Postulate and Recent Development of Atomic Theory,” Nature, 121 (3050), 580 (1928)
[2] C. Jönsson, “Elektroneninterferenzen an mehreren künstlich hergestellten Feinspalten,” Z. Phys. 161, 454 (1961)
[3] G. I. Taylor, Proc. Camb. Philos. Soc. 15, 114 (1909)
[4] P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed., Oxford U. Press (1974), p.9
[5] C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044 (1987)
[6] D. C. Burnham and D. L. Weinberg, Phys. Rev. Lett. 25, 84 (1970)
[7] R. J. Glauber, “Coherent and Incoherent States of the Radiation Field”, Phys. Rev. 131, 2766 (1963)
[8] C. Gerry and P. Knight, Introductory Quantum Optics, Cambridge University Press, New York, (2005)
[9] Brańczyk, A. M. “Hong-Ou-Mandel Interference,” arXiv:quant-ph/1711.00080v1 (2017)
[10] B. J. Smith and M. G. Raymer, New J. Phys. 9, 414 (2007)
[11] P. A. M. Dirac, “The quantum theory of the electron,” Proc. R. Soc. Lond. A 117, 610 (1928)
[12] P. A. M. Dirac, “The quantum theory of the electron II,” Proc. R. Soc. Lond. A 118, 351 (1928)
[13] L. D. Landau and R.J. Peierls, “Quantenelektrodynamik im konfigurationsraum,” Z. Phys. A 62, 188 (1930)
[14] T. D. Newton and E. P. Wigner, “Localized states for elementary systems,” Rev. Mod. Phys. 21, 400 (1949)
[15] J. E. Sipe, “Photon wave functions,” Phys. Rev. A 52, 1875 (1995)
[16] M. V. Fedorov, M.A. Efremov, A. E. Kazakov, K. W. Chan, C. K. Law and J. H. Eberly, “Spontaneous emission of a photon: wave-packet structures and atom-photon entanglement,” Phys. Rev. A 72, 032110 (2005)
[17] I. Bialynicki-Birula, “On the wave function of the photon,” Acta Phys. Pol. A 86, 97 (1994)
[18] I. Bialynicki-Birula, “On the photon wave function,” Coherence and Quantum Optics VII (New York: Plenum) (1995), p.313
[19] I. Bialynicki-Birula, “Photon wave function,” Progress in Optics XXXVI (1996), p.245
[20] U. M. Titulaer and R. J. Glauber, “Density operators for coherent fields,” Phys. Rev. 145, 1041 (1966)
[21] Umran S. Inan and Aziz Inan, Electromagnetic Waves, Prentice Hall, New Jersey (2000)
[22] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method 3rd Ed., Artech House, June (2005)
[23] K. Umashankar, A. Taflove, “A Novel Method to Analyze Electromagnetic Scattering of Complex Objects”, IEEE Trans. Electrom. Compat., 24, 397 (1982)
[24] R. Lopes, A. Imanaliev, A. Aspect, M. Cheneau, D. Boiron and C. I. Westbrook, “Atomic Hong–Ou–Mandel experiment Nature,” 520, 7547 (2015)