中文摘要
本研究的主要目的為利用光子的波函數描述洪-歐-孟德爾效應(Hong-Ou- Mandel effect)在量子光學中的現象。在洪-歐-孟德爾效應的實驗中，若有兩個不可分辨的光子通過一個50% / 50%分光鏡(beam splitter)，則結果顯示這兩個光子只會同時在其中一邊一起出現並且被偵測到，而各邊各出現一個光子的可能性為0。
光子的波方程式可由Einstein的能量與動量方程式得出，若將光子的波函數指定為E + icB，如此便可得出Maxwell equations。在量子光學中，光子於k-ω頻域空間的單頻平面波模態，可透過適當的basis映至r-t時空間的波包模態。另外為了恰當地定義機率，我們引進dual mode，使其波函數的機率有正確的物理意義即為光子的能量密度。
在50% / 50%分光鏡的設計上，我們找出此介質的折射率需為√2+1，且厚度為1/4波長的倍數。對此分光鏡，古典下的電磁場，往分光鏡兩邊出來的場有相同的強度。若藉由正確之真空態(vacuum state)的數學表示，並利用光子的波函數在時間空間下傳遞通過分光鏡，利用與符合機率定義的dual mode來定義光子出現的機率，我們模擬得到類似洪-歐-孟德爾效應的結果。

SUMMARY
In this work, we propose to use photon wave function to simulate the quantum optics phenomena of Hong-Ou-Mandel effect. Hong-Ou-Mandel effect is that when two identical photons entering a 50/50 beam splitter, only two photons will be detected in either side, and there is no probability of detecting one photon on each side. The photon wave equation is derived from Einstein energy momentum equation. The photon wave function is chosen to be E+icB such that the photon wave equation reduced to the Maxwell equations. We derive the proper basis for the photon mode in x-t real space and time from the quantum optics photon mode in k-ω space. To have proper probability definition, we need to define a dual mode to calculate the photon wave function probability which is regarded as the photon energy. By designing a 50/50 beam splitter from a dielectric slab, we found the refractive index of the medium is √2+1, and the thickness should be a multiple of quarter wavelength. For the classical field, this beam splitter will have same intensity coming out of two output side. By using the photon wave function and its proper dual mode probability definition, we simulate the Hong-Ou-Mandel like effect.

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