## 2 Two-Photon Interference in the Polarizing Beam Splitter
\begin{array}{lcl}\left|\psi_{\text{out}}\right\rangle&=&\left|c_{\text{V}}c_{ \text{V}}\right\rangle+\left|c_{\text{H}}c_{\text{H}}\right\rangle+\left|d_{ \text{V}}d_{\text{V}}\right\rangle+\left|d_{\text{H}}d_{\text{H}}\right\rangle +\left|c_{\text{V}}c_{\text{H}}\right\rangle\\ &&+\left|d_{\text{V}}d_{\text{H}}\right\rangle+\left|c_{\text{V}}d_{\text{V}} \right\rangle+\left|c_{\text{H}}d_{\text{H}}\right\rangle+\left|c_{\text{H}}d_ {\text{V}}\right\rangle+\left|c_{\text{V}}d_{\text{H}}\right\rangle,\end{array}  \tag{6}\]  
where, for example,  
\[\left|c_{\text{V}}d_{\text{V}}\right\rangle=\int\int\text{d}t_{1}\text{d}t_{2 }\xi_{\text{a}}(t_{1})\xi_{\text{b}}(t_{2})\sin\theta_{\text{a}}\sin\theta_{ \text{b}}\left(T^{2}\hat{c}_{V}^{\dagger}(t_{1})\hat{d}_{V}^{\dagger}(t_{2})+R ^{2}\hat{d}_{V}^{\dagger}(t_{1})\hat{c}_{V}^{\dagger}(t_{2})\right)\left|0\right\rangle \tag{7}\]  
corresponds to the outcome in which one photon is counted in detector \(c_{\text{V}}\) and one photon is counted in detector \(d_{\text{V}}\). The probability of the outcome \(c_{\text{V}}d_{\text{V}}\) is then  
\[P(c_{\text{V}}d_{\text{V}})=\langle c_{\text{V}}d_{\text{V}}|c_{\text{V}}d_{ \text{V}}\rangle. \tag{8}\]  
The dependence of the outcome probabilities on the photon wave functions can be expressed in terms of the overlap integral [16]  
\[\left|J\right|^{2}=\left|\int dt\xi_{\text{a}}^{*}\left(t\right)\xi_{\text{b}} \left(t\right)\right|^{2}. \tag{9}\]  
With \(\left|J\right|^{2}\) as defined in (9), the probabilities for each of the detection events are  
\[ \begin{array}{lcl}P(c_{\text{V}}c_{\text{V}})&=&P(d_{\text{V}}d_{\text{V}}) =\sin^{2}\theta_{\text{a}}\sin^{2}\theta_{\text{b}}\left|T\right|^{2}\left|R \right|^{2}\left(1+\left|J\right|^{2}\right)\\ P(c_{\text{H}}c_{\text{H}})&=&P(d_{\text{H}}d_{\text{H}})=\cos^{2}\theta_{ \text{a}}\cos^{2}\theta_{\text{b}}\left|T\right|^{2}\left|R\right|^{2}\left(1 +\left|J\right|^{2}\right)\\ P(c_{\text{V}}c_{\text{H}})&=&P(d_{\text{V}}d_{\text{H}})=\left|T\right|^{2} \left|R\right|^{2}\left(\sin^{2}\theta_{\text{a}}\cos^{2}\theta_{\text{b}}+ \cos^{2}\theta_{\text{a}}\sin^{2}\theta_{\text{b}}+2\sin\theta_{\text{a}}\cos \theta_{\text{a}}\sin\theta_{\text{b}}\cos\theta_{\text{b}}\left|J\right|^{2} \right)\\ P(c_{\text{V}}d_{\text{H}})&=&\sin^{2}\theta_{\text{a}}\cos^{2}\theta_{\text{b }}\left|T\right|^{4}+\cos^{2}\theta_{\text{a}}\sin^{2}\theta_{\text{b}}\left| R\right|^{4}-2\cos\theta_{\text{a}}\sin\theta_{\text{a}}\cos\theta_{\text{b}}\sin \theta_{\text{b}}\left|T\right|^{2}\left|R\right|^{2}\left|J\right|^{2}\\ P(c_{\text{H}}d_{\text{V}})&=&\sin^{2}\theta_{\text{a}}\cos^{2}\theta_{\text{b }}\left|R\right|^{4}+\cos^{2}\theta_{\text{a}}\sin^{2}\theta_{\text{b}}\left|T \right|^{4}-2\cos\theta_{\text{a}}\sin\theta_{\text{a}}\cos\theta_{\text{b}} \sin\theta_{\text{b}}\left|T\right|^{2}\left|R\right|^{2}\left|J\right|^{2}\\ P(c_{\text{V}}d_{\text{V}})&=&\sin^{2}\theta_{\text{a}}\sin^{2}\theta_{\text{b }}\left(1-2\left|T\right|^{2}\left|R\right|^{2}\left(1+\left|J\right|^{2}\right) \right)\\ P(c_{\text{H}}d_{\text{H}})&=&\cos^{2}\theta_{\text{a}}\cos^{2}\theta_{\text{b }}\left(1-2\left|T\right|^{2}\left|R\right|^{2}\left(1+\left|J\right|^{2} \right)\right).\end{array}  \tag{10}\]  
In order to make specific predictions, we model the photon wave functions \(\xi(t)_{\text{a,b}}\) as Gaussian wave packets with center angular frequency \(\omega_{\text{a,b}}\), bandwidth \(\Delta\omega_{\text{a,b}}\), and beam splitter arrival time \(t_{\text{a,b}}\).