## 6 Photon Bunching and Anti-Bunching in Photon Propagation  
We can also study the effect of photon propagation on second order two-point correlations as we did for spontaneous generation. In order to study a different state, we will consider as initial state a \(N00N\) state, \(\left|N00N\right\rangle=\left(\left|N0\right\rangle+\left|0N\right\rangle \right)/\sqrt{2}\), in order to see negative values of the two-point Mandel parameter at least for the initial state,  
\[q^{\left(N00N\right)}(\zeta) = \frac{n_{1212}^{\left(N00N\right)}(\zeta)}{n_{1}^{\left(N00N \right)}(\zeta)n_{2}^{\left(N00N\right)}(\zeta)}-1, \tag{35}\]  
where we have defined the following second order two-point correlation function,  
\[n_{1212}^{\left(N00N\right)}(\zeta)=e^{4\beta\zeta}\left\langle N00N|\hat{b}_{1 }^{\dagger}(\zeta)\,\hat{b}_{2}^{\dagger}(\zeta)\,\hat{b}_{1}\left(\zeta \right)\,\hat{b}_{2}\left(\zeta\right)\left|N00N\right\rangle, \tag{36}\]  
and we take the mean photon numbers as defined in Equation (27). For the two-photon \(N00N\) state, \(N=2\), the mean photon numbers at the waveguides are provided by the following expressions,  
\[ \begin{array}{rcl}n_{1}^{\left(2002\right)}(\zeta)&=&e^{2\beta\zeta}\left| \hat{U}_{11}\left(\zeta\right)\right|^{2}+e^{2\beta\zeta}\left|\hat{U}_{12} \left(\zeta\right)\right|^{2}+n_{1}^{\left(00\right)}(\zeta),\\ n_{2}^{\left(2002\right)}(\zeta)&=&e^{2\beta\zeta}\left|\hat{U}_{21}\left( \zeta\right)\right|^{2}+e^{2\beta\zeta}\left|\hat{U}_{22}\left(\zeta\right) \right|^{2}+n_{2}^{\left(00\right)}(\zeta),\end{array}  \tag{37}\]  
Figure 4: Instantaneously renormalized mean photon number, \(\hat{n}_{j}^{\left(10\right)}(\zeta)\), along different realizations of the effective \(\mathcal{PT}\)-symmetric dimer. The first row, (**a–c**), shows balanced gain-loss, the second row, (**d–f**), shows gain-gain, the third row, (**g–i**), shows gain-passive, and the fourth row, (**j–l**), shows both passive-loss and loss-loss configurations in the \(\mathcal{PT}\)-symmetric regime, the first column with \(\left|\gamma\right|=0.5\), the Kato point with \(\left|\gamma\right|=1\), the second column, and the broken symmetry regime, and the third column with \(\left|\gamma\right|=1.2\). Values for the first and second waveguides are shown with a solid blue and a dashed red lines, in that order. where the first and second terms corresponds to the stimulated generation, and the third term is related to the spontaneous generation and can be recovered from Section 3 for each and every configuration. The general expression for the two-point correlation is summarized in the following,  
\[ \begin{array}{rcl}n_{1212}^{(2002)}(\zeta)&=&e^{4\beta\zeta}\big{|}\hat{U}_{11 }\hat{U}_{21}+\hat{U}_{12}\hat{U}_{22}\big{|}^{2}+n_{1}^{(00)}(\zeta)n_{2}^{(00 )}(\zeta)+|n_{12}^{(00)}(\zeta)|^{2}\\ &+&e^{2\beta\zeta}\left[n_{1}^{(00)}(\zeta)\left(\big{|}\hat{U}_{21}\big{|}^{2} +\big{|}\hat{U}_{22}\big{|}^{2}\right)+n_{2}^{(00)}(\zeta)\left(\big{|}\hat{U}_ {11}\big{|}^{2}+\big{|}\hat{U}_{12}\big{|}^{2}\right)\right]\\ &+&2\;e^{2\beta\zeta}\;\Re\left[n_{12}^{(00)}(\zeta)\left(\hat{U}_{21}^{*}\hat {U}_{11}+\hat{U}_{22}^{*}\hat{U}_{12}\right)\right].\end{array}  \tag{38}\]  
For the **balanced gain-loss** dimer, we use the spontaneous generation terms, \(n_{j}^{(00)}(\zeta)\), provided by Equation (13), and the first order two-point correlation, \(n_{12}^{(00)}(\zeta)\), from Equation (21). In the **gain-gain** dimer, the expressions for \(n_{j}^{(00)}(\zeta)\) and \(n_{12}^{(00)}(\zeta)\) are given by Equations (14) and (22), in that order. In the **gain-passive** dimer, we only need the definitions provided by Equations (15) and (23) for the spontaneous generation and the first order two-point correlation, respectively.