## 4 Photon Bunching in Spontaneous Generation  
We can also look at the signatures provided by the probability to detect simultaneously one photon at each of the waveguides output in the different configurations. This can be written in the following form for the spontaneous generation of photons,  
\[ \begin{array}{lcl}\left\langle 0,0|\hat{a}_{1}^{\ddagger}\left(\zeta\right) \hat{a}_{2}^{\ddagger}\left(\zeta\right)\hat{a}_{1}\left(\zeta\right)\hat{a}_{2 }\left(\zeta\right)|0,0\right\rangle&=&\left\langle 0,0|\hat{a}_{1}^{\ddagger}\left( \zeta\right)\hat{a}_{1}\left(\zeta\right)|0,0\right\rangle\left\langle 0,0|\hat{a}_{2}^{ \ddagger}\left(\zeta\right)\hat{a}_{2}\left(\zeta\right)|0,0\right\rangle\\ &+&\left\langle 0,0|\hat{a}_{1}^{\ddagger}\left(\zeta\right)\hat{a}_{2}\left( \zeta\right)|0,0\right\rangle\left\langle 0,0|\hat{a}_{2}^{\ddagger}\left( \zeta\right)\hat{a}_{1}\left(\zeta\right)|0,0\right\rangle\\ &=&n_{10}^{\left(0\right)}(\zeta)n_{20}^{\left(0\right)}(\zeta)+\left|n_{12}^{ \left(0\right)}(\zeta)\right|^{2},\end{array}  \tag{19}\]  
where he have used the Gaussian nature of Langevin forces [15] to simplify this probability using the expressions derived in the last section and defining the following first order two-point correlation,  
\[ \begin{array}{lcl}n_{12}^{\left(00\right)}(\zeta)&=&\left\langle 0,0|\hat{a}_{1}^ {\ddagger}(\zeta)\hat{a}_{2}(\zeta)|0,0\right\rangle,\\ &=&e^{2\beta\zeta}\langle 0,0|\hat{b}_{1}^{\ddagger}(\zeta)\hat{b}_{2}( \zeta)|0,0\rangle.\end{array}  \tag{20}\]  
Note that the detection probability will always be positive independently of the form taken by the first order two-point correlation for the different configurations: **balanced gain-loss** dimer,  
\[ \begin{array}{lcl}n_{12}^{\left(00\right)}(\zeta)&=&-2\gamma\int_{0}^{\zeta }\hat{U}_{11}^{*}\left(t\right)\hat{U}_{21}\left(t\right)dt,\end{array}  \tag{21}\]  
Figure 2: Instantaneously renormalized spontaneous generation, \(\hat{n}_{j}^{\left(00\right)}(\zeta)\), along different realizations of the effective \(\mathcal{PT}\)-symmetric dimer. The first row, (**a–c**), shows balanced gain-loss, second row, (**d–f**), shows gain-gain, and third row, (**g–i**), shows gain-passive configurations in the \(\mathcal{PT}\)-symmetric regime, the first column with \(|\gamma|=0.5\), the Kato point with \(|\gamma|=1\), the second column, and broken symmetry regime, and the third column with \(|\gamma|=1.2\). Values for the first and second waveguides are shown with a solid blue and a dashed red lines, in that order. Note the oscillatory behavior of the spontaneous generation inside and its asymptotic behavior outside the \(\mathcal{PT}\)-symmetric regime. gain-gain dimer,  
\[n_{12}^{(00)}(\zeta) = 2\left(\beta-\gamma\right)\int_{0}^{\zeta}\hat{U}_{11}^{*}\left(t \right)\hat{U}_{21}\left(t\right)e^{2\beta t}dt\ +\ 2\left(\beta+\gamma\right)\int_{0}^{\zeta}\hat{U}_{12}^{*}\left(t \right)\hat{U}_{22}\left(t\right)e^{2\beta t}dt, \tag{22}\]  
gain-passive dimer,  
\[n_{12}^{(00)}(\zeta) = -4\gamma\int_{0}^{\zeta}\hat{U}_{11}^{*}\left(t\right)\hat{U}_{21 }\left(t\right)e^{-2\gamma t}dt, \tag{23}\]  
and, obviously, for the **passive-loss** and **loss-loss** dimer,  
\[n_{12}^{(00)}(\zeta)=0. \tag{24}\]  
Again, we have to be careful to consider the appropriate parameters \(\gamma\) and \(\beta\) for each configuration summarized in Table 1.  
In order to visualize the information, we can use a quantity similar to Mandel Q-parameter [19],  
\[ \begin{array}{rcl}q^{(00)}(\zeta)&=&g_{2}^{(00)}(\zeta)-1,\\ &=&\frac{1}{n_{12}^{(00)}(\zeta)|^{2}},\\ \end{array} \qquad\zeta>0, \tag{25}\]  
given in terms of the second order two-point correlation function,  
\[g_{2}^{(00)}(\zeta)=1+\frac{|n_{12}^{(00)}(\zeta)|^{2}}{n_{1}^{(00)}(\zeta)n_{ 2}^{(00)}(\zeta)},\qquad\zeta>0, \tag{26}\]  
that provides us with the probability of simultaneously detecting a photon in each waveguide output. The values of this two-point parameter are always positive, \(q^{(00)}(\zeta)\geq 0\), thus, the different configurations only show photon bunching.