## 5 Photon Propagation
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 \(|\gamma|=1.2\). and realize that the spontaneous generation terms are identical to those in the spontaneous generation in Equation (13). The same occurs to all the dimer configurations. In the **gain-gain** dimer,  
\[ \begin{array}{l}n_{1}^{\left(10\right)}\left(\zeta\right)=e^{2\beta\zeta} \left|\hat{U}_{11}\left(\zeta\right)\right|^{2}\;+\;2\left(\beta-\gamma\right) \int_{0}^{\zeta}\left|\hat{U}_{11}\left(t\right)\right|^{2}e^{2\beta t}dt\;+\;2 \left(\beta+\gamma\right)\int_{0}^{\zeta}\left|\hat{U}_{12}\left(t\right) \right|^{2}e^{2\beta t}dt,\\ n_{2}^{\left(10\right)}\left(\zeta\right)=e^{2\beta\zeta}\left|\hat{U}_{21} \left(\zeta\right)\right|^{2}\;+\;2\left(\beta-\gamma\right)\int_{0}^{\zeta} \left|\hat{U}_{21}\left(t\right)\right|^{2}e^{2\beta t}dt+\;2\left(\beta+\gamma \right)\int_{0}^{\zeta}\left|\hat{U}_{22}\left(t\right)\right|^{2}e^{2\beta t} dt,\end{array}  \tag{30}\]  
we recover the spontaneous generation terms from Equation (14). In the **gain-passive** dimer,  
\[ \begin{array}{l}n_{1}^{\left(10\right)}\left(\zeta\right)=e^{2\beta\zeta} \left|\hat{U}_{11}\left(\zeta\right)\right|^{2}\;-\;4\gamma\int_{0}^{\zeta} \left|\hat{U}_{11}\left(t\right)\right|^{2}e^{-2\gamma t}dt,\\ n_{2}^{\left(10\right)}\left(\zeta\right)=e^{2\beta\zeta}\left|\hat{U}_{21} \left(\zeta\right)\right|^{2}\;-\;4\gamma\int_{0}^{\zeta}\left|\hat{U}_{21} \left(t\right)\right|^{2}e^{-2\gamma t}dt,\end{array}  \tag{31}\]  
we recover those from the spontaneous generation in Equation (15). Finally, in the **passive-loss** and **loss-loss** dimers, the intensity only depends on the initial state and decays due to the nature of the auxiliary \(\beta\) parameter,  
\[ \begin{array}{l}n_{1}^{\left(10\right)}\left(\zeta\right)=e^{2\beta\zeta} \left|\hat{U}_{11}\left(\zeta\right)\right|^{2},\\ n_{2}^{\left(10\right)}\left(\zeta\right)=e^{2\beta\zeta}\left|\hat{U}_{21} \left(\zeta\right)\right|^{2}.\end{array}  \tag{32}\]  
In these expressions, it is easier to identify that the spontaneous generation component is identical to the one in the vacuum propagation case and the stimulated component, in the specific case of single photon propagation, will be the same than in the classical dimer, as expected. Figure 4 shows the renormalized mean photon number for the propagation of a single photon in the first waveguide,  
\[\hat{n}_{j}^{\left(10\right)}(\zeta)=\frac{n_{j}^{\left(10\right)}(\zeta)}{n _{1}^{\left(10\right)}(\zeta)+n_{2}^{\left(10\right)}(\zeta)}. \tag{33}\]  
Note that the initial state is not excluded, as in the spontaneous generation case, because now there will always be a nonzero probability that the dimer will have a photon propagating through it. Now, as the renormalized mean photon number will have a spontaneous and stimulated component, it is possible to see that the strongest differences in the \(\mathcal{PT}\)-symmetric regime will occur at small propagation distances, and the oscillation frequency will be larger than in the spontaneous generation case, Figure 4a,d,g. In the Kato exceptional point, Figure 4b,e,h, and the broken symmetry regime, Figure 4c,f,i, the same will happen. The strongest deviation from the spontaneous generation signature will occur for small propagation distances and it will take slightly longer propagation distances to reach an asymptotic limit identical in value to that of the spontaneous generation,  
\[ \begin{array}{lcl}\lim_{\zeta\rightarrow\infty}\hat{n}_{1}^{\left(10\right) }(\zeta)&=&\frac{1}{2\gamma}\left(\gamma+\sqrt{\gamma^{2}-1}\right)^{-1},\\ \lim_{\zeta\rightarrow\infty}\hat{n}_{2}^{\left(10\right)}(\zeta)&=&\frac{1}{2 \gamma}\left(\gamma+\sqrt{\gamma^{2}-1}\right).\end{array}