## 5 Photon Propagation  
In general, we can study the propagation of any given initial photon state through our dimer,  
\[n_{j}^{(\phi_{0})}(\zeta) = \langle\psi_{0}|\hat{a}_{j}^{\dagger}(\zeta)\hat{a}_{j}(\zeta)| \psi_{0}\rangle,\] \[= e^{2\beta\zeta}\langle\psi_{0}|\hat{b}_{j}^{\dagger}(\zeta)\hat{b}_ {j}(\zeta)|\psi_{0}\rangle,\]  
and realize that the mean photon number in each waveguide,  
\[ \begin{array}{rcl}n_{1}^{(\phi_{0})}\left(\zeta\right)&=&e^{2\beta\zeta} \left|\hat{U}_{11}\left(\zeta\right)\right|^{2}\left\langle\psi_{0}\right|\hat {a}_{1}^{\dagger}\left(0\right)\hat{a}_{1}\left(0\right)|\psi_{0}\rangle+e^{2 \beta\zeta}\hat{U}_{11}^{*}\left(\zeta\right)\hat{U}_{12}\left(\zeta\right) \left\langle\psi_{0}\right|\hat{a}_{1}^{\dagger}\left(0\right)\hat{a}_{2} \left(0\right)|\psi_{0}\rangle\\ &+&e^{2\beta\zeta}\hat{U}_{12}^{*}\left(\zeta\right)\hat{U}_{11}\left(\zeta \right)\left\langle\psi_{0}|\hat{a}_{2}^{*}\left(0\right)\hat{a}_{1}\left(0 \right)|\psi_{0}\rangle+e^{2\beta\zeta}\left|\hat{U}_{12}\left(\zeta\right) \right|^{2}\left\langle\psi_{0}|\hat{a}_{2}^{*}\left(0\right)\hat{a}_{2}\left( 0\right)|\psi_{0}\rangle\\ &+&e^{2\beta\zeta}\int_{0}^{\zeta}\int_{0}^{\zeta}\hat{U}_{11}^{*}\left(\zeta -t^{\prime}\right)\hat{U}_{11}\left(\zeta-t\right)e^{in\sigma_{0}^{t}t^{\prime }}e^{-in_{0}t}\left\langle\psi_{0}|\hat{f}_{1}^{\dagger}\left(t^{\prime} \right)\hat{f}_{1}\left(t\right)|\psi_{0}\right\rangle dt^{\prime}dt\\ &+&e^{2\beta\zeta}\int_{0}^{\zeta}\int_{0}^{\zeta}\hat{U}_{12}^{*}\left(\zeta -t^{\prime}\right)\hat{U}_{12}\left(\zeta-t\right)e^{in\sigma_{0}^{t}t^{\prime }}e^{-in_{0}t}\left\langle\psi_{0}|\hat{f}_{2}^{\dagger}\left(t^{\prime} \right)\hat{f}_{2}\left(t\right)|\psi_{0}\right\rangle dt^{\prime}dt,\\ n_{2}^{(\phi_{0})}\left(\zeta\right)&=&e^{2\beta\zeta}\left|\hat{U}_{21}\left( \zeta\right)\right|^{2}\left\langle\psi_{0}|\hat{a}_{1}^{\dagger}\left(0 \right)\hat{a}_{1}\left(0\right)|\psi_{0}\rangle+e^{2\beta\zeta}\hat{U}_{21}^ {*}\left(\zeta\right)\hat{U}_{22}\left(\zeta\right)\left\langle\psi_{0}|\hat {a}_{1}^{\dagger}\left(0\right)\hat{a}_{2}\left(0\right)|\psi_{0}\rangle\\ &+&e^{2\beta\zeta}\hat{U}_{22}^{*}\left(\zeta\right)\hat{U}_{21}\left(\zeta \right)\left\langle\psi_{0}|\hat{a}_{2}^{*}\left(0\right)\hat{a}_{1}\left(0 \right)|\psi_{0}\rangle+e^{2\beta\zeta}\left|\hat{U}_{22}\left(\zeta\right) \right|^{2}\left\langle\psi_{0}|\hat{a}_{2}^{*}\left(0\right)\hat{a}_{2}\left( 0\right)|\psi_{0}\rangle\\ &+&e^{2\beta\zeta}\int_{0}^{\zeta}\int_{0}^{\zeta}\hat{U}_{21}^{*}\left(\zeta -t^{\prime}\right)\hat{U}_{21}\left(\zeta-t\right)e^{in\sigma_{0}^{t}t^{\prime }}e^{-in_{0}t}\left\langle\psi_{0}|\hat{f}_{1}^{\dagger}\left(t^{\prime} \right)\hat{f}_{1}\left(t\right)|\psi_{0}\right\rangle dt^{\prime}dt\\ &+&e^{2\beta\zeta}\int_{0}^{\zeta}\int_{0}^{\zeta}\hat{U}_{22}^{*}\left(\zeta -t^{\prime}\right)\hat{U}_{22}\left(\zeta-t\right)e^{in\sigma_{0}^{t}t^{\prime }}e^{-in_{0}t}\left\langle\psi_{0}|\hat{f}_{2}^{\dagger}\left(t^{\prime} \right)\hat{f}_{2}\left(t\right)|\psi_{0}\right\rangle dt^{\prime}dt,\end{array}  \tag{28}\]  
will have a component related to spontaneous generation, those terms with the integrals, and another to stimulated generation, the rest. As a practical example, let us use as the initial state a single photon impinging the first waveguide, \(|\psi_{0}\rangle=|10\rangle\). Again, we can calculate the mean photon number in the **balanced gain-loss** dimer,  
\[ \begin{array}{l}n_{1}^{(10)}\left(\zeta\right)=\left|\hat{U}_{11}\left(\zeta \right)\right|^{2}-2\gamma\int_{0}^{\zeta}\left|\hat{U}_{11}\left(t\right) \right|^{2}dt,\\ n_{2}^{(10)}\left(\zeta\right)=\left|\hat{U}_{21}\left(\zeta\right)\right|^{2}-2 \gamma\int_{0}^{\zeta}\left|\hat{U}_{21}\left(t\right)\right|^{2}dt,\end{array}  \tag{29}\]  
Figure 3: Photon bunching shown in terms of the \(q^{(00)}(\zeta)\) parameter for different realizations of the effective \(\mathcal{PT}\)-symmetric dimer.