## 3 Spontaneous Generation of Photons  
The first signature that differentiates a quantum from a classical \(\mathcal{PT}\)-symmetric dimer is the spontaneous generation of photons due to the gain medium in presence of vacuum input fields,  
\[ \begin{array}{rcl}n_{j}^{(00)}(\zeta)&=&\langle 0,0|\hat{a}_{j}^{\dagger}( \zeta)\hat{a}_{j}(\zeta)|0,0\rangle,\\ &=&e^{2g\zeta}\langle 0,0|\hat{b}_{j}^{\dagger}(\zeta)\hat{b}_{j}(\zeta)|0,0 \rangle.\end{array}  \tag{12}\]  
In other words, vacuum fluctuations are enough to make the linear active media spontaneously generate photons [17]; something that is lacking in the classical model. The signatures available through the spontaneous generation of photos in diverse configurations are provided in the following for the standard balanced **gain-loss** dimer,  
\[ \begin{array}{rcl}n_{1}^{(00)}(\zeta)&=&-2\gamma f_{0}^{\zeta}\left|\hat{U}_ {11}\left(t\right)\right|^{2}\!\!dt,\\ n_{2}^{(00)}(\zeta)&=&-2\gamma f_{0}^{\zeta}\left|\hat{U}_{21}\left(t\right) \right|^{2}\!\!dt,\end{array}  \tag{13}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Realization** & \(n_{1}\) & \(n_{2}\) & \(n\) & \(n_{0}\) & \(\gamma\) & \(\beta\) \\ \hline
**Gain-loss** & \(n_{R}-in_{I}\) & \(n_{R}+in_{I}\) & \(-i\frac{n_{I}}{g}\) & \(\frac{n_{R}}{g}\) & \(-\frac{n_{I}}{g}\) & 0 \\
**Gain-gain** & \(n_{R}-in_{I,1}\) & \(n_{R}-in_{I,2}\) & \(i\frac{-n_{I,1}+n_{I,2}}{2g}\) & \(\frac{n_{R}}{g}-i\frac{n_{I,1}+n_{I,2}}{2g}\) & \(\frac{-n_{I,1}+n_{I,2}}{2g}\) & \(\frac{n_{I,1}+n_{I,2}}{2g}\) \\
**Gain-passive** & \(n_{R}-in_{I}\) & \(n_{R}\) & \(-i\frac{n_{I}}{2g}\) & \(\frac{n_{R}}{g}+i\frac{n_{I}}{2g}\) & \(-\frac{n_{I}}{2g}\) & \(\frac{n_{I}}{2g}\) \\
**Passive-loss** & \(n_{R}\) & \(n_{R}+in_{I}\) & \(-i\frac{n_{I}}{2g}\) & \(\frac{n_{R}}{g}+i\frac{n_{I}}{2g}\) & \(-\frac{n_{I}}{2g}\) & \(-\frac{n_{I}}{2g}\) \\
**Loss-loss** & \(n_{R}+in_{I,1}\) & \(n_{R}+in_{I,2}\) & \(i\frac{n_{I,2}-n_{I,2}}{2g}\) & \(\frac{n_{I}+in_{I,2}+n_{I,2}}{2g}\) & \(\frac{n_{I,1}-n_{I,2}}{2g}\) & \(\frac{-n_{I,1}-n_{I,2}}{2g}\) \\ \hline \end{tabular}
\end{table} 
Table 1: A summary of the parameters involved in the different feasible experimental realizations of the \(\mathcal{PT}\)-symmetric dimer. where the (\(i\),\(j\))-th component of the two by two propagation matrix has been written as \(\hat{U}_{ij}(\zeta)\), and it is important to note that these integrals can be solved analytically but yield expressions too long to write here. The **gain-gain** dimer yields the following expressions,  
\[ \begin{array}{rcl}n_{1}^{(00)}(\zeta)&=&2\left(\beta-\gamma\right)\int_{0}^{ \zeta}\left|\hat{U}_{11}\left(t\right)\right|^{2}\!e^{2\beta t}dt\;+\;2\left( \beta+\gamma\right)\int_{0}^{\zeta}\left|\hat{U}_{12}\left(t\right)\right|^{2 }\!e^{2\beta t}dt,\\ n_{2}^{(00)}(\zeta)&=&2\left(\beta-\gamma\right)\int_{0}^{\zeta}\left|\hat{U}_ {21}\left(t\right)\right|^{2}\!e^{2\beta t}dt\;+\;2\left(\beta+\gamma\right) \int_{0}^{\zeta}\left|\hat{U}_{22}\left(t\right)\right|^{2}\!e^{2\beta t}dt. \end{array}  \tag{14}\]  
For the **gain-passive** dimer, we can write the spontaneous generation as:  
\[ \begin{array}{rcl}n_{1}^{(00)}(\zeta)&=&-4\gamma\int_{0}^{\zeta}\left|\hat{U }_{11}\left(t\right)\right|^{2}\!e^{-2\gamma t}dt,\\ n_{2}^{(00)}(\zeta)&=&-4\gamma\int_{0}^{\zeta}\left|\hat{U}_{21}\left(t\right) \right|^{2}\!e^{-2\gamma t}dt,\end{array}  \tag{15}\]  
and, obviously, there is not spontaneous generation in the **passive-loss** and **loss-loss** dimer,  
\[n_{1}^{(00)}(\zeta)=n_{2}^{(00)}(\zeta)=0. \tag{16}\]  
While the expressions for the spontaneous generation are complicated, it is straightforward to realize from the analytic expressions that they will present different signatures through propagation in the dimer. The signatures from the different configurations can be seen in Figure 2, where we show an instantaneously renormalized spontaneous generation of photons,  
\[ \begin{array}{l}\bar{n}_{j}^{(00)}(\zeta)=\frac{n_{j}^{(00)}(\zeta)}{n_{1}^{ (00)}(\zeta)+n_{2}^{(00)}(\zeta)},\qquad\zeta>0,\end{array}