## 2 Quantum Model and Configurations  
In the laboratory, we can think about a more general realization of the effective quantum \(\mathcal{PT}\)-symmetric dimer provided by the quantum non-Hermitian dimer,  
\[-i\frac{d}{dz}\left( \begin{array}{c}\hat{a}_{1}(z)\\ \hat{a}_{2}(z)\end{array} \right)=\left( \begin{array}{cc}n_{1}&g\\ g&n_{2}\end{array} \right)\left( \begin{array}{c}\hat{a}_{1}(z)\\ \hat{a}_{2}(z)\end{array} \right)+\left( \begin{array}{c}\hat{f}_{1}(z)\\ \hat{f}_{2}(z)\end{array} \right). \tag{4}\]  
Again, the field annihilation operators in the first and second waveguides are given by the operators \(\hat{a}_{1}(z)\) and \(\hat{a}_{2}(z)\), in that order. The effective refractive indices of the optical waveguides are given by the complex numbers \(n_{1}\) and \(n_{2}\), the effective coupling between the optical modes is given by the real positive parameter \(g\). Finally, the Gaussian fluctuations, due to the active linear media,  
Figure 1: Schematic showing the renormalized light intensity arising from the spontaneous generation of photons through two coupled waveguides in the \(\mathcal{PT}\)-symmetry regime with balanced gain-loss configuration. are described by the Langevin forces \(\hat{f}_{1}(z)\) and \(\hat{f}_{2}(z)\), such that the only nonzero mean values involving them are their second order correlations [17],  
\[\langle\hat{f}_{j}^{\dagger}(z)\hat{f}_{j}(z^{\prime})\rangle = 2\Im(n_{j})\delta(z-z^{\prime}),\quad\text{for gain media},\] \[\langle\hat{f}_{j}(z)\hat{f}_{j}^{\dagger}(z^{\prime})\rangle = 2\Im(n_{j})\delta(z-z^{\prime}),\quad\text{for loss media}. \tag{5}\]  
It is straightforward to write a formal propagator for this quantum optical system,  
\[\left( \begin{array}{c}\hat{a}_{1}(\zeta)\\ \hat{a}_{2}(\zeta)\end{array} \right)=e^{in_{0}\zeta}\left( \begin{array}{c} \hat{b}_{1}(\zeta)\\ \hat{b}_{2}(\zeta)\end{array} \right), \tag{6}\]  
where we have factorized an average refractive index, \(n_{0}=(n_{1}+n_{2})/(2g)\), that introduces a common phase, \(\Re(n_{0})\), and a scaling factor due to gain or loss, \(\Im(n_{0})\). We have also scaled the propagation variable by the effective coupling of the dimer, \(\zeta=gz\). Note that the average refractive index cannot be zero for experimental realizations. It becomes a pure real number, a phase factor, for passive materials, and a pure imaginary number, a scaling factor, for identical media with balanced gain and loss. The second term in the propagator,  
\[\left( \begin{array}{c}\hat{b}_{1}(\zeta)\\ \hat{b}_{2}(\zeta)\end{array} \right)=e^{i\hat{H}\zeta}\left( \begin{array}{c} \hat{b}_{1}(0)\\ \hat{b}_{2}(0)\end{array} \right)+\int_{0}^{\zeta}e^{i\hat{H}(\zeta-t)}e^{-in_{ 0}t}\left( \begin{array}{c}\hat{f}_{1}(t)\\ \hat{f}_{2}(t)\end{array} \right)dt, \tag{7}\]  
provides us with the effective dynamics of the system. Here, we need use the scaling property of Dirac delta, \(\delta(\zeta)=\delta(z)/|g|\), and have defined a complex effective refractive index \(n=(n_{1}-n_{2})/(2g)\) that provides us with an auxiliary effective non-Hermitian matrix,  
\[\hat{H}=\left( \begin{array}{cc}n&1\\ 1&-n\end{array} \right). \tag{8}\]  
Note that the case of waveguides with identical real part of their effective refractive indices, \(\Re(n_{1})=\Re(n_{2})\), yields a purely imaginary effective refractive index that we can rename as \(n=i\gamma\) in order to recover the standard quantum \(\mathcal{PT}\)-symmetric dimer [15]. Again, let us stress that the dynamics introduced by a more realistic model of linearly active media, where configurations beyond balanced gain-loss are easily obtained, will include a phase and scaling factor proportional to the average refractive index that are not taken into consideration in the ideal \(\mathcal{PT}\)-symmetric dimer configuration.