# Tunable non-Hermiticity in Coupled Photonic Crystal Cavities with Asymmetric Optical Gain
## 3 Results and Discussion
### Asymmetric Gain in the Doubly Coupled PhC Cavity  
The introduction of universal optical gain to an optical cavity system does not have a significant effect on the eigenmodes and eigenfrequency because the entire cavity structure and the correspondingly excited modes equally experience the gain in space. However, applying a spatially asymmetric optical gain to a coupled cavity system can introduce unanticipated behaviors with regard to the eigenmodes and eigenfrequencies, thereby entailing the concept of PT symmetry in a non-Hermitian optical system [10; 12; 15]. The optical gain with controlled levels of asymmetricity can affect the state of the non-Hermitian optical system, that is, whether the system is in the unbroken or broken PT-symmetric phase. For example, at moderate levels of asymmetric optical gain, the system remains in the unbroken PT-symmetric phase, supporting two non-degenerate eigenmodes, that is, the bonding and antibonding modes. However, when the asymmetricity reaches a critical point, known as the exceptional point (EP), the phase transition between the two PT-symmetric phases occurs. Importantly, at the EP, these two eigenmodes are degenerated, yielding the same eigenfrequency [15; 16; 17]. Furthermore, for higher levels of asymmetricity than that of the EP, the system enters the broken PT-symmetric phase and supports single degenerate super-modes. We apply this concept of the non-Hermitian optical system to our doubly coupled PhC cavity and achieve systematic manipulation of the eigenmodes and eigenfrequencies. In our non-Hermitian coupled cavity system, the eigenfrequencies are determined by the fundamental eigen-frequency (\(f_{0}\)), the inter-cavity coupling constant (\(J\)), and the individual optical gain and loss in component cavities. Considering all of these, the previous set of coupled differential equations is modified as follows:  
\[\left[ \begin{array}{cc}f_{0}+i(\gamma_{1}-\kappa)&J\\ J&f_{0}+i(\gamma_{2}-\kappa)\end{array} \right]\left[ \begin{array}{c}\phi_{1 }\\ \phi_{2}\end{array} \right]=J\left[ \begin{array}{c}\phi_{1}\\ \phi_{2}\end{array} \right] \tag{3}\]  
where \(\gamma_{1}\) and \(\gamma_{2}\) are optical gains in component cavities 1 and 2, respectively. We assumed that the two component cavities have the same optical loss, i.e., \(\kappa\), and the eigenmodes are in a steady state. Based on Equation (3), the eigenfrequencies of the two eigenmodes are given by  
\[f_{\pm}=f_{0}\pm\sqrt{\beta^{2}-\Delta\gamma^{2}}+i\left(\gamma_{\rm avg}- \kappa\right) \tag{4}\]where \(\Delta\gamma=(\gamma_{2}-\gamma_{1})/2\) and \(\gamma_{\rm avg}=(\gamma_{1}+\gamma_{2})/2\) are the gain contrast and average gain, respectively. We systematically change the value of \(\Delta\gamma\) to control the level of the asymmetricity in the optical gain. The result of Equation (4) reveals important features: First, the eigenfrequencies in Equation (4) are determined by the interplay between the coupling constant \(J\) and the gain contrast \(\Delta\gamma\). Subsequently, the frequency difference between \(f_{+}\) and \(f_{-}\) can be tuned based on the gain contrast level. Second, the real part of the eigenfrequencies has two different values when \(J^{2}>\Delta\gamma^{2}\) (the unbroken PT-symmetric phase), but unexpectedly, it has a single value when \(J^{2}<\Delta\gamma^{2}\) (the broken PT-symmetric phase). Third, the imaginary part of the eigenfrequencies can have either a positive or negative value depending on the relative magnitude of \((\Delta\gamma^{2}-J^{2})^{1/2}\) and \(\gamma_{\rm avg}-\kappa\) when the system is in the broken PT-symmetric phase. These results indicate that one can tune the real and imaginary parts of complex eigenfrequencies, determine the system symmetry phase, and consequently manipulate the non-Hermiticity by simply introducing different levels of gain in component cavities 1 and 2. Figure 3 shows our approach for exploring the control of non-Hermiticity.