# Tunable non-Hermiticity in Coupled Photonic Crystal Cavities with Asymmetric Optical Gain
## 3 Results and Discussion
### Asymmetric Gain in the Doubly Coupled PhC Cavity
As mentioned earlier, to independently control \(J\) while keeping \(f_{0}\) constant, we employed the doubly coupled PhC cavity in which one of the component cavities was vertically shifted (cavity 2). Then, we introduced different optical gains, \(\gamma_{1}\) (shaded yellow box) and \(\gamma_{2}\) (shaded blue box), to component cavities 1 and 2 by systematically changing the extinction coefficient \(K_{\rm slab}\) (Figure 3a). At the initial stage, we started with two passive component cavities, then slowly increased \(\gamma_{1}\) to the maximum gain \(\gamma_{\rm max}\) while keeping cavity 2 passive; finally, \(\gamma_{2}\) was increased to the maximum gain \(\gamma_{\rm max}\) while keeping \(\gamma_{1}=\gamma_{\rm max}\), as shown in Figure 3b. Figure 4 shows the results. First, we employed the doubly coupled PhC cavity without performing vertical airhole shifting and calculated the real (Re(\(f\)), black squares) and imaginary parts (Im(\(f\)), red circles) of the complex eigenfrequencies as a function of the gain trajectory presented in Figures 3b and 4a. Analytical solutions (black and red lines) obtained from Equation (4) were co-plotted, and they agree well with the numerical simulations. The model parameters such as \(\Delta\gamma\), \(\gamma_{\rm avg}\), \(\kappa\), and \(J\) were obtained from the numerical simulations by fitting them with Equation (4). The results clearly reveal the unique non-Hermiticity nature, exhibiting the evolution of the state from the unbroken to the broken PT-symmetric phase and vice versa with phase transitions occurring at the EPs. For example, the real part of the eigenfrequencies, Re(\(f\)), shows two distinct eigenmodes (i.e., bonding (Re(\(f_{+}\))) and antibonding (Re(\(f_{-}\))) modes) at \(\Delta\gamma=0\), as expected from Equation (4). As \(|\Delta\gamma|\) increases, the eigenfrequency difference between Re(\(f_{+}\)) and Re(\(f_{-}\)) reduces, and the system is in the unbroken PT symmetry phase. At the EP with \(|\Delta\gamma|=0.345\) THz, which is equal to \(J\), the two eigenmodes coalesce and become a single degenerate eigenmode. As \(|\Delta\gamma|\) further increases, the system enters the broken PT-symmetric phase, and the degenerate eigenmode holds until \(|\Delta\gamma|=1.106\) THz. In contrast, the imaginary part of the eigenfrequencies, Im(\(f\)), exhibits an opposite development. Im(\(f\)) of two eigenmodes are both equal to each other when \(|\Delta\gamma|<0.345\) THz, which indicates that the two  
Figure 3: Asymmetric optical gain in doubly coupled PhC cavity. (**a**) Asymmetric optical gain in doubly coupled PhC cavity with gain \(\gamma_{1}\) in cavity 1 (yellowish shaded area) and gain \(\gamma_{2}\) in cavity 2 (bluish shaded area). (**b**) Asymmetric gain trajectory in doubly coupled PhC cavity. The initial gain was set to zero. Next, \(\gamma_{1}\) was increased to the maximum gain \(\gamma_{\rm max}\) (yellow bar) and then \(\gamma_{2}\) was increased to \(\gamma_{\rm max}\) while \(\gamma_{1}\) was kept equal to \(\gamma_{\rm max}\) (blue bar). eigenmodes experience an equal amount of optical gain. However, at \(|\Delta\gamma|\) values larger than that at the EP, Im(\(f\)) is bifurcated, independently contributing to the two degenerate eigenmodes. Consequently, the degenerate eigenmodes are classified into two kinds of eigenmodes that experience positive (amplifying mode) and negative (decaying mode) optical gain. The maximum bifurcation of Im(\(f\)) is obtained when \(|\Delta\gamma|=|\Delta\gamma|_{\text{max}}\) (i.e., 1.106 THz). Figure 4b,c shows the field and intensity profiles of the eigenmodes in the unbroken and broken PT-symmetric phases. Figure 4b shows the field distributions of the out-of-phase antibonding and in-phase bonding modes at \(\gamma_{1}=\gamma_{2}=\gamma_{\text{max}}\). As \(|\Delta\gamma|\) increases, these field distributions slowly alter in response to the relative magnitude of \(\gamma_{1}\) and \(\gamma_{2}\).