Level attraction and $\mathcal{PT}$ symmetry in indirectly coupled microresonators

Benefiting from the tunability of their phase delay and gain, we show the possibility of realizing level attraction, parity-time ($\mathcal{PT}$)-, and anti-$\mathcal{PT}$-symmetric photonic structures with indirectly coupled active and passive microresonators mediated by drop-filter waveguides. We find that level attraction originates from two mechanisms, i.e., exceptional point and resonance, and it may emerge not only for dissipative coupling but also coherent coupling and hybrid coupling after considering the gain and loss of microresonators, which confirms the universality of level attraction in the coupled photonic system. On the other hand, the transition from $\mathcal{PT}$ to anti-$\mathcal{PT}$ symmetry or vice versa may be realizable by tuning their phase delay and gain; thus the indirectly coupled microresonators may provide an excellent platform for exhibiting the unique features of $\mathcal{PT}$ and anti-$\mathcal{PT}$ symmetry. Furthermore, we also reveal the intrinsic relationship of the line shape and phase behavior of the reflection spectrum with $\mathcal{PT}$ symmetry, anti-$\mathcal{PT}$ symmetry, and level attraction.

Figure 1

The schematic diagram of indirectly coupled active WGM microresonator $\mu R_1$ and passive WGM microresonator $\mu R_2$ mediated by drop-filter waveguides. $a_{ju\left(l\right),\text{in}}$ ($a_{ju\left(l\right),\text{out}}$) are the input (output) fields of $j\mathrm{th}$ microresonator and $L$ is the center-to-center distance between two microresonators.

Figure 2

The real and imaginary parts of the energy difference $\mathrm{\Delta }{\omega }_{+}$ (red solid) and $\mathrm{\Delta }{\omega }_{-}$ (blue dashed) versus frequency detuning of two microresonators for coherent coupling (a), (d), hybrid coupling (b), (e) and dissipative coupling (c), (f), respectively. The parameters of the system are ${\gamma }_1={\gamma }_2=\kappa /2$ and $\xi =0$. EP and RP are labeled by red stars and triangle, respectively.

Figure 3

The real and imaginary parts of the energy difference $\mathrm{\Delta }{\omega }_{+}$ (red solid) and $\mathrm{\Delta }{\omega }_{-}$ (blue dashed) versus frequency detuning of two microresonators for coherent coupling (a), (d); hybrid coupling (b), (e); and dissipative coupling (c), (f), respectively. The parameters of system are ${\gamma }_1={\gamma }_2=\kappa /2$ and $\xi =2\kappa$. EP and RP are labeled by red stars and triangles, respectively.

Figure 4

The real and imaginary parts of the energy difference $\mathrm{\Delta }{\omega }_{+}$ (red solid) and $\mathrm{\Delta }{\omega }_{-}$ (blue dashed) versus frequency detuning of two microresonators for coherent coupling (a), (d); hybrid coupling (b), (e); and dissipative coupling (c), (f), respectively. The parameters of system are ${\gamma }_1={\gamma }_2=\kappa /2$ and $\xi =4.5\kappa$. RP is labeled by red triangles.

Figure 5

The real and imaginary parts of the eigenfrequencies of supermodes ${\omega }_{+}$ (red solid) and ${\omega }_{-}$ (blue dashed) as a function of relative coupling strength $\kappa /{\kappa }_{PT}$ for $\mathcal{PT}$ symmetry (a), (d); passive $\mathcal{PT}$ symmetry (b), (e); and anti-$\mathcal{PT}$ symmetry (c), (f). The parameters of system are ${\gamma }_2={\gamma }_1/2, \kappa =5{\gamma }_1$, and $\xi =10{\gamma }_1$.

Figure 6

The line shape and phase of reflection spectrum for (a), (d) $\mathcal{PT}$ (red solid) and anti-$\mathcal{PT}$ symmetry (blue dashed); (b), (e) coherent coupling (red solid) and dissipative coupling (blue dashed); and (c), (f) dissipative coupling (red solid) and hybrid coupling (blue dashed). The parameters of system are ${\gamma }_2={\gamma }_1/2$ and $\xi =10{\gamma }_1$.