Bound states in the continuum in whispering gallery resonators

We report the formation of bound states in the continuum in a whispering gallery resonator coupled to a one-dimensional waveguide. We find that an incident photon wave packet is partially stored in the bound state in the continuum due to a roughness-induced symmetry breaking. We discuss quantum interference effects such as Fano in the transmission spectra and the photonic density of states. The conditions under which the bound state can be accessed through single-photon transport processes are investigated.

Figure 1

Schematic of the system. 1D waveguide coupled to a WGR. The blue and red solid arrows represent the WGMs $a$ and $b$, respectively. The green arrow denotes the coupling between the waveguide and the WGR, ${\xi }_1$. The orange line represents the hopping between two nearest-neighbor cavities in the discrete waveguide, ${\xi }_0$.

Figure 2

(a) Single-photon transmission spectra and (b) density of state in WGR vs $\omega =2{\xi }_{0}cos\left(k\right)$ for $\varphi =0$ (black solid line), $\varphi =\pi /50$ (red doted line), and $\varphi =\pi /6$ (blue dashed line). The parameters are chosen as ${\xi }_1=0.2$ and $t_{ab}=0.1$ (all in units of ${\xi }_0$). ${\omega }_c$ and ${\omega }_{ab}$ are resonance frequencies.

Figure 3

Probability density of wave packet as a function of system modes for $\varphi =\pi /6$. The temporal evolution is shown for (a) $t=0$, (b) $t=180$, (c) $t=330$, and (d) $t=610$ (units of $\hslash /{\xi }_0$). The parameters are ${\omega }_c={\omega }_{ab}$, ${\xi }_1=0.2$, $t_{ab}=0.1$ (units of ${\xi }_0$), and $N=2000$. See Supplemental Material [30].

Figure 4

Scattering properties of a Gaussian wave packet. Figure shows the probability density of the wave packet as a function of the position and the time for phases (a) $\varphi =0$, (b) $\varphi =\pi /50$, and (c) $\varphi =\pi /6$. The details in the wave-packet propagation are plotted in scale 0 to $3\times {10}^{-3}$. The linear color scale of the insets are (a) 0 to $3\times {10}^{-3}$, (b) 0 to $1\times {10}^{-2}$, and (c) 0–0.2. The parameters of initial condition are $\sigma =250$, $x_0=1000$, $N=2000$, and $k_0=arccos(t_{ab}/2{\xi }_0)$. Always set ${\omega }_c={\omega }_{ab}$, ${\xi }_1=0.2$, and $t_{ab}=0.1$ (units of ${\xi }_0$).

Figure 5

Occupation probability of symmetric (red solid line) and antisymmetric (blue dotted line) mode as a function of time for (a) $\varphi =0$, (b) $\varphi =\pi /50$, and (c) $\varphi =\pi /6$. Always set ${\omega }_c={\omega }_{ab}$, ${\xi }_1=0.2$, $t_{ab}=0.1$ (all is in units of ${\xi }_0$), and $N=2000$ (cavities modes).

Figure 6

Temporal evolution of a single photon initially prepared in antisymmetric BIC mode. Probability density as a function of time and position for (a) $\varphi =\pi /50$, (b) and (c) $\varphi =\pi /6$. Always set ${\omega }_c={\omega }_{ab}$, ${\xi }_1=0.2$, $t_{ab}=0.1$ (all is in units of ${\xi }_0$), and $N=2000$ (cavities mode).