Coherent interaction of a quantum emitter and the edge states in two-dimensional optical topological insulators

The two-dimensional optical topological insulators have exhibited a series of novel optical behaviors, especially the topologically protected edge states. We theoretically locate an initial excited quantum emitter (QE) in the two-dimensional ring resonator array and analyze the evolution of the QE in three kinds structures, i.e., topologically nontrivial, trivial, and graphene structure. We found that the dynamic evolution in the nontrivial topological structure is much different from that in the trivial and the graphene structures. When the QE excites edge states in nontrivial structures, the excited edge states can detour along the boundary and re-interact coherently with the QE, resulting in the recovery of the QE population. However, this repeated energy exchange is incomplete, depending on the coupling $g$ between the QE and the resonator. More importantly, we check the influence of the QE's position and the coupling $g$ (between the QE and the resonator) on dynamic evolution of the whole system. For the nontrivial structure, the effective excitation of edge states by the QE requires that: (1) the QE is located in the resonator in boundary and (2) $g$ is much less than coupling coefficient $J$ (between neighbor resonators). When $g$ is much larger than $J$, the QE cannot excite edge states anymore. We give the critical value $g_{\mathrm{c}}$ and find that it relates to band gap and size. Our paper is conducive to an in-depth understanding of the interaction between light and matter in optical topological insulators and provides a useful reference for quantum communication and quantum computing mediated by topological photonics.

Figure 1

(a) Scheme of the two-dimensional coupled ring resonator array. The orange dot represents the QE, which is placed in a ring resonator. (b) Schematics of the coupling between ring resonators. (c) Scheme of the optical graphene structure, and the dots represent identical ring resonators.

Figure 2

Energy-band diagram of three types of 2D infinite periodic structures. (a) Nontrivial topological structure (two-dimensional coupled ring resonator array with $M=0)$. (b) Trivial structure (two-dimensional coupled ring resonator array with $M=10J$). (c) Graphene-type structure. $a$ represents the lattice constant.

Figure 3

Eigenvalues $\omega -{\omega }_0$ and eigenstates $|{\psi }_N{|}^2$ of finite optical structures. (a) Eigenvalues and (b) eigenstates with $N=46$, 54, 58, and 66 marked by red $A, B, C$, and $D$ of the topologically nontrivial structure with $M=0$. (c) Eigenvalues and (d) eigenstates with $N=46$, 54, 58, and 66 marked by red $A, B, C$, and $D$ of the topologically trivial structure with $M=10J$. (e) Eigenvalues and (f) eigenstates with $N=138$, 148, 152, and 162 marked by red $A, B, C$, and $D$ of the finite graphene array. Other parameters are shown in the context.

Figure 4

The population evolution of the QE ${\left|w\left(t\right)\right|}^2$ in (a) the optical topological nontrivial finite structure, (c) the trivial finite structure, and (e) the optical graphene-type finite structure. Snaps of probability spatial distribution $|v_{x,y}\left(t\right){|}^2$ at $t=0,22J^{-1},44J^{-1},\mathrm{and}66J^{-1}$ in (b) the optical topological nontrivial finite structure, (d) the trivial finite structure, and (f) the optical graphene-type finite structure. The initially excited QE is located in the resonator at $\mathit{m}=(2,1)$, and the coupling between the QE and the resonator is $g=J$. Other parameters are shown in the context.

Figure 5

(a) Population evolution of the QE ${\left|w\left(t\right)\right|}^2$ and (b) snapshots of probability spatial distribution $|v_{x,y}\left(t\right){|}^2$ at $t=50J^{-1}$ when the QE is placed in the resonator at $\mathit{m}=\left(7,6\right)$. Other parameters are the same as those in Fig. 4.

Figure 6

The population evolution of the QE ${\left|w\left(t\right)\right|}^2$ in the optical topological nontrivial finite structure for $g=0.3J$ (blue dotted curve), $g=0.8J$ (orange dot-dashed curve), $g=1.0J$ (yellow dashed curve), and $g=1.5J$ (purple solid curve). Other parameters are the same as those in Fig. 4.

Figure 7

The probability on the QE for the $N\mathrm{th}$ eigenstate $|{\mathrm{\Psi }}_{\mathrm{QE},N}{|}^2$ of the whole system as a function of $g$. Other parameters are the same as those in Fig. 4.

Figure 8

Probability spatial distribution of the eigenstates $|{\mathrm{\Psi }}_{\mathrm{QE},N}{|}^2$ with (a) $N=7$, (b) $N=57$, and (c) $N=100$ of the whole system when $g=3.0J$. Other parameters are the same as those in Fig. 4. The QE is referred by a thick red arrow.

Figure 9

The probability on the QE for the $N\mathrm{th}$ eigenstate $|{\mathrm{\Psi }}_{\mathrm{QE},N}{|}^2$ for $g=1.500J$ when (a) $M=0$ and (b) $M=0.15J$. $\max \left(|{\mathrm{\Psi }}_{\mathrm{QE},N\in \mathrm{edge}}{|}^2\right)\text{−}\max (|{\mathrm{\Psi }}_{\mathrm{QE},N\in \mathrm{bulk}}{|}^2$ as a function of $g$ for (c) $M=0$ and (d) $M=0.15J$. Other parameters are the same as those in Fig. 4.