Photonic band structure of two-dimensional atomic lattices

Two-dimensional atomic arrays exhibit a number of intriguing quantum optical phenomena, including subradiance, nearly perfect reflection of radiation, and long-lived topological edge states. Studies of emission and scattering of photons in such lattices require complete treatment of the radiation pattern from individual atoms, including long-range interactions. We describe a systematic approach to perform the calculations of collective energy shifts and decay rates in the presence of such long-range interactions for arbitrary two-dimensional atomic lattices. As applications of our method, we investigate the topological properties of atomic lattices both in free space and near plasmonic surfaces.

Figure 1

Constituent atoms in the 2D atomic lattice. Each atom is assumed to have one ground $|g\rangle$ and three excited states $|{\sigma }_{+}\rangle$, $|{\sigma }_{-}\rangle$, and $|\pi \rangle$ that can be Zeeman split by a magnetic field.

Figure 2

(a) Non-Bravais square lattice of atoms in free space. Atoms marked in red (green) have resonant frequency ${\omega }_A^{\left(1\right)}$ (${\omega }_A^{\left(2\right)}={\omega }_A^{\left(1\right)}+\delta \omega$). Each atom has two excited states $|{\sigma }_{+}\rangle$ and $|{\sigma }_{-}\rangle$ that are Zeeman shifted by $\pm \mu B$ in the presence of an out-of-plane magnetic field $B$. (b) Part of the photonic band structure of the lattice for $B=0$. Green dashed lines mark the edges of the light cone. Modes that have quasimomentum $k_B<{\omega }_{{\mathbf{k}}_B}$ (green shaded region) can couple to free-space modes of the same momentum, making them short lived. Modes that have quasimomentum $k_B>{\omega }_{{\mathbf{k}}_B}$ cannot couple to free-space modes and are long lived. Decay rates of the modes are color coded. Bands are degenerate near the midpoint of the line joining the $\mathbf{X}$ and $\mathrm{\Gamma }$ points. (c) An out-of-plane magnetic field ($\mu B=20{\mathrm{\Gamma }}_0$) opens a gap ($\mathrm{\Delta }=7{\mathrm{\Gamma }}_0$) between the bands (gray shaded region). The bands acquire nontrivial Chern numbers (purple numbers). Relevant parameters are $\lambda =790\text{nm}$, $\delta \omega =30{\mathrm{\Gamma }}_0$, ${\mathrm{\Gamma }}_0=2\pi \times 6\text{MHz}$, and $a=0.054\lambda$.

Figure 3

(a) Periodic strip of atoms in a non-Bravais square lattice. Each edge has edge modes propagating only in a single direction. (b) Modes that propagate on the upper (lower) edge are marked by squares (diamonds) in the band-structure diagram. Extended bulk modes are marked with dots. There are two sets of edge modes on each boundary. Decay rates of modes are color coded. The edge modes cross the gap with quasimomentum $k_B>{\omega }_{{\mathbf{k}}_B}$ and thus are long lived. Parameters are the same as in Fig. 2. The spectrum was obtained for a $80\times 40$ lattice of atoms with periodic boundary conditions along one direction. A state is classified as an edge state on the (upper) lower boundary if the excitation probability on the top (bottom) four rows is at least 15 times the excitation probability on the bottom (top) four rows.

Figure 4

Snapshot of the time evolution (at $t=11.1{\mathrm{\Gamma }}_0^{-1}$) of a finite non-Bravais square lattice in free space with an irregular defect. An atom on the edge (marked by the red star) is driven by a laser (see inset). The color code depicts the excitation probability $|\langle \psi \left(t\right)|{\sigma }_{+}^i\rangle {|}^2+{\left|\langle \psi \left(t\right)|{\sigma }_{-}^i\rangle \right|}^2$ at each site. Approximately 96% of the emitted excitation is coupled into the forward direction. Emission into bulk modes and the backward edge modes is suppressed. The excitation goes around the corners and routes around the large, irregular defect, which demonstrates the robustness of the topological excitation to disorder. Relevant parameters are $N=1576$, $\lambda =790\text{nm}$, $\delta \omega =30{\mathrm{\Gamma }}_0$, ${\mathrm{\Gamma }}_0=2\pi \times 6\text{MHz}$, $a=0.054\lambda$, and $\mu B=20{\mathrm{\Gamma }}_0$. The strength of the drive is $\mathrm{\Omega }=0.1{\mathrm{\Gamma }}_0$ and the driving frequency is ${\omega }_{\text{L}}={\omega }_A+18{\mathrm{\Gamma }}_0$. The driving laser is adiabatically switched on with a Gaussian profile $\mathrm{\Omega }\left(t\right)=\mathrm{\Omega }exp(-{[t-4.5{\mathrm{\Gamma }}_0^{-1}]}^2/\left[1.35{\mathrm{\Gamma }}_0^{-2}\right])$ for $t<4.5{\mathrm{\Gamma }}_0^{-1}$.

Figure 5

(a) Triangular lattice of atoms with interatomic spacing $a$ in free space. Each atom has two excited states $|{\sigma }_{+}\rangle$ and $|{\sigma }_{-}\rangle$ with Zeeman splitting $\pm \mu B$ due to an out-of-plane magnetic field $B$. (b) Photonic band structure for $B=0$ and pointlike atoms pinned to their lattice sites. Green dashed lines mark the light cone edges. Modes with quasimomentum $k_B<{\omega }_{{\mathbf{k}}_B}$ (green shaded region) couple to free space, making them short lived. Modes with quasimomentum $k_B>{\omega }_{{\mathbf{k}}_B}$ are long lived. Decay rates of the modes are color coded. Bands are degenerate at the $\mathrm{\Gamma }$ and $\mathbf{K}$ points. (c) A transverse magnetic field ($\mu B=0.5{\mathrm{\Gamma }}_0$) opens a small gap ($\mathrm{\Delta }=0.5{\mathrm{\Gamma }}_0$) between the bands (gray shaded region). The bands acquire nontrivial Chern numbers (purple numbers). (d) When the atoms fluctuate around the lattice sites, the size of the gap is reduced. Here the amplitude of fluctuations is $a_{\text{ho}}=0.4a$. Relevant parameters for all three plots are $\lambda =790\text{nm}$, ${\mathrm{\Gamma }}_0=2\pi \times 6\text{MHz}$, and $a=\lambda /2$.

Figure 6

(a) Square lattice of atoms with interatomic spacing $a$ and distance $h$ from a silver surface. Each atom has three transitions to the $|{\sigma }_{+}\rangle$, $|\pi \rangle$, and $|{\sigma }_{-}\rangle$ states. (b) Band structure of the lattice. Green dashed lines mark the edges of the free-space light cone (green shaded region); purple dashed lines show the plasmonic dispersion of the metal surface. The decay rate of the modes is color coded. Due to the strong coupling of the $|\pi \rangle$ state to the plasmons, a nontopological gap opens in the spectrum (gray shaded region). Relevant parameters are $\lambda =737\text{nm}$, ${\mathrm{\Gamma }}_0=2\pi \times 0.95\text{MHz}$, $a=\lambda /3$, and $h=\lambda /10$.

Figure 7

(a) Triangular lattice of atoms with transitions to the $|{\sigma }_{+}\rangle$ and $|{\sigma }_{-}\rangle$ states near a metal surface. (b) Band structure of the lattice for $B=0$. Green dashed lines mark the edges of the free-space light cone (green shaded region); purple dashed lines show the plasmonic dispersion of the metal surface. The decay rate of the modes is color coded. Bands are degenerate at the $\mathrm{\Gamma }$ and $\mathbf{K}$ points. (c) A transverse magnetic field ($\mu B=0.5{\mathrm{\Gamma }}_0$) opens two small gaps between bands that have nontrivial Chern numbers. The relevant parameters are $\lambda =437\text{nm}$, ${\mathrm{\Gamma }}_0=2\pi \times 0.95\text{MHz}$, $a=\lambda /1.95$, and $h=\lambda /15$.