Topological Quantum Optics in Two-Dimensional Atomic Arrays

We demonstrate that two-dimensional atomic emitter arrays with subwavelength spacing constitute topologically protected quantum optical systems where the photon propagation is robust against large imperfections while losses associated with free space emission are strongly suppressed. Breaking time-reversal symmetry with a magnetic field results in gapped photonic bands with nontrivial Chern numbers and topologically protected, long-lived edge states. Due to the inherent nonlinearity of constituent emitters, such systems provide a platform for exploring quantum optical analogs of interacting topological systems.

Figure 1

(a) Honeycomb lattice of atomic emitters with interatomic spacing $a$. Each atom has a $V$-type level structure with optical transitions to the $|{\sigma }_{+}⟩$ and $|{\sigma }_{-}⟩$ states. A magnetic field breaks the degeneracy via the Zeeman splitting. (b) Band structure of the lattice with $B=0$. Green dashed lines indicate the edges of the free-space light cone. Modes with quasimomentum $k_{\mathrm{B}}<{\omega }_{{\mathbf{k}}_B}/c$ couple to free-space modes and are short lived (green shaded region). Decay rates of the modes are color coded. Bands are degenerate at the symmetry points $\mathbf{}\mathbf{K}$ and $\mathbf{}\mathbf{\Gamma }$. (c) A transverse magnetic field ($\mu B=12{\mathrm{\Gamma }}_0$) opens a gap (grey shaded region) between topological bands with nontrivial Chern numbers. Relevant parameters are $\lambda =790\mathrm{nm}$, ${\mathrm{\Gamma }}_0=2\pi \times 6\mathrm{MHz}$, and $a=0.05\lambda$.

Figure 2

(a) Size of the gap between topological bands (blue line) as a function of magnetic field for $a=\lambda /20$. (b) The maximum gap size ${\mathrm{\Delta }}_{\max }$ (blue dotted line) as a function of the interatomic spacing $a$. The solid magenta line shows the dipolar interaction strength $J$ between two atoms with parallel dipole moments. The dashed green line is a phenomenological $J\sim 1/r^3$ fit. For $a\ll \lambda$, ${\mathrm{\Delta }}_{\max }$ scales as $1/a^3$.

Figure 3

Topological edge states on the (a) bearded and (b) armchair edges of periodic stripes of atoms. Each edge supports only one unidirectional mode. Modes propagating on the upper (lower) edges of the stripes are marked by diamonds (squares) in the band diagrams. Bulk modes are marked with dots. Decay rates of the modes are color coded. Modes of the bearded (armchair) edges cross the gap with quasimomentum $k_{\mathrm{B}}>{\omega }_{{\mathbf{k}}_B}/c$ ($k_{\mathrm{B}}<{\omega }_{{\mathbf{k}}_B}/c$) making them long (short) lived. Parameters are the same as in Fig. 1. The spectrum was obtained for the bearded (armchair) edges from a $40\times 42$ ($40\times 41$) lattice of atoms with periodic boundary conditions along the first dimension. States for which the ratio of the total amplitude on the top (bottom) four atom rows to the bottom (top) four rows is greater than 15 are classified as edge states.

Figure 4

Snapshot of the time evolution (at $t=5.7{\mathrm{\Gamma }}_0^{-1}$) of the system as an atom on the edge (red star) is driven by a laser (inset). The color code shows the excitation probability $|⟨\psi (t)|{\sigma }_{+}^i⟩{|}^2+|⟨\psi (t)|{\sigma }_{-}^i⟩{|}^2$ at each atomic site $i=1,\dots ,N$. Approximately 96% of the emitted excitation is coupled into the forward direction and scattering into bulk and backward edge modes is strongly suppressed. The excitation goes around corners and routes around the large lattice defect. Relevant parameters are $N=1243$, $\lambda =790\mathrm{nm}$, ${\mathrm{\Gamma }}_0=2\pi \times 6\mathrm{MHz}$, $a=0.05\lambda$, and $\mu B=12{\mathrm{\Gamma }}_0$. The strength of the drive is $\mathrm{\Omega }=1/5{\mathrm{\Gamma }}_0$ and the driving frequency is ${\omega }_{\mathrm{L}}={\omega }_A+15{\mathrm{\Gamma }}_0$. The driving laser is adiabatically switched on with a Gaussian profile $\mathrm{\Omega }(t)=\mathrm{\Omega }\exp (-[t-1.5{\mathrm{\Gamma }}_0^{-1}{]}^2/[0.15{\mathrm{\Gamma }}_0^{-2}])$ for $t<1.5{\mathrm{\Gamma }}_0^{-1}$.