Chirality-induced emergent spin-orbit coupling in topological atomic lattices

Spin-orbit coupling is of fundamental interest in both quantum optical and condensed matter systems alike. In this work, we show that optically induced electronic excitations in lattices of V-type atoms exhibit an emergent spin-orbit coupling when the geometry is chiral. This spin-orbit coupling arises naturally from the electric dipole interaction between the atomic sites and leads to a nontrivial topology for the lattice band structure. Using a general quantum optical model, we determine analytically the conditions that give rise to spin-orbit coupling and characterize the behavior under various symmetry transformations. We demonstrate that chirality-induced spin-orbit coupling can result from either the chirality of the underlying lattice geometry or the combination of an achiral lattice with a suitably chosen external quantization axis. We then discuss how these results are influenced by dissipation, which breaks time-reversal symmetry and illuminates the distinction between true and false chirality. Our results demonstrate that chiral atom arrays are a robust platform for realizing spin-orbit-coupled topological states of matter.

Figure 1

Schematic of the V-type level structure considered in this work. The excited-state orbitals have resonant frequency ${\omega }_0$ and spontaneous emission rate ${\mathrm{\Gamma }}_0$. Emitters are coupled with coherent and dissipative hopping rates $J_{ij}^{\sigma {\sigma }^{\prime }}$ and ${\mathrm{\Gamma }}_{ij}^{\sigma {\sigma }^{\prime }}$, respectively.

Figure 2

(a) Schematic of the helical lattice geometry for pitch $a$ and $\mathcal{N}=3$. The gray dashed rectangle denotes one unit cell, whereas the yellow solid rectangle denotes one sublattice. (b) Top-down view of the helical lattice. The sublattices are arranged with radial position $r_0$ and azimuthal separation $2\pi /\mathcal{N}$.

Figure 3

(a) Band structures for the left-handed (left) and right-handed (right) helices when $\widehat{\mathbf{q}}=\widehat{\mathbf{z}}$. The spin bands are nontrivially antisymmetric about $\mathbf{k}={\mathrm{\Gamma }}^i$, irrespective of $\mathcal{T}$ invariance. The system exhibits true chirality with a quantized nontrivial Zak phase. Additional parameters are $\mathcal{N}=3, r_0=0.05{\lambda }_0$, and $a=0.175{\lambda }_0$. (b) Top-down view of the helical unit cell for $\widehat{\mathbf{q}}=\widehat{\mathbf{z}}$. The $\mu =2$ sublattice serves as an anti-inversion for the lattice, and the corresponding transformation $\overline{\mathcal{P}}:(\widehat{\mathbf{k}},\sigma )\to (-\widehat{\mathbf{k}},\overline{\sigma })$ enforces antisymmetric Bloch bands. If dissipation is neglected, the spin antisymmetry is also protected by $\mathcal{T}$ symmetry.

Figure 4

(a) Top-down view of the helical unit cell for $\mathbf{q}$ oriented along the ${\varphi }_0$ axis. The $\mu =1$ and $\mu =3$ sublattices are separated by an angle $\pi$ in the polarization plane such that the $\mu =2$ sublattice serves as a true inversion center. Inversion symmetry transforms $\mathcal{P}:(\widehat{\mathbf{k}},\sigma )\to (-\widehat{\mathbf{k}},\sigma )$, enforcing spin symmetric Bloch bands. If the Hamiltonian is also $\mathcal{T}$ invariant, the additional relation $\mathcal{T}:(\widehat{\mathbf{k}},\sigma )\to (-\widehat{\mathbf{k}},\overline{\sigma })$ requires $\langle {\mathcal{S}}_z\rangle =0$. (b) Band structures for the ${\varphi }_0$-polarized helix. The system is $\mathcal{P}$ invariant with a quantized Zak phase. When $\mathcal{T}$ symmetry is broken (left), the spin bands are symmetric about $\mathbf{k}={\mathrm{\Gamma }}^i$ and the Zak phase is topologically nontrivial. If $\mathcal{T}$ symmetry is restored (right), the combined $\mathcal{P}\mathcal{T}$ symmetry causes both the spin textures and the Zak phase to vanish. The system is therefore falsely chiral. The helix parameters are the same as in Fig. 3.

Figure 5

Band structures for the transversely polarized helix in the absence of both $\mathcal{P}$ and $\overline{\mathcal{P}}$ invariance. When $\mathcal{T}$ symmetry is broken (left) the spin bands are neither symmetric nor antisymmetric about $\mathbf{k}={\mathrm{\Gamma }}^i$. When $\mathcal{T}$ symmetry is restored (right) the spin bands become antisymmetric. In both cases, the Zak phase is not quantized. The helix parameters are the same as in Figs. 3 and 4.

Figure 6

(a) Lattice geometry for an oblique triangular prism. Each triangular face represents a single unit cell. The faces are assumed to be isosceles with the $x\text{−}z$ plane chosen as the mirror plane. (b) Top-down view of the oblique triangular prism unit cell for $\widehat{\mathbf{q}}=\widehat{\mathbf{x}}$ (top) and $\widehat{\mathbf{q}}=\widehat{\mathbf{y}}$ (bottom). (c) Band structures for the oblique triangular prism. When $\mathbf{q}$ is oriented in the mirror plane (left), the spin textures vanish irrespective of the behavior under $\mathcal{T}$. If $\mathbf{q}$ is instead oriented orthogonal to the mirror plane, then the SO coupling can be nonzero. In both cases, the Zak phase is not quantized. Additional geometrical parameters are the same as in Figs. 3, 4, 5.