Dynamic optical superlattices with topological bands

We introduce an all-optical approach to producing high-flux synthetic magnetic fields for neutral atoms or molecules by designing intrinsically time-periodic optical superlattices. A single laser source, modulated to generate two frequencies, suffices to create dynamic interference patterns which have topological Floquet energy bands. We propose a simple laser setup that realizes a tight-binding model with uniform flux and well-separated Chern bands. Our method relies only on the particles' scalar polarizability and far-detuned light.

Figure 1

Two topological tight-binding models (a) and their realization in dynamic optical superlattices (b),(c): (I) Haldane-like model on a honeycomb lattice, formed by three running-wave beams and a circularly polarized coupling laser. (II) Triangular lattice with uniform flux, formed by two (or three) retroreflected lattice beams and a linearly polarized coupling laser. In (b), shading indicates the static potential $V$, while arrows indicate the magnitude and phase of the dynamic modulation $\tilde{V}$. In (c), thick red (dashed orange) arrows represent the lattice beams' $p$ ($s$) polarization components; orange labels indicate the retardances ${\alpha }_i$. In (II)(c), dotted green lines indicate an optional third standing wave, offset in frequency to avoid interference with the first two, that improves the isotropy of the triangular lattice. Bold black phases ${\phi }_{-2,+3}$ must be stabilized relative to ${\phi }_{\pm 1,+2,-3}=0$.

Figure 2

Peierls-like phases for hopping between sites (along the direction indicated by the black arrows on the bonds) in the tight-binding descriptions of the honeycomb (a) and triangular (b) lattice models. Arrows on the lattice sites show the phase ${\varphi }_{\mathbf{R}}$ of the drive potential. Gray (white) circles denote $A$ ($B$) sites as defined in the main text.

Figure 3

(a) Floquet-Bloch bands for the honeycomb lattice with increasing modulation frequency $\Omega$. The lattice parameters are (a) $V_0=10E_R$, $V_1=0.5E_R$, and ${\tilde{V}}_H=0.4E_R$, where $E_R={\hslash }^2{k}^2/2m$. Contour plots show the Berry curvature (b) and dispersion (c) of the ground band for $\Omega =1.71E_R$.

Figure 4

Floquet-Bloch bands (top) and the corresponding Berry curvature and dispersion relation of the lower band (bottom) for the triangular lattice with $\Phi =\pi /2$. In (d)–(f) the optional pair of laser beams [green dashed line in Fig. 1] has been left out, whereas the lattice shown in (a)–(c) is approximately sixfold symmetric. The lattice parameters $(V_0,V_1,\hslash \Omega )$ are (a)–(c) $(6,0.6,0.85)E_R$ [(b)–(d) $(6,0.6,0.51)E_R$]. These result in a width of the lowest band $W=0.0045E_R$ [$W=0.02E_R$] and a ratio of band gap to bandwidth $\Delta /W\sim 1.9$ [$\Delta /W\sim 1.3$].