Synthetic Gauge Fields in Synthetic Dimensions

We describe a simple technique for generating a cold-atom lattice pierced by a uniform magnetic field. Our method is to extend a one-dimensional optical lattice into the “dimension” provided by the internal atomic degrees of freedom, yielding a synthetic two-dimensional lattice. Suitable laser coupling between these internal states leads to a uniform magnetic flux within the two-dimensional lattice. We show that this setup reproduces the main features of magnetic lattice systems, such as the fractal Hofstadter-butterfly spectrum and the chiral edge states of the associated Chern insulating phases.

Figure 1

(a) Proposed experimental layout with ${}^{87}\mathrm{Rb}$. A pair of counterpropagating $\lambda =1064\mathrm{nm}$ lasers provide a $5E_L$ deep optical lattice lattice with period $a=\lambda /2$. A pair of “Raman” laser beams with wavelength ${\lambda }_R=790\mathrm{nm}$, at angles $\pm \theta$ from ${\mathbf{e}}_x$, couple the internal atomic states with recoil wave vector $k_R=2\pi \cos (\theta )/{\lambda }_R$. The laser beams’ polarizations all linear—are marked by symbols at their ends. (b) Raman couplings in the $F=1$ manifold. The Raman lasers, far detuned from the electronic excited state, connect to the ground electronic states as indicated with the green and blue arrows [each coupling resulting from the like-colored laser beam in Fig. 1]. (c) Synthetic 2D lattice with magnetic flux $\mathrm{\Phi }=\gamma /2\pi$ per plaquette ($\gamma =2k_{R}a$).

Figure 2

Spectrum for open boundary conditions: $F=1$ and $\mathrm{\Phi }=\gamma /2\pi =1/2\pi$ flux per plaquette. Colors specify the spin state $m$, i.e., blue, green, and red correspond to $m=-1,0,1$, respectively. The ground-state branch displays “edges” corresponding to $m=\pm 1$.

Figure 3

Edge-state dynamics. (a) Initial condition: A Fermi gas is trapped in the central region $x\in [-13a,13a]$ of a lattice with ${\mathrm{\Omega }}_0=0.5t$, $\mathrm{\Phi }=1/2\pi$, and the Fermi energy $E_{\mathrm{F}}=-1.4t$ is chosen to populate only the lowest energy band. The occupied edge states localized at $m=\pm F$ have opposite group velocities (for simplicity we sketch the $F=1$ case). (b) Time-evolving density $\rho$ after releasing the cloud into a harmonic potential $V_{\text{harm}}(x)=t(x/50a{)}^2$. The confinement limits the edge-state propagation and leads to chiral dynamics around the synthetic 2D lattice. Dashed white lines represent the Fermi radius $R_{\mathrm{F}}$ at which the edge states localized at $m=\pm F$ jump to the opposite edge $m=\mp F$.

Figure 4

Edge-state transmission probability. Short dashes (black): A spin-independent impurity. Solid (blue): Only $m=0$ scatters. Long dashes (red): Only $m=1$ scatters. Parameters are the same as in Fig. 2 and the scattering strength is $U=-t$.

Figure 5

The spectrum of Eq. (5) on an infinite 1D chain, for a three-level system with closed boundary conditions has the typical Hofstadter-butterfly characteristics.