Experimental Realization of Strong Effective Magnetic Fields in an Optical Lattice

We use Raman-assisted tunneling in an optical superlattice to generate large tunable effective magnetic fields for ultracold atoms. When hopping in the lattice, the accumulated phase shift by an atom is equivalent to the Aharonov-Bohm phase of a charged particle exposed to a staggered magnetic field of large magnitude, on the order of 1 flux quantum per plaquette. We study the ground state of this system and observe that the frustration induced by the magnetic field can lead to a degenerate ground state for noninteracting particles. We provide a measurement of the local phase acquired from Raman-induced tunneling, demonstrating time-reversal symmetry breaking of the underlying Hamiltonian. Furthermore, the quantum cyclotron orbit of single atoms in the lattice exposed to the magnetic field is directly revealed.

Figure 1

Experimental setup. (a) The experiment consists of a 2D array of 1D potential tubes with spacing $|{\mathbf{d}}_x|=|{\mathbf{d}}_y|={\lambda }_s/2$. While bare tunneling occurs along the $y$ direction with amplitude $J$, it is inhibited along $x$ owing to a staggered potential offset $\Delta$. A pair of Raman lasers with wave vectors ${\mathbf{k}}_{1,2}$ and frequency difference ${\omega }_1-{\omega }_2=\Delta /\hslash$, induces a resonant tunnel coupling of magnitude $K$ whose phase depends on position. This realizes an effective flux $\pm \varphi$ per plaquette with alternating sign along $x$. (b) Spatial distribution of the phase of the Raman-induced tunnel coupling realized in the experiment. The gray shaded area highlights the magnetic unit cell.

Figure 2

Momentum distribution measured after a time of flight of 20 ms for (a) simple cubic lattice, (b) $J/K=1.0(1)$ and (c) $J/K=2.5(1)$. The latter two are compared with theoretical profiles obtained by an exact diagonalization of Hamiltonian (1) on a $31\times 31$ lattice with harmonic confinement [15]. Red squares in the theoretical profiles indicate the magnetic Brillouin zone and the crosses mark the center. (d) Projection along $y$ of the momentum peaks located at $k_x=+k_s/4$, as a function of $J/K$. For $J/K<\sqrt{2}$ the peaks are located at $k_y=k_s/4$, while for $J/K>\sqrt{2}$ the peaks are split due to the emergent ground-state degeneracy (see insets). The solid lines correspond to the minima of the lowest Bloch band for a translationally invariant system [15].

Figure 3

Dispersion relation of the lowest Bloch band, calculated in the tight-binding approximation [15] for (a) $J/K=1$ and (b) $J/K=2.5$ and plotted for the first magnetic Brillouin zone. The red crosses mark the positions of the corresponding momentum peaks together with the ones shifted due to the intrinsic structure of the wave function. (c) Histogram of the measured fraction of atoms in peaks corresponding to the lower momentum state for $J/K=2.5$. The measurement was performed over 172 identical experimental runs. (d)–(e) Spatial distribution of the phase and atomic density (color brightness) for the ground-state wave function [15]. The vortices with different chirality in the phase distribution for $J/K=1$ (d) are illustrated by the rotation of the white arrows. While in this case the atomic density is uniform, it exhibits a charge density wave for $J/K=2.5$ (e). For the second degenerate ground state we observe similar behavior but the density pattern is shifted by one lattice site.

Figure 4

Time-reversal symmetry breaking and cyclotron orbits. (a) Phase evolution of the double-slit pattern along $y$ (integrated along $x$), as a function of time for $\hslash \omega =\Delta$ (blue) and $\hslash \omega =-\Delta$ (gray). The inset shows the Fourier transformation for $\hslash \omega =\Delta$ depicting two frequency components at 0.24(6) kHz and 0.62(13) kHz, in good agreement with theory (vertical lines). (b) Cyclotron orbits of the average particle position obtained from the mean atom positions $⟨X⟩/d_x$ (c) and $⟨Y⟩/d_y$ (d) for $J/h=0.50(2)\mathrm{kHz}$, $K/h=0.27(1)\mathrm{kHz}$, and $\Delta /h=5.05(2)\mathrm{kHz}$. Each data point is an average over three measurements. The inset in (b) shows the theoretical curve calculated for $\varphi =0.80\times \pi /2$ and a $1/e$-damping time of 13 ms obtained from damped sine fits to $⟨X⟩/d_x$ and $⟨Y⟩/d_y$.