Band Gap Closing in a Synthetic Hall Tube of Neutral Fermions

We report the experimental realization of a synthetic three-leg Hall tube with ultracold fermionic atoms in a one-dimensional optical lattice. The legs of the synthetic tube are composed of three hyperfine spin states of the atoms, and the cyclic interleg links are generated by two-photon Raman transitions between the spin states, resulting in a uniform gauge flux $\varphi$ penetrating each side plaquette of the tube. Using quench dynamics, we investigate the band structure of the Hall tube system for a commensurate flux $\varphi =2\pi /3$. Momentum-resolved analysis of the quench dynamics reveals a critical point of band gap closing as one of the interleg coupling strengths is varied, which is consistent with a topological phase transition predicted for the Hall tube system.

Figure 1

Realization of a synthetic Hall tube with neutral atoms. (a) Schematic of the experimental setup. Fermionic ${}^{173}\mathrm{Yb}$ atoms are confined in an optical lattice and illuminated by three Raman laser beams $R_{1,2,3}$. A magnetic field $B$ and an additional laser light (LB) are applied along $\widehat{z}$ to control the energy levels of the spin states. (b) The three lowest spin states of ${}^{173}\mathrm{Yb}$ are coupled to each other via two-photon Raman transitions by $R_{1,2,3}$. (c) Synthetic three-leg Hall tube with a uniform gauge flux $\varphi$ on each side plaquette. The three legs are formed by the three spin states of the atoms in the 1D optical lattice (black lines) and the interleg tunneling with complex amplitude (gray lines) is provided by the cyclic Raman couplings between the spin states.

Figure 2

Quench dynamics of the three-leg Hall tube for $\varphi =2\pi /3$. (a) Illustration of the atomic motion in the Hall tube. Atoms are initially prepared in the $\mathrm{spin}\text{−}|1⟩$ leg and the interleg couplings are suddenly activated. (b) Time evolution of the lattice momentum distribution $n(k,t)$ of the sample, $n_2(k,t)$ of the atoms in $|2⟩$, and $n_3(k,t)$ of the atoms in $|3⟩$. Time evolution of (c) the fractional spin populations, (d) the average lattice momentum $⟨k⟩$ of the sample, and the difference $\mathcal{C}(t)=⟨k_2⟩-⟨k_3⟩$ between the momenta of the two legs $|2⟩$ and $|3⟩$. Each data point comprises five measurements of the same experiment, and the error bar is their standard deviation. The solid and dashed lines in (d) show the numerical simulation results for $\mathcal{C}$ and $⟨k⟩$, respectively [26].

Figure 3

Quench dynamics of (a) two-leg and (b) three-leg ladders with open boundaries for $\varphi =2\pi /3$. Time evolution of (c),(d) the fractional spin populations and (e),(f) the average lattice momentum $⟨k⟩$. The solid lines display the numerical simulation results for $⟨k⟩$ [26]. Each data point comprises five measurements of the same experiment, and the error bar is their standard deviation.

Figure 4

Observation of band gap closing in the Hall tube. (a) Phase diagram in the plane of ${\mathrm{\Omega }}_{12}$ and ${\mathrm{\Omega }}_{31}$. The shaded area indicates a topologically nontrivial phase with a nonzero Zak phase $Z=1$ of the lowest band. The solid dots indicate the positions explored in the experiment. (b) Band structures calculated for various ${\mathrm{\Omega }}_{31}$ with ${\mathrm{\Omega }}_{12}={\mathrm{\Omega }}_{23}\approx 12.3t_x$. A topological phase transition occurs together with band gap closing at $q_c=\pm \pi$. (c) Quench evolution of $n_2(k_{2c})$ and $n_3(k_{3c})$, where $k_{2c}$ and $k_{3c}$ are the lattice momenta of $|2⟩$ and $|3⟩$, respectively, corresponding to $q_c$. Each data point is obtained by averaging five measurements and the error bar is their standard deviation. The time ${\tau }_2$ (${\tau }_3$) for the first maximum of $n_2(k_{2c})$ [$n_3(k_{3c})$] is determined by fitting the experimental data to an asymmetric parabola function (solid line). (d) ${\tau }_2$ and ${\tau }_3$ as functions of ${\mathrm{\Omega }}_{31}$. The red and blue dashed lines show the numerical results for ${\tau }_2$ and ${\tau }_3$, respectively. ${\tau }_2={\tau }_3$ indicates the critical point of band gap closing.