Realization of a Synthetic Hall Torus with a Spinor Bose-Einstein Condensate

TL;DR

The paper reports the first experimental realization of a synthetic Hall torus using a spinor Bose-Einstein condensate confined in a ring trap. By cyclically coupling the spin states $|m_F\rangle=\{-1,0,1\}$ with Raman and microwave fields, they impose periodic boundary conditions in the synthetic dimension, creating a torus with a synthetic magnetic flux that induces azimuthal density modulations; the modulation positions are controlled by the microwave phase $\theta_{\rm mw}$, enabling a torus analogue of Thouless charge pumping. The authors demonstrate topology-driven density modulations, study the transition from cylindrical to toroidal topology, and perform in situ imaging to observe robust twofold azimuthal patterns, whose location shifts linearly with $\theta_{\rm mw}$ with a characteristic nonsymmorphic symmetry. Complementary GP simulations and quench dynamics provide insight into branch mixing, non-adiabatic effects, and the nonequilibrium evolution when topology is changed, underscoring the platform’s potential for exploring quantum Hall physics and topological phenomena in synthetic curved spaces.

Abstract

We report the first experimental realization of a synthetic Hall torus using a spinor Bose-Einstein condensate confined in a ring-shaped trap with in situ imaging. By cyclically coupling three hyperfine spin states via Raman and microwave fields, we impose a periodic boundary condition in the synthetic dimension, which together with a real-space ring trap, realizes a toroidal geometry with a synthetic magnetic flux. This flux induces azimuthal density modulations in the condensate, whose periodicity is uniquely determined by the quantized toroidal magnetic flux-a hallmark of the Hall torus geometry. By varying the relative phase between the couplings across repeated experimental runs, we control the location of the density extrema, emulating the behavior of Thouless charge pump in a toroidal geometry. We further investigate the onset of these modulations as the system transitions from a cylindrical to a toroidal topology. Our results establish a versatile platform for investigating quantum Hall physics and topological phenomena in synthetic curved spaces.

Figures (6)

• Figure 1: (a) Experimental setup. (b) The torus combines a ring trap in real space with cyclically coupled spin states in the synthetic dimension. (c) Schematic of the cyclic coupling implemented by Raman (red) and microwave fields (blue). (d) Cyclic coupling displayed in the spin and OAM space. Solid (open) symbols denote atoms in torus branch 1(2).
• Figure 2: Atomic optical density (OD) of $|m_F=1,0,-1\rangle$ before turning on the microwave coupling $\Omega_{\rm mw}$ (a) and after turning on $\Omega_{\rm mw}$ in 50 ms with $\theta_{\rm mw} = 0$ and $t_h\lesssim 0.1$ ms (c). The color scale for the density profile is given by the maximum value in each panel. The field of view is $72 \times 72 {\ \mu{\rm m}}^2$. Azimuthal profiles of the ring (b) and torus (d) are displayed as the effective 1D density $\tilde{n}_{m_F=1}$ vs. $\phi$. (b) shows the average and standard deviation of 10 identical experimental realizations. Black (red) curve in (d) denotes the experimental data (fit). The images in (a) and (c) are single-shot.
• Figure 3: (a) Atomic OD of $|m_F=1\rangle$ vs. microwave phase $\theta_{\rm mw}$ with $t_{\rm on}=50$ ms and $t_h\lesssim 0.1$ ms. The field of view is $72 \times 72 {\ \mu{\rm m}}^2$. (b) Blue (red) symbols denote the azimuthal peak position $\mu_1~(\mu_2)$ vs. $\theta_{\rm mw}$. Lines denote the respective linear fits, both with slope $0.51\pm 0.01$.
• Figure 4: Dynamics after a sudden transition from a cylindrical to a toroidal geometry by abruptly turning on $\Omega_{\rm mw}$. Azimuthal widths $\sigma_1,\sigma_2$ of the torus peaks centered at $\phi=\mu_1,\mu_2$ vs. hold time $t_h$ are displayed in the top and bottom panel, respectively.
• Figure S1: (a) Optical density profiles of the torus in branch 1 for the ideal system with a perfect cylindrical symmetry. Branch 2 has identical density profiles as branch 1. (b) The area ratio $r_a$ as a function of the relative phase $\theta_{12}$ in the mixing of the two ground state branches of the torus. Branch mixtures of $(0.9,0.1)$ and $(0.98,0.02)$ are displayed.