Chiral edge transport, the frictionless, directional propagation of particles at the boundary of topological materials, is a key phenomenon in condensed matter physics. It underpins the integer and fractional quantum Hall effects, showcasing remarkable robustness against disorder due to the quantization of Hall conductivity. While theoretically significant, the controllable injection, imaging, and dynamic study of edge modes pose significant challenges. Existing condensed matter platforms often suffer from issues like insufficient spatial resolution, limited control over wall geometry, and unwanted probe-sample coupling. Ultracold quantum gases offer a potentially superior platform due to their high degree of tunability and the ability to directly image their dynamics. Artificial magnetic fields can be created using various techniques, including rotation of the trapped gas, which provides a direct analogy between the Lorentz and Coriolis forces. Bosonic atoms, unlike fermions, occupy a single wavefunction in the mean-field quantum Hall regime, offering unique microscopic insights. However, previous studies using lattice systems with synthetic dimensions have faced difficulties in exploring the effects of interactions and wall structure. This work aims to overcome these limitations by directly observing and characterizing chiral edge modes in a rapidly rotating bosonic superfluid, providing a highly controllable platform to study the fundamental aspects of edge physics and its dependence on wall structure and disorder.

The integer quantum Hall effect's discovery led to the realization that the quantized electrical conductivity could be explained by either bulk states or edge channels, showcasing the bulk-edge correspondence. This correspondence links edge mode properties to bulk topological invariants. Edge modes subsequently emerged as ubiquitous features in various topological systems, including fractional quantum Hall fluids, spin Hall fluids, topological insulators, photonic platforms, and exotic superfluids and superconductors. However, the interplay of edge disorder, interactions, and wall geometry in real materials significantly complicates the study of edge transport, making it crucial to utilize clean, tunable experimental platforms with precise control and direct imaging capabilities to disentangle these effects and observe the fundamental physics.

The experiment employs a condensate of approximately 8 x 10⁵ ²³Na atoms in a time-orbiting-potential trap, creating a synthetic magnetic field through rotation. An azimuthally symmetric optical wall confines the atoms. Chiral edge modes are injected using a rotating anisotropy in the harmonic trap, creating a static scalar saddle potential in the rotating frame. This saddle potential induces a radial flow towards the wall. Atoms near the edge acquire an energy that increases with their wavevector along the wall. The azimuthal impulse from the saddle injects atoms into states with non-zero group velocity, initiating chiral propagation along the boundary. The wavevector is 'frozen' by turning off the saddle potential, allowing observation of constant-velocity propagation. The edge mode speed is determined from the dispersion relation, specifically the lowest band E₀(k), which approximates the quadratic dispersion of a chiral free particle for a hard wall. The effect of wall sharpness is investigated by defocusing the optical boundary, modeling it as a piecewise potential. An obstacle is created using a co-rotating Gaussian laser beam to demonstrate the robustness of edge propagation against disorder. Theoretical calculations and Gross-Pitaevskii (GP) simulations are used to complement the experimental results, including the calculation of the dispersion relation for a piecewise linear wall potential and the simulation of the condensate evolution under experimental conditions. The experimental setup includes detailed calibration of wall steepness using a secondary microscope objective, deconvolving the observed intensity profile from the known point-spread-function to determine the actual wall potential shape. The oscillation frequency of the edge mode center-of-mass is also analyzed to infer the energy gap between edge bands.

The experiment successfully distilled chiral edge modes at the boundary of a rapidly rotating quantum gas. The measured edge mode speed closely matches the theoretical prediction for chiral free particles, particularly in the hard-wall limit. The analysis of the transverse density profile reveals that the atoms predominantly occupy the lowest band, consistent with the chemical potential being smaller than the energy gap between the ground and first excited bands. A clear crossover between soft wall behavior (speed proportional to wall steepness) and the hard wall regime is observed as the wall sharpness is varied. This crossover is explained by comparing the typical wavevector of the edge mode to the inverse wall steepness. When the wavevector is much larger than the inverse steepness, the atoms experience a linear potential, resulting in isopotential drift and a speed proportional to wall steepness. When the wavevector is smaller, atoms experience the force discontinuity at the wall, leading to skipping motion and a constant speed, regardless of wall steepness. The energy gap between the ground and first excited bands is observed to change from the bulk Landau level splitting for soft walls to a larger value for sharper walls, consistent with the theoretical predictions. The robustness of edge propagation against disorder is demonstrated by showing that atoms smoothly circumvent a co-rotating obstacle at the boundary. The experimental results are well-captured by the theoretical model and GP simulations, including the calculated dispersion relation and condensate dynamics.

The findings directly address the central question of realizing and characterizing chiral edge modes in a highly tunable system. The observed crossover between soft and hard wall regimes highlights the crucial role of boundary conditions in shaping edge mode properties. The agreement between experimental results and theoretical predictions validates the model used, reinforcing the understanding of the underlying physics. The robustness against disorder demonstrates the topological nature of the observed modes. The ability to controllably inject and image chiral edge modes opens possibilities for exploring various aspects of topological phenomena, such as the influence of interactions and disorder dynamics, leading to a better understanding of fundamental aspects of topological materials. This system offers a unique platform to study one-dimensional systems with speeds either proportional to or independent of the confining force, potentially enabling robust atomic waveguides.

This paper presents a significant advance in the experimental study of chiral edge modes by demonstrating their direct observation in a rapidly-rotating ultracold gas. The precise control of wall sharpness, the ability to directly image and characterize the edge modes, and the demonstration of their robustness against disorder establish a new platform for investigating topological phenomena. Future research could focus on exploring the effects of varying disorder length scales and dynamics, and investigating the role of interactions, potentially leading to the observation of a chiral Lieb-Liniger gas at the boundary. The inherent properties of these edge modes, notably their speed characteristics, could pave the way for robust atomic waveguides.

While the study successfully demonstrates chiral edge transport, limitations exist. The current study focuses on a specific type of bosonic system; generalizability to other systems might require further investigation. The analysis assumes a simplified model for the wall potential, neglecting potential complexities due to finite resolution and other experimental imperfections. Further research is needed to fully quantify the influence of these imperfections on the observed dynamics.