Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms (1407.4205v2)

Abstract: Sixty years ago, Karplus and Luttinger pointed out that quantum particles moving on a lattice could acquire an anomalous transverse velocity in response to a force, providing an explanation for the unusual Hall effect in ferromagnetic metals. A striking manifestation of this transverse transport was then revealed in the quantum Hall effect, where the plateaus depicted by the Hall conductivity were attributed to a topological invariant characterizing Bloch bands: the Chern number. Until now, topological transport associated with non-zero Chern numbers has only been revealed in electronic systems. Here we use studies of an atomic cloud's transverse deflection in response to an optical gradient to measure the Chern number of artificially generated Hofstadter bands. These topological bands are very flat and thus constitute good candidates for the realization of fractional Chern insulators. Combining these deflection measurements with the determination of the band populations, we obtain an experimental value for the Chern number of the lowest band $\nu_{\mathrm{exp}} =0.99(5)$. This result, which constitutes the first Chern-number measurement in a non-electronic system, is facilitated by an all-optical artificial gauge field scheme, generating uniform flux in optical superlattices.

• The paper demonstrates the first direct measurement of a Chern number in a non-electronic system using ultracold bosonic atoms.
• The experimental approach employs a 2D optical lattice with laser-assisted tunneling to realize a homogeneous magnetic flux mimicking the Harper-Hofstadter model.
• The observed transverse atomic drift yielded a Chern number near unity, validating theoretical predictions and opening paths for exploring topological quantum phases.

Measuring the Chern Number of Hofstadter Bands with Ultracold Bosonic Atoms

The paper "Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms" presents a paper that bridges experimental physics and topological phenomena, specifically focusing on the measurement of Chern numbers in non-electronic systems. This work introduces a novel approach to measure topological invariants in ultracold bosonic atoms trapped in optical lattices, demonstrating a unique method to probe the Hofstadter model's Chern numbers.

Experimental Approach

The paper constructed an experimental setup using ultracold gases of $^{87}$Rb atoms within a two-dimensional optical lattice framework. The lattice potential was configured to emulate the behavior predicted by the Harper-Hofstadter Hamiltonian, which describes a system subjected to a uniform magnetic flux. The artificial magnetic flux was realized by employing laser-assisted tunneling methods, following the theoretical framework set by Jaksch and Zoller. This methodology allowed the precise control of the Peierls phases, leading to the realization of a homogeneous flux of $\pi/2$ per plaquette.

A critical component of this experimental methodology was the ability to control the band populations within the Hofstadter bands using an auxiliary superlattice potential. This approach involved staggered potentials that effectively modified the coupling between adjacent lattice sites, enabling the loading and manipulation of atoms into desired Hofstadter bands. As a result, the experimental setup achieved a direct observation of the transverse atomic motion, a phenomenon attributed to the non-zero Chern numbers associated with the band structure.

Measurement and Results

The Chern number, a topological invariant representing the global features of band structures, was extracted from the transverse drift observed in the atomic cloud when subjected to an applied optical gradient. The transverse offset in cloud position, in response to the directional force, was indicative of the non-trivial topological properties in the Hofstadter bands. The experiments demonstrated the successful measurement of the Chern number for the lowest band, achieving a value near unity: $\nu_{\mathrm{exp}} = 0.99(5)$.

This result was significant as it marked the first direct measurement of a Chern number in a non-electronic system, utilizing ultracold atoms. The paper highlights that the engineered flat bands in the system are excellent candidates for realizing fractional Chern insulators, where interaction-driven topological phenomena can manifest.

Implications and Future Work

The implications of this paper are twofold: firstly, it provides experimental validation for theoretical models related to topological band structures in cold atom systems, reinforcing our understanding of topological states of matter. Secondly, it establishes a versatile platform for exploring strongly correlated topological phases, such as those predicted for fractional quantum Hall states.

Looking forward, the research opens up avenues for further experimentation with complex lattice structures and higher Chern numbers, potentially incorporating interaction effects among atoms to explore many-body topological phases. Additionally, this methodology could be extended to other quantum simulators, such as ion traps and photonic lattices, where engineered gauge fields could be similarly applied to investigate topological characteristics in diverse systems.

In conclusion, the paper successfully demonstrates a clear pathway for probing topological invariants using ultracold atomic systems, laying the groundwork for future explorations in the field of topological quantum matter.