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arXiv:1602.01696 · Physical review. B./Physical review. B

Measurement of Chern numbers through center-of-mass responses

Hannah M. Price, Oded Zilberberg, Tomoki Ozawa, Iacopo Carusotto, Nathan Goldman

Published Jun 6, 2016• 86 citations
arXivHTMLPDF

Abstract

The remarkable quantization of the Hall conductivity is routinely observed in two-dimensional electron gases subject to strong magnetic fields. Such quantum transport experiments are performed by measuring the current that flows in 2D systems in response to an applied electric field, and result in a quantized conductance proportional to a topological index, known as the Chern number. Recently, a similar effect was proposed and observed in gases of ultracold atoms, when these are loaded into Bloch bands with a nonzero Chern number. In this context, the center of mass of the atomic gas performs a measurable differential drift, in direct analogy with the electronic Hall effect, from which the corresponding Chern number can be obtained. Such detection relies on the assumption that the center-of-mass drift of the atomic cloud is directly related to the current density, which itself depends on the Chern number of the band. Here, the authors carefully revisit and analyze the center-of-mass detection scheme. Interestingly, we find that the center-of-mass motion also depends on the particle density, which in the presence of magnetic perturbations, can be explicitly related to the topology of the band. Whereas this additional dependence offers a correction to existing experimental results, it opens up a rich variety of new possible quantized responses and detection schemes in 2D and 4D engineered systems, such as cold atoms and photonic lattices.

DOI
10.1103/physrevb.93.245113

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The remarkable quantization of the Hall conductivity is routinely observed in two-dimensional electron gases subject to strong magnetic fields. Such quantum transport experiments are performed by measuring the current that flows in 2D systems in response to an applied electric field, and result in a quantized conductance proportional to a topological index, known as the Chern number. Recently, a similar effect was proposed and observed in gases of ultracold atoms, when these are loaded into Bloch bands with a nonzero Chern number. In this context, the center of mass of the atomic gas performs a measurable differential drift, in direct analogy with the electronic Hall effect, from which the corresponding Chern number can be obtained. Such detection relies on the assumption that the center-of-mass drift of the atomic cloud is directly related to the current density, which itself depends on the Chern number of the band. Here, the authors carefully revisit and analyze the center-of-mass detection scheme. Interestingly, we find that the center-of-mass motion also depends on the particle density, which in the presence of magnetic perturbations, can be explicitly related to the topology of the band. Whereas this additional dependence offers a correction to existing experimental results, it opens up a rich variety of new possible quantized responses and detection schemes in 2D and 4D engineered systems, such as cold atoms and photonic lattices.
We propose a realistic scheme to detect the 4D quantum Hall effect using ultracold atoms. Based on contemporary technology, motion along a synthetic fourth dimension can be accomplished through controlled transitions between internal states of atoms arranged in a 3D optical lattice. From a semiclassical analysis, we identify the linear and nonlinear quantized current responses of our 4D model, relating these to the topology of the Bloch bands. We then propose experimental protocols, based on current or center-of-mass-drift measurements, to extract the topological second Chern number. Our proposal sets the stage for the exploration of novel topological phases in higher dimensions.
We propose a scheme to measure the quantized Hall conductivity of an ultracold Fermi gas initially prepared in a topological Chern insulating phase and driven by a constant force. We show that the time evolution of the center of mass, after releasing the cloud, provides a direct and clear signature of the topologically invariant Chern number. We discuss the validity of this scheme, highlighting the importance of driving the system with a sufficiently strong force to displace the cloud over measurable distances while avoiding band-mixing effects. The unusual shapes of the driven atomic cloud are qualitatively discussed in terms of a semiclassical approach.
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Realizing strongly-correlated topological phases of ultracold gases is a central goal for ongoing experiments. And while fractional quantum Hall states could soon be implemented in small atomic ensembles, detecting their signatures in few-particle settings remains a fundamental challenge. In this work, we numerically analyze the center-of-mass Hall drift of a small ensemble of hardcore bosons, initially prepared in the ground state of the Harper-Hofstadter-Hubbard model. By extracting the Hall conductivity in a wide range of the magnetic flux, we identify an emergent Hall plateau compatible with a fractional Chern insulator state: the width of the plateau agrees with the spectral and topological properties of the prepared ground state, while the Hall conductivity approaches a fractional value determined by the many-body Chern number. A comparison with a direct application of Streda's formula is also discussed. Our calculations suggest that fractional Chern insulators can be detected in cold-atom experiments, using available detection methods.
The topology of two-dimensional materials traditionally manifests itself through the quantization of the Hall conductance, which is revealed in transport measurements1,2,3. Recently, it was predicted that topology can also give rise to a characteristic spectroscopic response on subjecting a Chern insulator to a circular drive: comparing the frequency-integrated depletion rates associated with drives of opposite orientation leads to a quantized response dictated by the topological Chern number of the populated Bloch band4,5. Here we experimentally demonstrate this intriguing topological effect using ultracold fermionic atoms in topological Floquet bands. In addition, our depletion-rate measurements also provide an experimental estimation of the Wannier-spread functional, a fundamental geometric property of Bloch bands related to the quantum metric6,7. Our results establish topological spectroscopic responses as a versatile probe, which could be applied to access the geometry and topology of many-body quantum systems, such as fractional Chern insulators
Abstract Topological band structures can be designed by subjecting lattice systems to time-periodic modulations, as was proposed for irradiated graphene, and recently demonstrated in two-dimensional (2D) ultracold gases and photonic crystals. However, changing the topological nature of Floquet Bloch bands from trivial to non-trivial, by progressively launching the time-modulation, is necessarily accompanied with gap-closing processes: this has important consequences for the loading of particles into a target Floquet band with non-trivial topology, and hence, on the subsequent measurements. In this work, we analyse how such loading sequences can be optimized in view of probing the topology of 2D Floquet bands through transport measurements. In particular, we demonstrate the robustness of center-of-mass responses, as compared to current responses, which present important irregularities due to an interplay between the micro-motion of the drive and inter-band interference effects. The results presented in this work illustrate how probing the center-of-mass displacement of atomic clouds offers a reliable method to detect the topology of Floquet bands, after realistic loading sequences.