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arXiv:1212.0562 · Physical Review Letters

Interferometric Approach to Measuring Band Topology in 2D Optical Lattices

Dmitry A. Abanin, Takuya Kitagawa, Immanuel Bloch, Eugene Demler

Published Apr 18, 2013• 121 citations
arXivHTMLPDF

Abstract

Recently, optical lattices with nonzero Berry's phases of Bloch bands have been realized. New approaches for measuring Berry's phases and topological properties of bands with experimental tools appropriate for ultracold atoms need to be developed. In this Letter, we propose an interferometric method for measuring Berry's phases of two-dimensional Bloch bands. The key idea is to use a combination of Ramsey interference and Bloch oscillations to measure Zak phases, i.e., Berry's phases for closed trajectories corresponding to reciprocal lattice vectors. We demonstrate that this technique can be used to measure the Berry curvature of Bloch bands, the π Berry's phase of Dirac points, and the first Chern number of topological bands. We discuss several experimentally feasible realizations of this technique, which make it robust against low-frequency magnetic noise.

DOI
10.1103/physrevlett.110.165304

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Recently, optical lattices with nonzero Berry's phases of Bloch bands have been realized. New approaches for measuring Berry's phases and topological properties of bands with experimental tools appropriate for ultracold atoms need to be developed. In this Letter, we propose an interferometric method for measuring Berry's phases of two-dimensional Bloch bands. The key idea is to use a combination of Ramsey interference and Bloch oscillations to measure Zak phases, i.e., Berry's phases for closed trajectories corresponding to reciprocal lattice vectors. We demonstrate that this technique can be used to measure the Berry curvature of Bloch bands, the π Berry's phase of Dirac points, and the first Chern number of topological bands. We discuss several experimentally feasible realizations of this technique, which make it robust against low-frequency magnetic noise.
Ultracold atoms in optical lattices have recently emerged as promising candidates for investigating the geometric and topological aspects of band structures. In this thesis, we exploit the high degree of control available in these systems to directly probe the band geometry of an optical honeycomb lattice. In the first series of experiments, we realize an atomic interferometer in quasimomentum space to measure the singular Berry flux associated with a Dirac point. This technique enables us to determine the distribution of Berry curvature in the Brillouin zone with high quasimomentum resolution. Next, we realize strong-force dynamics that are described by matrix-valued Wilson lines, which are generalizations of the Berry phase to degenerate systems. In this strong-force regime, we show that the evolution in the band populations directly reveals the band geometry. This method enables the reconstruction of both the cell-periodic Bloch states at every quasimomentum and the eigenvalues of Wilson-Zak loops. Our techniques can be used to determine the topological invariants, such as the Chern and Z2 numbers, that characterize the band structure. Lastly, having established our ability to detect the band geometry, we present preliminary experiments on engineering band structures.
The geometric structure of a single-particle energy band in a solid is fundamental for a wide range of many-body phenomena and is uniquely characterized by the distribution of Berry curvature over the Brillouin zone. We realize an atomic interferometer to measure Berry flux in momentum space, in analogy to an Aharonov-Bohm interferometer that measures magnetic flux in real space. We demonstrate the interferometer for a graphene-type hexagonal optical lattice loaded with bosonic atoms. By detecting the singular π Berry flux localized at each Dirac point, we establish the high momentum resolution of this interferometric technique. Our work forms the basis for a general framework to fully characterize topological band structures.
Topological properties lie at the heart of many fascinating phenomena in solid-state systems such as quantum Hall systems or Chern insulators. The topology of the bands can be captured by the distribution of Berry curvature, which describes the geometry of the eigenstates across the Brillouin zone. Using fermionic ultracold atoms in a hexagonal optical lattice, we engineered the Berry curvature of the Bloch bands using resonant driving and show a full momentum-resolved measurement of the ensuing Berry curvature. Our results pave the way to explore intriguing phases of matter with interactions in topological band structures.
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Topological states of matter emerge as a new type of quantum phases, which can be distinguished by their associated topological invariants, e.g. , Chern numbers. Currently, there has been increasing interest toward the physical detection of the new predicted topological phases. Here, we propose an interferometric approach to directly measure the Chern number of the topological Haldane model in bosonic optical lattices via detecting the associated Zak phases. We show that this interferometric approach can distinguish Zak phases of from 0 in the first Brillouin zone, and thus it provides a new tool to directly detect the Chern number of topological systems. In addition, we demonstrate that this method is feasible under realistic experimental conditions and may generalize to detect topological systems with higher Chern numbers.
We propose a general method by which experiments on ultracold gases can be used to determine the topological properties of the energy bands of optical lattices, as represented by the map of the Berry curvature across the Brillouin zone. The Berry curvature modifies the semiclassical dynamics and hence the trajectory of a wave packet undergoing Bloch oscillations. However, in two dimensions these trajectories may be complicated Lissajous-like figures, making it difficult to extract the effects of Berry curvature in general. We propose how this can be done using a ``time-reversal'' protocol. This compares the velocity of a wave packet under positive and negative external force, and allows a clean measurement of the Berry curvature over the Brillouin zone. We discuss how this protocol may be implemented and explore the semiclassical dynamics for three specific systems: the asymmetric hexagonal lattice and two ``optical flux'' lattices in which the Chern number is nonzero. Finally, we discuss general experimental considerations for observing Berry curvature effects in ultracold gases.