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arXiv:1703.02813 · Physical Review Letters

Observation of Topological Bloch-State Defects and Their Merging Transition

Matthias Tarnowski, Marlon Nuske, Nick Fläschner, Benno S. Rem, Dominik Vogel, Lukas Freystatzky, K. Sengstock, Ludwig Mathey, Christof Weitenberg

Published Jun 15, 2017• 31 citations
arXivHTMLPDF

Abstract

Topological defects in Bloch bands, such as Dirac points in graphene, and their resulting Berry phases play an important role for the electronic dynamics in solid state crystals. Such defects can arise in systems with a two-atomic basis due to the momentum-dependent coupling of the two sublattice states, which gives rise to a pseudospin texture. The topological defects appear as vortices in the azimuthal phase of this pseudospin texture. Here, we demonstrate a complete measurement of the azimuthal phase in a hexagonal optical lattice employing a versatile method based on time-of-flight imaging after off-resonant lattice modulation. Furthermore, we map out the merging transition of the two Dirac points induced by beam imbalance. Our work paves the way to accessing geometric properties in optical lattices also with spin-orbit coupling and interactions.

DOI
10.1103/physrevlett.118.240403

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Topological defects in Bloch bands, such as Dirac points in graphene, and their resulting Berry phases play an important role for the electronic dynamics in solid state crystals. Such defects can arise in systems with a two-atomic basis due to the momentum-dependent coupling of the two sublattice states, which gives rise to a pseudospin texture. The topological defects appear as vortices in the azimuthal phase of this pseudospin texture. Here, we demonstrate a complete measurement of the azimuthal phase in a hexagonal optical lattice employing a versatile method based on time-of-flight imaging after off-resonant lattice modulation. Furthermore, we map out the merging transition of the two Dirac points induced by beam imbalance. Our work paves the way to accessing geometric properties in optical lattices also with spin-orbit coupling and interactions.
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.
When two or more energy bands become degenerate at a singular point in the momentum space, such singularity, or ``Dirac points", gives rise to intriguing quantum phenomena as well as unusual material properties. Systems at the Dirac points can possess topological charges and their unique properties can be probed by various methods, such as transport measurement, interferometry and momentum spectroscopy. While the topology of Dirac point in the momentum space is well studied theoretically, observation of topological defects in a many-body quantum systems at Dirac point remain an elusive goal. Based on atomic Bose-Einstein condensate in a graphene-like optical honeycomb lattice, we directly observe emergence of quantized vortices at the Dirac point. The phase diagram of lattice bosons at the Dirac point is revealed. Our work provides a new way of generating vortices in a quantum gas, and the method is generic and can be applied to different types of optical lattices with topological singularity, especially twisted bilayer optical lattices.
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.
Dirac points lie at the heart of many fascinating phenomena in condensed matter physics, from massless electrons in graphene to the emergence of conducting edge states in topological insulators. At a Dirac point, two energy bands intersect linearly and the particles behave as relativistic Dirac fermions. In solids, the rigid structure of the material sets the mass and velocity of the particles, as well as their interactions. A different, highly flexible approach is to create model systems using fermionic atoms trapped in an optical lattice, a method which so far has only been applied to explore simple lattice structures. In my talk I will report on the creation of Dirac points with adjustable properties in a tunable honeycomb optical lattice [1]. Using momentum-resolved interband transitions, we observe a minimum band gap inside the Brillouin zone at the position of the Dirac points. We exploit the unique tunability of our lattice potential to adjust the effective mass of the Dirac fermions by breaking the inversion symmetry of the lattice. Moreover, changing the lattice anisotropy allows us to move the position of the Dirac points inside the Brillouin zone. When increasing the anisotropy beyond a critical limit, the two Dirac points merge and annihilate each other. We map out this topological transition in lattice parameter space and find excellent agreement with ab initio calculations. Our results not only pave the way to model materials where the topology of the band structure plays a crucial role, but also provide the possibility to explore many-body phases resulting from the interplay of complex lattice geometries with interactions.
The geometry of Bloch bands affects many physical properties of crystalline solids and other spatially periodic systems. Direct experimental determination of such geometry is an active area of research. In this work, we focus on the fundamental connection between optical excitations and the relative geometry of pairs of Bloch bands, as characterized by the interband Berry connection. We simulate the response of electrons in solids to optical excitation by the response of ultracold fermionic atoms in optical lattices to periodic modulation of the lattice position. The strength of resonant excitation between bands, measured at each quasimomentum and for various lattice-shaking polarizations, directly maps out the interband Berry connection. We apply this method to the optical honeycomb lattice, driving excitations between the ground n=1n=1n=1 band and the excited n′={2,3,4}n'=\{2,3,4\}n′={2,3,4} bands. We observe transparency lines of quasimomenta at which the response to excitation of specific polarization is zero. Further, the interband Berry connection between bands 1 and 3 shows irreducible Dirac strings connecting the KKK and K′K'K′ points in the Brillouin zone, lines along which the interband Berry connection abruptly changes orientation. Our work establishes optical response as a powerful tool for characterizing geometrical and topological properties of band structure.
Ultracold gases in optical lattices have proved to be a flexible tool to simulate many different phenomena of solid state physics [1, 2]. Recently, optical lattices with complex geometries have been realized [3, 4, 5, 6, 7], paving the way to simulating more realistic systems. The honeycomb structure has recently become accessible in an optical lattice composed of mutually perpendicular laser beams. This lattice structure exhibits topological features in its band structure – the Dirac points. At these points, two energy bands intersect linearly and the particles behave as relativistic Dirac fermions. In optical lattices, Bloch oscillations [8] resolved both in time and in quasi-momentum space can be directly observed. We make use of such Bloch-Zener oscillations to probe the vanishing energy gap at the Dirac points as well as their position in the band structure. In small band gap regions, we observe Landau-Zener tunneling [7, 9] to the second band and the regions of maximum transfer can be identified with the position of the Dirac points.