An Aharonov-Bohm interferometer for determining Bloch band topology

Abstract

The geometric structure of an energy band in a solid is fundamental for a wide range of many-body phenomena in condensed matter and is uniquely characterized by the distribution of Berry curvature over the Brillouin zone. In analogy to an Aharonov-Bohm interferometer that measures the magnetic flux penetrating a given area in real space, we realize an atomic interferometer to measure Berry flux in momentum space. We demonstrate the interferometer for a graphene-type hexagonal lattice, where it has allowed us to directly detect the singular π Berry flux localized at each Dirac point. We show that the interferometer enables one to determine the distribution of Berry curvature with high momentum resolution. Our work forms the basis for a general framework to fully characterize topological band structures and can also facilitate holonomic quantum computing through controlled exploitation of the geometry of Hilbert space.More than thirty years ago, Berry [1] delineated the effects of the geometric structure of Hilbert space on the adiabatic evolution of quantum mechanical systems. These ideas have found widespread applications in science [2] and are routinely used to calculate the geometric phase shift acquired by a particle moving along a closed path-a phase shift that is determined only by the geometry of the path and is independent of the time spent en route. Geometric phases provide an elegant description of the celebrated Aharonov-Bohm effect [3], where a magnetic flux in a confined region of space influences the eigenstates everywhere via the magnetic vector potential. In condensed-matter physics, an analogous Berry flux in momentum space is responsible for various anomalous velocities and Hall responses [4] and lies at the heart of many-body phenomena ranging from quantum Hall physics [5] to topological insulators [6]. The Berry flux density (Berry curvature) is indeed essential to the characterization of an energy band and determines its topological invariants. However, mapping out the geometric structure of an energy band [7][8][9] has remained a major unresolved challenge for experiments.Here, we demonstrate a versatile technique for measuring geometric phases in reciprocal space using spin-echo interferometry with ultracold atoms [9, 10]. In contrast to typical solid state experiments, where all geometric effects are averaged over the Fermi sea, the use of a Bose-Einstein condensate (BEC) enables measurements with high momentum resolution. We exploit this resolution to directly detect the singular topological properties of an individual Dirac cone [11] in a graphene-type hexagonal optical lattice (see Fig. 1). Concentrated at the Dirac point is a π Berry flux, which is analogous to a magnetic flux generated by an infinitely narrow solenoid [12]. This localized flux gives rise to several striking properties of graphene, including the half-integer shift in the positions of quantum Hall plateaus [13,14], the phase of Shubnikov-de Haas oscillations [13,14], and the polarization dependence in photoemission spec...

Synthetic Dimensions and Spin-Orbit Coupling with an Optical Clock Transition

“…These results open new directions for the investigation of topological states of matter with ultracold atomic gases. DOI: 10.1103/PhysRevLett.117.220401 Ultracold atoms are emerging as a very versatile platform for the investigation of topological states of matter [1], thanks to the possibility of using laser light to synthesize artificial gauge fields [2,3] and to engineer lattices with topological band structures [4][5][6][7][8]. A prime element for the emergence of nontrivial topological properties is the presence of spin-orbit coupling (SOC) [9,10], locking the spin of the particles to their motion.…”

Quantum Anomalous Hall Effect

“…Effective topological band structure can be induced by the timedependent coupling, such as optical transition between different bands for an electron in external photon field. Beyond electron systems, the topological band structure leading to QAH effect has also been proposed and even realized in cold atom systems [105][106][107][108][109][110], photonic crystals [111][112][113][114] and magnonics [115][116][117].…”

Artificial magnetic fields in momentum space in spin-orbit-coupled systems

“…Thanks to recent advances, geometrically nontrivial energy bands have now been created in photonic systems [21][22][23][24] and ultracold gases [25][26][27][28], where the Berry curvature [29,30] and topological invariants [31,32] have been measured directly.…”