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arXiv:1401.8240 · Physical Review Letters

Tomography of Band Insulators from Quench Dynamics

Philipp Hauke, Maciej Lewenstein, André Eckardt

Published Jul 23, 2014• 135 citations
arXivHTMLPDF

Abstract

We propose a simple scheme for tomography of band-insulating states in one- and two-dimensional optical lattices with two sublattice states. In particular, the scheme maps out the Berry curvature in the entire Brillouin zone and extracts topological invariants such as the Chern number. The measurement relies on observing--via time-of-flight imaging--the time evolution of the momentum distribution following a sudden quench in the band structure. We consider two examples of experimental relevance: the Harper model with π flux and the Haldane model on a honeycomb lattice. Moreover, we illustrate the performance of the scheme in the presence of a parabolic trap, noise, and finite measurement resolution.

DOI
10.1103/physrevlett.113.045303

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We propose a simple scheme for tomography of band-insulating states in one- and two-dimensional optical lattices with two sublattice states. In particular, the scheme maps out the Berry curvature in the entire Brillouin zone and extracts topological invariants such as the Chern number. The measurement relies on observing--via time-of-flight imaging--the time evolution of the momentum distribution following a sudden quench in the band structure. We consider two examples of experimental relevance: the Harper model with π flux and the Haldane model on a honeycomb lattice. Moreover, we illustrate the performance of the scheme in the presence of a parabolic trap, noise, and finite measurement resolution.
We propose an experimental scheme to directly detect the Chern number of the topological Haldane model realized with optical lattices based on the time-of-flight imaging. The Haldane model can be realized with two-component fermionic atoms loaded in the optical lattices consisting of two sublattices and each of the sublattices hosts atoms with a specific component of the atoms. We find that the Bloch wave function of the lower band can be measured with the time-of-flight imaging and thus the Chern number can be directly obtained from the Bloch function. We numerically simulate the Chern number of the Haldane model and its modified version in both honeycomb lattice and brick-wall lattice, and show that the proposed approach is general and feasible in practical experiments.
Ultracold atoms in optical lattices form a clean quantum simulator platform which can be utilized to examine topological phenomena and test exotic topological materials. Here we propose an experimental scheme to measure the Chern numbers of two-dimensional multiband topological insulators with bosonic atoms. We show how to extract the topological invariants out of a sequence of time-of-flight images by applying a phase-retrieval algorithm to matter waves. We illustrate advantages of using bosonic atoms as well as efficiency and robustness of the method with two prominent examples: the Harper-Hofstadter model with an arbitrary commensurate magnetic flux and the Haldane model on a brick-wall lattice.
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.
Recent experiments began to explore the topological properties of quench dynamics, i.e. the time evolution following a sudden change in the Hamiltonian, via tomography of quantum gases in optical lattices. In contrast to the well established theory for static band insulators or periodically driven systems, at present it is not clear whether, and how, topological invariants can be defined for a general quench of band insulators. Previous work solved a special case of this problem beautifully using Hopf mapping of two-band Hamiltonians in two dimensions. But it only works for topologically trivial initial state and is hard to generalize to multiband systems or other dimensions. Here we introduce the concept of loop unitary constructed from the unitary time-evolution operator, and show its homotopy invariant fully characterizes the dynamical topology. For two-band systems in two dimensions, we prove that the invariant is precisely equal to the change in the Chern number across the quench regardless of the initial state. We further show that the nontrivial dynamical topology manifests as hedgehog defects in the loop unitary, and also as winding and linking of its eigenvectors along a curve where dynamical quantum phase transition occurs. This opens up a systematic route to classify and characterize quantum quench dynamics.
Recent experiments began to explore the topological properties of quench dynamics, i.e., the time evolution following a sudden change in the Hamiltonian, via tomography of quantum gases in optical lattices. In contrast to the well-established theory for static band insulators or periodically driven systems, at present it is not clear whether, and how, topological invariants can be defined for a general quench of band insulators. Previous work solved a special case of this problem beautifully using Hopf mapping of two-band Hamiltonians in two dimensions. However, it only works for a topologically trivial initial state and is hard to generalize to multiband systems or other dimensions. Here we introduce the concept of loop unitary constructed from the unitary time-evolution operator and show its homotopy invariant fully characterizes the dynamical topology. For two-band systems in two dimensions, we prove that the invariant is precisely equal to the change in the Chern number across the quench, regardless of the initial state. We further show that the nontrivial dynamical topology manifests as hedgehog defects in the loop unitary and also as winding and linking of its eigenvectors along a curve where dynamical quantum phase transition occurs. This opens up a systematic route to classify and characterize quantum quench dynamics.
Cold-atom experiments in optical lattices offer a versatile platform to realize various topological quantum phases. A key challenge in those experiments is to unambiguously probe the topological order. We propose a method to directly measure the characteristic topological invariants (order) based on the time-of-flight imaging of cold atoms. The method is generally applicable to detection of topological band insulators in one, two, or three dimensions characterized by integer topological invariants. Using detection of the Chern number for the two-dimensional anomalous quantum Hall states and the Chern-Simons term for the three-dimensional chiral topological insulators as examples, we show that the proposed detection method is practical, and robust to typical experimental imperfections such as limited imaging resolution, inhomogeneous trapping potential, and disorder in the system.