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Quantized Hall Conductance of a Single Atomic Wire: A Proposal Based on Synthetic Dimensions

Salerno, Grazia, Hannah M. Price, Martin Lebrat, Samuel Häusler, Tilman Esslinger, Laura Corman, Jean-Philippe Brantut, Nathan Goldman

Published Oct 1, 2019• 31 citations

Abstract

We propose a method by which the quantization of the Hall conductance can be directly measured in the transport of a one-dimensional atomic gas. Our approach builds on two main ingredients: (1) a constriction optical potential, which generates a mesoscopic channel connected to two reservoirs, and (2) a time-periodic modulation of the channel specifically designed to generate motion along an additional synthetic dimension. This fictitious dimension is spanned by the harmonic oscillator modes associated with the tightly confined channel, and hence, the corresponding “lattice sites” are intimately related to the energy of the system. We analyze the quantum-transport properties of this hybrid two-dimensional system, highlighting the appealing features offered by the synthetic dimension. In particular, we demonstrate how the energetic nature of the synthetic dimension combined with the quasienergy spectrum of the periodically driven channel allows for the direct and unambiguous observation of the quantized Hall effect in a two-reservoir geometry. Our work illustrates how topological properties of matter can be accessed in a minimal one-dimensional setup, with direct and practical experimental consequences.

DOI: 10.3929/ethz-b-000368844

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We study quantum transport in a periodically driven (Floquet) topological system coupled to static fermionic reservoirs. Using the Floquet nonequilibrium Green's-function (NEGF) formalism we show, from exact numerics for a strip geometry, that the two-terminal (longitudinal) conductance is quantized as

|W_{\varepsilon}|\,e^2/h

, while the Hall (transverse) conductance is quantized as

W_{\varepsilon}\,e^2/h

, where

W_{\varepsilon}

is the Floquet winding invariant associated with the quasienergy gap at

\varepsilon = 0

or

\varepsilon = Ω/2

. Quantization is achieved only after summing over the contribution of all Floquet sidebands. We provide an analytic understanding of this Floquet conductance sum rule, by considering the Hall conductance in the weak coupling limit. In that limit, we show that the Floquet Hall conductance gets contributions from the Floquet sidebands, which includes the signs of the velocities of the edge modes. Their sum yields exact quantization, as predicted by the Floquet sum rule. We find that in a wide range of parameter regime, the convergence is fast, making observation of the sum rule and Floquet winding numbers accessible to experiments.

We propose to simulate and detect quantum anomalous Hall phase with ultracold atoms in a one-dimensional optical lattice, with the other synthetic dimension being realized by modulating spin-orbit coupling. We show that the system manifests a topologically nontrivial phase with two chiral edge states which can be readily detected in this synthetic two-dimensional system. Moreover, it is interesting that at the phase transition point there is a flat energy band and this system can also be in a topologically nontrivial phase with two Fermi zero modes existing at the boundaries by considering the synthetic dimension as a modulated parameter. We also show how to measure these topological phases experimentally in ultracold atoms. Another model with a random Rashba and Dresselhaus spin-orbit coupling strength is also found to exhibit topological nontrivial phase, and the impact of the disorder to the system is revealed.

We suggest a means to calculate the dc conductance of a one-dimensional electron system described by the Luttinger model. Our approach is based on the ideas of Landauer and B\"uttiker on transport in ballistic channels and on the methods of current algebra. We analyze in detail the way in which the system can be coupled to external reservoirs. This determines whether the conductance is renormalized or not. We provide a parallel treatment of a quantum wire and a fractional quantum Hall system on a cylinder with two widely separated edges. Although both systems are described by the same effective theory, the physical electrons are identified with different types of excitations, and hence the coupling to external reservoirs is different. As a consequence, the conductance in the wire is quantized in integer units of

\frac{{e}^{2}}{h}

per spin orientation whereas the Hall conductance allows for fractional quantization.

Modern condensed matter physics relies on the concept of topology to classify matter, from quantum Hall systems to topological insulators. Engineered systems, benefiting from synthetic dimensions, can potentially give access to topological states predicted in dimensions D > 3. We report the realization of an atomic quantum Hall system evolving in four dimensions (4D), with two spatial dimensions and two synthetic ones encoded in the large spin of dysprosium atoms. We measure the nontrivial topological index of the ground band through a full characterization of the nonlinear electromagnetic response and observe the associated anisotropic hyperedge modes. We also excite nonplanar cyclotron motion, in contrast to the planar orbits in D ≤ 3. Our work may enable the investigation of strongly correlated topological liquids in 4D, generalizing fractional quantum Hall states.