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

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.

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Plain Language Summary

Limiting quantum particles to move in one, two, or three dimensions has led to the observation of many striking phenomena. A prime example is the quantization of the Hall conductance measured in 2D materials in a strong magnetic field. Nowadays, gases of ultracold atoms provide a powerful platform for easily controlling the dimensionality of quantum systems. However, it is challenging in these setups to measure conductance properties, and a “cold-atomic quantum Hall effect” is yet to be observed. Here, we propose a realistic scheme to achieve this goal.

Our proposal builds on recent experiments at the Swiss Federal Institute of Technology (ETH) in Zurich, where researchers observed the transport of atoms along a 1D wire. To measure the quantum Hall effect, one must somehow extend this setup to two dimensions and include the effects of an external magnetic field. We solve this by introducing a novel type of conductance measurement, which allows for the study of genuine 2D effects starting from a single 1D wire. The key idea is to extend the 1D channel with an additional synthetic dimension, which is designed simply by shaking the channel: In addition to traveling along the wire direction, atoms are driven to higher transverse vibrational states, hence mimicking motion along a transverse lattice.

This out-of-equilibrium approach not only increases the possibilities offered by atomic wires but also offers a particularly efficient probe for topological physics in quantum-engineered matter.