Encoding a topological gauge theory on a ring-shaped Raman-coupled Bose gas

TL;DR

This work addresses realizing a one-dimensional topological gauge theory on a ring by dimensionally reducing the Chern-Simons action to a chiral BF theory and encoding the gauge degrees of freedom into a matter field. The authors derive the encoded Hamiltonian with a density-dependent gauge field and a winding term, leading to a density-dependent magnetic flux $\tilde{\omega}$ that governs angular dynamics, and propose implementation in a ring-shaped, Raman-dressed Bose gas with Laguerre-Gauss beams. They validate the mapping via mean-field simulations and Bogoliubov analysis, predicting observable signatures such as quantized persistent currents and chiral sound velocities $V_{\pm}$ that depend on the mean density and ring radius. By connecting the topology of the gauge theory with the ring geometry, the work provides a flexible platform to explore density-controlled topological gauge fields and potential extensions to non-Abelian statistics in quasi-$1D$ settings.

Abstract

Topological gauge theories constitute a framework for understanding strongly correlated quantum matter in terms of weakly interacting composite degrees of freedom. Their topological properties become evident when these theories are realized on a space of non-trivial topology. Here, we propose a scheme to realize a one-dimensional topological gauge theory, the chiral BF theory, on a ring geometry. We obtain such a theory by dimensionally reducing Chern-Simons theory on a disk to the so-called chiral BF theory defined on the ring. Then, we encode the theory into a Hamiltonian with a coupling between angular momentum and density, and we propose and numerically benchmark its realization in an optically-dressed Bose gas confined in a ring-shaped trap. There, the topological properties of the underlying theory manifest themselves through a magnetic flux variable that is density-dependent. We quantify such density-dependent magnetic flux in terms of the ground-state angular momentum and the chiral properties of the system through a Bogoliubov analysis. Our proposal enables the observation of the interplay between the topology of the theory and that of the space.

Figures (3)

• Figure 1: Chiral BF theory on a ring: derivation from Chern-Simons theory, experimental scheme and signatures. a) A winding number $\omega\in\mathbb{Z}$ is equivalent to placing an infinitesimal solenoid in the origin; outside the magnetic field is everywhere zero but there is a flux $2\pi\omega$. b) Lagrangian on a disk and its encoded Hamiltonian. Chern-Simons theory is represented as a purple solenoid in the bulk and the BF term as the red-shaded half of the solenoid outside the boundary. It restores gauge invariance by completing the solenoid cut by the edge. Integrating Chern-Simons theory in the bulk gives rise to a chiral boson term on the boundary (skipping orbits). Coupling the theory to a scalar field (black sphere) yields a Hamiltonian with chiral interactions, i.e., the angular momentum coupled to a density-dependent magnetic flux (green solenoid), which manifest in different sound velocities for right and left movers ($V_{\pm}$). c)-d) Experimental implementation of the encoded Hamiltonian. c) A Laguerre-Gauss (orange) and a Gaussian (green) beams produce a Raman transition between internal states $\ket{\uparrow}$ and $\ket{\downarrow}$ of a Bose gas with single-particle detuning $\delta$ and a two-photon Rabi coupling $\Omega$. The laser scheme simultaneously produces a scalar ring potential. d) Density profile of the gas with an angular momentum transfer $\ell$, together with the beams' direction of propagation and the quantization axis given by a bias magnetic field $\mathbf{B} = B_0 \hat{x}$. e) Chiral BF theory as effective Hamiltonian. Dispersion bands of the single-particle Hamiltonian as a function of the quasi-angular momentum $q$. The colormap shows the spin polarization $P$ of the eigenstates. A Taylor expansion of the lower band around a center momentum $q_0$ yields a static gauge $A_S$ (the slope of the tangent) and an effective mass $M^*$ (the curvature). Chiral interactions arise from imbalanced intrastate interactions of $\ket{\uparrow}$ and $\ket{\downarrow}$, producing a density-dependent displacement of the lower band, which is depicted by particles repelling with different strengths at different momenta of the band (sketches with spheres and arrows). f) Experimental signatures of the theory. Change of the angular momentum $Q$ (upper panel) and the chiral sound velocities $V_{\pm}$ (lower panel) as a function of the density-dependent magnetic flux variable (see Section \ref{['sec:numerics']}).
• Figure 2: Density-dependent angular momentum. a) The density-dependent magnetic flux $\tilde{\omega}$ is depicted as a solenoid extended in the $\hat{z}$ direction. When its value increases by a unit, which is depicted as an anti-clockwise current in the $xy$-plane, the ground state picks up a phase winding in the same plane, so that its circulation follows the opposite direction of the current. b)- c) The angular momentum $Q$ as a function of the density $n$ in the ground state for both the chiral BF \ref{['eq:H_lab_effective']} (green circles) and the two-component Raman-coupled \ref{['eq:Hlab']} (blue crosses) models. We scan it by varying the particle number for fixed trapping configurations with two different Laguerre-Gauss modes $2\ell=40$(b), $80$(c). The points where the momentum changes are marked with vertical dotted blue (dashed green) lines for the Raman-coupled two-component model (chiral BF model), and they are measured in terms of $\tilde{\omega}$\ref{['eq:magnetic_flux_variable']}.
• Figure 3: Chiral sound velocities. a) Time evolution of the perturbed ground state spatial density profile as a function of $r_0\phi$ for $Q=-2$, and the density values $n$ given by the colormap. The horizontal black lines link the timestamp to the ring-shaped density profile for $t=0,\ 120\ \mathrm{ms}.$ The blue and red dashed angles indicate the left and right dip velocities, respectively. b)- c) Chiral sound velocities as a function of the mean density, for two Laguerre-Gauss modes $2\ell=40,\ 80$. The Raman-coupled two-component model \ref{['eq:Hlab']} (chiral BF model \ref{['eq:H_lab_effective']}) right velocity is marked with red crosses (dark red circles), whereas the left one is marked with blue crosses (violet circles). Inset: Linear part of the Bogoliubov energy spectrum of the chiral BF model \ref{['eq:H_lab_effective']} as a function of the quasi-angular momentum of the perturbation. Violet (dark red) dashed angles correspond to the chiral BF left (right) sound velocities.