Laughlin’s Topological Charge Pump in an Atomic Hall Cylinder

The quantum Hall effect occurring in two-dimensional electron gases was first explained by Laughlin, who developed a thought experiment that laid the groundwork for our understanding of topological quantum matter. His proposal is based on a quantum Hall cylinder periodically driven by an axial magnetic field, resulting in the quantized motion of electrons. We realize this milestone experiment with an ultracold gas of dysprosium atoms, the cyclic dimension being encoded in the electronic spin and the axial field controlled by the phases of laser-induced spin-orbit couplings. Our experiment provides a straightforward manifestation of the nontrivial topology of quantum Hall insulators, and could be generalized to strongly correlated topological systems.

Figure 1

Laughlin’s thought experiment. Scheme of a two-dimensional electronic system in a cylindrical geometry, with a radial magnetic field ${\mathbf{B}}_{\perp }$ producing a quantum Hall effect. The orange area, pierced by one magnetic flux quantum ${\mathrm{\Phi }}_0$, defines the length ${\ell }_{\mathrm{mag}}$ of the magnetic unit cell—each cell being filled with one electron in a quantum Hall insulator. Laughlin’s thought experiment consists of performing an adiabatic cycle by threading one flux quantum $\mathrm{\Delta }{\mathrm{\Phi }}_{\parallel }={\mathrm{\Phi }}_0$ through the cylinder. The cycle shifts electron occupations by one unit cell, such that a single electron is pumped from one edge to the other, or equivalently the center-of-mass position is displaced by ${\ell }_{\mathrm{mag}}$.

Figure 2

Emerging quantum Hall cylinder. (a) Sketch of the laser configuration involving two beams counterpropagating along $x$ and sent on a thermal sample of dysprosium atoms—one beam having two frequency components. (b) Scheme of the two-photon optical transitions resonantly driving first- and second-order spin transitions, labeled $a$ and $b$, respectively. (c) Representation of a nontrivial 3-cycle between magnetic sublevels induced by the light couplings. (d) Scheme of the Hall cylinder dynamics emerging at low energy, involving three spin states $|y⟩$ (with $y=0$, 1, and 2). The hopping amplitudes have a complex phase $\phi -2kx$, where $2\hslash k$ plays the role of a radial magnetic field $B_{\perp }$ and $\phi$ is linked to an axial field $B_{\parallel }$. The orange area, of length ${\ell }_{\mathrm{mag}}=\lambda /6$ is threaded by one unit of magnetic flux quantum ${\mathrm{\Phi }}_0$.

Figure 3

Ground band characterization. (a) Spin-resolved velocity distribution measured for a gas of mean quasimomentum $⟨q⟩\simeq 2k$. (b) Distribution of discrete velocity components $v_x=\hslash (q+2kp)/M$ (with integer $p$) for states of quasimomentum $q$. The red line shows the mean velocity $⟨v_x⟩$. (c) Spin projection probabilities ${\mathrm{\Pi }}_m$ measured as a function of $q$. The red line stands for the mean spin projection $⟨m⟩$. (d) Probabilities ${\mathcal{P}}_y$ of projection on $y=m(\text{mod}3)$. The blue circles, orange squares, and green diamonds correspond to $y=0$, 1, and 2, respectively. Statistical error bars, computed from a bootstrap random sampling analysis, are smaller than the symbols. The lines are calculated from the expected band structure.

Figure 4

Topological charge pumps. (a) Center-of-mass displacement $⟨\mathrm{\Delta }y⟩$ as a function of the quasimomentum shift $\mathrm{\Delta }q$ induced by a force $F_x$ (orange squares), together with a linear fit (orange line). (b) Evolution of the mean velocity $⟨v_x⟩$ immediately after switching off the force $F_x$ (orange squares), fitted with a damped sine (solid line). The velocity $⟨v_y⟩$ is obtained from the initial slope of the fit (dashed line). (c) Displacement of the center of mass $⟨\mathrm{\Delta }x⟩$ induced by an axial magnetic flux ${\mathrm{\Phi }}_{\parallel }$ and averaged over the magnetic Brillouin zone (blue circles). The blue line is a linear fit. (d) Berry curvature $\mathrm{\Omega }$ measured as a function of the mean quasimomentum $⟨q⟩$ (blue circles). The solid line is the expected Berry curvature, which is not distinguishable from the constant value $\mathrm{\Omega }(q)=1/(2k)$.