Measuring Hall voltage and Hall resistance in an atom-based quantum simulator

TL;DR

The paper reports a direct measurement of the Hall voltage and Hall resistance in a neutral-atom quantum simulator, using ultracold fermions in synthetic Hall-bars formed by ladder geometries with a synthetic magnetic flux. By quenching longitudinal and transverse fields and tuning a compensating transverse field to set the transverse polarization to zero, the authors extract the Hall voltage and demonstrate a universal 1/n scaling of the Hall resistance, $\rho_{ m H} = \frac{2}{n}\tan\left(\frac{\varphi}{2}\right)$, across two- and three-leg ladders in the strongly interacting, single-band metal regime. The results agree with mean-field and numerical simulations and show robustness against microscopic details, bridging analogue quantum simulations with solid-state Hall measurements. This establishes a versatile platform to study Hall transport in strongly correlated systems and paves the way toward exploring quantum Hall physics and topological transport with ultracold atoms.

Abstract

In the Hall effect, a voltage drop develops perpendicularly to the current flow in the presence of a magnetic field, leading to a transverse Hall resistance. Recent developments with quantum simulators have unveiled strongly correlated and universal manifestations of the Hall effect. However, a direct measurement of the Hall voltage and of the Hall resistance in a non-electronic system of strongly interacting fermions was not achieved to date. Here, we demonstrate a technique for measuring the Hall voltage in a neutral-atom-based quantum simulator. From that we provide the first direct measurement of the Hall resistance in a cold-atom analogue of a solid-state Hall bar and study its dependence on the carrier density, along with theoretical analyses. Our work closes a major gap between analogue quantum simulations and measurements performed in solid-state systems, providing a key tool for the exploration of the Hall effect in highly tunable and strongly correlated systems.

Figures (7)

• Figure 1: Measuring electrical quantities in a synthetic Hall bar.a, An analogue quantum simulation of a Hall bar is realized by trapping ultracold atoms in ladder geometries. An atomic current $J_x$ is subjected to a perpendicular effective magnetic field $B$ and to synthetic electric fields realized by energy gradients along both the longitudinal direction ($\hat{x}$) and the transverse direction ($\hat{y}$, encoded in the internal spin state according to the concept of synthetic dimension). b, The transverse voltage bias that compensates the bending of the carrier trajectories induced by the Hall effect (measured by the spin polarization $P_y$) provides a direct measurement of the Hall voltage $V_{\rm H}$, from which the Hall resistance is extracted.
• Figure 1: Time-averaged Hall polarization $\langle P_y\rangle$ for two-leg ladders, measured at $t_y=3.30t_x$ and $U=6.56t_x$, as a function of the atom number $N$ with different transverse field $E_y$. The horizontal and vertical error bars depict the measurement interval and the standard error of the mean obtained with a statistical Bootstrap method, respectively. The solid lines in corresponding colours are obtained from a weighted global linear fit based on the error bars of the experimental data, while colour shades represent the 95$\%$ confidence bands of the fit.
• Figure 2: Measurement of the Hall voltage. The time-averaged Hall polarization $\langle P_y\rangle$ for two-leg ladders, measured at $t_y=3.30t_x$ and $U=6.56t_x$, is shown as a function of the transverse field $E_y$ with different total atom number $N$; the error bars depict the standard error of the mean and are obtained with a statistical Bootstrap method. The solid lines in corresponding colours are obtained from a weighted global linear fit based on the error bars of the experimental data, while colour shades represent the 95$\%$ confidence bands of the fit.
• Figure 2: Time evolution of the longitudinal current $J_x$ for two-leg ladders, measured at $t_y=3.30t_x$ and $U=6.56t_x$, with different atom number $N$ and transverse field $E_y$. The error bars represent the standard error of the mean and are obtained with a statistical Bootstrap method.
• Figure 3: Dependence of the inverse Hall resistance on atom number and robustness with respect to ladder geometry. The data are measured in the single-band regime at $t_y=3.30t_x$ and $U=6.56t_x$. a, Hall field $E_{\rm H}$ as a function of the atom number $N$ for a two-leg ladder. The filled circles are obtained from the linear fit of the data in Fig. \ref{['fig2']}, and the vertical error bars represent the 95$\%$ confidence interval; the empty circles and their horizontal error bars are also extracted from the data reported in Fig. \ref{['fig2']}, when fitting $P_y$ as a function of $N$ for different transverse field values $E_y$. b, Averaged current $\langle J_x\rangle$ as a function of the atom number $N$ for a two-leg ladder. The vertical error bars denote the standard deviation of $\langle J_x\rangle=\langle J_x(\tau)\rangle_{\tau,{E_y}}$. c, Inverse Hall resistance as a function of the atom number $N$ for both two-leg (filled circles) and three-leg (empty diamonds) configurations. The dashed line indicates the universal relation of Eq. \ref{['eq:rhohexp']}. The vertical error bars are obtained with standard uncertainty propagation. The horizontal error bars in a (filled circles), b and c indicate the range of atom number considered for each point. The coloured areas in a, b, and c are the numerical simulations from a mean-field approximation, accounting for the distribution of atom numbers in the tubes and experimental temperature uncertainty $1.5\leq T/t_x\leq 3$. d, The time-averaged Hall polarization $\langle P_y\rangle$ as a function of the transverse field $E_y$ for three-leg ladders, with different total atom number $N=15{\rm k},\,25{\rm k}$, and $30{\rm k}$. The error bars denote the standard error of the mean, wherein the solid lines in corresponding colours are obtained from a weighted global linear fit with colour shades representing the 95$\%$ confidence bands of the fit.