Probing the Hall Voltage in Synthetic Quantum Systems

In the context of experimental advances in the realization of artificial magnetic fields in quantum gases, we discuss feasible schemes to extend measurements of the Hall polarization to a study of the Hall voltage, allowing for direct comparison with solid state systems. Specifically, for the paradigmatic example of interacting flux ladders, we report on characteristic zero crossings and a remarkable robustness of the Hall voltage with respect to interaction strengths, particle fillings, and ladder geometries, which is unobservable in the Hall polarization. Moreover, we investigate the site-resolved Hall response in spatially inhomogeneous quantum phases.

Figure 1

(a) Sketch of the flux-ladder ring. (b) Statically tilted ladder with open boundaries. (c) Linear-ramp scheme for the calculation of the Hall voltage $V_H$; see text.

Figure 2

Noninteracting spinless fermions, $\nu =0.1$, $t_y/t_x=1.6$, $M=2$. (a) Transient dynamics in the current $j_x$ and in the polarization $p_y$ induced by a statically tilted potential $V_x$ with ${\mu }_x/t_x={10}^{-2}$ for $\chi /\pi =0.3$. (b) Transient dynamics in the Hall polarization $P_H=p_y/j_x$. (c) Hall polarization $P_H$ versus magnetic flux $\chi$ as obtained from static-tilt simulations (tilt) and adiabatic ring-ladder calculations (pbc). (d) Hall voltage $V_H$ versus $\chi$ as obtained from static-tilt simulations, adiabatic ring-ladder calculations, and linear potential ramps (ramp). Note that the divergence of $P_H$ and the kink in $V_H$ indicate the Meissner-to-vortex transition.

Figure 3

Hall voltage $\nu V_H$ versus magnetic flux $\chi$ for an interacting bosonic ladder, $M=2$, $\nu =0.8$, $U/t_x=2$, $t_y/t_x=1.6$. (a) Symbols depict $\nu V_H$ as obtained from MPS based static-tilt simulations (tilt) and adiabatic ring-ladder calculations (pbc) in the Meissner phase and in the vortex lattice ${\mathrm{VL}}_{1/2}$ and ${\mathrm{VL}}_{1/3}$. The solid blue line shows the weak-coupling result (w.-c.). The upper inset (b) is a close-up of the ${\mathrm{VL}}_{1/3}$ data. The lower inset (c) shows the values of the generalized Hall coefficient $R_H^{p/q}$ for the Meissner (top dashed line) and VL phases ${\mathrm{VL}}_{1/2}$ (dashed-dotted), ${\mathrm{VL}}_{1/3}$ (dotted) obtained from the weak-coupling approach, showing quadratic scaling in accordance with Eq. (3); crosses depict the MPS based data.

Figure 4

Transient dynamics induced by a static tilt in the ${\mathrm{VL}}_{1/3}$, $M=2$, $\nu =0.8$, $U/t_x=2$, $t_y/t_x=1.6$, ${\mu }_x/t_x={10}^{-3}$, $\chi /\pi =0.75$, MPS simulation. (a) Snapshot of the ten most central rungs at time $\tau =10/t_x$ after the quench. The arrows show the strength of the local particle currents, the size of the dots depicts the site-local particle density, and the background shading indicates the local polarization $p_y$ of the individual rungs, using the color code from (b). (b) Rung-resolved time evolution of the polarization $p_y$. (c) Transient dynamics in $p_y$, considering the rungs $v$, $l$, $m$ indicated in (a). The solid red line shows the nearly linear increase of the current $j_x$.

Figure 5

Robust Hall voltage $V_H$ in the Meissner phase and in the ${\mathrm{VL}}_{1/2}$, $t_y/t_x=1.6$. (a) and (b) The Meissner phase, showing the Hall polarization $P_H$ and $V_H$ as a function of the magnetic flux $\chi$ for multileg ($M=2$, 3, 4) ladders, different fillings $\nu =N/(LM)$, and interactions strengths $U$. Note that the data in (b) are vertically offset by $0.1n+0.4$ (with $n=0,1,2,\dots$ for different values of $M$, $\nu$, and $U$) for the purpose of a clear presentation. (c) and (d) The ${\mathrm{VL}}_{1/2}$ phase. The data in (d) are also vertically offset by $0.05n+0.2$. $P_H$ and $V_H$ are obtained by means of static-tilt simulations (lines) and adiabatic ring-ladder calculations (open circles), as described in the text. Contrarily to $P_H$, the $\nu V_H$ data scale on top of each other for different $\nu$, $U$, and $M$.