Observation of the Aharonov-Bohm Effect in Pilot-Wave Hydrodynamics

TL;DR

This study realizes a classical analogue of the Aharonov–Bohm effect in a macroscopic hydrodynamic pilot-wave system by surrounding a walking droplet with a shielded vortex in an annular cavity. The authors combine velocity measurements, Wigner-like phase-space reconstructions via delay-embedding tomography, and controlled vortex-strength variation to extract a gauge-phase signature. They observe a flux-induced momentum shift $\Delta p$ and a rigid translation of the phase-space distribution $W(x,p)$, together with a linear dependence of orbital speed on the vortex rotation rate $\Omega$, all consistent with AB-type dynamics in a classical setting. This work establishes walking droplets as a platform for synthetic gauge fields and path-dependent geometric phases, enabling trajectory-resolved phase-space studies and future exploration of decoherence and dephasing effects.

Abstract

We report the results of an experimental study of an analog of the Aharonov-Bohm (AB) effect achieved with the hydrodynamic pilot-wave system. A walking droplet is confined to an annular cavity that encircles a shielded vortex, but lies outside its range of direct influence. While there is no vortex-induced flow in the immediate vicinity of the droplets, the vortex modifies the droplet's spatially extended pilot-wave field that guides its motion, producing a vortex-dependent bias in the droplet's orbital speed. High-speed tracking and delay-embedding reconstructions yield Wigner-like phase-space distributions for this hydrodynamic system that exhibits a rigid, flux-dependent translation, providing a force-free, gauge-like realization of an AB-type phase.

Figures (4)

• Figure 1: (a) Schematic of the Aharonov--Bohm (AB) effect for an electron constrained to a ring threaded by magnetic flux $\Phi$. (b) Hydrodynamic analogue: a walking droplet ("walker") travels along an annular cavity while a weak, centrally pinned vortex generates a synthetic magnetic flux. (c) Experimental images showing two representative trajectories of walkers circulating in the annulus while the central vortex rotates clockwise.
• Figure 2: (a) Overlay of the normalized longitudinal velocity $v_x(t)$ for three regimes: clockwise (CW), vortex-off baseline (VOFF), and counterclockwise (CCW). Experimental data (markers) are normalized to unit amplitude and aligned in cycles, and are compared directly with the Aharonov--Bohm theoretical reference $\langle v_x(t)\rangle=v_0\cos(\omega t)$, with $v_0=\frac{\hbar n}{mR}-\frac{q\Phi}{2\pi m R}$ and $\omega=v_0/R$ (solid curves). (b) Relative phase accumulation versus period (cycles, $\tau$). Instantaneous phases are extracted from the normalized $v_x$ traces; differences are formed (CW--VOFF in red; VOFF--CCW in blue) and binned uniformly (20 bins) with SEM error bars. Dashed lines show linear fits whose slopes give the net phase-accumulation rates. For the co-rotating case (red), the fit yields $\Delta\varphi_{\mathrm{CW}}(\tau)=0.068\,\tau-0.073$, corresponding to a positive, flux-induced phase drift per cycle. For the counter-rotating case (blue), the fit yields $\Delta\varphi_{\mathrm{CCW}}(\tau)=-0.062\,\tau+0.052$, demonstrating an opposite-sign phase accumulation of comparable magnitude. The vortex-off baseline remains consistent with zero phase drift, confirming that the observed phase accumulation arises from the synthetic gauge flux.
• Figure 3: Wigner-like phase-space distributions $W(x,p)$ reconstructed from droplet trajectories using delay-embedding tomography with lag $\tau = T_F/2$. Panels (a)–(c) correspond, respectively, to the counter-rotating state (lowest momentum), the vortex-off baseline ($\Gamma_v = 0$), and the co-rotating state (highest momentum). The baseline distribution exhibits an approximately isotropic Gaussian envelope centered at $(\bar{x},\bar{p})$, consistent with the theoretical form of a single-branch Gaussian packet. Activating the vortex produces a rigid translation of the distribution along the momentum axis by $\Delta p/p_0 \approx \pm 0.05$, with no measurable change in width or covariance. This flux-dependent displacement constitutes a direct phase-space analogue of the Aharonov--Bohm momentum shift and quantitatively matches the phase offset inferred from the velocity-based analysis in Fig. \ref{['fig:2-velandphase']}.
• Figure 4: Dependence of mean orbital speed $|v|$ on vortex rotation rate $\Omega$ for co-rotating (red) and counter-rotating (black) motion. Error bars denote SEM. Linear fits (dashed) show nearly antisymmetric slopes, indicating that the synthetic gauge flux is proportional to $\Omega$