Tunneling Time for Walking Droplets on an Oscillating Liquid Surface

TL;DR

This paper investigates the tunneling time of walking droplets on an oscillating liquid surface, revealing classical analogies to quantum tunneling and deriving analytic models that support the quantum-like behavior of these droplets.

Contribution

It introduces a revised multiple scattering theory and a 'skipping stone' model to analytically describe tunneling times, strengthening the analogy between walking droplets and quantum particles.

Findings

Tunneling time varies with barrier width similarly to quantum predictions.

Distribution of tunneling times shows consistency with Bohmian mechanics.

Analytic expressions successfully model the observed tunneling behavior.

Abstract

In recent years, Couder and collaborators have initiated a series of studies on walking droplets. Experimentally, they found that at frequencies and amplitudes close to the onset of Faraday waves, droplets on the surface of silicone oil can survive and walk at a roughly constant speed due to resonance. Droplets excite local ripples from the Faraday instability when they bounce from the liquid surface. This tightly coupled particle-wave entity, although a complex yet entirely classical system, exhibits many phenomena that are strikingly similar to those of quantum systems, such as slit interference and diffraction, tunneling probability, and Anderson localization. In this Letter, we focus on the tunneling time of droplets. Specifically, we explore (1) how it changes with the width of an acrylic barrier, which gives rise to the potential barrier when the depth of the silicone oil is reduced to prevent the generation of ripples that can feed energy back to the droplet, and (2) the distribution of tunneling times at the same barrier width. Both results turn out to be similar to the numerical outcome of the Bohmian mechanics, which strengthens the analogy to a quantum system. Furthermore, we successfully derive analytic expressions for these properties by revising the multiple scattering theory and constructing a ``skipping stone" model. Provided that the resemblance in tunneling behavior of walking droplets to Bohmian particles is not coincidental, we discuss the lessons for the Copenhagen interpretation of quantum mechanics that so far fails to explain both characteristics adequately.