Experimental investigation of walking drops: Wave field and interaction with slit structures.

While bouncing walking silicone oil droplets (walkers) do show many quantumlike phenomena, the original, most intriguing, double-slit experiment with walkers has been shown to be an overinterpretation of data. Several experiments and numerical simulations have proven that for at least some parameter region there is no randomness. Still, true randomness was claimed to be observed in an experiment on chaotically bouncing walkers. Also, most of the available phase space has not been investigated. The main goal of this paper is therefore to look for true interference and chaos in the entire phase space. Recently, we made an extensive investigation of drops interacting with slits, but still in a limited range. However, the outcome was always deterministic and only incidentally mimicked the statistics of the corresponding quantum case. We also showed that the extra interference, already seen by others, in the double-slit case was caused by reflection of waves from the outlet of the unused slit after passage and thus was not a true double-slit effect. A new theoretical treatment of bouncing drop dynamics has since given analytic relations for the associated wave field, leading to a proposal for criteria for the possible occurrence of true interference in the double-slit experiment. Satisfying these criteria, requires working close to the onset of the Faraday instability, with two spatial conditions favoring slow walkers, and a temporal condition favoring fast walkers. The regions of high velocity, where the walkers bounce periodically, and of very low velocity, with chaotically bouncing walkers, have not been comprehensively investigated so far. We have therefore looked at these regions, probing the limits for interaction with slits. Furthermore, noting that a short transit time is essential to fulfill the criteria, experiments were done using double-slit barriers only 0.5 and 2 mm broad. Nowhere was true interference or a chaotic response found. As the theory has implications for many of the observed quantumlike phenomena involving walkers as, e.g., tunneling and interaction between drops, we have measured the spatial and temporal decay of the wave field. A comparison with the theory shows very good agreement but leads to a reformulation of the above-mentioned criteria.

Interaction of wave-driven particles with slit structures.

Just over a decade ago Couder and Fort [Phys. Rev. Lett. 97, 154101 (2006)PRLTAO0031-900710.1103/PhysRevLett.97.154101] published a provocative paper suggesting that a classical system might be able to simulate the truly fundamental quantum mechanical single- and double-slit experiment. The system they investigated was that of an oil droplet walking on a vibrated oil surface. Their results have since been challenged by Andersen et al. [Phys. Rev. E 92, 013006 (2015)PLEEE81539-375510.1103/PhysRevE.92.013006] by pointing to insufficient statistical support and a lack of experimental control over critical parameters. Here we show that the randomness in the original experiment is an artifact of lack of control. We present experimental data from an extensive scan of the parameter space of the system including the use of different size slits and tight control of critical parameters. For the single-slit we find very diverse samples of interference-like patterns but all causal by nature. This also holds for the double-slit. However, an extra interference effect appears here. The origin of this is investigated by blocking either the inlet or the outlet of one slit. Hereby we show that the extra interference is solely due to back-scatter of the associated wave field from the outlet of the slit not passed by the droplet. Recently Pucci et al. [J. Fluid Mech. 835, 1136 (2018)JFLSA70022-112010.1017/jfm.2017.790] using a much broader slit also showed that the classical system is basically causal. They, too, observed the extra interference effect for the double-slit. However, the reason behind was not determined. Moreover they claimed the existence of a chaotic regime just below the cri- tical acceleration for spontaneous generation of Faraday surface waves. Our measurements do not support the validity of this claim. However, the drop dynamics turns out to have an interesting multifaceted interaction with the slit structure.

Double-slit experiment with single wave-driven particles and its relation to quantum mechanics.

In a thought-provoking paper, Couder and Fort [Phys. Rev. Lett. 97, 154101 (2006)] describe a version of the famous double-slit experiment performed with droplets bouncing on a vertically vibrated fluid surface. In the experiment, an interference pattern in the single-particle statistics is found even though it is possible to determine unambiguously which slit the walking droplet passes. Here we argue, however, that the single-particle statistics in such an experiment will be fundamentally different from the single-particle statistics of quantum mechanics. Quantum mechanical interference takes place between different classical paths with precise amplitude and phase relations. In the double-slit experiment with walking droplets, these relations are lost since one of the paths is singled out by the droplet. To support our conclusions, we have carried out our own double-slit experiment, and our results, in particular the long and variable slit passage times of the droplets, cast strong doubt on the feasibility of the interference claimed by Couder and Fort. To understand theoretically the limitations of wave-driven particle systems as analogs to quantum mechanics, we introduce a Schrödinger equation with a source term originating from a localized particle that generates a wave while being simultaneously guided by it. We show that the ensuing particle-wave dynamics can capture some characteristics of quantum mechanics such as orbital quantization. However, the particle-wave dynamics can not reproduce quantum mechanics in general, and we show that the single-particle statistics for our model in a double-slit experiment with an additional splitter plate differs qualitatively from that of quantum mechanics.

Amplitude equation and long-range interactions in underwater sand ripples in one dimension.

We present an amplitude equation for sand ripples under oscillatory flow in a situation where the sand is moving in a narrow channel and the height profile is practically one dimensional. The equation has the form ht = - epsilon(h-h)+((hx)2-1)hxx-hxxxx+delta((hx)2)xx which, due to the first term, is neither completely local (it has long-range coupling through the average height h) nor has local sand conservation. We argue that this is reasonable and show that the equation compares well with experimental observations in narrow channels. We focus in particular on the so-called doubling transition, a secondary instability caused by the sudden decrease in the amplitude of the water motion, leading to the appearance of a new ripple in each trough. This transition is well reproduced for sufficiently large delta (asymmetry between trough and crest). We finally present surprising experimental results showing that long-range coupling is indeed seen in the initial details of the doubling transition, where in fact two small ripples are initially formed, followed by global symmetry breaking removing one of them.

Counting function for a sphere of anisotropic quartz.

We calculate the leading Weyl term of the counting function for a monocrystalline quartz sphere. In contrast to other studies of counting functions, the anisotropy of quartz is a crucial element in our investigation. Hence we do not obtain a simple analytical form, but we carry out a numerical evaluation. To this end we employ the Radon transform representation of the Green's function. We compare our result to a previously measured unique data set of several tens of thousands of resonances.