Walking droplets represent an ideal playground to explore wave-particle duality at the macroscopic scale. The proper control and measurement of such system requires several experimental tools that are not often easily affordable at undergraduate level. This paper proposes a complete low-cost and open-source experimental setup for walking droplets with a performance characterization. The setup is tested by examining the behaviour of droplets in the stadium billiard, a two-dimensional concave cavity that yields chaotic trajectories. Drastic differences between classical and quantum particles behaviour were observed in such geometry. In particular, the long-term evolution of walking droplets in a stadium billiard presents clear scarring patterns, informing on the existence of preferred "probable positions" within the billiard.
Walking droplets represent an ideal playground to explore wave-particle duality at the macroscopic scale. The proper control and measurement of such system requires several experimental tools that are not often easily affordable at undergraduate level. This paper proposes a complete low-cost and open-source experimental setup for walking droplets with a performance characterization. The setup is tested by examining the behaviour of droplets in the stadium billiard, a two-dimensional concave cavity that yields chaotic trajectories. Drastic differences between classical and quantum particles behaviour were observed in such geometry. In particular, the long-term evolution of walking droplets in a stadium billiard presents clear scarring patterns, informing on the existence of preferred "probable positions" within the billiard.
Several recent experiments were devoted to walkers, structures that associate a droplet bouncing on a vibrated liquid with the surface waves it excites. They reveal that a form of wave-particle duality exists in this classical system with the emergence of quantum-like behaviours. Here we revisit the single particle diffraction experiment and show the coexistence of two waves. The measured probability distributions are ruled by the diffraction of a quantumlike probability wave. But the observation of a single walker reveals that the droplet is driven by a pilot wave of different spatial structure that determines its trajectory in real space. The existence of two waves of these types had been proposed by de Broglie in his "double solution" model of quantum mechanics. A difference with the latter is that the pilot-wave is, in our experiment, endowed with a "path memory". When intrusive measurements are performed, this memory effect induces transient chaotic individual trajectories that generate the resulting statistical behaviour.
Abstract This study brings a recently discovered macroscopic phenomenon with wave-particle characteristics into the classroom. The system consists of a liquid droplet levitating over a vertically shaken liquid pool. The droplets allow visualization of a wave–particle system in a directly observable way. We show how to interpret this macroscopic phenomenon and how to set up and carry out this experiment. A class of students performed single slit diffraction experiments with droplets. By scoring individual droplet trajectories students find a diffraction pattern. This pilot application in the classroom shows that students can study and discuss the wave–particle nature of the bouncing droplet experiment. The experiment therefore provides a useful opportunity to show wave–particle behavior on the macroscopic level.
This chapter provides an overview of chaotic billiard lasers as a prominent branch of quantum chaos. These lasers offer an ideal experimental platform for demonstrating the principles of quantum chaos within a physical system. We begin by introducing the fundamental principles of chaotic ray dynamics in optical microcavities, where the transition from regular to fully chaotic dynamics fundamentally alters the underlying wavefunctions and lasing properties. A central focus is placed on "chaos-assisted light emission," which serves as a practical manifestation of "chaos-assisted tunneling" -- a hallmark phenomenon in the study of quantum chaos. We discuss both theoretical frameworks and experimental validations, demonstrating how chaotic orbits facilitate the coupling between evanescently localized modes and far-field emission. Furthermore, exploring how the presence of a gain medium influences established results from quantum chaos research remains a fundamental and intriguing problem in physics. To address this, we establish a rigorous and comprehensive derivation of the Maxwell-Bloch equations for two-dimensional microcavity lasers, specifically examining their application to fully chaotic, stadium-shaped billiard lasers. By bridging the gap between nonlinear lasing processes and chaotic wavefunctions, this chapter highlights the unique potential of chaotic billiards for controlling light-matter interactions and shaping the next generation of unconventional coherent light sources.
Quantum walks are, at present, an active field of study in mathematics, with important applications in quantum information and statistical physics. In this paper, we determine the influence of basic chaotic features on the walker behavior. For this purpose, we consider an extremely simple model consisting of alternating one-dimensional walks along the two spatial coordinates in bidimensional closed domains (hard wall billiards). The chaotic or regular behavior induced by the boundary shape in the deterministic classical motion translates into chaotic signatures for the quantized problem, resulting in sharp differences in the spectral statistics and morphology of the eigenfunctions of the quantum walker. Indeed, we found, for the Bunimovich stadium—a chaotic billiard—level statistics described by a Brody distribution with parameter δ≃0.1. This indicates a weak level repulsion, and also enhanced eigenfunction localization, with an average participation ratio (PR)≃1150 compared to the rectangular billiard (regular) case, where the average PR≃1500. Furthermore, scarring on unstable periodic orbits is observed. The fact that our simple model exhibits such key signatures of quantum chaos, e.g., non-Poissonian level statistics and scarring, that are sensitive to the underlying classical dynamics in the free particle billiard system is utterly surprising, especially when taking into account that quantum walks are diffusive models, which are not direct quantizations of a Hamiltonian.
Quantum walks are, at present, an active field of study in mathematics, with important applications in quantum information and statistical physics. In this paper, we determine the influence of basic chaotic features on the walker behavior. For this purpose, we consider an extremely simple model consisting of alternating one-dimensional walks along the two spatial coordinates in bidimensional closed domains (hard wall billiards). The chaotic or regular behavior induced by the boundary shape in the deterministic classical motion translates into chaotic signatures for the quantized problem, resulting in sharp differences in the spectral statistics and morphology of the eigenfunctions of the quantum walker. Indeed, we found, for the Bunimovich stadium-a chaotic billiard-level statistics described by a Brody distribution with parameter delta similar or equal to 0.1. This indicates a weak level repulsion, and also enhanced eigenfunction localization, with an average participation ratio (PR)similar or equal to 1150 compared to the rectangular billiard (regular) case, where the average PR similar or equal to 1500. Furthermore, scarring on unstable periodic orbits is observed. The fact that our simple model exhibits such key signatures of quantum chaos, e.g., non-Poissonian level statistics and scarring, that are sensitive to the underlying classical dynamics in the free particle billiard system is utterly surprising, especially when taking into account that quantum walks are diffusive models, which are not direct quantizations of a Hamiltonian.
We explore the effects of an imposed potential with both oscillatory and quadratic components on the dynamics of walking droplets. We first conduct an experimental investigation of droplets walking on a bath with a central circular well. The well acts as a source of Faraday waves, which may trap walking droplets on circular orbits. The observed orbits are stable and quantized, with preferred radii aligning with the extrema of the well-induced Faraday wave pattern. We use the stroboscopic model of Oza et al. [J. Fluid Mech. 737, 552–570 (2013)] with an added potential to examine the interaction of the droplet with the underlying well-induced wavefield. We show that all quantized orbits are stable for low vibrational accelerations. Smaller orbits may become unstable at higher forcing accelerations and transition to chaos through a path reminiscent of the Ruelle-Takens-Newhouse scenario. We proceed by considering a generalized pilot-wave system in which the relative magnitudes of the pilot-wave force and drop inertia may be tuned. When the drop inertia is dominated by the pilot-wave force, all circular orbits may become unstable, with the drop chaotically switching between them. In this chaotic regime, the statistically stationary probability distribution of the drop’s position reflects the relative instability of the unstable circular orbits. We compute the mean wavefield from a chaotic trajectory and confirm its predicted relationship with the particle’s probability density function.