A hydrodynamic analog of Friedel oscillations

Abstract

We present a macroscopic analog of an open quantum system, achieved with a classical pilot-wave system. Friedel oscillations are the angstrom-scale statistical signature of an impurity on a metal surface, concentric circular modulations in the probability density function of the surrounding electron sea. We consider a millimetric drop, propelled by its own wave field along the surface of a vibrating liquid bath, interacting with a submerged circular well. An ensemble of drop trajectories displays a statistical signature in the vicinity of the well that is strikingly similar to Friedel oscillations. The droplet trajectories reveal the dynamical roots of the emergent statistics. Our study elucidates a new mechanism for emergent quantum-like statistics in pilot-wave hydrodynamics and so suggests new directions for the nascent field of hydrodynamic quantum analogs.

Hydrodynamic quantum field theory: the free particle

“…The system has extended the range of classical physics to include many features previously thought to be exclusively quantum [7][8][9]. Hydrodynamic quantum analogs achieved with this system now include tunneling [10][11][12], Landau levels [13,14], Zeeman splitting [15], and Friedel oscillations [16,17]. Quantized orbits arise for droplets walking in either a rotating frame [13,14,18] or a simple harmonic potential [19][20][21].…”

“…Doing so has led to the discovery of hydrodynamic spin states [28,29], and rich two-particle dynamics [30]. Durey et al [31] examined the stability of the self-propelling state in this general classical pilot-wave framework, showing the propensity for in-line oscillations and emergent statistical behavior with a wavelength corresponding to that of the pilot wave [17]. More adventurous still was Fort and Couder's [32] theoretical abstraction of inertial walkers, which exhibited an analog of the Bohr-Sommerfeld quantization rule.…”

“…When the system becomes chaotic, the droplet switches intermittently between these weakly unstable periodic orbits, giving rise to a multimodal statistical signature whose precise form reflects the relative instability of the unstable quantized orbits [14,19,21]. In the second paradigm, the droplet speed oscillates with the wavelength of its pilot wave [52,53], leading to a statistical signature with this same wavelength [16,17]. This mechanism is most clearly demonstrated when a walker interacts with a submerged well: the drop is first drawn inwards at uniform speed along a spiral, then exits the well radially with speed oscillations induced by its interaction with the well [17].…”

Hydrodynamic Quantum Field Theory: The Onset of Particle Motion and the Form of the Pilot Wave

“…Moreover, it will allow for the analysis of the stability of various dynamical states, including the free self-propelling state [19] and the oscillatory particle motion arising in the presence of a harmonic potential [45]. Of particular interest is the stability of the free self-propelling state to speed oscillations with the de Broglie wavelength [39], as may result in a commensurate statistical signature [46]. The extension of our analysis to two dimensions, as outlined in section 3.4, follows through a similar procedure, and has allowed for a comparison between the wave forms in HQFT and those arising in pilotwave hydrodynamics.…”

Emergence of superwalking droplets

“…Intriguingly, such walkers have been shown to mimic several peculiar behaviours that were previously thought to be exclusive to the quantum world. These include orbital quantisation in rotating frames (Fort et al 2010;Oza et al 2014) and harmonic potentials (Perrard et al 2014a,b;Labousse et al 2016), Zeeman splitting in rotating frames (Eddi et al 2012;Oza, Rosales & Bush 2018), wave-like statistical behaviour in confined geometries (Harris et al 2013;Gilet 2016;Durey, Milewski & Wang 2020) as well as in an open system (Sáenz, Cristea-Platon & Bush 2020) and tunnelling across submerged barriers (Eddi et al 2009;Nachbin, Milewski & Bush 2017;Tadrist et al 2020). They have also been predicted to show anomalous two-droplet correlations Valani, Slim & Simula 2018).…”