Quantized orbital-chasing liquid metal heterodimers directed by an integrated pilot-wave field

Abstract: A millimetric bouncing droplet sustained on a vibrating bath becomes a moving wave source (particle) through periodically interacting with the local wave field it generates during the droplet-bath impact. By virtue of such particle-wave duality, the macroscopic hydrodynamic system imitates enigmatic behaviors of the quantum realm. Here we show that it is possible to create an integrated pilot-wave field to better prescribe the droplet trajectories, via amplified bath capillarity. This is demonstrated with a liquid metal droplet-bath system in which the local wave field generated by droplet bouncing is superposed by the global wave field induced by bath meniscus oscillation. The resulting dual pilot-wave configuration enables a class of directional chasing motions of two bound dissimilar droplets (heterodimers) in multilevel hydrodynamic traps (orbits), featuring two quantized regime parameters, namely the interdroplet binding level and the orbit level. We investigate the dynamics of the vibrating liquid metal bath, with its level-split ring wave field and its peculiar vortex field being highlighted. We also rationalize the exotic droplet motions by considering the interdroplet particle-wave interactions mediated by the integrated pilot-wave field. It is revealed that a temporal bouncing phase shift between the two droplets in the heterodimers, due to size mismatch, gives rise to their horizontal propulsion, while their spatial binding regime exclusively determines the collective chasing direction. It is further evidenced that the horizontal in-orbit chasing motion is directly related to vertical droplet bouncing. Our findings unveil the integrated pilot-wave field as a trail towards improved droplet guiding, thereby extending the hydrodynamic particle-wave analogy to optical systems and beyond.