Mechanism for orbital quantization of walking droplets in a rotating frame

Millimetric liquid droplets may walk on the surface of a vibrating fluid bath thanks to a resonant interaction with their self-excited wave field. This system constitutes a macroscopic realization of particle-wave duality, exhibiting a range of exotic behaviors previously thought exclusive to quantum particles. In particular, placing walking droplets, or `walkers', in a rotating frame to mimic the effect of a magnetic field leads to quantized circular orbits, with radii taking on values in a discrete set. Though captured by numerical simulations, a theoretical identification of the precise mechanism responsible for the walkers' orbital quantization remains elusive. We here use asymptotic methods to rigorously demonstrate that, owing to wave interference, the force responsible for orbital quantization originates from stationary points on the walker's past trajectory. Moreover, we derive a new minimal quantization model that contains the essential ingredients to reproduce orbital quantization and examine its validity in a wide parameter regime. We discuss the potential of our minimal model to rationalize the mechanism of other hydrodynamic quantum analogs.