# Complex collisions of ultracold molecules: a toy model

## TL;DR

This paper introduces a simplified one-dimensional model for ultracold molecular collisions, revealing insights into the transition from chaotic to non-chaotic potentials and establishing scaling laws for bound state densities relevant to real three-dimensional systems.

## Contribution

The study presents a toy model that captures key features of ultracold molecular collisions and derives scaling laws for bound state densities, bridging simplified models and real systems.

## Key findings

- Adiabatic potential distribution transitions from chaotic to non-chaotic with separation.
- Four-body bound states show no visible chaos in energy level spacing.
- Scaling laws accurately predict bound state density dependence on parameters.

## Abstract

We introduce a model to study the collisions of two ultracold diatomic molecules in one dimension interacting via pairwise potentials. We present results for this system, and argue that it offers lessons for real molecular collisions in three dimensions. We analyze the distribution of the adiabatic potentials in the hyperspherical coordinate representation as well as the distribution of the four-body bound states in the adiabatic approximation (i.e. no coupling between adiabatic channels). It is found that while the adiabatic potential distribution transitions from chaotic to non-chaotic as the two molecules are separated, the four-body bound states show no visible chaos in the distribution of nearest-neighbor energy level spacing. We also study the effects of molecular properties, such as interaction strength, interaction range, and atomic mass, on the resonance density and degree of chaos in the adiabatic potentials. We numerically find that the dependence of the four-body bound state density on these parameters is captured by simple scaling laws, in agreement with previous analytic arguments, even though these arguments relied on uncontrolled approximations. This agreement suggests that similar scaling laws may also govern real molecular collisions in three dimensions.