# Dynamical Fractal in Quantum Gases with Discrete Scaling Symmetry

## TL;DR

This paper demonstrates that quantum systems with discrete scaling symmetry can exhibit fractal structures in their dynamics, and shows how such phenomena can be observed in ultracold atomic gases through numerical simulations.

## Contribution

It establishes general conditions for dynamical fractals in quantum systems and verifies their realizability in cold atom experiments.

## Key findings

- Dynamical fractals can arise in quantum systems with discrete scaling symmetry.
- Numerical simulations show these fractals are observable within realistic experimental time scales.
- The predictions are directly verifiable in current cold atom experimental setups.

## Abstract

Inspired by the similarity between the fractal Weierstrass function and quantum systems with discrete scaling symmetry, we establish general conditions under which the dynamics of a quantum system will exhibit fractal structure in the time domain. As an example, we discuss the dynamics of the Loschmidt amplitude and the zero-momentum occupation of a single particle moving in a scale invariant $1/r^2$ potential. In order to show these conditions can be realized in ultracold atomic gases, we perform numerical simulations with practical experimental parameters, which shows that the dynamical fractal can be observed in realistic time scales. The predication can be directly verified in current cold atom experiments.