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

Dynamical Fractal in Quantum Gases with Discrete Scaling Symmetry