# Time fractals and discrete scale invariance with trapped ions

## TL;DR

This paper demonstrates how a chain of trapped ions can be engineered to exhibit discrete scale invariance and fractal-like time dependence, enabling exploration of exotic quantum phenomena with current technology.

## Contribution

It introduces a method to realize and control discrete scale invariance and time fractals in trapped ion systems, extending to multi-body states with complex scaling structures.

## Key findings

- Discrete scale invariance can be controlled with two tunable parameters.
- The system exhibits fractal-like time dependence.
- Extension to n-body states shows exotic scaling structures.

## Abstract

We show that a one-dimensional chain of trapped ions can be engineered to produce a quantum mechanical system with discrete scale invariance and fractal-like time dependence. By discrete scale invariance we mean a system that replicates itself under a rescaling of distance for some scale factor, and a time fractal is a signal that is invariant under the rescaling of time. These features are reminiscent of the Efimov effect, which has been predicted and observed in bound states of three-body systems. We demonstrate that discrete scale invariance in the trapped ion system can be controlled with two independently tunable parameters. We also discuss the extension to n-body states where the discrete scaling symmetry has an exotic heterogeneous structure. The results we present can be realized using currently available technologies developed for trapped ion quantum systems.