# Time crystal behavior of excited eigenstates

## TL;DR

This paper explores the possibility of realizing time crystals in excited eigenstates of quantum systems, demonstrating that spontaneous breaking of time translation symmetry can occur under certain conditions, especially in ultra-cold atomic gases.

## Contribution

It shows that time crystal behavior can be observed in excited eigenstates, extending the concept beyond ground states and providing a pathway for experimental realization.

## Key findings

- Time crystal behavior occurs in excited eigenstates.
- Symmetry breaking can be induced by measurements.
- The lifetime of the symmetry broken state is analyzed.

## Abstract

In analogy to spontaneous breaking of continuous space translation symmetry in the process of space crystal formation, it was proposed that spontaneous breaking of continuous time translation symmetry could lead to time crystal formation. In other words, a time-independent system prepared in the energy ground state is expected to reveal periodic motion under infinitely weak perturbation. In the case of the system proposed originally by Frank Wilczek, spontaneous breaking of time translation symmetry can not be observed if one starts with the ground state. We point out that the symmetry breaking can take place if the system is prepared in an excited eigenstate. The latter can be realized experimentally in ultra-cold atomic gases. We simulate the process of the spontaneous symmetry breaking due to measurements of particle positions and analyze the lifetime of the resulting symmetry broken state.