# Trial wave functions for ring-trapped ions and neutral atoms: Microscopic description of the quantum space-time crystal

## TL;DR

This paper develops a theoretical framework for quantum space-time crystals in ring-trapped particles, showing how superpositions of specific many-body states produce undamped, periodic crystalline behavior in space and time, with potential experimental realization.

## Contribution

It introduces a method to construct many-body wave functions that exhibit quantum space-time crystal behavior through symmetry breaking and restoration techniques.

## Key findings

- Existence of low-lying states with space-time crystal behavior.
- Superpositions of magic angular momentum states produce broken-symmetry crystals.
- Predicted experimental conditions near energy crossings for realization.

## Abstract

A constructive theoretical platform for the description of quantum space-time crystals uncovers for $N$ interacting and ring-confined rotating particles the existence of low-lying states with proper space-time crystal behavior. The construction of the corresponding many-body trial wave functions proceeds first via symmetry breaking at the mean-field level followed by symmetry restoration using projection techniques. The ensuing correlated many-body wave functions are stationary states and preserve the rotational symmetries, and at the same time they reflect the point-group symmetries of the mean-field crystals. This behavior results in the emergence of sequences of select magic angular momenta $L_m$. For angular-momenta away from the magic values, the trial functions vanish. Symmetry breaking beyond mean field can be induced by superpositions of such good-$L_m$ many-body stationary states. We show that superposing a pair of adjacent magic angular momenta states leads to formation of special broken-symmetry states exhibiting quantum space-time-crystal behavior. In particular, the corresponding particle densities rotate around the ring, showing undamped and nondispersed periodic crystalline evolution in both space and time. The experimental synthesis of such quantum space-time-crystal wave packets is predicted to be favored in the vicinity of ground-state energy crossings of the Aharonov-Bohm-type spectra accessed via an externally applied magnetic field. These results are illustrated here for Coulomb-repelling fermionic ions and for a lump of contact-interaction attracting bosons.