# Period-$n$ discrete time crystals and quasicrystals with ultracold bosons

## TL;DR

This paper explores novel period-$n$ discrete time crystals and quasicrystals in ultracold bosons on a ring lattice, revealing complex dynamical phases including time quasicrystals, with potential experimental realization.

## Contribution

It introduces a Floquet-driven model of interacting bosons exhibiting period-$n$ time crystals and quasicrystals, including a new incommensurate timescale and a detailed phase diagram.

## Key findings

- Discovery of period-$n$ discrete time crystals in bosonic ring lattices.
- Identification of a 'time quasicrystal' phase with incommensurate timescale.
- Rich phase diagram including thermal and oscillatory phases.

## Abstract

We investigate the out-of-equilibrium properties of a system of interacting bosons in a ring lattice. We present a Floquet driving that induces clockwise (counterclockwise) circulation of the particles among the odd (even) sites of the ring which can be mapped to a fully connected model of clocks of two counter-rotating species. The clock-like motion of the particles is at the core of a period-$n$ discrete time crystal where $L=2n$ is the number of lattice sites. In presence of a "staircase-like" on-site potential, we report the emergence of a second characteristic timescale in addition to the period $n$-tupling. This new timescale depends on the microscopic parameters of the Hamiltonian and is incommensurate with the Floquet period, underpinning a dynamical phase we call 'time quasicrystal'. The rich dynamical phase diagram also features a thermal phase and an oscillatory phase, all of which we investigate and characterize. Our simple, yet rich model can be realized with state-of-the-art ultracold atoms experiments.