Period-n Discrete Time Crystals and Quasicrystals with Ultracold Bosons.

Andrea Pizzi,Johannes Knolle,Andreas Nunnenkamp

doi:10.1103/physrevlett.123.150601

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 counterrotating species. The clocklike 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 the presence of a "staircaselike" 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.

Highlights

Introduction.— Symmetries pervade most fields of modern physics, ranging from nuclear physics to relativity and condensed matter physics
We find that three dynamical phases are possible: (I) period-n discrete time crystal (DTC): for small ≈ 0 the clock-like circulation of the particles is rigidified by the interaction U = 0 and signaled by Z ≈ 1
We emphasize that the observation of the DTC is not exclusive of the initial condition we considered here but is rather valid for a generic initial Fock state featuring a macroscopic imbalance, thanks to the underlying clock-like particle circulation

Summary

The eigenvalue problem reads

Yno , yne )T is a 2n-dimensional column vector. We rewrite the eigenvalue problem (15) component by component as (F y)2j. = ωj γyjo + αyje + γyjo+1 + βyje+1 = λyje (F y)2j−1 = ωj γyje + αyjo + γyje−1 + βyjo−1 = λyjo. (16) by ωj and introducing the Fourier transform yke/o =. N) we get γyko αyke γ ωk yko β ωk yke λyke−1 γyke + αyko + γωkyke + βωkyko = λyko−1,. Which is itself a 2 × 2-dimensional eigenvalue problem for the eigenvalue λn. M1 has determinant 1, and its eigenvalues can be written

For δ

Exact diagonalization