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

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.

Figure 1

Two entwined, counterrotating clocks on a ring lattice. As a concrete example we show the $n=3$ case. (a) Our Floquet driving scheme alternates (i) hopping between sites $2j$ and $2j-1$ ($J_1$, single line), (ii) hopping between sites $2j$ and $2j+1$ ($J_2$, double line), and (iii) on-site interaction and on-site potential, see Eq. (1). For $J_1=J_2=\pi /2$ (i.e., $\epsilon =0$) the particles in the odd (red) and even (blue) sites move clockwise and counterclockwise, respectively. (b) Fourier transform of the generalized imbalance $\tilde{\mathcal{I}}$ in Eq. (3). In the presence of interaction ($U=0.05$) and a staircaselike potential ($\delta =0.96$) we observe a DTC (for $\epsilon =0.01$) and a DTQC (for $\epsilon =0.05$). The former is characterized by a single sharp peak locked at frequency $2\pi /n$, whereas the latter also features sharp peaks at two frequencies ${\omega }_n^{1,2}$, which are incommensurate with the driving frequency.

Figure 2

Period-$n$ discrete time crystal, exemplified for $n=L/2=3$. For vanishing potential ($h_j=0$) we characterize the semiclassical dynamics via (b) the distance between two slightly different initial states $d^2(t)$ in Eq. (5), (c) the real part of the time crystal order parameter $Z(t)$ in Eq. (4), and (d) the Fourier transform of the generalized imbalance $\tilde{\mathcal{I}}(\omega )$ in Eq. (3). On-site interactions ($U$) rigidify the DTC, with $Z(t)\approx 1$ and $\tilde{\mathcal{I}}(\omega )$ sharply peaked at $\omega =2\pi /n$ for small $\epsilon$ (yellow lines). For larger $\epsilon$ and small interactions we find an oscillatory dynamical phase (OS), characterized by a few characteristic frequencies at which $\tilde{\mathcal{I}}(\omega )$ is peaked (red lines). The thermal phase (TH) has $Z$ oscillating chaotically, $\tilde{\mathcal{I}}(\omega )$ with no dominant peak, and $d^2$ growing to a finite value $\sim 1$ indicating sensitivity to the initial conditions (blue lines). The dynamical phases are identified as a function of $\epsilon$ and $U$ via $⟨Z{⟩}_t$ (e) and $⟨d^2{⟩}_t$ (f), which we compute as time averages over $M={10}^4$ Floquet periods. The dashed lines are the same in (e) and (f), the crosses correspond to the parameters considered in (b)–(d). We observe that interactions rigidify the DTC, which expands from $\epsilon =0$ for increasing $U$.

Figure 3

Discrete time quasicrystal. The on-site staircaselike potential leads to a DTQC, here exemplified for $L=4$. (a) The dynamical phases are characterized by the Fourier transform of the generalized imbalance $\tilde{\mathcal{I}}(\omega ,\epsilon )$ for $\delta =0.9$, $U=2$, and 500 Floquet periods. For a small $|\epsilon |$ we observe the DTC, corresponding to a single sharp peak locked at frequency $2\pi /n$. For larger $\epsilon$ we observe the DTQC, reflected by a sharp peak locked at frequency $2\pi /n$ and two further peaks at frequencies ${\omega }_n^{1,2}\approx \pi \pm 2\epsilon$ (dashed blue lines). (b) The dynamical phases are characterized via the quantities $⟨Z{⟩}_t=\tilde{\mathcal{I}}(2\pi /n)$ and $⟨d^2{⟩}_t$ (inset) computed over $10^4$ Floquet periods and for $\epsilon =0.1$. The DTC, DTQC, and oscillatory phases correspond to $⟨Z⟩\approx 1,0.5,0$, respectively, and thermalization is signaled by a finite $⟨d^2{⟩}_t\sim 1$. Dashed lines serve as a reference and are the same in the main plot and in the inset. The DTQC is rigidified by interactions, expanding from $\delta =1$ for increasing $U$. For sufficiently large $U$, the system enters a DTC. (c) We characterize the system as a function of $\epsilon$ and $U$ for $\delta =0.9$. The minimum interaction required to enter the DTC phase increases with $|\epsilon |$, and no DTC is possible at all if $|\epsilon |$ is too large.