Time Crystal Platform: From Quasicrystal Structures in Time to Systems with Exotic Interactions

Time crystals are quantum many-body systems that, due to interactions between particles, are able to spontaneously self-organize their motion in a periodic way in time by analogy with the formation of crystalline structures in space in condensed matter physics. In solid state physics properties of space crystals are often investigated with the help of external potentials that are spatially periodic and reflect various crystalline structures. A similar approach can be applied for time crystals, as periodically driven systems constitute counterparts of spatially periodic systems, but in the time domain. Here we show that condensed matter problems ranging from single particles in potentials of quasicrystal structure to many-body systems with exotic long-range interactions can be realized in the time domain with an appropriate periodic driving. Moreover, it is possible to create molecules where atoms are bound together due to destructive interference if the atomic scattering length is modulated in time.

Figure 1

Time quasicrystal: a particle bouncing on a vibrating mirror in a gravitational field in a one-dimensional model. The $1:1$ resonance condition is assumed. Left: the effective potential $V_{\mathrm{eff}}(\mathrm{\Theta })$ with the quasicrystal structure corresponding to the seventh Fibonacci number. Right: Fourier components of the periodic vibration of the mirror, $(f(t)/{\omega }^2)=\sum _k(f_k^c\cos k\omega t+f_k^s\sin k\omega t)$, that result in $V_{\mathrm{eff}}(\mathrm{\Theta })$ presented in the left panel. Full symbols are related to $f_k^c$, open symbols to $f_k^s$. The inset of this panel shows $\mathbf{(}f(t)/{\omega }^2\mathbf{)}$ over one period.

Figure 2

System with exotic interactions: ultracold atoms bouncing on a harmonically oscillating mirror in a one-dimensional model. The $20:1$ resonance condition is fulfilled and the many-body system is described by the Hamiltonian (1). The left panel shows the interaction coefficients $U_{ij}$ corresponding to the scattering length $g_0(t)$ that is presented in the right panel. The frequency $\omega =2.8$ of the mirror oscillations and $\lambda =0.1$ which result in $J=3.7\times {10}^{-5}$ and the gap of $302J$ between the lowest and first excited energy bands. Temporary interaction coefficients $(s2\pi /\omega )|g_0(t)|u_{ij}(t)\le 85J$.

Figure 3

Time crystal with properties of a two-dimensional space crystal: ultracold atoms bouncing between two perpendicular and harmonically oscillating mirrors with $\omega =1.05$, $\lambda =0.02$, and $\phi =(\pi /2)$. The $5:1$ resonance conditions are fulfilled and the many-body system is described by a two-dimensional version of the Bose-Hubbard Hamiltonian (1). For $g_0(t)=\mathrm{const}$, the on-site interactions are dominant, i.e., $U_{ii}/g_0\in [0.2,0.3]J$, where different values correspond to different classical trajectories, while $U_{ij\ne i}/g_0<0.07J$. The left panel shows 25 Wannier wave packets, i.e., $\rho (x,y,t)=\sum _{\mathbf{j}}|W_{\mathbf{j}}(x,y,t){|}^2$, at $t=(4/\omega )$ and trajectories along which they propagate—dots indicate positions of the centers of the wave packets. Two perpendicular mirrors are located at $x=0$ and $y=0$ and they form a $(\pi /4)$ angle with respect to the gravitational force ${\overrightarrow{F}}_g$. Right panels present $\rho (x,y,t)$ at $(x,y)=(90,90)$ (a) and (82,82) (b) versus $t$: these plots reflect cuts of a square lattice described by the Bose-Hubbard model in the moving frame. Numbers in parentheses indicate which lattice sites $\mathbf{j}=(j_x,j_y)$ are located along the cuts.

Figure 4

Two atoms bound together due to destructive interference. The left panel shows a schematic plot of an experiment where two distinguishable atoms move on a ring in opposite directions with momenta $\pm \omega$. Resonant modulation of atomic scattering length allows one to create a molecule where the atoms are bound together due to destructive interference; i.e., in the moving frame eigenstates of the system are Anderson localized. The right panel presents probability densities for detection of the atoms at ${\theta }_1={\theta }_2$ in the laboratory frame versus $t$ for two eigenstates related to energies $E=6020$ (orange line) and 10460 (blue line) for $\lambda =5660$ and $k_0=500$.