Discrete time crystal in a finite chain of Rydberg atoms without disorder

We study the collective dynamics of a clean Floquet system of cold atoms, numerically simulating two realistic setups based on a regular chain of interacting Rydberg atoms driven by laser fields. In both cases, the population evolution and its Fourier spectrum display clear signatures of a discrete time crystal (DTC), exhibiting the appearance of a robust subharmonic oscillation which persists on a timescale increasing with the chain size, within a certain range of control parameters. We also characterize how the DTC stability is affected by dissipative processes, typically present in this atomic system even though the Rydberg state is very long lived.

Figure 1

Schematic chain of eight equally spaced Rydberg atoms trapped in optical potentials. The atoms interact through a nearest-neighbor coupling $V$, and are driven by a common laser field of Rabi frequency $\mathrm{\Omega }$. Such driving field is periodically switched on ($T_1$) and off ($T_2$).

Figure 2

The average population difference $P\left(n\right)$ (left panels) and corresponding Fourier spectra $S\left(\nu \right)$ (right panels) for a closed chain of $L=8$ Rydberg atoms, in the case of zero detuning ($\mathrm{\Delta }=0)$. The various panels stand for [$\left({\mathrm{a}}_1\right)$ and $\left({\mathrm{b}}_1\right)$] $\epsilon =V=0$; [$\left({\mathrm{a}}_2\right)$ and $\left({\mathrm{b}}_2\right)$] $\epsilon =0.1\mathrm{MHz}$ and [$\left({\mathrm{a}}_3\right)$ and $\left({\mathrm{b}}_3\right)$] $V=0$; $\epsilon =V=0.1\mathrm{MHz}$, and $T_2=10\mu \mathrm{s}$. Here and in the next figures, the order parameter $Q\left(n\right)$ is plotted in colorbars over the left panels, with $Q\left(n\right)=-1$ and $Q\left(n\right)=1$ shown in blue and yellow, respectively.

Figure 3

Same as in Fig. 2, but for $\mathrm{\Delta }=0.6\mathrm{MHz}$ and in the case of zero perturbation of the Rabi frequency $\epsilon =0$ (i.e., $\mathrm{\Omega }{T}_1=\pi /2)$. Panels $\left({\mathrm{a}}_1\right)$ and $\left({\mathrm{b}}_1\right)$ are for $V=0$, while panels $\left({\mathrm{a}}_2\right)$ and $\left({\mathrm{b}}_2\right)$ are for $V=0.1\mathrm{MHz}$ and $T_2=10\mu \mathrm{s}$.

Figure 4

$\left({\mathrm{a}}_1\right)$–$\left({\mathrm{a}}_3\right)$: The average population difference in various Floquet time intervals, for a closed chain of $L=14$ Rydberg atoms with $V=0.09\mathrm{MHz}$. (b) The critical oscillation number $n_c$ against the size $L$ in a semilog scale for different values of $V$; we show here both data with only nearest-neighbor interactions (dashed lines with empty symbols) and including next-nearest-neighbor ($n.n.n.)$ interactions (solid lines with full symbols). The datapoint evidenced with a dark circle indicates the situation highlighted in the three upper panels. The other parameters are chosen as $\mathrm{\Delta }=0.6\mathrm{MHz}$, $\epsilon =0$, and $T_2=15\mu \mathrm{s}$.

Figure 5

The average population difference and the critical oscillation number for different values of $\epsilon$, $\mathrm{\Delta }$, and $V$. The three upper panels are for $\epsilon =0.4\mathrm{MHz}$, while the three lower panels are for $\epsilon =-0.4\mathrm{MHz}$. $\left({\mathrm{a}}_1\right)$–$\left({\mathrm{a}}_2\right)$ The population difference $P\left(n\right)$ for $\mathrm{\Delta }=-0.782\mathrm{MHz}$ and $\mathrm{\Delta }=0.578\mathrm{MHz}$, respectively, as indicated by arrows in panel $\left({\mathrm{c}}_1\right)$. $\left({\mathrm{b}}_1\right)$–$\left({\mathrm{b}}_2\right)$: The population difference $P\left(n\right)$ for $\mathrm{\Delta }=-0.285\mathrm{MHz}$ and $\mathrm{\Delta }=0.452\mathrm{MHz}$, respectively, as indicated by arrows in panel $\left({\mathrm{c}}_2\right)$. $\left({\mathrm{c}}_1\right)$–$\left({\mathrm{c}}_2\right)$: The critical oscillation number $n_c$ as a function of the detuning $\mathrm{\Delta }$. Green data sets stand for $V=0.1\mathrm{MHz}$, as is the case for all the four panels on the left (arrows point to specific values of $\mathrm{\Delta }$ on the green data sets). The red data sets stand for $V=-0.1\mathrm{MHz}$. In this figure we consider a chain of $L=8$ atoms and fixed $T_2=15\mu \mathrm{s}$.

Figure 6

Same as in Fig. 4, but for the alternative model that compensates the term in $\mathrm{\Delta }$ appearing in Eq. (2) [cf., Eq. (7)]. All the various parameters are set as in Fig. 4. Notice the reduced sensitivity of $n_c$ to the vdW potential $V$, as compared to the previous model of Sec. 2.

Figure 7

Same as in Fig. 5, but for the improved Rydberg-atom model [cf., Sec. 4]. Upper panels are for $\epsilon =0.4\mathrm{MHz}$, while lower panels are for $\epsilon =-0.4\mathrm{MHz}$. Panels $\left({\mathrm{a}}_1\right)$ and $\left({\mathrm{b}}_1\right)$ are for $\mathrm{\Delta }=0\mathrm{MHz}$; panels $\left({\mathrm{a}}_2\right)$ and $\left({\mathrm{b}}_2\right)$ are for $\mathrm{\Delta }=-0.5\mathrm{MHz}$; panels $\left({\mathrm{a}}_3\right)$ and $\left({\mathrm{b}}_3\right)$ are for $\mathrm{\Delta }=0.5\mathrm{MHz}$. All the plots with $P\left(n\right)$ correspond to specific points in the green curves of panels $\left({\mathrm{c}}_1\right)$ and $\left({\mathrm{c}}_2\right)$ indicated by arrows. Green data sets in panels $\left({\mathrm{c}}_1\right)$ and $\left({\mathrm{c}}_2\right)$ stand for $V=0.1\mathrm{MHz}$; red data sets stand for $V=-0.1\mathrm{MHz}$. In this figure we consider a chain of $L=8$ atoms and fixed $T_2=15\mu \mathrm{s}$.

Figure 8

(a) The critical oscillation number $n_c$ against the chain size $L$, for the model of Sec. 2 with $\mathrm{\Delta }=0.16\mathrm{MHz}$ (black circles), $0.155\mathrm{MHz}$ (red squares), $0.15\mathrm{MHz}$ (green diamonds), $0.145\mathrm{MHz}$ (blue triangles up), $0.143\mathrm{MHz}$ (cyan triangles right), and $0.142\mathrm{MHz}$ (orange stars), respectively. The other parameters are chosen as $\epsilon =0.4\mathrm{MHz}$, $V=0.1\mathrm{MHz}$, and $T_2=15\mu \mathrm{s}$. $\left({\mathrm{b}}_1\right)$–$\left({\mathrm{b}}_3\right)$ Finite-size phase diagram of the simplified model in Appendix pp2 [$\left({\mathrm{b}}_1\right)$ $\mathrm{\Delta }=0$, $V=0$ in $H_1$, and $V=0.1\mathrm{MHz}$ in $H_2$], the alternative model of Sec. 4 [$\left({\mathrm{b}}_2\right)$ $\mathrm{\Delta }=0.8\mathrm{MHz}$ in $H_1$, $\mathrm{\Delta }=0$ in $H_2$, and $V=0.1$ MHz], and the original model of Sec. 2 [$\left({\mathrm{b}}_3\right)$ $\mathrm{\Delta }=0.8\mathrm{MHz}$ and $V=0.1\mathrm{MHz}$], plotted against the chain size $L$ and the perturbation $\epsilon$. The yellow, green, and blue regions, respectively, correspond to $\delta n_c>0$, $\delta n_c=0$, and $\delta n_c<0$.

Figure 9

The modulus of the average population imbalance $\left|P\left(n\right)\right|$ against the Floquet cycle number $n$ in semilog scale, with (a) $\mathrm{\Gamma }=1.0\mathrm{kHz}$ and (b) $\mathrm{\Gamma }=10\mathrm{kHz}$. Black, red, green, and blue lines correspond to $L=4$, $L=6$, $L=8$, and $L=10$, respectively. The other parameters are chosen as $\epsilon =-0.1\mathrm{MHz}$, $V=0.1\mathrm{MHz}$, $\mathrm{\Delta }=0$, $T_1=1.0\mu \mathrm{s}$, and $T_2=15\mu \mathrm{s}$. The two left insets of $\left({\mathrm{a}}_1\right)$ and $\left({\mathrm{b}}_1\right)$ show the same time traces of the corresponding main frames for $L=10$, without taking the modulus of $P\left(n\right)$. The right inset of $\left({\mathrm{a}}_2\right)$ shows the decay rate of the population imbalance as a function of $L$, obtained from an exponential fit $\left|P\left(n\right)\right|\sim e^{-\alpha n}$ (see dashed lines for $n>75)$ of the numerical data for (a) $\mathrm{\Gamma }=1.0\mathrm{kHz}$.