Observation of Discrete-Time-Crystal Signatures in an Ordered Dipolar Many-Body System

A discrete time crystal (DTC) is a robust phase of driven systems that breaks the discrete time translation symmetry of the driving Hamiltonian. Recent experiments have observed DTC signatures in two distinct systems. Here we show nuclear magnetic resonance observations of DTC signatures in a third, strikingly different system: an ordered spatial crystal. We use a novel DTC echo experiment to probe the coherence of the driven system. Finally, we show that interactions during the pulse of the DTC sequence contribute to the decay of the signal, complicating attempts to measure the intrinsic lifetime of the DTC.

Figure 1

After preparing the ${}^{31}\mathrm{P}$ spins in a $+z$ state, we measure the signal after the pulse sequence $\{\tau -X_{\pi }{\}}^N$. Green dots are for $N$ odd, blue are for $N$ even, and lines are to guide the eye. (a) For $\tau =12.5\mu \mathrm{s}$ and $\theta =0.995\pi$, the response is an oscillation at half the drive frequency, corresponding to a single peak in the Fourier transform at normalized frequency $\tilde{\nu }=1/2$. (b) At the same $\tau =12.5\mu \mathrm{s}$ but with $\theta =1.054\pi$, the signal exhibits a beat pattern, corresponding to a splitting of the Fourier peak. (c) For increased $\tau =392.5\mu \mathrm{s}$ and $\theta =1.060\pi$, the sharp peak at $\tilde{\nu }=1/2$ is restored, despite the deviation of $\theta$ from $\pi$.

Figure 2

To explore the onset and robustness of oscillations at half the drive frequency, we examine the crystalline fraction $f$ using decoupling to turn the ${}^1\mathrm{H}$ off (left) and on (right). [(a) and (d)] The crystalline fraction varies smoothly as a function of $\theta /\pi$, fitting to Gaussians. Crystalline fractions from $\tau =12.5$ (open triangles) and $392.5\mu \mathrm{s}$ (closed circles) are shown. The horizontal dotted line shows $f=0.1$. [(b) and (c)] Cutoff at $f=0.1$ from the Gaussian fits, corresponding to the region in $\theta$ and $\tau$ within which we observe persistent DTC oscillations. With the ${}^1\mathrm{H}$ turned off, the boundary of this region shows structure near $\tau =1\mathrm{ms}$ (see [16] for further discussion). We also show $|\theta -\pi |=W^{\mathrm{P},\mathrm{P}}\tau$ [black dashes, (b) and (c), and (e) and (f)], where $W^{\mathrm{P},\mathrm{P}}$ is the typical interaction scale of the ${}^{31}\mathrm{P}\text{−}{}^{31}\mathrm{P}$ coupling (Table 1). For $\tau >10\mathrm{ms}$, the decoupling power begins to heat the tank circuit, skewing results, and preventing exploration of the $\tau >10\mathrm{ms}$ region for ${}^1\mathrm{H}$ off. [(d)–(f)] Without decoupling, we can avoid heating and explore further in $\tau$ with ${}^1\mathrm{H}$ on. In (f), data span the range $0.03<W^{\mathrm{P},\mathrm{P}}\tau <3200$ radians.

Figure 3

A decay envelope ${\cos }^N(\epsilon )$ (solid black line) bounds the magnitude of our DTC oscillations (closed red triangles) for long $\tau$ and large $\epsilon$. To show that the observed decay is not dominated by an irreversible process, we devise an approximate “unwrapping” sequence to create an echo above the classical decay envelope. After $N$ cycles of $\{\tau -X_{\theta }\}$ (closed red triangles), we switch to the second part of the echo sequence (see text) and increment $N^{\prime }$. The results of this $N^{\prime }$ sequence are shown for $\tau =192.5\mu \mathrm{s}$, where clear echoes are observable for both $\theta =1.08\pi$ (a) and $\theta =1.16\pi$ (b), with $N=\{2,6,10\}$ (open blue circles, green triangles, and yellow diamonds). The corresponding arrow and filled marker show the predicted time of the DTC echo maximum, i.e., where $N^{\prime }=N$. For the case of the DTC sequence at $\theta =\pi$ (blue dots), we observe decay where none is expected in the perfect, delta-function pulse model; the echoes do not rise above this $\theta =\pi$ decay envelope. We examine this effect in Fig. 4.

Figure 4

Experimental pulses are of nonzero duration, which allows the internal Hamiltonian to act. These actions, while small, produce significant effects, especially after many Floquet cycles. At $\tau =20\mu \mathrm{s}$, pulse sequences $\{X,X\}$ (black open squares) and $\{Y,Y\}$ (red open circles) decay much faster than the sequence $\{X,Y\}$ (green open triangles). Signals are sampled after each repeating block $\{\alpha ,\beta \}$. Here, our $T^{*}=\tau +t_p$.