Observation of a Prethermal $U(1)$ Discrete Time Crystal

A time crystal is a state of periodically driven matter that breaks discrete time-translation symmetry. Time crystals have been demonstrated experimentally in various programmable quantum simulators, and they exemplify how nonequilibrium, driven quantum systems can exhibit intriguing and robust properties absent in systems at equilibrium. These robust driven states need to be stabilized by some mechanism, with the preeminent candidates being many-body localization and prethermalization. This introduces additional constraints that make it challenging to experimentally observe time crystallinity in naturally occurring systems. Recent theoretical work has developed the notion of prethermalization without temperature, expanding the class of time-crystal systems to explain time-crystalline observations at (or near) infinite temperature. In this work, we conclusively observe the emergence of a prethermal $U(1)$ time-crystalline state at quasi-infinite temperature in a solid-state NMR quantum emulator by verifying the requisites of prethermalization without temperature. In addition to observing the signature period-doubling behavior, we show the existence of a long-lived prethermal regime whose lifetime is significantly enhanced by strengthening an emergent $U(1)$ conservation law. Not only do we measure this enhancement through the global magnetization, but we also exploit on-site disorder to measure local observables, ruling out the possibility of many-body localization and confirming the emergence of long-range correlations.

Popular Summary

Under periodic driving, quantum systems can enter into novel states of out-of-equilibrium matter. Unlike equilibrium states, out-of-equilibrium states display long-lived oscillations between two states. Such a phenomenon could be the basis of a robust quantum memory, among other applications. Here, we experimentally realize such an out-of-equilibrium state, called a time crystal, in a system at room temperature. In contrast, many previous observations of time crystals have been made in highly engineered systems at ultracold temperatures.

Time crystals are a phase of matter that breaks time-translation symmetry: They have a repeating structure not in space but rather in time. In our experiments, we used solid-state nuclear magnetic resonance to periodically control nuclear spins and alter their effective interaction strength. Importantly, changing the interaction strength allowed us to construct a phase diagram for the conditions under which a time crystal would either exist or “melt.”

In cases where time-crystalline order emerged, we found that the quantum system persisted for very long times, which is unusual for systems out of equilibrium at relatively high temperatures. Indeed, this behavior confirms the existence of “prethermalization without temperature,” a recently proposed theoretical mechanism in which high-temperature quantum systems can persist for long times out of equilibrium with their environment. To further confirm this mechanism, we used a new technique to look at local observables and rule out the possibility of many-body localization, which is not expected to coexist with a system undergoing prethermalization without temperature.

Creating long-lived quantum states is a vital task for the design of future quantum processors. Time crystals, which we have shown take a very long time to thermalize even at effectively infinite temperature, are therefore a good candidate for exploring near-term implementations of a robust quantum memory.

Figure 1

Crystal structure of fluorapatite, ${\mathrm{Ca}}_5({\mathrm{PO}}_4{)}_3\mathrm{F}$, with an emphasis on the spin-full nuclei. Intrachain fluorine (blue) bonds and nearest-neighbor fluorine-phosphorus (blue-red) bonds are shown. Additionally, the spin-less nuclei of oxygen (yellow) and calcium (green) are included with no bonds shown. The spin-full isotopes of calcium and oxygen are less than 1% natural abundance and neglected. The vertical crystal $c$-axis length is 6.805 Å (with two ${}^{19}\mathrm{F}$ per unit cell), while the horizontal $a,b$ axes are 9.224 Å long [23].

Figure 2

Magnetization of noninteracting spin ensembles under repeated kicking. Each Floquet cycle is $T=120\mathrm{\mu }\mathrm{s}$ long, comprising evolution under the time-suspension sequence followed by an $R_y(\theta )$ unitary, with $\theta \in \{170,180\}$ degrees. Perfect $\pi$ kicking gives trivial period doubling, with a stretched exponential ($b=0.70$) decay with a timescale of $\tau \approx 6.573\mathrm{ms}$ [Eq. (7), $a\approx 1$, $c\approx 0$]. The $\theta =(1-2\epsilon )\pi$ kicking results in beating with a frequency of $\frac{1}{2}\pm \epsilon$ decaying on a similar timescale.

Figure 3

Time traces of the magnetization response for noncrystalline (left panel) and time-crystalline (right panel) systems under $\theta =170°$ driving. The interaction strength is set by the physical interaction time, $T=u{T}_0$. Notice the dramatic order-of-magnitude improvement in signal lifetime of the time crystal over the melted crystal.

Figure 4

Fourier transform of the magnetization signal, $\mathcal{F}\mathbf{(}{m}_z(t)\mathbf{)}$, for various interaction magnitudes $\gamma =u{J}_0{T}_0$ under 170° kicking. The interaction magnitude is set by physically varying the time between consecutive $R_y(\theta )$ unitary kicks. For small $\gamma$, the signals show beating consistent with the $\theta \ne \pi$ drive. At large enough $\gamma$ ($\gamma >\pi /8$), the beating is replaced with a single “period-doubled” peak, often taken to herald a system’s transformation to a time crystal.

Figure 5

Difference in selected Fourier peak magnitudes of the observed magnetization under variable interaction magnitude ($\gamma$) and deviation from perfect $\pi$ kicking ($\epsilon$). Blue (red) shaded regions indicate a larger (smaller) period-doubled response than beating. Experimental data were collected for points at vertices of the grid lines; peak differences are smoothed within each grid for easier visualization. Using smoothed interpolations of the data, we estimate the intersection of beating and period doubling for each instance of $\epsilon$. The resulting intersections lead to the fitted phase boundary, shown as a dashed black line.

Figure 6

Decay of local and global magnetization signals for $\theta =170°$ kicking with $\gamma =\pi /8$ ($u=0.1$.) The locally magnetized spin signal decays much more rapidly than the global one, as local magnetization is not conserved by the prethermal Hamiltonian. Furthermore, there is no oscillatory behavior outside of period doubling, indicating that eigenstates compatible with MBL are induced. As usual, a Floquet period corresponds to $120\mathrm{\mu }\mathrm{s}$ of real-time evolution.

Figure 7

Plots of the log-transformed magnetization, $-\log (|⟨{\widehat{S}}_z(nT){\widehat{S}}_z|⟩)$ as a function of Floquet kicking periods, $n$, for $\theta =170°$, and select interaction magnitudes $\gamma$, in which period-doubled behavior is dominant and the driving frequency is not too slow. Notably, we see the existence of two exponential timescales shown via the linear fits in black. A short timescale, ${\tau }_1\approx 7.2$ Floquet cycles (dashed line), present at early experimental times, shown on the left, gives way to a longer-lived timescale, ${\tau }_2\approx 30.8$ Floquet cycles (dot-dashed line), after an initial prethermalization period, shown on the right. The partition between early and late times is made to optimize the quality of the linear fits in both regimes. As usual, a Floquet period corresponds to $120\mathrm{\mu }\mathrm{s}$ of real-time evolution.

Figure 8

Left panel: exemplary fitting results for variable interaction-time DTC experiments, plotted in nondimensional Floquet periods instead of evolution time for a more uniform comparison. For each fit, the relevant extracted parameters are the nondimensional decay timescale ${\tau }_F=\tau /T$ (units of Floquet periods) and the exponential stretching parameter $b$. Right panel: interpolation of the signal lifetimes extracted from fittings as shown on the left panel, as a function of the Floquet period $T$, $\tau (T)=aT-b$. The fitted parameters are $a=37.11$ (nondimensional) and $b=453.39\mathrm{\mu }\mathrm{s}$.

Figure 9

Two-timescale fitting for the $T=15\mathrm{\mu }\mathrm{s}$ DTC experiment and the $T=30\mathrm{\mu }\mathrm{s}$ DTC experiment in nondimensional units of Floquet periods.

Figure 10

Results of local $z$ state experiments under periodic driving with interactions turned off via Peng24. In the top panel, signals are shown for kicking angles $\theta =0°,170°,180°$. We notice that the decay envelope is similar for all three cases, as in the global $z$ case, albeit decaying more rapidly. In the bottom panel, we show the fitting of the $\theta =180°$ signal to the four-parameter model given in Eq. (7).

Figure 11

Two-timescale fitting results for various applied stroboscopic $z$ fields, in nondimensional units of Floquet periods. The data generated by this procedure are used to populate the entries of Table 1.

Figure 12

Effects of increasing disorder on time crystallinity for $\theta =175°$, $\gamma =\pi /16$, and $\theta =170°$, $\gamma =\pi /8$. On the left, we show the measured magnetization signal for each instance of scaled disorder strength, $v\in \{0.0,0.1,0.2,0.3,0.4\}$. Here, $v$ multiplies the disordered field in the Floquet engineered Hamiltonian as generated by Wei16. The Hamiltonian for variable $v$ is given in Eq. (f1). On the right, we take the difference between experiments with disorder and experiments with no disorder. Notice that the disordered field does not stabilize the signal; increasing disorder simply leads to increased variability in the measured signal.