Main

The quest to define notions of order and disorder as organizing principles of the natural world is one of the oldest endeavours of science and philosophy. The existence of water in solid, liquid and gaseous forms is a matter of everyday experience, but it also illustrates the complexity of such notions. At high pressure, the liquid and gas phases lose their sharp distinction. The solid phase—ice—exhibits both long-range spatial order and short-range disorder: although the oxygen ions form a crystalline structure with long-range order, the protons bonded to the oxygen ions exhibit a high degree of disorder over shorter length scales1,2.

The modern theory of phases and phase transitions, associated with the names of Landau, Ginzburg and Wilson, has the notion of symmetry and its breaking at its core3,4; adding topological forms of order5, invisible to local symmetry, completes our current understanding. For instance, through its ordered crystalline lattice of oxygen ions, ice breaks the translational symmetry of space. Yet, breaking translational symmetry in time6 (now known as ‘time crystallinity’) is forbidden in thermal equilibrium7,8.

Although thermal equilibrium essentially precludes the appearance of time crystals, this need not be the case for non-equilibrium settings. These provide a flexible setting for entirely new phenomena, which are being intensely studied at the moment; in particular, instances of time crystallinity have been proposed. A prominent example is the discrete time crystal (DTC) in periodically driven (Floquet) systems9,10,11,12,13,14,15,16,17,18,19,20,21,22,23: DTCs exhibit long-range order in both space and time. In arguably their most robust form, the many-body localized DTC10, this spatiotemporal phenomenon is a manifestation of a new notion of order, known as eigenstate order16,24, which can appear in spatially disordered and interacting quantum systems. The plethora of known spatial orderings raises the question about the existence of spatiotemporal order beyond the Floquet paradigm.

In this work, we reveal the existence of new types of temporal order, arising from non-periodic but structured drives. We propose a classification of temporal order based on the continuous Fourier spectrum of its generating drive (Methods), similar to spatial quasicrystals characterized by diffraction patterns in momentum space25. Figure 1a shows a genealogy of spatiotemporal order originating from external drives identified by their Fourier spectra. In the absence of an external drive (i), only spatial but no temporal order can occur, comprising conventional magnetic or crystalline states of matter. By contrast, periodic drives (ii), with discrete Fourier spectra supported at multiples of the drive frequency, can induce non-equilibrium spatiotemporal time-crystalline order.

Fig. 1: Conceptualization and experimental realization of a time rondeau crystal.
figure 1

a, Tree diagram giving an overview of parametrically (meta)stable equilibrium and non-equilibrium quantum phases of matter; tree branches show different types of drive characterized by their spectral decomposition (square boxes) (Methods). The family of RMDs (see also Fig. 2) we implement interpolates between structured random and aperiodic drives (orange frame). A comparison of DTC (green rectangle) and time rondeau crystal (orange rectangle): although both temporal orders show period-doubling dynamics at stroboscopic times, t = MT, the micromotion dynamics of the time rondeau crystal is (tunably) disordered. b, System comprising randomly placed dipolar interacting 13C nuclear spins in a diamond. The dashed lines indicate relevant dipole–dipole interactions. c–e, Example dataset of the experimental observation of rondeau order. Data represent a single-shot measurement of the 13C nuclei polarization projected onto the x axis of its rotating frame. τ = 74.4 μs, N+ = 200, N− = 100, θx = π/2 at γy = 0.98π for a 1-RMD sequence. Figure 2 provides details of the experimental sequence. c, Full 16-s dataset comprising 720 pulses. The 1/e lifetime, Te, exceeds 170 periods, corresponding to 4 s or JTe ≈ 2.6 × 103. d, Zoomed-in view into a window comprising 20 full stroboscopic cycles. The signal flips sign after each full cycle; however, the point within one cycle in which the signal flips is random, clearly indicating the coexistence of long-range temporal order and short-range temporal disorder. e, DFT amplitude for disordered micromotion (brown) and ordered stroboscopic (blue) dynamics; in stark contrast to the δ peak in the stroboscopic dynamics (blue arrow), the micromotion (brown) is almost flat in the entire frequency window of 0 ≤ ν < π with a (linear) suppression for ν→π.

Full size image

Our experiments show that breaking the periodicity of the drive can lead to new exotic forms of partial temporal order; these range from deterministic aperiodic drives (iii)—with discrete (but potentially dense) Fourier spectra—that can give rise to quasicrystalline temporal order26,27,28,29 at one end, to entirely structureless random drives (v)—characterized by a flat Fourier spectrum—associated with complete temporal disorder at the opposite extreme. In between lie structured random drives (iv), realizing the coexistence of temporal disorder on short timescales and temporal order on long timescales; stroboscopic order in the drive cycle coexists with—a tunable degree of—temporal disorder. As previously mentioned, such a coexistence of long-range order and short-range disorder is observed spatially in solid crystals, particularly in ice, and was, until recently, believed impossible to exist in time30. A pattern comprising a repeating theme (here stroboscopic order) that alternates with a contrasting variation theme (here short-time temporal disorder) is known in classical music as a rondeau. Perhaps one of the more famous examples of a rondeau in music is Mozart’s Rondo alla Turca (Turkish March); hence, we refer to this type of temporal order as a rondeau order.

A key aspect of this work is the explicit experimental demonstration of the existence of rondeau order in a single macroscopic system of interacting, hyperpolarized 13C nuclear spins at room temperature (Fig. 1b). Leveraging a new spin control architecture based on an arbitrary waveform generator (AWG) with extensive sequence memory (Methods), we are able to accurately implement stable long-drive protocols that realize a wide variety of non-equilibrium time-dependent drives, including structureless and structured random, aperiodic, quasiperiodic and periodic sequences. Such capability permits us to experimentally investigate random multipolar drives (RMDs)30—a family of structured random protocols with controllable randomness and heating channels—allowing for long spin polarization lifetimes. RMDs feature disorder in time, rather than space. During the long-lived metastable (prethermal) regime, we continuously monitor the x polarization of 13C nuclear spins. At stroboscopic times (t = ℓT for integer ℓ), the system exhibits rigid periodic oscillations, characteristic of long-time order, similar to conventional Floquet DTCs. However, at half-integer times (\(t=(\ell +\frac{1}{2})T\)), the system undergoes random fluctuations in x polarization, indicating short-time disorder (Fig. 1c,d). We refer to observations at these half-integer times as micromotion.

For a family of structured RMDs, including the aperiodic Thue–Morse sequence, we observe robust stroboscopic DTC order across a large parameter range, with long prethermal lifetimes comprising well over 100 (~170) cycles, corresponding to >4 s (Fig. 1c). The Fourier spectrum of micromotion reveals distinctive dynamical features of the time rondeau crystal compared with ordinary DTCs: randomness in the time traces of observables gives rise to a smooth Fourier spectrum in stark contrast to isolated peaks observed for periodic or quasiperiodic drives (Fig. 1d,e). We demonstrate a parametrically controlled lifetime of the prethermal temporal order, by changing the drive period and analysing imperfections of the applied pulses. We find little to no dependence of lifetime on the details of the drive, in agreement with numerical simulations and analytical predictions. This enables us to freely engineer the form of non-deterministic micromotion dynamics, without compromising the stability of the coexisting long-range temporal DTC order.

The existence of rondeau order challenges our current understanding by demonstrating that spatiotemporal order is not confined to time (quasi)crystals. This encourages the investigation of the diversity of non-equilibrium states of matter, beyond the scope of Floquet systems. Rondeau order sharply contrasts with time-crystalline order, where stroboscopic measurements at any point in the drive cycle yield the same temporal order (Fig. 1a, inset). As we demonstrate, rondeau-ordered states offer remarkable flexibility to shape the temporal response of observables, unlike the rigid periodic micromotion found in time-crystalline systems. This flexibility can open new avenues for practical applications, making them a suitable platform for advancements in time-sensitive technologies and novel material properties.

System

Our experiment is performed at room temperature on a single-crystal diamond doped with ~1 ppm of nitrogen-vacancy (NV) centres, and hosting a natural abundance (1.1%) of 13C nuclei. We utilize the randomly positioned network of 13C nuclear spins, described by the Hamiltonian

$${H}_{{\rm{dd}}}=\sum _{k < l}{B}_{kl}\left[3{I}_{k}^{z}{I}_{l}^{z}-{{\boldsymbol{I}}}_{k}\cdot {{\boldsymbol{I}}}_{l}\right],$$
(1)

where Ik is the vector of spin-1/2 operators for nuclear spin k, and the long-range dipole–dipole interaction decays as \({B}_{kl}\propto 1/{r}_{kl}^{3}\), with rkl being the distance between the two spins. The interaction strength between 13C nuclear spins can be characterized by their median coupling J = 〈Bkl〉 = 0.66 kHz (ref. 31). By applying a chirped microwave drive, we transfer polarization from optically pumped NV centres to the 13C nuclear spins, hyperpolarizing them. The density matrix of the initial state is then ρ0 ≈ μIz (\({I}^{z}=\sum _{k}{I}_{k}^{z}\)) with the z-polarization fraction μ ≈ 1%, enhanced ~998-fold over its thermal equilibrium value32.

The experimental sequence is detailed in Fig. 2: after 60 s of hyperpolarization (Methods), the spins are tipped onto the x–y plane by applying a \({(\frac{\uppi }{2})}_{y}\) pulse; we then deploy a two-tone sequence consisting of a ‘fast’ and ‘slow’ drive (Fig. 2a,b). The fast drive comprises short spin-lock θx pulses (length τx), interleaved with free evolution governed by Hdd for time tfree; the duration of one spin-lock cycle is τ = τx + tfree (Fig. 2e). We measure the polarization of the spins in the x–y plane inductively through a radio-frequency (RF) coil during the free evolution after each x pulse (Methods); such a read-out scheme allows us to track the polarization of spins non-destructively through repeated weak measurements33. As a result, spin evolution can be traced over long times and hundreds of thousands of pulses quasicontinuously and in a single shot, a unique feature of our experiments compared with other quantum simulation platforms34,35,36. Applying the fast drive imprints an emergent U(1) symmetry associated with the conservation of Ix polarization; this quasiconservation law enhances the lifetime of the x-polarized initial state by over four orders of magnitude (compared with the bare nuclear \({T}_{2}^{* }\approx 1/J=1.5\,\) ms)31,33 (Methods).

Fig. 2: Experimental implementation.
figure 2

a,b, Full experiment sequence (a) consists of the initial hyperpolarization of the nuclear spins (b) via nearby NV centres (Methods); the polarization is then flipped into the \(\hat{{\bf{x}}}\) direction via a \(\frac{\uppi }{2}\) \(\hat{{\bf{y}}}\) pulse (yellow box). This is followed by the subsequent application of n-pole sequences (pink or cyan box) with equal probability. c, An n-pole sequence is constructed from the systematic arrangement of 2n monopoles, ⊕ (red circle) and ⊖ (blue circle). d, ⊕ (⊖) is defined as a train of N = N− + N+ spin-locking pulses interrupted after N+ (N−) spin locks by \(\hat{{\bf{y}}}\) pulses of angle γy. We choose N+ = 200, N− = 100 and N = 300 throughout. e, Each spin-locking pulse (duration τ) consists of an \(\hat{{\bf{x}}}\) pulse of angle θx (yellow box), followed by a read-out window of time tfree during which only the dipole–dipole interactions (equation (1)) act.

Full size image

Temporal DTC order is realized using the slow drive, which consists of y pulses of angle γy = π + ε (length τy; Fig. 2d (dark yellow blocks)) with free evolution time tfree after every γy pulse, interspersed between trains of fast spin-lock cycles (grey blocks) at multiples of τ. To implement a structured random drive, we use RMD following the proposal in ref. 30. We define two RMD monopoles (n = 0), corresponding to two unitary operations, as follows: both ⊕/⊖ (Fig. 2d, red/blue ovals) consist of a train of N spin-lock cycles interspersed with a single y pulse applied after N+/N− spin locks, respectively. Higher-order n-multipole pairs, for example, dipoles (n = 1: ⊕ ⊖, ⊖ ⊕), quadrupoles (n = 2: ⊕ ⊖ ⊖ ⊕, ⊖ ⊕ ⊕ ⊖) and so on, can be recursively constructed by anti-aligning (n − 1)-multipole pairs together (Fig. 2c, magenta/teal boxes); the n→∞ limit corresponds to a deterministic aperiodic Thue–Morse drive37,38,39. Therefore, increasing the multipole order n can be used to decrease the degree of temporal disorder. Finally, for a fixed n, the complete structured RMD is built out of placing the two sequences of a multipole pair randomly in time (Fig. 2a, magenta/teal pulses; main protocol is discussed in the Methods).

To be precise, an instance of a drive protocol is defined by an ordered list of ⊕/⊖, indicating the order in which the ⊕/⊖ sequences are applied. For example, the periodic Floquet drive is given by the repeated application (⊕, ⊕, ⊕…) of ⊕ (or equivalently ⊖); the 0-RMD drives are obtained by drawing, for each element of the list, one of the two monopoles randomly with equal probability, for example, (⊕, ⊖, ⊖, ⊖, ⊕, ⊕, ⊖…); for the 1-RMD drive, one of the two dipoles is drawn with equal probability, determining two consecutive elements in the list, for example, ([⊕, ⊖], [⊕, ⊖], [⊖, ⊕]…), and so on for higher n-RMD drives. Although structured RMD drives are disordered as we illustrate below, the characteristic timescale T remains fixed and, hence, defines a ‘period’ for these drives.

Characteristics of rondeau order

We begin by analysing the two-tone drive in the fine-tuned case γy = π, where each y pulse fully inverts the polarization Ix↦−Ix. In between two y pulses, the x polarization is protected due to the emergent U(1) symmetry irrespective of its sign. Since both monopoles include exactly one y pulse, the system flips its polarization deterministically with period 2T, irrespective of the specific choice of monopole pairs. Hence, it establishes a long-range temporal order like conventional DTCs. By contrast, micromotion dynamics (t = (ℓ + 1/2T)) inherits randomness from the spin-flip operation: the polarization, which either flips sign or remains unchanged, depends on the monopole that has been applied.

In our experiment, we first confirm the coexistence of temporal order and tunable micromotion disorder for n = 1, and show that they persist away from the fine-tuned limit. We introduce imperfect polarization inversions by moving away from the fine-tuned point, γy = π + ε, by a small but finite deviation (ε ≠ 0). As shown in Fig. 1c,d, the system exhibits stroboscopic period-doubling behaviour together with a disordered micromotion for exceptionally long times: the signal persists even after M > 500 monopole sequences (Jt ≈ 104), corresponding to a physical lifetime of more than 10 s. We also observe a similar dynamical behaviour for the entire family of n-RMD protocols.

The discrete Fourier transform (DFT) of polarization micromotion (Fourier frequency ν) allows us to experimentally verify the characteristic features of RMDs (to remove the influence of prethermal decay on the Fourier transform, we consider the digitized signal of micromotion: positive (negative) values of the signal S > 0 (S < 0) are mapped to +1 (–1)). Although for n = 0, the DFT spectrum is flat since the drive and micromotion are both temporally disordered, for 2-RMD, the DFT spectrum is structured and smoothly distributed over all frequencies (Fig. 3a). The DFT spectrum of the micromotion signal closely reflects that of the 2-RMD used to generate it. The two DFT spectra are shifted with respect to each other (Fig. 3a,b), since polarization inversion introduces a (−1)M phase between the signal and the drive, resulting in a π-shifted DFT30. This feature is difficult to observe for n = 0 since both spectra are trivially flat; however, for more structured n-RMDs with n ≥ 1, multipolar correlation imprints an algebraic suppression νn for ν→0 in their characteristic spectrum (Methods). This suppression shifts to ν→π in the DFT of the micromotion, making the distinctive π-shift feature experimentally measurable.

Fig. 3: Characteristics of rondeau order.
figure 3

a, Amplitude of DFT of the polarization micromotion dynamics for Floquet DTC (pink) and 2-RMD (blue). In contrast to Floquet DTCs, which feature only a single δ peak, the micromotion of the time rondeau crystal has finite support on the entire frequency spectrum. b, Amplitude of DFT of the 2-RMD sequence (Methods) that generated the data in a. The Fourier amplitudes of the drive and micromotion are mirror images of one another with respect to the ν = π/2 axis (referred to as π shifted). c, Fourier amplitudes for RMD sequences n = 0, 1 and 2 on a log–log scale; the Floquet data point would appear at (π − ν) = 0 and is, thus, not shown. The dashed lines are linear fits to the data with slopes α0 = (0 ± 0.1), α1 = (1.0 ± 0.1) and α2 = (1.9 ± 0.1) for n = 0, 1 and 2, respectively. The different multipole orders show distinctive high-frequency suppression, in good agreement with the theoretical predictions (αn = n). The RMD data point represents the mean over 20 realizations of the random drive; error bars indicate the standard deviation; all datasets are normalized by the same normalization constant, here chosen as the maximum value of the n = 2 Fourier amplitude; we set ε = 0.03π; and the other parameters are the same as those in Fig. 1.

Full size image

To test the predicted frequency law, we expose the system to an n-RMD with n = 0, 1 and 2, and plot the DFT spectrum of the micromotion signal on a log–log plot against a π-shifted ν axis (Fig. 3c). The three datasets can be fitted to a good agreement by straight lines, and confirm the anticipated (π − ν)n scaling behaviour. This behaviour of the micromotion DFT in the time rondeau crystal comes in stark contrast to Floquet DTCs, where the micromotion is trivially period-doubled as is the stroboscopic dynamics, leading to a single δ peak in the corresponding DFT spectrum (Fig. 3a,b, pink arrow). Such a comparison, thus, serves as a smoking gun for observing novel types of temporal order beyond the conventional Floquet DTC paradigm30. Conceptually, the above analysis shows that the generalization of temporal order can be conveniently understood in Fourier space.

Stability and robustness

Since the time rondeau crystal is metastable and eventually melts, it is essential to analyse its lifetime and stability. The versatility of the driving protocols we implement allows us to efficiently scan over large parameter regimes and test the rigidity of the stabilized order against perturbations; we also have the ability to tune its lifetime parametrically over a large time window, as we now demonstrate.

Notice first that the long-range temporal order and short-range disorder actually share the same prethermal timescale, as evident from the long-time behaviour of the polarization dynamics (Fig. 1c). Let us, therefore, focus on the stroboscopic dynamics. To map out the phase diagram of the time rondeau crystal, we repeat the experiment for different values of the deviation parameter ε from the perfect kick angle π, γy = π + ε, keeping the period T fixed. Note that the parameter ε indicates deviation from the fine-tuned point γy = π and does not affect the randomness of the drive. We then calculate the DFT spectrum of the stroboscopic dynamics obtained from a fixed but long time window that comprises 720 pulses. Figure 4a,b shows the stroboscopic Fourier spectrum for a wide range of kick angles (0.5π ⪅ γy ⪅ 1.15π) for a 0-RMD drive. We find a dominant narrow peak centred around half the frequency (ν = π/T) that spans over a finite range of kick angles (∣ε∣ ⪅ 0.1π). This confirms the rigidity of rondeau order against small perturbations within this sufficiently long time window. For larger perturbations, the peak gradually fades away, suggesting a crossover from rondeau to trivial order. These experimental results are in excellent quantitative agreement with numerical simulations (Supplementary Section 2C). Remarkably, we observe similar behaviour for other multipolar orders: n = 0, 1, 2, ∞ (Supplementary Section 4B); hence, we find that the rondeau order is robust across the entire family of RMD protocols. In particular, the Thue–Morse sequence (n→∞) allows us to experimentally observe a robust prethermal time aperiodic crystal (Fig. 4c,d), which shows the suitability of our experimental platform to explore a wide range of temporal orders across non-equilibrium matter (Fig. 1a).

Fig. 4: Occurrence of prethermal rondeau order.
figure 4

Normalized Fourier amplitudes of stroboscopic dynamics as a function of flip angle γy around γy = π and frequency ν. a,b, Fourier intensities for monopole (n = 0) sequence (a) and two-dimensional projection (b). c,d, Same data as a (c) and b (d), but for the deterministic Thue–Morse (n = ∞) sequence. The prethermal temporal order shows a stable stroboscopic period-doubling response over a large parameter regime, as indicated by the strong response of the Fourier intensity at half the drive frequency. Remarkably, the stability of the stroboscopic temporal order is independent of the RMD order n (Supplementary Fig. 8 shows the data for n = 1, 2). Experimental 0-RMD data are averaged over ten drive realizations. The other parameters are as those in Fig. 3.

Full size image

Next, we quantify the parametric dependence of decay rate Γe of the spin polarization against changes in period T and deviation ε in the kick angle. We define Γe as the inverse 1/e lifetime of the polarization, that is, the time when the absolute value of the signal first drops below 1/e of its initial value. Note that even for perfect kicks (ε = 0), the polarization can still decay at a rate Γ0 due to the approximate character of the emergent U(1) quasiconservation law. Γ0 can be systematically suppressed by, for instance, increasing the number of spin-lock trains N per monopole31. Our data show that deviations from this limiting case enhance the decay quadratically, Γe − Γ0 ∝ ε2 (Fig. 5a). At the same time, we observe a linear suppression of Γe for small JT (Fig. 5b), regardless of the multipolar order n used in the drive (a deviation for small periods JT is observed due to uncertainty in the calibration of the γy pulse (Supplementary Section 1A)). Both decay laws match our numerical simulations (grey dashed line) with good accuracy (Supplementary Fig. 5), which can also be analytically justified by modelling the dynamics in a dephasing limit (Supplementary Section 2D). The independence on the multipolar order of this scaling and the above observed stability of rondeau order are a key result of the experiments operating in the low-frequency regime; in particular, these result starkly contrast the strong dependence on the multipolar order observed at high frequencies30 (Supplementary Section 2D)

Fig. 5: Controllable prethermal heating rate.
figure 5

a, Dependence of heating rate Γe − Γ0 (Γ0 = Γe∣ε=0) on deviation ε = γ − π for monopole (n = 0, green circles) and Thue–Morse (n = ∞, pink triangles) sequences; the dashed line indicates the ∝ε2 power law predicted by numerical simulations (Supplementary Fig. 5). b, Heating rate Γe against period T, as the deviation ε is simultaneously changed linearly in the period ε = BT to keep the ratio ε/JT fixed. We choose B/(Jπ) ≈ 5.9 × 10−4, and the other parameters are the same as those in a; the dashed line indicates the ∝T1 power law predicted by simulations. The non-equilibrium heating processes are systematically suppressed with decreasing deviation ε and period Nτ, across the entire family of multipole orders (Supplementary Fig. 9 shows the data for n = 1, 2). Experimental 0-RMD data are averaged over 20 drive realizations. The experimental data are consistent with power-law suppression of heating as ∝ε2 and ∝T, as predicted in the dephasing limit (Supplementary Section 2D).

Full size image

Micromotion engineering

So far, we have focused on RMD to realize structured random drives with the degree of disorder being controlled by the multipole order n. However, an important feature of our experimental platform is that it allows us to realize arbitrary structured drives, to investigate the corresponding ordered states.

Specifically, in Fig. 6, we demonstrate versatile micromotion engineering via a data encoding scheme, where the sign of the rigid x-polarization values + and – are interpreted as classical states to represent bits 0 and 1, respectively. Using this scheme, the drive sequence is characterized by the binary information encoded in the sequential arrangement of the two monopole drives ⊕ and ⊖. Importantly, this information can be extracted directly from measuring the micromotion of x polarization: since N+ > N−, measuring the x polarization at half-integer periods (2ℓ + 1)T/2 (\(\ell \in {\mathbb{N}}\)) is sensitive to polarization values before/after the γy kick for the ⊕/⊖ unitary; this allows us to read the sign of the corresponding monopole. To demonstrate the high capability of control, we encoded this paper’s title within the micromotion dynamics of a time rondeau crystal in a 7-bit encoding system (Fig. 6a). In the experiment, which operates in the dephasing limit (Supplementary Section 2D), we find that the heating rate of the string encoding drive aligns closely with that of the Thue–Morse sequence (Fig. 6b). Hence, this example demonstrates the capability to fully manipulate the micromotion without compromising the lifetime of the underlying spatiotemporal order.

Fig. 6: ‘Experimental observation of a time rondeau crystal. Temporal Disorder in Spatiotemporal Order’ encoded inside the micromotion of a time rondeau crystal using the ASCII encoding scheme.
figure 6

a, Micromotion that encodes the word ‘Disorder’ in the ASCII encoding scheme is comprehensively shown: ‘D’ corresponds to 1000100, ‘i’ corresponds to 1101001 and so on. The light-green lines are x-projection signals of 13C nuclear spin polarization measured at half-integer periods (see the main text). The brown (orange) dots indicate that the micromotion is positive (negative) and that the bit encoded is 1 (0). We can artificially engineer the order of ⊕/⊖ pulses to encode arbitrary information into the micromotion, as the heating is well suppressed even for a structureless random drive. b, Full dataset of the x-projection signal of the string encoding sequence (green). The lifetime of the engineered micromotion is comparable to the lifetime of the aperiodic Thue–Morse sequence (grey), demonstrating full controllability of micromotion without reducing the lifetime of the signal. We used γy = 0.98π (ε = 0.02π), with all the other parameters as those in Fig. 1c–e.

Full size image

This also demonstrates the robustness of rondeau order for a broader class of non-periodic but structured drives. Note that in contrast to the n-RMD sequences, encoding specific information in the micromotion necessitates a deterministic drive; at the same time, the drive is non-periodic and the lack of a repetitive structure closely resembles the randomness of the RMD sequences. The corresponding prethermal lifetime of polarization in the high-frequency regime may depend on the amount of structure (that is, information) in the micromotion. We leave this open question for future exploration.

Although Fig. 6 represents a proof of concept, the encoded sequence can be substantially longer, continuing as long as the signal remains above the noise floor, here for t ≈ 36.2 s, allowing the realization of sequences containing >1.3 × 103 bits or >1.9 × 102 characters (Supplementary Fig. 2).

Conclusion and outlook

Our experiments open a promising new avenue to investigate temporal order, demonstrating the long-lived stable coexistence of long-range temporal order and micromotion disorder at short timescales. Going beyond state-of-the-art techniques for controlling and probing non-equilibrium quantum matter, we are able to identify and implement random structureless and structured and aperiodic and periodic drives that give rise to a wide range of temporal orders, including time-crystalline, time-aperiodic-crystalline and rondeau orders—all in a single quantum simulation platform. The versatile structure of our drive protocol allows us to map out the stability diagram and explore the robustness against external perturbations.

Unlike ordinary time-crystalline order, rondeau order allows for great tunability of the temporal spectral micromotion response, at a moderate cost on the lifetime of the temporal order. In fact, in our experiments, we observe no dependence of the lifetime on the details of the driving sequence. Therefore, we can engineer arbitrary micromotion dynamics, beyond the RMD sequences considered before, without loss in signal quality (Fig. 6). This enhanced tunability can boost potential applications of temporal order, like quantum sensing40,41,42,43, cat state preparation44 or topological transport45. Specifically, the tunability of the power spectrum in our experiment may facilitate the creation of frequency-selective, DTC-based quantum sensors42. Moreover, spin-lock lifetimes here are influenced by relaxation from NV electrons46; we, instead, anticipate significantly longer rondeau lifetimes in alternate systems based on photoexcited triplet electrons47,48. Finally, although our experiment focused on nuclear spins in a diamond, the underlying concept is immediately applicable to a wide swathe of quantum simulator platforms.

Methods

Setup

Experiments here used a single crystal diamond with ~1-ppm NV centres and natural abundance (1.1%) of 13C nuclei. The diamond sample, immersed in water, is mounted in an 8-mm 7-inch glass sample tube, such that the [100] face is parallel to an external magnetic field with magnitude B0. The tube is, in turn, attached to a carbon-fibre rod, using two O-rings, and the rod is mounted on a belt-drive actuator (Parker) that ‘shuttles’ the sample rapidly between fields used for 13C hyperpolarization (B0 = 38 mT) and 13C interrogation (B0 = 7 T).

The 13C nuclei are hyperpolarized for tpol = 60 s via NV centres at a low magnetic field (38 mT) via a continuous-wave laser illumination and chirped microwave protocol described in ref. 49, and following a spin-ratchet polarization transfer mechanism described previously in refs. 50,51. The hyperpolarization setup uses multilaser excitation and has been described previously52. At high field (B0 = 7 T), the 13C nuclei are subsequently subjected to the RMDs, with the 13C Larmor precession being sampled in windows between the spin-locking sequences (Fig. 1c).

Spin control architecture

A particular innovation in the current experiments is the design of a new spin control infrastructure that facilitates versatile spin control. Hundreds of thousands of pulses are typically applied. Although in previous experiments33, all the pulses needed to be identical due to memory and other technical limitations, here we significantly lift this constraint. We accomplish this by constructing a new nuclear magnetic resonance spectrometer fully based on a high-speed large-memory AWG (Tabor P9484M). The AWG is used to construct the RF pulses, which are then amplified by a travelling-wave tube (Herley) amplifier and delivered to the RF coil via a cross-diode-based transmit/receive transcoupler (Tecmag). The rapid sampling rate (up to 9 GS s−1) and substantial onboard memory (16 GB) of the AWG, along with the ability to use onboard numerically controlled oscillators, provide a versatile control toolbox; in principle, any sequence of RF pulses can be applied to the spins, comprising over 64,000 unique building blocks, and any larger combination thereof.

In addition, the AWG is also used as a fast digitizer to rapidly sample (here up to 2.7 GS s−1) the 13C Larmor precession between the pulses. The inductively measured signal is amplified via a chain of low-noise amplifiers (ARR and Pasternack) through the transmit/receive transcoupler before digitization. Using the same onboard numerically controlled oscillators can now downshift the precession signals to obtain in-phase and quadrature components. This allows us to interrogate the x and y projections of the 13C nuclear spins directly in their rotating frame.

To create the rondeau order using this new capability, we first create two waveforms, which correspond to ⊕ and ⊖, defined as N∓ \({\frac{\uppi }{2}}_{x}\) pulses, y pulse of angle γy and N± \({\frac{\uppi }{2}}_{x}\) pulses, with free evolution time tfree after every pulse (N+ = 200, N− = 100), respectively; arbitrary other choices of \({N}_{\pm }\in {\mathbb{N}}\) are also possible. We then generated the sequence by placing the two waveforms, ⊕ and ⊖, in the desired order of application. To read-out the signal of the 13C nuclear spins after each pulse, we first wait for tring-down ≈ 12 μs to account for any pulse ring-down, followed by inductive detection for tacq = 12 μs. Thus, in total, the spacing in between the pulses is tfree = tring-down + tacq.

RMDs

Temporal order is realized using the slow drive, which follows a structured RMD sequence. More precisely, the protocol includes two elementary building blocks \({U}_{0}^{\pm }\) of equal duration T:

$${U}_{0}^{\pm }={({U}_{x}{U}_{{\rm{dd}}})}^{N-{N}_{\pm }}{U}_{y}{U}_{{\rm{dd}}}{({U}_{x}{U}_{{\rm{dd}}})}^{{N}_{\pm }}\,,$$
(2)

where Ux = exp(−iθxIx), Uy = exp(−iγyIy) and Udd = exp(−iτHdd). We neglect the dipole–dipole interaction when x and y pulses are applied, as the Rabi frequency Ω is much larger than the median coupling J: Ω > 10J. In the main text, we use ⊕/⊖ to denote \({U}_{0}^{+}\)/\({U}_{0}^{-}\) for simplicity. UxUdd implements a spin-lock pulse and a free evolution governed by Hdd, and Uy implements the polarization inversion. Trains of spin-lock cycles generate the emergent U(1) symmetry required for the quasiconservation of polarization. Note that different numbers of spin-lock pulses are deployed before Uy in these two trains. Hence, polarization flips at different times if different blocks are applied to the system.

For Floquet protocols, the system propagates deterministically with, for instance, only \({U}_{0}^{+}\). By contrast, for 0-RMD, the two operators (or monopoles) are randomly selected to evolve our system. Since this selection is random in time, its DFT is trivially flat. Note that the specific construction of \({U}_{0}^{\pm }\) already embeds a certain structure in the protocol, for instance, polarization flips precisely once within a period. For comparison, in completely structureless random drives, the polarization flip may happen at any time, which normally melts the long-range temporal order rapidly.

Higher-order multipolar operators of order n can be recursively constructed as \({U}_{n}^{\pm }={U}_{n-1}^{\mp }{U}_{n-1}^{\pm }\) by anti-aligning (n − 1)-multipole pairs together. The length of an n-multipole sequence grows exponentially in n as 2nT. In complete analogy, the n-RMD protocol consists of the sequential application of a random selection of \({U}_{n}^{\pm }\) with equal probability. In the limit n→∞, the protocol becomes deterministic and aperiodic in time. It indeed corresponds to the Thue–Morse sequence, which has also been extensively studied in the context of, for instance, quasicrystals and number theory37,53.

Prethermal order

Generic time-dependent many-body systems do not obey the energy conservation law. Therefore, they tend to absorb energy from the external drive, and eventually heat up towards a featureless state at infinite temperature. Although in generic closed systems, Floquet heating cannot be avoided entirely, it can be significantly suppressed if there is a notable mismatch between the local energy scale and the external driving frequency, for instance, in the high-frequency regime54. This can result in an exceptionally long-lived prethermal regime before notable heating happens55,56,57,58. Within this regime, dynamics can be approximated by a static quasiconserved effective Hamiltonian Heff, which can be perturbatively constructed, for instance, by using a Floquet–Magnus expansion or high-frequency expansion59,60. To avoid the eventual heat death of periodically driven systems entirely, non-ergodic, periodically driven many-body localized systems need to be considered10,24; however, their experimental realization—requiring strongly disordered one-dimensional systems61,62—is challenging in practice.

For ergodic interacting many-body systems, the existence of the effective Hamiltonian implies that during the prethermal regime, the local properties of the system can be captured by a prethermal canonical ensemble \({\rho }_{{\rm{pre}}} \approx {\rm{e}}^{-{\beta }_{{\rm{eff}}}{H}_{{\rm{eff}}}}\); here βeff denotes the prethermal temperature that is determined by the energy density of the initial state56. If βeff is sufficiently low and Heff allows for spontaneous symmetry breaking to occur at a finite temperature, ρpre can exhibit equilibrium spatial ordering. Additionally, if regular polarization inversion is further introduced by the drive, as described in the ‘Non-periodic deterministic drives: aperiodic and quasiperiodic sequences’ section, prethermal non-equilibrium time-crystalline order can form63,64.

This prethermal phenomenon can be generalized to other time-dependent protocols even in the absence of strict temporal periodicity. One typical example is quasiperiodically driven systems in which at least two drive frequencies are incommensurate with each other26,27,28,29,65,66,67,68,69. Prethermalization also occurs in RMD systems, where the driving spectrum has continuous support over the entire range of frequencies due to temporal randomness70.

The temporal multipolar correlation of RMD protocols significantly modifies the Fourier spectrum of the random drive sequence: ref. 70 shows that the envelope of the spectrum follows \(\mathop{\prod }\nolimits_{j = 1}^{n}{[1-\cos \left({2}^{j-1}\nu \right)]}^{1/2}\), where ν is the Fourier frequency. This expression indicates an algebraic suppression νn for ν→0. Since it is these low-frequency modes that normally produce the dominant contribution to heating, heating can be algebraically suppressed in the high-frequency regime. The algebraic scaling exponent also has an explicit dependence on multipolar order. One can also construct effective Hamiltonians by generalizing the Floquet–Magnus expansion and use the linear response theory to analyse the heating behaviour more systematically39. Let us emphasize that although prethermalization is readily extended beyond Floquet systems, to the best of our knowledge, the suppression of heating in many-body localized systems driven with random structured drives has not been demonstrated so far. Investigating possible routes to avoid heating in random structured drives is an interesting question for future work, and would open the possibility for an absolutely stable rondeau phase of matter and further tools to investigate them71,72,73.

If the temporal randomness only weakly perturbs the system, the RMD protocol indeed leads to a prethermal plateau, which can feature either equilibrium or non-equilibrium ordering, very similar to Floquet systems. However, the RMD protocol introduced in the ‘Non-periodic deterministic drives: aperiodic and quasiperiodic sequences’ section is designed such that temporal randomness in polarization inversion strongly changes the behaviour of micromotion of the system. As illustrated in the main text, this RMD protocol results in a rondeau order, beyond the conventional Floquet paradigm.

It is worth noting that although those prethermal orders eventually melt, their stability and lifetime can be parametrically controlled, permitting direct experimental observation with our current nuclear spin setup. Throughout, we estimate the lifetime via the 1/e decay time Te defined as the time in which the absolute value of the signal is closest to 1/e of its initial value S0, that is,

$${T}_{e}={\rm{argmin}}t > 0\,| | S(t)| -{S}_{0}/e| \,.$$
(3)

This feature may also be exploited to study interesting applications (see the main text).

DFT

In the main text, we use two different DFTs. For micromotion, we consider the DFT of the digitized micromotion extracted at times (2ℓ + 1)T/2 (\(\ell \in {\mathbb{N}}\)):

$${{\rm{DFT}}}_{{\rm{micro}}}({\nu }_{k})=\frac{1}{M}\mathop{\sum }\limits_{\ell =0}^{M-1}{\rm{e}}^{-{\rm{i}}{\nu }_{k}\ell }{\rm{sgn}}\left(S((2\ell +1)T/2)\right)\,,$$
(4)

where νk = k2π/M, k = 0…M − 1 and sgn(S(t)) = +1 or sgn(S(t)) = –1 depending on whether S(t) > 0 or S(t) < 0, respectively. Note that \(t=(\ell +\frac{1}{2})T\) was chosen for convenience, any other times t = ℓT + t0 with t0 such that the \(\hat{{\bf{y}}}\) pulse has already occurred for ⊕ but not for ⊖, that is, N+τ < t0 < N−τ; note also that this ‘disordered’ regime depends on the choice of N±, which are adjustable experimental parameters.

For the stroboscopic evolution, we consider the DFT of the signal extracted at stroboscopic times ℓT (\(\ell \in {\mathbb{N}}\)):

$${{\rm{DFT}}}_{{\rm{strobo}}}({\nu }_{k})=\frac{1}{M}\mathop{\sum }\limits_{\ell =0}^{M-1}{{\rm{e}}}^{-{\rm{i}}{\nu }_{k}\ell }S(\ell T)\,;$$
(5)

similarly, times \(t=\ell T+{t}_{0}^{{\prime} }\) with \({t}_{0}^{{\prime} } < {N}_{+}\tau\) or \({t}_{0}^{{\prime} } > {N}_{-}\tau\) are also possible. Note that the Fourier amplitude is defined as the absolute value of the DFT (Fourier amplitude = ∣DFT∣), in contrast to the Fourier intensity (that is, the power spectrum), which is defined as the absolute value squared.

Classification of drives and their Fourier transforms

Integer representation and DFT

A similar DFT can be defined for the RMD drives considered in the main text. Specifically, all RMD drives can be described as an ordered sequence of ⊕ and ⊖ (Fig. 2); therefore, by identifying ⊕ and ⊖ with numbers +1 and –1, respectively, any drive instance is uniquely determined by a sequence \({\mathcal{S}}={s}_{n}\), n = 1, 2…sn ∈ {+1, –1} (for example, Extended Data Fig. 1, top). Instead, we consider the DFT of the drive sequence given by

$${{\rm{DFT}}}_{{\rm{drive}}}({\nu }_{k})=\frac{1}{M}\mathop{\sum }\limits_{\ell =0}^{M-1}{{\rm{e}}}^{-{\rm{i}}{\nu }_{k}\ell }{s}_{\ell }\,,$$
(6)

where νk = k2π/M, k = 0…M − 1. In analogy to the signal DFTs, the Fourier amplitude is defined as the absolute value of the DFT; this definition of the drive Fourier amplitude, using the integer representation of the drive, is used in Fig. 3.

Note that this DFT should not be confused with the continuous Fourier spectrum of its generating drive, which is sketched in Fig. 1a. In particular, by assigning integers to the ⊕/⊖ sequences, we coarse-grain over the details of the monopoles. This results in a loss of information that can alter the resulting discrete Fourier spectrum. For instance, the Floquet sequence (+1, +1, +1…) has a DFT that is indistinguishable from that of a static (time-independent) drive, and the n = 0 sequence (for example, (+1, –1, –1, –1, +1, +1, –1…)) has an entirely flat spectrum, suggesting that this drive corresponds to a completely structureless drive; however, both these observations are wrong since the inner structure of the ⊕/⊖ are ignored: the Floquet sequence is a periodic time-dependent sequence since the \(\hat{{\bf{y}}}\) pulses within one ⊕/⊖ are not always on, and the n = 0 sequence is structured because only one of two structured sequences is applied.

Continuous Fourier transform

To restore the entire information of the drives, the Fourier transform of the analogue drive should be considered, which we refer to as continuous Fourier transform. In our case, the drive relevant for the temporal order is the \(\hat{{\bf{y}}}\) field; more generally, we can consider an arbitrary driving field, d(t), that couples to the system and can induce temporal order: H(t) = H0 + d(t)V. In Extended Data Fig. 1, we display the analogue drives and the corresponding continuous Fourier spectra for some of sequences considered in this work, where, for simplicity, we consider \(\hat{{\bf{y}}}\) to follow a square pulse. The different drives display the distinct features sketched in Fig. 1a(ii)–(iv); to obtain the two missing features (i) and (v), a constant drive d(t) = d and completely random drive d(t) = ξt, with white noise \({\mathbb{E}}({\xi }_{t})=0\) and \({\mathbb{E}}({\xi }_{t}{\xi }_{{t}^{{\prime} }})=\delta (t-{t}^{{\prime} })\cdot {\rm{const}}\) (where \({\mathbb{E}}\) denotes the expectation value), can be used, respectively. However, neither of these two realizes a stable temporal order.

Non-periodic deterministic drives: aperiodic and quasiperiodic sequences

In the phylogenetic tree of drives (Fig. 1a), we reported on the experimental observation of a time aperiodic crystal arising from the Thue–Morse sequence. Here, for completeness, we also report on the experimental observation of a time quasicrystal using the famous Fibonacci sequence (Extended Data Fig. 2). The Fibonacci sequence is defined via the recursive relation (n ≥ 2)

$${U}_{n}={U}_{n-2}{U}_{n-1}\,,$$
(7)

which is initialized by the elementary building blocks U1 = ⊕ and U0 = ⊖.

Quasiperiodic drives form a special subclass of aperiodic drives, which can be characterized by a finite number of incommensurate frequencies. Hence, their Fourier spectra consists of only finitely many independent frequencies ν1…νk (k < ∞) and integer linear combinations of these, that is, P(ν) ≠ 0 only if \(\nu =\mathop{\sum }\nolimits_{j=1}^{k}{\ell }_{j}{\nu }_{j}\), where \({\ell }_{j}\in {\mathbb{Z}}\). Although there have been various numerical and analytical studies on the heating properties of specific aperiodic39 and quasiperiodic28 drives, as of now, a systematic classification of the heating property in generic aperiodically driven many-body systems is missing.