Sensing with discrete time crystals

Abstract

Prethermal discrete time crystals are non-equilibrium states of matter with long-range spatiotemporal order and a subharmonic response stabilized by many-body interactions under periodic driving. The robustness of time-crystalline order to perturbations in the drive protocol makes these systems attractive for quantum sensing. Here we exploit the sensitivity of prethermal discrete time crystal order to deviations in its order parameter to implement the frequency-selective detection of time-varying magnetic fields in a system of strongly driven, dipolar-coupled ¹³C nuclear spins in a diamond. Incorporating an oscillating field into the time crystal dynamics extends its lifetime exponentially, producing a sharp resonant response in the order parameter. The sensor linewidth is set by the time crystal lifetime alone, as strong interspin interactions help stabilize the time-crystalline order. The device operates in the 0.5–50-kHz range—a challenging frequency regime for sensors based on atomic vapour or electronic spins—and achieves competitive sensitivity. The sensing principle we demonstrate is robust to drive errors and sample inhomogeneities, and is applicable across a range of physical platforms including superconducting circuits, neutral atoms and trapped ions.

Main

Non-equilibrium matter has emerged as a frontier in modern many-body physics, displaying novel phenomena beyond restrictions imposed by thermal equilibrium. A milestone is the demonstration^{1,2,3,4,5,6,7,8,9,10,11,12,13} of discrete time crystals (DTCs)^{14,15,16,17,18,19,20,21,22}, a new form of non-equilibrium matter that breaks time-translation symmetry, akin to ordinary crystals breaking spatial symmetry. A hallmark of DTCs is their robust period-doubling response, stabilized by many-body interactions of mean strength J, making them resilient to errors in the protocol creating them. Most observed time-crystalline states rely on Floquet prethermalization^{23,24,25,26,27,28}, where periodically driven quantum states are preserved for durations parametrically controlled by the drive frequency, resulting in lifetimes ${T}_{2}^{{\prime} }$ far exceeding the system’s interaction-dominated intrinsic decay time ${T}_{2}^{* }$ (∝J⁻¹).

The robustness and long lifetimes of DTCs in the presence of interactions make them promising for quantum technologies, such as simulating complex systems²⁹, topologically protected quantum computation³⁰ and robust generation of entangled states³¹. Experimental work has demonstrated the use of continuous time crystals as a d.c. field sensor³². Separately, theoretical proposals have suggested using DTCs for enhanced quantum sensing^{33,34,35,36,37}. However, experimental realizations using DTCs as quantum sensors have been challenging due to the necessity of using strongly correlated states³³ or fine-tuned systems^34,35,36.

In this work, we develop a new approach for using DTCs to construct highly frequency-selective quantum sensors for time-varying (a.c.) magnetic fields, and demonstrate it experimentally in an ensemble of randomly positioned, hyperpolarized ¹³C nuclear spins in diamond. The sensor operates in the 0.5–50-kHz range—typically a challenging frequency regime for sensors based on atomic vapour³⁸ or electronic spins³⁹—and achieves competitive sensitivity. The scheme leverages the robustness of DTC order, requires no preparation of strongly correlated states and is broadly applicable in platforms exhibiting prethermal U(1) DTC order^{2,4,5,7,10,12}.

Our approach is based on the observation that under certain conditions, the stability of DTC order can be remarkably enhanced by the presence of an a.c. field; this stabilization is highly frequency selective, enabling the DTC to be exploited for sensing. Specifically, the a.c. field B_a.c.(t) couples to the DTC order parameter only when its frequency ${f}_{{\rm{a}}.{\rm{c}}}={f}_{{\rm{r}}{\rm{e}}{\rm{s}}}$ matches the DTC oscillations, protecting the DTC order from symmetry-breaking perturbations and exponentially enhancing its lifetime ${T}_{2}^{{\prime} }$ (ref. ⁴⁰). We demonstrate that the lifetime extension can be as much as three orders of magnitude, yielding a record for coupling-normalized DTC lifetimes $J{T}_{2}^{{\prime} }\approx 14,051$, and importantly that it is a strongly resonant effect, producing a narrow a.c. frequency response, with linewidths under 70 mHz around ${f}_{res}$, determined solely by ${({T}_{2}^{{\prime} })}^{-1}$.

We leverage this, using a DTC excited via two-tone Floquet drive¹⁰, to construct a noise-rejected, continuously interrogated a.c. sensor that operates for extended periods without reinitialization. Unlike traditional quantum sensing⁴¹, which avoids interactions between sensor spins⁴², the DTC sensor here intimately relies on these interactions, as well as thermalization, to establish a ${T}_{2}^{{\prime} }$-limited sensor linewidth, and remaining robust against drive errors and on-site disorder (Supplementary Section 7). We additionally demonstrate that the a.c.-field-mediated lifetime extension applies equally to DTCs excited along both transverse and longitudinal axes, suggesting wide applicability across diverse platforms including spin systems^{2,4,5,7,10,12}, superconducting qubits^11,31,43 and cold atoms^1,3.

Results

Principle: prethermal DTC lifetime extension by a.c. fields

The system consists of a diamond crystal with ¹³C nuclear spins hyperpolarized by optically pumped nitrogen-vacancy (NV) centres (Fig. 1a). The ¹³C nuclei, at natural abundance, are randomly distributed and influenced by fluctuating fields from lattice NV and P1 centres^44,45. Spins interact via magnetic dipole interactions, ${{\mathcal{H}}}_{dd}={\sum }_{k < l}{J}_{kl}(3{I}_{k}^{z}{I}_{l}^{z}-{{\bf{I}}}_{k}\cdot {{\bf{I}}}_{l})$, with spin–spin coupling strengths J_kl, and median coupling strength J ≈ 0.6 kHz (ref. ⁴⁶) determined via free induction decay; ${I}_{k}^{\alpha }$ are spin-1/2 operators for ¹³C nuclear spin k and the total polarization is ${I}^{\alpha }={\sum }_{k}{I}_{k}^{\alpha }$, α = x, y, z.

Fig. 1: System and principle.

a, System consists of dipolar interacting ¹³C nuclear spins, hyperpolarized by NV centres using optical and chirped microwave excitation. b, Protocol. (i) Evolution under native dipole–dipole interactions, ${{\mathcal{H}}}_{dd}$ (see the main text) is interrupted by a concatenated two-tone drive with N spin-lock (pink) $\theta (\widehat{{\bf{x}}})$ pulses separated by τ, interspersed with spin-flip$\gamma (\widehat{{\bf{y}}})$ pulses (yellow). This time block (total period, T) is repeated M times. The protocol causes switching between $\widehat{{\bf{x}}}\leftrightarrow -\widehat{{\bf{x}}}$ every t = Nτ, remaining robust against deviations γ_y = π + ϵ, forming a PDTC. Additionally, a $\widehat{{\bf{z}}}$-oriented a.c. field (green) with amplitude B_a.c. and frequency f_a.c. is applied; the resonant case ${f}_{AC}={f}_{res}=1/(2T)$ is shown. (ii) Net spin magnetization I is monitored during acquisition time t_acq ≈ 13.6 μs between the θ_x pulses. Projection S onto the ¹³C nuclear spin’s rotating frame $\widehat{{\bf{x}}}-\widehat{{\bf{y}}}$ plane and its phase ϕ are measured. c, Main result. Magnetization 〈I^x〉 for the PDTC protocol without (blue) and with (red) applied resonant a.c. field. Here N = 16; pulse separation τ (36 μs); (π/2)_y and θ_x pulses (~50.25 μs); γ_y pulse (~98.5 μs); B_a.c. = 82.4 μT with ${f}_{res}=330$ Hz. The top axis indicates the number of flips M. The dashed line indicates the 1/e intercept, yielding lifetimes of ${T}_{2}^{{\prime} }=80\,ms$ without an a.c. field and ${T}_{2}^{{\prime} }=4.51\,s$ with an a.c. field. Data show M > 13,000 〈I^x〉 flips sustained over t = 20 s. (i) Zoomed-in view of the data in a small 20-ms window at t = 0.76 s, displaying magnetization switching from $-\widehat{{\bf{x}}}$ to $\widehat{{\bf{x}}}$. Lifetime extension under a.c. field is evident from the increased amplitude of the red data points. (ii) Tracked phase ϕ for data from the main panel, displaying a coherent signal far beyond the 1/e decay time. $\widehat{{\bf{x}}}\,(-\widehat{{\bf{x}}})$ rails correspond to phases 0 (π), respectively. Decoherence in the non-a.c. case leads to ϕ spread uniformly in [−π, π] (blue data points). The a.c. field leads to a notable lifetime increase (red).

A U(1) prethermal DTC (PDTC) is created using the two-tone drive protocol from ref. ¹⁰ (Fig. 1b(i)). Hyperpolarization initializes the ¹³C nuclear spins in the $\widehat{{\bf{x}}}$-polarized state ρ₀ ∝ I^x, after which the two-tone drive is activated. The first spin-lock drive, consisting of $\widehat{{\bf{x}}}$-oriented θ pulses separated by period τ, realizes an effective Hamiltonian H_eff with emergent U(1) symmetry: [H_eff, I^x] = 0.

The second drive (period T) establishes the PDTC order via superimposed $\widehat{{\bf{y}}}$ pulses of angle γ_y (~π), applied after every Nth spin-lock pulse. The ¹³C nuclear spins are inductively interrogated between pulses via a radio-frequency cavity; a downsampling technique⁴⁷ (Methods) enables the quasi-continuous monitoring of their projection onto the $\widehat{{\bf{x}}}\,-\,\widehat{{\bf{y}}}$ plane directly in the rotating frame (Fig. 1b(ii)). Net projection is denoted as S, and the phase in the $\widehat{{\bf{x}}}\,-\,\widehat{{\bf{y}}}$ plane is ϕ (Fig. 1b(ii)). Continuous interrogation with the two-tone drive allows full time-trace read-out in a single shot (key for quantum sensing) and distinguishes it from single-tone drives commonly used in other systems^4,8,12 (Supplementary Section IIC).

PDTC order, arising from emergent U(1) symmetry, is characterized by robust period doubling, seen in the long-lived oscillation of polarization 〈I^x〉 with period 2T (ref. ¹⁰). This is shown in Fig. 1c (blue data points), with the number of cycles M of the DTC γ_y drive on the top axes. The decay of 〈I^x〉 has a characteristic 1/e time, ${T}_{2}^{{\prime} }=79\,ms$, which is much longer than ${T}_{2}^{* }=1.5\,ms$ (ref. ⁴⁶). The purple data points in Fig. 1c(i) (inset) provide a zoomed-in view at t = 0.75 s.

The PDTC decay can be understood by noting that the initial state (${\rho }_{0} \sim {\mathbb{1}}+\mu {I}^{x}$) corresponds to a zero-energy state with respect to the effective Hamiltonian, ${\langle {H}_{e\mathrm{ff}}\rangle }_{{\rho }_{0}}=0$. The eigenstate thermalization hypothesis (ETH)^48,49,50,51 implies that without conservation laws, the system prethermalizes to a featureless infinite-temperature (${\mathcal{T}}=\infty$) state, ${\rho }_{{\mathcal{T}}=\infty } \sim {\mathbb{1}}$. For U(1) quasi-conservation, prethermalization is restricted to states with the same polarization. However, small symmetry-breaking perturbations restore prethermalization to an infinite temperature. In particular, higher-order corrections to H_eff in the two-tone drive break U(1) conservation, leading to an inverse decay time (heating rate) Γ_e = 1/T_e ∝ (JT)² (refs. ^10,46; Supplementary Section 6).

We now consider the effect of a resonant a.c. magnetic field, B_a.c.(t)I^z, with frequency ${f}_{res}$, aligned along $\widehat{{\bf{z}}}$ and locked to the DTC γ_y kicks (Fig. 1b(i), green line). We show (Supplementary Section 6) that it exponentially extends the lifetime of the U(1) PDTC order by the Floquet engineering of a finite energy density⁴⁰, forming the basis for the sensor operation.

To understand this, note that the a.c. field induces an effective coupling to the PDTC order parameter (−1)^ℓI^x, that is, H_eff → H_eff,ℓ,a.c. = H_eff + (−1)^ℓBI^x, which, like the DTC response, alternates in sign for each T-period ℓ, where the the coupling B ∝ B_a.c. is proportional to the a.c. field strength. Considering the effective Hamiltonian every even period, H_eff,a.c. = H_eff + BI^x, the DTC-ordered initial state, ${\rho }_{0} \sim {\mathbb{1}}+\mu {I}^{x}$, acquires a finite energy density, ${\langle {H}_{{\rm{e}}\mathrm{ff},{\rm{a}}.{\rm{c}}}\rangle }_{{\rho }_{0}}={\langle {H}_{{\rm{e}}\mathrm{ff}}\rangle }_{{\rho }_{0}}+{\langle {I}^{x}\rangle }_{{\rho }_{0}}\propto \mu B$, controlled by the a.c. field (Supplementary Section 6).

Thus, the a.c. field creates a finite energy density from the PDTC order, leading the PDTC to prethermalize to a finite-temperature state, ${\rho }_{{\mathcal{T}}}\propto \exp (-{H}_{e\mathrm{ff},AC}/{\mathcal{T}})$, even with symmetry-breaking perturbations (we set Boltzmann’s constant to unity). This enhances its robustness, energetically protecting the PDTC state from prethermalization to an infinite temperature, and results in a Floquet heating rate that is now exponentially suppressed in the driving period T, ${\varGamma }_{{\rm{e}}}^{\,\text{a.c}}\propto \exp (-1/JT)$. Note that experimentally observing this exponential extension of the lifetime is challenging due to technical limitations (Supplementary Section 6C).

Figure 1c (red data points) shows the PDTC under an a.c. field with B_a.c. = 82.4 μT and ${f}_{{\rm{a}}.{\rm{c}}}={f}_{{\rm{r}}{\rm{e}}{\rm{s}}}=330.023$ Hz. The 1/e lifetime is extended more than 50 fold to ${T}_{2}^{{\prime} }=4.51\,s$, and corresponding to over $M=2,900\,\widehat{{\bf{x}}}\leftrightarrow -\widehat{{\bf{x}}}$ spin flips. This manifests also in the larger signal in the zoomed-in view (Fig. 1c(i)). In particular, lifetime extension is not limited to the specific case of γ_y = π but applies throughout the stability regime of the DTC order (Supplementary Section 3 and Supplementary Fig. 2).

Spin evolution remains observable far beyond the value naively suggested by the 1/e time, as shown by the phase signal ϕ remaining coherent for several seconds (Fig. 1c(ii)). The $\widehat{{\bf{x}}}$ and $-\widehat{{\bf{x}}}$ rails correspond to phase values ϕ = 0, π, with each point tracking ϕ after every θ pulse (total, >500,000). Heating of the conventional PDTC (blue data points) towards the infinite-temperature state, ${\rho }_{{\mathcal{T}}=\infty }={\mathbb{1}}$, is evident as ϕ disperses across the [−π, π] phase space within ~2 s. By contrast, under the a.c. field (Fig. 1c(ii), red data points), the PDTC signal remains stable for over 20 s and 544,000 pulses (spin lock plus γ kick). We also note that with the a.c. field applied, micromotion among the interpulse spacings within a single period of the DTC sequence causes an apparent splitting of the signal into multiple strands. As described in ref. ⁵², this is due to the distinct prethermal plateaus corresponding to each stroboscopic frame within the period of the Floquet cycle. Similar micromotion is also evident in the signal shown in Supplementary Fig. 4.

Robust, high-resolution a.c. magnetic field sensing

Lifetime enhancement (Fig. 1c(i)) also yields a change in the measured signal at every fixed time t compared with the case without an additional a.c. field and, hence, enables a means to sense the a.c. field. We now consider how this extension applies to the a.c. field characteristics (Fig. 2a(i)), ${B}_{{\rm{a}}.{\rm{c}}}(t)={B}_{{\rm{a}}.{\rm{c}}}\sin (2\pi {f}_{{\rm{a}}.{\rm{c}}}t+{\Phi }_{{\rm{a}}.{\rm{c}}})$, that is, its (1) phase Φ_a.c., (2) amplitude B_a.c. and (3) frequency f_a.c..

Fig. 2: PDTC lifetime extension under resonant a.c. fields.

a, Effect of a.c. field phase Φ_a.c.. (i) Schematic: a.c. field phase Φ_a.c. is measured relative to the application of γ_y pulses, with Φ_a.c. = ±π/2 indicating that the troughs and peaks of the a.c. field align with the centre of γ_y kicks. (ii) Lifetime extension fidelity F (equation (1)) as a function of the a.c. field phase Φ_a.c. at a fixed amplitude of B_a.c. = 8.24 μT on resonance. Signal increase is the strongest at Φ_a.c. = π/2; for Φ_a.c. = 0, there is minimal lifetime increase over bare PDTC (dashed line). Blue (green) data points show cases for γ_y pulses on (ϵ = 2% away from) the γ_y = π PDTC stable point. Qualitatively, the same behaviour is observed, indicating PDTC robustness. (π/2)_y and θ_x pulses (~50.25 μs), π pulse (~100.5 μs), τ (~36 μs) and 0.98π pulse (~98.5 μs). b, Effect of a.c. field amplitude B_a.c.. Resonant a.c. amplitude B_a.c. versus fidelity F at Φ_a.c. = π/2. Points show the mean of n = 5 independent measurements, where each measurement corresponds to the diamond being separately rehyperpolarized and run through the full pulse sequence. Error bars denote mean ± s.d. The solid line is a spline-fit guide to the eye. The right axis shows a normalized lifetime extension (>3,500-fold ${T}_{2}^{{\prime} }$ increase for 82.4 μT). Inset: time-domain profiles of the representative points in b: (i) no a.c. field, (ii) intermediate field B_a.c. = 8.24 μT and (iii) B_a.c. = 82.4 μT.

To quantify the signal enhancement, we devise a fidelity metric that remains accurate even when the signal approaches the noise floor:

$$F=\frac{1}{{N}^{{\prime} }}\mathop{\sum }\limits_{i=1}^{{N}^{{\prime} }}\langle {I}^{x}({t}_{i})\rangle P({t}_{i}),$$

(1)

where P(t) represents the ideal DTC toggling response, alternating between ±1 as spins flip between the $\pm \widehat{{\bf{x}}}$ axes, and the normalized sum is carried out over all ${N}^{{\prime} }$ time points ${t}_{i}\in [{t}_{1},{t}_{{N}^{{\prime} }}]$; F yields the largest value when the DTC oscillations are the strongest and most stable. Formally, F corresponds to a weighted summation over the Fourier harmonics $\ell {f}_{res}$ ($\ell \in {{\mathbb{N}}}_{\, > \,0}$); when approaching the noise floor, it is more robust than the standard approach of estimating the PDTC response from only the period-doubling (that is, ${f}_{res}$) component. Lacking an analytical model for the DTC decay, the fidelity metric provides a simple, profile-agnostic measure of stability by integrating the observed magnetization. A more detailed understanding of the form of the DTC decay and its response to external magnetic field could provide a more optimized metric to improve the sensitivity of our approach.

Using this metric, Fig. 2a(ii) examines the impact of the a.c. field phase Φ_a.c. on resonance ${f}_{{\rm{a}}.{\rm{c}}}={f}_{{\rm{r}}{\rm{e}}{\rm{s}}}$ and γ_y = π (blue data points). Maximum lifetime extension occurs at Φ_a.c. = π/2, where the a.c. field peaks align with the centre of the γ_y pulses, as predicted theoretically (Supplementary Section 7). When the a.c. field nodes coincide with the γ_y pulses, there is a minimal effect on the PDTC lifetime. Optimal sensing, therefore, occurs when Φ_a.c. = π/2. We observe a slight phase shift in the response from the expected maximum at Φ_a.c. = π/2, due to unaccounted pulse transients when setting the a.c. field phase based on the pulse length (Fig. 2a(i)). This apparent shift arises because the actual pulse applied to the probe is slightly longer than the one generated by the arbitrary waveform transceiver⁵³.

Additionally, the blue data points in Fig. 2a(ii) show the response to slight deviations from the small point, here γ_y = 0.98π. The data confirm the robustness of the PDTC order. In Supplementary Fig. 2, we display the entire experimentally mapped PDTC phase diagram for all γ_y values, demonstrating a large stable region around γ_y = π, independent of t.

Figure 2b studies the effect of the a.c. field strength B_a.c., set at resonance with Φ_a.c. = π/2. The fidelity profile shows a gradual increase, followed by a linear rise and eventually plateaus at higher field strengths when B_a.c. becomes comparable with the Rabi field of the θ_x Floquet pulses. The right vertical axis shows the corresponding ${T}_{2}^{{\prime} }$ lifetimes; the maximum extension, corresponding to ${T}_{2}^{{\prime} }=21.3\,s$, is >3,000 fold. Here data are for N = 4; Supplementary Section 4A discusses the extensions obtained as a function of N. Figure 2b illustrates the time profiles of $\langle {I}^{x}\rangle$ for three cases: (i) no field B_a.c. = 0, (ii) B_a.c. = 8.24 μT and (iii) B_a.c. = 82.4 μT. Figure 2b indicates that this can be used for sensing at appropriately chosen bias points. The sensitivity is determined by comparing the response of the fidelity F to perturbation in the a.c. field amplitude B_a.c. (Fig. 2b), with the fluctuations in F obtained for multiple initializations at fixed B_a.c. (Methods and Supplementary Section 4D). We obtain a sensitivity of 880 pT Hz^−1/2 with an optimum bias field of B_a.c. = 415 nT.

A distinguishing feature of the a.c.-field-mediated lifetime extension is its strongly resonant nature. Figure 3a examines the fidelity F across a range of a.c. field frequencies f_a.c. under identical conditions. Off-resonant frequencies have a negligible impact on the DTC lifetime, matching the bare PDTC (B_a.c. = 0, F ≈ 0). By contrast, a notable lifetime increase is observed on resonance ${f}_{res}$ (Fig. 3a). A zoomed-in view (Fig. 3a(i)) reveals a narrow linewidth of Δf ≈ 70 mHz set by the maximum integration time (${t}_{{N}^{{\prime} }}$ in equation (1)), positively correlated with the inverse of the PDTC lifetime ${({T}_{2}^{{\prime} })}^{-1}$. We also note a weak additional response at subharmonics, especially ${f}_{res}/2$, at large B_a.c.. This is discussed in Supplementary Section 4C (Supplementary Fig. 5).

Fig. 3: PDTC-based a.c. magnetic field sensing.

a, Narrow-linewidth a.c. sensing: fidelity F (blue data points) is measured by sweeping the frequency f_a.c. with B_a.c. = 8.24 μT and N = 4, keeping other parameters consistent with those in Fig. 1c. A sharp increase in the PDTC lifetime (and hence, fidelity) occurs at the resonance condition ${f}_{AC}={f}_{res}$. By contrast, the spin-lock sensing scheme introduced in ref. ⁵⁴ (grey data points) lacks frequency selectivity. (i) Zoomed-in view into the resonance feature, showing a narrow linewidth Δℓ ≈ 70 mHz, determined by ${({T}_{2}^{{\prime} })}^{-1}$. Points (b)–(d) are marked on the spectral wing. (ii) Time-domain PDTC profiles of $\langle {I}^{x}\rangle$ at points (a)–(d) in a and (i) at various offset frequencies from resonance. (a) Far off-resonance: fast signal decay, similar to bare PDTC case. (b) On resonance: substantially extended PDTC lifetime. (c) and (d) Slightly off-resonance, showing long lifetimes with beat pattern at frequency $\delta f={f}_{AC}-{f}_{res}$, resulting in ${T}_{2}^{{\prime} }$-limited a.c. sensing. Supplementary Section 4E discusses exploiting this for noise-rejected sensing: (π/2)_y and θ_x pulses (~51.5 μs); γ_y pulse (~103 μs); τ (~36 μs). b, Robustness of the resonance feature to deviations in γ_y kick angle, γ_y = π − ϵ (colour bar). The fidelity baselines for different γ_y kick angles were offset by 2 × 10⁻² to prevent overlap. Data show that the spectral width Δℓ remains independent of ϵ. Similar experimental mapping of the PDTC phase diagram with respect to γ_y is shown in Supplementary Fig. 2. (π/2)_y and θ_x pulses (~52.25 μs); τ (~36 μs); γ_y pulse length scales linearly with its angle, with a π pulse (~104.5 μs). c, Tunable sensor profile for two-frequency sensing. Inset: sequence with two interspersed γ_y pulse blocks, leading to two resonance conditions: ${f}_{res}^{(1)}$ and ${f}_{res}^{(2)}$. B_a.c. = 32.96 μT. Main panel: measured frequency response (similar to a), showing a two-tone response centred at 208 Hz and 250 Hz, with a narrow linewidth Δℓ ≈ 5.5 Hz. (π/2)_y and θ_x pulses (~50.25 μs); γ_y pulse (~100.5 μs). t_acq ≈ 13.6 μs for c.

The sharp ${f}_{res}$ response is further clarified using the representative points marked in Fig. 3a(i). The corresponding time-domain PDTC profiles are shown in Fig. 3a(ii). Far off-resonance (point (a)), the dynamics remain unaffected by the a.c. field. Exactly on resonance (point (b)), a notable lifetime increase is observed. Slightly off-resonance (points (c) and (d)), a distinctive beating in the fidelity F appears, reflecting the frequency offset $\delta f={f}_{{\rm{a}}.{\rm{c}}}-{f}_{{\rm{r}}{\rm{e}}{\rm{s}}}$. The integration of this beating pattern over time leads to the ${({t}_{N}^{{\prime} })}^{-1}$ linewidth. In sensing applications, this can enable a precise reconstruction of unknown signals within the narrow resonance band Δf via a Fourier transform of the DTC temporal dynamics. The sensor bandwidth itself is determined by the shortest possible pulse lengths, and could span the 0.5–50-kHz range (Supplementary Fig. 1).

The dynamics of the tracked phase ϕ corresponding to Fig. 3a(ii) is presented in Supplementary Section 4B. The data (Supplementary Fig. 4) reveal intricate micromotion dynamics and demonstrate the ability to measure it for periods well beyond the 1/e lifetimes, exceeding 60,000 γ kicks, with high clarity.

We emphasize that the lifetime-limited linewidth is a feature of DTC-based sensing, distinguishing it from other methods such as magnetometry using spin-locked prethermal states introduced in ref. ⁵⁴. Figure 3a (grey data points) illustrates the response of the spin-lock sensing scheme over the same frequency range in which we measured the DTC’s response. Unlike DTC-based sensing, the spin-lock sensing scheme can detect multiple frequencies without modifying the pulse sequence parameters, but as indicated by the data, it exhibits monotonically increasing response with frequency, lacking frequency selectivity. Even on resonance, the linewidth of the spin-lock sensing scheme can extend to several hundred hertz—at least four orders of magnitude broader—primarily dominated by interspin couplings, and largely independent of ${T}_{2}^{{\prime} }$. A detailed analysis of the spin-lock sensing data is provided in Supplementary Section 4H. By contrast, the narrow linewidth of the DTC sensing scheme enables tuning into specific fields that meet the resonance condition, effectively rejecting non-resonant fields (Supplementary Fig. 7). More broadly, compared with conventional quantum sensors based on electronic spins⁵⁵, the two-tone cavity-interrogated nuclear PDTC allows single-shot, quasi-continuous sensing for $> 5{T}_{2\,}^{{\prime} \,}( > 100\,{\rm{s}})$ (Fig. 1c(ii) and Supplementary Fig. 4) without sensor reinitialization, with the resonant lifetime extension enhancing the sensor precision.

Another consequence of the PDTC order is that the narrow sensing linewidth remains highly robust to pulse errors ϵ in the γ_y pulses away from γ_y = π. This is shown in Fig. 3b with γ_y denoted by the colour bar. The sensor linewidth (zoomed-in view shown in Fig. 3a(i)) remains unaffected by these errors. Additionally, the system exhibits a remarkable tolerance to on-site disorder (Supplementary Section 7D) and fluctuations in the spin-lock θ_x drive, evidenced in the capacity of reliably applying >10⁶ θ_x pulses even with realistic imperfections (due to Rabi frequency heterogeneity) in these experiments.

The two-tone PDTC discussed so far (Fig. 1b(i)) hosts a single resonance frequency ${f}_{res}$, tunable via the sequence parameter T. However, it is possible to expand the number of resonance frequencies and adjust the DTC sensing spectrum by modifying the PDTC sequence. For instance, Fig. 3c(i) introduces a three-tone PDTC, establishing two resonance conditions at ${f}_{res}^{(1)}$ and ${f}_{res}^{(2)}$, achieved through two different interleaved periods for the γ_y pulses, interspersed with spin-lock θ_x pulses. The experimental response in Fig. 3c shows two distinct frequencies separated by ~42 Hz. We observe asymmetric spectra with a stronger x component when the period T is decreased, leading to a more frequent overlap between the a.c. field antinodes and the γ_y pulses. This increased overlap enhances the x-component response (Supplementary Section 6B). The two-frequency linewidths, around 5 Hz, remain appreciably narrower than the sensor’s linewidth without DTC order (~223 Hz)⁵⁴, although single-frequency linewidths (Fig. 3a) are narrower due to longer ${T}_{2}^{{\prime} }$ lifetimes in the two-tone case.

a.c.-field-mediated lifetime extension for single-tone PDTC

Lifetime enhancement from a.c. field-mediated finite energy density applies broadly to all U(1) PDTCs, not just the two-tone PDTC. To demonstrate this, we consider a conventional single-tone DTC that alternates between the $+\widehat{{\bf{z}}}$ and $-\widehat{{\bf{z}}}$ states on the Bloch sphere. This approach is widely used across platforms, including superconducting qubits^8,11, cold atoms⁵⁶ and nuclear magnetic resonance (NMR)^4,5,12. Unlike the two-tone DTC, which enables non-destructive inductive read-out in the $\widehat{{\bf{x}}}$–$\widehat{{\bf{y}}}$ plane to monitor the decay dynamics in a single shot, the single-tone DTC requires restarting the experiment for each data point.

The sequence is shown in Fig. 4a(i), and consists of M spin-flip γ_y pulses along $\widehat{{\bf{y}}}$ (Fig. 4a(ii) describes this using a schematic). We utilize the exact U(1) symmetry of the dipole–dipole Hamiltonian H_dd, which conserves $\widehat{{\bf{z}}}$ magnetization: $[{H}_{dd},{I}^{z}]=0$. For efficient read-out, spins are tipped onto the $\widehat{{\bf{x}}}\,-\,\widehat{{\bf{y}}}$ plane and spin locked using a train of θ_x pulses along $\widehat{{\bf{x}}}$. Unlike the two-tone case shown in Fig. 1, the data here are collected point by point for different values of M.

Fig. 4: Single-axis PDTC extension under a.c. field.

a, (i) Protocol: PDTC sequence (green) consists of a train of M 100.5-μs γ_y = π pulses along $\widehat{{\bf{y}}}$, spaced by 300 μs, flipping the spins between $+\widehat{{\bf{z}}}$ and $-\widehat{{\bf{z}}}$, robust to deviations from γ_y = π (ref. ¹⁰). Spin-lock trains before and after the DTC sequence read the initial and final spin population $\langle {I}_{z}\rangle$, enhancing the signal-to-noise ratio. (ii) Spin motion on a Bloch sphere. PDTC oscillates along the same axis as the applied a.c. field ($\widehat{{\bf{z}}}$), unlike the two-tone case in which the axes are orthogonal. b, PDTC lifetime extension. Black data points: normalized single-axis PDTC signal $\langle {I}_{z}\rangle$ without an a.c. field, decaying with ${T}_{2}^{{\prime} }=0.01\,s$ or M = 35 flips. Green data points: DTC under a resonant a.c. field with B_a.c. = 1 μT, showing a substantially extended lifetime of ${T}_{2}^{{\prime} }=20\,s$, or M = 70,000 spin flips—a 2,000-fold increase. Each data point requires restarting the experiment.

The results are shown in Fig. 4b. Without an a.c. field (black data points), we observe robust period-doubling dynamics with a 1/e decay time $T{\prime} \approx 0.01\,s$ and $M=40\,\widehat{{\bf{z}}}\leftrightarrow -\widehat{{\bf{z}}}$ flips. With a resonant a.c. field of B_a.c. = 40 μT (green data points), the lifetime is markedly prolonged, extended by more than three orders of magnitude to M = 70,000 and ${T}_{1}^{{\prime} }=20\,s$. The phase response to Φ_a.c. is opposite to that shown in Fig. 2a(ii), as the a.c. field direction aligns with the PDTC oscillation axis; here it is the maximum near Φ_a.c. = 0 (Supplementary Fig. 11). Although the quasi-continuous measurement from the two-tone drive is more suitable for sensing applications, the data in Fig. 4b demonstrate that this lifetime extension mechanism applies broadly to U(1) PDTCs.

Discussion

This work introduces several novel features. The key conceptual result is the resonant extension of the PDTC lifetime via an a.c. field, exponentially suppressing the heating rate relative to the driving period T (Supplementary Section 6). As shown in Fig. 2b, we extend the PDTC 1/e lifetime to ${T}_{2}^{{\prime} } > 20\,s$ and M = 40,000 spin-flipping Floquet cycles. Compared with previous work^{1,2,3,7,10,11,12}, this sets a new record for both parameters, representing over two orders of magnitude improvement in the total number of DTC spin flips. Supplementary Table 1 provides a detailed comparison, highlighting the remarkably increased DTC lifetime in our case, despite similar interaction strengths and pulsing rates to previous works.

The methodology introduced here is not limited to nuclear spins and can be applied to a wide range of quantum and classical systems. We anticipate immediate applications to cavity-interrogated NV centres⁵⁷ for quantum sensing in the range of 1 MHz to 1 GHz (ref. ⁵⁸). The underlying principles themselves are also applicable to superconducting qubits^8,11, cold atoms⁵⁶ and ions³.

Crucially, sensing based on many-body DTC order naturally tolerates strong interspin couplings, leading to a lifetime-limited linewidth ($\sim ({T}_{2}^{{\prime} })$) rather than being dominated by couplings (~J⁻¹). This enables sensing at high sensor densities, extending beyond the conventional regime of dilute, non-interacting sensors⁴²; all else being identical, this can yield substantial sensitivity improvements⁵⁹. Moreover, sensing here inherits the robustness of PDTC order against pulse sequence errors and sample inhomogeneities; we demonstrate this in Supplementary Section 7D, showing the resilience of a.c. sensing to strong on-site disorder. Finally, this approach does not require the preparation of strongly entangled states or fine tuning to a critical point. All these factors indicate broad applicability to a wide range of systems.

Methods

Materials

The sample used in this work is a single-crystal diamond measuring 3.4 × 3.2 × 2.1 mm³, containing a natural abundance of ¹³C nuclei and NV centres at ~1-ppm concentration. This same sample has been characterized in prior studies^10,46, allowing for direct comparisons with the lifetime extensions observed. The sample is oriented parallel to the B_pol magnetic field, ensuring the simultaneous hyperpolarization of the four NV-centre axes, and experiments are carried out underwater to provide more uniform illumination and aid in thermal management⁶⁰.

Experimental setup

Instrumentation for hyperpolarization and ¹³C read-out follows previous works, with detailed descriptions available in those studies^{10,45,46,47,54}. Data here are taken at room temperature and B₀ = 7 T. Polarization is carried out at a low field centre (B_pol ≈ 38 mT) located below the magnet, driven by optically excited NV centres and chirped microwave excitation. The polarization mechanism involves successive Landau–Zener anticrossings in the rotating frame^61,62. Sample shuttling to the B₀ high field occurs in under 1 s, with an NMR saddle coil used for inductive read-out of the ¹³C precession signal.

¹³C interrogation is performed using a custom-built NMR spectrometer based on a high-speed arbitrary waveform transceiver (Proteus P9484M)^47,63. The arbitrary waveform transceiver device generates NMR pulses and digitizes the nuclear precession directly at the Larmor frequency, eliminating insertion losses typically encountered with intermediate frequencies. The device’s high memory capacity (16 GB) and large sampling rate (up to 2.7 GS s⁻¹) enable a continuous interrogation of ¹³C spin precession in windows between pulses. In typical experiments, we apply 0.2–1 million pulses with a read-out window of ~f ≈ 13.6 μs. The entire Larmor precession can be sampled every 0.74 ns and mixed with an onboard numerically controlled oscillator at the Larmor frequency, allowing us to track both amplitude and phase of the spins directly in the rotating frame^47,52.

The receive chain amplifies the signal using low-noise preamplifiers (Advanced Receiver Research P75VDG and Pasternack PE15A1011), whereas the NMR pulse generation is via a Herley TWT amplifier through a Techmag transcoupler. For a.c. magnetometry, the spins are exposed to a weak magnetic field applied via a secondary coil, with the field parallel to $\widehat{{\bf{z}}}$ and positioned within the NMR probe. The field is applied via a Tektronix source, amplified by a Techron 7224 amplifier, and the field strength is calibrated through the voltage drop across a high-power 250-W 4-Ω resistor.

Classification of temporal order

To place our work in the broader context of temporal order in closed systems, let us recall the different ways in which temporal order can be realized. In closed periodically driven systems, so far, three mechanisms are known to realize temporal order, namely, (1) many-body localized DTCs, (2) PDTCs and (3) U(1) PDTCs. The many-body localized DTCs require stable many-body localization^64,65, which may only exist in one-dimensional short-range quantum systems in the presence of strong disorder⁶⁶; under these conditions, the emerging spatiotemporal eigenstate order^15,67 is entirely robust out to infinite times⁶⁸. By contrast, the PDTCs require the existence of (pre)-thermal order, that is, a low-temperature symmetry-breaking state, namely, by the Mermin–Wagner theorem only possible in two and higher dimensions. The resulting prethermal spatiotemporal order has a finite lifetime determined by Floquet heating, and thus, exponentially suppressed in the driving period for short-range interacting systems^23,24. Finally, U(1) PDTCs only require a (quasi-)conserved U(1) symmetry and an initial state that breaks this symmetry, irrespective of the effective temperature of this state. However, melting of the prethermal temporal order is dominated by symmetry-breaking perturbations, which, in the case of an emergent symmetry, only lead to the power-law suppression of heating in the driving frequency.

Our work focuses on the last class, the U(1) PDTCs, demonstrating that the lifetime of temporal order can be exponentially enhanced by coupling the system to the order parameter. Thus, the a.c.-enriched U(1) PDTCs effectively mimic the behaviour of PDTCs in terms of lifetime. We emphasize that adding the a.c. field to the U(1) PDTC is not sufficient to realize a thermally ordered state required for the PDTC.

Sensitivity calculation

The sensitivity to a.c. magnetic fields based on the described effect of DTC extension is calculated by

$$\mathrm{Se}{\rm{n}}{\rm{s}}{\rm{i}}{\rm{t}}{\rm{i}}{\rm{v}}{\rm{i}}{\rm{t}}{\rm{y}}=\frac{{\sigma }_{F}}{\partial F/\partial {B}_{{\rm{a}}.{\rm{c}}}}\sqrt{{t}_{\mathrm{int}}},$$

(M1)

that is, the response of the signal metric (here fidelity F) to perturbation in the field strength B_a.c. is compared with the noise σ_F in the metric at fixed field strength, normalized for the noise reduction achievable by integrating the signal over the measurement time t_int. Sensitivity is determined via a scan of the amplitude B_a.c.. For each chosen amplitude, the DTC experiment is repeated several times. The fluctuation σ_F in the fidelity metric is determined by the standard deviation amongst trials at a fixed amplitude B_a.c.. The mean fidelity for trials at each amplitude is used in calculating the centred difference between adjacent amplitude points B_a.c. to estimate the response ∂F/∂B_a.c.. As shown in Fig. 2b, the amplitude dependence of the DTC coherence extension is nonlinear; the response ∂F/∂B_a.c. increases with B_a.c. up to ~8 μT. This could suggest that the sensor should be operated with a bias a.c. field near 8 μT for best performance. However, the fluctuations in fidelity are also found to grow with B_a.c., even more steeply than the response ∂F/∂B_a.c.. Thus, the best sensitivity is actually obtained for a small bias of 415 nT. Supplementary Fig. 6 shows the calculated sensitivity as a function of bias. The cause of the sharp rise in σ_F with an applied field is unclear; this could be due to an increased instability of the a.c. source itself or due to the increased susceptibility of the DTC to drifts in the instrument as a result of the bias field. The sensitivity and opportunities to improve it are further discussed in Supplementary Section 4D.

Floquet engineering of finite energy density

As discussed below, in both single-tone and two-tone DTC, the a.c.-induced signal enhancement is the result of coupling the system to the DTC order parameter via the added a.c. field. The complete derivation can be found in the Supplementary Information.

Single-tone DTC

For the single-tone DTC, the order parameter corresponds to the $\widehat{{\bf{z}}}$ magnetization (〈I^z〉) flipping sign every period T, that is, ${{\mathcal{O}}}_{1DTC}(\ell T)=\langle {(-1)}^{\ell }{I}^{z}\rangle$, thereby oscillating with a period 2T. Thus, naturally, an a.c. field in the $\widehat{{\bf{z}}}$ direction couples to the order parameter such that the system every full DTC period (2ℓT) is effectively described by H_a.c.,eff = H_dd + B_effI^z, with the effective field B_eff proportional to the a.c. amplitude, B_eff ∝ B_a.c..

Two-tone DTC

For the two-tone DTC, a key difference is that the order parameter oscillates in between $\widehat{{\bf{x}}}$ and $-\widehat{{\bf{x}}}$, ${{\mathcal{O}}}_{2DTC}(\ell T)=\langle {(-1)}^{\ell }{I}^{x}\rangle$, orthogonal to the a.c. field. However, crucially, the $\widehat{{\bf{y}}}$ pulses implementing the DTC sequence are of a finite time duration τ_y, that is, a finite a.c. field is present also during the application of the $\widehat{{\bf{y}}}$ pulses. Indeed, one can show (Supplementary Section 6B) that for the γ_y = π pulses, applying the I^y field and I^z simultaneously corresponds to applying an I^x and an I^y field separately, that is,

$${{\rm{e}}}^{-{\rm{i}}({\gamma }_{y}{I}^{y}+{(-1)}^{n}{B}_{n,z}{\tau }_{y}{I}^{z})}\approx {{\rm{e}}}^{-{\rm{i}}{\gamma }_{y}{I}^{y}}{{\rm{e}}}^{-{\rm{i}}\frac{{(-1)}^{n}{B}_{n,z}{\tau }_{y}}{{\gamma }_{y}}{I}^{x}},$$

up to an error scaling as $O({({B}_{z}{\tau }_{y}/{\gamma }_{y})}^{2})$, where n labels the $\widehat{{\bf{y}}}$ pulse and ${B}_{z,n}=| {\int }^{{\tau }_{y}}{B}_{{\rm{a}}.{\rm{c}}}(t){\rm{d}}t/{\tau }_{y}|$ is the amplitude of the a.c. field accumulated during the nth $\widehat{{\bf{y}}}$ pulse. Therefore, indeed, the $\widehat{{\bf{z}}}$ a.c. field effectively induces an $\widehat{{\bf{x}}}$ a.c. field. In particular, this induced a.c. field is the strongest if the minima and maxima of the a.c. field align with the $\widehat{{\bf{y}}}$ pulses, in agreement with the experimental results shown in Fig. 2; Figs. 2 and 3 and Supplementary Section 7 provide a derivation of the other effects. Thus, similar to the single-tone case, the system in the presence of an a.c. field is effectively described by H_a.c.,eff = H_eff + B_effI^x after every full DTC period (2ℓT), for the effective interacting Hamiltonian ${H}_{e\mathrm{ff}}\approx {\sum }_{k < l}{J}_{kl}({I}_{k}^{x}{I}_{l}^{x}-{{\bf{I}}}_{k}\cdot {{\bf{I}}}_{l})$ and effective field B_eff ∝ B_a.c.; Supplementary Section 6B provides the derivation.

Lifetime enhancement via ETH

In both cases, the system over a full DTC cycle (2T) is effectively described by an interaction part and an emergent static field with strength proportional to the a.c. field strength, H_a.c.,eff = H_int + B_effI^α, where α = z, x for single- and two-tone drives, respectively. Note that for small imperfections in the drive γ_y ≠ π, the ‘interaction’ terms H_int will break the U(1) conservation law, [H_int, I^α] ≠ 0. Therefore, the only conserved quantity is the quasi-conserved energy, 〈H_a.c.,eff〉, in the prethermal plateau. Thus, the ETH^48,49,50,51 predicts that a generic system starting in an initial state $| {\psi }_{0}\rangle$ will relax at ‘late’ times to a state $| {\psi }_{\infty }\rangle$, which is indistinguishable from a thermal state ${\rho }_{{\mathcal{T}}}\propto {e}^{-{H}_{{\rm{a}}.{\rm{c}},{\rm{e}}{\rm{f}}{\rm{f}}}/{\mathcal{T}}}$, with temperature ${\mathcal{T}}$ determined by energy conservation. This means, for spatially local observables A, like energy or magnetization, the expectation value in the late-time state is equal to the thermal expectation value: ${\langle A\rangle }_{{\psi }_{\infty }}={\langle A\rangle }_{{\rho }_{{\mathcal{T}}}}$. The same holds for the initial mixed state ρ₀.

Importantly, in the absence of an a.c. field, the energy of the initial state ${\rho }_{0} \sim {\mathbb{1}}+\mu {I}^{\alpha }$ for both single- and two-tone DTC vanishes (${\langle {H}_{int}\rangle }_{{\rho }_{0}}=0$), thereby leading to (pre)thermalization to a featureless infinite-temperature state ${\rho }_{{\mathcal{T}}=\infty }\propto {\mathbb{1}}$. By contrast, in the presence of an a.c. field, the initial energy becomes finite, ${\langle {H}_{{\rm{e}}\mathrm{ff},{\rm{a}}.{\rm{c}}}\rangle }_{{\rho }_{0}}\propto \mu {B}_{{\rm{a}}.{\rm{c}}}$; thus, the system prethermalizes to a finite-temperature state ${\rho }_{{\mathcal{T}}\ne \infty }\propto {e}^{-{H}_{{\rm{a}}.{\rm{c}},{\rm{e}}{\rm{f}}{\rm{f}}}/{\mathcal{T}}}$, with finite magnetization ${\langle {I}^{\alpha }\rangle }_{{\rho }_{{\mathcal{T}}\ne \infty }}\ne 0$. This prethermal value persists until the ultimate melting of the prethermal plateau, which is exponentially suppressed in the period T, ${\varGamma }_{{\rm{e}}}^{{\rm{a}}.{\rm{c}}}\propto \exp (-c/JT)$ for some constant c. Thus, the lifetime can be exponentially enhanced by introducing an additional a.c. field (Supplementary Section 6).

Overview of requirements for a.c.-induced signal enhancement

Although the above analysis focused on dipole-coupled nuclear spins in diamond, the mechanism applies more generally. In a nutshell, the key ingredients are (1) (emergent) symmetry-protected period-doubling response; (2) the system should (pre)thermalize in agreement with ETH; (3) a high-temperature initial state such that the lifetime is limited by symmetry-breaking terms; and (4) the ability to (effectively) couple the system to the DTC order parameter, for example, via Floquet engineering. Note that since the DTC order parameter itself oscillates in time with period 2T, inducing a coupling to the order parameter necessarily requires adding an additional time-varying (a.c.) field. Then, by adding this time-varying field, one can exponentially extend the lifetime of the PDTC order using the procedure presented above.