Floquet Prethermalization with Lifetime Exceeding 90 s in a Bulk Hyperpolarized Solid

We report the observation of long-lived Floquet prethermal states in a bulk solid composed of dipolar-coupled ${}^{13}\mathrm{C}$ nuclei in diamond at room temperature. For precessing nuclear spins prepared in an initial transverse state, we demonstrate pulsed spin-lock Floquet control that prevents their decay over multiple-minute-long periods. We observe Floquet prethermal lifetimes $T_2^{\prime }\approx 90.9\mathrm{s}$, extended $>60000$-fold over the nuclear free induction decay times. The spins themselves are continuously interrogated for $\sim 10\min$, corresponding to the application of $\approx 5.8\times {10}^6$ control pulses. The ${}^{13}\mathrm{C}$ nuclei are optically hyperpolarized by lattice nitrogen vacancy centers; the combination of hyperpolarization and continuous spin readout yields significant signal-to-noise ratio in the measurements. This allows probing the Floquet thermalization dynamics with unprecedented clarity. We identify four characteristic regimes of the thermalization process, discerning short-time transient processes leading to the prethermal plateau and long-time system heating toward infinite temperature. This Letter points to new opportunities possible via Floquet control in networks of dilute, randomly distributed, low-sensitivity nuclei. In particular, the combination of minutes-long prethermal lifetimes and continuous spin interrogation opens avenues for quantum sensors constructed from hyperpolarized Floquet prethermal nuclei.

Figure 1

System. (a) Dipolar lattice of ${}^{13}\mathrm{C}$ nuclei in diamond. Optically pumped NV centers are employed to hyperpolarize the ${}^{13}\mathrm{C}$ nuclei (blue arrows). (b),(c) Signal gains from hyperpolarization, demonstrated by comparing single-shot ${}^{13}\mathrm{C}$ NMR spectra to conventional 7 T (thermal) NMR. Data are shown in (b) linear and (c) log scales; line is a fit. Here, optical pumping was for 2 min at 36 mT, and thermal measurement was taken after 4 hr in the magnet.

Figure 2

Floquet driving and lifetime extension. (a) Conventional ${}^{13}\mathrm{C}$ free induction decay (FID) with $T_2^{*}\approx 1.5\mathrm{ms}$. (b) Floquet drive consists of a train of $\theta$ pulses applied spin-locked with the ${}^{13}\mathrm{C}$ nuclei. Spins are interrogated in $t_{\mathrm{acq}}$ windows between the pulses (blue lines), the nuclear precession is sampled every 1 ns. Pulse repetition rate $\omega ={\tau }^{-1}$, and sequence not drawn to scale. (c) Minutes-long lifetimes of the transverse state result from the Floquet sequence ($\theta \approx \pi /2$). Data (blue points) shows single-shot measurement of survival probability in the state ${\rho }_I$. Here $t_{\mathrm{acq}}=2\mu \mathrm{s}$, $t_p=40\mu \mathrm{s}$ and $\tau =99.28\mu \mathrm{s}$, and the 573 s period corresponds to $\approx 5.8\times {10}^6$ pulses (upper axis). We neglect here the first 100 ms for clarity [see Fig. 3]. Inset I: raw data showing measurement of the ${}^{13}\mathrm{C}$ spin precession, here at 1 s into the decay. Inset II: data enlarged $200\times$ in a 1 s window. Using a $1/e$ proxy yields $T_2^{\prime }\approx 90.9\mathrm{s}$. This corresponds to a $>60000$-fold extension compared to the FID.

Figure 3

Floquet thermalization regimes (a) Log-scale visualization of the full data in Fig. 2. Points are experiment, there are $\approx 5.8\times {10}^6$ data points here. Upper axis denotes number of pulses applied, here $J\tau \approx 0.066$. Green points are the FID. We observe distinct, yet smoothly transitioning (shaded), thermalization regimes (I–IV): a $\approx 10\mathrm{ms}$ oscillatory approach (I) to the Floquet prethermal plateau (II), followed by unconstrained thermalization (III). Infinite temperature regime (IV) is not reached in these measurements up to 573 s. (b) Semilog plot of the experimental data shows a dynamic change of thermalization rate. (c) Semilog plot against $\sqrt{t}$ yields an approximately linear dependence (dashed line) for $\sim 500\mathrm{s}$. Cusp (marked) at $\approx 19.4\mathrm{ms}$ marks transition to the prethermal plateau (regime II, see also Fig. 5).

Figure 4

Exponential dependence of Floquet prethermal lifetimes. (a) Variation with $J\tau$. Data (points) show measured signal probing thermalization dynamics in regimes II and III for representative $J\tau$ values (color bar). Here $\theta \approx \pi /2$ and $t_{\mathrm{acq}}=32\mu \mathrm{s}$ and there are a high number ($\sim {10}^3-{10}^6$) points per line [49]. Data are normalized at the transition points to the prethermal plateau [following Fig. 3]. The Floquet prethermal decay rates reduce considerably with decreasing $J\tau$. See full data at Ref. [53]. (b) Extracted decay rates focusing on the region where decay follows $\sim \exp (-t^{1/2})$. Plotted in a semilog scale against $J\tau$, the dashed line reveals an approximately exponential scaling of the decay rates at high drive periods $\tau$ (dashed line is a linear fit). At low $\tau$ we observe sharp narrow features in the prethermal decay rates. (c) Log-log plot of the extracted transverse state lifetimes $T_2^{\prime }$ against $\omega /J$ (dashed line is a linear fit).

Figure 5

Transient approach to prethermal plateau. (a) Oscillations in the approach to prethermal plateau seen for data enlarged in region I, in a 1-s-long window [see Fig. 3]. Data (points) correspond to $\theta \approx \{\pi /2,\pi /4\}$, respectively. Solid line is data, with moving average filtering applied over the entire region [see Fig. 3]. (b) Enlarged region I shows the transient approach with high SNR. It is evident that the oscillations are at higher frequency for $\theta \approx \pi /2$. Solid line is a spline fit to guide the eye. (c) Fourier transforms of panels in (b) allows identification of the frequency components constituting the oscillations as a function of $\omega ={\tau }^{-1}$. Harmonics are represented by numbers. It is clear that primary oscillation frequency is higher for $\theta \approx \pi /2$, where we extract the primary harmonic position at $\approx 0.26{\tau }^{-1}$. (d) Variation with flip angle $\theta$. Data show the position of the oscillation frequency for the primary and higher harmonics (numbered). See full data at Ref. [56]. Solid lines are linear fits, while dashed line is an extrapolation. Slopes are in the ratio expected.