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Emergent hydrodynamics in a strongly interacting dipolar spin ensemble


Conventional wisdom holds that macroscopic classical phenomena naturally emerge from microscopic quantum laws1,2,3,4,5,6,7. However, despite this mantra, building direct connections between these two descriptions has remained an enduring scientific challenge. In particular, it is difficult to quantitatively predict the emergent ‘classical’ properties of a system (for example, diffusivity, viscosity and compressibility) from a generic microscopic quantum Hamiltonian7,8,9,10,11,12,13,14. Here we introduce a hybrid solid-state spin platform, where the underlying disordered, dipolar quantum Hamiltonian gives rise to the emergence of unconventional spin diffusion at nanometre length scales. In particular, the combination of positional disorder and on-site random fields leads to diffusive dynamics that are Fickian yet non-Gaussian15,16,17,18,19,20. Finally, by tuning the underlying parameters within the spin Hamiltonian via a combination of static and driven fields, we demonstrate direct control over the emergent spin diffusion coefficient. Our work enables the investigation of hydrodynamics in many-body quantum spin systems.

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Fig. 1: Nanoscale spin diffusion in a long-range interacting quantum system.
Fig. 2: Probing local spin-polarization dynamics using the NV centre.
Fig. 3: Controlling emergent hydrodynamics by engineering the microscopic Hamiltonian.

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Source data are provided with this paper. Further data are available from the corresponding author upon reasonable request. Source data are provided with this paper.


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We acknowledge the insights of and discussions with J. Choi, E. Davis, M. Dupont, D. Gangloff, S. Gopalakrishnan, A. Jayich, P. Stamp, R. Walsworth and H. Zhou. This work was supported as part of the Center for Novel Pathways to Quantum Coherence in Materials, an Energy Frontier Research Center funded by the US Department of Energy, Office of Science, Basic Energy Sciences under award number DE-AC02-05CH11231. A.J. acknowledges support from the Army Research Laboratory under Cooperative Agreement number W911NF-16-2-0008. S.H. acknowledges support from the National Science Foundation Graduate Research Fellowship under grant number DGE1752814. N.Y.Y. acknowledges support from the David and Lucile Packard foundation and the W. M. Keck foundation. The work of D.B. is supported by the EU FET-OPEN Flagship Project ASTERIQS (action 820394 and the Cluster of Excellence ‘Precision Physics, Fundamental Interactions, and Structure of Matter’ (PRISMA+ EXC 2118/1) funded by the German Research Foundation (DFG) within the German Excellence Strategy (project ID 39083149). C.R.L. acknowledges support from the NSF through PHY-1752727. S.C. acknowledges support from the Miller Institute for Basic Research in Science.

Author information

Authors and Affiliations



C.Z., T.M., S.H. and P.B. performed the experiments. F.M., B.Y., S.C., B.K., C.R.L., J.E.M. and N.Y.Y. developed the theoretical models and methodology. C.Z., F.M., and B.Y. performed the data analysis. F.M. and B.Y. performed the numerical simulations. M.M., D.T., A.J. and D.B. prepared and provided the diamond substrates. C.R.L., J.E.M. and N.Y.Y. supervised the project. C.Z., F.M., B.Y., C.R.L., J.E.M. and N.Y.Y wrote the manuscript with input from all authors.

Corresponding author

Correspondence to N. Y. Yao.

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The authors declare no competing interests.

Additional information

Peer review information Nature thanks Nir Bar-Gill, Benjamin Doyon and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data figures and tables

Extended Data Fig. 1 Experimental sequence to measure the coherence of P1 ensemble.

For XY-8 and interaction decoupling sequences, we fix the interval between pulses to be τ = 10 ns, and increase the number of repetition N. The P1 π(π/2) pulse duration is set to 36 ns (18 ns). In the interaction decoupling sequences, the pulses at bottom side correspond to rotations along \(-\hat{x}\) (blue) and \(-\hat{y}\) (orange) axes.

Extended Data Fig. 2 Extraction of diffusion constants for sample S1.

Fitting of the depolarization data in sample S1 for different groups ν (different panels) and different pumping times τp (different colours). For each group, we fit the experimental data across all τp data to a diffusive model in equation (7) with an additional background Pbg. From this procedure, we extract both D and b, as well as, a τp-dependent Γ, which captures the reduction in efficiency of the NV–P1 polarization transfer owing to the saturation of polarization near the NV

Source data.

Extended Data Fig. 3 Measurement of late time extrinsic decay time T1 for different samples and under driving.

a, b, Extraction of the extrinsic depolarization time of samples S1 and S2 at room and low temperature (25 K) and after polarizing for τp = 1,000 μs (a) and τp = 30 μs (b). The late time behaviour follows an exponential decay with timescale given by 1.0 ± 0.1 ms and 2.6 ± 0.2 ms, respectively. c, To extract the modified intrinsic depolarization time \({T}_{1}^{{\rm{dr}}}\) of ν = 1/4 P1 subgroup with the presence of a strong microwave driving Ω = (2π) × 11.7 MHz on the other ν = 1/4 subgroup, we apply the following pulse sequence: after a laser pumping time τp = 1000 μs, we wait for 1 ms so that the initial P1 spatial polarization profile diffuses to a nearly homogeneous background which decays with intrinsic depolarization time of P1 centres. We then turn on a continuous microwave driving on the other ν = 1/4 P1 subgroup, and measure the resulting background decay; the resulting timescale is given \({T}_{1}^{{\rm{dr}}}=0.9\pm 0.2\) ms

Source data.

Extended Data Fig. 4 Determination of the extrinsic decoherence rate γ and on-site random field distribution δi.

a, We estimate the extrinsic decoherence rate γ in the rate equation using the measured spin-echo coherence time of the NV. After polarizing the NV centre via a green laser, a π/2 pulse prepares the NV spin into a coherent superposition of \(|{m}_{{\rm{s}}}=0\rangle \) and \(|{m}_{{\rm{s}}}=-1\rangle \), which is allowed to dephase during a time t. A π pulse at the centre of the sequence ‘echos’ out the on-site random field generated by the nearby P1 centres, thus provides a direct estimation of the extrinsic decoherence time of a single spin in the system. We fit the spin-echo decay using a form \({{\rm{e}}}^{-{(t/{T}_{2}^{{\rm{echo}}})}^{1.5}}\)(ref. 57) and extract \({T}_{2}^{{\rm{echo}}}=1.9\pm 0.1{\rm{\mu }}{\rm{s}}\) \((\gamma \approx 1/{T}_{2}^{{\rm{echo}}}\approx 0.5{{\rm{\mu }}{\rm{s}}}^{-1})\). b, The distribution of on-site random fields δi is directly determined using the intrinsic linewidth of the NV spin state. After polarizing the NV centre via a green laser, we apply a microwave π pulse and sweep its frequency ω across the NV \(|0\rangle \) to \(|-1\rangle \) transition. To avoid microwave power broadening of NV transition, we choose a sufficient weak microwave π pulse with duration 2 μs. Note that the measured linewidth is dominated by interactions with the dense P1 ensemble (W ≈ (2π) × 4.5 MHz) (Supplementary Information). The presence of nuclear 13C spins leads to a much smaller contribution to the linewidth of about (2π) × 0.3 MHz (ref. 58). Crucially, both effects are taken into account in our analysis by sampling δi directly from the measured spectrum

Source data.

Extended Data Fig. 5 Agreement between semiclassical model and experimentally observed dynamics.

Given the approximately equal P1 density of both the sample S1 and S2, we simulate the the dynamics of a single NV defect surrounded by NP1s = {300, 225, 75} P1 centres for the groups ν = {1/3, 1/4, 1/12}, respectively. In the polarization protocol, we choose Γp = 0.1 μs−1 for S1 and Γp = 0.25 μs−1 for S2. The subsequent polarization dynamics of the NV centre is given by the difference in populations between the \(|0\rangle \) and \(|-1\rangle \) states. For the ν {1/3, 1/4} groups of S1, we observe excellent agreement with the experimental data for over four orders of magnitude in τp and throughout then entire experimental timescale using γ = 0.5 μs−1. For the ν = 1/12 group of S1, we observe good agreement, albeit with a smaller range of τp and using γ = 1.5 μs−1. We believe this discrepancy arises from a much larger separation between the strength of the on-site fields and the flip-flop rate of the ensemble. For the ν = 1/3 group of S2, we also observe excellent agreement throughout the entire dynamics, where we use γ = 0.3 μs−1. The agreement observed in the NV polarization decay in both samples gives us confidence that our semiclassical model can capture the polarization dynamics in the sample and provide an accurate calculation of the diffusive properties of the spin ensemble

Source data.

Extended Data Fig. 6 Summary of extraction of diffusion coefficient.

ac, Extraction of diffusion coefficient of sample S2 at low temperature. a, Growth of \(\langle {r}^{2}\rangle \) for different system sizes N and the infinite system scaling (black line). b, Finite size scaling of \(\langle {r}^{2}\rangle \) to N → ∞ assuming a linear in L−1N−1/3 correction for representative values of t. c, Fitting the early-time growth of \(\langle {r}^{2}\rangle \) up to different times Tmax [30, 300] leads to slightly different values of the diffusion coefficient, whether including a constant offset (light blue) or not (dark blue). Considering the fit without an offset, the final diffusion coefficient is taken to be the average with an uncertainty given by half the range of diffusion coefficients. d, For the different experimental conditions using the parameters discussed in Methods, we extract the diffusion coefficient from the growth of \(\langle {r}^{2}\rangle \), which is in great agreement with the experimentally extracted values after correcting for the non-Gaussian polarization profile (Table 1)

Source data.

Extended Data Fig. 7 Determination of the length scale \({\ell }\).

Extracted \({\ell }\) for different samples and different P1 groups as a function of the early time cut-off tmin. Averaging over the last three data points, where the \({\ell }\) are consistent, yields the reported value of \({\ell }\). The red dashed line corresponds to the final value and the shaded area is the associated uncertainty

Source data.

Extended Data Fig. 8 Long-range modification to conventional diffusion.

The presence of a long-range k3-term parametrically modifies the approach, \({A}_{{\rm{p}}}(t)={S}_{{\rm{p}}}(t)-{(4{\rm{\pi }}Dt)}^{-3/2}\), to the late-time Gaussian fixed point, as high-lighted in a three-dimensional, disorder-less numerical simulation, with lattice constant a and diffusion coefficient D

Source data.

Extended Data Fig. 9 Fitting of experimental data with different modifications to diffusion equation.

Fitting of the diffusive description with different terms and fixed T1 = 2.6 ms in sample S2 with τp = 30 μs. Different columns represent fitting to a different range of the data (highlighted by the red shaded region). The inclusion of more terms in the diffusive description allows for a better fit of the data; however, the improvement in the fitting range is only significant when the fitting regimes includes early time data (30 μs), as highlighted in the second row of the relative residuals. All data are presented with logarithmically spaced y axis, except in the grey shaded region where a linear regime is used to highlight the fluctuations of the residuals around 0. Fits in Fig. 1b correspond to the third column

Source data.

Extended Data Table 1 Extracted \({\ell }\) from the spin polarization dynamics for the different sample considered (S1 and S2) and the different P1 subgroups

Supplementary information

Supplementary Information

This file contains Supplementary Discussion, including Supplementary Figs. 1–10, and additional references.

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Zu, C., Machado, F., Ye, B. et al. Emergent hydrodynamics in a strongly interacting dipolar spin ensemble. Nature 597, 45–50 (2021).

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