Abstract
The most direct approach for characterizing the quantum dynamics of a strongly interacting system is to measure the time evolution of its full manybody state. Despite the conceptual simplicity of this approach, it quickly becomes intractable as the system size grows. An alternate approach is to think of the manybody dynamics as generating noise, which can be measured by the decoherence of a probe qubit. Here we investigate what the decoherence dynamics of such a probe tells us about the manybody system. In particular, we utilize optically addressable probe spins to experimentally characterize both static and dynamical properties of strongly interacting magnetic dipoles. Our experimental platform consists of two types of spin defects in nitrogen deltadoped diamond: nitrogenvacancy colour centres, which we use as probe spins, and a manybody ensemble of substitutional nitrogen impurities. We demonstrate that the manybody system’s dimensionality, dynamics and disorder are naturally encoded in the probe spins’ decoherence profile. Furthermore, we obtain direct control over the spectral properties of the manybody system, with potential applications in quantum sensing and simulation.
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Main
Understanding and controlling the interactions between a single quantum degree of freedom and its environment represents a fundamental challenge within the quantum sciences^{1,2,3,4,5,6,7,8,9}. Typically, one views this challenge through the lens of mitigating decoherence—enabling one to engineer a highly coherent qubit by decoupling it from the environment^{2,3,4,5,10,11,12}. However, the environment itself may consist of a strongly interacting manybody system, which naturally leads to an alternate perspective, namely using the decoherence dynamics of the qubit to probe the fundamental properties of the manybody system^{6,7,13,14,15,16,17,18}. Discerning the extent to which such ‘manybody noise’ can provide insight into transport dynamics, lowtemperature order and generic correlation functions of an interacting system remains an essential open question.
The complementary goals of probing and eliminating manybody noise have motivated progress in magnetic resonance spectroscopy for decades, including seminal work exploring the decoherence of paramagnetic defects in solids^{6,7,13,14,15,16,19}. More recently, many of the developed techniques have reemerged in the study of solidstate spin ensembles containing optically polarizable colour centres. The ability to prepare spinpolarized pure states enables fundamentally new prospects in quantum science, from the exploration of novel phases of matter^{20} to the development of new sensing protocols^{21}.
Prospects for optically polarizable spin ensembles in quantum sensing and simulation could be further enhanced by moving to twodimensional systems, which represents a longstanding engineering challenge for the colour centre community^{22,23,24}. Despite continued advances in fabrication, the stochastic nature of defect generation strongly constrains the systems one can create. The potential rewards are substantial enough to merit repeated engineering efforts: Twodimensional, longrange interacting spin systems are known to host interesting groundstate phases such as spin liquids^{25,26,27}. Moreover, twodimensional spin ensembles enable improved sensing capabilities owing to increased coherence times and uniform distance from the target (Supplementary Information).
In this article, we investigate manybody noise generated by a thin layer of paramagnetic defects in diamond. Specifically, we combine nitrogen delta doping during growth with local electron irradiation to fabricate a diamond sample (S1) where paramagnetic defects are confined to a layer whose width is, in principle, smaller than the average spin defect spacing (Fig. 1a,b)^{22,23,24}. This layer contains a hybrid spin system consisting of two types of defects: spin 1 nitrogen vacancy (NV) centres and spin 1/2 substitutional nitrogen (P1) centres. The dilute NV centres can be optically initialized and read out, making them a natural probe of the manybody noise generated by the strongly interacting P1 centres. In addition, we demonstrate a complementary role for the NV centres, as a source of spin polarization for the optically dark P1 centres. In particular, by using a Hartmann–Hahn protocol, we directly transfer polarization between the two spin ensembles.
We experimentally characterize the P1 system’s manybody noise via the decoherence dynamics of NV probe spins. To elucidate our results, we first present a theoretical framework that unifies and generalizes existing work, predicting a nontrivial temporal profile that exhibits a crossover between two distinct stretched exponential decays (for the average coherence of the probe spins) (Fig. 1)^{6,13,14,15,16,19,28}. Beyond solidstate spin systems, the framework naturally extends to a broader class of quantum simulation platforms, including trapped ions, Rydberg atoms and ultracold polar molecules^{29}. Crucially, we demonstrate that the associated stretch powers contain a wealth of information about both the static and dynamical properties of the manybody spin system.
We focus on three such properties. First, the stretch power contains a direct signature revealing the dimensionality of the disordered manybody system. Unlike previous work on lowerdimensional ordered systems in magnetic resonance spectroscopy^{30,31,32}, we cannot leverage conventional methods such as Xray diffraction to characterize our disordered spin ensemble. To the best of our knowledge, studying the decoherence dynamics provides the only robust method to determine the effective dimensionality seen by the spins.
The stretch power of the NV centres’ decoherence can also distinguish between different forms of spectral diffusion, shedding light on the nature of local spin fluctuations. In particular, we demonstrate that the P1 spinflip dynamics are inconsistent with the conventional expectation of telegraph noise but rather follow that of a Gauss–Markov process (Table 1). Understanding the statistical properties of the manybody noise and the precise physical settings where such noise emerges remains the subject of active debate^{4,6,16,33,34,35,36,37,38,39}. Finally, the crossover in time between different stretch powers allows one to extract the manybody system’s correlation time. We demonstrate this behaviour by actively controlling the correlation time of the P1 system via polychromatic driving, building upon techniques previously utilized in broadband decoupling schemes^{40}.
Theoretical framework for decoherence dynamics induced by manybody noise
We first outline a framework, building upon classic results in NMR spectroscopy, for understanding the decoherence dynamics of probe spins coupled to an interacting manybody system. This will enable us to present a unified theoretical background for understanding the experimental results in subsequent sections^{6,14,15,19,28,41,42,43}. The dynamics of a single probe spin generically depend on three properties: (1) the nature of the system–probe coupling, (2) the system’s manybody Hamiltonian H_{int} and (3) the measurement sequence itself. Crucially, by averaging across the dynamics of many such probe spins, one can extract global features of the manybody system (Fig. 1b). We distinguish between two types of ensemble averaging that give rise to distinct signatures in the decoherence: (1) an average over manybody trajectories (that is, both spin configurations and dynamics) that yields information about the microscopic spin fluctuations (for simplicity, we focus our discussion on the infinitetemperature limit, but the analysis can be extended to finite temperature) and (2) an average over positional randomness (that is, random locations of the system spins) that yields information about both dimensionality and disorder.
To be specific, let us consider a single spin 1/2 probe coupled to a manybody ensemble via longrange, 1/r^{α} Ising interactions:
where r_{i} is the distance between the probe spin \({\hat{s}}_{\mathrm{p}}\) and the ith system spin \({\hat{s}}_{i}\), and the Ising coupling strength J_{z} implicitly includes any angular dependence. Such powerlaw interactions are ubiquitous in solidstate, atomic and molecular quantum platforms (for example, Ruderman–Kittel–Kasuya–Yosida interactions, electric/magnetic dipolar interactions, van der Waals interactions, etc.).
Physically, the system spins generate an effective magnetic field at the location of the probe (via Ising interactions), which can be measured with Ramsey spectroscopy (Fig. 1e, inset)^{7}. In particular, we envision initially preparing the probe in an eigenstate of \({\hat{s}}_{\mathrm{p}}^{z}\) and subsequently rotating it with a π/2 pulse such that the initial normalized coherence is unity, \(C\equiv 2\left\langle {\hat{s}}_{\mathrm{p}}^{x}\right\rangle =1\). The magnetic field, which fluctuates due to manybody interactions, causes the probe to Larmor precess (Fig. 1b, inset and Supplementary Information). The phase associated with this Larmor precession can be read out via a population imbalance, after a second π/2 pulse.
Average over manybody trajectories
For a manybody system at infinite temperature, \(C(t)=2{{{\rm{Tr}}}}[\rho (t){\hat{s}}_{\mathrm{p}}^{x}]\), where ρ(t) is the full density matrix that includes both the system and the probe. The spin fluctuations are determined by the microscopic details of the manybody dynamics whose full analysis is intractable. To make progress, we approximate each spin as a stochastic classical variable \({\hat{s}}_{i}^{z}(t)\to {s}_{i}^{z}(t)\). The statistical properties of such variables and their resulting ability to capture the experimental observations provide important insights into the nature of fluctuations in strongly interacting spin systems.
The phase of the Larmor precession is given by \(\phi (t)= \int\nolimits_{0}^{t}\mathrm{d}\)\({t}^{{\prime} }\,{J}_{z}{\sum }_{i}{s}_{i}^{z}({t}^{{\prime} })/{r}_{i}^{\alpha }\). Assuming that ϕ(t) is Gaussian distributed, one finds that the average probe coherence decays exponentially as \(C(t)=\left\langle {{{\rm{Re}}}}[\mathrm{e}^{\mathrm{i}\phi (t)}]\right\rangle \approx \mathrm{{e}}^{\left\langle {\phi }^{2}\right\rangle /2}\), where \(\left\langle {\phi }^{2}\right\rangle = {\sum }_{i}{J}_{z}^{2}\chi (t)/{r}_{i}^{2\alpha }\) (refs. ^{4,13,39,44}; see the Supplementary Information for supporting derivations). Here, χ(t) encodes the response of the probe spins to the noise spectral density, S(ω), of the manybody system:
where f(ω; t) is the filter function associated with a particular pulse sequence (for example, Ramsey spectroscopy or spin echo) of total duration t (Fig. 1e).
Intuitively, S(ω) quantifies the noise power density of spin flips in the manybody system. It is the Fourier transform of the autocorrelation function, \(\xi (t)\equiv 4\left\langle {s}_{i}^{z}(t){s}_{i}^{z}(0)\right\rangle\), and captures the spin dynamics at the level of twopoint correlations^{45}. For Markovian dynamics, \(\xi (t)={\mathrm{e}}^{ t /{\tau }_{\mathrm{c}}}\), where τ_{c} defines the correlation time after which a spin, on average, retains no memory of its initial orientation. In this case, S(ω) is Lorentzian and one can derive an analytic expression for χ (refs. ^{15,19,37,41}; see the Supplementary Information for supporting derivations).
A few remarks are in order. First, the premise that manybody Hamiltonian dynamics produce Gaussiandistributed phases ϕ(t)—while often assumed—is challenging to analytically justify^{6,15,16,33,46}. Indeed, a wellknown counterexample of nonGaussian spectral diffusion occurs when the spin dynamics can be modelled as telegraph noise, that is, stochastic jumps between discrete values \({s}_{i}^{z}=\pm {s}_{i}\) (refs. ^{16,34}). The precise physical settings where such noise emerges remain the subject of active debate^{4,6,16,33,34,35,36,37,38,39}. Second, we note that our Markovian assumption is not necessarily valid for a manybody system at early times or for certain forms of interactions, which can also affect the decoherence dynamics.
Average over positional randomness
The probe’s decoherence depends crucially on the spatial distribution of the spins in the manybody system. For disordered spin ensembles, explicitly averaging over their random positions yields a decoherence profile:
where a is a dimensionless constant and N is the number of system spins in a Ddimensional volume V at a density n ≡ N/V (see the Supplementary Information for supporting derivations)^{19}. By contrast, for spins on a lattice or for a single probe spin, the exponent of the coherence scales as \(\sim {J}_{z}^{2}\chi (t)\) (Supplementary Information).
A resonance counting argument underlies the appearance of both the dimensionality and the interaction power law in equation (3). Roughly, a probe spin is only coupled to system spins that induce a phase variance larger than some cutoff ϵ. This constraint on the minimum variance defines a volume of radius \({r}_{\max } \approx {[\,{J}_{z}^{2}\chi (t)/\epsilon ]}^{1/2\alpha }\) containing \({N}_{\mathrm{s}} \approx n{r}_{\max }^{D}\) spins, implying that the total variance accrued at any given time is \(\epsilon {N}_{\mathrm{s}} \approx {[\,{J}_{z}^{2}\chi (t)]}^{D/2\alpha }\). Thus, the positional average simply serves to count the number of spins to which the probe is coupled.
Decoherence profile
The functional form of the probe’s decoherence, C(t), encodes a number of features of the manybody system. We begin by elucidating them in the context of Ramsey spectroscopy. First, one expects a somewhat sharp crossover in the behaviour of C(t) at the correlation time τ_{c}. For early times, t ≪ τ_{c}, the phase variance accumulates as in a ballistic trajectory with χ ∼ t^{2}, while for late times, t ≫ τ_{c}, the variance accumulates as in a random walk with χ ∼ t (refs. ^{15,28,41}). This leads to a simple prediction, namely that the stretch power, β, of the probe’s exponential decay (that is, \(\log C \propto \langle \phi^2 \rangle \sim t^\beta\)) changes from D/α to D/2α at the correlation time (Fig. 1f).
Second, moving beyond Ramsey measurements by changing the filter function, one can probe more subtle properties of the manybody noise. In particular, a spin echo sequence filters out the leadingorder DC contribution from the manybody noise spectrum, allowing one to investigate higherfrequency correlations of the spinflip dynamics. Different types of spinflip dynamics naturally lead to different phase distributions. For the case of Gaussian noise, one finds that (at early times) χ ∼ t^{3}. However, in the case of telegraph noise, the analysis is more subtle since higherorder moments of ϕ(t) must be taken into account. This leads to markedly different earlytime predictions for β, dependent on both the measurement sequence as well as the manybody noise (Table 1).
At late times, however, one expects the probe’s coherence to agree across different pulse sequences and spinflip dynamics. For example, in the case of spin echo, the decoupling π pulse (Fig. 1e, inset) is ineffective on timescales larger than the correlation time, since the spin configurations during the two halves of the free evolution are completely uncorrelated. Moreover, this same loss of correlation implies that the phase accumulation is characterized by incoherent Gaussian diffusion, regardless of the specific nature of the spin dynamics (for example, Markovian versus nonMarkovian or continuous versus telegraph).
Experimentally probing manybody noise in strongly interacting spin ensembles
Our experimental samples contain a high density of spin 1/2 P1 centres (Fig. 1b, blue spins) which form a strongly interacting manybody system coupled via magnetic dipole–dipole interactions:
where J_{0} = 2π × 52 MHz nm^{3}, r_{ij} is the distance between P1 spins i and j and \(c,\tilde{c}\) capture the angular dependence of the dipolar interaction (Supplementary Information). We note that H_{int} contains only the energyconserving terms of the dipolar interaction.
The probes in our system are spin 1 NV centres, which can be optically initialized to \(\left\vert {m}_{\mathrm{s}}=0\right\rangle\) using 532 nm laser light. An applied magnetic field B along the NV axis splits the \(\left\vert {m}_{\mathrm{s}}=\pm 1\right\rangle\) states, allowing us to work within the effective spin 1/2 manifold \(\{\left\vert 0\right\rangle ,\left\vert 1\right\rangle \}\). Microwave pulses at frequency ω_{NV} are used to perform coherent spin rotations (that is, for Ramsey spectroscopy or spin echo) within this manifold (Fig. 1c).
Physically, the NV and P1 centres are also coupled via dipolar interactions. However, for a generic magnetic field strength, they are highly detuned, that is, ∣ω_{NV} − ω_{P1}∣ is on the order of gigahertz, owing to the zerofield splitting of the NV centre (Δ_{0} = 2π × 2.87 GHz) (Fig. 1c). Since typical interaction strengths in our system are on the order of megahertz, direct polarization exchange between an NV and P1 is strongly offresonant. The strong suppression of spinexchange interactions between NV and P1 centres simplifies the full magnetic dipole–dipole Hamiltonian to a system–probe Ising coupling of precisely the form given by equation (1) with α = 3 (Supplementary Information).
Deltadoped sample fabrication
Sample S1 was grown via homoepitaxial plasmaenhanced chemical vapour deposition using isotopically purified methane (99.999% ^{12}C)^{22}. The deltadoped layer was formed by introducing naturalabundance nitrogen gas during growth (5 sccm, 10 min) in between nitrogenfree buffer and capping layers. To create the vacancies necessary for generating NV centres, the sample was electron irradiated with a transmission electron microscope set to 145 keV (ref. ^{23}) and subsequently annealed at 850 °C for 6 h.
Twodimensional spin dynamics
We begin by performing double electronelectron resonance (DEER) measurements on sample S1. While largely analogous to Ramsey spectroscopy (Table 1), DEER has the technical advantage that it filters out undesired quasistatic fields (for example, from hyperfine interactions between the NV and host nitrogen nucleus)^{7,24}. As shown in Fig. 2a (inset, blue data), the NV’s coherence decays on a timescale of approximately 5 μs.
To explore the functional form of the probe NV’s decoherence, we plot the negative logarithm of the coherence, \(\log C(t)\), on a log–log scale, such that the stretch power, β, is simply given by the slope of the data. At early times, the data exhibit β = 2/3 for over a decade in time (Fig. 2a, blue data). At a timescale of approximately 3 μs (vertical dashed line), the data cross over to a stretch power of β = 1/3 for another decade in time. This behaviour is in excellent agreement with that expected for twodimensional spin dynamics driven by dipolar interactions (Fig. 1f and Table 1).
For comparison, we perform DEER spectroscopy on a conventional threedimensional NV–P1 system (sample S2; Methods). As shown in Fig. 2a (orange), the data exhibit β = 1 for a decade in time, consistent with the prediction for threedimensional dipolar interactions (Table 1). However, the crossover to the latetime ‘random walk’ regime is difficult to experimentally access because the larger earlytime stretch power causes a faster decay to the noise floor.
Characterizing microscopic spinflip dynamics
To probe the nature of the microscopic spinflip dynamics in our system, we perform spin echo measurements on three dimensional samples (S3 and S4 (type IB)), which exhibit a much higher P1toNV density ratio (Methods). For lower relative densities (that is, samples S1 and S2), the spin echo measurement contains a confounding signal from interactions between the NVs themselves (Methods).
In both samples (S3 and S4), we find that the coherence exhibits a stretched exponential decay with β = 3/2 for well over a decade in time (Fig. 2b). Curiously, this is consistent with Gaussian spectral diffusion where β = 3D/2α = 3/2 but patently inconsistent with the telegraph noise prediction of β = 1 + D/α = 2. While in agreement with prior measurements on similar samples^{38}, this observation is actually rather puzzling and related to a question in the context of dipolar spin noise^{4,6,7,13,14,15,16,19,28,33,34,35,36,37,38,39,47,48,49}. In particular, one naively expects that spins in a strongly interacting system should be treated as stochastic binary variables, thereby generating telegraph noise. For the specific case of dipolar spin ensembles, this expectation dates back to seminal work from Klauder and Anderson^{6}. The intuition behind this noise model is most easily seen in the language of the master equation—each individual spin ‘sees’ the remaining system as a Markovian bath. The resulting local spin dynamics are then characterized by a series of stochastic quantum jumps that flip the spin orientation and give rise to telegraph noise. Alternatively, in the Heisenberg picture, the same intuition can be understood from the spreading of the operator \({\hat{s}}_{i}^{z}\). This spreading hides local coherences in manybody correlations, leading to an ensemble of telegraphlike, classical trajectories (Supplementary Information).
We conjecture that the observation of Gaussian spectral diffusion in our system is related to the presence of disorder, which strongly suppresses operator spreading^{50}. To illustrate this point, consider the limiting case where the operator dynamics are constrained to a single spin. In this situation, the dynamics of \({\hat{s}}_{i}^{z}(t)\) follow a particular coherent trajectory around the Bloch sphere and the rate at which the probe accumulates phase is continuous. Averaging over different trajectories of the coherent dynamics naturally leads to Gaussian noise.
Controlling the P1 spectral function
Next, we demonstrate the ability to directly control the P1 noise spectrum for both two and threedimensional dipolar spin ensembles (that is, samples S1 and S2). In particular, we engineer the shape and linewidth of S(ω) by driving the P1 system with a polychromatic microwave tone^{40}. This drive is generated by adding phase noise to the resonant microwave signal at ω_{P1} to produce a Lorentzian drive spectrum with linewidth δω (Fig. 3c). While such techniques originated in the context of broadband noise decoupling^{40}, here we directly tune the correlation time of the P1 system and measure a corresponding change in the crossover timescale between coherent and incoherent spin dynamics^{15,49}.
Microscopically, the polychromatic drive leads to a number of physical effects. First, tuning the Rabi frequency, Ω, of the drive provides a direct knob for controlling the correlation time, τ_{c}, of the P1 system. Second, since the manybody system inherits the noise spectrum of the drive, one has provably Gaussian statistics for the spin variables \({s}_{i}^{z}\) (Supplementary Information). Third, our earlier Markovian assumption is explicitly enforced by the presence of a Lorentzian noise spectrum. Taking these last two points together allows one to analytically predict the precise form of the NV probe’s decoherence profile, \(\log C(t) \sim \chi {(t)}^{D/2\alpha }\), for either DEER or spin echo spectroscopy:
We perform both DEER and spin echo measurements as a function of the power (∼Ω^{2}) of the polychromatic drive for our twodimensional sample (S1) (Fig. 3a). As expected, for weak driving (Fig. 3a, top), the DEER signal (blue) is analogous to the undriven case, exhibiting a crossover from a stretch power of β = 2/3 at early times to a stretch power of β = 1/3 at late times. For the same drive strength, the spin echo data (red) also exhibit a crossover between two distinct stretch powers, with the key difference being that β = 3D/2α = 1 at early times. This represents an independent (spin echo based) confirmation of the twodimensional nature of our deltadoped sample.
Recall that, at late times (that is t ≳ τ_{c}), one expects the NV’s coherence C(t) to agree across different pulses sequences (Fig. 1f). This is indeed borne out by the data (Fig. 3). In fact, the location of this latetime overlap provides a proxy for estimating the correlation time and is shown as the dashed grey lines in Fig. 3a. As one increases the power of the drive (Fig. 3a), the noise spectrum, S(ω), naturally broadens. In the data, this manifests as a shortened correlation time, with the location of the DEER/echo overlap shifting to earlier timescales (Fig. 3a).
Analogous measurements on a threedimensional spin ensemble (sample S2) reveal much the same physics (Fig. 3b), with stretch powers again consistent with a Gauss–Markov prediction (Table 1). For weak driving, C(t) is consistent with the earlytime ballistic regime for over a decade in time (Fig. 3b, top). However, it is difficult to access late enough timescales to observe an overlap between DEER and spin echo. Crucially, by using the drive to push to shorter correlation times, we can directly observe the latetime randomwalk regime in three dimensions, where β = 1/2 (Fig. 3b, middle and bottom).
Remarkably, as evidenced by the dashed curves in Fig. 3a,b our data exhibit excellent agreement—across different dimensionalities, drive strengths and pulse sequences—with the analytic predictions presented in equation (5). Moreover, by fitting χ^{D/2α} simultaneously across spin echo and DEER datasets for each Ω, we quantitatively extract the correlation time, τ_{c}. Up to an \({{{\mathcal{O}}}}(1)\) scaling factor, we find that the extracted τ_{c} agrees well with the DEER/echo overlap time. In addition, the behaviour of τ_{c} as a function of Ω also exhibits quantitative agreement with an analytic model that predicts τ_{c} ∼ δω/Ω^{2} in the limit of strong driving (Fig. 3d,e).
We emphasize that, although one observes β = 3D/2α in both the driven (Fig. 3a,b) and undriven (Fig. 2b) spin echo measurements, the underlying physics is extremely different. In the latter case, Gaussian spectral diffusion emerges from isolated, disordered, manybody dynamics, while in the former case, it is imposed by the external drive.
A twodimensional solidstate platform for quantum simulation and sensing
Our platform offers two distinct paths towards quantum simulation and sensing using strongly interacting, twodimensional, spinpolarized ensembles. First, treating the NV centres themselves as the manybody system directly leverages their optical polarizability. However, given their relative diluteness, it is natural to ask whether one can access regimes where the NV–NV interactions dominate over other energy scales. Conversely, treating the P1 centres as the manybody system takes advantage of their higher densities and interaction strengths, with the key challenge being that these dark spins cannot be optically pumped. Here, we demonstrate that both of these paths are viable for sample S1: (1) we show that the dipolar interactions among NV centres can dominate their decoherence dynamics, using advanced dynamical decoupling sequences, and (2) we demonstrate direct polarization exchange between NV and P1 centres, providing a mechanism to spin polarize the P1 system.
Interacting NV ensemble
To demonstrate NV–NV interactiondominated dynamics, we compare the decoherence timescales between spin echo, XY8 and disorderrobust interaction decoupling (DROID) dynamical decoupling sequences^{51}. The spin echo effectively decouples static disorder, while the XY8 sequence further decouples NV–P1 interactions. As depicted in Fig. 4a, XY8 pulses extend the spin echo decay time (defined as the 1/e time) by approximately a factor of two. With NV–P1 interactions decoupled, our hypothesis is that the dynamics are now driven by dipolar interactions between the NV centres. To test this, we perform a DROID decoupling sequence, which eliminates the dipolar dynamics between NV centres^{51} (Methods). Remarkably, this extends the coherence time by nearly an order of magnitude, demonstrating that NV–NV interactions are, by far, the dominant source of manybody dynamics in this regime. Moreover, the XY8 decoherence thus provides an estimate of an average NV spin–spin spacing of 15 nm.
Interacting P1 ensemble
The polarization of the optically dark P1 ensemble can be realized by either (1) working at low temperatures and large magnetic fields^{52} or (2) using NV centres to transfer polarization to the P1 centres. Here, we focus on the latter. While NV–P1 polarization transfer has previously been demonstrated^{53,54}, it has not been measured in a twodimensional system. Indeed, conjectures about localization in such systems indicate that polarization transfer could be highly suppressed^{55,56}.
To investigate, we employ a Hartmann–Hahn sequence designed to transfer polarization between NV and P1 spins in the rotating frame^{53,54}. In particular, we drive the NV and P1 spins independently, with Rabi frequencies Ω_{NV} and Ω_{P1}. When only the NV centres are driven, we are effectively performing a socalled spinlocking measurement^{57}. For Ω_{NV} = 2π × 5 MHz, we find that the NV centres depolarize on a timescale of T_{1ρ} = 1.05(3) ms (Fig. 4b, orange). The data are cleanly fit by a simple exponential and consistent with phononlimited decay (Fig. 4b, inset). By contrast, when the driving satisfies the Hartmann–Hahn condition, Ω_{NV} = Ω_{P1}, the NV and P1 spins can resonantly exchange polarization. To characterize this, we fix Ω_{NV} = 2π × 5 MHz and choose a spinlocking duration t_{s} = 200 μs. By sweeping the P1 Rabi frequency, we indeed observe a resonant polarization exchange feature centred at Ω_{P1} = 2π × 5 MHz with a linewidth of ∼ 2π × 1.2 MHz (Fig. 4c), consistent with the intrinsic P1 linewidth. As illustrated in Fig. 4b, on resonance, the NV depolarization is notably enhanced via polarization transfer to the P1 centres and the data exhibit a threefold decrease in the decay time. Moreover, the data are well fitted with a stretch power of β = 1/3 (Fig. 4b, inset), which is also indicative of interactiondominated decay^{47} (Methods).
Conclusion and outlook
Our results demonstrate the diversity of information that can be accessed via the decoherence dynamics of a probe spin ensemble. For example, we shed light on a longstanding debate about the nature of spinflip noise in a strongly interacting dipolar system^{4,6,16,33,34,35,36,37,38,39,48,49}. Moreover, we directly measure the correlation time of the manybody system and introduce a technique to probe its dimensionality. This technique is particularly useful for disordered spin ensembles embedded in solids^{58,59}, where a direct, nondestructive measurement of nanoscale spatial properties is challenging with conventional toolsets.
One can imagine generalizing our work in a number of promising directions. First, the ability to fabricate and characterize strongly interacting, twodimensional dipolar spin ensembles opens the door to a number of intriguing questions within the landscape of quantum simulation. Indeed, dipolar interactions in 2D are quite special from the perspective of localization, allowing one to experimentally probe the role of manybody resonances^{55,56}. In the context of groundstate physics, the longrange, anisotropic nature of the dipolar interaction has also been predicted to stabilize a number of exotic phases, ranging from supersolids to spin liquids^{25,26}. Connecting this latter point back to noise spectroscopy, one could imagine tailoring the probe’s filter function to distinguish between different types of groundstate order.
Second, dense ensembles of twodimensional spins also promise a number of unique advantages with respect to quantum sensing^{21,22,24}. For example, a 2D layer of NVs fabricated near the diamond surface would exhibit a pronounced enhancement in spatial resolution (set by the depth of the layer) compared with a threedimensional ensemble at the same density, ρ (refs. ^{22,60}). In addition, for samples where the coherence time is limited by spin–spin interactions, a lower dimensionality reduces the coordination number and leads to an enhanced T_{2} scaling as n^{−α/D} (Supplementary Information).
Third, one can probe the relationship between operator spreading and Gauss–Markov noise by exploring samples with different relaxation rates, interaction power laws, disorder strengths and spin densities^{33,49}. One could also utilize alternate pulse sequences, such as stimulated echo, to provide a more finegrained characterization of the manybody noise (for example, the entire spectral diffusion kernel)^{28,33}.
Finally, our framework can also be applied to longrangeinteracting systems of Rydberg atoms, trapped ions and polar molecules. In such systems, the ability to perform imaging and quantum control at the singleparticle level allows for greater freedom in designing methods to probe manybody noise. As a particularly intriguing example, one could imagine a nondestructive, timeresolved generalization of manybody noise spectroscopy, where one repeatedly interrogates the probe without projecting the manybody system.
Methods
Sample preparation and characterization
Sample S1
Sample fabrication
Here, we add to the details provided in Section Deltadoped sample fabrication. Sample S1 was grown on a commercially available Element6 electronic grade (100) substrate, polished by Syntek^{61} to a surface roughness less than 200 pm. Throughout the plasmaenhanced chemical vapour deposition growth process^{22}, we used 400 sccm of hydrogen gas with a background pressure of 25 Torr, and a microwave power of 750 W. The sample temperature was held at 800 °C.
NV density
We estimate the NV areal density in sample S1 via the XY8 decoherence profile^{62}, which is dominated by intragroup NV interactions (that is, within the NV group aligned with the applied magnetic field B) (Fig. 4a). We therefore treat the XY8 data as a Ramsey measurement of the average NV–NV coupling, which we convert to a density using the dipolar interaction strength J_{0} = 2π × 52 MHz nm^{3}. We compare the XY8 data with numerically computed Ramsey decoherence, which we calculate as follows: We consider a central probe NV interacting with a bath of other NVs, placed randomly in a thin slab of thickness w with density \({n}_{{{{\rm{3D}}}}}^{{{{\rm{NV}}}}}/4\) (one NV group). After selecting a random spin configuration for the bath NVs, we compute the Ramsey signal \(\sim \cos (\phi )\) for the probe NV. We then average over many such samples, and the resulting curve exhibits a stretched exponential decay of the form \(C(t)=\mathrm{e}^{{(t/{T}_{2})}^{2/3}}\). This functional form matches our expectation for the earlytime ballistic regime (Table 1), because we have not included flipflop dynamics in the numerical model. We treat the decoherence as arising only from intragroup Ising interactions, which is correct at short times when the NV centres are spin polarized. With the above prescription, we compute a set of Ramsey signals (Extended Data Fig. 1a, dashed lines) as a function of areal density \({n}_{{{{\rm{3D}}}}}^{{{{\rm{NV}}}}}w\), which we compare against the XY8 data (Extended Data Fig. 1a, orange points). The estimated areal density is thus \({n}_{{{{\rm{3D}}}}}^{{{{\rm{NV}}}}} w =19\pm 2\,{{{\rm{ppm}}}}\, {{{\rm{nm}}}}\), corresponding to a density \({n}_{{{{\rm{3D}}}}}^{{{{\rm{NV}}}}}=3.2\pm 0.3\,{{{\rm{ppm}}}}\) assuming a layer with thickness of w = 6 nm.
P1 density
We estimate the P1 density by using a similar procedure but with DEER data instead of XY8 data. We first remove the contribution due to NV–NV interactions from the DEER signal by subtracting an interpolation of the XY8 data (Extended Data Fig. 1a) from the raw DEER data. Then, we compare the measured earlytime dynamics with numerically computed curves for a range of P1 densities \({n}_{{{{\rm{3D}}}}}^{{{{\rm{P1}}}}}/3\) (Extended Data Fig. 1b). Here, we include a factor of 1/3 in the P1 density because the microwave tone ω_{P1} addresses only onethird of the P1 spins (the ‘P11/3 group’) in our DEER measurement, which are separated by ∼ 100 MHz from the four other groups due to the hyperfine interaction^{63,64}. By comparing the data and theory curves, we estimate an areal density of \({n}_{{{{\rm{3D}}}}}^{{{{\rm{P1}}}}} w=85\pm 10\,{{{\rm{ppm}}}}\, {{{\rm{nm}}}} \approx 1.4(1)\times 1{0}^{2}\,{{{{\rm{nm}}}}}^{2}\).
At fixed areal density, the numerics indicate that the DEER decoherence profile depends on the layer thickness (Extended Data Fig. 1d). The same dependence is not present in the XY8 dynamics due to the relatively small density of NV centres (Extended Data Fig. 1c). Although this method does not yield nanometre resolution, our observations are inconsistent with a layer with w > 6 nm, placing a more stringent bound on the thickness of the layer. The areal density \({n}_{{{{\rm{3D}}}}}^{{{{\rm{P1}}}}} w=85\pm 10\,{{{\rm{ppm}}}}\, {{{\rm{nm}}}}\) corresponds to \({n}_{{{{\rm{3D}}}}}^{{{{\rm{P1}}}}}=14\pm 2\,{{{\rm{ppm}}}}\), assuming a layer with thickness of w = 6 nm.
Other spin 1/2 paramagnetic defects in diamond^{65} may have the same resonant frequency as the P11/3 group, causing a possible systematic error in our method for estimating the P1 density. To determine whether such defects are present in sample S1, we measured the P1 spectrum and compared the relative integrated areas under the peaks for the P11/3, 1/4 and 1/12 groups, thus obtaining an estimate of the relative densities between P1 groups. As shown in Extended Data Fig. 2, the results agree with the expected ratios 1:0.75:0.25 and are consistent with a negligible contribution of nonP1 defects to the DEER signal.
Sample S2
A detailed characterization of the threedimensional sample S2 is given in ref. ^{24} (sample C041). Here, we describe the key properties relevant for the present study. The sample was grown by depositing a 32 nm diamond buffer layer, followed by a 500 nm nitrogendoped layer (99%^{15}N), and finished with a 50 nm undoped diamond capping layer. Vacancies were created by irradiating with 145 keV electrons at a dosage of 10^{21} cm^{−2}, and vacancy diffusion was activated by annealing at 850 °C for 48 h in an Ar/Cl atmosphere. The resulting NV density is ∼0.4 ppm, obtained through instantaneous diffusion measurements^{24}. The P1 density is measured to be ∼20 ppm through a modified DEER sequence^{24}. The average spacing between P1 centres (∼4 nm) is much smaller than the thickness of the nitrogendoped layer, ensuring threedimensional behaviour of the spin ensemble.
Samples S3 and S4
Samples S3 and S4 used in this work are synthetic type Ib single/crystal diamonds (Element Six) with intrinsic substitutional ^{14}N concentration of ∼100 ppm (calibrated with an NV linewidth measurement^{63}). To create NV centres, the samples were first irradiated with electrons (2 MeV energy and 1 × 10^{18} cm^{−2} dosage) to generate vacancies. After irradiation, the diamonds were annealed in vacuum (∼10^{−6} Torr) with temperature >800 °C. The NV densities for both samples were measured to be ∼0.5 ppm using a spinlocking measurement^{63}.
Experimental methods
Experimental details for sample S1
The deltadoped sample S1 was mounted in a scanning confocal microscope. For optical pumping and readout of the NV centres, about 100 μW of 532 nm light was directed through an oilimmersion objective (Nikon Plan Fluor 100×, NA 1.49). The NV fluorescence was separated from the green 532 nm light by using a dichroic filter and collected on a fibrecoupled singlephoton counter. A magnetic field B was produced by using a combination of three orthogonal electromagnetic coils and a permanent magnet, and aligned along one of the diamond crystal axes. The microwaves used to drive magnetic dipole transitions for both NV and P1 centres were delivered via an Omegashaped stripline with typical Rabi frequencies of ∼2π × 10 MHz.
DROID and Hartmann–Hahn sequences
Here, we describe the pulse sequences used to perform the measurements shown in Fig. 4. In Fig. 4a, we compare the coherence times across different dynamical decoupling sequences, demonstrating that the longest coherence times are achieved when we decouple both onsite disorder and dipolar NV–NV dynamics using a DROID sequence proposed by Choi et al.^{62}. To achieve the best decoupling, we experimented with a few variations on the wellknown DROID60 sequence^{51}. These measurements are plotted in Extended Data Fig. 3. The DROID60 data exhibit a pronounced coherent oscillation (purple points), which we attribute mainly to errors in composite pulses formed by sequential π/2 rotations along different axes. The data exhibiting the longest coherence time (red points) are obtained using socalled sequence H (fig. 9 of ref. ^{62}). We hypothesize that sequence H behaves more predictably precisely because it eliminates composite pulses.
In Fig. 4b,c, we demonstrate polarization transfer between NV and P1 spins using a Hartmann–Hahn sequence. Following an initial π/2 pulse, the NV centres are spinlocked with Rabi frequency Ω_{NV}, while the P1 centres are simultaneously driven with Rabi frequency Ω_{P1}, for a duration t_{s}. A final π/2 pulse is applied before detection. When the two Rabi frequencies are equal, that is, Ω_{NV} = Ω_{P1}, the spins are resonant in the rotating frame, and spinexchange interactions enhance the depolarization rate. The resonant depolarization data (Fig. 4b, green curve) are well fitted by the functional form
where the first factor captures phononlimited exponential decay and the second factor captures the independent depolarization channel driven by spinexchange interactions, with stretch power β = 1/3 (ref. ^{47}). We determine T_{1ρ} = 1.05(3) ms from the NV spinlocking measurement (Fig. 4b, orange curve).
Experimental details for sample S2
Sample S2 was mounted in a confocal microscope. For optical initialization and readout, about 350 μW of 532 nm light was directed through an air objective (Olympus UPLSA 40×, NA 0.95). The NV fluorescence was similarly separated from the 532 nm light by using a dichroic mirror and directed onto a fibrecoupled avalanche photodiode. A permanent magnet produced a field of about 320 G at the location of the sample. The field was aligned along one of the NV axes, and alignment was demonstrated by maximizing the ^{15}N nuclear polarization^{66}. Microwaves were delivered with a freespace rf antenna positioned over the sample.
Experimental details for samples S3 and S4
Samples S3 and S4 were mounted in a confocal microscope. For optical initialization and readout, about 3 mW of 532 nm light was directed through an air objective (Olympus LUCPLFLN, NA 0.6). The NV fluorescence was separated from the 532 nm light by using a dichroic mirror and directed onto a fibrecoupled photodiode (Thorlabs). The magnetic field was produced with an electromagnet with field strength of ∼174 G (∼275 G) for sample S3 (S4). The field was aligned along one of the NV axes, and microwaves were delivered using an Omegashaped stripline with typical Rabi frequencies of ∼2π × 10 MHz.
Polychromatic drive
The polychromatic drive was generated by phase modulating the resonant P1 microwave tone^{67}. A random array of phase jumps Δθ was pregenerated and loaded onto an arbitrary waveform generator controlling the IQ modulation ports of a signal generator. The linewidth of the drive δω was controlled by fixing the s.d. of the phase jumps \(\sigma =\sqrt{\updelta \omega \updelta t}\) in the pregenerated array, where 1/δt = 1 gigasample per second was the sampling rate of the arbitrary waveform generator. The power in the drive was calibrated by measuring Rabi oscillations of the P1 centres without modulating the phase, that is, by setting δω = 0.
Spin echo for samples S1 and S2 without polychromatic driving
In Section Characterizing microscopic spinflip dynamics, we discussed spin echo measurements limited by NV–P1 interactions (as one would naively expect) and which exhibit an earlytime stretch power of β = 3D/2α = 3/2. These measurements were performed on samples S3 and S4 that exhibit a P1toNV density ratio of ∼200. By contrast, spin echo measurements on samples S1 and S2, with P1toNV density ratios of ∼ 10 and ∼ 40, respectively, exhibit an earlytime stretch of β = D/α (Extended Data Fig. 4), consistent with the prediction for a Ramsey measurement (Table 1). Here, we are discussing a ‘canonical’ spin echo measurement with no polychromatic drive (Fig. 2b, inset), and thus these data are not in contradiction with those presented in Fig. 3.
A possible explanation for the observed earlytime stretch β = D/α is that the spin echo signal is limited by NV–NV interactions rather than by NV–P1 interactions. To understand this limitation, it is important to realize that the measured spin echo signal actually contains at least two contributions: (1) the expected spin echo signal from NV–P1 interactions, arising because the intermediate π pulse decouples the NVs from any quasistatic P1 contribution and (2) a Ramsey signal from NV interactions with other NVs, arising because these NVs are flipped together by the π pulse, and the intragroup Ising interactions are not decoupled.
Our hypothesis that NV–NV interactions limit the spin echo coherence in sample S1 is supported by the fact that a stretch power of β = 3D/2α = 1 can in fact be observed in spin echo data if the environment is made noisier, for example, by reversibly worsening the quality of the diamond surface (Extended Data Fig. 4b, green points). A subsequent threeacid clean restores the original β = 2/3 stretch power (Extended Data Fig. 4a, red points).
Data analysis and fitting
Normalization of decoherence data
The coherence of the NV spins is read out via the population imbalance between \(\{\left\vert 0\right\rangle ,\left\vert 1\right\rangle \}\) states. The maximum measured contrast ≲8% is proportional (not equal) to the normalized coherence C(t). To see a physically meaningful stretch power in our log–log plots of the data (Figs. 2 and 3), it is necessary to normalize the data by an appropriate value that captures our best approximation of the t = 0 time point for the DEER and spin echo measurements.
Samples S1, S3 and S4: t = 0 measurement
For a given pulse sequence (for example, Ramsey or spin echo) and fixed measurement duration t, we perform a differential readout of the populations in the \(\left\vert 0\right\rangle\) and \(\left\vert 1\right\rangle\) spin states of the NV, which mitigates the effect of NV and P1 charge dynamics induced by the laser initialization and readout pulses. As depicted schematically in Extended Data Fig. 5, we allow the NV charge dynamics to reach steady state (I) before applying an optical pumping pulse (II). Subsequently, we apply microwave pulses to both the NV and P1 spins (for example Ramsey or spin echo pulse sequences as shown in Figs. 2 and 3) (III). Finally, we detect the NV fluorescence (IV) to measure the NV population in \(\left\vert 0\right\rangle\), obtaining a signal S_{0}. We repeat the same sequence a second time, with one additional π pulse before detection to measure the NV population in \(\left\vert 1\right\rangle\), obtaining a signal S_{−1}. The raw contrast, C_{raw}, at time t is then computed as C_{raw}(t) ≡ [S_{0}(t) − S_{−1}(t)]/S_{0}(t), and is typically ≲8%. We normalize the raw contrast to the t = 0 measurement to obtain the normalized coherence, C(t), defined above as
Sample S2: t = 0 measurement
For sample S2, we have an earlytime, rather than a t = 0, measurement at t = 320 ns for spin echo and DEER sequences. Because the DEER signal decays on a much faster timescale than the spin echo signal, we normalize both datasets to the earliesttime spin echo measurement.
Data analysis for Fig. 3
We separate our discussion of the data analysis relevant to Fig. 3 into two parts. First, we discuss how comparing the D = 2 and D = 3 best fits to the DEER measurements enables us to identify the dimensionality of the underlying spin system. Second, armed with the fitted dimensionality, we fit spin echo and DEER measurements simultaneously to equation (5) to extract the correlation time τ_{c} of the P1 system. We note that, except for the t = 0 normalization point (Section Normalization of decoherence data in the Methods), we only consider data at times t > 0.5 μs, to mitigate any effects of earlytime coherent oscillations caused by the hyperfine coupling between the NV and its host nitrogen nuclear spin.
Determining the dimensionality of the system
To determine the dimensionality of the different samples S1 and S2, we focus on the DEER signal, where the stretch power is given by β = D/α in the earlytime ballistic regime and β = D/2α in the latetime randomwalk regime. Employing both Gaussian and Markovian assumptions, a closed form for the decoherence can be obtained as
where χ^{DEER} is defined in equation (5) (Supplementary Information).
Armed with equation (8), we consider the decoherence dynamics for different powers of the polychromatic drive for both D = 2 and D = 3 (with α = 3, as per the dipolar interaction). We compare the reduced \({\chi }_{{{{\rm{fit}}}}}^{2}\) goodnessoffit parameters for the two values of D, and demonstrate that the stretch power analysis above indeed agrees with the dimensionality that best explains the observed DEER data. Changing the dimension D does not change the number of degrees of freedom in the fit, so a direct comparison of \({\chi }_{{{{\rm{fit}}}}}^{2}\) is meaningful. Our results are summarized in Extended Data Fig. 6, where we observe that for sample S1 indeed the D = 2 fitting leads to a smaller \({\chi }_{{{{\rm{fit}}}}}^{2}\), while for sample S2 the data are best captured by D = 3 (Extended Data Fig. 6). Independently fitting both the extracted signal C(t) as well to its negative logarithm \(\log C(t)\) yields the same conclusions. This analysis complements the discussion above in terms of the earlytime and latetime stretch power of the decay.
Extracting the correlation time τ_{c}
Having determined the dimensionality of samples S1 and S2, we now turn to characterizing the correlation times of the P1 spin systems in these samples. To robustly extract τ_{c}, we perform a simultaneous fit to both the DEER signal with equation (8) and the spin echo signal with
assuming a single amplitude A and correlation time τ_{c} for both normalized signals. Here, χ^{DEER/SE} depends on τ_{c} as defined in equation (5).
To carefully evaluate the uncertainty in the extracted correlation time, we take particular care to propagate the uncertainty in the t = 0 data used to normalize the raw contrast, that is, C_{raw}(t = 0) (Section Normalization of decoherence data in the Methods). Owing to the two normalization methods for samples S1 and S2 (Section Normalization of decoherence data in the Methods), we estimate the uncertainty in two different ways:

For samples S1, S3 and S4, we consider fluctuations of the normalization value, C_{raw}(t = 0), by ±10%. This is meant to account for a possible effect of the hyperfine interaction in this data point, as well as any additional systematic error.

For sample S2, we first compute a linear interpolation of the earlytime spin echo decoherence to t = 0. We then sample the normalization uniformly between this extrapolated value and the earliest spin echo value.
By sampling over the possible values of C_{raw}(t = 0), we build a distribution over the extracted values of τ_{c} fitting to both the coherence, C(t), and its logarithm, \(\log C(t)\). The reported values in Fig. 3d,e correspond to the mean and s.d. evaluated over this distribution.
We end this section by commenting that, as the drive strength is reduced, the spin echo signal looks increasingly similar to the undriven spin echo data (Extended Data Fig. 4), that is, the earlytime stretch changes from β = 3D/2α to β = D/α. Our explanation for this observed stretch is given in Section Spin echo for samples S1 and S2 without polychromatic driving in the Methods. The deviation from the expected functional form for the decoherence leads to a large uncertainty in the extracted correlation time. The data also deviate from the model for larger drive strengths, for example, Ω = 2π × 4.05 MHz, δω = 2π × 20 MHz, where our assumption that δω ≫ Ω is no longer valid (Extended Data Fig. 7).
Data availability
Data supporting the findings of this paper are available from the corresponding authors upon request. Source data are provided with this paper. Source data for Figs. 1–4 and Extended Data Figs. 1–7 are provided with this paper.
Code availability
Code developed for the data analysis and visualization is available from the corresponding author upon request.
References
Purcell, E. M. in Confined Electrons and Photons (eds Burstein, E. & Weisbuch, C.) 839–839 (Springer, 1995).
Viola, L., Knill, E. & Lloyd, S. Dynamical decoupling of open quantum systems. Phys. Rev. Lett. 82, 2417–2421 (1999).
Houck, A. et al. Controlling the spontaneous emission of a superconducting transmon qubit. Phys. Rev. Lett. 101, 080502 (2008).
De Lange, G., Wang, Z., Riste, D., Dobrovitski, V. & Hanson, R. Universal dynamical decoupling of a single solidstate spin from a spin bath. Science 330, 60–63 (2010).
Tyryshkin, A. M. et al. Electron spin coherence exceeding seconds in highpurity silicon. Nat. Mater. 11, 143–147 (2012).
Klauder, J. & Anderson, P. Spectral diffusion decay in spin resonance experiments. Phys. Rev. 125, 912 (1962).
Schweiger, A. & Jeschke, G. Principles of Pulse Electron Paramagnetic Resonance (Oxford University Press on Demand, 2001).
Kofman, A. & Kurizki, G. Acceleration of quantum decay processes by frequent observations. Nature 405, 546–550 (2000).
Romach, Y. et al. Spectroscopy of surfaceinduced noise using shallow spins in diamond. Phys. Rev. Lett. 114, 017601 (2015).
Kleppner, D. Inhibited spontaneous emission. Phys. Rev. Lett. 47, 233 (1981).
Kotler, S., Akerman, N., Glickman, Y., Keselman, A. & Ozeri, R. Singleion quantum lockin amplifier. Nature 473, 61–65 (2011).
BarGill, N. et al. Suppression of spinbath dynamics for improved coherence of multispinqubit systems. Nat. Commun. 3, 1–6 (2012).
Herzog, B. & Hahn, E. L. Transient nuclear induction and double nuclear resonance in solids. Phys. Rev. 103, 148–166 (1956).
Kubo, R., Toda, M. & Hashitsume, N. Statistical Physics II: Nonequilibrium Statistical Mechanics, Vol. 31 (Springer Science & Business Media, 2012).
Salikhov, K., Dzuba, S.A. & Raitsimring, A. M. The theory of electron spinecho signal decay resulting from dipole–dipole interactions between paramagnetic centers in solids. J. Magn. Reson. (1969) 42, 255–276 (1981).
Chiba, M. & Hirai, A. Electron spin echo decay behaviours of phosphorus doped silicon. J. Phys. Soc. Jpn. 33, 730–738 (1972).
Altman, E., Demler, E. & Lukin, M. D. Probing manybody states of ultracold atoms via noise correlations. Phys. Rev. A 70, 013603 (2004).
Hofferberth, S. et al. Probing quantum and thermal noise in an interacting manybody system. Nat. Phys. 4, 489–495 (2008).
Fel’dman, E. B. & Lacelle, S. Configurational averaging of dipolar interactions in magnetically diluted spin networks. J. Chem. Phys. 104, 2000–2009 (1996).
Choi, S. et al. Observation of discrete timecrystalline order in a disordered dipolar manybody system. Nature 543, 221–225 (2017).
Sushkov, A. et al. Magnetic resonance detection of individual proton spins using quantum reporters. Phys. Rev. Lett. 113, 197601 (2014).
Ohno, K. et al. Engineering shallow spins in diamond with nitrogen deltadoping. Appl. Phys. Lett. 101, 082413 (2012).
McLellan, C. A. et al. Patterned formation of highly coherent nitrogenvacancy centers using a focused electron irradiation technique. Nano Lett. 16, 2450–2454 (2016).
Eichhorn, T. R., McLellan, C. A. & Bleszynski Jayich, A. C. Optimizing the formation of depthconfined nitrogen vacancy center spin ensembles in diamond for quantum sensing. Phys. Rev. Mater. 3, 113802 (2019).
Yao, N. Y., Zaletel, M. P., StamperKurn, D. M. & Vishwanath, A. A quantum dipolar spin liquid. Nat. Phys. 14, 405–410 (2018).
Chomaz, L. et al. Longlived and transient supersolid behaviors in dipolar quantum gases. Phys. Rev. X 9, 021012 (2019).
Semeghini, G. et al. Probing topological spin liquids on a programmable quantum simulator. Science 374, 1242–1247 (2021).
Anderson, P. W. & Weiss, P. R. Exchange narrowing in paramagnetic resonance. Rev. Mod. Phys. 25, 269–276 (1953).
Georgescu, I. M., Ashhab, S. & Nori, F. Quantum simulation. Rev. Mod. Phys. 86, 153 (2014).
Engelsberg, M., Lowe, I. & Carolan, J. Nuclearmagneticresonance line shape of a linear chain of spins. Physical Review B 7, 924 (1973).
Cho, G. & Yesinowski, J. P. H and 19f multiplequantum NMR dynamics in quasionedimensional spin clusters in apatites. J. Phys. Chem. 100, 15716–15725 (1996).
Cho, H., Ladd, T. D., Baugh, J., Cory, D. G. & Ramanathan, C. Multispin dynamics of the solidstate NMR free induction decay. Phys. Rev. B 72, 054427 (2005).
Mims, W. Phase memory in electron spin echoes, lattice relaxation effects in CaWO_{4}: Er, Ce, Mn. Phys. Rev. 168, 370 (1968).
Abe, E., Itoh, K. M., Isoya, J. & Yamasaki, S. Electronspin phase relaxation of phosphorus donors in nuclearspinenriched silicon. Phys. Rev. B 70, 033204 (2004).
Zhong, M. et al. Optically addressable nuclear spins in a solid with a sixhour coherence time. Nature 517, 177–180 (2015).
de Sousa, R. & Sarma, S. D. Theory of nuclearinduced spectral diffusion: spin decoherence of phosphorus donors in si and gaas quantum dots. Phys. Rev. B 68, 115322 (2003).
Wang, Z.H. & Takahashi, S. Spin decoherence and electron spin bath noise of a nitrogenvacancy center in diamond. Phys. Rev. B 87, 115122 (2013).
Bauch, E. et al. Decoherence of ensembles of nitrogenvacancy centers in diamond. Phys. Rev. B 102, 134210 (2020).
Hanson, R., Dobrovitski, V., Feiguin, A., Gywat, O. & Awschalom, D. Coherent dynamics of a single spin interacting with an adjustable spin bath. Science 320, 352–355 (2008).
Ernst, R. R. Nuclear magnetic double resonance with an incoherent radiofrequency field. J. Chem. Phys. 45, 3845–3861 (1966).
Hu, P. & Hartmann, S. R. Theory of spectral diffusion decay using an uncorrelatedsuddenjump model. Phys. Rev. B 9, 1–13 (1974).
Cucchietti, F. M., Paz, J. P. & Zurek, W. H. Decoherence from spin environments. Phys. Rev. A 72, 052113 (2005).
de Sousa, R. in Electron Spin Resonance and Related Phenomena in LowDimensional Structures (ed Fanciulli, M.) 183–220 (Springer, 2009).
Yang, W., Ma, W.L. & Liu, R.B. Quantum manybody theory for electron spin decoherence in nanoscale nuclear spin baths. Rep. Prog. Phys. 80, 016001 (2017).
Kogan, S. Electronic Noise and Fluctuations in Solids (Cambridge Univ. Press, 2008).
Witzel, W. & Sarma, S. D. Quantum theory for electron spin decoherence induced by nuclear spin dynamics in semiconductor quantum computer architectures: spectral diffusion of localized electron spins in the nuclear solidstate environment. Phys. Rev. B 74, 035322 (2006).
Choi, J. et al. Depolarization dynamics in a strongly interacting solidstate spin ensemble. Phys. Rev. Lett. 118, 093601 (2017).
Zhidomirov, G. & Salikhov, K. Contribution to the theory of spectral diffusion in magnetically diluted solids. Sov. J. Exp. Theor. Phys. 29, 1037 (1969).
Glasbeek, M. & Hond, R. Phase relaxation of photoexcited triplet spins in cao. Phys. Rev. B 23, 4220 (1981).
Witzel, W. M., Carroll, M. S., Cywiński, Ł. & Sarma, S. D. Quantum decoherence of the central spin in a sparse system of dipolar coupled spins. Phys. Rev. B 86, 035452 (2012).
Zhou, H. et al. Quantum metrology with strongly interacting spin systems. Phys. Rev. X 10, 031003 (2020).
Takahashi, S., Hanson, R., Van Tol, J., Sherwin, M. S. & Awschalom, D. D. Quenching spin decoherence in diamond through spin bath polarization. Phys. Rev. Lett. 101, 047601 (2008).
Belthangady, C. et al. Dressedstate resonant coupling between bright and dark spins in diamond. Phys. Rev. Lett. 110, 157601 (2013).
Laraoui, A. & Meriles, C. A. Approach to dark spin cooling in a diamond nanocrystal. ACS Nano 7, 3403–3410 (2013).
Bruin A. Manybody delocalization in a strongly disordered system with longrange interactions: finitesize scaling. Phys. Rev. B 91, 094202 (2015).
Yao, N. Y. et al. Manybody localization with dipoles. Phys. Rev. Lett. 113, 243002 (2014).
Hartmann, S. & Hahn, E. Nuclear double resonance in the rotating frame. Phys. Rev. 128, 2042 (1962).
Cappellaro, P., Ramanathan, C. & Cory, D. G. Dynamics and control of a quasionedimensional spin system. Phys. Rev. A 76, 032317 (2007).
Lukin, D. M., Guidry, M. A. & Vučković, J. Integrated quantum photonics with silicon carbide: challenges and prospects. PRX Quantum 1, 020102 (2020).
Rosskopf, T. et al. Investigation of surface magnetic noise by shallow spins in diamond. Phys. Rev. Lett. 112, 147602 (2014).
Syntek. Products 1: various industrial diamonds. Syntek http://www.syntek.co.jp/en/products/ (2023).
Choi, J. et al. Robust dynamic Hamiltonian engineering of manybody spin systems. Phys. Rev. X 10, 031002 (2020).
Zu, C. et al. Emergent hydrodynamics in a strongly interacting dipolar spin ensemble. Nature 597, 45–50 (2021).
Hall, L. T. et al. Detection of nanoscale electron spin resonance spectra demonstrated using nitrogenvacancy centre probes in diamond. Nat. Commun. 7, 10211 (2016).
Grinolds, M. et al. Subnanometre resolution in threedimensional magnetic resonance imaging of individual dark spins. Nat. Nanotechnol. 9, 279–284 (2014).
Jacques, V. et al. Dynamic polarization of single nuclear spins by optical pumping of nitrogenvacancy color centers in diamond at room temperature. Phys. Rev. Lett. 102, 057403 (2009).
Joos, M., Bluvstein, D., Lyu, Y., Weld, D. M. & Jayich, A. B. Protecting qubit coherence by spectrally engineered driving of the spin environment. npj Quantum Inf. 8, 47 (2022).
Acknowledgements
We gratefully acknowledge the insights of and discussions with M. Aidelsburger, D. Awschalom, B. Dwyer, C. Laumann, J. Moore, E. Urbach and H. Zhou. This work was support by the Center for Novel Pathways to Quantum Coherence in Materials, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences (materials growth, sample characterization and noise spectroscopy), the US Department of Energy (BES grant no. DESC0019241) for driving studies and the Army Research Office through the MURI programme (grant no. W911NF2010136) for theoretical studies, the W. M. Keck foundation, the David and Lucile Packard Foundation and the A. P. Sloan Foundation. E.J.D. acknowledges support from the Miller Institute for Basic Research in Science. S.A.M. acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC) (funding reference no. AID 5167042018) and the NSF Quantum Foundry through QAMASEi programme award DMR1906325. D.B. acknowledges support from the NSF Graduate Research Fellowship Programme (grant DGE1745303) and The Fannie and John Hertz Foundation.
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E.J.D., W.W., T.M, W.S., M.J., Y.L., Z.W. and C.Z. performed the experiments. B.Y., F.M., B.K., D.B. and S.C. developed the theoretical models and methodology. E.J.D., F.M. and W.W. performed the data analysis. S.M. and A.B.J. prepared and provided the diamond substrates. A.B.J and N.Y.Y. supervised the project. E.J.D, B.Y., F.M., C.Z. and N.Y.Y wrote the manuscript, with input from all authors.
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Extended data
Extended Data Fig. 1 Defect densities.
(ab) NV and P1 areal densities. Computed dynamics (dashed lines) for earlytime Ramsey decoherence caused by NVNV interactions (a) and NVP1 interactions (b), as a function of the NV areal density \({n}_{3{{{\rm{D}}}}}^{{{{\rm{NV}}}}} w\) and the P1 areal density \({n}_{3{{{\rm{D}}}}}^{{{{\rm{P}}}}1}w\), respectively. We compare these numerical results against the measured decoherence dynamics obtained via the XY8 sequence (orange points, a) and the DEER sequence [after removing the NV contribution via the red interpolation in (a)] (purple points, b) to obtain the areal density of defects in sample S1. We estimate the areal density of NV centers to be \({n}_{3{{{\rm{D}}}}}^{{{{\rm{NV}}}}} w=19\pm 2\,{{{\rm{ppm}}}}\cdot {{{\rm{nm}}}}\) and the P1 density to be \({n}_{3{{{\rm{D}}}}}^{{{{\rm{P}}}}1} w=85\pm 10\,{{{\rm{ppm}}}}\cdot {{{\rm{nm}}}}\). At late times (grey shaded regions), the noise dynamics approach an incoherent random walk and should not be used to compute the density within this analysis, because the numerics do not include flipflop interactions. (cd) Effect of finitethickness layer. At fixed areal density 19ppm ⋅ nm, choosing different layer widths w does not affect the computed dynamics (dashed curves, c). For higher density P1 centers \({n}_{3{{{\rm{D}}}}}^{{{{\rm{P}}}}1}=85\,{{{\rm{ppm}}}}\cdot {{{\rm{nm}}}}\), the finite thickness of the layer can induce a sizable effect on the DEER decoherence dynamics (purple points, d) at early times. Numerical calculations are plotted as dotted lines.
Extended Data Fig. 2 Relative density extraction for three of the five different P1 groups.
The P1 spectrum is fit to a sum of three Lorentzian curves (dashed line). The relative areas of the three dips are 1 : 0.74 : 0.26, which is consistent with the expected ratio 1 : 0.75 : 0.25.
Extended Data Fig. 3 The measured decoherence profiles of variations on DROID60.
Due to imperfections in our composite microwave pulses, pulse error accumulates coherently in the DROID60 (ref. ^{62}) sequence and a pronounced oscillation is observed (purple points). To avoid such oscillations, we implement sequences that do not require composite pulses, see for example Seqs. A, H, G in Fig. 9 of ref. ^{62}. The data exhibiting the longest coherence time (Seq. H, τ_{p} = 100ns, red points) are also shown in Fig. 4(a) of the main text.
Extended Data Fig. 4 Undriven DEER and spin echo data.
(a) In sample S1, with a clean surface, both the DEER (blue) and spin echo (red) data exhibit a stretch power β = 2/3. (b) After worsening the surface quality, the spin bath becomes noisier and we observe the expected β = 1 stretch power in the echo data (green); in the DEER data (purple) the correlation time τ_{c} increases but the stretch power is unchanged. (c) As in panel (a), the spin echo data (teal) for sample S2, presumably limited by NVNV interactions rather than the bath, exhibit the same stretch power β = 1 as the DEER data. The DEER data in (a, c) are also plotted in Fig. 2(a) of the main text.
Extended Data Fig. 5 Experiment sequence schematic for differential measurement.
The pulses for the laser (green), photodetector readout (blue), and microwaves addressing the NV (orange) and P1 (red) transitions are shown. The differential measurement subtracts the fluorescence obtained from two sequences (before and after the dotted vertical line), which are identical except for an additional πpulse on the NV spins.
Extended Data Fig. 6 Reduced \({\chi }_{{{{\rm{fit}}}}}^{2}\) for fits to the DEER measurements on samples S1 (a) and S2 (b) for four different fit models.
Reduced \({\chi }_{{{{\rm{fit}}}}}^{2}\) for fits to the DEER measurements on samples S1 (a) and S2 (b) for four different fit models.
Extended Data Fig. 7 DEER and spin echo for sample S1 (a) and sample S2 (b), under a fast incoherent drive.
DEER and spin echo for sample S1 (a) and sample S2 (b), under a fast incoherent drive. The Rabi frequencies for panels (a) and (b) are Ω = 2π × 3.45MHz and Ω = 2π × 4.05MHz, respectively. The data obtained for sample S2 plotted in (b) does not exhibit the correct “randomwalk” regime stretch power of 1/2, even though the DEER and spin echo signals overlap at all measured times.
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Davis, E.J., Ye, B., Machado, F. et al. Probing manybody dynamics in a twodimensional dipolar spin ensemble. Nat. Phys. 19, 836–844 (2023). https://doi.org/10.1038/s41567023019445
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DOI: https://doi.org/10.1038/s41567023019445
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