## INTRODUCTION

Electron spin qubits in semiconductors are prominent candidates for building blocks of scalable quantum computers, thanks to a combination of small physical size, long quantum coherence times, and potential manufacturability using industry-standard processes (*1*, *2*). Demonstrating a universal set of quantum gates with fidelities beyond the fault-tolerance threshold remains the main focus of current research in the field.

Microscopically, one of the main sources of decoherence and gate errors is the coupling between the electron spin qubit and the mesoscopic bath of nuclear spins present in the host semiconductor material. The fluctuating polarization of the nuclear spin bath produces an effective magnetic field noise, which results in a random component of the instantaneous value of the qubit resonance frequency. This randomness affects the qubit coherence time, since the precise rate of phase accumulation is no longer accurately known, and the quantum gate fidelities, since the frequency of the classical control fields may be off-resonance with the instantaneous precession frequency of the spin qubit.

Silicon, the material underpinning all of the modern microelectronic industry, also has the key property of having a large natural abundance of the spin-zero nuclear isotope ^{28}Si, whereas only 4.7% ^{29}Si nuclei carry a spin *I* = 1/2. Isotopic enrichment methods compatible with wafer-scale fabrication have provided ^{29}Si concentrations of orders 500 to 800 parts per million (ppm) (*3*, *4*). As was known for decades (*5*), and more recently studied in detail in ensemble experiments on ^{31}P donors in silicon (*6*, *7*), the electron spin coherence (as measured by a Hahn echo) in these enriched ^{28}Si substrates is substantially improved from the values found in natural silicon. There is excellent agreement between measurements of an electron spin Hahn echo decay in donor ensembles and their predicted behavior based on the theoretical description of ^{29}Si nuclear spin dynamics (*8*, *9*).

A series of experiments has been carried out in recent years on single–donor ion–implanted spin qubit devices in enriched silicon with 800-ppm residual ^{29}Si. These experiments have qualitatively confirmed that the enrichment of spin-zero isotopes improves spin coherence times (*10*) and quantum gate fidelities (*11*, *12*). The Hahn echo decay time (see the “Experimental evidence of nuclear freezing by hyperfine interactions” section for a description of the Hahn echo method),

μs in natural silicon (*13*), was shown to increase by a factor of ∼5 in enriched ^{28}Si (*10*). However, the “pure dephasing” time

obtained in a Ramsey experiment went from

${T}_{2}^{*}\approx 55$ns in natural silicon to

${T}_{2}^{*}\approx 270$ μs in 800-ppm material, an improvement of over three orders of magnitude. Notably, this surpasses by nearly two orders of magnitude what would be predicted, assuming an ergodic behavior of the spin polarization of the residual ^{29}Si nuclei.

The expected value of

${T}_{2}^{*}$ for electron spin qubits coupled to a nuclear spin bath can be obtained by the following simple consideration (*14*, *15*). Suppose that, over a sequence of experiments on an electron spin qubit, the system samples with equal probability every possible configuration of the nuclear spins coupled to the qubit. The total electron-nuclear interaction is described by the Hamiltonian

(1)where *A _{j}* is a hyperfine coupling coefficient (dominated by the Fermi contact term) between the electron and the

*j*th nucleus,

is the *z* component of the spin operator for nucleus *j*, and *S ^{z}* is the

*z*component of the spin operator for the electron. In a semiclassical picture, the hyperfine Hamiltonian above can be thought of resulting in a longitudinal magnetic field

*B*

_{HF}coupling to the electron (Overhauser field). This Hamiltonian assumes a large applied magnetic field

*B*

_{0}in the

*z*direction, which allows neglecting the terms that do not conserve Zeeman energy (our experiments use

*B*

_{0}= 1.4 T, corresponding to an electron Larmor frequency of ∼40 GHz, far exceeding the values of

*A*/

_{j}*h*, typically of order megahertz or less). If we could presume a sufficiently long averaging time to ensure that the fluctuations in the nuclear bath eventually lead the system to sample all possible nuclear spin configurations, then the variance of the Overhauser field

*B*

_{HF}would be found as the sum of the variances of equal binomial random variables for each nuclear spin, giving a total variance σ

_{ω}in the Larmor frequency ω

*of the electron spin*

_{e}(2)

As a result, electron coherences would follow a Gaussian decay of the form exp [ − (σ_{ω}*t*)^{2}/2]. These presumptions describe the ergodic limit for

, here notated as

${T}_{2\infty}^{*}$, which takes the value

${T}_{2\infty}^{*}=\sqrt{2}/{\mathrm{\sigma}}_{\mathrm{\omega}}$ according to Eq. 2. The underlying noise process that drives the system to its ergodic limit is often further captured by describing it via spin diffusion resulting from the nuclear dipole-dipole interaction, leading to a local power spectral density (PSD) proportional to 1/ω^{2} (*16*–*18*).

In the case of a ^{31}P donor in silicon, the hyperfine constants for nearby ^{29}Si, *A _{j}*, are known, and tabulated in (

*19*). In a radius of 3.5-nm sphere around the

^{31}P,

*A*has a mean value across all sites of ≈160 kHz, an SD of ≈340 kHz, and a maximum of 6 MHz. An important difference between donors in silicon and paramagnetic impurities in crystals with a simpler band structure is the fact that

_{j}*A*does not vary monotonically with the distance between each

_{j}^{29}Si nucleus and the

^{31}P donor. This is due to the sixfold degenerate conduction band minima of silicon, causing valley interference in the wave function of bound electron states. Nonetheless, we may use these tabulated values and evaluate a large ensemble of random

^{29}Si placements at 800-ppm concentration to find that

is log-normal distributed as 10^{0.4 ± 0.7} (rad/s)^{2}. The probability of a random placement of nuclear spins providing

μs is therefore less than 1 in 50 million. This begs the question as to why observed dephasing times are two orders of magnitude larger than a reasonable value for

${T}_{2\infty}^{*}$. The question is particularly pressing in view of the fact that, also in the case of pure dephasing times (or its inverse, the inhomogenous linewidth), theory and experiments agree very well in the case of spin ensembles (*20*). It is also pressing because the fidelity of two-qubit logic operations, or their control resources for noise compensation, depends more crucially on the pure dephasing time than on the Hahn echo time (*21*).

Understanding the microscopic reason why the single-spin experimental results consistently deviate from the ergodic model is the aim of this work. We present a simple experiment, designed to provide a clear insight into what freezes the dynamics of the ^{29}Si spin bath. The experiment shows the ability to control the dynamics of the nuclear bath and, in turn, approaching the spin dephasing values expected from an ergodic assumption. We have repeated the key parts of the experiment on three nominally identical single-donor devices (labeled devices A, B, and C in the rest of the paper) to verify that, despite the random and uncontrollable placement of the sparse ^{29}Si nuclei around each donor, our observation have general validity.

We then provide a detailed and nearly parameter-free theoretical model that puts our observation in a quantitative framework and reveals important details on the interplay between qubit measurement and nuclear bath dynamics. By using the ^{31}P donor system in 800-ppm enriched silicon, the small (of order 10 within the electron Bohr radius) number of ^{29}Si nuclei involved allows us to treat the problem in a brute-force, numerically accurate way.

Our work provides insights into decoherence processes that could not, even in principle, be obtained from experiments on spin ensembles. By definition, the local Overhauser field averaged over a large spin ensemble will exhibit statistics that reflect the distribution of all possible ^{29}Si nuclear spin configurations (unless some hyperpolarization method is applied). Therefore, only a single-spin experiment can unveil the precise statistics and time scales over which the Overhauser field actually explores the whole range allowed by the nuclear spin concentration. In addition, our theoretical model includes explicitly the backaction of the electron spin measurement on the nuclear bath. This turned out to be a key ingredient to fully describe the qubit and bath dynamics, also unavailable from ensemble magnetic resonance experiments.

## RESULTS

### Experimental evidence of nuclear freezing by hyperfine interactions

The design of our experiment aims at testing the “nuclear freezing” hypothesis, which postulates the existence of a “frozen core” of nuclear spins located within a “spin diffusion barrier” close to the electron spin (*22*, *23*). The dynamics of these nuclear spins is frozen by the presence of electron-nuclear hyperfine interactions that vary strongly at each lattice site, causing the nuclear spins to have markedly different energy splittings. As a consequence, energy-conserving nuclear flip-flop processes are strongly suppressed, since the nuclear dipole-dipole interaction is much weaker than the energy difference caused by the local variation of hyperfine couplings. Experimental evidence for the existence of a diffusion barrier for nuclear spins was given by Wolfe in 1973 (*24*). Key signatures of nuclear freezing effects also appear in combined optical and magnetic resonance studies of single defects (*25*), decoherence in molecular magnets (*26*), and in dynamic nuclear polarization (*27*, *28*).

Here, we aim at providing direct evidence of the impact of nuclear freezing on the dephasing of a single electron spin qubit in silicon, using a novel method where the barrier to nuclear spin diffusion can be forcibly removed.

Beside the novelty of our experimental method, we note a couple features particular to silicon. First, the standard models of “diffusion barriers” should be used with some care when studying electronic spins in silicon. This is because the hyperfine couplings have an oscillatory character as a function of the distance from the impurity. The frozen nuclear spins are not contained within a well-defined radius beyond which spin diffusion is allowed, as indicated in prior studies (*24*–*28*).

Second, in other materials such as GaAs, additional mechanisms exist for freezing the nuclear spin dynamics, caused by the nonzero quadrupole moment of the nuclei (e.g., ^{69}Ga, ^{71}Ga, and ^{75}As), which can couple to strain (*29*) and electric field gradients (*30*) to induce local shifts in individual nuclear energy splittings. Since the *I* = 1/2 ^{29}Si nuclei have zero quadrupole moment, these freezing mechanisms do not exist in silicon. Therefore, our system provides a clean and simple platform in which to study nuclear spin freezing by hyperfine and dipole-dipole effects alone.

The major distinction in the present experiment relative to other systems previously studied is the deliberate use of our ability to ionize single phosphorus donors in gated nanostructures (*31*), allowing us to “unfreeze” nuclear spins on demand (Fig. 1). Figure 2 (A and E) shows the basic schematic of the experimental protocol, in which we perform measurements of

using the Ramsey technique while maintaining the donor in different ionization states between each individual run. First, the donor system is initialized in the ground electron ∣ ↓ 〉 state. The ^{31}P nuclear spin is set in the ∣ ⇑ 〉 state and remains unchanged during the entire duration of the experiment. From here onward, the ^{31}P nucleus will be ignored, since it plays no role other than providing a constant hyperfine frequency shift, i.e., causing the nominal electron spin Larmor frequency to be ω_{e} = γ_{e}*B*_{0} + *A*_{P}/2, where γ_{e}/2π ≈ 28 GHz/T is the electron gyromagnetic ratio, and *A*_{P}/2π ≈ 100 MHz is the hyperfine coupling to the ^{31}P nucleus. Then, a microwave electron spin resonance (ESR) π/2 pulse at frequency ω_{MW} establishes an electron spin coherence of the form

(we can set φ_{0} = 0 as the result of the first pulse). This coherence evolves freely for a time τ; if the electron Larmor frequency had been exactly ω_{MW} during the free evolution time τ, then the initial superposition state

would remain unchanged when observed in the rotating frame of the microwave source. Therefore, a second π/2 pulse would cause the final state of the spin to be simply ∣ ↑ 〉, verifiable by a subsequent single-shot measurement of the electron spin (*32*).

Including now the effect of the bath nuclear spins, described in a simple picture by a time-dependent Overhauser field *B*_{HF}(*t*), the instantaneous electron Larmor frequency becomes ω_{e} + δω_{e}(*t*), where δω_{e}(*t*) = γ_{e}*B*_{HF}(*t*). After the first π/2 pulse at time *t*, and the free evolution time τ, the electron coherence becomes

, where the accumulated phase is

${\mathrm{\varphi}}_{\mathrm{\tau}}={\int}_{t}^{t+\mathrm{\tau}}dt\prime {\mathrm{\delta}\mathrm{\omega}}_{\mathrm{e}}(t\prime )$. This accumulated phase determines the final state after the second π/2 pulse: for example, if ϕ_{τ} = π, then the second pulse has the effect of flipping the electron spin back to ∣ ↓ 〉.

We call a “single run” the following sequence of operations: initialize ∣ ↓ 〉 – π/2 pulse, wait τ, π/2 pulse, and single-shot spin measurement. The probability

${P}_{\uparrow}^{i}$ of measuring ∣ ↑ 〉 at the end of the *i*th run is thus

. We perform 100 inidividual runs at each value of τ to extract the average electron spin-up fraction *P*_{↑}(τ). The Ramsey experiment is completed by repeating the above operations for 100 different values of τ. Choosing the microwave frequency with a deliberate detuning from the Larmor frequency, Δω = ω_{e} − ω_{MW}, causes the appearance of Ramsey fringes due to the phase accumulation ϕ_{τ} = Δω · τ, which is reflected in the oscillatory behavior of

. As long as Δω (and therefore, *B*_{HF}) is constant, however, the amplitude of the Ramsey fringes does not decay with τ. This would be true even in the presence of an a priori unknown, but constant, Overhauser field. What causes Ramsey decay is the time-dependent fluctuations of δω_{te}(*t*) between individual experimental runs. If *B*_{HF}(*t*) changes from run to run, then so does

in each run. When the range of variation of *B*_{HF} causes

to take any possible value between 0 and 1, the averaged Ramsey fringes decay to *P*_{↑} = 0.5. The 1/*e* decay time of the Ramsey fringes as a function of τ, extracted via fit to Gaussian decay, is used as the definition of

.

The key novelty in our protocol is the insertion of a wait time *T*_{wait} before each individual run. We perform two sets of identical Ramsey experiments, but in one set, the wait time *T*_{wait} is spent with the donor in the neutral charge state (Fig. 2A), while in the other set, the donor is kept ionized (Fig. 2E). This protocol intends to test the hypothesis that, while the donor is in the neutral state, its nuclear spin bath is “frozen” by the presence of the electron. The extra wait time is thus unlikely to cause a change in the Overhauser field between runs. In the ionized case, instead, the absence of the electron allows the bath nuclear spins to undergo energy-conserving flip-flop dynamics mediated by their mutual magnetic dipole interaction. Once the electron is reintroduced to take the next run, it is likely to find the bath in a different state from the previous run. This is expected to lead to a drastic reduction in the observed

.

In Fig. 2, we report the individual Ramsey decays obtained by repeating the protocol for three different wait times *T*_{wait} = 100, 300, and 1000 ms. The results qualitatively confirm our hypothesis: With the donor in the neutral state during the wait time, the dephasing time remained constant at around

μs, whereas the donor ionization caused

${T}_{2}^{*}$ to drop to 51, 19, and 17 μs for *T*_{wait} = 100, 300, and 1000 ms, respectively. The above values were obtained on device A; the systematic drop in

with increasing *T*_{wait} in the ionized state was further verified on three different single-donor devices (see below in the section ‘Parameter-free numerical model of nuclear freezing and unfreezing’).

The hypothesis of nuclear freezing by hyperfine interactions can be further cross-checked by replacing the Ramsey experiment with a Hahn echo sequence. The Hahn echo includes a refocusing π pulse between the two π/2 pulses, which has the effect of canceling out the effect of the static randomness of the instantaneous Larmor frequency. This is because the phase accumulated during the first half of the free evolution,

${\mathrm{\varphi}}_{\mathrm{\tau}/2}^{i}$, is unwound during the second half by the application of the π-pulse, resulting in a final state that has always

${\mathrm{\varphi}}_{\mathrm{\tau}}^{i}=0$, unless the phase accumulated between τ/2 and τ is different from the one accumulated between 0 and τ. The decay of the Hahn echo signal thus reveals the presence of random variations of *B*_{HF} within one run. Therefore, introducing a wait time with the donor in the ionized state before the echo sequence should make no difference for the Hahn echo decay. Figure 3 confirms this expectation: The Hahn echo decay time

remains approximately constant (within the experimental errors) around

${T}_{2}^{\text{Hahn}}\approx 1.6$ms, regardless of the length of the ionized-donor wait time introduced before each run.

### Parameter-free numerical model of nuclear freezing and unfreezing

We now address whether the reduction of

${T}_{2}^{*}$ agrees quantitatively with reasonable models for nuclear dipole-dipole dynamics. Since the number of ^{29}Si nuclei within the target core of 3.5-nm radius is typically fewer than 10 at 800-ppm concentration, we are able to calculate exactly the evolution of this small spin bath under magnetic dipole-dipole interactions. To do so, we generate baths of randomly located ^{29}Si nuclei at 800-ppm concentration for a simulated set of single phosphorus impurities. We use the bulk values (*19*) for the hyperfine coupling constants *A _{j}* and closely mimic the timing of the experiment. The nuclear spin bath evolution is then interrupted by a simulated measurement, which is modeled as the sudden appearance of a spin-down electron spin sometime during the load period (chosen from an exponential distribution with a random tunnel-in lifetime at average 2 ms), and the sudden disappearance of the electron during the ionization period. The ionization can result either from a spin-up measurement (

*32*) or from the forcible removal of the electron at the start of the wait period of the next cycle. Millisecond-scale timing details of the pulsing and tunneling have no discernible impact on our simulation results. The important physics in the simulation is the interplay between the hyperfine coupling with the electron and the slow nuclear dipole-dipole interaction, which occurs on much longer time scales. The projected electron spin is logged as a perfect single-shot measurement, and the averaged results are fit to Gaussian decay as a function of τ, just as in the Ramsey experiment. Further details appears in Materials and Methods.

Figure 4 (A to E) shows examples of simulated traces of the time-dependent Overhauser shift, defined as

${\mathrm{\delta}\mathrm{\omega}}_{\mathrm{e}}(t)={\sum}_{j}{A}_{j}\u3008{I}_{j}^{z}(t)\u3009$, which acts as a drifting noise on the electron spin qubit. As a result of this drift, simulated Ramsey experiments show varying

${T}_{2}^{*}$ decay time scales, depending on the wait time *T*_{wait} preceding each run. Example fits to the Ramsey decay for five simulated wait times are shown in Fig. 4 (F to J).

Figure 5A summarizes the values of

${T}_{2}^{*}$ as a function of the wait time with an ionized donor, obtained experimentally in three different devices, alongside the numerical simulations (Fig. 5B) obtained for six different random placements of ^{29}Si nuclei. Similar simulations for many more configurations were also performed, and it was found that these do not significantly contribute to the spread of

values shown. The qualitative and even quantitative trend of

${T}_{2}^{*}$ as a function of *T*_{wait} shows good agreement between theory and model, especially for long *T*_{wait} where the nuclear spin dynamics provide the dominant source of dephasing. This is rather remarkable, considering that the model is essentially parameter free. The only variable in the model, and also in experimental results obtained from different samples, is simply the random placement of the ^{29}Si nuclei.

While our model captures key elements of the frozen-core hypothesis and its experimental test, it omits some factors that can have an impact on the experimental results. Intrinsic factors are the ensemble of nuclear spins outside the inner electron core, while an extrinsic one is, e.g., the magnetic noise arising from the superconducting solenoid that produces the *B*_{0} field (*10*). These omitted factors are likely to be the main contribution to the finite experimental values of

as *T*_{wait} → 0, since the model predicts a completely frozen nuclear bath when the electron is present. Other omitted factors include the effects of anisotropic hyperfine interactions, for which preliminary numerical investigations indicate negligible effect at the high magnetic fields used here, but would have increasing importance at much lower magnetic fields, as recently observed in metal-oxide semiconductor quantum dots in the same ^{28}Si material with 800-ppm ^{29}Si (*33*).

### Hyperfine noise spectrum

The simulation also provides a glimpse into the plausible low-frequency noise spectrum of nuclear Overhauser field that couples to the electron qubit. Figure 6A shows example noise PSDs of the frequency shift δω_{e}(*t*) caused by Overhauser fields, each calculated for five different values of *T*_{wait} for one isotopic configuration. With frequent electron interactions (short *T*_{wait}), the spectrum approaches a 1/ω^{2} power law; in this case, the noise is dominated by a Brownian random walk due to “ionization shock” (*31*, *34*), i.e., the sudden reconfiguration in effective magnetic field seen by each nuclear spin due to the diabatic appearance or disappearance of the neutralizing electron. The forcible removal of the hyperfine-coupled electron is unique to our experiment; however, similar “sudden shock” effects occur also in conventional pulsed-electron paramagnetic resonance, when a fast rotation of the electron spin causes a sudden reorientation of the hyperfine field on the nearby nuclei (*35*).

More notably, when making infrequent measurements (long *T*_{wait}), the spectrum has a reduced slope approaching a 1/ω power law, reminiscent of noise in spin glasses (*36*). Note that these spectra, as shown in Fig. 6A, are not samples of one large, measurement-insensitive spectral density. The amplitude and exponent for the noise vary drastically, depending on how often electron interactions interrupt coherent dipole-dipole evolution, indicating that the relative amount of backaction plays a key role in determining the spectral noise properties for this system. This is a critical feature, as it indicates strong deviations from a classical Markovian model: Since we must interact the electron with the nuclear bath to measure it, and that interaction strongly perturbs the nuclear bath, the noise spectrum depends critically on our measurement of it [we note that measurement backaction effects in spin echo, even for the much larger nuclear ensembles of GaAs, have been well studied (*37*)].

The simplicity and small Hilbert space size of the ^{31}P donor system in a bath of ^{29}Si nuclei at 800-ppm concentration have thus allowed us to provide numerically accurate estimates of the hyperfine noise PSD. However, this brute-force approach would quickly become inapplicable to larger systems or denser nuclear baths. We may therefore ask what classical noise models could serve as a suitable approximation to the nuclear dipolar dynamics. For this, we use the language of filter functions, in which the application of control pulses to the electron spin qubit is likened to applying a filter to an independent noise process that couples to the qubit (*38*, *39*). The precise spectral properties of the filter depend on the sequence and timing of the control pulses. The filter function for a Ramsey experiment is mostly sensitive to the “zero-frequency” component of the noise spectral density, which is often modeled under ergodic, “quasistatic” assumptions as some run-to-run variation in a static frequency offset. However, when we abandon assumptions of ergodicity, we must ask how the process of averaging over many repetitions (for a total duration of several seconds) affects the sensitivity to very low–frequency noise components. A derivation provided in Materials and Methods yields an approximate filter function for *N* averages with wait time *T* between measurements as

(3)

In the presence of an input magnetic field noise with spectral density *S _{B}*(ω), the variance of the magnetic field noise observed in an experiment can be estimated as

(4)

Following (*18*), the simplest noise model we may use is the Ohrenstein-Uhlenbeck process, which likens nuclear drift to Brownian motion with friction. This provides a Gaussian, Markovian, stationary process characterized by a simple exponential autocorrelation function with correlation time τ* _{c}*. Hence, the model for the PSD of the noise, integrating to a total noise power

, would, in this case, be

$${S}_{B}(\mathrm{\omega})=\frac{4{\mathrm{\sigma}}_{0}^{2}}{{\mathrm{\tau}}_{c}}\frac{1}{1+{(\mathrm{\omega}{\mathrm{\tau}}_{C})}^{2}}$$(5)

Inserting this in Eq. 4, the anticipated

${T}_{2}^{*}$ measured as a function of the duration of each run, *T* ≈ *T*_{wait}, becomes

(6)

This functional dependence of

${T}_{2}^{*}$ on *T* ≈ *T*_{wait} may be fit with the experimental data of Fig. 5A. Using *N* = 100 for the number of single runs at each value of τ, fits to that data provide correlation times varying between half a second and half an hour for the three samples. This is consistent with the simulation results in (*18*), which uses cluster-expansion techniques to similarly simulate bath dynamics in six different nuclear configurations, fitting the autocorrelation functions to exponentials and finding a broad range of correlation times.

However, many possible models may fit with the experimental data. A key role of the numerical simulation data is to provide a higher density of data without error sources beyond those expected from the noise model in question, and for the data in Fig. 5B, we find that Eq. 6 is a poor fit. This should not be unexpected for two key reasons: First, the noise source is not independent of measurement, as we have indicated, and second, when measurement is infrequent, the Overhauser noise PSDs of Fig. 6A poorly match Eq. 5, appearing instead to follow a more general 1/ω^{α} power law with low-frequency roll-off.

These observations suggest that the Ohrenstein-Uhlenbeck process may not be an adequate description on the nuclear bath dynamics in a system undergoing infrequent measurement. It may be more appropriate to instead characterize the simulated noise using the empirical function

$${S}_{B}(\mathrm{\omega})=\frac{2\mathrm{\pi}{K}^{2}}{{\mathrm{\omega}}_{0}}{\left[\frac{{\mathrm{\omega}}_{0}}{{\mathrm{\omega}}_{1}}{\text{tan}}^{-1}\right(\frac{{\mathrm{\omega}}_{1}}{\mathrm{\omega}}\left)\right]}^{\mathrm{\alpha}}$$(7)and find an effective amplitude *K*, slope α, and low-frequency roll-off ω_{1} by fitting to simulated PSD functions; examples of which are shown in Fig. 6A. The parameter ω_{0} is taken as 2π rad/s to maintain units of seconds^{−1} for *K*. The fit values of *K* and α for six simulated samples over a range of *T*_{wait} are shown in Fig. 6 (B and C). We note that the low-frequency roll-off ω_{1} occurs in our simulations largely due to the finite number of nuclei considered, leading to a finite lower bound to the dipolar Hamiltonian, and likely does not well inform the case of a real donor in an extended crystal with thousands of weakly coupled ^{29}Si. If we numerically integrate Eq. 7 with Eq. 4, we then find a

varying with *T*_{wait} in good correspondence with the simulated measurements, indicating that treating the underlying Overhauser fluctuations as a classical noise bath with general 1/ω^{α} power law provides a useful model, but noting that *K* and α depend sensitively on how often the nuclei interact with the measuring electron spin.

Of course, the 1/ω^{α} noise from the ^{29}Si nuclear bath cannot persist at frequencies higher than allowed by the dipole-dipole dynamics driving the noise. Already at a kilohertz range, the noise must roll off, and a dynamical decoupling experiment such as Hahn echo should be insensitive to this wait time–dependent drift, as confirmed by the experimental data in Fig. 3. The echo decay at rate

is dominated by residual noise in the kilohertz range. Here, ensemble experiments (*6*) [and supporting theory; (*9*)] show that the intrinsic

for this ^{29}Si concentration is tens of milliseconds. The much shorter

ms reported here and in other single-donor experiments suggests that other sources of noise (*10*, *40*), not accounted for in the present simulations, must be responsible for the observed Hahn echo decay.

### Comparison to other physical systems

Standard models of spin diffusion, lacking nuclear freezing effects, predict an Overhauser field “brown noise” spectrum with α = 2 (*16*–*18*). Conversely, our experiments and simulations unveiled a PSD resembling more a “pink noise” of the form 1/ω^{α}, with α < 2. As seen in Fig. 6C, at *T*_{wait} → 0, the spectrum certainly behaves as an α = 2 power law, but here, it is due to the telegraph noise–like influence of the dephasing from frequent measurement rather than spin diffusion effects. Spin diffusion effects become more important as the nuclear bath is unfrozen, i.e., with increasing *T*_{wait}, for which the exponent α decreases. The resulting pink noise for mostly empty donors is consistent with the direct observation of 1/ω nuclear noise in isotopically enriched Si/SiGe quantum dots (*41*). There, the dots are empty or contain two hyperfine-inactive singlet electrons during most of the experiment, allowing free nuclear dipole-dipole interaction for the duration of the experiment, and

times are close to the ergodic limit. Spectral drift due to nuclear spins in isotopically enriched metal-oxide-semiconductor (MOS) dots also shows 1/ω^{α} spectral characteristics (*21*), suggesting that this is a common behavior in silicon, where the nuclear spins appear in a dilute, disordered lattice, in contrast to widely studied dense nuclear spin crystals such as GaAs or CaF_{2}. Theoretical validation of 1/ω dipolar dynamics in larger quantum dot systems cannot be achieved with the brute-force numerical methods adopted here and will require more advanced simulation techniques such as coupled cluster expansions (*18*).

Another interesting open question is whether, in general, the whitening of the Overhauser dipolar noise corresponds to a change in the resonance line shape (*16*). It is well known that a dense ensemble of dipolarly coupled spins results in a Gaussian line shape, which turns into a Lorentzian if the spins become sparse upon spatial dilution (*20*). The same effect can be found as a function of temperature, since the spin excitations become more sparse when the thermal energy is lower than the energy splitting of the individual spins (*42*). Here, we have only considered the second moment of the resonance line (the Fourier transform of the Ramsey decay) via the noise spectral density, and we have assumed a Gaussian decay, which is not necessarily the best possible fit to all experimental or numerical data. Since our concern in this study is the lowest-order autocorrelation of the noise, these higher-order line shape effects fall outside the scope of the present study, but we envisage that this line of investigation might in the future shed light on the general relationship between noise spectra and line shapes.

A 1/ω noise spectrum has been observed in multiple recent experiments on isotopically enriched silicon devices, each including deliberate magnetic field gradients generated by locally fabricated micromagnets (*43*, *44*). As these studies verify, in these cases, the 1/ω noise results from device electric-field noise transduced into magnetic-field noise due to the strong gradient. Of course, there are certainly nuclear freezing effects in these experiments due to not only the hyperfine field of the electrons at play but also the micromagnetic gradients. Because of these additional complications, the present results are difficult to apply to these systems, requiring further simulation work incorporating classical, macroscopic field gradients.

## DISCUSSION

The experiments and theoretical analysis presented here provide conclusive evidence that the presence of a hyperfine-coupled electron can markedly slow down the dynamics of a bath of nuclear spins. This observation can have important consequences on the operation of a spin-based quantum computer in semiconductors, especially in the context of achieving quantum gate fidelities, surpassing some fault-tolerance threshold.

At finite isotopic purification, high gate fidelities in the presence of fluctuating Overhauser fields will quite certainly demand the use of dynamically corrected gates and/or frequent recalibration of the qubit frequency. Examples already exist, where dynamical recalibration and compensation sequences have substantially extended

${T}_{2}^{*}$ and reduced gate errors in quantum dot systems, both in GaAs (*45*, *46*) and in isotopically purified silicon (*21*, *47*). While these methods are known to be effective at the one- or two-qubit level, the operation of a large-scale spin-based quantum processor is likely to benefit from the ability to achieve high-fidelity gates without the need for frequency recalibration.

In this context, our work provides a simple recommendation: Wherever possible, introduce a hyperfine-coupled electron to freeze out the nuclear bath. This may be especially pertinent in singlet-triplet qubit systems using isotopically enriched Si, where the frequent presence of empty dots or hyperfine-inactive singlets lead to

${T}_{2}^{*}$ at the ergodic limit and introduces associated limits to control fidelity (*48*). The experiments and models presented here suggest that a redesign of the experimental protocols in these systems to maximize the time each dot spends containing an odd number of hyperfine-coupled electrons may improve overall fidelity and reduce the need for frequent recalibration. Hence, we believe that our explorations into the very low–frequency character of nuclear dipole dynamics in single phosphorus impurities may assist the engineering of high-fidelity quantum logic gates in a broad class of semiconductor-based future quantum processors.

More broadly, our observations of controllable freezing of a nuclear spin bath may, in the future, provide an interesting platform in which to study many-body localization and thermalization in disordered systems with random interactions (*49*).

## MATERIALS AND METHODS

### Experimental methods

Details on the fabrication and operation of our implanted single-donor devices may be found in (*10*). The devices were mounted on the mixing chamber of a dilution refrigerator with a typical electron temperature of 100 mK and subjected to a static magnetic field *B ^{z}* ≈ 1.4 T created by a superconducting solenoid. Coherent electron spin control was achieved by irradiating the spin with oscillating magnetic fields at microwave frequencies (typically around 40 GHz) delivered by an on-chip coplanar waveguide terminated by a short circuit (

*50*). The electron spin state at the end of each experimental run was measured in a single shot using the well-established spin-dependent tunneling method. This is achieved by tunnel coupling the donor to a cold charge reservoir and detecting the spin-dependent ionization of the donor with a single-electron transistor (SET). The SET is fabricated in proximity of the ion-implantation region (

*32*), where a small number of

^{31}P donors was implanted with energies of 10 (device A) or 14 keV (devices B and C) per ion. In our devices, the cold charge reservoir is the SET island itself.

### Ramsey data fitting

Although the Ramsey experiment is a common procedure, the extracted values of

${T}_{2}^{*}$can depend not only on averaging times and wait times but also on the method chosen for curve fitting. For both the experiment and the theory, finite averaging effects result in notable variation in

${T}_{2}^{*}$, depending on how the decaying sinusoidal Ramsey fringes are fit. One option for fitting is to attempt a direct nonlinear fit to the decay function

${P}_{\uparrow}(\mathrm{\tau})={C}_{0}+{C}_{1}\text{cos}(\mathrm{\Delta}\mathrm{\omega}\xb7\mathrm{\tau})\text{exp}[-{(\mathrm{\tau}/{T}_{2}^{*})}^{2}]$, as used in Fig. 2 (A to C and J to K). However, for large variations of the average detuning δω_{e}(*t*), this fit may easily fail. A more robust option when focusing on

is to ignore the oscillating component of the Ramsey fringes and fit instead the envelope of the observed decay. We do so, both for experiments and simulations, by taking multiple traces of the Ramsey oscillation curve and finding a least-squares fit between the data at each delay τ and a normal distribution with SD σ(τ), constrained so that σ(τ) follows an offset Gaussian decay

$\mathrm{\sigma}(\mathrm{\tau})={\mathrm{\sigma}}_{0}+{\mathrm{\sigma}}_{1}\text{exp}[-{(\mathrm{\tau}/{T}_{2}^{*})}^{2}]$. In Figs. 2 (E to H) and 4 (F to J), the fit outcomes σ(τ) and 2σ(τ) are shown, shaded according to a confidence interval using the 1 − σ SE, resulting from the Levenberg-Marquardt nonlinear least-squares algorithm. This fitting procedure is also insensitive to errors due to imperfect initialization and measurement visibility. Notably, the parameter

${T}_{2}^{*}$in this and in most other studies tracks the decay of an averaged sinusoid but discards the information of the sinusoid’s center frequency. This results in an effective insensitivity to the lowest-frequency drifts in the system and hence, provides, in conjunction with the averaging time over the full experiment, the effective low-frequency cutoff of the noise spectrum to which the experiment is sensitive.

### Numerical simulations

The complete Hamiltonian of the simulated model is

$$H={H}_{\text{dipolar}}+n(t){H}_{\text{HF}}-n(t)g{\mathrm{\mu}}_{B}{B}^{z}{S}^{z}-{\displaystyle \sum _{k}}\mathrm{\hslash}\mathrm{\gamma}{B}^{z}{I}_{k}^{z}$$(8)

The latter two terms here are the electron and nuclear Zeeman energies, which we may subsume into a rotating frame. The secular term of the nuclear-nuclear dipolar interaction in this rotating frame (*16*) is then

(9)where **I*** _{k}* is the spin operator for the

*k*th spin-1/2

^{29}Si nucleus, γ/(2π) = − 8.5 MHz/T is the

^{29}Si gyromagnetic ratio, θ

*is the angle between the vector, connecting nuclear spins*

_{jk}*j*and

*k*and the applied magnetic field

*B*, and

^{z}*r*is the distance between those spins. The coupling constants here range from a few millihertz to about a hertz. The secular hyperfine interaction

_{jk}*H*

_{HF}was discussed in Introduction as Eq. 1. The crucial component of the model is to account for the donor charge via the variable

*n*(

*t*). Upon donor ionization,

*n*(

*t*) = 0 removes the hyperfine and electron Zeeman terms from the Hamiltonian.

We simulate the experiment in which a Ramsey sequence is performed, but between each single run, the donor electrochemical potential is chosen to ensure that it is ionized [*n*(*t*) = 0] for a time *T*_{wait}. During this time, the nuclear spins undergo free evolution according to *H*_{dipolar}. Then, a ∣ ↓ 〉 electron is loaded [*n*(*t*) = 1] to begin the run. Each simulated Ramsey measurement subroutine includes the following components:

(i) The calculation of the Overhauser shift extracts

${\mathrm{\delta}\mathrm{\omega}}_{\mathrm{e}}={\sum}_{k}{A}_{k}\u3008\mathrm{\psi}\mid {I}_{k}^{z}\mid \mathrm{\psi}\u3009$directly from the simulation data.

(ii) The electron spin measurement probability distribution for a Ramsey experiment is calculated as

$${P}_{\uparrow ,\downarrow}=\frac{1}{2}\pm \frac{1}{2}\text{cos}\{[\mathrm{\Delta}{\mathrm{\omega}}_{\mathrm{e}}+{\mathrm{\delta}\mathrm{\omega}}_{\mathrm{e}}]\mathrm{\tau}\}$$(10)where *P*_{↑} (corresponding to +) and *P*_{↓} (corresponding to −) are the spin-up and spin-down probabilities, respectively. The fixed detuning Δω_{e} includes a deliberate offset from the Larmor frequency, plus any initial Overhauser shift from the start of the simulation. Using sufficient Δω_{e} to see fringe contrast assists curve fitting, but we find that the precise value of it has no discernible impact on our results.

(iii) The simulated single-shot readout result is randomly chosen by drawing a binomial random variable according to *P*_{↑, ↓}; we notate the random result as *m*, which takes values +1/2 for spin up and −1/2 for spin down.

(iv) To determine a dephasing operator corresponding to initialization and measurement, we note that the principal source of backaction randomness is not the measurement itself but rather the uncertain duration of time the electron occupies the donor between tunnel events to and from the reservoir. The essence of the randomness is to assume that during the interval in which an electron is initialized, *t*_{init} = 100 ms, the actual time it takes for the electron to tunnel in is exponentially distributed with time scale *t*_{tunnel}, which we estimate as 2 ms, and similarly, for measurement. The corresponding

variance results in a corresponding hyperfine phase variance (*A _{j}t*

_{tunnel})

^{2}on nucleus

*j*. Hence, the simulated measurement and reinitialization process proceeds as follows: The simulation draws a random tunnel-in time

*t*

_{1}for an electron to occupy the donor from the cumulative distribution function (CDF) 1 − exp ( −

*t*

_{1}/

*t*

_{tunnel}). The duration of the two ESR π/2 pulses and the Ramsey free evolution is treated as negligible for the nuclear dynamics. Once loaded, the electron remains on the donor for a time

*t*

_{2}, which may be one of two choices, depending on

*m*. If

*m*= 1/2, then

*t*

_{2}is a tunneling time drawn from the CDF 1 − exp ( −

*t*

_{2}/

*t*

_{tunnel}).

If *m* = − 1/2, then *t*_{2} is the sum of the 100-ms read time and a random tunnel-out time, similarly sampled. The backaction on the nuclear spins for this process is then captured by applying the dephasing operator

(11)where *t*_{1}, *t*_{2}, and *m* are random numbers from shot to shot, hence causing phase scrambling within the nuclear ensemble. We have verified that our results do not depend critically on the *m* dependence of the model; these spin-projection details are in place to mimic the experiment, but a simpler model that only captures a random electron interaction time could be used to obtain similar results, possibly allowing simpler analytic approximations.

This simulated measurement subroutine is repeated *N*_{meas} times. It is embedded into the full nuclear simulation as follows:

1) Randomly populate a sphere of unstrained silicon lattice sites about 3.5 nm around a donor with random ^{29}Si nuclei at 800-ppm density.

2) Choose a random nuclear spin state ∣ψ〉 via coin flipping.

3) Choose a *T*_{wait} value and calculate

(12)

4) For *N*_{τ} × *N*_{meas} cycles, incrementing the value of τ every *N*_{meas} cycles,

(i) Apply *U*_{evolve} to ∣ψ〉.

(ii) Simulate the Ramsey experiment, as described above.

(iii) Record the average of those *N*_{meas} single-shot measurements.

5) Curve fit the averaged measurements at each τ to a Gaussian decay to extract

${T}_{2}^{*}$.

This whole process is then repeated *N*_{ens} different times and for an array of different wait times *T*_{wait}. The independent *N*_{ens} ensemble members use the same configuration of nuclei, but with different initial nuclear spin states (and independent statistics for simulating single-shot measurements). We note that *N*_{meas} and *N*_{τ} dictate the amount of time averaging affecting the observed

and so are chosen to mimic the experiment in the simulations presented here. In contrast, *N*_{ens} ensemble members “reset” the clock and fully scramble nuclei. Parallelized simulations use *N*_{ens} = 20, while serialized experiments use *N*_{ens} = 2.

### Filter function across many averages

In this section, we consider the general problem of measuring the frequency of a signal when that frequency drifts substantially over a time scale much slower than the duration of a single measurement but faster than the overall time it takes to make an ensemble of measurements. We approach the problem using a filter-function formalism. While this formalism is well known, a formal treatment of handling noise correlations from measurement to measurement, as opposed to noise correlations during a single pulse sequence, is rarely considered.

Consider a zero-mean angular frequency shift *x* (e.g., an Overhauser shift), which we seek to measure via a Ramsey-like measurement, meaning that an ensemble of measurements in which a probe (e.g., an electron spin) precesses at drifting frequency ω_{0} + *x*(*t*) for a varying time τ, generating a single-shot measurement probability

(13)

Here, *A* and *B* are irrelevant linear scale parameters, capturing measurement details not presently of interest. If our random frequency shift *x*(*t*) was stationary and ergodic, then after “sufficient” averaging, we would ignore the absolute time *t* and assume that after averaging some *N* ≫ 1 measurements at different times *t _{n}*

(14)where *S _{x}*(ω) is the noise spectral density of

*x*(

*t*), and the sampled mean value of

*x*(

*t*) is

(15)

The integral in the decay in 14 exhibits the standard filter function, sinc^{2}(ωτ/2), for a Ramsey-type frequency measurement. This filter is sharply peaked at ω = 0, suggesting that this measurement is sensitive to the “noise at DC,” which is poorly defined, and can place excessive reliance on a poorly known or possibly nonexistent low-frequency roll-off of *S _{x}*(ω).

In a slow-drift scenario, *x _{n}* cannot be considered ergodic. To make a more appropriate filter in this case, we focus on the notion that each individual measurement queries

*x*(

*t*) at some absolute time

*t*for sampling instance

_{n}*n*; abbreviate

*x*(

*t*) =

_{n}*x*. We essentially assume that

_{n}*x*is constant but random for the duration of a Ramsey experiment, similar to the quasistatic limit, but we further allow that

_{n}*x*is not fully randomized from run to run. Instead, it undergoes drifts according to a random stationary process defined by the very low–frequency part of

_{n}*S*(ω). The key insight is that the extraction of the mean,

_{x}according to 15, provides the low-frequency cutoff to the filter function. Our averaging process measures, under an assumption of Gaussian noise

$$\u3008\frac{1}{N}{\displaystyle \sum _{n=1}^{N}}\text{cos}[({\mathrm{\omega}}_{0}+{x}_{n})\mathrm{\tau}]\u3009=\text{cos}[(\mathrm{\omega}+\overline{x})\mathrm{\tau}]{e}^{-\u3008{y}_{n}^{2}\u3009{\mathrm{\tau}}^{2}/2}$$(16)where

$${y}_{n}={x}_{n}-\stackrel{\u0304}{x}$$(17)

Our critical assumption is that although *x _{n}* is not ergodic, the deviation

*y*from the randomly sampled mean

_{n} may be considered as such. This means that the ensemble-averaged variance of *y _{n}* should be equal to its time-averaged variance over each of the

*n*measurements, i.e.

(18)

The result of this assumption is that the low-frequency components of the fluctuations of *x*(*t*), which prevent ergodicity in a finite time–averaged experiment, are “absorbed” into the random mean

. The information of

$\stackrel{\u0304}{x}$ is, in principle, observable as a mean shift relative to ω_{0}, but since the true value of ω_{0} may be unknown (or irrelevant), the information of the sampled

is lost (or ignored), and the experimentally extracted

${T}_{2}^{*}$may be taken as

$\sqrt{2/\u3008{y}_{n}^{2}\u3009}$.

We may now cast this simple characterization as a filter function by introducing again the noise spectral density of *x*(*t*) via the Wiener-Khinchin theorem and the assumption that the time between sampling *x _{n}* and

*x*is

_{m}*T*× (

*m*−

*n*) for single-measurement time

*T*, giving the variance

(19)where

$$F(\mathrm{\omega};N,T)=\frac{(2N-1)\text{sin}(\mathrm{\omega}T/2)-\text{sin}[(2N-1)\mathrm{\omega}T/2]}{2N\text{sin}(\mathrm{\omega}T/2)}$$(20)

We have, hence, derived a “filter function” *F*(ω; *N*, *T*) for the long-time diffusion in this experiment. Usually, we choose *N* ≫ 1, in which case it simplifies to Eq. 3 of the main text. This equation effectively describes a high-pass filter, with the stop band at frequencies below 1/*NT*. This may be added to a decay function using the standard τ-dependent Ramsey filter function sinc^{2}(ωτ/2) but with an appropriately chosen low-frequency cutoff for the noise.

**Acknowledgments: **We acknowledge helpful conversations with M. Grace and W. Witzel. **Funding:** The research was funded by the Australian Research Council Centre of Excellence for Quantum Computation and Communication Technology (grant no. CE170100012) and the U.S. Army Research Office (contract nos. W911NF-17-1-0200 and W911NF-17-1-0198). We acknowledge support from the Australian National Fabrication Facility (ANFF) and the AFAiiR node of the NCRIS Heavy Ion Capability for access to ion-implantation facilities. K.M.I. acknowledges support from a Grant-in-Aid for Scientific Research by MEXT. T.D.L. acknowledges support from the Gordon Godfrey Bequest Sabbatical grant. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the ARO or the U.S. Government. **Author contributions:** M.T.M. and F.E.H. fabricated the devices, with A.M.’s and A.S.D.’s supervision. A.M.J., B.C.J., D.N.J., and J.C.M. designed and performed the ion implantation. K.M.I. supplied the isotopically enriched ^{28}Si wafer. A.M. and A.L. designed the nuclear freezing experiment. M.T.M. and A.L. performed the measurements and analyzed the data. T.D.L. designed and performed the theoretical modeling. M.T.M., T.D.L., and A.M. wrote the manuscript, with input from all coauthors. **Competing interests:** All authors declare that they have no competing interests. **Data and materials availability:** All data needed to evaluate the conclusions in the paper are present in the paper. Additional data related to this paper may be requested from the authors.