Few-second-long correlation times in a quantum dot nuclear spin bath probed by frequency-comb nuclear magnetic resonance spectroscopy

Abstract

One of the key challenges in spectroscopy is the inhomogeneous broadening that masks the homogeneous spectral lineshape and the underlying coherent dynamics. Techniques such as four-wave mixing and spectral hole-burning are used in optical spectroscopy^1,2,3, and spin-echo⁴ in nuclear magnetic resonance (NMR). However, the high-power pulses used in spin-echo and other sequences^4,5,6,7,8 often create spurious dynamics^7,8 obscuring the subtle spin correlations important for quantum technologies^{5,6,9,10,11,12,13,14,15,16,17}. Here we develop NMR techniques to probe the correlation times of the fluctuations in a nuclear spin bath of individual quantum dots, using frequency-comb excitation, allowing for the homogeneous NMR lineshapes to be measured without high-power pulses. We find nuclear spin correlation times exceeding one second in self-assembled InGaAs quantum dots—four orders of magnitude longer than in strain-free III–V semiconductors. This observed freezing of the nuclear spin fluctuations suggests ways of designing quantum dot spin qubits with a well-understood, highly stable nuclear spin bath.

Main

Pulsed magnetic resonance is a diverse toolkit with applications in chemistry, biology and physics. In quantum information applications, solid state spin qubits are of great interest and are often described by the so-called central spin model, where the qubit (central spin) is coupled to a fluctuating spin bath (typically interacting nuclear spins). Here microwave and radiofrequency (rf) magnetic resonance pulses are used for the initialization and readout of a qubit¹⁸, dynamical decoupling⁵ and dynamic control⁶ of the spin bath.

However, the most important parameter controlling the central spin coherence^9,11,19—the correlation time τ_c of the spin bath fluctuations—is very difficult to measure directly. The value of τ_c is determined by the spin exchange (flip-flops) of the interacting nuclear bath spins. By contrast, pulsed NMR reveals the spin bath coherence time T₂, which characterizes the dynamics of the transverse nuclear magnetization^7,8,20 and is much shorter than τ_c. The problem is further exacerbated in self-assembled quantum dots, where quadrupolar effects lead to inhomogeneous NMR broadening exceeding 10 MHz (refs 21,22), so that pulsed NMR requires practically unattainable rf field amplitudes exceeding 1 T.

Here we develop an alternative approach to NMR spectroscopy: we measure non-coherent depolarization of nuclear spins under weak noise-like rf fields. Contrary to intuitive expectation, we show that such measurement can reveal the full homogeneous NMR lineshape describing the coherent spin dynamics. This is achieved by employing rf excitation with a frequency-comb profile (widely used in precision optical metrology²³). We then exploit non-resonant nuclear–nuclear interactions: the homogeneous NMR lineshape of one isotope measured with frequency-comb NMR is used as a sensitive non-invasive probe of the correlation times τ_c of the nuclear flip-flops of the other isotope. Although initial studies^9,17,19 suggested τ_c ∼ 100 μs for nuclear spins in III–V semiconductors, it was recently recognized^20,24,25 that quadrupolar effects may have a significant impact in self-assembled quantum dots. Here, for the first time, we obtain a quantitative measurement of extremely long τ_c ≳ 1 s, revealing strong freezing of the nuclear spin bath due to the inhomogeneous strain—a crucial advantage for quantum information applications of self-assembled quantum dots.

The experiments were performed on individual neutral self-assembled InGaAs/GaAs quantum dots at magnetic field B_z = 8 T. All measurements of the nuclear spin depolarization dynamics employ the pump-depolarize-probe protocol shown in Fig. 1a. Here we exploit the hyperfine interaction of the nuclei with the optically excited electron^17,21,22 both to polarize the nuclei (pump pulse) and to measure the nuclear spin polarization in terms of the Overhauser shift ΔE_hf in the quantum dot photoluminescence spectrum (probe pulse). The rf magnetic field depolarizing nuclear spins is induced by a small copper coil. (Further experimental details can be found in Methods and Supplementary Note 1.)

Figure 1: Frequency-comb technique for homogeneous NMR lineshape measurement.

a, Timing diagram of the experimental cycle consisting of nuclear spin optical pumping (duration t_pump), frequency-comb rf excitation (t_rf), and optical probing of the nuclear spin state (t_probe). b, The green line shows schematically a quantum dot NMR spectrum consisting of a central transition (CT) peak and two satellite transition (ST) bands. The inhomogeneous lineshape (width Δν_inh) is the sum of a large number (10⁴–10⁵) of nuclear spin transitions with homogeneous linewidths Δν_hom (shown with red lines). The black line shows the spectrum of the frequency-comb excitation with mode spacing f_MS and total width Δν_comb exceeding Δν_inh. c,d, Demonstrate how experiments with varying f_MS can reveal the width Δν_hom of the NMR homogeneous lineshape (red lines). When Δν_hom > f_MS (c), all individual nuclei are uniformly excited by the frequency comb (shown with black lines). In the opposite case Δν_hom < f_MS (d), some of the nuclei (dashed line) are not excited, resulting in a slow-down of nuclear spin dynamics. The transition between the two cases takes place when f_MS ∼ Δν_hom, allowing Δν_hom to be measured.

All isotopes in the studied dots possess non-zero quadrupolar moments. Here we focus on the spin I = 3/2 nuclei ⁷¹Ga and ⁷⁵As. The strain-induced quadrupolar shifts result in an inhomogeneously broadened NMR spectrum^21,22, as shown schematically by the green line in Fig. 1b. The spectrum consists of a central transition (CT) −1/2 ↔ +1/2 and two satellite transition (ST) ±1/2 ↔ ±3/2 peaks. All nuclei in the dot are coupled by dipole–dipole interactions: an exact description of such a system is a very complex many-body quantum mechanical problem. Here we use a simplified semiclassical description where nuclear–nuclear interactions are taken into account by introducing a phenomenological homogeneous broadening of each nuclear spin transition. Thus, the NMR spectrum in Fig. 1b is an inhomogeneous distribution (with width Δν_inh) of individual nuclear spin transitions (shown by the red lines) with much smaller homogeneous linewidth Δν_hom.

Radiofrequency excitation is considered to be weak when nuclear spin precession is slower than transverse relaxation, resulting in non-coherent depolarization (that is without Rabi oscillations) of the nuclei²⁶. To make such depolarization sensitive to the homogeneous NMR lineshape, we use rf excitation with a frequency-comb spectral profile. As shown in Fig. 1b (black line), the frequency comb has a mode spacing of f_MS and a total comb width Δν_comb exceeding Δν_inh. The key idea of the frequency-comb technique is described in Fig. 1c, d, where two possible cases are shown. If the spacing is small (f_MS ≪ Δν_hom, Fig. 1c) all nuclear transitions are excited by a large number of rf modes. As a result, all nuclear spins are depolarized at the same rate and we expect an exponential decay of the total nuclear spin polarization. In the opposite case of large mode separation (f_MS > Δν_hom, Fig. 1d), some of the nuclear transitions are out of resonance and are not excited (for example, the one shown by the dashed red line). As a result we expect a slowed-down non-exponential nuclear depolarization. The experiments are performed at different f_MS; the f_MS for which a slow-down in depolarization is observed gives a measure of the homogeneous linewidth Δν_hom.

Experimental demonstration of this technique is shown in Fig. 2a. The Overhauser shift variation ΔE_hf of ⁷¹Ga is shown as a function of the depolarizing rf pulse duration t_rf for different f_MS. For small values of f_MS = 80 and 435 Hz an exponential depolarization is observed. However, when f_MS is increased the depolarization becomes non-exponential and slows down significantly. The detailed dependence ΔE_hf(t_rf, f_MS) is shown as a colour-coded plot in Fig. 2b. The threshold value of f_MS (marked with a white arrow) above which the nuclear spin dynamics becomes sensitive to the discrete structure of the frequency comb provides an estimate of Δν_hom ∼ 450 Hz. Such a small homogeneous linewidth is detected in NMR resonances with inhomogeneous broadening of Δν_inh ∼ 6 MHz (ref. 21), demonstrating the resolution power of frequency-comb non-coherent spectroscopy.

Figure 2: Measurement of the homogeneous NMR lineshape in self-assembled quantum dots using frequency-comb excitation.

a, The change in the polarization of the ⁷¹Ga nuclear spins (measured in terms of the change in the Overhauser shift ΔE_hf) is shown as a function of the rf pulse duration t_rf (symbols) at B_z = 8 T and different mode spacings f_MS. Lines show model fitting (see text). b, A full 2D plot of ΔE_hf as a function of t_rf and f_MS in the same experiment as in a. There is a clear slow-down of the nuclear spin depolarization at f_MS > 450 Hz (shown with a white arrow), providing an estimate of the homogeneous linewidth of Δν_hom.

The information revealed by frequency-comb spectroscopy is not limited to rough estimates. An accurate determination of the linewidth and the shape of the tails of the full homogeneous lineshape is achieved with spin dynamics modelling based on rate equations (see details in Methods and Supplementary Note 2). We use the following two-parameter phenomenological model for the homogeneous lineshape:

where Δν_hom is the homogeneous full-width at half-maximum and k is a roll-off parameter that controls the tails (the behaviour of L(ν) at large ν). For k = 1 the lineshape corresponds to Lorentzian, whereas for k → ∞ it tends to Gaussian: in this way equation (1) seamlessly describes the two most common lineshapes. Using Δν_hom and k as parameters we calculate the model ΔE_hf(t_rf, f_MS) dependence and fit it to the experimental ΔE_hf(t_rf, f_MS) to find a phenomenological description of the homogeneous NMR lineshape in self-assembled quantum dots.

The solid line in Fig. 3a shows the best-fit lineshape (Δν_hom ≍ 221 Hz and k ≍ 1.67) for the measurement shown in Fig. 2a, b. The dashed and dashed-dotted lines in Fig. 3a show for comparison the Lorentzian (k = 1) and Gaussian (k → ∞) lineshapes with the same Δν_hom. The difference in the lineshape tails is seen clearly in Fig. 3b, where a logarithmic scale is used. The model ΔE_hf(t_rf, f_MS) dependence calculated with the best-fit parameters is shown in Fig. 3c and with lines in Fig. 2a—there is excellent agreement with experiment. By contrast, modelling ΔE_hf(t_rf, f_MS) with Lorentzian (Fig. 3d) and Gaussian (Fig. 3e) lineshapes shows a pronounced deviation from the experiment, demonstrating the excellent sensitivity of the frequency-comb spectroscopy to the tails of the homogeneous spectral lineshape.

Figure 3: Homogeneous lineshape modelling.

a,b, Model homogeneous NMR lineshapes of ⁷¹Ga nuclei shown on linear (a) and logarithmic (b) scales. Solid lines show the best-fit lineshape with a full-width at half-maximum Δν_hom ≍ 221 Hz and a roll-off parameter k ≍ 1.67. Dashed and dashed-dotted lines show for comparison Lorentzian and Gaussian lineshapes with the same Δν_hom. c–e, The calculated ΔE_hf(t_rf, f_MS) dependences for the lineshapes in a,b. For the fitted lineshape (c) an excellent agreement with the experiment in Fig. 2b is found, whereas calculations with Lorentzian (d) and Gaussian (e) lineshapes give markedly different results, demonstrating the sensitivity of the frequency-comb technique.

We have also performed frequency-comb NMR spectroscopy on ⁷⁵As nuclei (Fig. 4a). Despite their larger inhomogeneous broadening, Δν_inh ∼ 18 MHz, the model fitting reveals even smaller Δν_hom ≍ 117 Hz and k ≍ 1.78. From the Δν_hom derived here we can estimate the nuclear spin coherence times T₂ ≍ 1/(πΔν_hom) ∼ 1.4 and 2.7 ms for ⁷¹Ga and ⁷⁵As, respectively. These values are in good agreement with the corresponding spin-echo T₂ ≍ 1.2 and 4.3 ms derived in ref. 20. However, spin-echo could be measured only on the central transitions, for which Δν_inh is relatively small. Moreover, pulsed NMR does not allow determination of the full homogeneous lineshape, which for dipole–dipole interactions typically has a ‘top-hat’-like (Gaussian) profile²⁷. And, most importantly, owing to parasitic effects such as ‘instantaneous diffusion’⁷ and spin locking⁸, pulsed NMR does not reveal the characteristic correlation time τ_c of the spin exchange flip-flops. In particular, owing to the ‘instantaneous diffusion’, the slow flip-flop dynamics become obscured by the much faster and trivial spin dynamics arising from a combined effect of secular spin–spin interactions and strong rf pulses⁷.

Figure 4: Probing the correlation times of the nuclear spin bath fluctuation.

a,b, Frequency-comb measurement on ⁷⁵As nuclei at B_z = 8 T without (a) and with (b) additional frequency-comb excitation ‘heating’ the ⁷¹Ga nuclei. The increase of Δν_hom of ⁷⁵As under the ⁷¹Ga ‘heating’ reveals the strongly suppressed nuclear spin fluctuations of ⁷¹Ga. c,d, Symbols show rf-induced depolarization time τ_As of ⁷⁵As nuclei at fixed f_MS = 903 Hz and four different amplitudes of the ⁷⁵As comb β_{1, As} as a function of the amplitude β_{1, Ga} of the additional rf excitation ‘heating’ either the central transition (CT, c) or the satellite transition (ST, d) of ⁷¹Ga. Error bars show 95% confidence intervals. The top scales show β_{1, Ga} expressed in terms of the rf-induced spin-flip time τ_Ga of ⁷¹Ga. The lines show model fitting from which we deduce the ⁷¹Ga nuclear spin correlation times τ_{c, CT} ≍ 1.0 ± 0.2 s and τ_{c, ST} ≍ 6.2 ± 2.5 s for the CT and ST transitions.

As we now show, the non-Gaussian lineshapes can be understood and τ_c can be derived using experiments with two frequency combs exciting nuclei of two isotopes (⁷⁵As and ⁷¹Ga). We again excite ⁷⁵As nuclei with a frequency comb to measure their homogeneous lineshape. The difference is that now we simultaneously apply a second comb exciting the ⁷¹Ga spins. Importantly, in this experiment the ⁷¹Ga nuclei are first fully depolarized after the optical nuclear spin pumping and before the two-comb excitation (see Supplementary Fig. 3)—in this way the excitation of ⁷¹Ga has no direct effect on the measured hyperfine shift ΔE_hf. By contrast, it leads to ‘heating’ of the ⁷¹Ga spins, which has only an indirect effect on ΔE_hf by changing the ⁷⁵As lineshape via dipolar coupling between ⁷¹Ga and ⁷⁵As spins. The result of the two-comb experiment is shown in Fig. 4b: a clear increase of Δν_hom for ⁷⁵As is observed. From model fitting we find that ⁷¹Ga ‘heating’ leads to a three times broader homogeneous linewidth Δν_hom ≍ 355 Hz of ⁷⁵As and its homogeneous lineshape is modified towards Gaussian, observed as an increase in k ≍ 2.32.

To explain this result we note that the NMR lineshape is a statistical distribution of NMR frequency shifts of each nucleus produced by its dipolar interaction with all possible configurations of the neighbouring nuclear spins. However, in a frequency-comb experiment the evolution of the ⁷⁵As spin is limited in time by its depolarization. If the nuclear spin environment of each ⁷⁵As nucleus does not go through all possible configurations during the ⁷⁵As characteristic depolarization time, the frequency shifts are effectively static, and hence are eliminated from the lineshape, as for any other inhomogeneous broadening.

Thus, we conclude that the narrowed, non-Gaussian (k ≍ 1.6–1.8) homogeneous NMR lineshape arises from the ‘snapshot’ nature of the frequency-comb measurement, probing the strongly frozen nuclear spin configuration. When the additional ⁷¹Ga ‘heating’ excitation is applied it ‘thaws’ the ⁷¹Ga spins, detected as broadening of the ⁷⁵As lineshape (as demonstrated in Fig. 4a, b). If all isotopes were fully ‘thawed’ we would have observed a close-to-Gaussian (k ≫ 1) lineshape. By contrast, in the case of completely frozen flip-flops, the lineshape should be close to Lorentzian (k = 1), ultimately limited by the longitudinal nuclear spin relaxation. In a real experiment some spin fluctuations are always present: in particular the spins depolarized by the frequency comb influence one another via homonuclear dipolar interaction, resulting in a homogeneously broadened lineshape with some intermediate k > 1. This interpretation is further supported by a microscopic Bloch equation model of the frequency-comb experiments, as discussed in Supplementary Note 5C.

We use the ⁷⁵As lineshape as a probe to measure the dynamics of the ⁷¹Ga equilibrium spin bath fluctuations. Rather than performing lengthy measurements of the entire ⁷⁵As lineshape, we excite its spins with a frequency comb with a fixed f_MS = 903 Hz, for which the ⁷⁵As depolarization dynamics is most sensitive to the ⁷¹Ga ‘heating’. We find that the resulting ⁷⁵As depolarization is well described by a stretched exponential function ∝exp[−(t_rf/τ_As)^r] with r ≍ 0.6–0.9 (see Methods); the measured value of τ_As is used to characterize the response of ⁷⁵As to ⁷¹Ga ‘heating’. Furthermore, we now use selective ‘heating’ of either the −1/2 ↔ +1/2 CT or the + 1/2 ↔ +3/2 ST of ⁷¹Ga. The amplitude β_{1, Ga} of the ‘heating’ frequency comb is varied—the resulting dependences of the ⁷⁵As depolarization time τ_As at different ⁷⁵As comb amplitudes β_{1, As} are shown in Fig. 4c, d for CT and ST ‘heating’, respectively. For analysis we also express β_{1, Ga} in terms of the rf-induced ⁷¹Ga spin-flip time τ_Ga ∝ β_{1, Ga}⁻² (top scales in Fig. 4c, d, see details in Methods).

For vanishing ⁷¹Ga excitation (β_{1, Ga} < 2 nT Hz^−1/2, τ_Ga > 20 s) the rf-induced spin-flip time τ_Ga of ⁷¹Ga is much larger than the correlation time τ_c of the ⁷¹Ga intrinsic spin flip-flops (τ_Ga > τ_c). By contrast, strong ‘heating’ (β_{1, Ga} > 200 nT Hz^−1/2, τ_Ga < 2 ms) completely ‘thaws’ the ⁷¹Ga spins (τ_Ga < τ_c), broadens the ⁷⁵As lineshape (via heteronuclear interaction), and is observed as a reduction of τ_As. Another clear trend in Fig. 4c, d is that the τ_As(β_{1, Ga}) dependence becomes less pronounced at small β_{1, As}. This is because ⁷⁵As nuclei probe ⁷¹Ga spin fluctuations on a timescale of the ⁷⁵As spin depolarization ∝β_{1, As}⁻². At very small β_{1, As} the ⁷⁵As spin evolution time becomes longer than τ_c, hence the ⁷⁵As lineshape is broadened by intrinsic ⁷¹Ga fluctuations, and additional ⁷¹Ga heating leads only to a minor reduction in τ_As. As we now show, this effect allows τ_c to be deduced from the measurements with variable β_{1, As} and β_{1, Ga} shown in Fig. 4c, d.

First, as we show in Supplementary Note 5B, the heating-induced spin flips (with correlation time τ_Ga) and intrinsic dipolar flip-flops (with correlation time τ_c) of ⁷¹Ga have the same effect on ⁷⁵As. Therefore, τ_As depends only on the effective correlation time (τ_Ga⁻¹ + τ_c⁻¹)⁻¹ of ⁷¹Ga fluctuations. Second, as discussed above, ⁷⁵As homogeneous broadening is determined by fluctuations of ⁷¹Ga only during the characteristic ⁷⁵As depolarization time. Consequently, the ⁷⁵As homogeneous linewidth, and hence τ_As, depend only on the ratio of the ⁷¹Ga effective correlation time and the ⁷⁵As characteristic evolution time (∝β_{1, As}⁻²). This ratio is proportional to β_{1, As}²(τ_Ga⁻¹ + τ_c⁻¹)⁻¹ and τ_As is its function. Although the analytical expression for this function is not known, it can be written as a Taylor series, from which we find that τ_As depends linearly on (τ_Ga⁻¹ + τ_c⁻¹)⁻¹ in the limit of small β_{1, As} (see details in Supplementary Note 4). Taking into account that (τ_Ga⁻¹ + τ_c⁻¹)⁻¹ ∈ (0, τ_c) and (τ_Ga⁻¹ + τ_c⁻¹)⁻¹ = τ_c/2 when τ_c = τ_Ga, we find that τ_c can be estimated as the value of τ_Ga for which τ_As is a mean of its minimum and maximum values. From the measurements with the smallest β_{1, As} = 3.4 nT Hz^−1/2 we estimate in a straightforward way τ_{c, CT} ∼ 0.5 s and τ_{c, ST} ∼ 1.5 s for CT and ST respectively. A more accurate derivation of τ_c is achieved by fitting the entire τ_As data set with a stretched exponential function of β_{1, As}²(τ_Ga⁻¹ + τ_c⁻¹)⁻¹, as shown by the lines in Fig. 4c, d (see details in Methods and Supplementary Note 4). Fitting yields correlation times τ_{c, CT} ≍ 1.0 ± 0.2 s for CT and τ_{c, ST} ≍ 6.2 ± 2.5 s for ST. Furthermore, in Supplementary Note 5 we reproduce the experimental data of Fig. 4c and d with a numerical model based on Bloch equations. We then successfully verify the fitting procedure used here to derive τ_c by applying it to numerical model results with known τ_c.

The observed τ_c ≳ 1 s greatly exceeds typical nuclear dipolar flip-flop times in strain-free III–V solids τ_c ∼ 100 μs (refs 9,10,17). We attribute the extremely long τ_c in self-assembled quantum dots to the effect of inhomogeneous nuclear quadrupolar shifts making nuclear spin flip-flops energetically forbidden^20,24. This interpretation is corroborated by the observation of τ_{c, ST} > τ_{c, CT}, because quadrupolar broadening of the ST transitions is much greater than that of the CT transitions (ref. 21). Furthermore, the ⁷¹Ga spins examined here have the largest gyromagnetic ratio γ and the smallest quadrupolar moment Q, so we expect that all other isotopes in InGaAs have even longer τ_c, resulting in the overall τ_c ≳ 1 s of the entire quantum dot nuclear spin bath. This implies that in high magnetic fields the spin-echo coherence times of the electron and hole spin qubits in self-assembled dots are not limited by the nuclear spin bath up to sub-second regimes^{9,10,12,28,29}. Therefore, the effort aimed at achieving III–V semiconductor quantum dot spin qubits with long coherence should be focused on understanding other mechanisms of central spin dephasing, such as electron-mediated nuclear–nuclear interactions, nuclear spin dynamics induced by electron spin manipulation, or charge fluctuations^13,14,15.

Because the frequency-comb technique is not limited by artefacts in the spin dynamics hampering pulsed magnetic resonance, it allows detection of very slow spin bath fluctuations. Such sensitivity of the method can be used, for example, to investigate directly the effect of the electron or hole on the spin bath fluctuations in charged quantum dots—arising, for example, from hyperfine-mediated nuclear spin interactions. The experiments are well understood within a classical rate equation model and confirmed by a microscopic Bloch equation model, although further advances in frequency-comb spectroscopy can be expected with the development of a full quantum mechanical model. Furthermore, the simple and powerful ideas of frequency-comb NMR spectroscopy can be readily extended beyond quantum dots: as we show in Supplementary Note 3, the only essential requirement is that the longitudinal relaxation time T₁ should be larger (by about two orders of magnitude) than the transverse relaxation time T₂, which is usually the case in solid state spin systems. Finally, our approaches in the use of frequency combs can go beyond NMR, enriching, for example, the techniques in optical spectroscopy.

Methods

Sample structures and experimental techniques.

The experiments were performed on individual neutral self-assembled InGaAs/GaAs quantum dots. The sample was mounted in a helium-bath cryostat (T = 4.2 K) with a magnetic field B_z = 8 T applied in the Faraday configuration (along the sample growth and light propagation direction Oz). A radiofrequency (rf) magnetic field B_rf perpendicular to B_z was induced by a miniature copper coil. Optical excitation was used to induce nuclear spin magnetization exceeding 50%, as well as to probe it by measuring hyperfine shifts in photoluminescence spectroscopy²¹.

Two sample structures have been studied, both containing a single layer of InGaAs/GaAs quantum dots embedded in a weak planar microcavity with a Q-factor of ∼250. In one of the samples the dots emitting at ∼945 nm were placed in a p–i–n structure, where application of a large reverse bias during the rf excitation ensured the neutral state of the dots. The results for this sample are shown in Fig. 2. The second sample was a gate-free structure, where most of the dots emitting at ∼914 nm are found in a neutral state, although the charging can not be controlled. Excellent agreement between the lineshapes of both ⁷¹Ga and ⁷⁵As in the two structures was found, confirming the reproducibility of the frequency-comb technique. The results for this second sample are shown in Fig. 4.

Rate equation model for homogeneous lineshape derivation.

Let us consider an ensemble of spin I = 1/2 nuclei with gyromagnetic ratio γ and inhomogeneously broadened distribution of nuclear resonant frequencies ν_nuc. We assume that each nucleus has a homogeneous absorption lineshape L(ν), with normalization ∫ _−∞^+∞L(ν)dν = 1. A small amplitude (non-saturating) rf field will result in depolarization, which can be described by a differential equation for population probabilities p_±1/2 of the nuclear spin levels I_z = ±1/2

For frequency-comb excitation the decay rate is the sum of the decay rates caused by each rf mode with magnetic field amplitude B₁, and can be written as:

where the summation goes over all modes with frequencies ν_j = ν₁ + jf_MS (ν₁ is the frequency of the first spectral mode).

The change in the Overhauser shift E_hf produced by each nucleus is proportional to p_+1/2 −p_−1/2 and, according to equation (2) has an exponential time dependence ∝exp(−W(ν_nuc)t). The quantum dot contains a large number of nuclear spins with randomly distributed absorption frequencies. Therefore, to obtain the dynamics of the total Overhauser shift we need to average over ν_nuc. Because the spectrum of the rf excitation is periodic and Δν_inh ≫ f_MS, the averaging can be done over one period f_MS. Furthermore, because the total width of the rf frequency comb Δν_comb is much larger than f_MS and Δν_hom, the summation in equation (3) can be extended to ±∞. Thus, the following expression is obtained for the time dependence ΔE_hf(t, f_MS), describing the dynamics of the rf-induced nuclear spin depolarization:

Equation (4) describes the dependence ΔE_hf(t, f_MS) directly measurable in experiments such as shown in Fig. 2b. ΔE_hf(t → ∞) is the total optically induced Overhauser shift of the studied isotope and is also measurable, whereas f_MS and B₁ are parameters that are controlled in the experiment. We note that in the limit of small mode spacing, f_MS → 0, the infinite sum in equation (4) tends to the integral ∫_−∞^+∞L(ν)dν = 1 and the Overhauser shift decay is exponential (as observed experimentally) with a characteristic time

Equation (4) is a Fredholm integral equation of the first kind on the homogeneous lineshape function L(ν). This is an ill-conditioned problem: as a result finding the lineshape requires some constraints to be placed on L(ν). Our approach is to use the model lineshape of equation (1). After substituting L(ν) from equation (1), the right-hand side of equation (4) becomes a function of the parameters Δν_hom and k, which we then find by least-squares fitting of equation (4) to the experimental dependence ΔE_hf(t, f_MS).

This model is readily extended to the case of I > 1/2 nuclei. Equation (2) becomes a tri-diagonal system of differential equations, and the solution (equation (3)) contains a sum of multiple exponents under the integral. These modifications are straightforward but tedious, and can be found in Supplementary Note 2.

Derivation of the nuclear spin bath correlation times.

Accurate lineshape modelling is crucial in revealing the ⁷⁵As homogeneous broadening arising from ⁷¹Ga ‘heating’ excitation (as demonstrated in Fig. 4a, b). However, because a measurement of the full ΔE_hf(t, f_MS) dependence is time consuming, the experiments with variable comb amplitudes β_{1, Ga} and β_{1, As} (Fig. 4c, d) were conducted at a fixed f_MS = 903 Hz well in excess of the ⁷⁵As homogeneous linewidth Δν_hom ≍ 117 Hz. To extract the ⁷⁵As depolarization time τ_As we fit the ⁷⁵As depolarization dynamics ΔE_hf(t) with the following formula: ΔE_hf(t_rf) = ΔE_hf(t_rf → ∞) × (1 − exp[−(t_rf/τ_As)^r]), using r as a common fitting parameter and τ_As independent for measurements with different τ_Ga. We find r ≍ 0.6–0.9, depending on β_{1, As}, and the dependence of τ_As on β_{1, Ga} obtained from the fit is shown in Fig. 4c, d by the symbols with error bars corresponding to 95% confidence intervals.

The spacing of the ⁷¹Ga ‘heating’ frequency comb is kept at a small value, f_MS = 150 Hz, ensuring uniform excitation of all nuclear spin transitions. The amplitude of the ‘heating’ comb is defined as , where B₁ is magnetic field amplitude of each mode in the comb (further details can be found in Supplementary Note 1). To determine the correlation times we express β_{1, Ga} in terms of the rf-induced spin-flip time τ_Ga. The τ_Ga is defined as the exponential time of the ⁷¹Ga depolarization induced by the ‘heating’ comb, and is derived from an additional calibration measurement. The values of β_{1, Ga} shown in Fig. 4c are calculated using equation (5) as , where γ_Ga is the ⁷¹Ga gyromagnetic ratio and τ_Ga is experimentally measured. The additional factor of 4 in the denominator is due to the matrix element of the CT of spin I = 3/2. For experiments on ST the β_{1, Ga} values shown in Fig. 4d are obtained with the same formulae, but multiplied by .

The effect of ⁷¹Ga heating on ⁷⁵As (probed with τ_As) is controlled by the ratio of the ⁷¹Ga effective correlation time (τ_Ga⁻¹ + τ_c⁻¹)⁻¹ and the characteristic time of ⁷⁵As rf-induced depolarization 2/(γ_As²β_{1, As}²) given by equation (5). The experimental data of Fig. 4c, d are fitted with the function τ_As = τ_{As, min}(Δτ_As + (1 − Δτ_As)exp[−(γ_As²β_{1, As}²(τ_Ga⁻¹ + τ_c⁻¹)⁻¹/(2θ))^s]), where θ ≍ 0.4, s ≍ 0.6 are dimensionless parameters describing the modified (stretched) exponential function, Δτ_As ≍ 2 is the ratio of τ_As without and with strong ⁷¹Ga heating, τ_{As, min} is τ_As in the limit of large β_{1, Ga}, dependent on β_{1, As}, and τ_c is the correlation time of the ⁷¹Ga spin fluctuations. Further details of the analysis used to derive τ_c from experimental data can be found in Supplementary Note 4. Moreover, we have verified this fitting procedure by applying it to numerically simulated spin dynamics with known τ_c, as described in Supplementary Note 5D.