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

One of the key challenges in spectroscopy is inhomogeneous
broadening that masks the homogeneous spectral lineshape and the
underlying coherent dynamics. A variety of techniques including
four-wave mixing and spectral hole-burning are used in optical
spectroscopy ^{1, 2, 3}
while in nuclear magnetic resonance (NMR) spin-echo
^{4} is the most common way to counteract
inhomogeneity. 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 that play a crucial role in quantum information
applications
^{5, 6, 9, 10, 11, 12, 13, 14, 15, 16, 17}.
Here we develop NMR techniques that allow the correlation times of
the fluctuations in a nuclear spin bath of individual quantum dots
to be probed. This is achieved with the use of frequency comb
excitation which allows the homogeneous NMR lineshapes to be
measured avoiding high-power pulses. We find nuclear spin
correlation times exceeding 1 s in self-assembled InGaAs quantum
dots - four orders of magnitude longer than in strain-free III-V
semiconductors. The observed freezing of the nuclear spin
fluctuations opens the way for the design of quantum dot spin
qubits with a well-understood, highly stable nuclear spin bath.

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
radio-frequency (rf) magnetic resonance pulses are used for the
initialization and readout of a qubit ^{18}, dynamic
decoupling ^{5} and dynamic control
^{6} of the spin bath.

However, the most important parameter controlling the central spin
coherence ^{19, 9, 11} - the correlation
time of the spin bath fluctuations is very
difficult to measure directly. The is determined
by the spin exchange (flip-flops) of the interacting nuclear bath
spins. By contrast pulsed NMR reveals the spin bath coherence time
, which characterizes the dynamics of the transverse nuclear
magnetization ^{7, 8, 31}
and is much shorter than . The problem is further
exacerbated in self-assembled quantum dots where quadrupolar
effects lead to inhomogeneous NMR broadening exceeding 10 MHz
(Refs. ^{30, 22}), making rf field amplitudes
required for pulsed NMR practically unattainable.

Here we develop an alternative approach to NMR spectroscopy: we
measure non-coherent depolarisation 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
when rf excitation has a frequency comb profile (widely used in
precision optical metrology ^{23}). 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
of the nuclear flip-flops of the other isotope.
While initial studies ^{19, 9, 17}
suggested s for nuclear spins in III-V
semiconductors, it was recently recognized
^{24, 31, 38} that quadrupolar effects
may have a significant impact in self-assembled quantum dots. Here
we for the first time obtain a quantitative measurement of
extremely long s revealing strong
freezing of the nuclear spin bath - 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
T. All measurements of the nuclear spin
depolarisation dynamics employ the pump-depolarise-probe protocol
shown in Fig. 1a. Here we exploit the hyperfine
interaction of the nuclei with the optically excited electron
^{17, 30, 22} both to polarise the nuclei
(pump pulse) and to measure the nuclear spin polarisation in terms
of the Overhauser shift in the QD
photoluminescence spectrum (probe pulse). The rf magnetic field
depolarising nuclear spins is induced by a small copper coil.
(Further experimental details can be found in Methods and
Supplementary Note 1.)

All isotopes in the studied dots possess non-zero quadrupolar
moments. Here we focus on the spin nuclei Ga and
As. The strain-induced quadrupolar shifts result in an
inhomogeneously broadened NMR spectrum
^{30, 22} as shown schematically by the green
line in Fig. 1b. The spectrum consists of a
central transition (CT) and two
satellite transition (ST) peaks.
The NMR spectrum with inhomogeneous linewidth
consists of individual nuclear spin
transitions (shown with red lines) with much smaller homogeneous
linewidth .

To make a non-coherent depolarisation experiment sensitive to the homogeneous NMR lineshape rf excitation with a frequency comb spectral profile is used. As shown in Fig. 1b (black line) the frequency comb has a period of and a total comb width exceeding . The key idea of the frequency comb technique is described in Figs. 1c and d where two possible cases are shown. If the comb period is small (, Fig. 1c) all nuclear transitions are excited by a large number of rf modes. As a result all nuclear spins are depolarised at the same rate and we expect an exponential decay of the total nuclear spin polarisation. In the opposite case of large comb period (, Fig. 1d) some of the nuclear transitions are out of resonance and are not excited (e.g. the one shown by the dashed red line). As a result we expect a slowed-down non-exponential nuclear depolarisation. The experiments are performed at different ; the for which a slow-down in depolarisation is observed gives a measure of the homogeneous linewidth .

Experimental demonstration of this technique is shown in Fig.
2a. The Overhauser shift variation of Ga is shown as a function of the
depolarising rf pulse duration for different
. For small 80 and 435 Hz an
exponential depolarisation is observed. However, when
is increased the depolarisation becomes
non-exponential and slows down dramatically. The detailed
dependence
is shown as a colour-coded plot in Fig. 2b. The
threshold value of (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
450 Hz. Such a small homogeneous
linewidth is detected in NMR resonances with inhomogeneous
broadening of MHz (Ref.
^{30}) demonstrating the resolution power of
frequency-comb non-coherent spectroscopy.

The information revealed by frequency-comb spectroscopy is not limited to linewidth estimates. An accurate determination of the full homogeneous lineshape is achieved with modeling based on solving an integral equation (see details in Methods and Supplementary Note 2). We use the following two-parameter phenomenological model for the homogeneous lineshape:

(1) |

where is the homogeneous full width at half maximum and is a roll-off parameter that controls the tails of the lineshape (the behavior of at large ). For the lineshape corresponds to Lorentzian, while for it tends to Gaussian: in this way Eq. 1 seamlessly describes the two most common lineshapes. Using and as parameters we calculate the model dependence and fit it to the experimental to find an accurate phenomenological description of the homogeneous NMR lineshape in self-assembled quantum dots.

The solid line in Fig. 3a shows the best-fit lineshape ( Hz and ) for the measurement shown in Figs. 2a, b. The dashed and dashed-dotted lines in Fig. 3a show for comparison the Lorentzian () and Gaussian () lineshapes with the same . The difference in the lineshape tails is seen clearly in Fig. 3b where a logarithmic scale is used. The model 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 with Lorentzian (Fig. 3d) and Gaussian (Fig. 3e) lineshapes show a pronounced deviation from the experiment, demonstrating the excellent sensitivity of the frequency-comb spectroscopy to accurately probe the homogeneous spectral lineshape.

We have also performed frequency comb NMR spectroscopy on
As nuclei (Fig. 4a). Despite their larger
inhomogeneous broadening MHz the
model fitting reveals even smaller
Hz and . The
frequency-comb measurements are in agreement with the previous
findings based on spin-echo NMR measurements ^{31}:
indeed, from derived here we can
estimate the nuclear spin coherence time
and ms for
Ga and As, in good agreement with the corresponding
spin-echo and ms. On the other hand,
spin-echo could only be measured on central transitions for which
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 (Guassian) profile ^{26}. And most
importantly, due to the parasitic effects such as ”instantaneous
diffusion” ^{7} and spin locking
^{8} pulsed NMR does not reveal the
characteristic correlation time of the spin
exchange (spin flip-flop) between the nuclei in the absence of rf
excitation.

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

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 neighboring nuclear spins. However, the frequency comb experiment is limited in time (up to 100 s as shown in Figs. 4a, b). If the nuclear spin environment of each As nucleus does not go through all possible configurations during the measurement 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 (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 Ga ”heating” excitation is applied it ”thaws” the Ga spins, detected as broadening of the As lineshape (as demonstrated in Figs. 4a, b). We use such sensitivity of the As lineshape to measure the dynamics of the Ga equilibrium spin bath fluctuations. Based on the results of Fig. 4 the As spins are now excited with a frequency comb with a fixed kHz for which the As depolarisation dynamics is most sensitive to the Ga ”heating”. Furthermore, we now use selective ”heating” of either the CT or the ST of Ga. The amplitude of the ”heating” frequency comb is varied – the resulting dependencies of the As depolarisation time are shown in Fig. 4c by the squares and triangles for CT and ST ”heating” respectively. For analysis we also express in terms of the rf-induced Ga spin-flip time (bottom scale in Fig. 4c, see details in Methods).

It can be seen that for vanishing Ga excitation () the As depolarisation time is constant. In this ”frozen” regime the rf-induced spin-flip time of Ga is larger than the correlation time of the Ga intrinsic spin flip-flops (). As a result is determined only by the rf excitation of As itself. However, when is increased to 1–10 nT Hz the rf induced spin-flips of Ga nuclei become faster than their intrinsic flip-flops (). Such ”thawing” of Ga broadens the As lineshape (via heteronuclear interaction), and is observed as a reduction of . Thus the transition from the ”frozen” to ”thawed” regimes takes place when , allowing to be determined. In Fig. 4c we extrapolate graphically (dashed lines) the power-law dependence in the ”thawed” regime. The points of the intersections with the limiting value of in the ”frozen” regime (solid horizontal line) yield correlation times s for CT and s for ST.

The observed s exceeds very strongly
typical nuclear dipolar flip-flop times in strain-free III-V
solids s
^{10, 9, 17}. We attribute the extremely
long in self-assembled quantum dots to the
effect of inhomogeneous nuclear quadrupolar shifts making nuclear
spin flip-flops energetically forbidden
^{24, 31}. This interpretation is corroborated
by the observation of ,
since quadrupolar broadening of the ST transitions is much larger
than that of the CT ^{30}. Furthermore, the Ga
spins examined here have the largest gyromagnetic ratio
and the smallest quadrupolar moment , so we expect that all
other isotopes in InGaAs have even longer ,
resulting in the overall 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
^{10, 27, 9, 12}. Provided that other
mechanisms of central spin dephasing, such as charge fluctuations
^{13, 14, 15} are eliminated, this
would open the way for optically active spin qubit networks in
III-V semiconductors with coherence properties previously
achievable only in nuclear-spin-free materials
^{28, 29}.

Since the frequency-comb technique is not limited by artifacts 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 can be well understood within a classical rate equation model, while 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 should be larger (by about two orders of magnitude) than the transverse relaxation time , which is usually the case in solid state spin systems. Finally our approaches in the use of frequency combs can go beyond NMR, and for example enrich the techniques in optical spectroscopy.

## I Methods Summary

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 (=4.2 K) with a magnetic field T
applied in the Faraday configuration (along the sample growth and
light propagation direction ). Radio-frequency (rf) magnetic
field perpendicular to 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
spectroscopy^{30}.

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 nm were placed in a 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 nm are found in a neutral state, although the charging can not be controlled. Excellent agreement between the lineshapes of both Ga and As in the two structures was found, confirming the reproducibility of the frequency-comb technique.

Homogeneous lineshape theoretical model. Let us consider an ensemble of spin nuclei with gyromagnetic ratio and inhomogeneously broadened distribution of nuclear resonant frequencies . We assume that each nucleus has a homogeneous absorption lineshape , with normalization . A small amplitude (non-saturating) rf field will result in depolarisation, which can be described by a differential equation for population probabilities of the nuclear spin levels

(2) |

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

(3) |

where the summation goes over all modes with frequencies ( is the frequency of the first spectral mode).

The change in the Overhauser shift produced by each nucleus is proportional to and according to Eq. 2 has an exponential time dependence . 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 , which can be done over one period since the spectrum of the rf excitation is periodic. Furthermore, since the total width of the rf frequency comb is much larger than and , the summation in Eq. 3 can be extended to . Thus, the following expression is obtained for the time dependence , describing the dynamics of the rf-induced nuclear spin depolarisation:

(4) |

Equation 4 describes the dependence directly measurable in experiments such as shown in Fig. 2b. is the total optically induced Overhauser shift of the studied isotope and is also measurable, while and are parameters that are controlled in the experiment. We note that in the limit of small comb period the infinite sum in Eq. 4 tends to the integral and the Overhauser shift decay is exponential (as observed experimentally) with a characteristic time

(5) |

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

This model is readily extended to the case of nuclei. Eq. 2 becomes a tri-diagonal system of differential equations, and the solution (Eq. 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 modeling is crucial in revealing the As homogeneous broadening arising from Ga ”heating” excitation (as demonstrated in Figs. 4a, b). However, since a measurement of the full dependence is time consuming, the experiments with variable Ga excitation amplitude (Fig. 4c) were conducted at fixed kHz exceeding noticeably the As homogeneous linewidth Hz. To extract the arsenic depolarisation time we fit the arsenic depolarisation dynamics with the following formulae: , using as a common fitting parameter and independent for measurements with different . We find , while the dependence on obtained from the fit is shown in Fig. 4c with error bars corresponding to 95% confidence intervals.

The period of the Ga ”heating” frequency comb is kept at a small value Hz ensuring uniform excitation of all nuclear spin transitions. The amplitude of the ”heating” comb is defined as , where 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 in terms of the rf-induced spin-flip time . The is defined as the exponential time of the Ga depolarisation induced by the ”heating” comb and is derived from an additional calibration measurement. The values of shown in Fig. 4c correspond to the experiment on the CT and are calculated using Eq. 5 as , where is the Ga gyromagnetic ratio and is experimentally measured. The additional factor of 4 in the denominator is due to the matrix element of the CT of spin . For experiments on ST the values shown in Fig. 4c must be multiplied by .

ACKNOWLEDGMENTS The authors are grateful to K.V. Kavokin for useful discussions. This work has been supported by the EPSRC Programme Grant EP/J007544/1, ITN SNANO. E.A.C. was supported by a University of Sheffield Vice-Chancellor’s Fellowship. I.F. and D.A.R. were supported by EPSRC.

ADDITIONAL INFORMATION Correspondence and requests for materials should be addressed to A.M.W (a.waeber@sheffield.ac.uk) or E.A.C. (e.chekhovich@sheffield.ac.uk).

## Supplementary Information

a, Homogeneous lineshape experiments (Fig. 2): | ||||
---|---|---|---|---|

Isotope | As | Ga | ||

Frequency comb | As-full | Ga-full | ||

Comb central frequency (MHz) | 58.81 | 104.80 | ||

Comb spectral width (MHz) | 18 | 9 | ||

Comb period (Hz) | varied | varied | ||

Rf field density () | 65.5 | 39.1 | ||

b, Lineshape broadening experiments (Figs. 4a, 4b): | ||||

Isotope | As | Ga | ||

Frequency comb | As-full | Ga-full | ||

Comb central frequency (MHz) | 58.81 | 104.80 | ||

Comb spectral width (MHz) | 18 | 8 | ||

Comb period (Hz) | varied | 159 | ||

Rf field density () | 30.5 | 27.6 | ||

c, Correlation time experiments (Fig. 4c): | ||||

Isotope | As | Ga | Ga | Ga |

Frequency comb | As-full | Ga-full | Ga-ST | Ga-CT |

Comb central frequency (MHz) | 58.81 | 104.80 | 106.10 | 104.80 |

Comb spectral width (MHz) | 18 | 8 | 2.5 | 0.05 |

Comb period (Hz) | 1466 | 159 | 150 | 150 |

Rf field density () | 30.5 | 27.6 | varied | varied |

## Appendix Supplementary Note 1 Details of experimental techniques

### Supplementary Note 1.1 Frequency combs

The key novel findings of this work are based on the use of radiofrequency (rf) excitation with a frequency comb spectral profile. Here we give detailed parameters of the frequency combs used in the experiments on self-assembled quantum dots.

Supplementary Figure 1a shows a schematic NMR
spectrum of a self-assembled InGaAs quantum dot at T,
based on results of the inverse NMR measurements
^{30}. Four sharp peaks arise from the central
transitions (CTs) of each isotope. All
spin-3/2 nuclei (As, Ga and Ga) have two
satellite transitions (STs) and
observed as inhomogeneously broadened
bands on both sides of the CTs. The spin-9/2 In nuclei
have a total of eight STs. For clarity these are shown as four
bands on each side of the CT, although in experimental NMR spectra
the peaks arising from different STs merge forming two
inhomogeneously broadened bands on both sides of the In CT
^{30}. As shown in Supplementary Figure
1a the spectral contributions from the
In and Ga nuclei overlap significantly;
furthermore, there is a small overlap between the As and
In NMR resonances.

In the experiments we use rf frequency combs that are designed to influence only the chosen transition(s) of one isotope without affecting the other isotopes as demonstrated in Supplementary Figures 1b-d. The frequency-comb labeled Ga-full has a total width of 8 or 9 MHz. This comb entirely covers the inhomogeneous lineshape of Ga and thus uniformly excites all nuclear spin transitions of this isotope. Since Ga has a large resonance frequency the Ga-full comb does not affect the polarisation of the other isotopes. To achieve selective excitation of the Ga ST () we use frequency comb Ga-ST with MHz (Supplementary Fig. 1c). Similarly, selective excitation of Ga CT is achieved with a narrow comb Ga-CT with a width of kHz as shown in Supplementary Figs. 1c, d.

In the case of As we use a frequency comb As-full that excites the entire inhomogeneouse resonance line. This comb inevitably excites some of the In nuclear transitions, mostly STs. However, excitation of the ST alone has a negligible effect on the overall change in the nuclear polarisation of the spin-9/2 nuclei of In. Thus the As-full comb with the optimum width MHz is used in experiments for selective depolarisation of As.

The central frequencies , the widths and the comb periods (spectral separation between the adjacent modes) of all the combs used in experiments are listed in Supplementary Table 1. Depending on the experiment the is either varied or kept constant. Typically the values of ranging from 30 Hz to 21 kHz are employed, so that the total number of modes in the comb ranges from 330 to 600000.

The phases of individual modes of the frequency comb are chosen in a way that minimizes the peak power for a given average power (i.e. a waveform with the minimum crest-factor). In particular the following expression satisfies this criterion:

(1) |

where the summation goes over all modes, and is the frequency of the first mode of the comb. In experiments the frequency-comb signal is generated by an arbitrary waveform generator equipped with a 64 million points memory.

Each mode of the frequency comb has the same amplitude of the rf oscillating magnetic field. In experiments where the comb period is varied has to be adjusted to maintain the same total power of the frequency comb excitation. Since the power is proportional to the ratio has to be kept constant. Thus the amplitude of the frequency comb can be conveniently characterized by the magnetic field density .

The amplitude of the frequency comb rf magnetic field can be
calibrated from an additional pulsed Rabi oscillation experiment
on the CT of a selected isotope ^{31}. The Rabi
oscillation circular frequency can be
measured experimentally and is proportional to the rf field
amplitude in the rotating frame:

(2) |

where is nuclear gyromagnetic ratio and the factor
originates from the dipole transition matrix element of the CT of
a spin-3/2 nucleus ^{32}. Throughout all experiments,
we monitored the applied rf fields via a pick-up coil that was
placed close to the sample and connected to a spectrum analyzer.
By comparing the voltages induced by the frequency comb modes with
the voltage associated with a field in the Rabi
oscillation experiment, we derived the values of and
of the comb. The values of
used in different experiments are given in Supplementary Table 1
- these correspond to the rotating frame, i.e. the physical values
of the magnetic field induced by the coil are twice as large.

### Supplementary Note 1.2 Optical pump-probe techniques for frequency-comb NMR

Supplementary Fig. 2 shows the time sequence of optical and rf excitation pulses used in a frequency comb measurement of the equilibrium nuclear spin bath fluctuations of the Ga isotope (the results are shown in Fig. 4c of the main text). As explained in the main text this experiment is based on measuring the rf induced dynamics of the As spins in the presence of additional rf excitation of Ga spins. The experimental cycle consists of the following four stages described below.

Optical nuclear spin pumping. At the start of each new
measurement cycle, the nuclear spin bath is reinitialized
optically. This is achieved with optically induced dynamic nuclear
polarisation (DNP) ^{33, 34}: under high power,
circularly polarised laser excitation, spin polarised electrons
are created. These electrons can efficiently transfer their
polarisation to the nuclear spin bath via hyperfine interaction
^{35}. We use polarised excitation with a
pump laser operating at 850 nm, in resonance with the QD
wetting layer. By using sufficiently long pumping times
( s) and high powers (where is the saturation power of
the QD ground states), we create a reproducible and high degree of
nuclear polarisation.

Depolarisation of Ga isotope. Optical spin pumping polarises the nuclei of all isotopes. However, in order to probe the equilibrium fluctuations of Ga spins their longitudinal relaxation has to be excluded from the measured dynamics. For that Ga nuclear polarisation has to be erased, which is achieved by exciting the spins with a Ga-full frequency comb for a sufficiently long time s. To simplify experimental implementation the depolarising rf is kept on during the nuclear spin pumping stage as well, which has no effect on the experimental results.

Frequency comb rf excitation. Following the spin bath preparation (optical DNP and Ga depolarisation), the main rf excitation (variable duration ) is applied. For the spin bath fluctuation measurement (Fig. 4c) this excitation is a sum of the As-full frequency comb and either the Ga-ST or the Ga-CT comb. Since Ga is completely depolarised by the previous pulse, all changes in the total nuclear polarisation at this stage are solely due to the As depolarisation. In this way we ensure that it is the depolarisation dynamics of As that is measured, while the Ga-ST or Ga-CT ”heating” excitation only induces nuclear spin-flips of the corresponding Ga transition.

Optical probing of the nuclear spin polarisation. At the end of the experiment cycle, a short probe laser pulse is applied and the resulting photoluminescence spectrum is collected by a 1 m double spectrometer with a CCD. The changes in the quantum dot Zeeman splitting (the Overhauser shift ) are used to probe the nuclear spin state. The probe laser is non-resonant ( nm), yet unlike the pump laser it is linearly polarised and the probe power and duration are chosen such that no noticeable DNP is induced and the final nuclear spin polarisation is measured accurately. Typical probe parameters used in the experiments were ms and for QDs in the diode sample and ms and in the gate-free structure. Depending on the QD photoluminescence intensity the experiment cycle was repeated times to achieve the optimum signal-to-noise ratio.

For the purpose of data analysis we are interested in measuring the rf-induced change in the nuclear spin polarisation (rf-induced change in the Overhauser shift ), for that we perform a control measurement where As-full comb is off, and subtract the resulting QD Zeeman splitting from the Zeeman splittings obtained in the measurements with As excitation. In this way corresponds to no nuclear spin depolarisation induced by the rf.

The diagram of Supplementary Fig. 2 also describes the other types of frequency comb NMR measurements presented in the main text with the following modifications: For the line broadening measurements shown in Figs. 4a, b the main excitation is a sum of the As-full and Ga-full frequency combs (the Ga-full is off for the measurement in Fig. 4a and is on for Fig. 4b). The homogeneous lineshape measurement (Fig. 2) is performed without the additional rf depolarisation excitation, while for the main rf excitation the Ga-full comb is used.

## Appendix Supplementary Note 2 Theoretical model for nuclear spin dynamics under frequency-comb excitation

The Methods section of the main text describes the model for spin nuclei. Here we consider a more general case of nuclear spins with gyromagnetic ratio . We consider the Overhauser shifts of nuclei of only one isotope, which is justified since the polarisation of other isotopes stays constant during the measurement and can be neglected. In an external magnetic field the nuclear spin state is split into states with spin projections . In our classical rate equation model we assume that each nuclear spin has a probability to be found in a state with with normalization condition

(3) |

We also assume that at optical pumping initializes all nuclear spins into a Boltzman distribution

(4) |

so that nuclear spins can be characterized by a temperature . When optical nuclear spin pumping is used, is very small compared to the spin bath temperature , so the equilibrium nuclear polarisation can be neglected.

Application of the radiofrequency (rf) excitation leads to the changes in population probabilities. The rf excites only dipole-allowed transitions for which changes by . In the experiment we use weak, non-saturating radiofrequency fields. Thus instead of the full Bloch equations for nuclear magnetization, the evolution of the population probabilities of the states with can be described with the following first-order differential equation (see further details in Supplementary Note 3):

(5) |

Here the first equation describes the states with . Its first term is due to nuclei with making transitions into the and states, whereas the second and third terms describe the opposite case of nuclei transitioning into the state from the and states respectively. The second and third equations correspond to the case of and respectively. Taken for all , for which , the Supplementary Eq. 5 yields a system of first-order ordinary differential equations (ODEs) for time-dependent variables with initial conditions given by Supplementary Eq. 4. Due to the normalization condition of Supplementary Eq. 3, only variables and equations are independent. This system of ODEs has a tridiagonal matrix with coefficients determined by the rf induced transition rates which satisfy a symmetry condition .

Similar to the case of , each transition rate resulting from the frequency-comb excitation is a sum of transition rates caused by individual rf modes each having magnetic field amplitude . We assume that each nuclear transition has the same broadening described by the homogeneous lineshape function , with normalization . Due to the inhomogeneous quadrupolar shifts the NMR transition frequency is generally different for each pair of spin levels and (even for one nucleus). Thus for the transition rates we can write:

(6) |

where the summation goes over all modes with frequencies
, and is
extended to since the total width of the rf excitation
comb is much larger than
and the homogeneous linewidth . The
factor arises from the dipolar transition matrix
element ^{32}. We note that Supplementary Eq.
5 does not involve any explicit nuclear-nuclear
interactions. Instead such interactions are introduced in
Supplementary Eq. 6 phenomenologically via the
homogeneous broadening described by the lineshape function
. On the other hand the presence of finite homogeneous
broadening is essential in order to use the limit of weak rf
fields ^{36} and transform the Bloch equations into
rate equations (Supplementary Eq. 5). The
validity of the weak rf field approximation is discussed and
verified experimentally in Supplementary Note 3.

Since Supplementary Eq. 5 is a system of linear first-order equations, the solution is a multiexponential relaxation towards the fully depolarised state where all nuclear spin states have equal populations . The solution has the general form:

(7) |

where are the non-zero eigenvalues of the ODE system matrix of Supplementary Eq. 5. The values of depend on all transition rates from Supplementary Eq. 6, while the coefficients depend both on and the initial probabilities from Supplementary Eq. 4.

Non-zero nuclear spin polarisation along the magnetic field ( axis) changes the spectral splitting of the quantum dot Zeeman doublet. Such change known as the Overhauser shift is measured experimentally using photoluminescence spectroscopy. The time evolution of the Overhauser shift for the fixed values of nuclear transition frequencies reads as:

(8) |

where is the hyperfine constant and we used Supplementary Eqns. 4, 6, 7 so that is dependent on , , , the homogeneous lineshape function and all nuclear transition frequencies as parameters.

Each QD contains a large number of nuclear spins with randomly distributed absorption frequencies. Thus to describe the experiment on nuclear spins in a self-assembled quantum dot we need to average over all , which can be done over one period since the spectrum of the frequency-comb rf excitation is periodic. Similarly to the case of , the following expression is obtained for the time dependence of the Overhauser shift, describing the dynamics of rf-induced nuclear spin depolarisation:

(9) |

The quantity measured in the experiment is the rf-induced variation of the Overhauser shift:

(10) |

The values of and are the parameters that are controlled in the experiment. The nuclear spin temperature can be determined using the known hyperfine constant and the measured total Overhauser shift . Thus for a given homogeneous NMR lineshape the nuclear spin depolarisation dynamics can be fully predicted from Supplementary Eqns. 3–10. Conversely, Supplementary Eq. 9 can be treated as an integral equation on the unknown homogeneous lineshape function . Since Fredholm’s integral equation of the first kind is an ill-conditioned problem, some constrains on are required. Our approach is to use a model lineshape with two parameters and (Eq. 1 of the main text). Upon substituting this model lineshape the Overhauser shift variation of Supplementary Eq. 10 becomes . We then perform least-square fitting to the experimental dependence using , the homogeneous linewidth and the roll-off parameter as fitting parameters and using the nuclear spin temperature obtained from the experiment.

The ODE system of Supplementary Eq. 5 can be solved analytically for , however it turns out to be more practical to perform numerical diagonalization in order to obtain the eigenvalues and coefficients which are then used in Supplementary Eq. 7. Similarly we use numerical integration to evaluate Supplementary Eq. 9.

## Appendix Supplementary Note 3 Applicability of the frequency comb technique and the rate equation model.

The evolution of the nuclear magnetization under rf excitation can
be described by the Bloch equations ^{37}. In this model
the solution under resonant monochromatic excitation is determined
by the three important parameters: the amplitude of the resonant
field , and the relaxation times characterizing the
system, the longitudinal and the transverse . In
self-assembled quantum dots is extremely long (few hours
^{38, 39}), so that the longitudinal relaxation
can be neglected. Thus the nuclear spin dynamics is determined by
the relation between and . Two cases are
possible ^{36}. If rf magnetic field is strong
() the nuclear magnetization has
oscillatory behaviour (Rabi oscillations are observed). By
contrast, for weak rf excitation ()
there are no oscillations, and any nuclear magnetization
along the external field decays exponentially to its steady state
value ^{36}. The exponential dynamics in the weak rf
excitation regime allow for the problem to be simplified and for
the rate equation model described by Supplementary Eqns.
5 to be used. The validity of the rate equation
model is essential for the determination of the homogeneous
lineshape and thus sets the applicability limit for the frequency
comb technique itself.

To verify the applicability of the rate equation model we performed frequency comb spectroscopy measurements at different rf amplitudes. Supplementary figure 3 shows the results for Ga measured at high rf field density nT Hz and low rf field density nT Hz – in all other respects the conditions in these experiments were the same as in the experiment with medium nT Hz shown in Fig. 2 of the main text.

The Supplementary figure 3a shows by symbols the nuclear spin depolarisation dynamics at small comb period Hz. At low rf field amplitude nT Hz (triangles) the decay can be described very well by a single exponential decay ( ms) shown with a solid line. By contrast, at high amplitude nT Hz (squares) the dynamics shows signatures of oscillations, and there is a clear deviation from the exponential behaviour (best exponential fit is for ms).

Supplementary Figures 3b and c show the full dependencies measured at high (b) and low rf (c) field densities. The profiles are in good agreement for the two experiments (except for the rescaling along the axis). Using model fitting we find Hz, for nT Hz and Hz, for nT Hz. This is also in good agreement with Hz, found for the measurement at nT Hz shown in Fig. 2 of the main text.

We thus conclude that the discrepancy between the results in Supplementary Figs. 3b and c becomes significant only at small comb period Hz (as also demonstrated in Supplementary Figure 3a). This can be explained as follows: At high rf excitation amplitude the nuclear spin depolarisation takes place on a shorter time scale . If the rf pulses are shorter than , the spectral profile of the frequency comb becomes distorted. Thus if the nuclear polarisation decay timescales are shorter than , the rate equation model is no longer applicable since the rf excitation can not be described as a frequency comb. Thus it is required that . Furthermore, to measure the homogeneous lineshape and linewidth we only need to use frequency combs with comb periods comparable to or larger than , so it is required that . Combining and we find the following condition on the frequency comb technique applicability:

(11) |

which restricts the rf amplitude, characterized by the depolarisation time .

Another requirement, arising from the applicability of the weak rf field limit of the Bloch equations, is that the rf induced depolarisation time must be longer than the transverse relaxation time . However, is related to the homogeneous linewidth as . Thus the requirement leads to the same condition as that of Supplementary Eq. 11. Furthermore, since the frequency comb technique relies on the measurement of the longitudinal nuclear magnetization, the depolarisation time must be shorter than the nuclear spin times. In combination with Supplementary Eq. 11 this leads to the following condition:

(12) |

This condition has a dual role: it sets the boundaries for the rf excitation amplitude (characterized by ) and sets the limitation on the properties of the nuclear spin system that can be studied with the frequency comb technique. This latter condition can be rewritten as

(13) |

From the measurements at different rf amplitudes we find that the parameter must be or larger in order for the frequency comb technique to work reliably. This however is a rather weak condition and is satisfied for a large class of solid-state nuclear spin systems where . This demonstrates the wide applicability of the frequency comb spectroscopy technique developed here.

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