# Nuclear spin diffusion in the central spin system of a GaAs/AlGaAs quantum
dot

Peter Millington-Hotze  Department of Physics and Astronomy, University of
Sheffield, Sheffield S3 7RH, United Kingdom  Santanu Manna  Institute of
Semiconductor and Solid State Physics, Johannes Kepler University Linz,
Altenbergerstr. 69, 4040 Linz, Austria  Saimon F. Covre da Silva  Institute of
Semiconductor and Solid State Physics, Johannes Kepler University Linz,
Altenbergerstr. 69, 4040 Linz, Austria  Armando Rastelli  Institute of
Semiconductor and Solid State Physics, Johannes Kepler University Linz,
Altenbergerstr. 69, 4040 Linz, Austria  Evgeny A. Chekhovich
<EMAIL_ADDRESS>Department of Physics and Astronomy, University
of Sheffield, Sheffield S3 7RH, United Kingdom

###### Abstract

The spin diffusion concept provides a classical description of a purely
quantum-mechanical evolution in inhomogeneously polarized many-body systems
such as nuclear spin lattices. The central spin of a localized electron alters
nuclear spin diffusion in a way that is still poorly understood. In contrast
to previous predictions, we show experimentally that in GaAs/AlGaAs quantum
dots the electron spin accelerates nuclear spin diffusion, without forming any
Knight field gradient barrier. Such acceleration is present even at high
magnetic fields, which we explain as a result of electron spin-flip
fluctuations. Diffusion-limited nuclear spin lifetimes range between 1 and 10
s, providing plenty of room for recent proposals seeking to store and process
quantum information using quantum dot nuclear spins.

###### pacs:

##  I Introduction

Interacting many-body spin ensembles exhibit a variety of phenomena such as
phase transitions [2, 3] spin waves [4, 5] and emergent thermodynamics [6, 7].
Spin diffusion [8, 9] is one of the earliest studied phenomena, where unitary
quantum-mechanical evolution results in an irreversible dissipation of a
localized spin polarization that is well described by the classical diffusion
model. Pure spin diffusion in homogeneous solids has been observed in a few
notable examples [10, 11]. However, most systems of interest are inhomogeneous
by nature. In particular, magnetic (hyperfine) interaction with the central
spin of a localized electron [Fig. 1(a)] causes shifts (known as the Knight
shifts[12, 13]) in the nuclear spin energy levels [Fig. 1(b)]. The resulting
nuclear spin dynamics are complicated, as observed in a wide range of solid
state impurities [14, 15, 16, 17, 18, 19, 20] and semiconductor nanostructures
[21, 22, 18, 23, 24, 25, 26]. It is still an open question whether the
inhomogeneous Knight shifts accelerate [27, 24, 28] or suppress [29, 30, 17,
31, 28, 26] spin diffusion between the nuclei. Resolving this dilemma is both
of fundamental interest and practical importance for the recent proposals to
use nuclear spins as quantum memories and registers [32, 33, 34], since spin
diffusion would set an ultimate limit to the longevity of any useful quantum
state.

Figure 1:  Schematic of a central spin model, sketched for the one-dimensional
case, along the growth axis $z$ of a GaAs/AlGaAs structure. (a) Wavefunction
density $|\psi_{\rm{e}}|^{2}$ of an electron ($e$, ball and arrow) localized
in GaAs. (b) Energy levels of the nuclei, that are depicted for simplicity as
spins 1/2, and can occupy states with $+1/2$ and $-1/2$ spin projections (up
and down arrows). Dashed lines show the bulk nuclear spin energies dominated
by the external magnetic field $B_{\rm{z}}$. These bulk energies are generally
different in GaAs ($z\leq 4$) and AlGaAs ($z\geq 5$) due to the difference in
chemical shifts and homogeneous strain. The energies of the individual nuclei
are further shifted by the electron Knight field (mainly in GaAs) and by the
atomic-scale strain disorder in the AlGaAs alloy. Magnetic dipole interaction
between the nuclei can result in spin exchange via a flip-flop process,
sketched by the curved arrows for nuclei at $z=-1$ and $z=0$ as an example. If
energy mismatch is larger than the nuclear spin level homogenous broadening,
for example for nuclei at $z=4$ and $z=5$, spin exchange becomes prohibited,
suppressing nuclear spin diffusion.

Figure 1 sketches the central spin model where an electron can be trapped in a
GaAs layer surrounded by the AlGaAs barriers, and for simplicity spin-1/2
particles are used to describe the energy levels of the nuclei subject to the
strong external magnetic field $B_{\rm{z}}$. Any two nuclear spins $i$ and $j$
are coupled by the dipole-dipole interaction $\propto
2\hat{I}_{{\rm{z}},i}\hat{I}_{{\rm{z}},j}-(\hat{I}_{{\rm{x}},i}\hat{I}_{{\rm{x}},j}+\hat{I}_{{\rm{y}},i}\hat{I}_{{\rm{y}},j})$,
where $\hat{I}_{{\rm{x}},i}$, $\hat{I}_{{\rm{y}},i}$ and
$\hat{I}_{{\rm{z}},i}$ are the Cartesian components of the spin operator
${\bf{I}}_{i}$ of the $i$-th nucleus. The last two terms of the dipole-dipole
interaction describe a flip-flop spin exchange process (dashed arrows at
$z=-1$ and $0$ in Fig. 1(b)), responsible for the transfer of spin
polarization in space, known as spin diffusion. The electric quadrupolar
moments of the spin-3/2 nuclei make them sensitive to electric field gradients
(EFGs), which can be induced by the GaAs/AlGaAs interfaces roughness ($z=4.5$)
or atomic-scale strains arising from random positioning of the aluminium atoms
[35, 36] in AlGaAs ($z\geq 5$). The resulting energy splitting mismatch
between the adjacent nuclei can impede nuclear spin diffusion.

When an electron is added, its spin $\bf{s}$ couples to the nuclear spin
ensemble via hyperfine interaction:

$\displaystyle\mathcal{\hat{H}}_{\rm{hf}}=\sum_{j}{A_{j}(\hat{s}_{\rm{x}}\hat{I}_{{\rm{x}},j}+\hat{s}_{\rm{y}}\hat{I}_{{\rm{y}},j}+\hat{s}_{\rm{z}}\hat{I}_{{\rm{z}},j})},$
(1)

where the summation goes over all nuclei $j$, and the coupling constants
$A_{j}$ are proportional to the electron density
$|\psi_{\rm{e}}({\bf{r}}_{j})|^{2}$ at the nuclear sites ${\bf{r}}_{j}$. On
the one hand, through the term $\hat{s}_{\rm{z}}\hat{I}_{{\rm{z}}}$, the
electron spin can produce a further diffusion barrier [29, 30, 17, 31, 28,
26], at the points of strong Knight shift gradient ($z=3$ in Fig. 1(a)). On
the other hand, the electron spin can mediate spin flip-flops between two
distant nuclei with similar energy splitting (e.g. $z=-2$ and $z=2$),
potentially opening a new channel for spin diffusion, especially at low
magnetic fields [27, 24, 28].

Here, we examine electron-controlled nuclear spin diffusion in high quality
epitaxial GaAs/AlGaAs quantum dots (QDs), which emerged recently as an
excellent platform for quantum light emitters [37, 38, 39] as well as spin
qubits [34, 40] and quantum memories [33]. Crucially, we design experiments
where nuclear spin dynamics are examined either in absence or in presence of
the electron central spin, but under otherwise identical initial nuclear spin
state. In this way, we distinguish with high accuracy the effects specific to
the electron spin, and demonstrate that no observable Knight field barrier is
formed. Instead, the nuclear-nuclear interactions mediated by the electron
spin accelerate nuclear spin diffusion up to unexpectedly high magnetic fields
– we attribute this to the previously neglected impact of the electron spin
flips. Our results answer a long-standing question in spin physics, and
provide practical guidelines for the design and optimization of quantum dot
electron-nuclear spin qubits and quantum memories.

Figure 2:  (a) Atomic force microscopy (AFM) profile of the AlGaAs surface
after nanohole etching. (b) Surface level profiles taken along the horizontal
and vertical lines through the center of the nanohole in (a). (c) Schematic
(not to scale) of the sample structure. GaAs quantum dots (QDs) are formed by
infilling of the in-situ etched nanoholes in the bottom Al0.33Ga0.67As
barrier. The bottom (top) Al0.15Ga0.85As layer is $n$ ($p$) type doped to form
a $p-i-n$ diode structure. External gate bias $V_{\rm{Gate}}$ is applied for
deterministic QD charging with electrons. (d) Photoluminescence (PL) spectra
of a negatively charged trion $X^{-}$ following $\sigma^{+}$ (triangles) and
$\sigma^{-}$ optical pumping which induces nuclear spin polarization that
manifests in hyperfine shifts of the Zeeman doublet spectral splitting $\Delta
E_{\rm{PL}}$. (e) Optically detected NMR of the 75As spin-3/2 nuclei measured
in a single QD. Strain-induced quadrupolar shifts of the nuclear spin-3/2
levels (left inset) give rise to an NMR triplet with splitting
$\nu_{\rm{Q}}\approx 24$ kHz, observed in an empty QD (0$e$, diamonds).
Charging the QD with a single electron (1$e$, circles) induces inhomogeneous
Knight shifts observed as NMR spectral broadening. The measurement is
conducted using “inverse NMR” signal amplification technique [41], with
spectral resolution shown by the horizontal bars (smaller for 0$e$ and larger
for 1$e$). The measurement pump-probe cycle is shown in the top inset. The
bias $V_{\rm{Gate}}$ is tuned to 0$e$ charge state for optical pumping of the
nuclear spins and to 1$e$ state for their optical probing. The radio frequency
(RF) pulse is applied in the dark under either 0$e$ or 1$e$ bias.

##  II Sample and experimental techniques

The studied heterostructure is grown by in-situ etching of nanoholes [42, 43]
in the AlGaAs surface [Figs. 2(a,b)], which are then infilled with GaAs to
form the quantum dots (QDs). The structure is processed into a $p-i-n$ diode
[Fig. 2(c)] where an external bias $V_{\rm{Gate}}$ is applied to charge QDs
deterministically with individual electrons (See details in Supplementary
Section 1). In this way, it is possible to study nuclear spin dynamics in an
empty (0$e$) or single-electron (1$e$) state. A static magnetic field
$B_{\rm{z}}$ is applied along the growth axis $z$ (Faraday geometry) and the
sample is kept at liquid helium temperature of 4.2 K. We use confocal
microscopy configuration where QD photoluminescence (PL) is excited and
collected through an aspheric lens with a focal distance of 1.45 mm and
numerical aperture of 0.58. The collected PL is dispersed in a two-stage
grating spectrometer, and recorded with a charge-coupled device (CCD) camera.

The changes in the PL spectral splitting $\Delta E_{\rm{PL}}$ of a negatively
charged trion $X^{-}$ [see Fig. 2(d)] is the hyperfine shift $E_{\rm{hf}}$,
which gives a measure of an average nuclear spin polarization degree within
the QD [13]. The hyperfine shifts (also known as Overhauser shifts) arise from
the $\hat{s}_{\rm{z}}\hat{I}_{{\rm{z}}}$ term of the hyperfine interaction
Hamiltonian (Eq. 1). Large nonequilibrium nuclear spin polarization is
generated on demand by exciting the QD with a circularly polarized pump laser,
which repeatedly injects spin-polarized electrons into a QD, and causes
nuclear spin polarization build up via electron-nuclear spin flip-flops
described by the
$\hat{s}_{\rm{x}}\hat{I}_{{\rm{x}}}+\hat{s}_{\rm{y}}\hat{I}_{{\rm{y}}}$ part
of Eq. 1. A small copper wire coil is placed near the sample to produce
radiofrequency (RF) oscillating magnetic field perpendicular to the static
magnetic field. Application of the RF field allows for the energy spectrum of
the nuclear spins to be probed via nuclear magnetic resonance (NMR). Moreover,
the RF field can be used to depolarize the nuclear spins on-demand. Further
details can be found in the Supplementary Section 2, including sample growth
details, PL spectra, characterization of QD charge state control, and
additional results at an elevated temperature of 15.2 K.

##  III Experimental results and discussion

###  III.1 Nuclear spin system of a GaAs quantum dot

Figure 2(e) shows NMR spectra of 75As in a single GaAs QD, measured using
“inverse NMR” technique with an optical Pump-RF-Probe cycle shown in the top
inset. For an empty QD (open symbols), an NMR triplet is observed [44],
corresponding to the three magnetic-dipole transitions between the four
Zeeman-split states $I_{\rm{z}}=\\{-3/2,-1/2,+1/2,+3/2\\}$ of a spin-3/2
nucleus (left inset). The central resolution-limited peak originates from the
$-1/2\leftrightarrow+1/2$ NMR transition that is weakly affected by strain.
The two satellite transition peaks $\pm 1/2\leftrightarrow\pm 3/2$ are split
from the central transition peak by the strain-induced EFGs. The average
splitting $\nu_{\rm{Q}}\approx 24$ kHz between the triplet components
corresponds to an average elastic strain of $\approx 2.6\times 10^{-4}$ (Refs.
[45, 46]). The satellite transitions are inhomogeneously broadened, with non-
zero NMR amplitudes detected approximately in a range of
$\nu_{\rm{Q}}\in[10,50]$ kHz, indicating that elastic strain varies within the
nanoscale volume of the QD.

When a single electron occupies the QD, it induces inhomogeneous Knight shifts
that exceed the quadrupolar shifts, leading to a broadened NMR peak [solid
symbols in Fig. 2(e)]. From the NMR peak width, the Knight shifts,
characterizing the typical coupling strength between the electron spin and an
individual nuclear spin, are estimated to be $A_{j}/h\approx 50$ kHz, where
$h$ is the Planck’s constant.

These NMR characterization results indicate a complex interplay of dipolar,
quadrupolar and hyperfine interactions governing the nuclear spin dynamics,
which we now investigate experimentally.

###  III.2 Nuclear spin relaxation in a GaAs quantum dot

In the nuclear spin relaxation (NSR) experiment [see timing diagram in Fig.
3(a)] any remnant nuclear spin polarization is first erased by saturating the
75As, 69Ga and 71Ga NMR resonances in the entire heterostructure [47]. This is
followed by a variable-duration ($T_{\rm{Pump}}$) optical pumping [16, 21, 18,
23] with photon energies below the AlGaAs barrier bandgap, which prepares
nuclear spin polarization localized around the QD nanoscale volume. After the
pump laser is turned off, the gate bias $V_{\rm{Gate}}$ is set to a desired
level for a dark time $T_{\rm{Dark}}$ – this way evolution under 0$e$ or 1$e$
QD charge state is studied for nominally identical initial nuclear spin
polarizations. Finally, the remaining polarization within the QD volume is
probed through an optically detected hyperfine shift $E_{\rm{hf}}$.

Figure 3:  (a) NSR measurement cycle starting with a radio frequency erase
pulse, followed by circularly polarized ($\sigma^{+}$) optical pumping and
then optical probing of the QD nuclear spin polarization after dark evolution
delay $T_{\rm{Dark}}$ (see details in Supplementary Section 2). The sample
gate bias $V_{\rm{Gate}}$ is varied allowing to choose between 0$e$ (dashed
line) and 1$e$ (dotted line) QD charge state during $T_{\rm{Dark}}$. (b) Dark
time dependence of the hyperfine shift $E_{\rm{hf}}$, which probes the average
polarization of $\approx 10^{5}$ QD nuclear spins, weighted by the QD electron
density $|\psi_{\rm{e}}|^{2}$. The nuclear spin decay is measured (symbols) at
$B_{\rm{z}}=0.39$ T for different pumping times $T_{\rm{Pump}}$ while keeping
the QD empty (0$e$, open symbols) or charged with one electron (1$e$, solid
symbols) during the dark time. Lines show numerical solution of the spin
diffusion Eq. 2. (c) Fitted QD nuclear spin half-life times $T_{\rm{1,N}}$
(right scale) and the corresponding NSR rates
$\varGamma_{\rm{N}}=1/T_{\rm{1,N}}$ (left scale) at magnetic field
$B_{\rm{z}}=0.39$ T. (d) same as (c) for $B_{\rm{z}}=9.82$ T. All results are
for the same individual dot QD1. Error bars are 95% confidence intervals.

Figure 3(b) shows the average QD nuclear spin polarization as a function of
the pump-probe delay $T_{\rm{Dark}}$ during which the sample is kept in the
dark. The decay is non-exponential, thus we characterize the NSR timescale
$T_{\rm{1,N}}$ by the half-life time over which the QD hyperfine shift
$E_{\rm{hf}}$ decays to 1/2 of its initial value. The NSR rate is then defined
as $\varGamma_{\rm{N}}=1/T_{\rm{1,N}}$. When the pumping time $T_{\rm{Pump}}$
is increased, $T_{\rm{1,N}}$ notably increases, as can be seen in Figs.
3(c,d). Such dependence of $T_{\rm{1,N}}$ on $T_{\rm{Pump}}$ is observed both
in an empty (0$e$) and charged (1$e$) QD states, and in a wide range of
magnetic fields.

###  III.3 Nuclear spin diffusion

In order to explain the results of Fig. 3, we note that nuclear spin dipole-
dipole interactions conserve the nuclear spin polarization for any magnetic
field exceeding the dipolar local field, typically $\lesssim 1$ mT. Therefore,
at high magnetic field the decay of nuclear spin polarization can proceed via
two routes: either via spin-conserving diffusion to the surrounding nuclei, or
spin transfer to external degrees of freedom, including quadrupolar coupling
to lattice vibrations [48, 17] or a hyperfine interaction with a charge spin
[17, 49, 50, 51] that is in turn coupled to the lattice or other charges. Spin
diffusion can only take place if the spatial profile of the initial nuclear
spin polarization is inhomogeneous. By contrast, direct spin-lattice and
hyperfine interactions have no explicit dependence on the spin polarization
spatial profile. Optical pumping that is short compared to spin diffusion
timescales creates nuclear spin polarization localized to the QD volume and
the resulting short $T_{\rm{1,N}}$ is therefore a clear indicator of spin
diffusion as the dominant NSR mechanism [16, 21, 18, 23]. Conversely, long
pumping provides enough time for nuclear polarization to diffuse from the QD
into the surrounding AlGaAs barriers, suppressing any subsequent spin
diffusion out of the QD and increasing $T_{\rm{1,N}}$, as observed in Figs.
3(c,d).

In order to complement our experimental investigation we model the
spatiotemporal evolution of the nuclear spin polarization degree
$P_{\rm{N}}(t,z)$ by solving numerically the one dimensional spin diffusion
equation

$\displaystyle\frac{\partial P_{\rm{N}}(t,z)}{\partial
t}=D(t)\frac{{\partial}^{2}P_{\rm{N}}(t,z)}{\partial z^{2}}+$
$\displaystyle+w(t)|\psi_{\rm{e}}(z)|^{2}(P_{\rm{N},0}-P_{\rm{N}}(t,z)),$ (2)

where the last term describes optical nuclear spin pumping with a rate
proportional to electron density $|\psi_{\rm{e}}(z)|^{2}$ and the time-
dependent factor $w(t)$ equal to 0 or $w_{0}$ when optical pumping is off or
on, respectively. Correspondingly, the spin diffusion coefficient $D(t)$ takes
two discrete values $D_{\rm{Dark}}$ or $D_{\rm{Pump}}$ when optical pumping is
off or on, respectively. $P_{\rm{N},0}$ is a steady state nuclear spin
polarization degree that optical pumping would generate in the absence of spin
diffusion. Eq. 2 is solved numerically and the parameters such as
$D^{(ne)}_{\rm{Dark}}$, $w_{0}(B_{\rm{z}})$, $D_{\rm{Pump}}(B_{\rm{z}})$ are
varied to achieve the best fit to the entire experimental datasets of
$E_{\rm{hf}}(T_{\rm{Pump}},T_{\rm{Dark}})$ measured at $B_{\rm{z}}=0.39,9.82$
T for empty ($n=0$) and charged ($n=1$) QD states. The best-fit calculated
dynamics are shown by the lines in Fig. 3(b) and capture well the main
features of the experimentally measured nuclear spin decay, confirming the
validity of the spin diffusion picture. The one-dimensional character of
diffusion, occurring predominantly along the sample growth $z$ direction, is
justified by the large ratio of the QD diameter $\approx 70$ nm to QD height
$<9~{}$nm, and is further verified by modeling two-dimensional spin diffusion
(see Supplementary Section 4).

###  III.4 Effect of central spin on nuclear spin diffusion

Dividing the typical Knight shift of $\approx 50$ kHz by half the QD thickness
(4.5 nm) we calculate the gradient and roughly estimate the Knight shift
difference of $\approx 4.4$ kHz for the two nearest-neighbor spins of the same
isotope separated by $a_{0}/\sqrt{2}$, where $a_{0}=0.565$ nm is the lattice
constant. Such difference significantly exceeds the energy that can be
exchanged with the dipole-dipole reservoir for a spin flip-flop to happen [26]
(the dipole-dipole energy is on the order of $\approx h/T_{2,{\rm{N}}}$, where
$T_{2,{\rm{N}}}\in[1,5]$ ms is the nuclear spin-echo coherence time [52, 34]).
Therefore, one may naively expect a Knight field gradient barrier to form and
suppress spin diffusion in an electron-charged QD (since the flip-flops would
be limited to the few nuclear spin pairs whose vector differences are nearly
orthogonal to the Knight field gradient). By contrast, Figs. 3(c,d) show that
in experiment the NSR is faster when the QD is occupied by a single electron
(1$e$, solid symbols) for all studied $T_{\rm{Pump}}$, demonstrating that no
significant Knight field barrier is formed. However, in order to quantify the
effect of the central spin on nuclear spin diffusion we must distinguish it
from other non-diffusion NSR mechanisms introduced by the electron spin. To
this end, we examine the magnetic field dependence shown in Fig. 4.

Figure 4:  (a) Nuclear spin relaxation (NSR) rate $\varGamma_{\rm{N}}$ as a
function of $B_{\rm{z}}$ measured in 0$e$ (open symbols) and 1$e$ (solid
symbols) states upon long pumping $T_{\rm{Pump}}=990$ s. Top horizontal axis
shows the electron Zeeman splitting at zero nuclear spin polarization. (b)
Ratio $\varGamma_{\rm{N}}^{(1e)}/\varGamma_{\rm{N}}^{(0e)}$ of the NSR rates
in 1$e$ and 0$e$ charge states as a function of $B_{\rm{z}}$ measured under
long pumping $T_{\rm{Pump}}=990$ s (squares) and short pumping
$T_{\rm{Pump}}\in[0.08,0.48]$ s (triangles). All results are for the same
individual dot QD1, except for the additional data from QD2 at
$B_{\rm{z}}=0.5$ T in (b). Error bars are 95% confidence intervals.

First, we examine a case where long optical pumping is used to suppress spin
diffusion and highlight the non-diffusion mechanisms. Fig. 4(a) shows the
experimental dependence $\varGamma_{\rm{N}}(B_{\rm{z}})$ for long
$T_{\rm{Pump}}=990$ s. In an empty QD (0$e$) spin diffusion is still the
dominant NSR mechanisms – indeed, the observed rates
$\varGamma_{\rm{N}}^{(0e)}\in[3\times 10^{-3},6\times 10^{-3}]$ s-1 are
considerably higher than $\varGamma_{\rm{N}}\in[6\times 10^{-5},1\times
10^{-3}]$ s-1, found in bulk crystals [48] such as semi-insulating GaAs [17],
where spin diffusion is negligible. The electron-induced rates under long
pumping $\varGamma_{\rm{N}}^{(1e)}\in[4\times 10^{-3},2\times 10^{-2}]$ s-1
are nearly independent of $B_{\rm{z}}$, and exceed the 0$e$ rates by no more
than a factor of $\varGamma_{\rm{N}}^{(1e)}/\varGamma_{\rm{N}}^{(0e)}<4$
[squares in Fig. 4(b)]. Such small effect of the electron is explained by the
small strain of the GaAs/AlGaAs structures, which reduces the efficiency of
the non-diffusion NSR mechanisms related to phonon and electron cotunneling.
This is in stark contrast to the large magnetic-field-induced variation
$\varGamma_{\rm{N}}^{(1e)}\in[5\times 10^{-4},1\times 10^{1}]$ s-1 in
Stranski-Krastanov self-assembled InGaAs QDs [51], where phonon and
cotunneling mechanisms dominate, both enabled by the noncollinear hyperfine
interaction [49, 51], arising in turn from the large strain-induced nuclear
quadrupolar shifts. Overall, the absolute QD NSR rates shown in Fig. 4(a) are
nearly constant, with some residual irregular dependence on magnetic field
which we ascribe to uncontrollable parameters, such as charge state of the
nearby impurities, or the initial spatial profile of the nuclear spin
polarization defined by the optical nuclear spin pumping rate. By contrast,
for any given $B_{\rm{z}}$ and $T_{\rm{Pump}}$ the ratio
$\varGamma_{\rm{N}}^{(1e)}/\varGamma_{\rm{N}}^{(0e)}$ shown in Fig. 4(b) gives
a reliable measure, which we use to examine the electron spin’s effect on spin
diffusion.

In order to discriminate the diffusion-related effect of the QD electron spin,
in addition to the long-pumping measurements [squares in Fig. 4(b)], we choose
for each magnetic field a short pumping time, typically
$T_{\rm{Pump}}\in[0.08,0.48]$ s, that yields initial QD nuclear spin
polarization at $\approx 1/2$ of the steady-state long-pumping polarization.
The resulting short-pumping ratio
$\varGamma_{\rm{N}}^{(1e)}/\varGamma_{\rm{N}}^{(0e)}$ is shown by the
triangles in Fig. 4(b) – its excess over the long-pumping ratio
$\varGamma_{\rm{N}}^{(1e)}/\varGamma_{\rm{N}}^{(0e)}$ is ascribed to spin
diffusion alone, discriminating it from any non-diffusion mechanisms
introduced by the electron spin. The electron-spin-induced acceleration of the
nuclear spin diffusion is seen to be particularly pronounced at low magnetic
fields $B_{\rm{z}}\lesssim 0.5$ T, consistent with the influence of the
electron-mediated nuclear-nuclear spin interaction [27, 24, 28]. Such pairwise
indirect interaction of nuclei $j$ and $k$ is derived from second order
perturbation expansion of Eq. 1:

$\displaystyle\mathcal{H}_{\rm{hf},j,k}^{\rm{ind}}\propto\frac{A_{j}A_{k}}{\Delta
E_{\rm{e}}}\hat{s}_{\rm{z}}\hat{I}^{(+)}_{j}\hat{I}^{(-)}_{k},$ (3)

where $\hat{I}_{j}^{(\pm)}=\hat{I}_{{\rm{x}},j}\pm i\hat{I}_{{\rm{y}},j}$ and
$\Delta E_{\rm{e}}=\mu_{\rm{B}}g_{\rm{e}}B_{\rm{z}}+E_{\rm{hf}}$ is the
electron spin splitting due to both the Zeeman effect and the nuclear-spin-
induced hyperfine shift $E_{\rm{hf}}$. In our experiments both contributions
are negative, so that any nuclear spin polarization increases $|\Delta
E_{\rm{e}}|$. The rate of the indirect nuclear-nuclear spin flip-flops scales
as $\propto\Delta E_{\rm{e}}^{-2}$. Consequently, the resulting acceleration
of nuclear spin diffusion in gate-defined GaAs QDs was previously found to be
limited to the low fields $B<0.02-0.75~{}$T (Refs. [24, 28, 53]). By contrast,
Fig. 4(b) shows that such acceleration persists at magnetic fields well above
$B_{\rm{z}}\gtrsim 2$ T, with short- and long-pumping
$\varGamma_{\rm{N}}^{(1e)}/\varGamma_{\rm{N}}^{(0e)}$ ratios converging only
at the maximum field $B_{\rm{z}}=9.82$ T.

One contributing factor is the smaller electron $g$-factor
$g_{\rm{e}}\approx-0.1$ (see Supplementary Section 2) and an order of
magnitude smaller number of nuclei in the studied epitaxial QDs, which result
in a smaller $|\Delta E_{\rm{e}}|$ and larger $A_{j}$, respectively, when
compared to the gate-defined QDs with $g_{\rm{e}}\approx-0.4$. While these
factors lead to a stronger hyperfine-mediated couplings in the studied QDs,
they do not explain the magnetic field dependence: At high field
$B_{\rm{z}}=9.82$ T the electron spin Zeeman splitting is $|\Delta
E_{\rm{e}}|\approx 58~{}\mu$eV, whereas at low field $B_{\rm{z}}=0.39$ T we
take into account both the Zeeman splitting $\approx-2.3~{}\mu$eV and the
time-averaged hyperfine shift $E_{\rm{hf}}\approx-2.5~{}\mu$eV (half of the
initial $E_{\rm{hf}}\approx-5~{}\mu$eV under the shortest used
$T_{\rm{Pump}}\approx 8~{}$ms) to estimate $|\Delta E_{\rm{e}}|\approx
5~{}\mu$eV. This would correspond to a factor of $(58/5)^{2}\approx 130$
reduction in the hyperfine mediated rates, while the measured short-pumping-
limit NSR rate reduces only by a factor of $\approx 6$ from
$\varGamma_{\rm{N}}^{(1e)}\approx 0.74~{}{\rm{s}}^{-1}$ at low field [Fig.
3(c)] to $\varGamma_{\rm{N}}^{(1e)}\approx 0.12~{}{\rm{s}}^{-1}$ at high field
[Fig. 3(d)]. Prompted by these observations, we point out that Eq. 3 treats
electron spin as isolated, while in a real system the electron is coupled to
external environments such as phonons and other charges. A fluctuating
electron spin can accelerate nuclear spin diffusion, provided there is a
frequency component in the time-dependent Knight field that equals the energy
mismatch of a pair of nuclei [54, 55] – this contribution has been considered
for deep impurities [15], but has been previously ignored in the context of
III-V semiconductor nanostructures. For similar GaAs/AlGaAs QDs [38] the
electron spin lifetime was reported to be 48 $\mu$s, while for InGaAs/GaAs QDs
with a similar tunnel coupling it was found to vary with magnetic field
between $\approx$50 $\mu$s and a few milliseconds [51]. These lifetimes
correspond to fluctuation frequencies in a $[1,10]$ kHz range, indeed matching
the typical differences in the nuclear spin energies as revealed by NMR
spectra of Fig. 2(e). Thus we speculate that the intrinsic electron spin
flips, governed e.g. by the phonon relaxation and cotunneling coupling to the
electron Fermi reservoir of the $n$-doped layer [56, 57, 51], contribute to
acceleration of nuclear spin diffusion in the studied GaAs QDs, especially at
high magnetic fields.

###  III.5 Comparison with previous results on nuclear spin diffusion

In order to understand what controls the rate of spin diffusion we first make
a comparison with Stranski-Krastanov InGaAs/GaAs and InP/GaInP self-assembled
QDs, where quadrupolar shifts are so large (MHz range [58, 41]) that all
nuclear spins are essentially isolated from each other, eliminating spin
diffusion and resulting in very long nuclear spin lifetimes
$T_{\rm{1,N}}^{(0e)}>10^{4}~{}$s in empty (0$e$) QDs [30, 59, 27, 60, 49, 49,
51]. Even in presence of the electron spin (1$e$) the nuclear spin diffusion
takes place only inside the QD [27, 49], without diffusion into the
surrounding material.

In the lattice-matched GaAs QDs the strain-induced effects are smaller but not
negligible, characterized by quadrupolar shifts $\nu_{\rm{Q}}$ ranging
approximately between 10 and 50 kHz within the QD, as revealed by NMR spectra
in Fig. 2(e). Nuclei in $I_{\rm{z}}=\pm 1/2$ and $|I_{\rm{z}}|>1/2$ states
must be considered separately. The central transition between the
$I_{\rm{z}}=-1/2$ and $+1/2$ spin states is affected only by the second order
quadrupolar shifts, which scale as $\propto\nu_{\rm{Q}}^{2}/\nu_{\rm{L}}$ and
are within a few kHz for the studied range of nuclear spin Larmor frequencies
$\nu_{\rm{L}}\in[1,130]$ MHz. These second order quadrupolar shifts are
comparable to the homogeneous nuclear spin linewidth $\propto
1/T_{2,{\rm{N}}}$, and therefore spin diffusion in GaAs/AlGaAs QDs is expected
to be unimpeded for the nuclei in the $I_{\rm{z}}=\pm 1/2$ states. By
contrast, the $I_{\rm{z}}=\pm 3/2$ spin states experience first order
quadrupolar shifts $\nu_{\rm{Q}}$, which are tens of kHz, significantly
exceeding the homogeneous NMR linewidths in the studied GaAs QDs. The
resulting dynamics of the $I_{\rm{z}}=\pm 3/2$ nuclei is therefore sensitive
to nanoscale inhomogeneity of the strain-induced $\nu_{\rm{Q}}$. From the NSR
experiments [Fig. 3(b)] we observe that nuclear spin polarization relaxes to
zero, even in an empty QD (0$e$). This can only happen if spin diffusion is
unimpeded not only for the $I_{\rm{z}}=\pm 1/2$ states, but also for the
$I_{\rm{z}}=\pm 3/2$ states that are subject to the larger first order
quadrupolar shifts. Our interpretation is that strain in the studied
GaAs/AlGaAs QDs is a smooth function of spatial coordinates: for nearly each
QD nucleus it is possible to find some neighboring nuclei with a strain
variation small enough to form a chain that conducts spin diffusion out of the
GaAs QD into the AlGaAs barriers.

Similarly fast NSR was observed previously in neutral QDs formed by monolayer
fluctuations in GaAs/AlGaAs quantum wells [23]. However, the opposite scenario
was realized in QDs with nanoholes etched in pure GaAs [44] where nuclear spin
polarization in an empty QD (0$e$) was preserved for over
$T_{1,{\rm{N}}}>5000$ s, suggesting that some of the nuclei were frozen in the
$I_{\rm{z}}=\pm 3/2$ states, akin to quadrupolar blockade of spin diffusion in
self-assembled QDs. This contrast is rather remarkable since the average
strain, characterized by the average $\nu_{\rm{Q}}\approx 20-30$ kHz, is very
similar for QDs grown in nanoholes etched in AlGaAs (studied here) and in GaAs
(Ref. [44]). This comparison suggests that nuclear spin dynamics are sensitive
to QD morphology down to the atomic scale, and could be affected by factors
such as QD shape, as well as GaAs/AlGaAs interface roughness and intermixing
[61, 62, 63]. One possible contributing factor is the QD growth temperature,
which was 610∘ C in the structures used here, considerably higher than 520∘ C
in the structures studied previously [43, 44]. Further work would be required
to elucidate the role of all the underlying growth parameters. Conversely, NSR
can be a sensitive probe of the QD internal structure.

We now quantify the spin diffusion process and compare our results to the
earlier studies in GaAs-based structures. The best fit of the experimental NSR
dynamics [lines in Fig. 3(b)] yields $D^{(0e)}_{\rm{Dark}}=2.2^{+0.7}_{-0.5}$
nm2 s-1 for the diffusion coefficient in an empty QD and in the absence of
optical excitation, in agreement with $D=1.0\pm 0.15$ nm2 s-1 measured
previously for spin diffusion between two GaAs quantum wells across an
Al0.35Ga0.65As barrier [35]. This is approximately an order of magnitude
smaller than the first-principle estimate [64, 65, 66] of
$D^{(0e)}_{\rm{Dark}}\approx 19$ nm2 s-1 for bulk GaAs (see Supplementary
Section 3) and the $D=15.0\pm 7$ nm2 s-1 value measured in pure AlAs [67]. The
reduced diffusion in the AlGaAs alloy can be explained by the quadrupolar
disorder, arising from the random positioning of the aluminium atoms [35].
Charging of the QD with a single electron accelerates spin diffusion: we find
$D^{(1e)}_{\rm{Dark}}(9.82~{}{\rm{T}})=4.7^{+1.2}_{-1.0}$ nm2 s-1, which
increases to $D^{(1e)}_{\rm{Dark}}(0.39~{}{\rm{T}})=7.7\pm 1.9$ nm2 s-1 at low
magnetic fields where hyperfine-mediated nuclear-nuclear spin exchange is
enhanced in accordance with Eq. 3. While experimental data can be well
described by the spin diffusion Eq. 2, it is worth noting the limited nature
of the model, which ignores the spatial variations of the nuclear-nuclear
couplings, the dependence of the electrons spin splitting $\Delta E_{\rm{e}}$
on the instantaneous nuclear spin polarization, the isotopic difference
between 75As, 69Ga and 71Ga, as well as neglecting any spin diffusion
orthogonal to the sample growth $z$ direction. As such, the diffusion
coefficients $D$ should be treated as a coarse-grained description,
aggregating the numerous lattice-constant-scale parameters of the many-body
spin ensemble evolution.

##  IV Discussion and Outlook

The GaAs/AlGaAs QDs grown by nanohole infilling combine excellent optical
properties with low intrinsic strain, allowing for nuclear spin qubit and
quantum memory designs [33, 34, 40]. The key performance characteristic is the
nuclear spin coherence time, which can be extended up to
$T_{2,{\rm{N}}}\approx 10~{}$ms (Ref. [34]), but is ultimately limited by the
longitudinal relaxation time $T_{\rm{1,N}}$. Moreover, it is the state
longevity of the nuclei interfaced with the QD electron spin that is relevant.
Thus, one should consider the NSR time in the regime of short pumping, found
here to range from $T_{\rm{1,N}}^{(1e)}\approx 1~{}$s at low magnetic fields
to $T_{\rm{1,N}}^{(1e)}\approx 10~{}$s at high fields. For nuclear spin
quantum computing with the typical 10 $\mu$s coherent control gates [34], a
large number of operations $\gtrsim 10^{5}$ would be possible without the
disruptive effect of spin diffusion.

In conclusion, we have addressed the long-standing dilemma of whether the
central spin of an electron accelerates or suppresses diffusion in a nuclear
spin lattice. We have used variable-duration optical pumping [16, 21, 18, 23]
to identify nuclear spin diffusion as the dominant NSR mechanism. In contrast
to previous studies of nuclear spin diffusion [16, 17, 21, 22, 25, 26], we use
a charge tunable structure and probe nuclear spin dynamics with and without
the electron under otherwise identical conditions – importantly, our QD charge
control is achieved without reverting to optical pumping [25, 26], thus
eliminating the unwanted charge fluctuations. Combining these two aspects, we
conclude that in a technologically important class of lattice matched
GaAs/AlGaAs nanostructures the electron spin accelerates the nuclear spin
diffusion, with no signature of a Knight-field-gradient barrier. We expect
these findings to be relevant for a range of lattice-matched QDs [23, 24, 28,
53] and shallow impurities [17], whereas an efficient spin diffusion barrier
can arise from an electron with sub-nanometer localization [15]. Future work
can examine reduction of spin diffusion in low-strain nanostructures. The
proximity of the $n$-doped layer, acting as a sink for nuclear polarization,
as well as QD morphology can be optimized. Alternatively, pure AlAs barriers
can be used to grow GaAs QDs with well isolated Ga nuclei, potentially
offering long-lived spin memories and qubits.

###### Acknowledgements.

Acknowledgements: P.M-H. and E.A.C. were supported by EPSRC through a doctoral
training grant and EP/V048333/1, respectively. E.A.C. was supported by a Royal
Society University Research Fellowship. A.R. acknowledges support of the
Austrian Science Fund (FWF) via the Research Group FG5, I 4320, I 4380, I
3762, the European Union’s Horizon 2020 research and innovation program under
Grant Agreements No. 899814 (Qurope) and No. 871130 (Ascent+), the Linz
Institute of Technology (LIT), and the LIT Secure and Correct Systems Lab,
supported by the State of Upper Austria. E.A.C is grateful to René Dost for
advice on sample processing. Author contributions: S.M., S.F.C.S and A.R.
developed, grew and processed the quantum dot samples. P.M-H. and E.A.C.
conducted the experiments. E.A.C. drafted the manuscript with input from all
authors. E.A.C. coordinated the project.

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## Supplementary Material

##  Supplementary Section 1 Sample structure

The sample is grown using molecular beam epitaxy (MBE) on a semi-insulating
GaAs (001) substrate. The growth starts with a layer of Al0.95Ga0.05As
followed by a single pair of Al0.2Ga0.8As and Al0.95Ga0.05As layers acting as
a Bragg reflector in optical experiments. Then, a 95 nm thick layer of
Al0.15Ga0.85As is grown. The rest of the structure follows the schematic shown
in Fig. 2(c) of the main text beginning with a 95 nm thick layer of
Al0.15Ga0.85As doped with Si at a volume concentration of $1.0\times 10^{18}$
cm-3. The low Al concentration of $0.15$ in the Si doped layer mitigates the
issues caused by the deep DX centers [3, 4, 5]. Under optical excitation this
Al0.15Ga0.85As:Si gives rise to broad photoluminescence at around 730 nm as
observed in Supplementary Fig. 5(a). The $n$-type doped layer is followed by
the electron tunnel barrier layers: first a 15 nm thick Al0.15Ga0.85As layer
and then a 15 nm thick Al0.33Ga0.67As layer. Aluminium droplets are grown on
the surface of the Al0.33Ga0.67As layer and are used to etch the nanoholes [6,
7]. An atomic force microscopy (AFM) image of a similar sample in Fig. 2(a) of
the main text shows a typical nanohole with a depth of $\approx 6.5$ nm and
$\approx 70$ nm in diameter. Next, a 2.1 nm thick layer of GaAs is grown to
form QDs by infilling the nanoholes as well as to form the quantum well (QW)
layer. Thus, the maximum height of the QDs in the growth $z$ direction is
$\approx 9$ nm. Low temperature PL of QDs and QW is observed [Supplementary
Fig. 5(a)] at 785 nm and 690 nm, respectively. The GaAs layer is followed by a
268 nm thick Al0.33Ga0.67As barrier layer. Finally, the $p$-type contact
layers doped with C are grown: a 65 nm thick layer of Al0.15Ga0.85As with a
$5\times 10^{18}$ cm-3 doping concentration, followed by a 5 nm thick layer of
Al0.15Ga0.85As with a $9\times 10^{18}$ cm-3 concentration, and a 10 nm thick
layer of GaAs with a $9\times 10^{18}$ cm-3 concentration.

The sample is processed into a $p-i-n$ diode structure. Mesa structures with a
height of 250 nm are formed by etching away the $p$-doped layers and
depositing Ni(10 nm)/AuGe(150 nm)/Ni(40 nm)/Au(100 nm) on the etched areas.
The sample is then annealed to enable diffusion down to the $n$-doped layer to
form the ohmic back contact. The top gate contact is formed by depositing
Ti(15 nm)/Au(100 nm) on to the $p$-type surface of the mesa areas. The sample
gate bias $V_{\rm{Gate}}$ is the bias of the $p$-type top contact with respect
to the grounded $n$-type back contact. By changing $V_{\rm{Gate}}$ the
equilibrium charge state of the quantum dot is tuned using the Coulomb
blockade effect. Due to the large thickness of the top Al0.33Ga0.67As layer,
the tunneling of the holes is effectively blocked, whereas tunnel coupling to
the $n$-type layer enables deterministic charging of the quantum dots with
electrons.

Supplementary Figure 5:  (a) Broad range photoluminescence (PL) spectra
measured under 532 nm laser excitation at different gate biases
$V_{\rm{Gate}}$. Spectra are offset in a vertical direction (log scale) by a
factor of 10 for clarity. Spectral features arising from the different parts
of the sample are labeled accordingly. (b) High resolution PL spectra of a
negatively charged $X^{-}$ trion following $\sigma^{+}$ (triangles) and
$\sigma^{-}$ (squares) circularly polarized optical pumping, which creates
$s_{\rm{z}}=-1/2$ ($\downarrow$) and $s_{\rm{z}}=+1/2$ ($\uparrow$) spin
polarized electrons, respectively. The electrons transfer their spin to the
nuclei via magnetic (hyperfine) interaction, resulting in a build up of
negative or positive net nuclear spin polarization, respectively. Through the
same hyperfine interaction, the average nuclear spin polarization shifts the
$s_{\rm{z}}=-1/2$ and $s_{\rm{z}}=+1/2$ electron spin energy levels in the
opposite directions. These (Overhauser) shifts $E_{\rm{hf}}$ lead to the
observed change in the spectral splitting of the trion PL, where the two
components of the doublet correspond to an electron-hole recombination in
presence of another electron in a $s_{\rm{z}}=-1/2$ or $s_{\rm{z}}=+1/2$
state. (c) Energy level diagram. The electron ground state is split by the
Zeeman energy $\mu_{\rm{B}}g_{\rm{e}}B_{\rm{z}}$ and the hyperfine shift
$E_{\rm{hf}}$. The $X^{-}$ trion energy includes the QD bandgap energy
$E_{\rm{gap}}$ and the Zeeman splitting of the unpaired hole with a positive
($\Uparrow$) or negative ($\Downarrow$) momentum projection. The valence band
hole hyperfine effect can be neglected due to its smaller magnitude [8]. The
electron and hole $g$-factors are $g_{\rm{e}}$ and $g_{\rm{h}}$, respectively,
with $|g_{\rm{h}}|\gg|g_{\rm{e}}|$ in the studied QDs. Solid arrows depict the
two optically allowed transitions responsible for the spectral doublet in (b).
The dashed lines show the two forbidden “diagonal” transitions.

##  Supplementary Section 2 Experimental details and additional results

The sample is placed in a liquid helium bath cryostat. A superconducting coil
is used to apply magnetic field up to $B_{\rm{z}}=10$ T. The field is parallel
to the sample growth direction and the optical axis $z$ (Faraday geometry). We
use confocal microscopy configuration. An aspheric lens with a focal distance
of 1.45 mm and NA=0.58 is used as an objective for optical excitation of the
QD and for photoluminescence (PL) collection. The excitation laser is focused
into a spot with a diameter of $\approx 1~{}\mu$m. The collected PL is
dispersed in a two-stage grating spectrometer, each stage with a 0.85 m focal
length, and recorded with a charge-coupled device (CCD) camera. The changes in
the spectral splitting of a negatively charged trion $X^{-}$, derived from the
PL spectra, are used to measure the hyperfine shifts $E_{\rm{hf}}$
proportional to the nuclear spin polarization degree.

Investigation of spin diffusion relies on the ability to prepare a
reproducible spatial distribution of the nuclear spin polarization. This is
achieved with a radiofrequency (RF) erase pulse (Fig. 3(a) of the main text)
which effectively resets the nuclear spin polarization to zero in the entire
sample by saturating the nuclear magnetic resonance of the As and Ga isotopes.
The required oscillating magnetic field $B_{\rm{x}}\perp z$ is produced by a
coil placed at a distance of $\approx 0.5$ mm from the QD sample. The coil is
made of 10 turns of a 0.1 mm diameter enameled copper wire wound on a $\approx
0.4$ mm diameter spool in 5 layers, with 2 turns in each layer. The coil is
driven by a class-A RF amplifier (rated up to 20 W) which is fed by the output
of an arbitrary waveform generator. The spectrum of the RF excitation consists
of three bands, each 340 kHz wide and centered on the NMR frequency of the
corresponding As or Ga isotope. Each band is generated as a frequency comb [8]
with a mode spacing of 120 Hz, much smaller than the homogeneous NMR
linewidth. The RF power density in the comb is chosen to be low enough and the
RF pulse duration long enough (ranging between 0.1 and 10 s depending on
magnetic field) to achieve noncoherent exponential depolarization of the
nuclear spin ensemble.

Optical pumping of the QD nuclear spin polarization is achieved using the
emission of a 690 nm circularly polarized diode lasers, which is resonant with
the GaAs QW states, as seen in Supplementary Fig. 5(a). It is therefore
possible that optical pumping results in dynamical nuclear polarization not
only in the QDs but also in the adjacent parts of the QW. On the other hand,
the pump laser photon energy is well below the bandgap of the AlGaAs barriers.
For that reason we assume that dynamic nuclear polarization in AlGaAs is
induced only through spin diffusion from the GaAs layer of the QW and QDs.
During the pump the sample gate is set to a large reverse bias, typically
$V_{\rm{Gate}}=-2$ V. The pump power is $\approx 300~{}\mu$W, which is two
orders of magnitude higher than the ground-state PL saturation power. The
resulting hyperfine shifts do not exceed $|E_{\rm{hf}}|<50~{}\mu$eV,
corresponding to initial nuclear spin polarization degree within
$|P_{\rm{N},0}|\lesssim 0.4$. While polarization as high as
$P_{\rm{N},0}\approx 0.8$ is possible [9], we deliberately use lower values to
ensure linear regime of spin diffusion, free from hyperpolarization regime
corrections [10]. For optical probing of the nuclear spin polarization we use
a diode laser emitting at 640 nm. Sample forward bias, typically $+0.5$ V, and
the probe power are chosen to maximize (saturate) PL intensity of the ground
state $X^{-}$ trion. The duration of the probe pulse, typically 20 ms, is
selected to ensure minimal (few percent) depolarization of the nuclear spins
during probing. Supplementary Fig. 5(b) shows $X^{-}$ PL probe spectra
measured at $B_{\rm{z}}=1$ T following optical pumping with $\sigma^{+}$
(triangles) and $\sigma^{-}$ (squares) circular polarization. The difference
in spectral splitting of the $X^{-}$ trion doublet reveals the hyperfine
shifts $E_{\rm{hf}}$ [see energy level diagram in Supplementary Fig. 5(c)].
These shifts are used to monitor the average QD nuclear spin polarization in
NSR experiments such as shown in Fig. 3(b) of the main text.

The energy splitting of the two electron spin states $\Delta E_{\rm{e}}$ is a
sum of the Zeeman splitting $\mu_{\rm{B}}g_{\rm{e}}B_{\rm{z}}$ (where
$\mu_{\rm{B}}$ is the Bohr magneton) and the hyperfine splitting
$E_{\rm{hf}}$, arising from the nuclear spin polarization. We quantify the
$g$-factor $g_{\rm{e}}$ of a resident electron using photoluminescence
spectroscopy of a negatively charged trion. In Faraday geometry, two out of
four optical transitions are forbidden, so that only the difference
$g_{\rm{h}}-g_{\rm{e}}$ of the heavy hole and electron $g$-factors can be
accessed. In order to derive the individual $g$-factors, we measure
photoluminescence in oblique field configuration, where the sample growth axis
is tilted by $\theta\approx 12^{\circ}$ away from the static magnetic field.
In this configuration the “diagonal” transitions, shown by the dashed lines in
Supplementary Fig. 5(c), become weakly allowed. Owing to the nearly vanishing
electron $g$-factor in this type of GaAs/AlGaAs QDs [11], all four $X^{-}$
transitions can be resolved in our setup only in high magnetic field $B=10$ T
and in presence of the optically induced hyperfine shifts. The experiment is
conducted using an optical pump-probe method. The probe PL spectra are shown
in Supplementary Fig. 6 as a function of the half-wave plate angle. The angle
is varied to control the degree of circular polarization of the pump laser and
the resulting hyperfine shift $E_{\rm{hf}}$. The two weak transitions (labeled
by the arrows) become visible when the splitting of the two bright transitions
is maximized by the hyperfine shift. We further measure the spectral splitting
of the two bright transitions after RF depolarization of the nuclei, which
results in $E_{\rm{hf}}\approx 0$. It is then possible to perform linear fit
of the PL energies of all four $X^{-}$ transitions and derive the $g$-factors.
For QD1 studied in the main text and examined in Supplementary Fig. 6 we find
the 95% confidence estimate $g_{\rm{e}}\approx-0.101\pm 0.007$ for the
$g$-factor of a single resident electron. From the hole spin splitting of
$X^{-}$ at $B=10$ T we estimate the hole $g$-factor in presence of two
electrons as $g_{\rm{h}}\approx+1.68$. This value should be treated as a rough
estimate because of the significant nonlinearity in hole Zeeman splitting for
this type of QDs [11]. We also measure the $g$-factors in a neutral exciton
$X^{0}$, using PL of the dark states: we find $g_{\rm{e}}\approx-0.090\pm
0.035$ for the electron in presence of one hole. It is notable that the
electron $g$-factor is nearly unaffected by the extra hole [12]. Using PL
spectroscopy of the $X^{-}$ trion state, we have measured $g$-factors in two
more QDs from the same sample to find $g_{\rm{e}}\approx-0.077\pm 0.018$ and
$g_{\rm{e}}\approx-0.107\pm 0.002$ for a resident electron in QD2 and QD3,
respectively. From the $X^{0}$ PL of QD2 we find $g_{\rm{e}}\approx-0.12\pm
0.01$ for an electron in presence of a hole, whereas no dark excitons could be
observed in QD3. The $g$-factors found here are in good agreement with the
previous studies on the samples where QDs were grown in nanoholes etched in
pure GaAs [11].

Supplementary Figure 6:  Photoluminescence spectra of a negatively charged
trion $X^{-}$ measured in oblique magnetic field $B=10$ T tilted by
$\theta\approx 12^{\circ}$ from the Faraday geometry. The measurement uses a
pump-probe protocol, where the angle of a half-wave plate on the pump laser is
varied, while the probe laser is used to detect the resulting changes in PL
spectrum. The two bright lines correspond to the two allowed transitions. When
the circularly polarized pump generates a sufficiently large hyperfine shift,
the two weakly allowed trion transitions, labeled by the arrows, become
resolved. Fitting of the PL energies reveals the electron and hole
$g$-factors. Other (broad) spectral features correspond to PL of excitons
charged with more than one electron.

The sample gate bias $V_{\rm{Gate}}$ is controlled by the output of an
arbitrary waveform generator connected through a 1.9 MHz low pass filter.
During the dark evolution time $T_{\rm{Dark}}$ the bias can be set to an
arbitrary value. For an empty dot regime (0$e$) we use large reverse bias
$V_{\rm{Gate}}=-1.3$ V. The bias corresponding to 1$e$ Coulomb blockade is
found by measuring the bias dependence of $\varGamma_{\rm{N}}(V_{\rm{Gate}})$
such as shown in Supplementary Fig. 7. In agreement with the previous studies
on InGaAs QDs [13, 14] we observe tunnelling peaks (at $\approx 0.43$ V and
$\approx 0.6$ V), where the electron Fermi reservoir energy matches the QD
charging energy. A bias at the middle of the Coulomb valley between the peaks,
0.517 V in this case, is used to charge the QD with one electron (1$e$).
Supplementary Fig. 7 shows that when the QD is charged with two electrons
(2$e$) the NSR rate is identical to the 0$e$ case, confirming that the NSR
acceleration produced by the single electron (1$e$) is related to its spin.

Supplementary Figure 7:  Bias dependence of the nuclear spin relaxation rate
$\varGamma_{\rm{N}}$.

Optical pump and probe pulses are formed by mechanical shutters with a
switching time of a few milliseconds. Under certain regimes in $B_{\rm{z}}$
and $V_{\rm{Gate}}$ this is comparable to the nuclear spin relaxation times
$T_{\rm{1,N}}$. However, the relaxation time in an empty QD (0$e$) is always
considerably longer. Thus we keep QD under 0$e$ bias during the shutter
switching times and the dark time $T_{\rm{Dark}}$ is implemented by pulsing
the gate bias to the target dark-state value $V_{\rm{Gate}}$ for a duration
$T_{\rm{Dark}}$.

Nuclear magnetic resonance (NMR) characterization (Fig. 2(e) of the main text)
is conducted using inverse NMR method [15] which enhances the signal for
$I>1/2$ spins and improves the signal to noise ratio. In this method the
nuclei are depolarized by a weak RF field, whose spectral profile is a
broadband frequency comb with a narrow gap of width $w_{\rm{gap}}$. The value
of $w_{\rm{gap}}$ controls the balance between NMR amplitude and spectral
resolution. In an empty QD (0$e$), NMR spectra of As and Ga measured with
$w_{\rm{gap}}=6$ kHz consist of well-resolved quadrupolar-split triplets,
consistent with previous observations for similar QD structures [11, 16]. The
spin of a single electron (1$e$) leads to inhomogeneous Knight shifts
comparable to the quadrupolar splitting. As a result, the NMR triplets are no
longer resolved (solid circles in Fig. 2(e) of the main text). Moreover, the
electron spin lifetime, which is on the order of milliseconds in the studied
QDs, is much shorter than the radio frequency burst (typically 0.18 s), and
the average electron spin polarization is therefore close to 0. Each nucleus
then experiences both positive and negative Knight shifts during the RF burst.
These dynamic spectral shifts disrupt the enhancement of the inverse NMR
method: for example, if a nuclear spin transition fits into the RF spectral
gap $w_{\rm{gap}}$ under one sign of the Knight shift, it may be moved out of
the gap and into resonance with the depolarizing RF field under the opposite
Knight shift. As a result, the NMR spectrum amplitude is reduced in the 1$e$
measurement. By varying the gap width $w_{\rm{gap}}$, we find that a spectrum
with a reasonable signal to noise ratio is obtained at $w_{\rm{gap}}=70$ kHz,
as shown by the circles in Fig. 2(e) of the main text. Although the
deterioration of the inverse NMR method precludes an accurate measurement of
the NMR lineshape in presence of the electron, the overall width $\approx 50$
kHz of the resonance still provides a valid order-of-magnitude estimate of the
Knight shifts experienced by the nuclear spins in the QD. More sophisticated
measurements, using pulsed NMR (to be reported separately elsewhere) confirm
this rough estimate based on inverse NMR measurement.

Supplementary Figure 8:  (a) Dark time dependence of the hyperfine shift
$E_{\rm{hf}}$, which probes average nuclear spin polarization weighted by the
QD electron density $|\psi_{\rm{e}}|^{2}$. Nuclear spin decay is measured
(symbols) at $B_{\rm{z}}=0.39$ T for different pumping times $T_{\rm{Pump}}$
while keeping QD empty (0$e$, open symbols) or electron-charged (1$e$, solid
symbols) during the dark time. Lines show numerical solution of the spin
diffusion equation. (b) Same as (a) but for $B_{\rm{z}}=9.82$ T.

The QD NSR curves measured at $B_{\rm{z}}=0.39$ T and shown in Fig. 3(b) of
the main text are reproduced in Supplementary Fig. 8(a), together with the
similar measurements carried out at high magnetic field $B_{\rm{z}}=9.82$ T
and shown in Supplementary Fig. 8(b). Similar to the low fields, at
$B_{\rm{z}}=9.82$ T shorter optical pumping time $T_{\rm{Pump}}$ results in
faster NSR through spin diffusion. However, the acceleration of NSR in
presence of a single electron (1$e$) is less pronounced at high magnetic
field, owing to the reduction of the hyperfine-mediated nuclear-nuclear spin
interaction. It is also worth noting that the optical spin pumping becomes
slower at high magnetic field. While $T_{\rm{Pump}}=0.018$ s at
$B_{\rm{z}}=0.39$ T is sufficient to achieve $\approx 1/4$ of the steady state
nuclear spin polarization, it takes an order of magnitude longer
$T_{\rm{Pump}}=0.17$ s to reach the same $\approx 1/4$ level at
$B_{\rm{z}}=9.82$ T. This difference limits the shortest $T_{\rm{Pump}}$ for
which NSR dynamics can be measured at high magnetic field, as can be seen in
Fig. 3(d) of the main text.

Supplementary Figure 9:  Fitted QD nuclear spin half-life times $T_{\rm{1,N}}$
(right scale) and corresponding NSR rates $\varGamma_{\rm{N}}=1/T_{\rm{1,N}}$
(left scale) measured for different pumping times $T_{\rm{Pump}}$ at
$B_{\rm{z}}=0.5$ T. Experiments are conducted at base sample temperature
($T=4.27$ K, squares) and an elevated temperature ($T=15.2$ K, triangles).

The experiments presented in the main text are conducted at the cryostat base
temperature, measured with a resistive sensor to be $T\approx 4.27$ K.
Additional measurements, similar to those shown in Fig. 3(c,d) of the main
text, have been conducted on an empty QD (0$e$) at an elevated temperature
$T=15.2$ K and are shown in Supplementary Fig. 9. We find that at high
temperature the relaxation rate follows the same trend of reduction at short
pumping times $T_{\rm{Pump}}$, consistent with NSR dominated by spin
diffusion. In case of a pure spin diffusion driven by nuclear dipole-dipole
interactions, one would expect the rate to be independent of the temperature.
From Supplementary Fig. 9 we find that for any given $T_{\rm{Pump}}$ the
relaxation is slightly accelerated at $T=15.2$ K. One possibility is the
temperature dependence of the optical nuclear spin pumping process [17]
creating different spatial distributions of the nuclear spin polarization for
the same $T_{\rm{Pump}}$. Contribution of the temperature-dependent non-
diffusion mechanisms, such as two-phonon quadrupolar relaxation [18, 19], is
also possible.

##  Supplementary Section 3 First principle estimate of the GaAs nuclear spin
diffusion coefficient

In the absence of free electrons, nuclear spin diffusion is driven by the
dipole-dipole magnetic nuclear spin interaction. The total dipole-dipole
Hamiltonian term is a sum of pairwise couplings:

$\displaystyle\mathcal{H}_{\rm{DD}}=\sum_{i<j}b_{i,j}\left(2\hat{I}_{{\rm{z}},i}\hat{I}_{{\rm{z}},j}-\hat{I}_{{\rm{x}},i}\hat{I}_{{\rm{x}},j}-\hat{I}_{{\rm{y}},i}\hat{I}_{{\rm{y}},j}\right){\rm{,}}$
$\displaystyle
b_{i,j}=\frac{\mu_{0}}{4\pi}\frac{\gamma_{i}\gamma_{j}}{2}\frac{1-3\cos^{2}{\theta_{i,j}}}{r_{i,j}^{3}},$
(S4)

where $\mu_{0}=4\pi\times 10^{-7}\;{\rm{NA}}^{-2}$ is the magnetic constant
and $r_{i,j}$ denotes the length of the vector, which forms an angle $\theta$
with the static magnetic field direction ($z$) and connects the two spins $i$
and $j$. The typical magnitude of the interaction constants for the nearby
nuclei in GaAs is $\max{(|b_{j,k}|)}/h\approx 100$ Hz. The Hamiltonian of Eq.
S4 has been truncated to eliminate all spin non-conserving terms, which is
justified for static magnetic fields exceeding $\gtrsim 1$ mT, as used in this
work. The evolution of a large nuclear spin ensemble can be described in terms
of spin diffusion with coefficient $D$. In crystalline solids the nuclear spin
diffusion coefficient $D$, is a rank-2 tensor which can be calculated from the
first principles using density matrix approach [20] or the method of moments
[21, 22]. The calculation involves a somewhat lengthy evaluation of the
various lattice sums. Here we use a more recent version of the method of
moments from Ref. [23]. We re-evaluate numerically the sums of Eqns. 8 and 10
from Ref. [23] using an FCC lattice of 6859 spins. Our results are in good
agreement with the those derived for 1330 neighboring spins previously [23].
We find the following values for the diagonal components of $D$:
$D_{xx}=D_{yy}\approx
0.2594\frac{\mu_{0}}{4\pi}\frac{\hbar\gamma^{2}}{a_{0}}\rho^{1/3}$ and
$D_{zz}\approx
0.3289\frac{\mu_{0}}{4\pi}\frac{\hbar\gamma^{2}}{a_{0}}\rho^{1/3}$, where
$\hbar$ is the reduced Planck’s constant, $a_{0}\approx 0.565$ nm is the GaAs
lattice constant, and $\gamma$ is the nuclear gyromagnetic ratio. Here we use
the coordinate system aligned with the cubic crystal axes $x\parallel[100]$,
$y\parallel[010]$, $z\parallel[001]$, and the strong magnetic field is
parallel to the $z$ direction. We have also introduced the correction factor
$\rho^{1/3}$ to account for the increase of the average internuclear distance
for the isotope whose abundance $\rho$ is less then unity.

In case of arsenic, 75As is the only stable isotope, so that $\rho=1$. For
gallium isotopes we have the natural abundances $\rho=0.601$ and $\rho=0.399$
for 69Ga and 71Ga, respectively. The gyromagnetic ratios $\gamma$ are known
[24] and, since we approximate the spin diffusion as a one-dimensional process
along the sample growth direction $z$, we are interested in the $D_{zz}$
component of the tensor. Substituting the numerical values we find
$D_{zz}\approx 13$, 21, 30 nm2 s-1 for 75As, 69Ga and 71Ga, respectively. The
experiments presented in this work do not resolve between spin diffusion of
the individual isotopes. As a simple approximation we can treat the observed
NSR dynamics as a result of spin diffusion within one type of nuclei but with
a weighted average diffusion constant. We use as weights the relative
contributions of the isotopes to the optically measured hyperfine shift
$E_{\rm{hf}}$. From the previous studies of the similar QDs [9] these
contributions are estimated as 0.49, 0.28 and 0.23 for 75As, 69Ga and 71Ga,
respectively, from where the average diffusion coefficient is approximated as
$D_{zz}\approx 19$ nm2.

##  Supplementary Section 4 Numerical simulation of nuclear spin diffusion
coefficient and additional analysis

The spatiotemporal evolution of the nuclear spin polarization degree
$P_{\rm{N}}(t,z)$ is modeled by solving the partial differential spin
diffusion equation

$\displaystyle\frac{\partial P_{\rm{N}}(t,z)}{\partial
t}=D(t)\frac{{\partial}^{2}P_{\rm{N}}(t,z)}{\partial
z^{2}}+w(t)|\psi_{\rm{e}}(z)|^{2}(P_{\rm{N},0}-P_{\rm{N}}(t,z)),$ (S5)

where the last term describes optical nuclear spin pumping with a rate
proportional to electron density $|\psi_{\rm{e}}(z)|^{2}$ and the time-
dependent factor $w(t)$ equal to 0 or $w_{0}$ when optical pumping is off or
on, respectively. The spin diffusion coefficient $D(t)$ also takes two
discrete values $D_{\rm{Dark}}$ or $D_{\rm{Pump}}$ when optical pumping is off
or on, respectively. $P_{\rm{N},0}$ is a steady state nuclear spin
polarization degree that optical pumping would generate in the absence of spin
diffusion. At each time point we assume the same diffusion coefficient $D$
across the entire structure. The equation describes a one-dimensional problem
where diffusion can take place only along the $z$ coordinate so that the
nuclear spin polarization degree $P_{\rm{N}}$ does not depend on $x$ or $y$.
The GaAs QD layer is modeled by taking a Gaussian profile for the electron
density $|\psi_{\rm{e}}(z)|^{2}\propto
2^{-\left(\frac{z-z_{0}}{h_{\rm{QD}}/2}\right)^{2}}$, where $h_{\rm{QD}}$ is
the full width at half maximum of the $|\psi_{\rm{e}}(z)|^{2}$ function and
the center of the QD is set to be $z_{0}=0$. We use Dirichlet boundary
condition $P_{\rm{N}}=0$ to model fast nuclear spin depolarization in presence
of the free carriers both in the $n$\- and $p$-type doped layers. The boundary
coordinates where Dirichlet conditions are enforced are chosen to match the
actual sample structure as described in Supplementary Section 1. We note that
the hyperfine interaction of the valence band holes is approximately 10 times
weaker than for the conduction band electrons [8]. Moreover, the $p$-type
layer is approximately 10 times further away from the QDs than the $n$-type
layer. As a result the dynamics of the nuclear spin polarization at the QD are
dominated by the $n$-type layer, while the exact boundary condition at the
$p$-type layer is less crucial, justifying the use of the Dirichlet condition
at both doped layers.

Supplementary Figure 10:  Calculated normalized nuclear spin polarization as a
function of the $z$ coordinate at different $T_{\rm{Dark}}$. (a) Calculations
for $T_{\rm{Pump}}=1$ ms. (b) Same calculations for $T_{\rm{Pump}}=1000$ s.

Supplementary Eq. S5 is solved numerically using the method of lines
implemented in wolfram mathematica 12.0. The initial condition is taken to be
$P_{\rm{N}}=0~{}\forall~{}z$, which models the result of the RF Erase pulse at
the start of each measurement cycle. Optical nuclear spin pumping starts at
$t=-T_{\rm{Pump}}$ and the equation is solved until $t=0$ with
$D=D_{\rm{Pump}}$ and $w=w_{0}$. At $t=0$ optical pumping is switched off by
setting $w=0$ and the equation is solved until $t=T_{\rm{Dark}}$ with
$D=D_{\rm{Dark}}$. Supplementary Fig. 10 shows the calculated spatial profiles
of the final nuclear spin polarization $P_{\rm{N}}(T_{\rm{Dark}},z)$
normalized by its steady-state value $P_{\rm{N},0}$. The results are shown for
several $T_{\rm{Dark}}$ values in case of a short pumping [(a),
$T_{\rm{Pump}}=1$ ms] and long pumping [(b), $T_{\rm{Pump}}=1000$ s]. Short
pumping results in a small-magnitude ($P_{\rm{N}}\ll P_{\rm{N},0}$) spatially-
narrow nuclear spin polarization, which quickly dissipates at $t>0$. By
contrast, long pumping leads to a steady-state spatial distribution where
polarization peaks at the quantum dot coordinate $z=0$ and reduces linearly
towards the doped layers which act as nuclear spin polarization sinks.
Interestingly, this calculation predicts that the maximum polarization
$P_{\rm{N},0}$ is not achieved because of the diffusion towards the doped
layers, especially the closely located $n$-type layer at $z<0$.

In order to compare simulations with the experimental results the final
spatial distribution $P_{\rm{N}}(T_{\rm{Dark}},z)$ is multiplied by
$|\psi_{\rm{e}}(z)|^{2}$ and integrated over $z$. This way we reproduce the
optical probing of the nuclear spin polarization, where the measured hyperfine
shift $E_{\rm{hf}}$ is effectively weighted by the electron envelope
wavefunction density $|\psi_{\rm{e}}(z)|^{2}$. The simulated hyperfine shift
is then derived as $E_{\rm{hf}}=E_{\rm{Z}^{\rm{X^{-}}}}+AIP_{\rm{N}}$, where
$I$ is the nuclear spin number, $A$ is the hyperfine constant and
$E_{\rm{Z}^{\rm{X^{-}}}}$ is the trion Zeeman splitting in the absence of
nuclear spin polarization. We then use a differential evolution algorithm to
vary the parameters such as $D^{(ne)}_{\rm{Dark}}(B_{\rm{z}})$,
$w_{0}(B_{\rm{z}})$ and $D_{\rm{Pump}}(B_{\rm{z}})$ and fit the simulated
$E_{\rm{hf}}$ dynamics to the entire experimental datasets of
$E_{\rm{hf}}(T_{\rm{Pump}},T_{\rm{Dark}})$ measured at $B_{\rm{z}}=0.39$ and
$9.82$ T for empty ($n=0$) and charged ($n=1$) QD states. As discussed in the
main text the best-fit diffusion coefficients in the dark are
$D^{(0e)}_{\rm{Dark}}=2.2^{+0.7}_{-0.5}$ nm2 s-1 independent of magnetic
field, $D^{(1e)}_{\rm{Dark}}(9.82~{}{\rm{T}})=4.7^{+1.2}_{-1.0}$ nm2 s-1 and
$D^{(1e)}_{\rm{Dark}}(0.39~{}{\rm{T}})=7.7\pm 1.9$ nm2 s-1. For spin diffusion
coefficients under optical pumping we find significantly larger values
$D^{(1e)}_{\rm{Pump}}(9.82~{}{\rm{T}})=96^{+44}_{-28}$ nm2 s-1 and
$D^{(1e)}_{\rm{Pump}}(0.39~{}{\rm{T}})=850^{+240}_{-220}$ nm2 s-1. This can be
ascribed to the spectrally broad fluctuations of the optically generated
electron spins which provide coupling between the distant nuclear spins, thus
facilitating the diffusion. The other best-fit parameters are
$w_{0}(0.39~{}{\rm{T}})=37^{+7}_{-5}$ s-1,
$w_{0}(9.82~{}{\rm{T}})=5.7^{+1.1}_{-0.9}$ s-1 and
$h_{\rm{QD}}=2.1^{+0.3}_{-0.2}$ nm. Previous studies on GaAs/AlGaAs QDs
emitting at a similar wavelength estimated that 0.92 of the electron density
resides in the GaAs layer [9], whose full width can then be estimated as
$h_{\rm{QD}}\frac{{\rm{erf}}^{-1}(0.92)}{\sqrt{\ln(2)}}=3.2^{+0.4}_{-0.3}$ nm.
This best-fit value somewhat underestimates the true QD thickness in $z$
direction, but is within the range bounded by the QW thickness (2.1 nm) and
the maximum QD thickness ($\approx 9$ nm) estimated from the nanohole depth in
AFM. The spatial profiles shown in Supplementary Fig. 10 are calculated with
the best-fit parameters for the 1$e$ case at $B_{\rm{z}}=0.39$ T.

Supplementary Figure 11:  Numerically simulated nuclear spin polarization
degree $P_{\rm{N}}$ weighted by the electron envelope wavefunction density
$|\psi_{\rm{e}}(z)|^{2}$ and normalized by the maximum nuclear spin
polarization $P_{\rm{N},0}$ in the absence of spin diffusion. The weighted
polarization is plotted as a function of the dark time $T_{\rm{Dark}}$ for
different $T_{\rm{Pump}}$. The results are shown for the case of one
dimensional diffusion (1D, solid lines) and two dimensional diffusion (2D,
dashed lines, $P_{\rm{N}}$ values multiplied by 1.5).

The use of the one dimensional spin diffusion model is motivated by the large
aspect ratio of the QD: the diffusion proceeds predominantly along the
direction of the strongest gradient in the nuclear spin polarization degree,
which is the growth $z$ direction. For numerical simulations, one dimensional
model is also advantageous as it requires significantly less computational
resources than a full three dimensional diffusion model. In order to evaluate
the limitations of the one dimensional model we run a simulation of a two
dimensional diffusion problem, where the equation now reads:

$\displaystyle\frac{\partial P_{\rm{N}}(t,x,z)}{\partial
t}=D(t)\left(\frac{{\partial}^{2}P_{\rm{N}}(t,x,z)}{\partial
x^{2}}+\frac{{\partial}^{2}P_{\rm{N}}(t,x,z)}{\partial
z^{2}}\right)+w(t)|\psi_{\rm{e}}(x,z)|^{2}(P_{\rm{N},0}-P_{\rm{N}}(t,x,z))$
(S6)

The electron density is taken to be $|\psi_{\rm{e}}(x,z)|^{2}\propto
2^{-\left(\frac{x-x_{0}}{d_{\rm{QD}}/2}+\frac{z-z_{0}}{h_{\rm{QD}}/2}\right)^{2}}$,
where $d_{\rm{QD}}$ is a full width at half maximum diameter of the QD, which
we set to $d_{\rm{QD}}=47$ nm in order to match the 0.92 electron wavefunction
density in a QD with a full diameter of $70~{}$nm. The same
$|\psi_{\rm{e}}(x,z)|^{2}$ is used to calculate the weighted nuclear spin
polarization degree, emulating the optical probing of the QD hyperfine shift
$E_{\rm{hf}}$. The computational domain is limited to $|x|<700$ nm and we
implement the additional Dirichlet boundary condition $P_{\rm{N}}(x=\pm
700~{}{\rm{nm}})=0$.

Supplementary Fig. 11 shows the simulated QD NSR dynamics in a one dimensional
(1D, solid lines) and two dimensional (2D, dashed lines) cases, following
nuclear spin pumping with different durations $T_{\rm{Pump}}$. One difference
in the resulting dynamics is the lower weighted nuclear spin polarization
degree within the QD volume in the 2D case. Consequently, all the 2D-case
$P_{\rm{N}}$ values in Supplementary Fig. 11 have been multiplied by 1.5, to
simplify comparison with the 1D case. At short $T_{\rm{Pump}}\leq 0.1$ s QD
nuclear spin polarization decays on the same timescale both in 1D and 2D
cases. This is expected: the spatial profile of the nuclear spin polarization
produced by short pumping in 2D case is proportional to
$\propto|\psi_{\rm{e}}(x,z)|^{2}$, so that the subsequent diffusion in the
dark proceeds predominantly along the direction of the highest gradient (the
growth $z$ direction), making diffusion essentially one dimensional. By
contrast, long pumping $T_{\rm{Pump}}\geq 0.1$ s in a 2D model makes the
polarization profile more isotropic in the $xz$ plane (for an unbounded
problem at $T_{\rm{Pump}}\rightarrow\infty$ the polarization will tend to a
profile with circular contour lines in the $xz$ plane). In other words, after
long pumping the system “forgets” the initial profile
$\propto|\psi_{\rm{e}}(x,z)|^{2}$ of the QD pumping source. The subsequent
diffusion in the absence of pumping (i.e. in the dark) is controlled by the
dimensionality of the unpolarized space, and is seen to be faster in the 2D
case. From this additional results we conclude that one dimensional model
captures the key aspects of QD NSR dynamics, such as slower relaxation
following long optical nuclear spin pumping. However, some quantitative
discrepancies arise, especially at long $T_{\rm{Pump}}$, where dimensionality
affects the diffusion dynamics. Such discrepancies are likely to introduce
systematic errors in the best fit values of the spin diffusion coefficient
$D$. However, in a real QD system $D$ is not constant, and the approximate
nature of the spin diffusion concept itself entails a range of systematic
errors. This justifies the use of a simplified one dimensional model to
describe our experimental results.

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