Current address: ]School of Electrical and Computer Engineering, University of
Oklahoma, Norman, OK 73019, USA

# Enhanced Electron Spin Coherence in a GaAs Quantum Emitter

Giang N. Nguyen<EMAIL_ADDRESS>Department of Physics, University of
Basel, 4056 Basel, Switzerland  Clemens Spinnler  Department of Physics,
University of Basel, 4056 Basel, Switzerland  Mark R. Hogg  Department of
Physics, University of Basel, 4056 Basel, Switzerland  Liang Zhai  Department
of Physics, University of Basel, 4056 Basel, Switzerland  Alisa Javadi  [
Department of Physics, University of Basel, 4056 Basel, Switzerland  Carolin
A. Schrader  Department of Physics, University of Basel, 4056 Basel,
Switzerland  Marcel Erbe  Department of Physics, University of Basel, 4056
Basel, Switzerland  Marcus Wyss  Swiss Nanoscience Institute, University of
Basel, 4056 Basel, Switzerland  Julian Ritzmann  Lehrstuhl für Angewandte
Festkörperphysik, Ruhr-Universität Bochum, 44780 Bochum, Germany  Hans-Georg
Babin  Lehrstuhl für Angewandte Festkörperphysik, Ruhr-Universität Bochum,
44780 Bochum, Germany  Andreas D. Wieck  Lehrstuhl für Angewandte
Festkörperphysik, Ruhr-Universität Bochum, 44780 Bochum, Germany  Arne Ludwig
Lehrstuhl für Angewandte Festkörperphysik, Ruhr-Universität Bochum, 44780
Bochum, Germany  Richard J. Warburton  Department of Physics, University of
Basel, 4056 Basel, Switzerland

(August 28, 2024)

###### Abstract

A spin-photon interface should operate with both coherent photons and a
coherent spin to enable cluster-state generation and entanglement
distribution. In high-quality devices, self-assembled GaAs quantum dots are
near-perfect emitters of on-demand coherent photons. However, the spin rapidly
decoheres via the magnetic noise arising from the host nuclei. Here, we
address this drawback by implementing an all-optical nuclear-spin cooling
scheme on a GaAs quantum dot. The electron-spin coherence time increases
156-fold from $T_{2}^{*}=$3.9\text{\,}\mathrm{ns}$$ to
$0.608\text{\,}\mathrm{\SIUnitSymbolMicro s}$. The cooling scheme depends on a
non-collinear term in the hyperfine interaction. The results show that such a
term is present even though the strain is low and no external stress is
applied. Our work highlights the potential of optically-active GaAs quantum
dots as fast, highly coherent spin-photon interfaces.

first keyword, second keyword, third keyword

## Introduction

Photonic quantum technologies require quantum emitters capable of high-
fidelity and high-rate operation. Of particular interest for quantum networks
[1, 2, 3, 4, 5, 6] and measurement-based quantum computing [7, 8, 9] are
quantum emitters that host a spin [10], allowing the creation of spin-photon
entanglement.

Self-assembled semiconductor quantum dots (QDs) are potential candidates for
spin-photon interfaces due to the deterministic photon emission at
exceptionally high quality and rates [11, 12, 13, 14] and the ability to load
a QD with a single electron or hole [15]. This has led to demonstrations of
spin-photon entanglement [16, 17, 18, 19], distant spin-spin entanglement [20,
21], and the creation of multi-photon cluster states [22, 23, 24]. However, in
these previous experiments, the short spin coherence times, $T_{2}^{*}\sim
1-$10\text{\,}\mathrm{ns}$$, limited the entanglement fidelity. The short
$T_{2}^{*}$ is a consequence of magnetic noise in the host nuclear spins,
coupling to the electron spin via the hyperfine interaction [25, 26, 27].

A powerful way to mitigate the short $T_{2}^{*}$ is to cool the nuclear spins
to ultralow temperatures in order to reduce the fluctuations. The nuclei can
be cooled via the electron spin itself, exploiting the hyperfine interaction
[28]. In an optical experiment, this was originally demonstrated on an
ensemble of QDs [29]. On single QDs, nuclear spin cooling was demonstrated on
gate-defined GaAs QDs via a measure-and-correct feedback loop [30, 31]. More
recently, the highly inhomogeneous nuclear spins of a self-assembled InGaAs QD
were cooled via an autonomous feedback [32]. Subsequently, a quantum sensing
protocol was employed, narrowing the nuclear distribution further, thereby
increasing $T_{2}^{*}$ to $300\text{\,}\mathrm{ns}$ [33]. For both schemes, a
non-collinear term in the hyperfine interaction is required to allow for the
cooling of the nuclei. In contrast to the collinear term from the contact
hyperfine interaction ($\propto S_{z}I_{z}$), the non-collinear term ($\propto
S_{z}I_{x}$) arises from nuclear quadrupolar fields in strained QDs; here
$S_{z}$ ($I_{z}$) is the electron (nuclear) spin operator along the direction
of the applied magnetic field [34, 28, 35].

The most studied QDs for spin-photon applications are QDs in the InGaAs/GaAs
system. InGaAs QDs are self-assembled via the strain-driven Stanski-Krastanov
mechanism. Self-assembled GaAs QDs in an AlGaAs matrix represent an
alternative platform. The strain is low such that these QDs are self-assembled
via an alternative mechanism, droplet-etching. Low-noise GaAs QDs have
excellent photonic properties, all at a convenient wavelength (around 780 nm).
In high-quality material, the optical linewidths are within 10% of the
transform limit [36]. Photons emitted by remote QDs have achieved a two-photon
interference visibility of 93% without spectral or temporal filtering [37].
The biexciton cascade generates entangled photon pairs with an extremely high
entanglement concurrence [38]. In terms of the nuclear spins, the lack of both
strain and spin-$\frac{9}{2}$ In atoms results in a homogeneous nuclear spin
ensemble [39], as demonstrated by the success of the Carr-Purcell-Meiboom-Gill
(CPMG) decoupling scheme in prolonging the electron spin $T_{2}$ from
$3.8\text{\,}\mathrm{\SIUnitSymbolMicro s}$ to
$113\text{\,}\mathrm{\SIUnitSymbolMicro s}$ [40]. However, as for InGaAs QDs,
noise in the nuclear spins limits $T_{2}^{*}$ to values of a few-ns. To date,
the possibility of feedback cooling the nuclear spins via the electron spin
has remained uncertain, due to the predicted absence of the strain-generated
non-collinear hyperfine interaction.

Figure 1: Coherent spin control of an electron in a droplet-etched GaAs QD.
(a) High-angle dark-field scanning transmission image of a droplet-etched GaAs
QD. The dashed line is a guide to the eye to describe the droplet shape. (b)
Schematic of the sample design: a layer of GaAs QDs is embedded in a diode
structure. A magnetic field perpendicular to the growth direction defines the
quantization axis. (c) Energy level diagram of a charged QD in an in-plane
magnetic field. The “vertical” transitions are $x$-polarised while the
“diagonal” transitions are $y$-polarised. A circularly polarised rotation
pulse detuned by $\Delta_{\text{L}}=$700\text{\,}\mathrm{GHz}$$ drives a Raman
transition between the electron spin states. The readout laser is on resonance
with the lower-frequency “vertical” transition and initialises the electron
into the $\ket{\uparrow}$-state. (d) Electron spin Rabi oscillations as a
function of drive time $t$. The solid line is an exponential fit to the data
with $T_{2}^{\text{Rabi}}=$73(5)\text{\,}\mathrm{ns}$$. (e) Full control of
the rotation axis about the Bloch sphere using two consecutive
$\frac{\pi}{2}$-pulses as a function of the phase $\phi$ of the second pulse.
The solid line is a sinusoidal fit to the data.

Here, we implement all-optical cooling schemes on low-noise GaAs QDs and
demonstrate an increase in the electron spin coherence time from
$T_{2}^{*}=$3.9\text{\,}\mathrm{ns}$$ to
$0.608\text{\,}\mathrm{\SIUnitSymbolMicro s}$. This is achieved with
autonomous feedback and without any external perturbation (such as strain
tuning). We demonstrate spin control with
$T_{2}^{*}=$0.608\text{\,}\mathrm{\SIUnitSymbolMicro s}$$, an extension of
$T_{2}$ with CPMG (with a scaling of $T_{2}^{\text{CPMG}}\propto N^{0.69}$
matching previous experiments [40]), fast spin rotations (Rabi frequencies
above 100 MHz), and high-fidelity spin control ($F_{\pi}>98\%$). Our results
establish GaAs QDs as an emitter of coherent photons and a host to a coherent
spin.

To create the QDs, droplet-etched nanoholes in an
$\text{Al}_{0.15}\text{Ga}_{0.85}\text{As}$ matrix are filled with GaAs and
capped by an $\text{Al}_{0.33}\text{Ga}_{0.67}\text{As}$ layer. The materials
are almost lattice-matched. Figure 1(a) shows a high-angle dark-field scanning
transmission (HAADF-STEM) image of a GaAs QD [41]. Notable is a thin, Al-rich
layer at the bottom surface of the QD [41]. The QD is embedded in a p-i-n
diode structure (see Fig. 1(b)) such that the QD charge is stabilised via the
Coulomb blockade. Individual QDs exhibit near-transform-limited optical
linewidths [41, 36, 37]. A $3.00\text{\,}\mathrm{T}$ magnetic field is applied
perpendicular to the growth direction (Voigt geometry), at an angle of
$45^{\circ}$ to the in-plane crystal axes. The electron Zeeman frequency is
$f_{\text{Z}}=$4.54\text{\,}\mathrm{GHz}$$ corresponding to a g-factor of
$g_{\text{e}}=-0.11$.

The spin is manipulated by a two-colour Raman pulse detuned from the excited
states by $\Delta_{\text{L}}=$700\text{\,}\mathrm{GHz}$$ (see Fig. 1(c)). This
pulse is created by amplitude-modulating circularly-polarised light with an
electro-optic modulator driven by an arbitrary waveform generator [41, 42]. A
laser resonant with the red “vertical” transition is used to read out the spin
(such that the $\ket{\downarrow}$-state is bright, the $\ket{\uparrow}$-state
is dark) and to prepare the spin in the $\ket{\uparrow}$-state via optical
spin pumping [41].

Driving the electron spin resonance (ESR) (Fig. 1(d)) shows clear Rabi
oscillations between $\ket{\uparrow}$ and $\ket{\downarrow}$ with increasing
drive time $t$. We find an exponential decay of the oscillations with
$T_{2}^{\text{Rabi}}=$73(5)\text{\,}\mathrm{ns}$$, corresponding to a quality
factor of $Q=2T_{2}^{\text{Rabi}}f_{\text{Rabi}}=$19(1)\text{\,}$$ and
$\pi$-pulse fidelity $f_{\pi}=\frac{1}{2}(1+e^{-1/Q})=$0.975(2)\text{\,}$$ at
$\Omega=2\pi\times$130\text{\,}\mathrm{MHz}$$. As has been observed for InGaAs
QDs [42], we find a strong modulation of the quality factor [41] when the
electron spin is driven close to the nuclear Larmor frequencies
$\omega_{\text{n}}$ (i.e., $\Omega\sim\omega_{\text{n}}$), a signature of an
electron-nuclei interaction via a Hartmann-Hahn resonance [43].

We access rotation around a second axis on the Bloch sphere by controlling the
phase of the microwave signal that is imprinted on the optical field. Fig.
1(e) shows the sinusoidal response after two consecutive
$\frac{\pi}{2}$-pulses on changing the phase $\phi$ of the second pulse,
thereby demonstrating rotation around an arbitrary axis on the equator of the
Bloch sphere.

Figure 2: Locking of electron spin resonance (ESR) and cooling of nuclei with
a Rabi drive. (a) Rabi oscillations versus detuning show locking of the ESR to
the drive within a window of frequencies and unstable Rabi oscillations
outside the window. (b) Top: Pulse sequence for Ramsey interferometry with
prior Rabi cooling. For Rabi cooling a
$T_{\text{c}}=$1\text{\,}\mathrm{\SIUnitSymbolMicro s}$$ long pulse at a Rabi
frequency of $\Omega_{\text{c}}=2\pi\times$17\text{\,}\mathrm{MHz}$$ is used.
The Ramsey experiment was performed at a larger Rabi frequency of
$2\pi\times$100\text{\,}\mathrm{MHz}$$. Bottom: Top and bottom envelopes of
the Ramsey interferometry with $100\text{\,}\mathrm{\SIUnitSymbolMicro s}$
pause (circles), zero pause (squares) and Rabi cooling (diamonds); the
extracted coherence times are $T_{2}^{*}=$3.9(2)\text{\,}\mathrm{ns}$$,
$T_{2}^{*}=$7.8(2)\text{\,}\mathrm{ns}$$, and
$T_{2}^{*}=$78(2)\text{\,}\mathrm{ns}$$, respectively. Counts are normalised
to $0.5$ for long delays. (c) Ramsey interferometry as a function of detuning
with respect to the cooling frequency $f_{\text{c}}$ of the Rabi drive. (d)
Linecut at $\Delta=$50\text{\,}\mathrm{MHz}$$ with
$T_{2}^{*}=$87(6)\text{\,}\mathrm{ns}$$ (dashed box in (c)). The solid lines
in (b) and (d) are Gaussian fits to the data.

On driving Rabi oscillations as a function of the detuning $\Delta$ with
respect to the Zeeman frequency ($\Delta=f_{\text{Z}}-f_{\text{probe}}$), we
find strong deviations from the typical chevron pattern expected for a two-
level system (see Fig. 2(a)). In a $\sim$200\text{\,}\mathrm{MHz}$$ window
around the Zeeman frequency, we find that the spin rotations lock to the probe
frequency $f_{\text{probe}}$, a clear signature of electron spin–nuclear spin
coupling [44, 45, 35, 46].

When the ESR is locked via the hyperfine interaction, cooling of the nuclei,
equivalently narrowing of the nuclear distribution, is predicted [47, 48].
This can be quantified by a reduction in $\sigma_{\text{OH}}$, the standard
deviation of the ESR frequency fluctuations due to the changing Overhauser
field. To probe this, we perform a free-induction decay (FID) experiment to
measure the electron coherence time $T_{2}^{*}$ in a Ramsey experiment, which
acts as a gauge of the temperature of the nuclear spin ensemble
($\sigma_{\text{OH}}\propto T_{2}^{*}$) [26, 27]. We compare the bare
$T_{2}^{*}$ to that obtained after locking the ESR (see Fig. 2(b)). We observe
a 20-fold increase from $T_{2}^{*}=$3.9(2)\text{\,}\mathrm{ns}$$ to
$78(2)\text{\,}\mathrm{ns}$ corresponding to a narrowing of
$\sigma_{\text{OH}}$ from $52(1)\text{\,}\mathrm{MHz}$ to
$2.90(5)\text{\,}\mathrm{MHz}$ following the Rabi drive. Remarkably, we
already find an enhancement in coherence time without a dedicated cooling
pulse when the Ramsey experiment is carried out with a high duty cycle:
repetitive Ramsey experiments lead to a $T_{2}^{*}$ of
$7.8(2)\text{\,}\mathrm{ns}$. To determine the bare electron coherence time,
we add a $100\text{\,}\mathrm{\SIUnitSymbolMicro s}$ buffer between each
cycle. This observation suggests that the repetitive application of spin
manipulation pulses as short as $4\text{\,}\mathrm{ns}$ already leads to a
narrowing of $\sigma_{\text{OH}}$.

We confirm the nuclear-spin cooling and locking of the ESR to the Rabi drive
by fixing the cooling frequency $f_{\text{c}}$ during Rabi cooling,
subsequently detuning the probe frequency $f_{\text{probe}}$ in a Ramsey
experiment. Oscillations arise at the detuning frequencies
$\Delta=f_{\text{c}}-f_{\text{probe}}$ as expected in a classic Ramsey
experiment (see Fig. 2 (c,d)), now with an increased coherence time.

To cool the nuclei further, we implement the recently developed quantum-
sensing-based cooling scheme [33]. In this protocol, each cooling cycle
consists of three steps (see Fig. 3(a, top)): (i) The electron spin is
initialised and then rotated to the equator with a $\frac{\pi}{2}$-pulse. A
period of free evolution $\tau_{\text{sense}}$ allows the electron to sense
the Overhauser field fluctuation that leads to a detuning $\Delta$ from the
target frequency $f_{\text{c}}$. (ii) A coherent electron-nuclei flip-flop
interaction arising from a non-collinear term in the hyperfine interaction is
activated through ESR driving at Hartmann-Hahn resonance
$\Omega\approx\omega_{\text{n}}$. The sign of the detuning $\Delta$ determines
the direction of the nuclear flops and thus leads to a reversal of the
measured fluctuation. (iii) A projective measurement of the spin state
transfers entropy from the nuclei and concludes one cycle of the cooling
scheme. Repeating this cycle with increasing sensing time $\tau$ results in a
narrower feedback function in each cycle and hence an increased sensitivity to
changes in $\sigma_{\text{OH}}$.

We find optimal parameters for the quantum-sensing-based cooling at $N=40$
cycles with a linearly increasing sensing time $\tau_{\text{sense}}$ from
$\tau_{\text{min}}=$20\text{\,}\mathrm{ns}$$ to
$\tau_{\text{max}}=$400\text{\,}\mathrm{ns}$$, and electron-nuclei drive time
$T_{\text{c}}=$125\text{\,}\mathrm{ns}$$ at a Rabi frequency
$\Omega_{\text{c}}=2\pi\times$17\text{\,}\mathrm{MHz}$$, followed by a spin
pumping pulse of $200\text{\,}\mathrm{ns}$ [41]. This preparation sequence
takes $\sim$22\text{\,}\mathrm{\SIUnitSymbolMicro s}$$ and is repeated before
each Ramsey cycle.

Figure 3: Quantum-sensing-based cooling and dynamical decoupling. (a) Top:
Pulse scheme for the quantum-sensing-based cooling consisting of (i) a sensing
step, (ii) a driven electron-nuclei interaction, and (iii) a reset. The last
reset pulse in the cooling scheme initialises the electron spin for the Ramsey
experiment performed at a Rabi frequency of
$2\pi\times$100\text{\,}\mathrm{MHz}$$. Bottom: Ramsey interferometry with
serrodyne frequency $\omega_{\text{serr}}=2\pi\times$20\text{\,}\mathrm{MHz}$$
($\phi(\tau)=\sin(\omega_{\text{serr}}\tau)$) following quantum-sensing-based
cooling gives $T_{2}^{*}=$0.608(13)\text{\,}\mathrm{\SIUnitSymbolMicro s}$$.
(b) Comparison of $T_{2}^{*}$ before cooling (squares), after Rabi cooling
(diamonds), and after quantum-sensing-based cooling (circles). Inset: fast
Fourier transform of the Ramsey visibilities gives
$\sigma_{\text{\text{OH}}}=$52(1)\text{\,}\mathrm{MHz}$$,
$\sigma_{\text{\text{OH}}}=$2.90(5)\text{\,}\mathrm{MHz}$$, and
$\sigma_{\text{\text{OH}}}=$0.355(4)\text{\,}\mathrm{MHz}$$, respectively. (c)
$T_{2}^{*}$ versus Rabi frequency during cooling ($\Omega_{\text{c}}$) for
Rabi cooling (diamonds) and quantum-sensing-based cooling (circles). Dashed
lines correspond to nuclear Larmor frequencies, from left to right:
$\Delta\omega=2\pi\times$17.08\text{\,}\mathrm{MHz}$$,
$\omega(\ce{{}^{75}As})=2\pi\times$21.9\text{\,}\mathrm{MHz}$$,
$\omega(\ce{{}^{69}Ga})=2\pi\times$30.7\text{\,}\mathrm{MHz}$$,
$\omega(\ce{{}^{27}Al})=2\pi\times$33.28\text{\,}\mathrm{MHz}$$,
$\omega(\ce{{}^{71}Ga})=2\pi\times$39.0\text{\,}\mathrm{MHz}$$. (d) Rabi
oscillations at $\Omega_{\text{c}}=2\pi\times$8.9\text{\,}\mathrm{MHz}$$ as a
function of detuning from $f_{\text{c}}$ following quantum-sensing-based
cooling. (e) Dynamical decoupling of the electron spin with a CPMG sequence.
The solid lines in (a,b) are Gaussian fits to the data. The solid line in (e)
is a power law fit to the data.

The electron coherence time $T_{2}^{*}$ increases from
$3.9(2)\text{\,}\mathrm{n}\mathrm{s}$ to
$0.608(13)\text{\,}\mathrm{\SIUnitSymbolMicro s}$ after application of the
protocol (see Fig. 3(a, b)). This constitutes a 156-fold increase in
$T_{2}^{*}$. The final $T_{2}^{*}$ is a factor of two larger than the previous
highest $T_{2}^{*}$ reported on an electron spin hosted by an InGaAs QD
($296\text{\,}\mathrm{ns}$ [33]) and just below the highest reported
$T_{2}^{*}$ of a single electron spin qubit in a gate-defined GaAs QD
($767\text{\,}\mathrm{ns}$ [31]). The enhancement corresponds to a narrowing
of the nuclear-spin ensemble from
$\sigma_{\text{OH}}=$52(1)\text{\,}\mathrm{MHz}$$ to
$0.355(4)\text{\,}\mathrm{MHz}$ (see Fig. 3(b, inset)).

Using hyperfine constants $A_{k}$ and abundancies $\eta_{k}$ of the nuclei
species $k\in\\{\ce{{}^{69}Ga},\ce{{}^{71}Ga},\ce{{}^{75}As}\\}$ we can
estimate the number of nuclei involved
$N=5/4\sum_{k}\eta_{k}A_{k}^{2}T_{2}^{*2}=1.4\cdot 10^{5}$ and estimate the
hyperfine interaction per nuclei $A_{\text{c}}=1/(\sqrt{5N/2}\pi
T_{2}^{*})=$0.13\text{\,}\mathrm{MHz}$$ [25, 33, 40]. This corresponds to a
distribution of $\sigma_{\text{OH}}/A_{\text{c}}\approx 376.8$ macrostates in
the uncooled state and 2.6 after quantum-sensing-based cooling, entering the
regime where just a few nuclei excitations remain.

For both the quantum-sensing-based and Rabi cooling schemes, the Rabi
frequency $\Omega_{\text{{c}}}$ is an important parameter (see Fig. 3(c)). The
maximum performance for both cooling schemes occurs at
$\Omega_{\text{{c}}}=2\pi\times$17\text{\,}\mathrm{MHz}$$, close to the
difference frequency of $\ce{{}^{71}Ga}$ and $\ce{{}^{75}As}$
($\Delta\omega=\omega(\ce{{}^{71}Ga})-\omega(\ce{{}^{75}As})=2\pi\times$17.08\text{\,}\mathrm{MHz}$$).
This result is in contrast to those on InGaAs QDs for which cooling was most
effective at a direct Hartmann-Hahn resonance [33]. Generally speaking, the
fact that cooling via an autonomous feedback process is effective on GaAs QDs
shows that a non-collinear term in the hyperfine interaction [35, 44, 28] must
be present even though the strain in the QDs is small.

Following cooling, a typical chevron pattern is observed on driving Rabi
oscillations as a function of detuning with respect to the cooling frequency
$f_{\text{c}}$ (Fig. 3(d)), using here a Rabi frequency below the Hartmann-
Hahn resonances. This demonstrates that in this case the electron spin is
isolated from the nuclear environment and behaves as a two-level system. In
addition, the quality factor of the oscillations now increases to
$Q=$30.0(14)\text{\,}$$ (corresponding to a $\pi$-pulse fidelity of
$98.4(1)\text{\,}\mathrm{\char 37\relax}$) [41], consistent with a reduction
of hyperfine-interaction-induced Rabi decay.

Recent experiments showed that the electron spin $T_{2}$ can be increased by
implementing a decoupling scheme, the CPMG protocol. As a final step, we
verify that this is also possible on the QD for which nuclear spin cooling was
highly effective (see Fig. 3(d)). By applying CPMG pulses, we extend $T_{2}$
from $T_{2}^{\text{HE}}=$2.93(6)\text{\,}\mathrm{\SIUnitSymbolMicro s}$$ using
a Hahn echo ($N_{\pi}$=1) to
$T_{2}^{\text{CPMG}}=$22(8)\text{\,}\mathrm{\SIUnitSymbolMicro s}$$, an order
of magnitude increase, with $N_{\pi}$=20 pulses. We extract a $T_{2}$ scaling
of $T_{2}^{\text{CPMG}}\propto N_{\pi}^{\gamma}$ with
$\gamma=$0.69(12)\text{\,}$$, consistent with recent results on droplet-etched
QDs [40] and gate-defined QDs [49]. This result confirms that the nuclear spin
ensemble is highly homogeneous. The application of more pulses is currently
limited by imperfect pulse calibrations and the electron spin relaxation time
$T_{1}\sim$40\text{\,}\mathrm{\SIUnitSymbolMicro s}$$ [41].

In conclusion, we have demonstrated fast and flexible optical control of an
electron spin confined to a self-assembled GaAs QD. We show that autonomous
feedback protocols to cool the nuclear spins are very effective even on an as-
grown, close-to-strain-free QD. Nuclear-spin cooling leads to a 156-fold
increase in the $T_{2}^{*}$ time,
$T_{2}^{*}=$0.608\text{\,}\mathrm{\SIUnitSymbolMicro s}$$. Furthermore, both
$T_{2}^{*}$ and $T_{2}$ can be extended on exactly the same QD, $T_{2}^{*}$ by
nuclear spin cooling, $T_{2}$ by dynamic decoupling. These results imply that
a small non-collinear term must be present in the hyperfine Hamiltonian.
Following nuclear spin cooling, $T_{2}^{*}$ becomes much longer than both the
time required to rotate the spin and the time required to generate a photon.
Together with recent results on the generation of indistinguishable photons
from remote GaAs QDs [37] performed on the same sample as used in this
experiment, our results highlight the promise of GaAs QDs for a coherent spin-
photon interface. Furthermore, the system represents an ideal testbed for
creating non-classical collective states within the nuclear spin ensemble
[50].

We thank Ming-Lai Chan and Peter Lodahl at the Niels-Bohr Institute, Leon
Zaporski and Mete Atatüre at the University of Cambridge, and Dorian Gangloff
at the University of Oxford for stimulating discussions.

The work was supported by SNF Project 200020_204069 and Horizon 2020 FET-Open
Project QLUSTER. LZ, GNN and AJ received funding from the European Union’s
Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-
Curie grant agreement No. 721394 (4PHOTON), No. 861097 (QUDOT-TECH), and No.
840453 (HiFig), respectively. HGB, JR, ADW and AL acknowledge financial
support from the grants DFH/UFA CDFA05-06, DFG TRR160, DFG project 383065199,
and BMBF Q.Link.X 16KIS0867.

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## Supplementary Information: Enhanced Electron Spin Coherence in a GaAs
Quantum Emitter

##  I Experimental Setup

The quantum dot (QD) sample is cooled down to $4.2\text{\,}\mathrm{K}$ in a
helium bath cryostat (see Fig. 4). A $B=$3.00\text{\,}\mathrm{T}$$ magnetic
field is applied perpendicular to the QD growth direction (Voigt geometry), at
$45^{\circ}$ to the crystal axes (along the [100] or [010] direction). QD
excitation, spin manipulation, and readout are performed all-optically with
the use of a dark-field microscope [51]. QD emission is filtered from the
excitation laser at frequency $f_{1}$ (Toptica DL Pro) using two polarising
beam-splitters (PBS). Spin initialisation and readout are triggered using an
acousto-optic modulator (AOM, Gooch&Housego 3200-124). Spin manipulation is
facilitated with a second laser at frequency $f_{2}$,
$\Delta_{\text{L}}=$700\text{\,}\mathrm{GHz}$$ red detuned from $f_{1}$. The
laser is amplitude-modulated with an electro-optic modulator (EOM, Jenoptik
AM785) that is driven at $f_{\text{AWG}}$ to generate sidebands at $\pm
f_{\text{AWG}}$. A second AOM (Gooch&Housego 3200-1113) is used to set the
amplitude of the spin control pulses. A quarter-wave plate (QWP) sets the
required circular polarisation and the pulse is sent down the axis of the
cryostat to the QD with a $30:70$ beam splitter. The reflected rotation laser
light is filtered from the QD emission with a
$25\text{\,}\mathrm{GHz}$-bandwidth grating filter. Control signals for spin
initialisation, manipulation and readout are generated by an arbitrary
waveform generator (AWG, Tektronix 7122C) and QD counts are detected with a
superconducting nanowire single photon detector (SNSPD, Single Quantum) and
processed with a time-tagger (Swabian Instruments Time Tagger Ultra). The QD
sample is the same as used in Ref. [36] and [37] and growth details can be
found in Ref. [52].

Figure 4: Experimental setup. Spin manipulation is achieved using optical
pulses at frequency $f_{2}$, detuned from $f_{1}$ by $\Delta_{\text{L}}$.
Optical sidebands at $f_{2}\pm f_{\text{AWG}}$ are generated by an EOM that is
modulated at half the Zeeman frequency ($f_{\text{AWG}}=f_{\text{Z}}/2$). A
readout laser at frequency $f_{1}$ is used for spin initialisation (via
optical spin pumping) and spin readout. Back-reflected laser light at $f_{1}$
is suppressed via a cross-polarisation microscope head consisting of two
polarising beam splitters (PBS), a linear polariser (LP), and a quarter-wave
plate (QWP). Laser light at $f_{2}$ is rejected with a grating filter with
$25\text{\,}\mathrm{GHz}$ bandwidth (BW) centred around $f_{1}$. A half-wave
plate (HWP) before the QD is used to match the polarisation of the excitation
to the “vertical” QD transition.

##  II Quantum dot

###  II.1 Structural properties and composition

We perform energy-dispersive X-ray spectroscopy (EDX) on a QD from the same
wafer to determine the spatial distribution of arsenic, gallium and aluminium
atoms. The sample preparation for EDX/scanning transmission electron
microscopy (STEM) was carried out in an FEI Helios NanoLab 650 DualBeam, a
combined scanning electron microscope (SEM) and focused ion beam (FIB). A
double layer of carbon is deposited to protect the QD from ion-induced damage.
The first C-layer was deposited using electron-induced deposition at a beam
energy of $5\text{\,}\mathrm{keV}$ and a beam current of
$3.2\text{\,}\mathrm{nA}$. The second C-layer was deposited with ion-induced
deposition at a beam energy of $30\text{\,}\mathrm{keV}$ and a beam current of
$83\text{\,}\mathrm{pA}$. Sample cutting and polishing were carried out with
the FIB at a beam energy of $30\text{\,}\mathrm{kV}$ and beam currents ranging
from $240\text{\,}\mathrm{pA}$ down to $83\text{\,}\mathrm{pA}$. The sample
thickness in the upper area, where the QD is located, was
$<$50\text{\,}\mathrm{nm}$$. The imaging of the TEM specimen was carried out
in a JEOL JEM-F200 operated in the STEM-mode at a beam energy of
$200\text{\,}\mathrm{kV}$. A high-angle dark-field scanning transmission image
(HAADF-STEM) of a QD is shown in Fig. 1 of the main text.

The arsenic EDX intensity is homogeneous as expected: the arsenic
concentration is constant throughout the growth of the QD layer and matrix
material (see Fig. 5(a)). The EDX intensity for gallium atoms shows a high
signal below the QD
($y\sim$50\text{\,}\mathrm{nm}$,x\sim$50\text{\,}\mathrm{nm}$$) and a low
signal above (see Fig. 5(b)). This is expected as the matrix material below
the QD is Al0.15Ga0.85As and the QD is grown on and capped with
Al0.33Ga0.67As. Accordingly, the QD alone would be expected to have an even
higher EDX intensity, as it is filled purely with GaAs. However, in the EDX
experiment the gallium signal at the QD is dominated by matrix material around
the QD. The EDX signal for aluminium atoms shows a low aluminium signal below
the QD and a high signal above (see Fig. 5(c)). In addition, a thin film of
high aluminium signal can be seen above the QD at
$y\sim$45\text{\,}\mathrm{n}\mathrm{m}$$ revealing the presence of a high-
aluminum content layer. This layer is formed from aluminium used in drilling
the nanoholes. Interestingly, this thin, high Al-content layer also forms at
the boundary of the nanohole, leading to the growth of GaAs not on
Al0.33Ga0.67As but on a thin layer with higher Al-content. This high Al-
content layer represents an increase in both alloying and strain with respect
to a pure GaAs QD [40].

Figure 5: Energy-dispersive x-ray spectroscopy. EDX intensity as a function of
x-and y- position around the QD for (a) arsenic, (b) gallium, and (c)
aluminium atoms.

###  II.2 Optical properties at zero magnetic field

The QD linewidth is measured by slowly scanning a narrow-band laser through
the $X^{-}$ resonance (see Fig. 6(a)). Low excitation power is used to avoid
power broadening. A Lorentzian fit to the data gives a FWHM of
$491\text{\,}\mathrm{MHz}$. This should be compared to the Fourier-limited
linewidth as inferred from an $X^{-}$ lifetime measurement. In a pulsed
experiment, the emission decay time is recorded in a histogram (see Fig.
6(b)). An exponential fit gives a $X^{-}$ lifetime of
$\tau=$392\text{\,}\mathrm{ps}$$, which corresponds to a Fourier-limited
linewidth of $\frac{1}{2\pi\tau}=$405\text{\,}\mathrm{MHz}$$. The linewidth is
therefore $22\text{\,}\mathrm{\char 37\relax}$ above the Fourier limit. This
is slightly higher than the average QD-linewidth on the sample [36].

Figure 6: Optical properties at zero magnetic field. (a) Scan of a narrow-band
laser through the resonance of the negatively charged exciton $X^{-}$. The
solid line is a Lorentzian fit with FWHM of $491\text{\,}\mathrm{MHz}$. (b)
Lifetime measurement of the $X^{-}$. The solid line is an exponential fit with
lifetime $\tau=$392\text{\,}\mathrm{ps}$$. This corresponds to a Fourier-
limited linewidth of $405\text{\,}\mathrm{MHz}$.

###  II.3 Optical properties at B = 3.00 T

We record a resonance fluorescence (RF) plateau map by scanning both
excitation laser frequency ($f$) and gate voltage ($V_{\text{g}}$) across the
$X^{-}$ transitions at $B=$3.00\text{\,}\mathrm{T}$$ (see Fig. 7(a)). A fine
scan across the plateau centre (see Fig. 7(b)) shows a single line, while a
linecut across the plateau edge (see Fig. 7(c)) shows four distinct peaks
corresponding to the four optical transitions. From a fit to a sum of four
Lorentzians, we can extract the ground-state and excited-state Zeeman
splittings to be $\omega_{\mathrm{e}}=2\pi\times$4.54\text{\,}\mathrm{GHz}$$
and $\omega_{\mathrm{h}}=2\pi\times$5.40\text{\,}\mathrm{GHz}$$, respectively,
corresponding to g-factors
($g_{\mathrm{e/h}}=\hbar\omega_{\mathrm{e/h}}/\mu_{\mathrm{B}}B$) of
$g_{\mathrm{e}}=-0.11$ and $g_{\mathrm{h}}=0.13$. (The assumption here is that
$g_{\mathrm{e}}$ is negative.) Vanishing signal is expected in the plateau
centre due to optical spin pumping. The single line we observe arises due to
repumping as the laser drives both of the near-degenerate “diagonal”
transitions. The “vertical” transitions are extinguished by spin pumping, as
expected. Optical spin pumping is ineffective at the plateau edges due to
cotunneling with the Fermi sea. Changing the rotation angle of the HWP in the
beam path of the QD excitation (cf. Fig. 4), we can either address the two
outer “vertical” transitions (x-polarised), the two inner “diagonal”
transitions (y-polarised), or all four transitions with “diagonal”
polarisation (see Fig. 7(d)). For the spin-manipulation experiments, we set
the HWP such that only the outer “diagonal” transitions are excited ($\sim
85^{\circ}$).

Figure 7: Optical properties at B = 3.00 T. (a) RF plateau map of the $X^{-}$.
(b) Zoom in on the plateau centre, area (i) in (a). The two outer “vertical”
transitions disappear due to spin pumping, while spin pumping for the inner
“diagonal” transitions is ineffective due to a too-small frequency splitting.
(c) Linescan at (ii) of (a) in the cotunneling regime shows four transitions.
Spin-pumping is ineffective due to cotunneling of the electron to the Fermi
sea. The solid line is a fit to a sum of four Lorentzians. (d) Linecut at (ii)
as a function of excitation polarisation given by the rotation angle of the
HWP.

##  III Spin properties

###  III.1 Data acquisition

For all experiments, a laser background was recorded by carrying out a
measurement with the $X^{-}$-transition out of resonance with the readout
laser. This was carried out by changing the voltage applied to the diode.
After subtracting the background signal, counts during the readout pulse were
integrated with an integration time depending on the duty cycle of the
experiment (typically $5\text{\,}$$-$$60\text{\,}\mathrm{s}$ for a signal to
noise ratio of $\sim 4$). For Ramsey and Carr-Purcell-Meiboom-Gill (CPMG)
experiments, two measurements were performed. In the second one, the electron
is initialised in one state but subsequently projected into the opposite spin-
state by adding a $\pi$-phase shift to the final $\frac{\pi}{2}$-pulse. This
gives the top and bottom envelopes of the experiment. Additionally, it avoids
a dynamic nuclear polarisation from building up [53, 54]. The counts from the
two envelopes ($c_{\downarrow},c_{\uparrow}$) are used to calculate the
visibility $C$ via:

$\displaystyle
C=\frac{c_{\downarrow}-c_{\uparrow}}{\text{max}(c_{\downarrow}-c_{\uparrow})}$
(1)

###  III.2 Spin relaxation and optical spin pumping

The electron spin lifetime $T_{1}$ is measured in a pump-probe experiment (see
Fig. 8(a)). A $200\text{\,}\mathrm{ns}$ pulse in resonance with the red
“vertical” transition pumps the spin into the $\ket{\uparrow}$-state. The same
pulse is applied after a delay $\tau$. For short delays, no signal is expected
from the second readout pulse. For longer delays spin relaxation processes
flip the spin back to the $\ket{\downarrow}$-state leading to the reappearance
of a signal. By fitting the data to $C=C_{0}(1-\exp(-\tau/T_{1}))$, we find an
electron spin lifetime of $T_{1}=$47(7)\text{\,}\mathrm{\SIUnitSymbolMicro
s}$$.

Capturing the time histogram of the counts during spin pumping, we find an
exponential decay characterising the spin-pumping time (see Fig. 8(b)). The
spin-pumping time decreases with increasing laser power. For all experiments
in this work, the spin pumping laser-power (equivalently, readout laser-power)
is set such that the spin pumping time is $\sim$17\text{\,}\mathrm{ns}$$. An
estimate of the spin-pumping fidelity is given by the residual counts
($c_{\infty}$) compared to the initial counts ($c_{0}$) via
$F_{\text{OSP}}=\sqrt{(1-c_{\infty}/c_{0})}=0.99$ [55].

Figure 8: Electron-spin relaxation time. (a) A pump-probe experiment
determines the electron relaxation time,
$T_{1}=$47(7)\text{\,}\mathrm{\SIUnitSymbolMicro s}$$. (b) Characteristic spin
pumping decay. The solid line is an exponential fit with spin pumping time
$\tau_{\text{OSP}}=$17.0(1)\text{\,}\mathrm{ns}$$ and spin-pumping fidelity
$F_{\text{OSP}}=0.99$.

###  III.3 Rabi frequency scaling

Figure 9(a) shows the Rabi frequency $\Omega/2\pi$ of the driven electron spin
as a function of laser power. Ideally, $\Omega/2\pi$ depends linearly on the
laser power [42]. We find a small deviation to a linear behaviour which we
attribute to a slightly nonlinear response of the AOM we used for power
control. The Rabi frequency scales inversely with detuning with respect to the
excited states $\Delta_{\text{L}}$. Figure 9(b) shows Rabi oscillations for
increasing $\Delta_{\text{L}}$. The Rabi frequency is extracted from an
exponential fit and shown in Figure 9(c). While Rabi frequencies of several
hundred MHz are possible, the quality factor falls at the highest laser powers
on account of a laser-induced spin flip [42].

Figure 9: Rabi frequency scaling. (a) Rabi frequency $\Omega/2\pi$ versus
optical power follows a linear increase. The laser power is measured after the
30:70 beam splitter in front of the optical window. (b) Rabi oscillations for
different laser detunings: from top to bottom
$\Delta_{\text{L}}=$200\text{\,}\mathrm{GHz}$$,
$\Delta_{\text{L}}=$300\text{\,}\mathrm{GHz}$$,
$\Delta_{\text{L}}=$500\text{\,}\mathrm{GHz}$$,
$\Delta_{\text{L}}=$700\text{\,}\mathrm{GHz}$$. (c) Rabi frequency as a
function of excited state detuning $\Delta_{\text{L}}$.

###  III.4 Laser-induced spin flip

The effect of a rotation-laser-induced spin flip can be measured by setting
$f_{\text{AWG}}$ off-resonance with respect to the electron spin resonance
(ESR) (see Fig. 10, diamonds). On increasing the drive time $t$ we see an
increase in counts, a consequence of a rotation-laser-induced spin flip. This
process limits the quality factor of the Rabi oscillations [42]. An
exponential fit $\exp(-\kappa t)$ gives a spin-flip rate of
$\kappa=2\pi\times$2.5\text{\,}\mathrm{MHz}$$ for
$\Omega=2\pi\times$130\text{\,}\mathrm{MHz}$$ and
$\kappa=2\pi\times$1.6\text{\,}\mathrm{MHz}$$ for
$\Omega=2\pi\times$65\text{\,}\mathrm{MHz}$$. We find a scaling factor of
$\kappa/\Omega=0.19$ and $0.25$, respectively. The mechanism responsible for
this unwanted process is unclear.

Figure 10: Laser-induced spin flip. Blue circles: Rabi oscillations. Orange
diamonds: rotation-laser-induced spin flip on driving Rabi oscillations far
off-resonance with respect to the ESR. (a)
$\Omega=2\pi\times$130\text{\,}\mathrm{MHz}$$, (b)
$\Omega=2\pi\times$64.6\text{\,}\mathrm{MHz}$$.

###  III.5 Hartmann-Hahn resonances

The quality factor $Q=2T_{2}^{\text{Rabi}}f_{\text{Rabi}}$ of the electron
spin rotations shows a nontrivial dependence on the Rabi frequency
$f_{\text{Rabi}}=\Omega/2\pi$ (see Fig. 11). Driving at
$\Omega>2\pi\times$50\text{\,}\mathrm{MHz}$$, the quality factor is constant
at $Q\approx 30$, limited by rotation-laser-induced spin-flips. For
$\Omega<2\pi\times$50\text{\,}\mathrm{MHz}$$ there is a complicated
$\Omega$-dependence. Hartmann-Hahn resonances are expected when the Rabi
frequency matched the nuclei frequencies $\omega_{\mathrm{n}}$ (i.e.,
$\Omega\approx\omega_{\mathrm{n}}$) [43]. Something akin to the Hartmann-Hahn
process is revealed here as $Q$ rises and falls in the frequency range where
$\Omega/2\pi$ and the various values of $\omega_{\mathrm{n}}/2\pi$ match.
However, the minima in $Q$ do not lie consistently at the exact values of
$\omega_{\mathrm{n}}/2\pi$ for the isotopes involved. Furthermore, we also
find a minimum in $Q$ close to the difference frequency 71Ga – 75As. Similar
structure was observed for InGaAs QDs [42] despite the very different strain
environments of the two systems and on gate-defined GaAs QDs [31].

Figure 11: Hartmann-Hahn resonances. Quality factor $Q$ as a function of Rabi
frequency $\Omega/2\pi$. For $\Omega>2\pi\times$50\text{\,}\mathrm{MHz}$$, $Q$
is constant, $Q\approx 30$. Dashed lines correspond to the nuclear Larmor
frequencies without quadrupolar shifts.

##  IV Cooling of the nuclear ensemble

###  IV.1 Rabi cooling

Figure 12(a) shows the nuclear cooling dependence on the duration of the Rabi
cooling pulse length $T_{\text{c}}$. We find a maximum of $T_{2}^{*}$ for
$T_{\text{c}}\approx$200\text{\,}\mathrm{ns}$$ and a slight drop of
$T_{2}^{*}$ for longer $T_{\text{c}}$. Note that a single Rabi cooling cycle
is not enough to reach $T_{2}^{*}=$78\text{\,}\mathrm{ns}$$ – it requires the
repetitive application of the Rabi cooling pulses and the Ramsey measurement.
Adding a waiting time $t_{\text{wait}}$ between Rabi cooling pulse and the
Ramsey experiment, we find a decay of the $T_{2}^{*}$ (see Fig. 12(b)).
Fitting the data to a single exponential decay, we find a decay time of
$39(8)\text{\,}\mathrm{\SIUnitSymbolMicro s}$. In both experiments, the
initialise/readout pulse is $200\text{\,}\mathrm{ns}$ in duration, the
$\frac{\pi}{2}$-pulses are $3\text{\,}\mathrm{ns}$ in duration.

Figure 12: Rabi cooling dependencies. (a) $T_{2}^{*}$ as a function of Rabi
drive duration $T_{\text{c}}$ before each Ramsey sequence. The entire sequence
is repeated many times for each value of $T_{\text{c}}$. The nuclear spins do
not return to their fully warm state ($T_{2}^{*}=$3.9\text{\,}\mathrm{ns}$$)
after each sequence. (b) $T_{2}^{*}$ decay for increased wait time
$t_{\text{wait}}$ between cooling pulse and Ramsey experiment. The solid line
is an exponential fit with a decay time of
$39(8)\text{\,}\mathrm{\SIUnitSymbolMicro s}$ for which we fix the decay to
the bare $T_{2}^{*}=$3.9\text{\,}\mathrm{ns}$$ for large values of
$t_{\text{wait}}$.

###  IV.2 Quantum-sensing-based cooling

Figure 13(a) shows the pulse sequence applied to the spin control laser, the
readout laser and the corresponding signal we measure on the SNSPD for the
quantum-sensing-based cooling protocol. Figure 13(b-e) summarises the
dependencies of the quantum-sensing-based cooling on $T_{\text{c}}$,
$N_{\text{pulses}}$, $\tau_{\text{max}}$, and $\tau_{\text{min}}$. Each of the
parameters was changed individually with the other parameters kept constant.
For $T_{\text{c}}$ and $\tau_{\text{max}}$ a clear maximum could be found at
$T_{\text{c}}\approx$125\text{\,}\mathrm{ns}$$ and
$\tau_{\text{max}}\approx$500\text{\,}\mathrm{ns}$$. Conversely, the
dependence on $N_{\text{pulses}}$ did not show a strong optimum, we chose to
use $N=40$ pulses. Remarkably, with just $N_{\text{pulses}}=5$ we reach a
$T_{2}^{*}$ of $500\text{\,}\mathrm{ns}$. This suggests that also the quantum-
sensing-based cooling protocol relies on the repetitive application of the
cycle in Fig. 13(a), i.e., that full cooling is not achieved with a single
cycle. $\tau_{\text{min}}$ is set such that the Ramsey envelope does not show
oscillations. This is most clearly visible on performing a fast Fourier
transform (FFT) on the Ramsey decay. For
$\tau_{\text{min}}<$30\text{\,}\mathrm{ns}$$ only a single peak is visible in
the FFT. For $\tau_{\text{min}}=$30\text{\,}\mathrm{ns}$$ side peaks appear,
suggesting that $\tau_{\text{min}}$ is set too large such that there are
multiple locking points [33]. We thus decided to work with
$\tau_{\text{min}}=$20\text{\,}\mathrm{ns}$$.

Figure 13: Quantum-sensing-based cooling dependencies. (a) Schematic of pulse
sequence for the quantum-sensing-based cooling. The initialise/readout/reset
pulse is $200\text{\,}\mathrm{ns}$ in duration, the $\frac{\pi}{2}$-pulses are
$3\text{\,}\mathrm{ns}$ in duration. The Rabi frequency during cooling is set
to $\Omega_{\text{c}}=2\pi\times$17\text{\,}\mathrm{MHz}$$. (b) $T_{2}^{*}$ as
a function of electron-nuclei interaction drive time $T_{\text{c}}$. (c)
$T_{2}^{*}$ as a function of the number of cooling pulses $N_{\text{pulses}}$.
(d) $T_{2}^{*}$ as a function of maximum sensing time $\tau_{\text{max}}$. (e)
FFT of the Ramsey decay as a function of the minimum sensing time
$\tau_{\text{min}}$. Black dashed lines are a guide to the eye to highlight
the appearance of side peaks at oscillation frequency $f$ in the Ramsey
experiment. (f) $T_{2}^{*}$ decay after optimum cooling. The solid line is an
exponential fit to the data with a decay time of
$41(4)\text{\,}\mathrm{\SIUnitSymbolMicro s}$ for which we fix the decay to
the bare $T_{2}^{*}=$3.9\text{\,}\mathrm{ns}$$ for large values of
$t_{\text{wait}}$.

Figure 13(f) shows $T_{2}^{*}$ as a function of the wait time
$t_{\text{wait}}$ between quantum-sensing-based cooling and the Ramsey
experiment, similar to Fig. 12(b). Fitting the data with an exponential decay,
we find a decay time of $41(4)\text{\,}\mu\mathrm{s}$. The decay time is short
compared to the nuclei diffusion times extracted from similar experiments on
InGaAs QDs [33] and nuclei diffusion times extracted from NMR experiments on
droplet-etched GaAs QDs [39, 56]. Our results on this GaAs QD may hint at the
importance of an electron cotunneling or an electron-mediated coupling between
separate nuclear spins [57, 56].

###  IV.3 Chevron after quantum-sensing-based cooling

On measuring Rabi oscillations as a function of ESR detuning
($\Delta=f_{\text{AWG}}-f_{\text{Z}}$) after quantum-sensing-based cooling at
$f_{\text{c}}=f_{\text{Z}}$ we find the typical chevron pattern (see Fig. 14).
Taking into account the effect of Gaussian-distributed Overhauser field noise
with effective ESR broadening of width $\sigma_{\text{OH}}$, the Rabi
oscillations can be modelled as:

$\displaystyle
C(t,\Delta)=\int_{-\infty}^{+\infty}\text{d}\delta\frac{1}{\sqrt{2\pi}\sigma_{\text{OH}}}\exp{-\frac{\delta^{2}}{2\sigma_{\text{OH}}^{2}}}\cdot\frac{\Omega^{2}}{\Omega^{2}+(\Delta-\delta_{\text{AC}}+\delta)^{2}}\sin^{2}(\sqrt{\Omega^{2}+(\Delta-\delta_{\text{AC}}+\delta)^{2})}t/2).$
(2)

Fitting the data to Eq. 2 we find a perfect match for
$\sigma_{\text{OH}}=$8.1\text{\,}\mathrm{MHz}$$ and a small frequency offset
of $\delta_{\text{AC}}=$-1.61\text{\,}\mathrm{MHz}$$ (see Fig. 14). The larger
broadening with respect to the fully cooled state can be explained by the fact
that the Rabi pulses have a much longer duration than the Ramsey sequence: the
narrowed nuclei distribution following quantum-sensing-based cooling is
disturbed by the Rabi experiment leading to an increase in
$\sigma_{\text{OH}}$ with respect to the maximally cooled case of
$\sigma_{\text{OH}}=$0.36\text{\,}\mathrm{MHz}$$ as extracted from a FFT of
the Ramsey experiment.

The frequency offset $\delta_{\text{AC}}$ arises as a consequence of the AC-
Stark effect, which can occur if the couplings of the two transitions of the
lambda system are slightly imbalanced (e.g. if the rotation laser polarisation
is not perfectly circular). During cooling, the frequency is locked at
$f_{\text{c}}+\delta_{\text{AC}}$, which leads to a small offset in detuning
[58]. Here, cooling was performed at
$\Omega=2\pi\times$17\text{\,}\mathrm{MHz}$$, while the chevron was recorded
at $\Omega=2\pi\times$8.9\text{\,}\mathrm{MHz}$$.

Note that the experimental data were acquired in two runs, the first for Rabi
oscillations up to $t=$400\text{\,}\mathrm{ns}$$, the second for Rabi
oscillations from $t=$400\text{\,}\mathrm{ns}$$ to $480\text{\,}\mathrm{ns}$
in order to resolve the fourth oscillation. This is the origin of the
deviation between the model and experiment for $t>$400\text{\,}\mathrm{ns}$$:
there is a slight change in power and cooling performance between the two runs
(see Fig. 14(b)).

Figure 14: Chevron after quantum-sensing-based cooling. (a) Left: experiment,
right: fit of data to Eq. 2. (b) Residual between data and fit.

###  IV.4 Rabi decay after quantum-sensing-based cooling

The decay envelope of the Rabi oscillations can take different forms [59, 60].
We fit the data to $\exp[(-t/T_{2}^{\text{Rabi}})^{\alpha}]$ with $\alpha$ a
free fit parameter. However, it is difficult to distinguish unambiguously
between a Gaussian ($\alpha=2$) and an exponential ($\alpha=1$) decay: the
residuals of the Gaussian and exponential fits are very similar. We decided to
use the exponential fit to make a comparison to previous experiments [42, 31].
Comparing Rabi oscillations with and without quantum-sensing-based cooling
(Fig. 15(a,b)) we find an increase in the quality factor from $Q=23.5$ to
$30.3$ consistent with a reduction of hyperfine-interaction-related decay.

Figure 15: Quality factor after quantum-sensing-based-cooling. Comparison of
the Rabi decay fitted to $\exp[-(t/T_{2}^{\text{Rabi}})^{\alpha}]$ for
Gaussian ($\alpha$=2) and exponential ($\alpha=1$) decay. Rabi oscillations
for $\Omega=2\pi\times$64.6\text{\,}\mathrm{MHz}$$ without cooling (top), and
after quantum-sensing-based cooling (bottom).

##  V Decoupling Sequences

The CPMG pulse sequence potentially allows the electron spin to be decoupled
from nuclear noise, enhancing the spin $T_{2}$ time [61]. The CPMG sequence is
shown in Fig. 16(a): after a $\frac{\pi}{2}_{x}$-pulse and $\tau/2$ delay,
$N_{\pi}$ $\pi_{y}$-refocusing pulses separated by $\tau$ were applied
followed by another $\tau/2$ delay and $\frac{\pi}{2}_{x}$-pulse. We fit the
experimental data (see Fig. 17) with stretched exponentials
$\exp[(-t/T_{2}^{\text{CPMG}})^{\alpha}]$ and plot $T_{2}^{\text{CPMG}}$
against the number of applied refocusing pulses $N_{\pi}$, where
$t=2N_{\pi}\tau$. Using a power-law fit, we find a $T_{2}$ time scaling of
$N_{\pi}^{\gamma}$ with $\gamma=$0.69(12)\text{\,}$$ (see Fig. 16(b,c)). The
power-law fit is insensitive to the exponent $\alpha$ of the stretched
exponential and gives insight into the noise sources in the environment [61].
$\gamma=$0.69(12)\text{\,}$$ is similar to the value obtained in Ref. [40]
($\gamma=0.75$) for which up to 81 decoupling pulses were applied to droplet-
etched GaAs QDs. Ref. [49] reports $\gamma=0.8$ on gate-defined GaAs QDs for
$N_{\pi}<16$ and a crossover to $\gamma=0.5$ for large $N_{\pi}$. On
increasing $N_{\pi}$, no change of $\gamma$ was observed in the experiments
presented here.

The application of more refocusing pulses was not possible due to a reduction
in readout contrast as more pulses were applied. This is due to imperfect
pulse calibrations and the limit imposed by the electron-spin relaxation time
$T_{1}$. These problems can be mitigated with more precise calibration and
with a new sample with a larger tunnel barrier to the back contact,
respectively.

Figure 16: CPMG decoupling pulse scaling. (a) Pulse sequence for the CPMG
experiment. (b) Electron spin decoupling using CPMG pulses. Solid line is a
power-law fit with $T_{2}$ time scaling $T_{2}^{\text{CPMG}}\propto
N_{\pi}^{\gamma}$ with $\gamma=$0.69(12)\text{\,}$$. (c) $\alpha$ extracted
from a stretched exponential fit to the data shown in Fig. 17.  Figure 17:
CPMG decoupling data. Electron spin decoupling using CPMG pulses. Solid lines
are fits to stretched-exponentials $\exp[(-t/T_{2}^{\text{CPMG}})^{\alpha}]$.

*


# Semi-Supervised Few-Shot Atomic Action Recognition

Xiaoyuan Ni
Hong Kong University of Science and Technology
<EMAIL_ADDRESS>
&Sizhe Song 11footnotemark: 1
Hong Kong University of Science and Technology
<EMAIL_ADDRESS>
&Yu-Wing Tai
Kauishou
<EMAIL_ADDRESS>
Chi-Keung Tang
Hong Kong University of Science and Technology
<EMAIL_ADDRESS>
Equal contribution.

###### Abstract

Despite excellent progress has been made, the performance on action
recognition still heavily relies on specific datasets, which are difficult to
extend new action classes due to labor-intensive labeling. Moreover, the high
diversity in Spatio-temporal appearance requires robust and representative
action feature aggregation and attention. To address the above issues, we
focus on atomic actions and propose a novel model for semi-supervised few-shot
atomic action recognition. Our model features unsupervised and contrastive
video embedding, loose action alignment, multi-head feature comparison, and
attention-based aggregation, together of which enables action recognition with
only a few training examples through extracting more representative features
and allowing flexibility in spatial and temporal alignment and variations in
the action. Experiments show that our model can attain high accuracy on
representative atomic action datasets outperforming their respective state-of-
the-art classification accuracy in full supervision setting.

##  1 Introduction

Significant achievements have been made in action recognition [1, 2, 3] thanks
to the development of 3D-convolutional networks (C3D) and Spatio-temporal
attention over videos. Recent sequential embedding networks including LSTM [4]
and temporal convolution (TCN) [5] have been applied for achieving better
temporal alignment of video action. However, the performance of state-of-the-
art action recognition models relies on large-scale training datasets which
are not easy to collect and annotate. In particular, the pertinent action may
not occupy the entire Spatio-temporal volume of the given video, i.e., the
action may occupy a subset of spatial and temporal volume of the given video
frames with intra-class variations in relative position and length. To further
complicate the problem, the relative sequences of sub-actions may vary, i.e.,
‘playing basketball’ may contain a different permutation of ‘dribbling’ and
‘passing’, which poses great challenges in temporal alignment.

Current methods either ignore alignment such as permutation-invariant
attention [6] or impose overly strict alignment such as dynamic time warp [7].
The flexibility within action also presents difficulty in the aggregation of
action features within a class. Naïve aggregation functions such as summation
may harm representation which may also be easily affected by outliers.

To tackle the above issues, this paper focuses on atomic or fine-grained
actions of duration typically less than 2 secs (e.g., dribbling and passing),
which sidestep the need for strict alignment while making loose alignment
sufficient. Atomic actions have shown promising performance gain for action
recognition over conventional methods trained using coarse-grained action
videos (e.g., playing basketball) [8]. We propose a novel semi-supervised
network for the few-shot atomic action classification, that supports action
recognition of long query videos under the $K$-way $N$-shot setting [9].
Specifically, our model features a better understanding of human-centric
atomic action with:

  1. 1.

Semi-supervised training. The video embedding module is trained in an
unsupervised manner, extracting representative video features and classifying
the action given only several examples.

  2. 2.

Loose action alignment. We adopt sliding windows over the temporal domain and
use connectionist temporal classification (CTC) loss [10] to train the video
embedding with relatively loose alignment, making the model more robust to
variations in the sub-action permutation.

  3. 3.

Multi-head video comparison. We develop a multi-head relational network to
consider both global and local similarity.

  4. 4.

Attention-based feature aggregation. Our model aggregates class features
through computing the mutual relationship between support and query videos.
Compared with traditional mean aggregation, the attention-based one extracts
the most related features for classification, thus reducing the representation
complexity for class features and improving classification efficiency.

Overall, this paper contributes to few-shot atomic action recognition with
semi-supervised learning. Extensive experiments over published datasets show
that our method outperforms the state-of-the-art accuracy achieved by models
trained in full supervision.

Figure 1: Overview. Our model provides fine-grained spatial and temporal video
processing with high length flexibility, which embeds the video feature and
temporally combines the features with TCN. Further, our models provides
attention pooling and compares the multi-head relation. Finally, the CTC and
MSE loss enables our model for time-invariant few shot classification
training.

##  2 Related Work

###  2.1 Few Shot Learning

Few Shot Learning (FSL) refers to a machine learning strategy that learns from
a small amount of labeled data where the data labeling cost on large datasets
can be prohibitively high [11]. Wide use of FSL includes multiple object
tracking and detection [12, 13] and gesture recognition [14, 15].

In this paper, we propose FSL with novel technical contributions in embedding
complex human actions through extending the original relational network [16]
into a multi-head relation network for robust feature comparison that
adequately takes into consideration the high variety of atomic actions, while
not requiring a large amount of human-annotated data.

###  2.2 Action Recognition

Significant progress has been made in action recognition and classification
with the recent development of 3D convolutional models, e.g., I3D [17], C3D
[18] and TRN [19]. All these models perform data-driven classification and
process fixed-size videos by combining the locality in either the temporal or
spatial domain. Their high accuracy is highly dependent on dependency on
datasets used in training and testing.

To address the data issue, we experimented with a number of FSL for action
recognition and found that almost all of these works attempt to align the
video in the temporal domain and matching the relative 2D frames rather than
3D videos [20, 7], or search for the temporal attention of video [6]. While
achieving temporal alignment and attention techniques, these methods partition
the given video into individual frames or tiny clips, thus introducing great
complexity in their alignment strategies and inevitably losing the generality
over datasets with distinct Spatio-temporal features. In contrast, our method
provides a simple model with holistic understanding over the entire video,
focusing on the human-centric video prediction without relying on any
background and object information.

###  2.3 Semi-Supervised Learning

Semi-supervised learning is the learning based on both labeled and unlabeled
data. In our task, although all the videos all have action labels, there are
no boxes to localize where the actions are taking place in individual frames.
Thus it is possible to divide the learning strategies into two stages: the
first stage is action classification with supervised learning and the second
stage is action localization in terms of spatial attention with unsupervised
learning. In [6], spatial attention, and temporal attention are trained with
unsupervised learning.

Typical issues in applying unsupervised learning in feature extraction include
limited dictionary size and inconsistent memory. Most recently, the Momentum
Contrast (MoCo) has been proposed for unsupervised visual representation
learning [21], which regards contrastive learning as dictionary-lookup and
builds a dynamic and consistent dictionary on-the-fly. In this paper, MoCo has
adopted to pretrain our encoder under an unsupervised setting.

##  3 Method

Figure 1 illustrates our model structure. We use a C3D+TCN encoder to embed a
video clip to obtain the input feature. The C3D extracts Spatio-temporal
features from videos and TCN processes the temporal information on a larger
scale. Next, we apply an attention-pooling module where the support features
are refined and integrated. With the query features and refined support
features of each class, we then compute the classification probability by a
multi-head relation network. Finally, the probability vector and ground truth
label are used to obtain a CTC loss and MSE loss to update the network.

###  3.1 Action Augmentation

We apply the following three augmentation methods:

  1. 1.

Human-centric Cropping and Normalization. Process the videos to produce human-
centric video tubes and normalize the figures to the center, expecting the
encoder can thus learn to lay more emphasis on human bodies in the videos, and
invariant to background information.

  2. 2.

Background Subtraction. Apply background subtraction where the moving object
is regarded as foreground and the rest is simply discarded.

  3. 3.

Usual Image Augmentation. Apply random flipping, blurring, color inverting and
rotation over each frame.

(a) Human-centric

(b) Background Subtraction

(c) Usual Image Augmentation

Figure 2: Three methods of action augmentation.

The last two methods are easy to implement. For the first method, see Figure 2
where we use a human tracking tool, such as real-time MOT [22] to obtain
human-centric video tubes, which are frame-wise bounding boxes of the human
body with individual’s identity labeled in each box. Given these video human
tubes, we crop the original frames to get normalized images that precisely
capture the movement of each individual in the video. While real-time MOT can
generate all the normalized frames on-the-fly, in our modular implementation
we generate all the tubes in advance.

###  3.2 Encoder Pretraining

We use MoCo to pretrain our C3D+TCN encoder in an unsupervised manner. The
original videos are input as support and videos augmented with 3.1 as the
query. Then MoCo trains the encoder by a contrastive loss [21] by comparing
the query with multiple supports, both positive and negative. This enables the
encoder to recognize and extract robust key features from the videos.

MoCo updates the encoder in a momentum way, so that the model is capable of
building a consistent memory bank of recently seen data:

$\displaystyle\theta_{k}=m\theta_{k}+(1-m)\theta_{q}$ (1)

where $m$ is the momentum coefficient, $\theta_{k}$ and $\theta_{q}$ are
parameters of the key encoder and query encoder. During back propagation, only
$\theta_{q}$ is updated, and then $\theta_{k}$ is updated using this momentum
mechanism.

###  3.3 Loose Action Alignment

For a video $V$ with undefined length, we utilize sliding windows to segment
the video $V$ to a set of windows of fixed-length
$\\{W_{1},W_{2},\cdots,W_{n}\\}$. After the video embedding, each query window
feature $\tilde{W^{q}_{k}}$ will be compared with a weighted aggregation of
support class windows. Thus the classification probability of few shot class
$\theta$ of $\tilde{W^{q}_{k}}$ will be represented as $P(\theta|W_{k})$:

$\displaystyle P(\theta|W_{k})$
$\displaystyle=g(\tilde{W^{q}_{k}},A(\tilde{W^{s}_{j}},j\in C_{\theta}))$ (2)

where $g(\cdot)$ is the window-wise relational convolution, $A(\cdot)$ is the
attention over windows per class which will be detailed in Section 3.4.

After obtaining the class probability for each window $S_{k}^{\theta}$, two
losses will be computed. The Connectionist temporal classification (CTC) loss
is computed for each query video, by aligning all the windows of the query
video sequentially, and take the negative likehood afterward. The standard MSE
loss over one-hot label will also be computed by adding up the each window’s
probability for each class:

$\displaystyle\mathcal{L}_{CTC}$
$\displaystyle=-P(V,l)=-\sum_{\Theta:\kappa(\Theta)=l}\prod_{t=1}^{T}P(\theta_{t}|W_{t})$
(3) $\displaystyle\mathcal{L}_{MSE}$
$\displaystyle=\frac{1}{T}\sum_{t=1}^{T}\sum_{\theta\in\Theta}(\mathbbm{1}(\theta,l)-P(\theta|W_{t}))^{2}$
$\displaystyle\mathcal{L}$
$\displaystyle=\mathcal{L}_{CTC}+\lambda*\mathcal{L}_{MSE}$

where $l$ refers to the correct label of the video $V$, and $\kappa(\cdot)$ is
the union of the window class trajectories leading to the right label.

The sliding windows trained with CTC loss can effectively solve the alignment
issue for the videos in the temporal domain, which is robust against object
occlusions, disocclusion, and other video instabilities. While the MSE loss
can accelerate the training at the beginning. A weighted combination of the
two loss provides optimized convergence of the model. At the test time, beam
search decoding is performed to retrieve the video label.

###  3.4 Attention Pooling

Since multiple windows are spread across the support videos to extract support
features, a pooling operation is necessary to generate fixed-size final
representations for the relation network where they will be compared with the
query.

Commonly used methods are straightforward max-pooling, average-pooling and
self-attention. In our work, we propose an attention pooling module. As
illustrated in Figure 3, our specially-designed attention pooling takes both
the support feature $\mathcal{S}\in\mathbb{R}^{S\times F}$ and query feature
$\mathcal{Q}\in\mathbb{R}^{Q\times F}$ as input and computes new support
features $\mathcal{S}^{\prime}$ as follows:

$\displaystyle\mathcal{S}^{\prime}$
$\displaystyle=f_{2}(f_{1}(\mathcal{Q}\cdot\mathcal{S}^{T})\cdot\mathcal{S})$
(4)

where $f_{i}(\cdot)$ are linear transformations. The idea is to introduce
query features to the pooling procedure by multiplying query features with
support features transpose, which will generate a weight matrix
$\mathcal{W}\in\mathbb{R}^{Q\times S}$. Each entry $\mathcal{W}_{i,j}$
represents the weight of $j^{th}$ support window to the $i^{th}$ query window.
Then the product of this weight matrix and original support features can be
seen as the refined support features. Two linear functions $f_{1}$ and $f_{2}$
are added to provide some learnable parameters.

In addition, to support feature refinement, we propose to refine query
features by support features. Specifically, the same Equation 4 can be applied
except we swap the support and query. This mutual refinement strategy can
enhance the prediction performance.

(a) Attention Pooling

(b) Multi-head Relation Network

Figure 3: Attention Pooling and Multi-head Relation Network

###  3.5 Multi-Head Relation

Since the few shot models cannot retain the action class features in the FC
layers, as the unseen classes keep coming, it’s crucial to compare the feature
similarity of the query video. The previous relation network [16] used in few
shot action recognition compares the video-wise similarity with a number of
convolutional layers, whose performance significantly as the length of video
increases. We extend this network to a multi-head relation network in two
ways.

First, we reduce the convolution kernel size to 1 and introduce the FC layer
in our multi-head relation since the extracted features no longer retain any
spatial structure. Second, we add one more computation layer on top of the
original Conv+FC layers, which is a window-wise vector product. This provides
a more localized comparison between the support and query. The final output of
multi-head relation network is the sum of the probabilities obtained from both
methods.

##  4 Experiments

###  4.1 Dataset

We test our model on three datasets including Human-Centric Atomic Action
(HAA), Finegym (Gym288), and Moments in Time (MIT). MIT is a general atomic
action dataset with over one million videos from diverse classes [23]. Finegym
is a recent dataset which focuses on fine-grained gymnastic action classes
[24]. HAA provides human-centric videos, with a high average of 69.7%
detectable joints [8]. These datasets are not constructed specifically for
few-shot learning, so we reorganize them to suit our few-shot tasks.

HAA & Gym288. Instead of splitting videos into training and test set, we split
action classes into training and test set for our few-shot tasks.
Consequently, we have 310/156 classes in our HAA training/test set, and 231/57
in our Gym288 respectively.

MIT. The total number of videos in the MIT dataset is huge so we build a mini-
MIT for our experiment. Each action class in mini-MIT has 60 videos, half from
the original training set, and another half from the original validation set.
Like HAA and Gym288, our mini-MIT has 272/67 classes for training and test
respectively.

###  4.2 Model Performance

Model |  HAA [8] |  Gym288 [24] |  mini-MIT [23]
---|---|---|---
Ours | 80.68 | 75.84 | 61.77
SOTA | 55.33 / 75 | 73.7 | 31.16 / 57.67
top1 / top3 | top1 / top3
Table 1: Top-1 accuracy (%) on three datasets under a 3-way 5-shot setting
compared with state-of-the-art performance in [8, 24, 23] where their models
are trained in full supervision.

Table 1 tabulates our results, showing that our few-shot model has a leading
performance compared with the state-of-the-art methods on all the three
datasets trained in full supervision. Note that our model is few-shot which
has access to only a very limited amount of data.

Further, the HAA and Gym288 are the atomic human-centric datasets, while the
former has similar background and the latter’s background is heterogeneous.
The MIT, on the other hand, doesn’t follow a strict human-centric manner. The
outperforming result over all 3 datasets presenting the potentials of our
innovative loose action alignment and attention-based feature aggregation over
a general understanding of atomic action. Besides, the multi-head relational
network’s improvement is not limited to human-centric datasets, showing the
importance of local comparison on action similarity.

###  4.3 Ablation Study

Model | HAA | Gym288 | mini-MIT
---|---|---|---
w/o pretraining | 81.84 | 72.42 | 42.48
w/o AP | 77.53 | 66.06 | 39.93
w/o RN | 63.22 | 69.30 | 37.37
Ours | 80.68 | 75.84 | 61.77
Table 2: Performances without the key features in our model.

Table 2 shows our ablation study result, and proves the effectiveness of our
semi-supervised training, attention-based feature aggregation and multi-head
video comparison.

Specifically, the unsupervised action encoder may experience relatively less
accuracy drop in human-centric datasets such as HAA where the action features
are better aligned. However, on more general datasets, the human-centric
augmentation in Section 3.1 shows greater importance and the ablation accuracy
drops significantly on mini-MIT. Besides, the performance on HAA and Gym288
drops less compared with that on mini-MIT, indicating our model’s better
representativeness over a general set of action data.

##  5 Conclusion

This paper introduces a novel semi-supervised few-shot atomic action
recognition model. We propose to use the sliding windows and CTC alignment to
make the model more robust to coarse temporal annotation. Operating in the
few-shot setting, our model can effectively alleviate human annotation effort.
Moreover, we propose the attention pooling and multi-head relation module to
achieve better feature refinement and comparison. By incorporating
unsupervised and contrastive video embedding, our few-shot model produces
state-of-the-art performance comparable to previous models trained in full
supervision.

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