Enhanced Electron Spin Coherence in a GaAs Quantum Emitter

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 ns to 0.608 $\mu$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.

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 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 (∝ S z I z ), the non-collinear term (∝ 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 [28,34,35].
The most studied QDs for spin-photon applications are QDs in the InGaAs/GaAs system.InGaAs QDs are selfassembled 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- 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 µs to 113 µs [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.
Here, we implement all-optical cooling schemes on lownoise GaAs QDs and demonstrate an increase in the electron spin coherence time from T * 2 = 3.9 ns to 0.608 µs.This is achieved with autonomous feedback and without any external perturbation (such as strain tuning).We demonstrate spin control with T * 2 = 0.608 µs, an extension of T 2 with CPMG (with a scaling of T CPMG matching previous experiments [40]), fast spin rotations (Rabi frequencies above 100 MHz), and high-fidelity spin control (F π > 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 Al 0.15 Ga 0.85 As matrix are filled with GaAs and capped by an Al 0.33 Ga 0.67 As layer.The materials are almost lattice-matched.Figure 1(a) shows a high-angle darkfield 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 [36,37,41].A 3.00 T magnetic field is applied perpendicular to the growth direction (Voigt geometry), at an angle of 45 • to the in-plane crystal axes.The electron Zeeman frequency is f Z = 4.54 GHz corresponding to a g-factor of g e = −0.11.
The spin is manipulated by a two-colour Raman pulse detuned from the excited states by ∆ L = 700 GHz (see Fig. 1(c)).This pulse is created by amplitudemodulating circularly-polarised light with an electrooptic 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 |↓state is bright, the |↑ -state is dark) and to prepare the spin in the |↑ -state via optical spin pumping [41].
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 π 2 -pulses on changing the phase φ of the second pulse, thereby demonstrating rotation around an arbitrary axis on the equator of the Bloch sphere.
On driving Rabi oscillations as a function of the detuning ∆ with respect to the Zeeman frequency (∆ = f Z − f probe ), we find strong deviations from the typical chevron pattern expected for a two-level system (see Fig. 2(a)).In a ∼ 200 MHz window around the Zeeman frequency, we find that the spin rotations lock to the probe frequency f probe , a clear signature of electron spin-nuclear spin coupling [35,[44][45][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 σ 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 (σ OH ∝ 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) ns to 78(2) ns corresponding to a narrowing of σ OH from 52(1) MHz to 2.90(5) 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) ns.To determine the bare electron coherence time, we add a 100 µs buffer between each cycle.This observation suggests that the repetitive application of spin manipulation pulses as short as 4 ns already leads to a narrowing of σ OH .
We confirm the nuclear-spin cooling and locking of the ESR to the Rabi drive by fixing the cooling frequency f c during Rabi cooling, subsequently detuning the probe frequency f probe in a Ramsey experiment.Oscillations arise at the detuning frequencies ∆ = f c − f 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 π 2 -pulse.A period of free evolution τ sense allows the electron to sense the Overhauser field fluctuation that leads to a detuning ∆ from the target frequency f 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 Ω ≈ ω n .The sign of the detuning ∆ 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 τ results in a narrower feedback function in each cycle and hence an increased sensitivity to changes in σ OH .
We find optimal parameters for the quantum-sensingbased cooling at N = 40 cycles with a linearly increasing sensing time τ sense from τ min = 20 ns to τ max = 400 ns, and electron-nuclei drive time T c = 125 ns at a Rabi frequency Ω c = 2π × 17 MHz, followed by a spin pumping pulse of 200 ns [41].This preparation sequence takes ∼ 22 µs and is repeated before each Ramsey cycle.The electron coherence time T * 2 increases from 3.9(2) ns to 0.608 (13) µs 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 ns [33]) and just below the highest reported T * 2 of a single electron spin qubit in a gatedefined GaAs QD (767 ns [31]).The enhancement corresponds to a narrowing of the nuclear-spin ensemble from σ OH = 52(1) MHz to 0.355(4) MHz (see Fig. 3(b, inset)).
Using hyperfine constants A k and abundancies η k of the nuclei species k ∈ { 69 Ga, 71 Ga, 75 As} we can estimate the number of nuclei involved [25,33,40].This corresponds to a distribution of σ OH /A c ≈ 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 Ω c is an important parameter (see Fig. 3(c)).The maximum performance for both cooling schemes occurs at Ω c = 2π × 17 MHz, close to the difference frequency of 71 Ga and 75 As (∆ω = ω( 71 Ga) − ω( 75 As) = 2π × 17.08 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 [28,35,44] 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 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)(corresponding to a π-pulse fidelity of 98.4(1) %) [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 HE 2 = 2.93(6) µs using a Hahn echo (N π =1) to T CPMG 2 = 22(8) µs, an order of magnitude increase, with N π =20 pulses.We extract a T 2 scaling of T CPMG 2 ∝ N γ π with γ = 0.69 (12), consistent with recent results on droplet-etched QDs [40] and gatedefined 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 ∼ 40 µs [41].
In conclusion, we have demonstrated fast and flexible optical control of an electron spin confined to a selfassembled 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.Nuclearspin cooling leads to a 156-fold increase in the T * 2 time, T * 2 = 0.608 µs.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].

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 ioninduced damage.The first C-layer was deposited using electron-induced deposition at a beam energy of 5 keV and a beam current of 3.2 nA.The second C-layer was deposited with ion-induced deposition at a beam energy of 30 keV and a beam current of 83 pA.Sample cutting and polishing were carried out with the FIB at a beam energy of 30 kV and beam currents ranging from 240 pA down to 83 pA.The sample thickness in the upper area, where the QD is located, was < 50 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 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 ∼ 50 nm, x ∼ 50 nm) and a low signal above (see Fig. 5(b)).This is expected as the matrix material below the QD is Al 0.15 Ga 0.85 As and the QD is grown on and capped with Al 0.33 Ga 0.67 As.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 ∼ 45 nm 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 Al 0.33 Ga 0.67 As 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].
Spin manipulation is achieved using optical pulses at frequency f2, detuned from f1 by ∆L.Optical sidebands at f2 ± fAWG are generated by an EOM that is modulated at half the Zeeman frequency (fAWG = fZ/2).A readout laser at frequency f1 is used for spin initialisation (via optical spin pumping) and spin readout.Back-reflected laser light at f1 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 f2 is rejected with a grating filter with 25 GHz bandwidth (BW) centred around f1.A half-wave plate (HWP) before the QD is used to match the polarisation of the excitation to the "vertical" QD transition.

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 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 τ = 392 ps, which corresponds to a Fourier-limited linewidth of 1 2πτ = 405 MHz.The linewidth is therefore 22 % above the Fourier limit.This is slightly higher than the average QD-linewidth on the sample [36].We record a resonance fluorescence (RF) plateau map by scanning both excitation laser frequency (f ) and gate voltage (V g ) across the X − transitions at B = 3.00 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 excitedstate Zeeman splittings to be ω e = 2π × 4.54 GHz and ω h = 2π × 5.40 GHz, respectively, corresponding to g-factors (g e/h = ω e/h /µ B B) of g e = −0.11and g h = 0.13.(The assumption here is that g 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 (∼ 85 • ).

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 5−60 s for a signal to noise ratio of ∼ 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 π-phase shift to the final π 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 ↓ , c ↑ ) are used to calculate the visibility C via:

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 ns pulse in resonance with the red "vertical" transition pumps the spin into the |↑ -state.The same pulse is applied after a delay τ .For short delays, no signal is expected from the second readout pulse.For longer delays spin relaxation processes flip the spin back to the |↓ -state leading to the reappearance of a signal.By fitting the data to C = C 0 (1 − exp(−τ /T 1 )), we find an electron spin lifetime of T 1 = 47 (7) µs.
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 ∼ 17 ns.An estimate of the spin-pumping fidelity is given by the residual counts (c ∞ ) compared to the initial counts (c 0 ) via F OSP = (1 − c ∞ /c 0 ) = 0.99 [55].

Rabi frequency scaling
Figure 9(a) shows the Rabi frequency Ω/2π of the driven electron spin as a function of laser power.Ideally, Ω/2π 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 ∆ L . Figure 9(b) shows Rabi oscillations for increasing ∆ 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].

Laser-induced spin flip
The effect of a rotation-laser-induced spin flip can be measured by setting f 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(−κt) gives a spin-flip rate of κ = 2π × 2.5 MHz for Ω = 2π × 130 MHz and κ = 2π × 1.6 MHz for Ω = 2π × 65 MHz.We find a scaling factor of κ/Ω = 0.19 and 0.25, respectively.The mechanism responsible for this unwanted process is unclear.

Hartmann-Hahn resonances
The quality factor Q = 2T Rabi 2 f Rabi of the electron spin rotations shows a nontrivial dependence on the Rabi frequency f Rabi = Ω/2π (see Fig. 11).Driving at Ω > 2π × 50 MHz, the quality factor is constant at Q ≈ 30, limited by rotation-laser-induced spin-flips.For Ω < 2π × 50 MHz there is a complicated Ω-dependence.Hartmann-Hahn resonances are expected when the Rabi frequency matched the nuclei frequencies ω n (i.e., Ω ≈ ω n ) [43].Something akin to the Hartmann-Hahn process is revealed here as Q rises and falls in the frequency range where Ω/2π and the various values of ω n /2π match.However, the minima in Q do not lie consistently at the exact values of ω n /2π for the isotopes involved.Furthermore, we also find a minimum in Q close to the difference frequency 71 Ga -75 As.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].

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 c , N pulses , τ max , and τ min .Each of the parameters was changed individually with the other parameters kept constant.For T c and τ max a clear maximum could be found at T c ≈ 125 ns and τ max ≈ 500 ns.Conversely, the dependence on N pulses did not show a strong optimum, we chose to use N = 40 pulses.Remarkably, with just N pulses = 5 we reach a T * 2 of 500 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.τ 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 τ min < 30 ns only a single peak is visible in the FFT.For τ min = 30 ns side peaks appear, suggesting that τ min is set too large such that there are multiple locking points [33].We thus decided to work with τ min = 20 ns. Figure 13(f) shows T * 2 as a function of the wait time t 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) µs.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 [56,57].

Chevron after quantum-sensing-based cooling
On measuring Rabi oscillations as a function of ESR detuning (∆ = f AWG − f Z ) after quantum-sensing-based cooling at f c = f 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 σ OH , the Rabi oscillations can be modelled as: ( Fitting the data to Eq. 2 we find a perfect match for σ OH = 8.1 MHz and a small frequency offset of δ AC = −1.61MHz (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 quantumsensing-based cooling is disturbed by the Rabi experiment leading to an increase in σ OH with respect to the maximally cooled case of σ OH = 0.36 MHz as extracted from a FFT of the Ramsey experiment.
The frequency offset δ 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 c + δ AC , which leads to a small offset in detuning [58].Here, cooling was performed at Ω = 2π × 17 MHz, while the chevron was recorded at Ω = 2π × 8.9 MHz.
Note that the experimental data were acquired in two runs, the first for Rabi oscillations up to t = 400 ns, the second for Rabi oscillations from t = 400 ns to 480 ns in order to resolve the fourth oscillation.This is the origin of the deviation between the model and experiment for t > 400 ns: there is a slight change in power and cooling performance between the two runs (see Fig.

2 ∝Figure 1 .
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 ∆L = 700 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 |↑ -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 Rabi 2 = 73(5) ns.(e) Full control of the rotation axis about the Bloch sphere using two consecutive π 2 -pulses as a function of the phase φ of the second pulse.The solid line is a sinusoidal fit to the data.

2 �Figure 2 .
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 Tc = 1 µs long pulse at a Rabi frequency of Ωc = 2π × 17 MHz is used.The Ramsey experiment was performed at a larger Rabi frequency of 2π × 100 MHz.Bottom: Top and bottom envelopes of the Ramsey interferometry with 100 µs pause (circles), zero pause (squares) and Rabi cooling (diamonds); the extracted coherence times are T * 2 = 3.9(2) ns, T * 2 = 7.8(2) ns, and T * 2 = 78(2) 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 fc of the Rabi drive.(d) Linecut at ∆ = 50 MHz with T * 2 = 87(6) ns (dashed box in (c)).The solid lines in (b) and (d) are Gaussian fits to the data.

Figure 5 .
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.

Figure 6 .
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 MHz.(b) Lifetime measurement of the X − .The solid line is an exponential fit with lifetime τ = 392 ps.This corresponds to a Fourier-limited linewidth of 405 MHz.

Figure 7 .
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.

Figure 9 .
Figure 9. Rabi frequency scaling.(a) Rabi frequency Ω/2π 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 ∆L = 200 GHz, ∆L = 300 GHz, ∆L = 500 GHz, ∆L = 700 GHz.(c) Rabi frequency as a function of excited state detuning ∆L.

Figure 13 .
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 ns in duration, the π 2 -pulses are 3 ns in duration.The Rabi frequency during cooling is set to Ωc = 2π × 17 MHz.(b) T * 2 as a function of electron-nuclei interaction drive time Tc.(c) T * 2 as a function of the number of cooling pulses N pulses .(d) T * 2 as a function of maximum sensing time τmax.(e) FFT of the Ramsey decay as a function of the minimum sensing time τmin.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) µs for which we fix the decay to the bare T * 2 = 3.9 ns for large values of twait. 14(b)).