SUPPLEMENTARY INFORMATIONDOI: 10.1038/NPHYS2078

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Ultrafast Coherent Control and Suppressed Nuclear

Feedback of a Quantum Dot Hole Qubit:

Supplementary Information

Kristiaan De Greve1, Peter L. McMahon1, David Press1, Thaddeus D. Ladd1,2,4,Dirk Bisping3, Christian Schneider3, Martin Kamp3, Lukas Worschech3,

Sven Hofling1,3, Alfred Forchel3 & Yoshihisa Yamamoto1,2

1E. L. Ginzton Laboratory, Stanford University, Stanford, California 94305, USA2National Institute of Informatics, Hitotsubashi 2-1-2, Chiyoda-ku,

Tokyo 101-8403, Japan3 Technische Physik and Wilhelm Conrad Rontgen Research Center for Complex Material

Systems, University of Wurzburg, Am Hubland, D97074 Wurzburg, Germany4 Currently at HRL Laboratories, LLC, 3011 Malibu Canyon Rd., Malibu, CA 90265, USA

SI. DEVICE DESIGN

Two different types of samples were studied: δ-doped samples, and charge-tuneable devices. The δ-doped samples contain about 1.5×1010 cm−2 self-assembledquantum dots, and the charge-tuneable samples about 7×109 cm−2. For both typesof samples, the quantum dots were grown using the Stranski-Krastanov method.The Indium flushing and partial capping technique used during the QD growth [1]leads to the formation of flattened QDs, with an approximate height of 2 nm, and abase length of ∼25 nm. The detailed layer structures are provided in Fig. SI-1. For

140 nm i‐GaAs

p‐GaAs (substrate)

69 nm p‐GaAs λ/4

82 nm p‐AlAs λ/4

…

113 nm p‐GaAs25 nm i‐GaAs

25 x

10 nm i‐GaAs

…110 nm i‐AlAs/GaAs superlattice (e‐ blocking layer)

25 nm n‐GaAs

69 nm n‐GaAs λ/4

…5 x

100 nm Al Shadow mask16 μm

i‐GaAs (substrate)

69 nm i‐GaAs λ/4

83 nm i‐AlAs λ/4

…

140 nm i‐GaAs

69 nm i‐GaAs λ/4

…

100 nm Al Shadow mask16 μm

10 nm

A B

140 nm i‐GaAs

83 nm i‐AlAs λ/4 82 nm n‐AlAs λ/4

Figure SI-1: Detailed layer structure. (A) Detailed layer structure of the δ-dopedsamples. The δ-doping layer (dashed line) is located 10 nm below the quantum dots. (B)Detailed layer structure of the charge-tuneable devices used in the hole spin experiment.Two DBR layer stacks form an asymmetric cavity, in which a p-i-n-diode is embedded. QDs(brown triangles) are in tunnel contact (25 nm i-GaAs) with a hole reservoir. The biasvoltage is applied over the Al shadow mask, and a bottom contact (not shown).

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both types of samples, the QDs are embedded in a planar microcavity, consisting ofDistributed Bragg Reflector (DBR) mirrors. The top and bottom mirrors consist of5 and 25 pairs of AlAs/GaAs λ/4 layers respectively. The resulting quality factor isaround 200, and helps both in increasing the signal strength (directing the emittedlight upward) and reducing the noise (enabling the use of lower laser power, andtherefore reducing the noise due to scattered laser light).

For the δ-doped samples, a carbon δ-doping layer is used, located 10 nm belowthe QDs. The δ-doping concentration is approximately 1.2×1011 cm−2, and leads toa fraction of the QDs being charged with a single hole; we perform magneto-PL mea-surements in order to identify those QDs that are charged. For the charge-tuneablesamples, deterministic charging occurs by embedding the QDs into a p-i-n-diodestructure. The bottom DBR, as well as part of the cavity, is p-doped (≥ 1018 cm−3),while the top DBR is n-doped (≥ 1018 cm−3). The i-layer consists of two parts: a

3500 3600 3700 3800 3900

-3

-2

-1

0

Position [nm]

Val

ence

ban

d en

ergy

[eV

]

3610 3620 3630 3640 3650 3660-0.4

-0.3

-0.2

-0.1

0

0.1

Position [nm]

Val

ence

ban

d en

ergy

[eV

]

-20 -10 0 10 20

0

0.5

1

1.5

Position (growth direction) [nm]

Ene

rgy

[eV

]

0 0.5 1 1.5 2 2.5

-0.1

-0.2

-0.3

-0.4

Position (growth direction) [nm]

Ene

rgy

[eV

]QDGaAs GaAsEC

EV (HH, LH)

a b

199 meV 185 meV

Top DBRBottomDBR

AlAssuper‐lattice

QD

EF EF

Vb=0: 230 meV

Vb>0: 199 meV

c d

Figure SI-2: Band line-up. (A) Calculated energy structure (3D) of the InAs QD. EC,V

are the conduction and valence bands respectively (the latter is calculated for the heavy (HH,black) and light hole (LH, blue) subbands). The two most tightly bound energy states in theconduction and valence bands are indicated by the red and green dashed lines. (B) Zoomed-in version of A. A flattened QD was assumed, with 2 nm height, 25 nm base length, andup to 50 % In-Ga intermixing. The most tightly bound HH state is located 199 meV abovethe GaAs valence band, and the second-highest state is split off by ∼14 meV. (C) Bandline-up for zero (black) and positive (blue) bias (note that only the p-i-n-diode region isshown, not the entire DBR microcavity). EF: Fermi-level. (D) Zoomed-in version of C: forzero applied bias, the GaAs valence band lies 230 meV below the Fermi-level, and the mosttightly bound HH state therefore lies below the Fermi-level. For a positive bias, the offsetbetween the Fermi-level and the GaAs valence band is reduced; when this offset equals the199 meV separation with the most tightly bound HH state, resonant tunneling can resultin hole-charging of the QD.

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25 nm i-GaAs layer acting as a tunnel barrier between the QDs and the p-layer [2, 3],and a 120 nm layer separating the QDs from the n-contact. Inside the latter, weincorporated a 110 nm i-AlAs/GaAs superlattice (20 layers) to prevent chargingfrom the n-layer [3]. A back contact (not shown) allows for biasing of the substrate,while a metal shadow mask also serves as a contact to the n-doped layer. Aperturesin the metal mask provide optical access to the QDs, at the expense of a reducednet bias over the QDs: given the relatively large width of the metal mask apertures(16 µm), the exact bias over a QD depends on the lateral position of that QD withinthe aperture shadow. In particular, for QDs near the center of the aperture, partof the applied bias voltage will result in a resistive voltage drop inside the n-layer,reducing the net bias over the QD. In addition, Schottky-barrier effects at the metalmask-DBR interface further reduce the effective QD bias for a given applied biasvoltage.

We calculated the band structure and energy levels of the QD in a full three-dimensional simulation, using the 3D simulation tool “nextnano” [4]. The resultsare shown in Figs. SI-2A and B. We assume an InAs QD with an approximateheight of 2 nm, and a base length of 25 nm, and account for a finite amount ofIn-Ga intermixing. The resulting QD emission wavelength is around 940 nm. Thetwo most tightly bound states in both the conduction band (electron charging) andvalence band (hole charging) were calculated and are indicated by the red and greendashed lines. We calculated both the HH and LH subbands, though the HH bandis by far the most important. The two most tightly bound HH states in the QD areseparated by ∼14 meV, and are located 199 meV above the GaAs valence band.

Next, the band structure of the charge-tuneable devices was calculated – seeFigs. SI-2C and D. Without externally applied bias, the built-in diode voltage leadsto a band bending (black curve), where the hole Fermi-level is located ∼230 meVabove the GaAs valence band right at the position of the QD. In order to keep ourcalculations tractable, we split the problem into two subproblems: a full 3D calcu-lation of the band line-up of the QD HH bound states, and a 1D calculation of theband bending of the entire device. Hence, without applied bias, the HH QD statesare located below the Fermi-level, making hole charging of the QD energeticallyunfavorable. At some positive bias, the offset between the Fermi-level and the HHQD state vanishes (blue curve), with resonant tunneling allowing for deterministiccharging of the QD. Numerically, this bias voltage is around 200 mV, though weemphasize that this is the real bias over the p-i-n-diode near the QD, which is oftenless than the applied voltage between the device contacts, especially for QDs locatednear the center of the mask aperture (DC Stark shifts and Schottky-barrier effectsaccount for an additional offset). Pauli- and Coulomb-blockade effects subsequentlylead to a stable voltage plateau where single-hole charging is possible.

In view of the relatively large QD density, about 50 QDs were located withinour diffraction-limited laser spot (charge-tuneable devices; for the δ-doped samples,some 150 QDs), and several of those were resonant with the microcavity. In Fig. 1Bwe show the photoluminescence (PL) as a function of applied bias voltage for aparticular QD from a charge-tuneable device (above-band excitation, λ = 785 nm).A magnetic field of 6 T in Voigt geometry splits the transitions, and identification ofthe respective lines is made easier through the particular fine structure of transitionsfrom charged and uncharged QDs [5]. The charged QDs display a fourfold split of thePL for large magnetic field – see Fig. 1C, where the magneto-PL of a hole-charged

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QD is shown (note that two of the four lines overlap due to limited spectrometerresolution, which can be seen as an apparent increase in brightness of the centerline); it is this same particular signature that also allows us to identify the chargedQDs in the δ-doped samples. The dependence on the pumping power allows usto separate excitonic emission lines from lines due to multi-excitonic complexes.The inhomogeneity in size and composition of the different QDs, together withthe expected spectral line-up of the different charge states of a single QD, allowus to identify the lines in Fig. 1B. As expected, increasing the QD bias leads toa transition from an uncharged to a charged state. However, and as reported inRef. 2, we see a significant overlap between the respective voltage plateaus of thecharged and uncharged QD state, which can be attributed to the relatively slowtunneling of the hole in our QDs. In addition, we notice that the exact position ofthe voltage plateaus depends on the amount of optical power used. Both above-bandand resonant CW-excitation (as well as below-band modelocked (ML) laser pulsesused for coherent spin rotations) can alter the bias voltage by as much as 0.1 - 0.2 V– we attribute this to residual absorption in the vicinity of the QD, which leads tothe generation of charged carriers that can shift the QD energy (“DC”-Stark shift).

Finally, we characterized the fine structure of the QDs in our experiment. With-out any magnetic field, the polarization of the emitted PL contains informationabout the hole spin eigenstates. In particular, strain and quantum dot asymmetryresult in a small amount of HH-LH mixing. The resulting hole spin ground states

Figure SI-3: Angular PL dependence. Polarization angle dependence of the emittedphotoluminescence (PL) of a hole-charged QD at 0 magnetic field. Blue dots: raw data; redcurve: least squares fit of the elliptical polarization. The distance from the origin indicatesthe relative intensity of the emitted PL for a particular polarization angle. D1 and D2 arethe main axes of the resulting elliptical polarization dependence (see text). Note that thedata were taken for polarization angles between 0 and 180 degrees, and copied for the 180 -360 degrees trajectory in view of the inversion symmetry of the system. The discontinuityat 0 degrees is a systematic experimental artifact.

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|⇑〉 and |⇓〉 can be modeled as:

|⇑〉 =[∣∣Ψ+3/2

⟩+ β+

∣∣Ψ−1/2

⟩]/

√1 + |β|2

|⇓〉 =[∣∣Ψ−3/2

⟩+ β−

∣∣Ψ+1/2

⟩]/

√1 + |β|2. (1)

Here,∣∣Ψ±3/2

⟩and

∣∣Ψ±1/2

⟩represent the HH and LH states respectively. LH inmix-

ing reflects itself in a slightly elliptical polarization of the hole-charged PL [6]. Werefer to Fig. SI-3. The ratio, η, between the two axes of the elliptical polarization(D1,2) is a measure for the amount of inmixing:

η = (√

3− |β|)2/(√

3 + |β|)2. (2)

For the results presented in this Letter, the particular QD had a value of β ∼ 17%.

SII. COHERENT ROTATION PULSE MODEL

The effect of a broadband, detuned optical pulse on the hole spin ground statescan be equivalently modeled as resulting from a stimulated Raman transition or asthe result of an AC-Stark shift – we elaborate here on the latter interpretation [7, 8].As the pulse duration of 3.67 ps is much shorter than the Larmor precession fre-quency δHH/2π = 30.2 GHz, one can look at the interaction in the basis of the lightpulse (z-basis as indicated in Fig. SI-4A; note that this is not the eigenbasis of thespins in the absence of any light pulses – that is the x-basis in our convention). Inthis basis, the magnetic field results in an off-diagonal coupling between the holespins, indicated by Bx in Fig. SI-4B. However, given that the pulse is much fasterthan the Larmor-precession, the z-basis spins can be considered as effectively de-generate, and the magnetic field can be approximately neglected in the remainderof the analysis.

For perfect selection rules and ideally circularly polarized light pulses, only oneof the z-basis hole spin states is coupled to the trion states; the other state is dark.For realistic quantum dots, imperfect selection rules and limited control over theexact polarization of the light pulse inside the cavity lead to both hole spin groundstates being coupled to the trion states. The coupling strengths Ω1,2 are indicated inFig. SI-4B; typically, one is much larger than the other. For a detuning ∆ (340 GHzin our case) larger than the pulse bandwidth, the pulse adiabatically mixes thehole spin ground state and its excited trion state. The effect of the time-dependentadiabatic mixing is a time-dependent AC-Stark-shift δ1,2(t), given by:

δ1,2(t) =12

√∆2 + |Ω1,2(t)|2 − ∆

2. (3)

A hole spin initialized in the x-basis due to the magnetic field can be written asa superposition of z-basis states with equal weight. The effect of the pulse is then toAC-Stark-shift these states by a different amount, leading to rotation pulse power(Prot) dependent Rabi oscillations with net rotation angle:

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θ =∫dt

[δ1(t)− δ2(t)

]=

12

∫dt

[√∆2 + |Ω1(t)|2 −

√∆2 + |Ω2(t)|2

]. (4)

Here, the integration is over the duration of a single rotation pulse.Fig. SI-4 textbfC illustrates the predicted Rabi oscillations in this AC-Stark frame-work. The data show an incoherent background (∼ P 0.65

rot ) which is shown asthe green curve in this figure. After subtracting the background, a least-squaresfit extracted the amplitude of the Rabi oscillations. The pulse shape was mod-elled as Gaussian, with a measured FWHM of 3.67 ps, and for the detuning,the measured value of 340 GHz was used. The best fit was obtained for a ratio

,,ΔΔ

z z

zz

Bx

Bx

δ1(t) δ2(t)

x

zB σ+

A B

C

Ω1(t) Ω2(t)

Prot

Figure SI-4: Coherent rotation modelling. (A) Geometry and axis convention usedin the experiment. The magnetic field is oriented along x, while the laser pulse is alignedto the growth direction z. (B) AC-Stark shift in the Z-basis: |⇓〉 and |⇑〉 are the holespin ground states, while |⇓⇑, ↓〉 and |⇓⇑, ↑〉 represent the trion states. ∆ represents thedetuning, and the circularly polarized laser pulse couples the ground states to the excitedstates (Ω1,2(t)), resulting in AC-Stark shifts δ1,2(t). (C) Rabi oscillations fit through theAC-Stark model. Blue circles: raw data; red: AC-Stark shift predicted Rabi oscillations, ontop of an incoherent background (green).

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|Ω1(t)|2/|Ω2(t)|2 = 3.7, and is indicated by the red curve. The model fits the datavery well, with the exception of the height of the first peak. This deviation canbe attributed to the still finite duration of the laser pulse, and our neglecting theLarmor precession in this model. In Ref. 9 we demonstrated how the combinedeffect of Larmor precession and pulse-induced Rabi oscillations leads to an effectiverotation axis that is in between the laser pulse (z-axis) and the magnetic field axis(x), leading to a reduced height of the first π pulse. A full time-dependent co-herent simulation can qualitatively reproduce the reduced height. The backgroundand upward trend, however, cannot be reproduced by this simulation. Its origin iscurrently unknown, although it might be related to a change in the optimum biasposition of the QD for high rotation pulse powers as reported above.

SIII. ROTATION PULSE FIDELITY

We model the fidelity of a π/2 pulse using the same methods as described inRef. 9. We assume that the Bloch vector after optical pumping starts with lengthL0, and shrinks by a factor Dπ/2 after a single π/2 pulse. The combined effect of twoπ/2 pulses separated by a variable Larmor precession delay τ is that the populationin |⇑〉 oscillates between (1 + L0D

2π/2)/2 and (1 − L0D

2π/2)/2 with period 2π/δHH.

We use a digital lock-in procedure which automatically subtracts a background of(1−L0)/2, leading to a net oscillation between (L0+L0D

2π/2)/2 and (L0−L0D

2π/2)/2.

With a scaling factor C connecting population in |⇑〉 to the measured counts, theresulting signal oscillates between CL0(1 +D2

π/2)/2 and CL0(1−D2π/2)/2.

From the sinusoidal fit to the π/2 Ramsey fringes in Fig. 2D, we obtain a valuefor Dπ/2 of 0.89, which implies a fidelity Fπ/2 = (1 + Dπ/2) = 0.945 as quoted inthe main text. This value is comparable to currently achieved values for electronspins (Fπ/2 = 0.94 as previously reported in Ref. 9, subsequently slightly improvedto about 0.97 by the incorporation of a microcavity and optimized detuning, see forexample Fig. 1D in Ref. 10). The fidelity is limited by the incoherent population,which can be seen in Fig. SI-4C.

A similar analysis method can be used for different pulse angles, and yields similarfidelities. For different dots, different fidelities are obtained, but are consistentlyfound to be upwards of 0.9, limited by the incoherent population induced by theoptical rotation pulse.

SIV. ELECTRON SPIN-NUCLEAR FEEDBACK

For an electron-charged QD, a simple and mathematically tractable nonlineardiffusion equation was derived in Ref. 11, describing the nonlinear feedback loopresulting from nuclear-dependent Larmor precession (Overhauser shift) and electron-spin dependent nuclear spin relaxation. This model can be extended to incorporatethe hysteretic and asymmetric curves obtained while scanning through the opticalresonance frequency of the QD (the |↓〉-|↓↑,⇑〉 transition, see Fig. 3C).

The devices used for the electron spin-nuclear feedback experiments have beenextensively described in Refs. 10, 11. They consist of self-assembled InAs QDs(density: roughly 2 × 109 cm−2), with a Si δ-doping layer 10 nm below the QDs

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300 400 500 600

1

2

Delay τ [ps]

Cou

nts

[a.u

.]

300 400 500 600

1

2

Delay τ [ps]

Cou

nts

[a.u

.]

a

b

π/2 τ π/2

τ

τ

Figure SI-5: Electron spin Ramsey fringe hysteresis. (A) Asymmetric and hystereticRamsey fringes for an electron spin (measured; from Ref. 11). The green and blue arrowsindicate the scanning direction of the pulse delay τ . (B) Asymmetric and hysteretic electronspin Ramsey fringes as predicted by the model in Ref. 11. Inset: scan direction and pulsetiming.

for stochastic QD charging (roughly 1 in 3 QDs were charged). The QDs wereembedded in a one-sided planar microcavity (Q ∼ 200) consisting of 24 (bottom)respectively 5 (top) AlAs/GaAs λ/4 layers.

SIV.1. Hysteretic and asymmetric electron spin Ramsey fringes

For a single electron spin in a QD, the strong contact hyperfine interaction makesthe electron spin Larmor precession frequency very sensitive to the net polarizationof the nuclear spin bath through the Overhauser shift. Conversely, in Ref. 11 it wasshown that the evolution of the nuclei in the quantum dot depends on the rate oftrions being generated, as the unpaired hole in the electron spin trion state allows

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for quasi energy-conserving nuclear spin flips to occur. This effect, together withbackground nuclear spin diffusion, can be modelled as a nonlinear diffusion equationfor the average Overhauser shift ω:

∂ω

∂t= −κω + α

∂C(ω, τ)∂ω

, (5)

where C(ω, τ) is the trion generation rate in the experiment. For Ramsey fringes,C(ω, τ) can be calculated by analyzing the pulse pattern (see Fig. SI-5B, inset). Inparticular, a CW optical pumping pulse is interspersed with two π/2 rotation pulses,separated by a variable delay τ . Denoting the average spin polarization by S, thespin-up probability by P(↑) (equal to (1 + S)/2) and the spin-down probability byP(↓), we can relate the spin polarization Safter right after optical pumping for a timeT to the polarization Sbefore before the arrival of the optical pumping pulse:

Safter =∑m=↑,↓

[12P (↑|m)Pbefore(m)− 1

2P (↓|m)Pbefore(m)

]

=(1− e−β(ω)T )

2+ Sbeforee

−β(ω)T . (6)

Here, β(ω) represents the optical pumping/absorption profile, assumed to be Gaus-sian. In addition, the interference of the two π/2 rotation pulses separated by adelay τ yields

Sbefore = − cos((ω0 + ω)τ)Safter, (7)

where ω0 represents the Larmor precession in the absence of any Overhauser effects.Together, this results in a trion generation rate C(ω, τ):

C(ω, τ) = Safter − Sbefore

=12

(1− e−β(ω)T )1 + cos[(ω0 + ω)τ ]1 + cos[(ω0 + ω)τ ]e−β(ω)T

. (8)

Eq. (5) is derived by invoking a separation of timescales [11], and can be solvedto yield steady-state solutions ωf(τ). The integration times used in the experiment(few seconds for each different value of τ) were found to be sufficient for reachingthese steady-state solutions. Due to the nonlinearity, the solutions depend on theinitial conditions, and therefore on the direction in which τ is varied. Figs. SI-5Aand B compare the experimentally obtained Ramsey fringes with those obtainedby numerically solving Eq. (5). In Fig. SI-5B, β(ω) was assumed to be Gaussian(β0e

−(ω2/2σ2) with σ/2π = 1.6 GHz), κ = 10 s−1 and κ/α = 104 ps2.

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939.02 939.03 939.04 939.05

1

2

Wavelength λ [nm]

Cou

nts

[a.u

.]

939.02 939.03 939.04 939.05

1

2

Wavelength λ [nm]

Cou

nts

[a.u

.]

a

b

πλ λ

λ

λ

Figure SI-6: Electron spin resonant scanning hysteresis. (A) Asymmetric andhysteretic resonance scan for an electron spin (experimental). The green and blue circlesindicate the wavelength scanning direction. (B) Asymmetric and hysteretic electron spinresonance scan, as predicted by an extension of the model in Ref. 11. Inset: scan directionand pulse timing.

SIV.2. Hysteretic and asymmetric electron spin CW resonance scanning

In Refs. 12, 13 hysteretic effects were observed while scanning a narrowband CWlaser through the QD optical resonance frequency. In Ref. 13, a single electron-charged QD was studied using Coherent Population Trapping in Voigt geometry.In order to avoid optical pumping into one of the electron spin ground states, onelaser was kept fixed, while another laser was scanned through the QD resonancewavelength. Trion-induced nuclear spin flips lead to a dragging of the resonancewavelength upon scanning the laser frequency, and the consequent hysteresis.

A similar effect was observed for a single electron-charged QD in our system.Instead of having two narrowband lasers, a π rotation pulse was used to compensatefor optical pumping, and a narrowband CW-laser (few MHz linewidth) was scanned

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through the QD resonance wavelength (|↓〉-|↓↑,⇑〉-transition [11]). Figure SI-6Aillustrates the hysteresis and asymmetry upon scanning the CW-laser in differentdirections.

Equation (5) can again be used to model the Overhauser shift ω. The differencewith the Ramsey fringe hysteresis lies in the trion generation rate C(ω, ωlas, θ),where ωlas stands for the laser frequency, and θ for the rotation angle of the singlepulse used in the experiment (π in our case). The pulse sequence used is shown inFig. SI-6B, inset. After optical pumping for a time T , we still have that

Safter =(1− e−β(ω,ωlas,ωres)T )

2+ Sbeforee

−β(ω,ωlas,ωres)T , (9)

where ωlas and ωres respectively stand for the laser frequency and the QD resonancefrequency in the absence of nuclear spin effects. However, the single rotation pulsewith angle θ has now a different effect on the spin polarization:

Sbefore = cos(θ)Safter. (10)

This results in a net trion generation rate C(ω, ωlas, θ):

C(ω, ωlas, θ) = Safter − Sbefore

=12

(1− e−β(ω,ωlas,ωres)T )[1− cos(θ)]1− cos(θ)e−β(ω,ωlas,ωres)T

. (11)

We can again obtain steady-state values ωf from Eq. (5). Whether or notsteady state is obtained, however, depends critically on the scan speed – a de-pendence we also notice experimentally. We assume a Lorentzian QD linewidth(β(ω) = β0/(1 + (ωlas − ωres − ω)2/σ2), σ/2π = 200 MHz). Other lineshapes (Gaus-sian, Voigt) yield qualitatively similar results. While the exact resulting lineshapedepends critically on the intial conditions and details of the QD and experiment(initial Overhauser shift ω0, scan speed, lineshape, etc.), the qualitative features arewell reproduced in this model; Fig. SI-6B shows the numerical solution to Eq. (5).κ is estimated as 8.5 s−1, and κ/α = 2.8×104 ps2.

SV. NUCLEAR FEEDBACK: COMPARISON BETWEEN ELECTRONAND HOLE SPINS

For the hysteretic effects of a single electron spin coupled to the nuclear spins inthe QD, the average Overhauser contribution to the Larmor precession frequency canbe extracted from the model described above. Figure SI-7A shows the Overhausershift for the hysteretic Ramsey fringes described above. As our model captures onlythe average Overhauser shift [8, 11], one conservative way of estimating the error onthe obtained values is to assume that there is no narrowing of the nuclear spin distri-bution due to the development of nuclear spin polarization [11]. In that case, the T ∗2value of 1.71 ns obtained in Ref. 10 can be used to estimate the variance on the Over-hauser shift due to time-ensemble effects, yielding σω/2π =

√2/(2πT ∗2 ) = 130 MHz.

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300 400 500

-2

0

Delay τ [ps]

Ove

rhau

ser s

hift

[GH

z]

A

B C

π/2 τ π/2

π/2 τ π/2

τ

ττ τ

Figure SI-7: Electron and hole spin Overhauser shifts compared. (A) Modeledaverage Overhauser shift for hysteretic Ramsey fringes of a single electron spin; the greenand blue circles indicate the wavelength scanning direction, as indicated by the arrows.(B) Time-averaged dephasing of a single hole spin; blue: raw data, red: fit to a sinusoid withGaussian envelope. No variance of the average Overhauser shift was observed. (C) Zoomed-in version of (B) – note that the phase of the fringes remains constant over the entire scanrange.

We can therefore estimate the maximum Overhauser shifts for a single electron spindue to the interaction with the nuclei at 3±0.13 GHz. The resonance scan modelpredicts a similar, or slightly reduced, maximum Overhauser shift.

A single hole spin does not display any hysteresis or nonlinearity in either theRamsey fringe or resonance scanning experiment. Moreover, compared to the in-direct method of extracting Overhauser shifts through modelling based on Eq. (5),a more accurate estimate of the Overhauser shift can be obtained from the phaseof the Ramsey fringes. That phase equals (ω0+ω)τ , and the Ramsey fringes areshown in Figs. SI-7B and C, together with a sinusoidal fit with Gaussian envelope(red curve). The raw data hardly deviate from the fit, except for very long delays,where noise effects dominate. Even with the noise, the deviation is at most 0.5 to1 radians for a total delay τ of 3.5 ns, leading to a maximum Overhauser shift ω/2πof 40±100 MHz. We may bound any possible hole Overhauser shifts by suppos-ing they are masked by experimental noise. Here, the width of the time-averagedLarmor precession frequency distribution leading to T ∗2 -decay results in an uncer-tainty σω/2π =

√2/(2πT ∗2 ) = 100 MHz, using our experimentally observed T ∗2 value

of 2.3 ns. We emphasize that this is a worst-case estimate, for the case in whichnuclear effects would limit the time-averaged dephasing, which we consider unlikely.

Comparing these two values, we see that the developed Overhauser shift for the

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hole spin is at least 30 times smaller than that for the electron spin, where thefactor of 30 is limited by experimental noise and T ∗2 effects of the hole spin. Whilethe Overhauser shift depends both on the developed nuclear spin polarization andthe sensitivity of the hole spin to that nuclear spin polarization, this significantreduction of the measured Overhauser shift illustrates the suppression of feedbackeffects between the nuclear spin bath and the hole spin.

SVI. COHERENCE DECAY OF THE HOLE QUBIT

The time-averaged dephasing data in Figs. 4A and B were obtained by movinga motorized stage on a rail. For each rail position, the motorized stage was scanned,resulting in a relative delay τrel (varying between 0 and 270 ps) between the twoπ/2 pulses. The distance between subsequent rail position was 270 ps. For each railposition, the resulting Ramsey fringes were fit to a sinusoid, where the amplitude,offset, period and phase were used as fit parameters. The resulting fringe heightsare plotted in Fig. 4B, and are best fit to a Gaussian. A slight upward trend canbe observed in Fig. 4A, due to an increased incoherent background and/or slightdeviations from the ideal π/2 pulse angle.

For the studies of the effects of electric bias variations on the hole qubit dephasing,the bias variations were calibrated such as to yield shifts in the QD CW-absorptionprofile similar to its inhomogeneous linewidth (see Fig. 3D), all the while verify-ing that the charge state of the QD was not affected by these bias changes. Themaximum change in bias used was ±50 mV, well below the width of the chargingplateau.

A similar procedure was used for the spin echo data in Figs. 4E and F. Here,the position of the motorized stage on the rail was kept fixed; however, a coarsedelay was added by having the EOMs pick pulses with a delay of 13 ns or multiplesthereof (see Fig. 4E, inset: 2T is a multiple of 13 ns). For each value of 2T , echofringes were obtained when scanning the middle π pulse. The fringes were fit to asinusoid, and the fringe heights are shown in Fig. 4F. Here, the best fit to the fringeheights is obtained by an exponential.

We studied the effects of charge noise by AC-modulating the bias voltage, withfrequencies varying between 100 kHz and several MHz (amplitude: 50-100 mV peak-to-peak). While RC-filtering effects of the wires and contacts inside the cryostatprevent accurate calibration of the resulting effective charge noise on the QD, wedid notice pronounced reductions in the visibility of the resulting echo-fringes, andreductions in the T2-decoherence time (data not shown).

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