# Deterministic preparation of spin qubits in droplet etched GaAs quantum dots
using quasi-resonant excitation

Caspar Hopfmann<EMAIL_ADDRESS>Nand Lal Sharma  Weijie Nie
Institute for Integrative Nanosciences, Leibniz IFW Dresden, Helmholtzstraße
20, 01069 Dresden, Germany  Robert Keil  Institute for Integrative
Nanosciences, Leibniz IFW Dresden, Helmholtzstraße 20, 01069 Dresden, Germany
Present address: Fraunhofer-Institut für Angewandte Festkörperphysik (IAF),
Tullastraße 72, 79108 Freiburg, Germany  Fei Ding  Institut für
Festkörperphysik, Leibniz Universität Hannover, Appelstraße 2, 30167 Hannover,
Germany  Oliver G. Schmidt  Institute for Integrative Nanosciences, Leibniz
IFW Dresden, Helmholtzstraße 20, 01069 Dresden, Germany  Material Systems for
Nanoelectronics, Technische Universität Chemnitz, 09107 Chemnitz, Germany
Nanophysics, Faculty of Physics and Würzburg-Dresden Cluster of Excellence
ct.qmat, TU Dresden, 01062 Dresden, Germany

###### Abstract

We present a first comprehensive study on deterministic spin preparation
employing excited state resonances of droplet etched GaAs quantum dots. This
achievement facilitates future investigations of spin qubit based quantum
memories using the GaAs quantum dot material platform. By observation of
excitation spectra for a range of fundamental excitonic transitions the
properties of different quantum dot energy levels, i.e. shells, are revealed.
The innovative use of polarization resolved excitation and detection in quasi-
resonant excitation spectroscopy facilitates determination of $85$ $\%$
maximum spin preparation fidelity - irrespective of the relative orientations
of lab and quantum dot polarization eigenbases. Additionally, the
characteristic non-radiative decay time is investigated as a function of
ground state, excitation resonance and excitation power level, yielding decay
times as low as $29$ ps for s-p-shell exited state transitions. Finally, by
time resolved correlation spectroscopy it is demonstrated that the employed
excitation scheme has a significant impact on the electronic environment of
quantum dot transitions thereby influencing its charge and coherence.

##  I Introduction

Optically accessible quantum memories are fundamental for practical
implementations of quantum networks as they facilitate synchronization
required for schemes of long distance quantum information exchange [1, 2, 3,
4]. While tremendous progress towards optical quantum networks using entangled
photon sources has been achieved in the past decade [5, 6, 7, 8, 9, 10, 11],
quantum systems that address all of the required aspects - namely: high source
efficiency, repetition rates and quantum coherence - are still sorely lacking
[4]. Atomic and diamond defect based systems feature long quantum storage
times required for memory applications, their repetition rate and internal
quantum efficiency, respectively, are however fundamentally limited [12, 13].
Due to their rapid technological advance, semiconductor quantum dots (QDs)
have also received significant attention [7, 14, 9, 10]. The main drawback of
these systems so far has been that the quantum coherence of their spin qubits
is limited by interaction with the nuclear magnetic environment of its
constituting atoms - i.e. the Overhauser field [15, 16, 17]. While nuclear
polarization and spin echo techniques may extend the coherence time of spin
qubits significantly [18, 19], the achievable values in commonly employed
InGaAs QDs are still limited by the presence of high spin (+9/2)
$\prescript{115}{}{\text{In}}$ and $\prescript{113}{}{\text{In}}$ isotopes. In
recent years droplet etched GaAs QDs have been established as potent sources
of entangled photon pairs [14, 9, 10, 20], featuring high quality on-demand
entangled photon pair generation at GHz clock rates suitable for entanglement
swapping operations. Due to the lack of In, these quantum dots are also
attractive candidates for extended spin qubit coherence - which has not been
investigated so far. The presented study aims at obtaining the foundational
knowledge to facilitate further quantum optical applications and
investigations based on GaAs QDs. This first exploration is achieved by
demonstrating that quasi-resonant (QR) excitation schemes can be used to
initialize fundamental QD spin qubits deterministically. To this end,
polarization resolved excitation spectroscopy and time resolved pump-probe
investigations are combined to obtain a comprehensive picture of the
properties of excited states in GaAs quantum dots and their possible
applications. This study therefore provides an ideal stepping stone towards
future investigations of quantum memories using GaAs QD spin qubits.

##  II Results and Discussion

Figure 1: (a) Typical spectrum of a single GaAs QD using above-band
excitation. Selected fundamental emission lines (c.f. Tab. 1) are annotated.
(b) and (c-e) depict $\text{X}_{S}^{0}$ two-photon auto- and cross-correlation
traces, respectively, as a function of the time delay $\tau$ between detection
events. Traces (fitted model, see text) recorded using above-band (AB) and
quasi-resonant (QR, $\Delta_{\text{ex}}=10.08$ meV, c.f. Fig. 2) excitation
are drawn in black (red) and blue (green), respectively. Cross-correlations
(c-e) are recorded between $\text{X}_{S}^{0}$ and
$\text{X}_{S}^{-}+\text{XX}_{S}^{0}$, $\text{X}_{S}^{+}$ as well as
$\text{XXX}_{S}^{0}$, respectively.

A typical above-band photoluminescence (PL) spectrum of a single QD is shown
if Fig. 1(a) in saturation of the neutral QD exciton $\text{X}_{S}^{0}$
transition. Though the general shape of GaAs quantum dot spectra has been
established previously [21], the exact energy differences between the emission
lines depend on the Al concentration of the QD matrix material. Contrary to
In(Ga)As-based quantum dots which are grown using the Stranski–Krastanov
process, the spectral properties - such as the energetic distribution of
fundamental emission lines - of droplet etched GaAs quantum are much more
consistent [22]. In Fig. 1(b) transitions of four basic excitonic QD charge
configurations, namely $\text{X}_{S}^{0}$, $\text{X}_{S}^{+}$ and
$\text{X}_{S}^{-}+\text{XX}_{S}^{0}$ are indicated. The transitions were
identified using a combination of polarization, power dependent PL and PL
excitation (PLE) spectroscopy, details are given in the following sections and
in the supplementary material. Note that $\text{X}_{S}^{-}+\text{XX}_{S}^{0}$
transition lines are not separable in PL spectra. Using PLE spectroscopy the
observed energetic difference between these lines is about $5$ $\mu$eV, which
is well within the resolution limit of the employed spectrometer. Spectral
overlap between $\text{X}_{S}^{-}$ and $\text{XX}_{S}^{0}$ is observed
consistently for matrix material Al-concentrations of $15$ $\%$. The
nomenclature of fundamental QD transitions used throughout this work is
summarized in Tab. 1.

Label | Transition |  $\Delta_{i-X_{S}^{0}}$ (meV)
---|---|---
$\text{X}_{S}^{0}$ | $(1e^{1})(1h^{1})\to 0$ | $0$
$\text{X}_{S}^{+}$ | $(1e^{1})(1h^{2})\to(1h^{1})$ | $-2.24$
$\text{XX}_{S}^{0}$ | $(1e^{2})(1h^{2})\to(1e^{1})(1h^{1})$ | $-3.47$
$\text{X}_{S}^{-}$ | $(1e^{2})(1h^{1})\to(1e^{1})$ | $-3.48$
$\text{XXX}_{S}^{0}$ | $(1e^{2}2e^{1})(1h^{2}2h^{1})\to(1e^{1}2e^{1})(1h^{1}2h^{1})$ | $-4.09$
Table 1: Description of various optical transitions of GaAs QDs embedded in
$\text{Al}_{0.15}\text{Ga}_{0.85}\text{As}$ matrix material using the
nomenclature established by Benny et. al. [23]. Relative emission energies
with respect to $\text{X}_{S}^{0}$ ($E_{\text{X}_{S}^{0}}=1.5880$ eV) are
determined by polarization and power dependent high resolution spectroscopy,
cf. Fig. 1(a) and Suppl. Fig. S8.

###  II.1 Correlation Spectroscopy

Figs. 1(b-e) depict two-photon auto- and cross-correlation traces $g(\tau)$
using continuous wave (cw) excitation of selected combinations of emission
lines. The curves are modeled, depending on the nature of the correlation,
according to:

$\displaystyle g^{(2)}(\tau)$ $\displaystyle=\left(g_{\text{anti}}(\tau)\times
g_{\text{blink}}(\tau)\right)\ast\mathcal{N}_{\text{Det}}(\delta_{\text{det}})\;,$
(1) $\displaystyle g_{\text{anti}}^{(2)}(\tau)$
$\displaystyle=(g_{0}-1)\,e^{-\frac{|\tau|}{T_{\text{corr}}}}+1\;,$ (2)
$\displaystyle g_{\text{blink}}^{(2)}(\tau)$
$\displaystyle=(\frac{1}{\beta}-1)\,e^{-\frac{\left|\tau\right|}{T_{\text{blink}}}}+1\;.$
(3)

Where $T_{\text{corr}}$ and $T_{\text{blink}}$ are the characteristic
correlation and blinking timescales, respectively, and $\beta$ is the blinking
on-off ratio. $g_{\text{blink}}(\tau)$ models spectral blinking of the
investigated line and is adopted from Jahn et. al. [24]. In the limit of an
adiabatic (coherent) excitation scheme $T$ is equivalent to the radiative
lifetime $T_{\text{corr}}\to T_{1}$ of QD excitonic complexes. In order to
take into account the limited temporal response of the detection system the
raw auto-correlation trace is convoluted with the bi-detector response
function $\mathcal{N}_{\text{Det}}(\delta_{\text{det}})$, modeled by a normal
distribution of FWHM $\delta_{\text{det}}=75$ ps. The latter is determined
independently. Eq. (1) can be adopted for cross-correlations by
differentiating between regimes $\tau_{i\to j}<0$ and $\tau_{j\to i}>0$,
effectively yielding two correlation timescales that represent switching
between emission events of two different QD excitonic complexes. A summary of
the modeled parameter values is shown in Tab. 2.

Fig. 1 | Transition | $g^{(2)}(0)$ | $T_{\text{corr}}(ns)$
---|---|---|---
|  $i$ |  $j$ |  $i\to j$ |  $j\to i$ |  $i\to j$ |  $j\to i$
b(AB) | $\text{X}_{S}^{0}$ | $0.023\pm 0.006$ | $2.15\pm 0.07$
b(QR) | $0.00+0.02$ | $0.32\pm 0.01$
c(AB) |  $\text{X}_{S}^{0}$ |  $\text{X}_{S}^{+}$ | $0.00+0.03$ |  $0.56\pm 0.07$ |  $2.3\pm 0.1$
c(QR) |  |  | $0.025\pm 0.004$ |  $21.6\pm 0.3$ |  $15.2\pm 0.3$
d(AB) |  $\text{X}_{S}^{0}$ |  $\text{XX}_{S}^{0}$ |  $0.00+0.04$ |  $9.4\pm 0.1$ |  $0.303\pm 0.006$ |  $1.70\pm 0.09$
e(AB) |  $\text{X}_{S}^{0}$ |  $\text{XXX}_{S}^{0}$ |  $0.00+0.04$ |  $3.8\pm 0.3$ |  $0.21\pm 0.03$ |  $1.9\pm 0.2$
Table 2: Estimated parameter values according to Eq. (1) and (2) of the auto-
and cross-correlation traces shown in Fig. 1(b-e), respectively. The
abbreviations AB and QR stand for above-band and quasi-resonant excitation
schemes, respectively. Errors are statistically estimated by $1\,\sigma$
confidence intervals.

Both $\text{X}_{S}^{0}$ auto-correlation traces of Fig. 1(b) exhibit strong
anti-bunching and are modeled according to Eq.(1). The estimated parameter
values related to the blinking are $\beta^{632nm}=0.86\pm 0.01$ and
$T_{\text{blink}}^{632nm}=(2.0\pm 0.1)$ ns, for quasi-resonant excitation
values of $\beta^{\text{QR}}=0.06\pm 0.01$ and
$T_{\text{blink}}^{\text{QR}}=(22.3\pm 0.3)$ ns are obtained. While the
$g^{(2)}(0)$ values for both excitation schemes are comparable, the modeled
timescales are very divergent. The difference in $T_{\text{corr}}$ can be
readily understood when considering that the quasi-resonant excitation is a
significantly more coherent excitation scheme than above-band pumping, in fact
$T_{\text{corr}}^{\text{QR}}\to T_{1}$ holds as confirmed below. This is a
first indication that quasi-resonant excitation schemes in GaAs quantum dots
can be coherent. Blinking of the $\text{X}_{S}^{0}$ emission line is much more
pronounced in quasi-resonant compared to above-band excitation, as can be
observed in the auto-correlation trace by the strong bunching and its decay,
similarly observed elsewhere [24]. We attribute this behavior to two effects:
Firstly, the fluctuating population of defect states in the matrix material
environment of the QD influence its energetic structure through the quantum-
confined Stark effect [25]. These defect states may get saturated in the case
of above-band excitation, but not in resonant pumping [26]. Secondly, the
average QD charge state may get reconfigured depending on the employed
excitation scheme [27]. We observe that the average QD charge becomes
generally more positive in quasi-resonant excitation.

In Fig. 1(c) traces of cross-correlations between $\text{X}_{S}^{+}$ and
$\text{X}_{S}^{0}$ are depicted for both above-band and quasi-resonant
excitation schemes. The two correlation traces provide information with
respect to the charge fluctuations between neutral and positively charged
states of the QD. Again, as in the case of the $\text{X}_{S}^{0}$ auto-
correlation, the charge related blinking behavior is very different in the two
excitation schemes. In above-band pumping the fluctuations take place on
shorter timescales compared to quasi-resonant excitation. This is a further
indication of the increased coherence in the quasi-resonant driving scheme, as
there are far fewer free carriers in the matrix material surrounding the QD -
which in turn enhances stability of the QD charge state. Under the assumption
that both $\text{X}_{S}^{0}$ and $\text{X}_{S}^{+}$ are pumped equivalently -
which is a valid assumption for above-band excitation -, the intensity ratio
between $\text{X}_{S}^{0}$ and $\text{X}_{S}^{+}$ lines can independently be
determined form the ratio
$T_{\text{corr}}^{632nm}(\text{X}_{S}^{0}\to\text{X}_{S}^{+})/T_{\text{corr}}^{632nm}(\text{X}_{S}^{+}\to\text{X}_{S}^{0})\simeq
4.3$. This value qualitatively matches the intensity ratio determined from the
PL spectrum of Fig. 1(a) of $2.8$, the difference can be attributed to the
dissimilar exciton powers of $4.4$ and $1.0$ $\mu$W, respectively. Due to the
presence of a double $\text{X}_{S}^{0}$ and $\text{X}_{S}^{+}$ resonance at
$\Delta_{E_{i}-E_{X_{S}^{0}}}=10.8$ meV, c.f. Fig. 2, the assumption of
equivalent excitation conditions for both lines can be extended also to the
quasi-resonant excitation scheme in this particular case. The resulting
intensity ratio
$T_{\text{corr}}^{\text{QR}}(\text{X}_{S}^{0}\to\text{X}_{S}^{+})/T_{\text{corr}}^{\text{QR}}(\text{X}_{S}^{+}\to\text{X}_{S}^{0})$
is about $0.7$ which demonstrates that the QR excitation scheme favors
positive charge states in the QD. This result is a consistent observation for
GaAs quantum dots, but the exact mechanism of this effect remains unclear and
is beyond the scope of this work.

In Fig. 1(d) the cross-correlation between the $\text{X}_{S}^{0}$ and
$\text{X}_{S}^{-}+\text{XX}_{S}^{0}$ emission lines is shown. As expected from
the $\text{XX}_{S}^{0}\to\text{X}_{S}^{0}$ cascade a strong bunching is
observed, while the $\text{X}_{S}^{0}\to\text{XX}_{S}^{0}$ process is anti-
bunched [28]. Due to the nature of the cascade for each emission of a
$\text{XX}_{S}^{0}$ photon the $\text{X}_{S}^{0}$ state is inherently
coherently prepared. Therefore
$T_{\text{corr}}(\text{XX}_{S}^{0}\to\text{X}_{S}^{0})$ is equivalent to the
$\text{X}_{S}^{0}$ lifetime $T_{1}^{\text{X}_{S}^{0}}=(0.303\pm 0.006)$ ns,
which matches the value $T_{\text{corr}}^{\text{QR}}=(0.32\pm 0.01)$ ns. This
confirms that the quasi-resonant excitation is largely coherent, at least in
the case of the specific resonance used - the latter is further discussed
using results for PLE and pump-probe experiments.

Fig. 1(e) depicts a cross-correlation trace between $\text{XXX}_{S}^{0}$ and
$\text{X}_{S}^{0}$ emission lines recorded using above-band excitation.
Analogous to the correlation trace of Fig. 1(d) a clear cascade behavior -
although with a lower bunching - is observed. We therefore attribute the
emission line to a tri-exciton s-shell emssion line, c. f. Tab. 1.

###  II.2 Excitation Spectroscopy

Figure 2:  Combined PLE (blue) and PL (black) spectra of selected s-shell
emission lines (c.f. Tab. 1) obtained by integration over s-shell lines as a
function of relative energy of cw excitation laser and $\text{X}_{S}^{0}$
emission energy $\Delta_{\text{ex}}$. Different resonances and excitation
shells are annotated, for details see Tab. 3 and text. Above-band spectra are
recorded at $\text{X}_{S}^{0}$ saturation excitation power of $1.5$ $\mu$W,
while the PLE spectra are recorded by employing a $1.2$ $\mu$W quasi-resonant
laser excitation.

Excited state resonances are characterized by cw PLE spectroscopy employing
excitation energies relative to the $\text{X}_{S}^{0}$ transition
$\Delta_{\text{ex}}$ in the range of $1$ to $25$ meV. The resulting PLE
spectra (blue lines) of several s-shell QD emission lines are depicted in Fig.
2 as a function of $\Delta_{\text{ex}}$. Reference s-shell emission spectra
(black lines) obtained using above-band excitation are shown for comparison.
The s-shell transitions, which are numerically integrated as a function of
$\Delta_{\text{ex}}$ in order to obtain the PLE spectra, are indicated by blue
arrows. The spectral integration window is equivalent to the full width half
maximum (FWHM) of the s-shell emission. The resolution of the PLE spectra is
better than $5$ $\mu$eV, limited only by the step size of the cw tunable laser
emission energy. As can readily be seen from the PLE spectra, a significant
number of resonances can be found in the presented $\Delta_{\text{ex}}$ range.
Resonances that are relevant for the further discussion are annotated in the
figure and detailed in Tab. 3. For InGaAs QDs grown in the Stranski-Krastanov
mode unambiguous identification of many excited state resonances has been
demonstrated [29, 30, 31, 23, 27]. Though the same would in principle also be
possible for excited states of GaAs QDs used in this study, the energetic
structure of these QDs is different as outlined below. Taking into account the
fact that there is presently no theory model for the energetic structure of
droplet etched GaAs QDs, the rigorous identification of the PLE resonances to
specific QD multi carrier excitations - beyond some general observations - is
outside the scope of this work and is left to future studies.

Resonances $R_{0}^{X_{S}^{0}}$ and $R_{0}^{XX_{S}^{0}}$ of PLE spectra in Fig.
2(a) and (c), respectively, are equivalent because every bi-excitonic photon
emission is necessarily followed by emission of $\text{X}_{S}^{0}$ due to the
neutral exciton cascade, c. f. Fig. 1(d). By using this the association
between the PLE spectra of $\text{X}_{S}^{-}+\text{XX}_{S}^{0}$ it is possible
to separate resonances of $\text{X}_{S}^{-}$ and $\text{XX}_{S}^{0}$ which are
intermingled in Fig. 2(c). The annotated resonances reflect this association.
By comparison with PLE scans performed on InGaAs QDs of Ref. [27] resonances
$R_{0}^{X_{S}^{0}}$ and $R_{0}^{XX_{S}^{0}}$ can be identified as the excited
state configuration $(1e^{1})(2h^{1})$. In the study of _Benny et. al._ this
excited state was identified at a $\Delta_{\text{ex}}$ of about $16$ meV,
which is an increase of about three times compared to the GaAs QDs used in
this study. It can therefore be concluded that the energetic splittings in the
confinement potential of GaAs quantum dots are significantly lower. We
attribute this difference to the fact that droplet etched GaAs QDs are larger
in their dimensions both parallel and perpendicular to the growth direction
[21]. Furthermore, compared to the InGaAs QDs of Ref. [27] the energetic
splitting of the heavy holes in GaAs QDs is significantly lower, which can be
observed in the comparatively smaller energy separation between PLE resonances
of Fig. 2(b) and (c). For example: The energetic splitting between
$R_{0}^{X_{S}^{+}}$ and $R_{1}^{X_{S}^{+}}$ is $(0.20\pm 0.02)$ meV, while for
$R_{0}^{X_{S}^{-}}$ and $R_{1}^{X_{S}^{-}}$ it equates to $(2.12\pm 0.02)$
meV, yielding a $10.7$ fold reduction of the $\text{X}_{S}^{+}$ compared to
$\text{X}_{S}^{-}$ excited state resonance splittings. The exact mechanism
causing the dissimilar energetic splittings of electronic sub-levels is
currently not known, we think it could be influenced significantly by the
difference of the confinement potential depths between electrons and holes.

S-shell |  Ex. Res. |  $\Delta_{\text{ex}}$ (meV) |  Shell |  $P_{\text{sat}}$ ($\mu$W) |  $T_{\text{NR}}$ (ps)
---|---|---|---|---|---
$\text{X}_{S}^{0}$ |  $R_{0}^{\text{X}_{S}^{0}}$ | $5.444$ |  s-p($h^{+}$) | $13$ | $163.0$
|  | $5.830$ |  |  |
|  $R_{1}^{\text{X}_{S}^{0}}$ | $10.100$ |  s-p($e^{-}$) | $8$ | $218$
|  |  |  | 0.04 $P_{\text{sat}}$ | $36.9$
|  $R_{2}^{\text{X}_{S}^{0}}$ | $16.472$ |  p | $14$ | $389$
|  $R_{3}^{\text{X}_{S}^{0}}$ | $22.324$ |  p | $6$ | $385$
$\text{X}_{S}^{+}$ |  $R_{0}^{\text{X}_{S}^{+}}$ | $4.264$ |  s-p($h^{+}$) |  |
|  $R_{1}^{\text{X}_{S}^{+}}$ | $4.464$ |  s-p($h^{+}$) |  |
|  $R_{2}^{\text{X}_{S}^{+}}$ | $6.017$ |  s-p($h^{+}$) | $6$ | $129$
|  |  |  | 0.03 $P_{\text{sat}}$ | $88.3$
|  $R_{3}^{\text{X}_{S}^{+}}$ | $7.781$ |  s-p($h^{+}$) |  |
|  $R_{4}^{\text{X}_{S}^{+}}$ | $10.179$ |  s-p($e^{-}$) | $9$ | $210$
|  $R_{5}^{\text{X}_{S}^{+}}$ | $19.591$ |  p | $1$ | $165$
$\text{X}_{S}^{-}$ |  $R_{0}^{\text{X}_{S}^{-}}$ | $4.900$ |  s-p($h^{+}$) | $10$ | $56.8$
|  |  |  | 0.04 $P_{\text{sat}}$ | $29.1$
|  $R_{1}^{\text{X}_{S}^{-}}$ | $7.040$ |  s-p($h^{+}$) |  |
|  $R_{2}^{\text{X}_{S}^{-}}$ | $7.786$ |  s-p($h^{+}$) | $20$ | $74.1$
|  $R_{3}^{\text{X}_{S}^{-}}$ | $11.119$ |  s-p($h^{+}$) |  |
|  $R_{4}^{\text{X}_{S}^{-}}$ | $12.305$ |  s-p($h^{+}$) |  |
|  $R_{5}^{\text{X}_{S}^{-}}$ | $20.464$ |  p | $4$ | $240$
$\text{XX}_{S}^{0}$ |  $R_{0}^{\text{XX}_{S}^{0}}$ | $5.444$ |  s-p($h^{+}$) |  |
|  | $5.834$ |  |  |
|  $R_{1}^{\text{XX}_{S}^{0}}$ | $10.027$ |  s-p($e^{-}$) |  |
|  $R_{2}^{\text{XX}_{S}^{0}}$ | $16.469$ |  p |  |
Table 3: List of annotated excitation resonances of Fig. 2(a-c). Excitation
resonance energies $\Delta_{\text{ex}}$ given relative to the s-shell
$\text{X}_{S}^{0}$ transition. The accuracy of $\Delta_{\text{ex}}$ is
estimated to $16$ $\mu$eV. Each excitation resonance is attributed to a
specific series of transitions, called shells. Details and definitions are
found in the text. Excitation saturation powers $P_{\text{sat}}$ and non-
radiative decay times $T_{\text{NR}}$ obtained by modeling of pump-probe
experiments, c.f. Fig. 3 and Suppl. Fig. S9, are shown for selected
resonances. The statistical standard deviation of $T_{\text{NR}}$ is about $1$
$\%$ at $P_{\text{sat}}$ and to about $5$ $\%$ at 0.04 $P_{\text{sat}}$.

While the exact identification of most excited state resonances is beyond the
scope of this study, it is clear that there should be different series of
excitation resonances - very much analogous to the spectral series (Lyman,
Balmer, Paschen, etc.) in hydrogen atoms [32] \- present in this system. Just
as in the case of hydrogen atoms the excitation resonance with the lowest
$\Delta_{\text{ex}}$ can be considered the fundamental transition, while the
series of excitation resonances extends towards increasing
$\Delta_{\text{ex}}$. In contrast to atoms in QDs the individual transitions
of a series originate not from higher order electronic orbitals but from
transitions between different excitonic (spin) complexes. In order to
accommodate this difference the series of excitation state resonances in QDs
are henceforth called shells. The relevant ones for the discussion of the PLE
spectra of Fig. 2 are: s-shell [$\left|0\right\rangle\to(1e^{1})(1h^{1})$],
s-p-shell($h^{+}$) [$\to(1e^{1})(2h^{1})$], s-p-shell($e^{-}$)
[$\to(2e^{1})(1h^{1})$] and p-shell [$\to(2e^{1})(2h^{1})$], defined in
accordance to [27]. The fundamental transitions of each shell are presented
schematically at the top of Fig. 2. Since the PLE trace of Fig. 2 was recorded
at about one third of the typical saturation intensity of most transitions of
about $4$ $\mu$W, it can be concluded that the relative intensities of the
different excitation resonances generally reflect the strength of the light-
matter interaction matrix element of the individual transitions. Due to the
nature of the different energetic orbitals of electronic states in QDs the
dipole moment between individual series of excitation resonances can be
significantly different [31, 23]. At the boundary between s- and s-p-shell
resonances we therefore expect to detect a significant change in the intensity
of the observed basic s-shell transitions (Fig. 2(a-c)). Consequently, we
attribute the first bright PLE resonances to the lower boundary of the s-p-
shell of $\Delta_{\text{ex}}\simeq 4.2$ meV. This boundary is indicated in
Fig. 2 by a red line.

In Fig. 2(d) the PLE spectrum of a spin-blockaded negative trion
($\text{X}_{S,T}^{-}$) is shown. This state is a negatively charged exciton in
which the electron spin configuration is such that a single electron spin is
locked in the p-shell by the Fermi exclusion principle due to equivalent spins
of s- and p-shell electrons. The details and properties of this state are
discussed elsewhere [33]. Due to the presence of an electron in the p-shell
the s-p-shell($h^{+}$) transitions are blocked, while the ones of the
electron, i.e. the s-p-shell($e^{-}$), are still allowed. The consequence can
be seen in the PLE spectrum: most resonances $\Delta_{\text{ex}}<14$ meV that
are observed in Fig. 2(a-c) are absent, which also confirms the identity of
the observed s-shell emission line. We therefore conclude that the absent
excitation resonances belong to the s-p-shell($h^{+}$), while
$R_{0}^{\text{X}_{S,T}^{-}}$ belongs to the s-p-shell($e^{-}$).

Fig. 2(e) depicts the PLE spectrum of a s-shell line we associate with the
s-shell transition of a tri-excitonic complex
[$(1e^{2}2e^{1})(1h^{2}2h^{1})$]. As a consequence both s-p-shells are absent
in its PLE spectrum. The first excitation resonance can therefore be found in
the p-shell, which enables the estimation of the lower p-shell boundary to
$\Delta_{\text{ex}}\simeq 14.1$ meV. This is indicated in Fig. 2 by a green
line. Around $\Delta_{\text{ex}}\simeq 20$ meV a broad maximum in the PLE
spectra, especially in the one of $\text{X}_{S}^{+}$, is observed. This can be
attributed to the overlap of p-shell transitions of both electrons and holes
and not to effects related due to phonon-enhanced absorption - which would be
expected beyond $\Delta_{\text{ex}}\gtrsim 32$ meV [34, 27].

###  II.3 Pump-Probe Correlation Spectroscopy

Figure 3:  Quasi-resonant pump-probe correlation measurements of
$\text{X}_{S}^{0}$, $\text{X}_{S}^{+}$ and $\text{X}_{S}^{-}$ fundamental QD
transitions and their respective excitation resonances (a) at excitation
saturation and (b) far below saturation power. The curves are modeled
according to Eq. (4) with $T_{\text{NR}}$ as the only free parameter. Specific
resonances, excitation conditions and estimated $T_{\text{NR}}$ values are
summarized in Tab. 1.

In order to investigate the non-radiative decay mechanism of the excitation
resonances, pump-probe experiments are conducted. In these investigations
specific excitation resonances are pumped by a pulsed laser while the time
resolved emission of the respective s-shell transition is monitored. The
results for selected $\text{X}_{S}^{0}$, $\text{X}_{S}^{+}$ and
$\text{X}_{S}^{-}$ transitions and resonances pumped at and far below
($\approx 0.04P_{\text{sat}}$) excitation saturation are shown in Fig. 3(a)
and (b), respectively. Further results of pump-probe experiments are plotted
in Suppl. Fig. S9. The intensity correlation data is modeled by a delayed
exponential decay function

$\displaystyle C(\tau)$
$\displaystyle=e^{-\frac{\tau}{T_{1}^{\text{X}_{S}^{0}}}}\theta(\tau)\ast\left(\frac{e^{-\frac{\tau}{T_{\text{NR}}}}}{T_{\text{NR}}}\ast\mathcal{N}_{\text{Det}}(\delta_{\text{det}})\right)\;,$
(4)

where the non-radiative decay time $T_{\text{NR}}$ is the only free parameter.
All other parameters are fixed to values determined previously. The delay
which is induced by a single particle non-radiative decay process is modeled
by an exponential distribution
$\frac{e^{-\frac{\tau}{T_{\text{NR}}}}}{T_{\text{NR}}}$. This simple
assumption fits the experimental data very well for most resonances, for the
curve shown in Suppl. Fig. S9(c) - the model shows deviations to the
experimental data. This can be attributed to the break down of the assumption
of a single stage decay process. A summary of the extracted $T_{\text{NR}}$
values can be found in Tab. 3. For all basic QD charge states the decay time
increases towards elevated values of $\Delta_{\text{ex}}$ and towards
excitation power saturation. As a consequence, the observed excitation
resonances with the lowest $T_{\text{NR}}$ \- and therefore the most coherent
excitation process - can be found in the s-p-shells. Note that no
$\pi$-pulses, a sign of coherent preparation of the QD s-shell states, are
observed for any excitation resonance [35, 36], the reason is that the
required condition $T_{\text{NR}}\ll T_{1}^{\text{X}_{S}^{0}}$ of the optical
Bloch equations is not fulfilled at saturation of the excitation resonances.
Far below saturation, where the influence of the excitation laser induced
increase of the decay time should be neglectable, the $T_{\text{NR}}$ values
of $(37\pm 2)$, $(88\pm 2)$ and $(29\pm 1)$ ps are determined for the QD
s-shell transitions $\text{X}_{S}^{0}$, $\text{X}_{S}^{+}$ and
$\text{X}_{S}^{-}$, respectively. The excitation resonances in which an
additional electron is present in the QD exhibit the fastest non-radiative
decay times, while if an additional hole is present the decay is about a
factor of $3$ slower. Consequently, the value of $T_{\text{NR}}$ in the
absence of additional carriers lies between the two former cases. By
considering the natural spectral line broadening induced by the lifetime of
the excited states limited by $T_{\text{NR}}$, these differences explain the
variation of the observed spectral line widths of the PLE resonances of Fig.
2.

###  II.4 Polarization Resolved Emission Spectroscopy

Figure 4:  Polarization resolved PL spectra of
$\text{X}_{S}^{-}+\text{XX}_{S}^{0}$, $\text{X}_{S}^{+}$ and
$\text{X}_{S}^{0}$ without applied external magnetic field (a) and with a
field of $0.2$ T (b), respectively. The spectra are recorded using above-band
excitation at $P\simeq P_{sat}/2$. The raw spectra are shown in the upper
panels, while in the lower ones the degree of polarization (PolDeg, see text)
between orthogonal bases are depicted.

In order to characterize the spin related properties of excitonic complexes in
QDs, a convenient method is to investigate the polarization characteristics of
the emission. This basic characterization is shown with and without applied
magnetic field in Fig. 4. To distinguish the spectral polarization dependence
the degree of polarization is defined as
$\text{PolDeg}(\text{I-J})=\frac{\rho(I)-\rho(J)}{\rho(I)+\rho(J)}$, where $I$
and $J$ represent a set of orthogonal polarization bases and $\rho$ are the
spectral intensities associated with the respective bases. In the absence of
an external magnetic field, $\text{X}_{S}^{0}$ features a finestructure
splitting due to its integer quasi-particle spin $j_{\text{X}_{S}^{0}}=\pm 1$
which is oriented along the H-V polarization axis, cf. Fig. 4(a). For finite
magnetic fields, depending on the field strength, the eigenstates are oriented
on a superposition axes of H-V and R-L (c.f. Fig. 4(b)), and are split
according to the Zeeman splitting [29]. $\text{X}_{S}^{+}$ does not exhibit a
fine structure in the absence of a magnetic field (${j_{\text{X}_{S}^{+}}=\pm
3/2,j_{\text{X}_{S}^{+}}=\pm 1/2}$) due to the Kramers degeneracy of half-
integer spin states [37, 29]. Its polarization eigenbase is therefore R-L. As
the excitation is not polarization selective it does not show any polarization
dependence in Fig. 4(a). In the presence of a magnetic field
$\text{X}_{S}^{+}$ exhibits Zeeman splitting, c.f. Fig. 4(b), which allows the
QD g-factor of $1.86\pm 0.02$ to be determined. Due to the overlap of the
$\text{X}_{S}^{-}+\text{XX}_{S}^{0}$ transitions, the signature of the
$\text{XX}_{S}^{0}$ finestructure splitting in the polarization resolved
spectra is masked.

The polarization resolved emission spectra of Fig. 4 are used to calibrate the
polarization axes of the QD eigenbases to the lab frame. In this process the
fundamental polarization quantization axes are used in the following way: The
finestructure splitting of $\text{X}_{S}^{0}$ is used to calibrate the D-A and
R-L axes to an orthogonal orientation with respect to H-V. The eigenbases
shift to R-L induced by a small magnetic field is used to minimize the
polarization degree in D-A axis. As the orthogonality between the axes is
preserved, this fully defines the rotation between the QD and lab frames. The
degree of polarization of Fig. 4 shows that this calibration is not fully
equivalent for all investigated transitions. This effect can be attributed to
the wavelength dependent birefringent behavior of the employed GaP microlenses
(see methods section) - as this relation is not present in samples without
these lenses.

###  II.5 Polarization Resolved Excitation Spectroscopy

Figure 5:  Exemplary polarization resolved PLE spectra and polarization
degrees (PolDeg) in 18 different combinations of excitation and detection
polarization bases (I,J) of the $\text{X}_{S}^{-}+\text{XX}_{S}^{0}$ emission
lines. The data corresponds to the third run of Fig. 6. The first (I) and
second (J) bases are of the excitation and detection, respectively. Upper
panels: Combinations with excitation polarization bases (a) H/V, (b) D/A and
(c) R/L. The respective excitation PolDegs are shown in the lower panels.
Additionally, total PolDegs of each excitation axes
$\left\langle\lVert\text{I-J}\rVert\right\rangle$ according to Eq. (5) are
drawn. Resonances corresponding to Tab. 3 are annotated.

Polarization resolved PLE spectra are depicted exemplarily in Fig. 5 for the
$\text{X}_{S}^{-}+\text{XX}_{S}^{0}$ transitions. Polarization resolved PLE
spectra of $\text{X}_{S}^{0}$ and $\text{X}_{S}^{+}$ emission lines are found
in Suppl. Figs. S10 and S11, respectively. As can readily be observed, the
behavior of the various resonances is not equivalent, c.f.
$R_{1}^{\text{X}_{S}^{-}}$ vs. $R_{2}^{\text{X}_{S}^{-}}$ vs.
$R_{3}^{\text{X}_{S}^{-}}$. There are two reasons for this, both of which can
be negated similarly to the polarization resolved PL by calibration of the
excitation polarization bases to the spin eigenstates of the specific
excitation resonance. Firstly, the polarization calibration of the excitation
changes significantly with wavelength due to the birefringent behavior of the
GaP microlens. Secondly, the excitation resonances origin from different
excitonic complexes with distinct spin configurations. Depending on whether
the excited and ground state spin symmetries match and if the non-radiative
decay process preserves the initial spin created in the excited state, the
polarization of the QD excitation and emission are correlated. In order to
extract this correlation, which constitutes the s-shell spin preparation
fidelity, in the presence of the birefringent GaP microlens the PLE spectra of
Fig. 5 are measured in 18 different combinations of polarization bases, i.e. 6
excitation and 3 detection bases. To avoid calibrating each excitation
resonance separately, the excitation polarization degree
$\left\langle\lVert\text{I-J}\rVert\right\rangle$ may be defined as

$\displaystyle\text{PolDeg}_{\text{ex}}(\text{I-J})$
$\displaystyle:=\left\langle\lVert\text{I-J}\rVert\right\rangle$ (5)
$\displaystyle=\sqrt{\frac{\sum_{K}\left.(\rho(\text{I})-\rho(\text{J}))^{2}\right|_{\text{K}}}{3\;\sum_{K}\left.(\rho(\text{I})+\rho(\text{J}))^{2}\right|_{\text{K}}}}\;,$
(6)

where K is the observant (detection) polarization base. Exemplary results of
$\left\langle\lVert\text{I-J}\rVert\right\rangle$ are shown in the lower
panels of Fig. 5 together with the individual polarization degrees
$\left.\text{PolDeg}(\text{I-J})\right|_{\text{K}}$. It can be observed that
$\left\langle\lVert\text{I-J}\rVert\right\rangle$ effectively constitutes the
vector norm of the polarized response of the QD via the excitation resonance.
In other words it renders the calibration of specific excitation resonances to
the fundamental QD eigenstates unnecessary.

In order to determine the total spin preparation fidelity of the excitation
resonances the polarization degree can be abstracted further. This can be done
irrespective of varying detection and excitation polarization bases
calibration in respect to specific eigenstates and excitation resonances. The
total excitation polarization degree, which is equivalent to the ground state
spin preparation fidelity $f_{\text{SpinPrep}}$, is therefore defined as

$\displaystyle f_{\text{SpinPrep}}$
$\displaystyle\equiv\lVert\text{PolDeg}_{\text{ex}}\rVert:=\sqrt{\sum_{\text{I-J}}\left\langle\lVert\text{I-J}\rVert\right\rangle^{2}}\;.$
(7)

Figure 6:  Spin preparation fidelity
$f_{\text{SpinPrep}}\equiv\lVert\text{PolDeg}_{\text{ex}}\rVert$ (c.f. Eq.
(7)) spectra for QD s-shell transitions $\text{X}_{S}^{0}$,
$\text{X}_{S}^{+}$, $\text{X}_{S}^{-}+\text{XX}_{S}^{0}$ and multiple
experimental runs, more details are given in the text. Resonances of Tab. 3
are annotated by labels and gray dashed guidelines. The $80$ $\%$ spin
preparation fidelity threshold is indicated in green.

Spectra of $f_{\text{SpinPrep}}$, extracted from the polarization resolved PLE
spectra in analogue to Fig. 5 of $\text{X}_{S}^{0}$, $\text{X}_{S}^{+}$ and
$\text{X}_{S}^{-}+\text{XX}_{S}^{0}$, are summarized in Fig. 6.

By employing this method the spin memory of different excitation resonances
can be compared effectively. For example the polarization response of
resonances $R_{2}^{\text{X}_{S}^{0}}$ and $R_{2}^{\text{XX}_{S}^{0}}$ does not
show any dependence on the excitation polarization, indicating that the
excited and ground states exhibit orthogonal eigenstates. Resonances
$R_{0}^{\text{X}_{S}^{+}}$, $R_{3}^{\text{X}_{S}^{+}}$,
$R_{3}^{\text{X}_{S}^{-}}$ and $R_{4}^{\text{X}_{S}^{-}}$ on the other hand
show $f_{\text{SpinPrep}}$ values up to $85$ $\%$, demonstrating matching
excited and ground spin configurations as well as spin preserving non-
radiative decay processes. For $\text{X}_{S}^{0}$ the excitation resonance
induced spin memory is limited to $75$ $\%$, c.f. $R_{0}^{\text{X}_{S}^{0}}$.
Other resonances show intermediate spin memory effects, which can be
attributed to either partial mismatch between excited and ground state
eigenstates or non-polarization preserving non-radiative decay processes. A
reason for the latter would be, if more than one particle is involved in the
decay process. The resonances with the highest spin preparation fidelities all
belong to the s-p-shell, c.f. Fig. 2, indicating that resonances of this shell
would be preferable for implementations of quantum spin memories based on
droplet etched GaAs QDs.

Polarization resolved PLE spectra are recorded in three separate experimental
runs which feature different experimental conditions. The first, second and
third runs are optimized for detection polarization bases of
$\text{X}_{S}^{0}$, $\text{X}_{S}^{+}$ and $\text{X}_{S}^{-}$, respectively.
Additionally, weak above-band excitation ($<2$ nW) is employed and the
excitation polarization bases are aligned using the reflection of a resonant
laser in the first and third runs, while no above-band excitation and
excitation calibration to $R_{1}^{\text{X}_{S}^{+}}$ is used in the second.
The resonant excitation powers of the three runs are $1.4$, $4$ and $4$
$\mu$W, respectively. Generally the results between different experimental
runs are very consistent - underling the validity and robustness of the method
of extraction of $f_{\text{SpinPrep}}$ presented in this study. Only the
resonance $R_{2}^{\text{X}_{S}^{-}}$ shows significant deviation between the
experimental runs, indicating the influence of above-band excitation for this
specific excitation resonance.

##  III Conclusions

For the first time droplet etched GaAs QDs are investigated comprehensively
using polarization resolved PL, PLE and correlation spectroscopy. It is found
that the GaAs QDs carrier dynamics depend drastically on the excitation
method, e.g. in continuous wave quasi-resonant excitation blinking of the
$\text{X}_{S}^{0}$ transition is very pronounced. Consequently, the on-off
ratio of $\text{X}_{S}^{0}$ in above-band excitation is about $8$ times
higher. We attribute this behavior to the defect states around the QDs,
similarly reported in Ref. [24]. Furthermore, the predominant charge of the
QDs is shifted towards excitations with excess holes in quasi-resonant
excitation schemes. The lifetime of the $\text{X}_{S}^{0}$ transition is
determined consistently using both quasi-resonant and above-band correlation
spectroscopy to $(303\pm 6)$ ps. Using this information, the non-radiative
decay processes of several different excitation resonances for
$\text{X}_{S}^{0}$, $\text{X}_{S}^{+}$ and $\text{X}_{S}^{-}$ transitions are
investigated by pump-probe experiments. The minimal characteristic non-
radiative decay times $T_{\text{NR}}$ are estimated to $37$, $88$ and $29$ ps,
respectively, all of which can be attributed to s-p-shell excitation
resonances. $T_{\text{NR}}$ is found to increase for excitation powers as well
as higher energy (e.g. p-shell) resonances. The energetic structure of the
excitation resonances is investigated using PLE spectroscopy. By employing
observed properties of various excitonic complexes specific resonances can be
attributed to different energetic shells. In order to match the polarization
eigenbases of the laboratory frame to the one of the QDs, its polarization
eigenstates both with and without external magnetic field are employed. Due to
the wavelength dependent birefringency of the GaP microlenes, used to enhance
the collection efficiency of the QD emission [14], the polarization
calibration is wavelength dependent. To effectively extract the spin
preparation fidelity $f_{\text{SpinPrep}}$ independently of the calibration of
excitation and detection polarization bases a novel method is proposed. This
procedure employs measurements in $18$ different excitation and detection
bases and the orthogonality between bases pairs to determine
$f_{\text{SpinPrep}}$ obtained using excitation resonances.
$f_{\text{SpinPrep}}$ of up to $85$ $\%$ for both $\text{X}_{S}^{+}$ and
$\text{X}_{S}^{-}$ as well as about $75$ $\%$ for $\text{X}_{S}^{0}$ are
achieved for s-p-shell type transitions.

In conclusion the presented comprehensive investigations, methods and findings
enable the directed usage of excited state resonances to deterministically
prepare fundamental spin states in GaAs quantum dots. These fundamental
abilities will pave the way to use these quantum dots in a large variety of
future quantum optical experiments and applications, such as QD based quantum
memories [38, 17, 39], quantum entanglement repeaters [9, 10], photon graph
[6] and cluster states [7] as well as more efficient entangled photon pair
sources [14, 20]. Additionally, the presented study increases the detailed
understanding of droplet etched GaAs QDs significantly. While not all aspects
are explored exhaustively, it provides an ideal starting point for more
detailed investigations in a variety of aspects. Examples of the latter are
identification of transitions of excitonic complexes, excitation scheme
dependent photonic and electronic coherences, the nature of the non-radiative
decay process and dependency of QD properties on matrix material composition.

##  IV Methods

Figure 7:  Simplified schematic drawing of the experimental setup. The
following abbreviations are used: Beam splitter (BS), superconducting single
photon nano wire detectors (SSPDs) and charge coupled device (CCD).

The presented study is performed using droplet etched GaAs quantum dots (QDs)
embedded in QD-nanomembranes and attached to GaP microlenes. Samples were
grown on [001] GaAs substrate using molecular beam epitaxy. The nanoholes are
etched through deposition of Al droplets onto the
$\text{Al}_{0.15}\text{Ga}_{0.85}\text{As}$ matrix material. Consequently, the
$15$ nm deep and $40$ nm wide nanoholes are filled with GaAs thereby forming
the QDs. The $380$ nm thick $\text{Al}_{0.15}\text{Ga}_{0.85}\text{As}$ matrix
material with the centrally embedded QDs is grown on-top of a AlAs sacrificial
layer, which is removed by selective hydrofluoric acid etching to yield the
QD-nanomembranes. In order to overcome the strong internal reflection of the
semiconductor material due to its high refractive index ($\simeq 3.5$) and
enhance the out-coupling efficiency the QD-nanomembranes are attached to GaP
microlenses using a $50$ nm thick layer of PMMA. This enhances the photon
extraction efficiency compared to unprocessed samples by a factor of about
$100$. A detailed description of the growth and processing steps can be found
in Refs. [40, 14].

The employed experimental setup is shown as a simplified sketch in Fig. 7. The
QD-devices consisting of GaP microlenses with attached QD-nanomembranes are
placed inside a dry closed-cycle cryostat. The cryogenic system features a
z-axis $9$ T superconducting magnet and a $3.8$ K base temperature. Sample
luminescence is collected using a aspheric lens of $0.64$ NA.
Photoluminescence (PL) spectroscopy is performed using a $2\times 0.750$ m
double spectrometer and gratings of either $1800$ or $1200$ lines/mm. The
maximal spectral resolution of this system is about $15$ $\mu$eV at $780$ nm.
The spectrometer can be configured as two independent monochromators, which is
employed in cross-correlation experiments. Spectroscopic investigations are
performed by a standard back-illuminated deep-depletion CCD. For PLE
spectroscopy a narrow-band ($100$ MHz) wavelength-tuneable and -stabilized cw
laser, in conjunction with the high resolution PL detection system, is
employed. Wavelengths are tuned and stabilized to an absolute accuracy of $2$
pm using a calibrated wavelength meter. In order to enhance the suppression of
the excitation laser and separate it from the QD emission tuneable band edge
filters in both excitation and detection are employed. Pump-probe experiments
are performed using a wavelength tuneable and pulsed optical parametric
oscillator (OPO) laser pumped by a pulsed frequency doubled fiber laser at
$516$ nm. The OPO laser system exhibits a repetition rate of $76.271$ MHz and
a shaped pulse width of $3.5$ ps. For above-band excitation a HeNe laser
featuring a $632.8$ nm emission wavelength is employed. In order to perform
polarization resolved PL and PLE spectroscopy the polarization of the QD
emission is projected onto one of the set of orthogonal polarization bases [H,
V, D, A, R, L]. This is achieved by using two liquid crystal variable
retarders (LCVRs) and a linear polarizer. Aforementioned configuration can be
calibrated to any orthogonal set of polarization bases on the Poincaré sphere
and enables the compensation of the mismatch between QD and lab polarization
eigenbases. The accuracy of LCVR calibration was determined to about $0.95$
using a polarimeter. Three independent polarization projector units are
employed: two in the detection arms and one in the excitation path, c.f. Fig.
7. Correlation spectroscopy is performed by time resolved correlation of
electronic signals from superconducting nanowire single photon detectors
(SSPDs). The time resolution of this system is about $53$ ps for one and $75$
ps for two-photon correlations.

###### Acknowledgements.

We thank Michael Zopf and Jingzhong Yang (LU Hannover) for fruitful
discussions. We acknowledge funding by the BMBF (Q.link.X) and the European
Research Council (QD-NOMS).

##  V Supplementary

###  V.1 Fine structure

Figure S8:  Relative energies of selected emission lines ($\text{X}_{S}^{0}$,
$\text{X}_{S}^{+}$, $\text{X}_{S}^{-}+\text{XX}_{S}^{0}$) as a function of the
polarization projection angle. Modeling to simple sine functions (see text)
are shown in red.

Fig. S8(a-c) depicts the relative emission energies of $\text{X}_{S}^{0}$,
($\text{X}_{S}^{-}+\text{XX}_{S}^{0}$) and $\text{X}_{S}^{+}$, respectively,
as a function of the polarization rotation angle. While $\text{X}_{S}^{0}$ as
well as $\text{X}_{S}^{-}+\text{XX}_{S}^{0}$ exhibit an anti-correlated
interdependence signaling the presence of the neural exciton finestructure
splitting, $\text{X}_{S}^{+}$ does not. The latter is a consequence of Kramers
degeneracy of half-integer spin states [37, 29]. The data is modeled using
sinusoidal functions: $\Delta
E(\alpha)=E_{0}+\Delta_{X^{i}}^{FSS}/2\>\text{sin}(2\pi\alpha+\alpha^{0})$,
where the two free parameters are the fine structure splitting
$\Delta_{X^{i}}^{FSS}$ and phase $\alpha^{0}$. The model yields a fine
structure splitting of $\Delta_{\text{X}_{S}^{0}}^{FSS}=(10.1\pm 0.3)$ $\mu$eV
for $\text{X}_{S}^{0}$, while there is no significant splitting for
$\text{X}_{S}^{+}$. Since the lines $\text{X}_{S}^{-}$ and $\text{XX}_{S}^{0}$
cannot be spectrally separated a reduced fine structure splitting
$\Delta_{\text{XX}_{S}^{0}}^{FSS}=(3.8\pm 0.1)$ $\mu$eV compared to
$\text{X}_{S}^{0}$ is observed. Note that
$\alpha^{0}_{\text{X}_{S}^{0}}=0.75\pm 0.06$ and
$\alpha^{0}_{\text{X}_{S}^{-}+\text{XX}_{S}^{0}}=0.26\pm 0.06$ have, as
expected, orthogonal phases. These observations also support the line
identifications of Tab. 1.

###  V.2 Pump-Probe Correlation Spectroscopy

Figure S9:  Quasi-resonant pump-probe correlation measurements using different
QD charge states and their respective excitation resonances. The curves are
modeled according to Eq. (4) with $T_{\text{NR}}$ as the only free parameter.
Specific resonances, corresponding to Tab. 3, are annotated in the graphs.

In Fig. 3 pump-probe experiments employing quasi-resonant excitation
resonances $R_{1}^{\text{X}_{S}^{0}}$, $R_{2}^{\text{X}_{S}^{+}}$ and
$R_{0}^{\text{X}_{S}^{-}}$ are shown for both saturation and low excitation
powers. Pump-probe investigations of additional resonances are depicted in
Fig. S9. These correlation traces are modeled according to Eq. (4), the
corresponding extracted characteristic non-radiative decay times are
summarized in Tab. 3. Except $R_{3}^{\text{X}_{S}^{0}}$ of Fig. S9(c) all
obtained correlation traces fit very well to the simple model provided by Eq.
(4). Since the model assumes a single stage decay process, it follows that
most non-radiative decay processes involve a single process - with
$R_{3}^{\text{X}_{S}^{0}}$ as an exception. Interestingly, a strong dependence
of $T_{\text{NR}}$ on the excitation power is only observed for
$R_{1}^{\text{X}_{S}^{0}}$, $R_{2}^{\text{X}_{S}^{+}}$ and
$R_{0}^{\text{X}_{S}^{-}}$ are affected to a reduced extend. The origin of
this effect is currently not known but could be a topic of future
investigations.

###  V.3 Polarization Resolved Excitation Spectroscopy

Figure S10:  Polarization resolved PLE spectra and polarization degrees
(PolDeg) in 18 different combinations of excitation and detection polarization
bases (I,J) of the $\text{X}_{S}^{0}$ emission line. The data corresponds to
the third run of Fig. 6. The first (I) and second (J) bases are of the
excitation and detection, respectively. Upper panels: Combinations with
excitation polarization bases (a) H/V, (b) D/A and (c) R/L. The respective
excitation PolDegs are shown in the lower panels. Additionally, total PolDegs
of each excitation axes $\left\langle\lVert\text{I-J}\rVert\right\rangle$
according to Eq. (5) are drawn. Resonances corresponding to Tab. 3 are
annotated. Figure S11:  Polarization resolved PLE spectra and polarization
degrees (PolDeg) in 18 different combinations of excitation and detection
polarization bases (I,J) of the $\text{X}_{S}^{+}$ emission line. The data
corresponds to the third run of Fig. 6. The first (I) and second (J) bases are
of the excitation and detection, respectively. Upper panels: Combinations with
excitation polarization bases (a) H/V, (b) D/A and (c) R/L. The respective
excitation PolDegs are shown in the lower panels. Additionally, total PolDegs
of each excitation axes $\left\langle\lVert\text{I-J}\rVert\right\rangle$
according to Eq. (5) are drawn. Resonances corresponding to Tab. 3 are
annotated.

Polarization resolved PLE spectroscopy of $\text{X}_{S}^{-}+\text{XX}_{S}^{0}$
is shown in Fig. 5 for the conditions of run $3$ discussed in Sec. II.5 of the
main text. Corresponding PLE spectra for $\text{X}_{S}^{0}$ and
$\text{X}_{S}^{+}$ transitions are depicted in Fig. S10 and S11, respectively.
The spin preparation fidelity plotted in Fig. 6 is determined numerically
according to Eqs. (5-7) using data of all three polarization resolved PLE
spectra. Resolved PLE traces relevant excitation resonances of Tab. 1 and Fig.
2 are annotated by $R^{\text{X}}_{i}$. It can be observed that the
polarization resolved response - including its polarization Eigenbase - is
very much dependent on the specific excitation resonance.. in question.

## References

  * Cirac  _et al._ [1997] J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, Quantum state transfer and entanglement distribution among distant nodes in a quantum network, Phys. Rev. Lett. 78, 3221 (1997).
  * Kimble [2008] H. J. Kimble, The quantum internet, Nature 453, 1023 (2008).
  * Simon [2017] C. Simon, Towards a global quantum network, Nat. Photonics 11, 678 (2017).
  * van Loock  _et al._ [2019] P. van Loock, W. Alt, C. Becher, O. Benson, H. Boche, C. Deppe, J. Eschner, S. Höfling, D. Meschede, P. Michler, F. Schmidt, and H. Weinfurter, Extending quantum links: Modules for fiber- and memory-based quantum repeaters, arXiv (2019), 1912.10123v1 .
  * Ritter  _et al._ [2012] S. Ritter, C. Nölleke, C. Hahn, A. Reiserer, A. Neuzner, M. Uphoff, M. Mücke, E. Figueroa, J. Bochmann, and G. Rempe, An elementary quantum network of single atoms in optical cavities, Nature 484, 195 (2012).
  * Wang  _et al._ [2016] X.-L. Wang, L.-K. Chen, W. Li, H.-L. Huang, C. Liu, C. Chen, Y.-H. Luo, Z.-E. Su, D. Wu, Z.-D. Li, H. Lu, Y. Hu, X. Jiang, C.-Z. Peng, L. Li, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, and J.-W. Pan, Experimental ten-photon entanglement, Phys. Rev. Lett. 117, 210502 (2016).
  * Schwartz  _et al._ [2016] I. Schwartz, D. Cogan, E. R. Schmidgall, Y. Don, L. Gantz, O. Kenneth, N. H. Lindner, and D. Gershoni, Deterministic generation of a cluster state of entangled photons, Science 354, 434 (2016).
  * Liao  _et al._ [2018] S.-K. Liao, W.-Q. Cai, J. Handsteiner, B. Liu, J. Yin, L. Zhang, D. Rauch, M. Fink, J.-G. Ren, W.-Y. Liu, Y. Li, Q. Shen, Y. Cao, F.-Z. Li, J.-F. Wang, Y.-M. Huang, L. Deng, T. Xi, L. Ma, T. Hu, L. Li, N.-L. Liu, F. Koidl, P. Wang, Y.-A. Chen, X.-B. Wang, M. Steindorfer, G. Kirchner, C.-Y. Lu, R. Shu, R. Ursin, T. Scheidl, C.-Z. Peng, J.-Y. Wang, A. Zeilinger, and J.-W. Pan, Satellite-relayed intercontinental quantum network, Phys. Rev. Lett. 120, 030501 (2018).
  * Zopf  _et al._ [2019] M. Zopf, R. Keil, Y. Chen, J. Yang, D. Chen, F. Ding, and O. G. Schmidt, Entanglement swapping with semiconductor-generated photons violates bell’s inequality, Phys. Rev. Lett. 123, 160502 (2019).
  * Basso Basset  _et al._ [2019] F. Basso Basset, M. B. Rota, C. Schimpf, D. Tedeschi, K. D. Zeuner, S. F. Covre da Silva, M. Reindl, V. Zwiller, K. D. Jöns, A. Rastelli, and R. Trotta, Entanglement swapping with photons generated on demand by a quantum dot, Phys. Rev. Lett. 123, 160501 (2019).
  * Bhaskar  _et al._ [2020] M. K. Bhaskar, R. Riedinger, B. Machielse, D. S. Levonian, C. T. Nguyen, E. N. Knall, H. Park, D. Englund, M. Lončar, D. D. Sukachev, and M. D. Lukin, Experimental demonstration of memory-enhanced quantum communication, Nature 580, 60 (2020).
  * Körber  _et al._ [2018] M. Körber, O. Morin, S. Langenfeld, A. Neuzner, S. Ritter, and G. Rempe, Decoherence-protected memory for a single-photon qubit, Nat. Photonics 12, 18 (2018).
  * Abobeih  _et al._ [2018] M. H. Abobeih, J. Cramer, M. A. Bakker, N. Kalb, M. Markham, D. J. Twitchen, and T. H. Taminiau, One-second coherence for a single electron spin coupled to a multi-qubit nuclear-spin environment, Nat. Commun. 9, 1 (2018).
  * Chen  _et al._ [2018] Y. Chen, M. Zopf, R. Keil, F. Ding, and O. G. Schmidt, Highly-efficient extraction of entangled photons from quantum dots using a broadband optical antenna, Nat. Commun. 9, 2994 (2018).
  * Bloembergen  _et al._ [1948] N. Bloembergen, E. M. Purcell, and R. V. Pound, Relaxation effects in nuclear magnetic resonance absorption, Phys. Rev. 73, 679 (1948).
  * Gammon  _et al._ [2001] D. Gammon, A. L. Efros, T. A. Kennedy, M. Rosen, D. S. Katzer, D. Park, S. W. Brown, V. L. Korenev, and I. A. Merkulov, Electron and nuclear spin interactions in the optical spectra of single GaAs quantum dots, Phys. Rev. Lett. 86, 5176 (2001).
  * Cogan  _et al._ [2018] D. Cogan, O. Kenneth, N. H. Lindner, G. Peniakov, C. Hopfmann, D. Dalacu, P. J. Poole, P. Hawrylak, and D. Gershoni, Depolarization of electronic spin qubits confined in semiconductor quantum dots, Phys. Rev. X 8, 041050 (2018).
  * Greilich  _et al._ [2006] A. Greilich, D. R. Yakovlev, A. Shabaev, A. L. Efros, I. A. Yugova, R. Oulton, V. Stavarache, D. Reuter, A. Wieck, and M. Bayer, Mode locking of electron spin coherences in singly charged quantum dots, Science 313, 341 (2006).
  * Högele  _et al._ [2012] A. Högele, M. Kroner, C. Latta, M. Claassen, I. Carusotto, C. Bulutay, and A. Imamoglu, Dynamic nuclear spin polarization in the resonant laser excitation of an InGaAs quantum dot, Phys. Rev. Lett. 108, 197403 (2012).
  * Hopfmann  _et al._ [2020] C. Hopfmann, W. Nie, N. L. Sharma, C. Weigelt, F. Ding, and O. G. Schmidt, Maximally entangled and Ghz-clocked on-demand photon pair source, arXiv (2020), 2010.11817 .
  * Huber  _et al._ [2017] D. Huber, M. Reindl, Y. Huo, H. Huang, J. S. Wildmann, O. G. Schmidt, A. Rastelli, and R. Trotta, Highly indistinguishable and strongly entangled photons from symmetric GaAs quantum dots, Nat. Commun. 8, 1 (2017).
  * Keil  _et al._ [2017] R. Keil, M. Zopf, Y. Chen, B. Höfer, J. Zhang, F. Ding, and O. G. Schmidt, Solid-state ensemble of highly entangled photon sources at rubidium atomic transitions, Nat. Commun. 8, 1 (2017).
  * Benny  _et al._ [2011] Y. Benny, Y. Kodriano, E. Poem, S. Khatsevitch, D. Gershoni, and P. M. Petroff, Two-photon photoluminescence excitation spectroscopy of single quantum dots, Phys. Rev. B 84, 075473 (2011).
  * Jahn  _et al._ [2015] J.-P. Jahn, M. Munsch, L. Béguin, A. V. Kuhlmann, M. Renggli, Y. Huo, F. Ding, R. Trotta, M. Reindl, O. G. Schmidt, A. Rastelli, P. Treutlein, and R. J. Warburton, An artificial rb atom in a semiconductor with lifetime-limited linewidth, Phys. Rev. B 92, 245439 (2015).
  * Miller  _et al._ [1984] D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, T. H. Wood, and C. A. Burrus, Band-edge electroabsorption in quantum well structures: The quantum-confined stark effect, Phys. Rev. Lett. 53, 2173 (1984).
  * Besombes  _et al._ [2003] L. Besombes, J. J. Baumberg, and J. Motohisa, Coherent spectroscopy of optically gated charged single InGaAs quantum dots, Phys. Rev. Lett. 90, 257402 (2003).
  * Benny  _et al._ [2012] Y. Benny, Y. Kodriano, E. Poem, D. Gershoni, T. A. Truong, and P. M. Petroff, Excitation spectroscopy of single quantum dots at tunable positive, neutral, and negative charge states, Phys. Rev. B 86, 085306 (2012).
  * Santori  _et al._ [2002] C. Santori, D. Fattal, M. Pelton, G. S. Solomon, and Y. Yamamoto, Polarization-correlated photon pairs from a single quantum dot, Phys. Rev. B 66, 045308 (2002).
  * Bayer  _et al._ [2002] M. Bayer, G. Ortner, O. Stern, A. Kuther, A. A. Gorbunov, A. Forchel, P. Hawrylak, S. Fafard, K. Hinzer, T. L. Reinecke, S. N. Walck, J. P. Reithmaier, F. Klopf, and F. Schäfer, Fine structure of neutral and charged excitons in self-assembled In(Ga)As/(Al)GaAs quantum dots, Phys. Rev. B 65, 195315 (2002).
  * Siebert  _et al._ [2009] E. Siebert, T. Warming, A. Schliwa, E. Stock, M. Winkelnkemper, S. Rodt, and D. Bimberg, Spectroscopic access to single-hole energies in InAs/GaAs quantum dots, Phys. Rev. B 79, 205321 (2009).
  * Warming  _et al._ [2009] T. Warming, E. Siebert, A. Schliwa, E. Stock, R. Zimmermann, and D. Bimberg, Hole-hole and electron-hole exchange interactions in single InAs/GaAs quantum dots, Phys. Rev. B 79, 125316 (2009).
  * Bohr [1954] N. Bohr,  _Rydberg’s discovery of the spectral laws_ (C.W.K. Gleerup, 1954).
  * Benny  _et al._ [2014] Y. Benny, R. Presman, Y. Kodriano, E. Poem, D. Gershoni, T. A. Truong, and P. M. Petroff, Electron-hole spin flip-flop in semiconductor quantum dots, Phys. Rev. B 89, 035316 (2014).
  * Adachi [1993] S. Adachi,  _Properties of aluminium gallium arsenide_ , 7 (IET, 1993).
  * Kamada  _et al._ [2001] H. Kamada, H. Gotoh, J. Temmyo, T. Takagahara, and H. Ando, Exciton rabi oscillation in a single quantum dot, Phys. Rev. Lett. 87, 246401 (2001).
  * Fox  _et al._ [2006] A. Fox, M. Fox, D. Fox, and O. U. Press,  _Quantum Optics: An Introduction_, Oxford Master Series in Physics (OUP Oxford, 2006).
  * Kramers [1930] H. Kramers, General theory of paramagnetic rotation in crystals,  _Proc. Acad. Sci. Amsterdam_ , 33, 959 (1930).
  * Stockill  _et al._ [2016] R. Stockill, C. Le Gall, C. Matthiesen, L. Huthmacher, E. Clarke, M. Hugues, and M. Atatüre, Quantum dot spin coherence governed by a strained nuclear environment, Nat. Commun. 7, 12745 (2016).
  * Chekhovich  _et al._ [2020] E. A. Chekhovich, S. F. C. da Silva, and A. Rastelli, Nuclear spin quantum register in an optically active semiconductor quantum dot, Nat. Nanotechnol. (2020).
  * Wang  _et al._ [2007] Z. M. Wang, B. Liang, K. Sablon, and G. Salamo, Nanoholes fabricated by self-assembled gallium nanodrill on GaAs (100), Applied physics letters 90, 113120 (2007).