††thanks: These authors contributed equally††thanks: These authors contributed
equally††thanks: These authors contributed equally

# Controlling the Photon Number Coherence of Solid-state Quantum Light Sources
for Quantum Cryptography

Yusuf Karli  Institut für Experimentalphysik, Universität Innsbruck,
Innsbruck, Austria  Daniel A. Vajner  Institute of Solid State Physics,
Technische Universität Berlin, 10623 Berlin, Germany  Florian Kappe  Institut
für Experimentalphysik, Universität Innsbruck, Innsbruck, Austria  Paul C. A.
Hagen  Theoretische Physik III, Universität Bayreuth, 95440 Bayreuth, Germany
Lena M. Hansen  University of Vienna, Faculty of Physics, Vienna Center for
Quantum Science and Technology (VCQ), Vienna, Austria,
Christian Doppler Laboratory for Photonic Quantum Computer, Faculty of
Physics, University of Vienna, Vienna, Austria  René Schwarz  Institut für
Experimentalphysik, Universität Innsbruck, Innsbruck, Austria  Thomas K.
Bracht  Condensed Matter Theory, Department of Physics, TU Dortmund, 44221
Dortmund, Germany  Institut für Festkörpertheorie, Universität Münster, 48149
Münster, Germany  Christian Schimpf  Institute of Semiconductor and Solid
State Physics, Johannes Kepler University Linz, Linz, Austria  Saimon F. Covre
da Silva  Institute of Semiconductor and Solid State Physics, Johannes Kepler
University Linz, Linz, Austria  Philip Walther  University of Vienna, Faculty
of Physics, Vienna Center for Quantum Science and Technology (VCQ), Vienna,
Austria,
Christian Doppler Laboratory for Photonic Quantum Computer, Faculty of
Physics, University of Vienna, Vienna, Austria  Armando Rastelli  Institute of
Semiconductor and Solid State Physics, Johannes Kepler University Linz, Linz,
Austria  Vollrath Martin Axt  Theoretische Physik III, Universität Bayreuth,
95440 Bayreuth, Germany  Juan C. Loredo  University of Vienna, Faculty of
Physics, Vienna Center for Quantum Science and Technology (VCQ), Vienna,
Austria,
Christian Doppler Laboratory for Photonic Quantum Computer, Faculty of
Physics, University of Vienna, Vienna, Austria  Vikas Remesh  Institut für
Experimentalphysik, Universität Innsbruck, Innsbruck, Austria  Tobias Heindel
Institute of Solid State Physics, Technische Universität Berlin, 10623 Berlin,
Germany  Doris E. Reiter  Condensed Matter Theory, Department of Physics, TU
Dortmund, 44221 Dortmund, Germany  Gregor Weihs  Institut für
Experimentalphysik, Universität Innsbruck, Innsbruck, Austria

(Date: August 28, 2024
XXX E-mail<EMAIL_ADDRESS>
###### Abstract

Quantum communication networks rely on quantum cryptographic protocols
including quantum key distribution (QKD) using single photons. A critical
element regarding the security of QKD protocols is the photon number coherence
(PNC), i.e. the phase relation between the zero and one-photon Fock state,
which critically depends on the excitation scheme. Thus, to obtain flying
qubits with the desired properties, optimal pumping schemes for quantum
emitters need to be selected. Semiconductor quantum dots generate on-demand
single photons with high purity and indistinguishability. Exploiting two-
photon excitation of a quantum dot combined with a stimulation pulse, we
demonstrate the generation of high-quality single photons with a controllable
degree of PNC. Our approach provides a viable route toward secure
communication in quantum networks.

††preprint: APS/123-QED

##  I Introduction

Single photons are an essential resource for future high-security
communication networks, with applications like measurement-based or
distributed quantum computing and quantum cryptography [1, 2, 3]. Every
quantum information protocol has its unique set of practical requirements [4].
While early quantum key distribution (QKD) protocols [5, 6] primarily relied
on high single-photon purity, more advanced schemes have further requirements
such as high indistinguishability, for example in quantum repeaters or
measurement-device-independent (MDI)-QKD, which relies on remote two-photon
interference [7, 8]. The search for efficient single-photon sources has led to
semiconductor quantum dots [9], thanks to their high single-photon purity
[10], brightness [11], indistinguishability [12], scalability [13] and above
all, versatility in emission wavelength selection.

Figure 1:  Overview of various quantum information protocols: The protocols
are sorted by their requirements on indistinguishability $\cal{I}$ and PNC,
specifically focusing on discrete variable cryptographic protocols using
polarization or time-bin encoding. Protocols that require low PNC and need no
high $\cal{I}$ are BB84 [5], decoy-BB84 [14], 6-state-protocol [15], SARG04
[16], LM05 [17], QBC: quantum bit commitment [18], UT: unforgeable quantum
tokens [19, 20], CF: quantum coin flipping [21, 22], OT: Oblivious Transfer
[23]; low PNC and high $\cal{I}$ is required by MDI-QKD [8], quantum repeaters
and entanglement swapping for QKD [7, 24], quantum teleportation for QKD [25],
DI-QKD with single photons [26]; high PNC and variable $\cal{I}$ are needed in
TF: twin-field QKD with single photons [27]. Note that carrying out protocols
from the low PNC column with phase encoding would require an initially defined
phase, before randomizing it in a reversible way, which requires PNC in the
beginning. To illustrate this we have also added MDI-QKD with phase encoding
to the diagram.

Photon number coherence (PNC) [28, 29] is another crucial quantity relevant
for the security of single photon quantum cryptography schemes. It must vanish
for most protocols [30], compromising security otherwise [31, 32] due to side-
channel attacks enabled by the fixed relative phase between different photon
number states [33, 34]. While there are more general security proofs that
allow a non-zero PNC, they lead to lower key rates, as some of the bits must
be devoted to compensate for the additional information leakage towards an
eavesdropper [35]. To achieve zero PNC in practice, actively phase-randomized
single photons can be used, as typically implemented for faint laser pulses
[36, 37]. Otherwise, a suitable excitation scheme without PNC must be chosen,
which might however deteriorate the other single photon properties [28, 30].

In this work, we achieve tailored degrees of PNC, on demand, maintaining high
purity and indistinguishability. We implement optical excitation protocols to
demonstrate the experimental single-photon generation from quantum dots. The
photon output in our scheme can be increased up to twice as high compared to
more commonly used methods like resonant excitation. We therefore set the
stage for the quantum dot platform to be used for advanced cryptographic
implementations.

Due to its versatility, the excitation scheme presented here covers the
requirements of a broad range of quantum cryptographic protocols. An overview
of various applications in the context of PNC and indistinguishability
requirements is given in Fig. 1a. For example, established protocols like BB84
[5], decoy-BB84 [14], 6-state-protocol [15], SARG04 [16], LM05 [17] and
primitives like strong quantum coin flipping [21, 22], unforgeable quantum
tokens [19, 20], quantum bit commitment [18] or quantum oblivious transfer
[38] require the absence of PNC to ensure, for instance, security in QKD or
fairness in coin-flipping protocols. On the other hand, there exist protocols
that benefit from a finite amount of initial PNC like MDI-QKD when done with
phase encoding [8] or twin-field QKD protocols [27] to know and set the
initial phase [30].

Figure 2:  Generating single photons with variable PNC: (a) Level scheme of a
quantum dot consisting of ground state $|g\rangle$, two linearly polarized
exciton states $|x_{V/H}\rangle$ and biexciton state $|xx\rangle$. Straight
lines indicate laser excitation, while dashed lines denote relaxation
processes with rate $\gamma$. Both schemes start with a two-photon excitation
(TPE) from $|g\rangle\to|xx\rangle$. In stiX, an additional H-polarized laser
pulse stimulates the transition $|xx\rangle\to|x_{H}\rangle$. We only collect
$H$-polarized photons. (b) Theoretically calculated biexciton ($|xx\rangle$)
occupation showing Rabi rotations as a function of the TPE pulse area (red
curve) and the corresponding coherence between $|g\rangle$ and $|xx\rangle$
(purple curve). (c) Exciton photon counts recorded under reX (blue dots) and
stiX (red dots) manifesting the enhancement of photon counts under
stimulation. (d) Sketch of the experimental setup: a Ti:Sapphire laser source
producing $\approx$$2\text{\,}\mathrm{ps}$-long laser pulses, with a spectral
FWHM of $0.5\text{\,}\mathrm{nm}$, is used to spectrally shape TPE and stim.
pulses at appropriate wavelengths $\lambda_{\text{TPE}}$ and
$\lambda_{\text{stim}}$ using two 4$f$ pulse shapers. A fiber-optic delay line
enables the time control of the stim. pulse with respect to the TPE pulse. An
electronic variable optical attenuator (VOA) helps sweep the laser power. The
two pulses meet at a 10:90 beamsplitter (BS) and propagate to the cryostat
which holds the quantum dot at $1.5\text{\,}\mathrm{K}$. Emitted single
photons from the quantum dot are spectrally filtered by a notch filter (NF)
and send to an unbalanced Mach-Zehnder interferometer with a freely evolving
phase on one arm (labeled as PNC setup). Two single-photon sensitive avalanche
photodiodes (APD1 and APD2) detect the single photon counts at the output arms
of the interferometer. pol: linear polarizer, HWP: half-wave plate, QWP:
quarter-wave plate, BS: beamsplitter, PBS: polarizing beam splitter, FBS:
fiber beam splitter.

Our excitation protocols are based on resonant two-photon excitation (TPE) of
a quantum dot from the ground state $|g\rangle$ into the biexciton state
$|xx\rangle$ [39, 40], yielding Rabi rotations. A simulation of TPE Rabi
rotations is shown in Fig. 2b (red dashed line) and experimental data in Fig.
2c (blue dots). From the biexciton state, the system relaxes into either
horizontally $|x_{H}\rangle$ or vertically $|x_{V}\rangle$ polarized exciton
state, from which we collect only horizontally ($H$) polarized photons. We
call this scheme _relaxation into the exciton_ (reX). The reX scheme is
advantageous over the direct, resonant excitation of the exciton, due to the
suppressed re-excitation and therefore provides high-purity photon states
[10]. Because the exciting laser energy is different from the emitted photon
energy, a challenging cross-polarization filtering is avoided and the photon
count rate can be increased up to a factor of two, which is also achieved by
several other recently-proposed excitation schemes [41, 42, 43, 44, 45, 46,
47]. However, the indistinguishability of the single photons via the reX
scheme suffers greatly from the spontaneous decay of the biexciton [48] and if
a specific polarization is required, the photon output is reduced due to the
two available decay channels.

An improved protocol to overcome these problems uses an additional stimulation
laser pulse following the TPE pulse [49]. This _stimulated preparation of the
exciton_ (stiX) scheme can generate higher indistinguishability exciton
photons due to the reduced time jitter [50, 51, 52]. Because the stimulation
pulse determines the polarization of the emitted photon, the photon counts in
that polarization state is also enhanced up to a factor of two (see also Fig.
2c). Although the presence of PNC under resonant excitation and reX has been
investigated before [28, 30], it remains to be seen if PNC exists in the stiX
scheme. Additionally, assessing the controllability of PNC is essential for
advancing optical preparation schemes of quantum dot states for quantum
cryptography applications.

##  II Results

### Definition of photon number coherence

In a pure state $|\Psi\rangle=\sum_{n=0}^{\infty}c_{n}|n\rangle$ in the photon
number Fock basis with eigenstates $|n\rangle$ and the complex coefficients
$c_{n}$, we define PNC as the absolute value of the coherence between the Fock
states. For QKD based on single photons, as considered in this paper, the PNC
refers to the coherence between the Fock states $|0\rangle$ and $|1\rangle$.
More generally, we employ a density matrix description using

$\rho=\left(\begin{array}[]{cc}\rho_{0,0}&\rho_{0,1}\\\
\rho_{1,0}&\rho_{1,1}\end{array}\right)\qquad\text{with}\qquad\text{PNC}=|\rho_{0,1}|,$
(1)

$\rho_{1,1}$ ($\rho_{0,0}$) being the occupation of the one (zero)-photon
state and $\rho_{0,1}$ being the coherence. We recall that it holds that
$|\rho_{0,1}|^{2}\leq\rho_{1,1}\,\rho_{0,0}$ with equality in case of a pure
state. The inequality implies that for $\rho_{1,1}=1$ or $\rho_{0,0}=1$ the
PNC vanishes, while for $\rho_{0,0}=\rho_{1,1}=1/2$ it is maximal.

There are several factors that affect the PNC. One aspect is the non-perfect
preparation of the photon state. The properties of photons from a quantum dot
depend on the preparation fidelity of the quantum dot electronic state. This
fidelity, in turn, is affected by the interaction with the environment of the
quantum dot, most strongly by the interaction with phonons [53]. Phonons can
also degrade the coherence properties of the photons [54] and therefore also
the PNC. Additional losses of photons into other modes, which are not
detected, further affect the photon properties.

It is likewise important to consider the measurement process. To detect PNC, a
phase-evolving Mach-Zehnder Interferometer (MZI) is employed [28]. The outputs
of the MZI are simultaneously recorded with two avalanche photodiodes (APDs).
The count rates $N_{1},N_{2}$ in the APD result in the visibility

$v_{i}=\frac{N_{i}^{\text{max}}-N_{i}^{\text{min}}}{N_{i}^{\text{max}}+N_{i}^{\text{min}}}\,.$
(2)

In the case of an ideal measurement and perfectly indistinguishable photons,
the visibility is connected to the PNC via

$v=\frac{|{\rho}_{0,1}|^{2}}{{\rho}_{1,1}}\,.$ (3)

In the MZI used in the experiment, the interference of subsequent single
photons takes place. Phase scrambling between subsequent emission events leads
to a further reduction of the visibility in addition to the aforementioned
phonon and loss effects. It should be kept in mind that a vanishing visibility
in the experiment can therefore result either from vanishing PNC, phase
scrambling, or a combined effect of both.

### Theoretical expectations

We perform theoretical simulations to estimate the PNC for both reX and stiX
for a quantum dot modeled as a four-level system driven by a (classical) laser
field and coupled to discrete photon modes. The simulation considered the
presence three photon modes, however, only the zero and one modes were found
to be noticeably populated. We further account for coupling with acoustic
phonons within a numerically exact path integral formalism [55]. In addition,
we include relaxation between the quantum dot states accounting for photons
not being emitted into the relevant modes (see Methods section for details of
the model and calculation). We assume an ideal detection, i.e., no phase
scrambling and perfect indistinguishability and model the visibility via Eq.
(3).

Figure 3: Theoretical predictions: Left: Dynamics of the four-level system
coupled to two photon modes including phonons and losses calculated via a
numerically exact path integral formalism. The exciting laser pulses with the
TPE pulse (orange) and stimulating pulse (red) shown in (c). The occupation of
the biexciton and exciton state for stiX are displayed in (b) with the dashed
line indicating the behaviour for reX. The PNC for stiX (red) and reX (blue)
is displayed in (a). Note the logarithmic scale. During the stimulating pulse
the exciton becomes occupied resulting in a rise of the PNC. Right: Time-
integrated coherence $\text{PNC}^{\text{calc.}}$ (e) and visiblity
$v^{\text{calc}}$ (d) as a function of TPE pulse area for both reX
(blue,magnified) and stiX (red). The TPE pulse areas of $\pi$ and $\pi/2$ are
marked by vertical lines. The time-integrated occupation of the one-photon
Fock state $\text{occ}^{\text{calc}}$ is shown as a green dashed line. Due to
the relaxation process the PNC is almost lost in the reX case. In stiX, we
find that the PNC is controlled via the TPE pulse area.

The time evolution of the biexciton and the $|x_{H}\rangle$ exciton occupation
together with the PNC $|\rho_{0,1}|$ is shown in Fig. 3a,b. Both schemes start
with excitation from the ground into the biexciton state induced by a
Gaussian-shaped laser pulse with a TPE pulse area of $\pi/2$. In the reX
scheme, the biexciton state then relaxes into the exciton states via the
emission of photons. However, these photons are at a different wavelength and
therefore ignored. The exciton state is transiently occupied because it
rapidly generates the desired photon relaxing further into the ground state.
The corresponding PNC (blue curve in Fig. 3a) is almost vanishing, too,
because of the incoherent biexciton-exciton relaxation destroying the
electronic coherence. The remaining PNC can be traced back to deviations from
the ideal case caused by phonon interaction, radiative losses, as well as
relaxation into other (undesired) states in the quantum dot.

In contrast, in the stiX scheme, the stimulating pulse brings the biexciton
coherently into $|x_{H}\rangle$ by the application of a $\pi$-pulse resonant
with the $|xx\rangle\to|x_{H}\rangle$ transition as evidenced in Fig. 3b. The
small oscillations on top of the population-exchange result from the off-
resonant driving of the complementary transition $|x_{H}\rangle\to|g\rangle$.
Because the transition to the exciton state $|x_{H}\rangle$ is coherent, the
electronic coherence, which translates to the PNC, is preserved. Accordingly,
in Fig. 3a (red curve), we see that as soon as the stimulating pulse sets in,
the PNC becomes very high. In other words, a timed stimulation preparation of
the exciton state recovers the PNC that is lost in the reX scheme.

By controlling the electronic coherence through the pulse areas of the
exciting pulses, we can thus manipulate the PNC. To ensure the best
comparability, we fix the stimulating pulse to a $\pi$ pulse and vary the
pulse area of the TPE pulse, which results in an oscillating coherence as
shown in Fig. 2c. The time-integrated occupation of the one-photon state
$\text{occ}^{\text{calc}}$ follows the Rabi rotations of the biexciton. We
have checked that under the present conditions, the higher Fock states always
have negligible occupations. The highest coherence is expected for pulse areas
$(2n+1)\pi/2$, where also the electronic coherence is maximal. However, due to
the incoherent relaxation process from the biexciton into the exciton, in reX
the PNC is close to zero for all pulse areas as expected. This is confirmed by
the numerical results in Fig. 3e. Only for large pulse areas, detrimental
processes due to phonons or losses lead to some residual PNC. Accordingly, for
the reX scheme also the measured visibility $v^{\text{calc}}$ in Fig. 3d is
vanishing.

For stiX, the coherence is preserved and we find an oscillating behaviour as a
function of the TPE pulse area with maxima of the PNC occurring for pulse
areas with $(2n+1)\pi/2$ and minima for pulse areas $n\pi$. Ideally, PNC
should be zero for pulse area $n\pi$. In the full simulation including finite
pulse lengths and losses, the TPE pulse does not fully invert the system,
leading to a residual PNC even for a TPE $\pi$-pulse. The visibility behaves
differently: While a clear minimum at $\pi$ is recovered, the PNC is not
maximal at $\pi/2$. Instead, due to its definition, $v^{\text{calc}}$
increases for the even smaller TPE pulse areas. Still, compared to reX,
visibility for stiX shows a strong dependence on the TPE pulse area.

### Experimental data

Figure 4: Controlled generation of PNC on a single quantum dot for reX (a,
left) and stiX (b, right): Top panel: Measured purity of the generated single
photons. For $g^{(2)}_{\text{reX}}(0)$ (blue), the TPE pulse power is kept at
$\pi$-power, and stim pulse is absent. For $g^{(2)}_{\text{stiX}}(0)$(red),
TPE and stim pulses are kept at $\pi$-power. For $\text{HOM}_{\text{reX}}$,
blue and gray shaded curves represent HOM coincidences recorded for parallel
and orthogonal polarizations respectively, for $\pi$ TPE pulse area. For
$\text{HOM}_{\text{stiX}}$, red and gray shaded curves represent HOM
coincidences recorded for parallel and orthogonal polarizations respectively,
TPE and stim pulses are kept at $\pi$-power. Middle panel: Extracted
visibilities $v^{\text{exp}}$ (dark-blue dots) and the reconstructed
$\text{PNC}^{\text{exp}}$ (yellow) at different TPE pulse areas alongside the
measured X counts (blue curve for $\text{X}_{\text{reX}}$ and red curve for
$X_{\text{stiX}}$). The X counts are normalized to their respective values at
$\pi$-power. Bottom panel: Exemplary time traces recorded at the two detector
outputs of the PNC setup at TPE pulse areas of $0.5\pi$, $\pi$, and $1.5\pi$.

We perform the reX and stiX experiments to test the theoretical prediction on
a single quantum dot in our setup displayed in Fig. 2d. We note that in stiX
we fix the time delay between the TPE pulse and stimulating pulse to
$7\text{\,}\mathrm{ps}$, where the photon count is maximal. A detailed
description on the experiment is provided in the Methods Section.

We start by quantifying the photon properties for reX and stiX, at various
powers (see SI Table S1) measuring the single photon purity in a Hanbury Brown
and Twiss (HBT) setup and the indistinguishability via Hong-Ou-Mandel (HOM)
measurements. At $\pi$ power of both reX and stiX we validate that the
generated photons have high purity with $g^{(2)}_{\text{reX}}(0)=0.0004(1)$
and $g^{(2)}_{\text{stiX}}(0)=0.0009(1)$. For the indistinguishability, the
HOM visibility reaches only 58(3)% under reX, while for stiX it increases to
95(6)%, in line with previous observations [50, 51, 52]. These results already
underline that stiX is a advantageous scheme compared to reX.

We then sweep the TPE pulse area under reX and stiX yielding Rabi rotations
for the exciton (X) photon counts (blue and red curves in Fig. 4) and
investigate the PNC. For each TPE pulse power, we analyze the spectrally
filtered X photons using a phase-evolving MZI [28]. Its outputs are
simultaneously recorded with two avalanche photodiodes (APDs) for
$20\text{\,}\mathrm{s}$ each. In the bottom panel of Fig. 4, we display
exemplary time traces (denoted by green and magenta curves, representing the
two detector outputs of the MZI, see Fig. 2d) at TPE powers $0.5\pi$, $1\pi$,
$1.5\pi$. From the time traces, we compute the visibility according to Eq. (3)
from the normalized detector counts taking the average of the two detectors as
$v^{\text{exp}}=(v_{1}+v_{2})/2$.

The visibility $v^{\text{exp}}$ as a function of pulse area is displayed
alongside the respective Rabi rotations in Fig. 4 as black dots. Under reX,
the visibility $v^{\text{exp}}$ is vanishing for all TPE pulse areas and no
clear dependence is found. This is in agreement with the exemplary time traces
(displayed in the bottom panel of Fig. 4a)., where indeed no oscillations are
seen for different pulse areas

In contrast, for stiX, the PNC shows a more interesting behaviour: in the
exemplary time traces of the MZI outputs (displayed in the bottom panel of
Fig. 4b), we observe clear oscillations for TPE powers $0.5\pi$, and $1.5\pi$
and almost no oscillations at $\pi$. Accordingly, the visibilities vary from
$\approx\,$0.6 at TPE power $0.5\pi$ to being minimal at $\pi$ and then rise
again until $1.5\pi$.

From the visibilities, using the formalism from Ref. [28], we extract the
$\text{PNC}^{\text{exp.}}$ shown as the yellow line in Fig. 4. The data
clearly confirms the trend expected from the theory: We find minima of PNC
when exciting with TPE pulses of pulse area $n\pi$ and maxima at
$(2n+1)\pi/2$. This behaviour is evident in stiX, while in reX only a small
modulation is found.

Hence, we conclude that the PNC is negligible in reX, while in stiX we have
tuneable PNC controlled via the TPE pulse area.

##  III Discussion

We now set our results in the context of finding the optimal photon source for
high-security quantum networks. As indicated before, purity,
indistinguishability, and PNC are the key parameters that must be known when
choosing an excitation scheme. We have shown that reX generates high-purity
photons, while indistinguishability and PNC are low, and also, if filtering
only a single polarization, the photon output is reduced. Looking back at Fig.
1a, we find that reX produces photons in the bottom left corner with low PNC
and low indistinguishability, which limits the amounts of applicable
protocols.

Within stiX, photons with high purity and high indistinguishability are
generated. More importantly, the PNC can be controlled via the pulse area. If
the TPE power is set to $(2n+1)\pi/2$, one obtains high PNC, enabling
protocols in the top right corner of the diagram in Fig. 1a. By changing the
pulse area to $n\pi$, the PNC is minimal which allows performing protocols in
the top left corner of the diagram in Fig. 1a. For all TPE powers, stiX is
suitable for protocols that require a high indistinguishability. Besides power
control, the time delay of the stimulating pulse also controls the PNC (see SI
Section E). The largest PNC is obtained when the time separation between the
TPE and the stimulating pulse is optimal.

In summary, we showed a controlled generation of single photons with variable
degrees of PNC as well as high purity, high indistinguishability, and high
brightness via a stimulated two-photon excitation. This is a big step forward
towards the realization of secure quantum networks based on single photons.

##  IV Methods

###  IV.1 Theoretical model

For the theoretical modelling, we set up the Hamiltonian consisting of the
quantum dot system $\hat{H}^{\text{QD}}$, the out-coupling to two-photon modes
$\hat{H}^{\text{photon}}$, the excitation of the TPE $\hat{H}^{\text{TPE}}$
and the stimulating laser pulse $\hat{H}^{\text{stim}}$, as well as the
coupling to phonons

$\hat{H}=\hat{H}^{\text{QD}}+\hat{H}^{\text{photon}}+\hat{H}^{\text{TPE}}+\hat{H}^{\text{stim}}+\hat{H}^{\text{phonon}}\,.$
(4)

In addition, we consider radiative decay and losses by a Lindblad operator
$\cal{L}$. In the following, we describe the individual terms in detail.

The quantum dot is modeled using four states (see also Fig. 2b) denoted by
$|g\rangle$ as the ground state, $|x_{H}\rangle$ and $|x_{V}\rangle$ as the
two excitons and $|xx\rangle$ as the biexciton. The ground-state energy is set
to zero, while both excitons have the same energy $\hbar\omega_{x}$, i.e., no
fine-structure splitting is assumed. The biexciton has a binding energy
$E_{B}$ such that its energy is given by
$\hbar\omega_{xx}=2\hbar\omega_{x}-E_{B}$.

$\begin{split}\hat{H}^{\text{QD}}=&\hbar\omega_{x}\left(|x_{H}\rangle\langle
x_{H}|+|x_{V}\rangle\langle x_{V}|\right)+\hbar\omega_{xx}|xx\rangle\langle
xx|\end{split}$ (5)

The quantum dot is coupled to two photon modes with polarisations $V$ and $H$
for the out-coupling of the photons, similar to positioning the quantum dot in
a photonic cavity. We model the photon modes by the Fock states
$|n_{H}\rangle$ and $|n_{V}\rangle$ with the frequency $\omega_{c}$ via the
annihilation (creation) operators $\hat{a}_{H/V}$($\hat{a}^{\dagger}_{H/V}$).
The photonic modes are coupled to the quantum dot transitions with the same
strength via the coupling constant $\hbar g=0.05\text{ meV}$, yielding

$\begin{split}\hat{H}^{\text{photon}}=&\hbar\omega_{c}\left(\hat{a}^{\dagger}_{H}\hat{a}_{H}+\hat{a}^{\dagger}_{V}\hat{a}_{V}\right)\\\
+&\hbar g\,\hat{a}_{H}\left(|x_{H}\rangle\langle g|+|xx\rangle\langle
x_{H}|\right)+h.c.\\\ +&\hbar g\,\hat{a}_{V}\left(|x_{V}\rangle\langle
g|+|xx\rangle\langle x_{V}|\right)+h.c.\\\
=&\hat{H}^{\text{photon}}_{0}+\hat{H}^{\text{photon}}_{\text{coupl.}}.\end{split}$
(6)

We use the Hamiltonian in a rotating frame with
${\omega=\omega_{l}=\omega_{x}-E_{B}/(2\hbar)}$, which corresponds to the
frequency of the TPE laser pulse. With this, the QD-photon Hamiltonian has the
form

$\begin{split}\hat{H}^{\text{QD-
photon}}=&\hbar\Delta\omega_{x-l}\left(|x_{H}\rangle\langle
x_{H}|+|x_{V}\rangle\langle x_{V}|\right)\\\ +&(\hbar
2\Delta\omega_{x-l}-E_{B})|xx\rangle\langle xx|\\\
+&\hbar\Delta\omega_{c-l}\left(\hat{a}^{\dagger}_{H}\hat{a}_{H}+\hat{a}^{\dagger}_{V}\hat{a}_{V}\right)\\\
+&\hat{H}^{\text{photon}}_{\text{coupl.}}.\end{split}$ (7)

The index convention of frequency differences is chosen such that the second
index is subtracted from the first, e.g.,
$\Delta\omega_{x-l}=\omega_{x}-\omega_{l}$. We choose the photon mode to be
resonant with the quantum dot transition from the ground to the excited state,
i.e., $\hbar\Delta\omega_{c-x}=\mbox{0\text{ meV}}$.

The TPE is modelled by an external classical laser field with diagonal
polarization in dipole and rotating wave approximation. We consider a resonant
TPE process and accordingly set the detuning
$\Delta\omega_{x-l}=E_{B}/(2\hbar)$. With this, the Hamiltonian reads

$\begin{split}\hat{H}^{\text{TPE}}(t)=-\frac{\hbar}{2}f^{\text{TPE}}(t)&(|g\rangle\langle
x_{H}|+|g\rangle\langle x_{V}|\\\ &+|x_{H}\rangle\langle
xx|+|x_{V}\rangle\langle xx|+h.c.).\end{split}$ (8)

Here, $f^{\text{TPE}}(t)$ denotes the instantaneous Rabi frequency as given by
the product of dipole moment and electric field. We use Gaussian pulses

$f^{\text{TPE}}(t)=\frac{\Theta_{\text{TPE}}}{\sqrt{2\pi}\,\sigma_{\text{TPE}}}\text{e}^{-\frac{t^{2}}{2\sigma_{\text{TPE}}^{2}}},$
(9)

with the pulse area $\Theta_{\text{TPE}}$ and the pulse width
$\sigma_{\text{TPE}}$. We assign the TPE pulse area $\pi$ in the plot (cf.
Fig. 3 to the one which results in the first maximum of the biexciton
occupation and the TPE pulse area of $\pi/2$ to the first maximum of the
electronic coherence. In the calculations, these values were determined
numerically.

We describe the stimulating laser with the same approximations, but assume it
to be horizontally polarized. Its frequency is set to match the
$|xx\rangle\to|x_{H}\rangle$ transition, such that

$\hat{H}^{\text{stim}}(t)=-\frac{\hbar}{2}f^{\text{stim}}(t)\,\left[\text{e}^{i\Delta\omega_{l}^{\text{stim}}t}\left(|g\rangle\langle
x_{H}|+|x_{H}\rangle\langle xx|\right)\right]+h.c..$ (10)

Here,
$\Delta\omega_{l}^{\text{stim}}=\omega_{l}^{\text{stim}}-\omega_{l}=-E_{B}/(2\hbar)$.
The stimulating laser’s envelope function $f^{\text{stim}}$ is delayed by a
time $\Delta t$ compared to the TPE laser. We also assume a Gaussian envelope
for the stimulating pulse

$f^{\text{stim}}(t)=\frac{\Theta_{\text{stim}}}{\sqrt{2\pi}\,\sigma_{\text{stim}}}\text{e}^{-\frac{(t-\Delta
t)^{2}}{2\sigma_{\text{stim}}^{2}}}.$ (11)

with the pulse area $\Theta_{\text{stim}}$ and the pulse length
$\sigma_{\text{stim}}$. Here, a "$\pi$-pulse" refers to a full inversion of
the resonantly driven transition for ideal conditions (without
losses/phonons).

In addition we consider the coupling to longitudinal-acoustic (LA) phonons via
the deformation potential coupling. Here, $\hat{b}_{\mathbf{k}}$
($\hat{b}^{\dagger}_{\mathbf{k}}$) annihilates (creates) a phonon of mode
$\mathbf{k}$ with energy $\omega_{\mathbf{k}}$. We consider the typical pure-
dephasing type coupling in the standard Hamiltonian [56, 57]

$\hat{H}^{\text{phonon}}=\hbar\sum_{\mathbf{k}}\omega_{\mathbf{k}}\hat{b}^{\dagger}_{\mathbf{k}}\hat{b}_{\mathbf{k}}+\hbar\sum_{{\mathbf{k}},S}\left(\gamma_{\mathbf{k}}^{S}\hat{b}_{\mathbf{k}}^{\dagger}+\gamma_{\mathbf{k}}^{S^{*}}\hat{b}_{\mathbf{k}}\right)|S\rangle\langle
S|,$ (12)

coupling each mode $\mathbf{k}$ to the quantum dot state $|S\rangle$, where
$S\in\\{x_{H},x_{V},xx\\}$. The coupling constant $\gamma_{\mathbf{k}}^{S}$
and the material parameters are taken to be the same as in Ref. [55].

Both, cavity and quantum dot, are subject to losses into the free photonic
field outside of the cavity. These losses are described by Lindblad-
superoperators, affecting the density operator $\hat{\rho}$

$\mathcal{L}_{\hat{O},\delta}[\hat{\rho}]=\delta\left(\hat{O}\hat{\rho}\,\hat{O}^{\dagger}-\frac{1}{2}\left[\hat{\rho},\hat{O}^{\dagger}\hat{O}\right]_{+}\right),$
(13)

where $\hat{O}$ is an operator, $\delta$ a rate and $[.,.]_{+}$ the anti-
commutator. We assume that the decay processes of the quantum dot take place
with rate $\gamma$ and losses of the photonic modes go with the rate $\kappa$,
such that Lindblad-superoperators are

$\begin{split}\mathcal{L}[\hat{\rho}]&:=\mathcal{L}_{\hat{a}_{H},\kappa}[\hat{\rho}]+\mathcal{L}_{\hat{a}_{V},\kappa}[\hat{\rho}]\\\
&+\mathcal{L}_{|g\rangle\langle
x_{H}|,\gamma}[\hat{\rho}]+\mathcal{L}_{|g\rangle\langle
x_{V}|,\gamma}[\hat{\rho}]\\\ &+\mathcal{L}_{|x_{H}\rangle\langle
xx|,\gamma}[\hat{\rho}]+\mathcal{L}_{|x_{V}\rangle\langle
xx|,\gamma}[\hat{\rho}]\,.\end{split}$ (14)

The rates are chosen such that we are in the weak coupling regime.

With the Hamiltonian and the Lindbladian terms we calculate the dynamics of
the system states via the Liouville-von Neumann equation

$\frac{\text{d}}{\text{d}t}\hat{\rho}=-\frac{i}{\hbar}\left[\hat{H}(t),\hat{\rho}\right]+\mathcal{L}[\hat{\rho}].$
(15)

As initial state we assume that the quantum dot is in its ground state and no
photonic excitation exists. For the numerical integration, we use a
numerically complete path-integral method, which is described in Refs. 55, 58
and the parameters from Tab. 1, to solve Eq. (15).

Table 1: Parameters used in the simulation. Material parameter are taken as in Ref. [55]. QD-cavity detuning | $\hbar\Delta\omega_{c-x}$ | 0 meV
---|---|---
QD-laser detuning | $\hbar\Delta\omega_{x-l}$ | 2 meV
detuning stim. pulse | $\hbar\Delta\omega_{l}^{\text{stim}}$ | 2 meV
duration stim. pulse | FWHM${}_{\text{stim}}$ | 3 ps
duration TPE pulse | FWHM${}_{\text{TPE}}$ | 4.5 ps
delay between pulses | $\Delta t$ | 15 ps
QD-cavity coupling | $\hbar g$ | 0.05 meV
Binding energy | $E_{B}$ | 4 meV
cavity loss rate | $\kappa$ | 0.577 ps-1
QD loss rate | $\gamma$ | 0.001 ps-1
QD size | $a$ | 3 nm
temperature | $T$ | 1.5 K

We obtain results for the full density matrix, from which we can obtain the
reduced density matrices for the quantum dots
$\rho^{\text{QD}}_{S,S^{\prime}}$ with $S\in\\{g,x_{H},x_{V},xx\\}$ and for
the photons $\rho^{\text{photon}}_{n_{i},n^{\prime}_{i}}$ with
$i\in\\{H,V\\}$, by tracing out the other degrees of freedom. We are
interested in the coherence $\rho_{0,1}=\rho^{\text{photon}}_{0_{H},1_{H}}$.
The absolute value of $\rho_{0,1}=\rho^{\text{photon}}_{0_{H},1_{H}}$ is
referred to as PNC.

As a measure for the overall PNC at a given pulse area, we introduce the time-
integrated absolute value of the instantaneous PNC

$\text{PNC}^{\text{calc}}\propto\tilde{\rho}_{0,1}=\int|\rho^{\text{photon}}_{0_{H},1_{H}}|dt\,.$
(16)

$\text{PNC}^{\text{calc}}$ is the calculated quantity that corresponds with
experimental quantity $\text{PNC}^{\text{exp}}$ below in Eq. 20.

Analogously, we define the time-integrated occupation of the one-photon number
states as

$\text{occ}^{\text{calc}}\propto\tilde{\rho}_{1,1}=\int\rho^{\text{photon}}_{1_{H},1_{H}}dt\,.$
(17)

We assume that the photonic space can be reduced to a two-level system
consisting of $|0_{H}\rangle$ and $|1_{H}\rangle$. This is reasonable because
the higher-order Fock states are not occupied. We then follow Ref. [28] to
calculate the visibility $v$ as measure in a MZI for a mixed state as

$v^{\text{calc}}=\frac{\tilde{\rho}_{0,1}^{2}}{\tilde{\rho}_{1,1}}\,.$ (18)

We stress that this is an estimate of the visibility, which does not account
for the imperfection of the beam splitter, higher photon states, phase
scrambling, or reduced indistinguishability. Nonetheless, we expect the
qualitative behaviour to agree with the experiment.

###  IV.2 Experimental setup

Our setup (Fig. 2d) consists of a Ti:Sapphire laser source (Tsunami 3950,
SpectraPhysics) producing $2.7\text{\,}\mathrm{p}\mathrm{s}$ pulses (measured
as intensity autocorrelation FWHM), that is tuned to
$793\text{\,}\mathrm{n}\mathrm{m}$, enabling spectral shaping of both the TPE
and stimulating (stim). pulses via two independent 4$f$ pulse shapers. The
intensities of the TPE and stim pulses are individually controlled via
electronic variable optical attenuators (VOA, V800PA, Thorlabs) and the
arrival time of the stimulating pulse is precisely controlled via a fiber
optic delay line (ODL-300, OZ Optics). The two beams are combined at a 10:90
beamsplitter near the optical window of a closed-cycle cryostat (base
temperature $1.5\text{\,}\mathrm{K}$, ICEOxford) where the quantum dot sample
is mounted on a three-axis piezoelectric stage (ANPx101/ANPz102, attocube
systems AG). The two beams are focused on a single quantum dot with a cold
objective (numerical aperture 0.81, attocube systems AG).

Our sample consists of GaAs/AlGaAs quantum dots with exciton emission centered
around 790 nm grown by the Al-droplet etching method [59, 60]. The dots are
embedded in the center of a lambda-cavity placed between a bottom (top)
distributed Bragg reflector consisting of 9 (2) pairs of $\lambda/4$ thick
Al0.95Ga0.05As/Al0.2Ga0.8As layers.

The quantum dot emission is collected via the same path as the excitation,
where the exciton (X) photons are spectrally separated from the scattered
laser light and phonon side-bands using a home-built monochromator equipped
with two narrow-band notch filters (BNF-805-OD3, FWHM
$0.3\text{\,}\mathrm{nm}$, Optigrate). To improve the suppression of the
reflected TPE pulse we employ a cross-polarized configuration in which two
orthogonal linear polarizers on excitation and collection paths block any
residual laser scattering. In fact, this would not be necessary for a
sufficiently narrow laser spectrum, as the TPE energy is detuned from the
exciton energy. To measure the spectra, collected photons are routed to a
single-photon sensitive spectrometer (Acton SP-2750, Roper Scientific)
equipped with a liquid Nitrogen cooled charge-coupled device camera (Spec10
CCD, Princeton Instruments). For lifetime measurements, we use an avalanche
photodiode (SPAD, Micro Photon Device) together with time-tagging electronics.

Phase scan HOM setup: To measure the indistinguishability, the filtered X
photons are sent through a Mach-Zehnder Interferometer (MZI) with a path-
length difference of $12.5\text{\,}\mathrm{ns}$, to interfere with
successively emitted photons from the quantum dot in a 50:50 fiber beam
splitter (TW805R5A2, Thorlabs) for HOM measurement. The two output ports of
the fiber beam splitter are monitored by avalanche photodiodes (SPCM-NIR,
Excelitas). The arrival times of the photons are recorded using a time tagger
(Time Tagger Ultra, Swabian Instruments), and coincidence counting is employed
to determine the correlation between the photons. In the HOM measurement, the
polarization in both MZI arms is controlled individually, enabling a
comparison between the co-polarized scenario with maximum indistinguishability
and the cross-polarized situation with distinguishable photons to obtain the
HOM visibility.

For PNC measurements, a phase shifter is placed into one of the arms of the
unbalanced MZI. The phase shifter consists of a motorized rotation stage
(ELL14K, Thorlabs) holding a half-wave plate positioned between two quarter-
wave plates that are oriented orthogonally with respect to each other’s fast
axis. This arrangement effectively acts as a variable phase shifter for
linearly polarised input light since:

$\begin{split}\mathrm{J(\theta)}&=\mathrm{QWP\left(\frac{\pi}{4}\right)}\cdot\mathrm{HWP\left(\theta\right)}\cdot\mathrm{QWP\left(-\frac{\pi}{4}\right)}\\\
&=-\frac{i}{2}\begin{bmatrix}1&-i\\\
-i&1\end{bmatrix}\begin{bmatrix}\cos^{2}\theta-\sin^{2}\theta&2\sin\theta\cos\theta\\\
2\sin\theta\cos\theta&\sin^{2}\theta-\cos^{2}\theta\end{bmatrix}\begin{bmatrix}1&i\\\
i&1\end{bmatrix}\\\ &=\begin{bmatrix}0&e^{-i2\theta}\\\
-e^{i2\theta}&0\end{bmatrix}.\end{split}$ (19)

Here $\theta$ is the orientation of the fast axis of the half-wave plate. By
rotating the half-wave plate at a fixed speed, the phase in one of the arms is
varied continuously without changing the polarization, while the phase in the
other arm remains constant on the timescale of the rotation. The two arms are
then recombined at the fiber beam splitter, where the interference occurs and
photons are directed towards two separate single-photon detectors. The
matching of the timing and relative polarization of the two arms was ensured
by interfering the excitation laser with itself and maximizing the contrast,
which yielded a visibility of 98$\,\%$.

###  IV.3 Extraction of the PNC from data

We follow Ref. [28] to compute the PNC from the visibility. We remind that we
only consider the $H$-polarized photons and stay in the approximation of the
two-level system composed of the $|0\rangle$ and $|1\rangle$ Fock state. From
the detector counts, we obtain the visibility $v^{\text{exp}}$, which is
proportional to the occupation $\rho_{0,0}$. In the next step, we decompose
the density matrix
$\rho=\lambda\rho_{\text{pure}}+(1-\lambda)\rho_{\text{mixed}}$ into a part
corresponding to a pure state and a part being a statistical mixture with the
off-diagonal elements being zero. Note that we are only interested in the
absolute value of the coherence and not in its phase. Following Ref.[28], the
visibility can be approximated by
$v\approx\lambda^{2}\rho_{0,0}\sqrt{V_{\text{HOM}}}$ with $0\leq\lambda\leq 1$
and $V_{\text{HOM}}$ being the photon indistinguishability. Considering the
slope of the visibility as a function of $\rho_{0,0}=(1-\rho_{1,1})$ allows us
to extract $\lambda$ (see SI Section F). Together with the knowledge of
$\rho_{1,1}$ via the photon counts, we can estimate the PNC as

$\text{PNC}^{\text{exp.}}=\lambda\sqrt{\rho_{1,1}(1-\rho_{1,1})}\,.$ (20)

##  V Acknowledgements

The authors gratefully acknowledge insightful discussions with Stefan Frick,
Robert Keil, Tommaso Faleo, Mathieu Bozzio and Serkan Ates. Nils Kewitz and
Bhavana Panchumarthi supported the early phases of the experiment. YK, FK, RS,
VR and GW acknowledge financial support through the Austrian Science Fund FWF
projects W1259 (DK-ALM Atoms, Light, and Molecules), FG 5, TAI-556N (DarkEneT)
and I4380 (AEQuDot). DAV and TH acknowledge financial support by the German
Federal Ministry of Education and Research (BMBF) via projects 13N14876
(‘QuSecure’) and 16KISQ087K (tubLAN Q.0). TKB and DER acknowledge financial
support from the German Research Foundation DFG through project 428026575
(AEQuDot). A.R. and SFCdS acknowledge the FWF projects FG 5, P 30459, I 4320,
the Linz Institute of Technology (LIT) and the European Union’s Horizon 2020
research, and innovation program under Grant Agreement Nos. 899814 (Qurope),
871130 (ASCENT+) and the QauntERA II Programme (project QD-E-QKD). LMH, PW and
JCL acknowledge financial support from the European Union’s Horizon 2020 and
Horizon Europe research and innovation programme under grant agreement No
899368 (EPIQUS), the Marie Skłodowska-Curie grant agreement No 956071
(AppQInfo), and the QuantERA II Programme under Grant Agreement No 101017733
(PhoMemtor); FWF through F7113 (BeyondC), and FG5 (Research Group 5); from the
Austrian Federal Ministry for Digital and Economic Affairs, the National
Foundation for Research, Technology and Development and the Christian Doppler
Research Association. For the purpose of open access, the author has applied a
CC BY public copyright licence to any Author Accepted Manuscript version
arising from this submission.

##  VI Author contributions

The experimental setup was built by Y.K., F.K., R.S., V.R., J.C.L., D.A.V.,
L.M.H., and the measurements were performed by Y.K., F.K., D.A.V. The
numerical calculations were done by P.C.A.H. The sample was provided by C.S.,
S.F.CdS, A.R. The first draft of the manuscript was written by D.A.V., F.K.,
Y.K., P.C.A.H., V.R., D.E.R.. Conceptual work and supervision was done by
G.W., P.W., V.M.A, D.E.R., V.R., J.C.L., T.H., A.R.. All authors discussed the
results and were involved in writing the manuscript.

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Supplementary information

##  Appendix A Quantum Dot Sample

The sample used contains GaAs/AlGaAs quantum dots (QDs) obtained by the Al-
droplet etching method [60] and was grown by molecular beam epitaxy. The QD
are embedded in the center of a $\lambda$-cavity placed between a bottom(top)
distributed Bragg reflector consisting of 9(2) pairs of $\lambda/4$-thick
$Al_{0.95}Ga_{0.05}As/Al_{0.20}Ga_{0.80}As$layers with respective thickness of
69/60 nm. The QDs are placed between to $\lambda/2$-thick
$Al_{0.33}Ga_{0.67}As$ layers. The QD growth process starts by depositing 0.5
equivalent monolayers of Al in the absence of arsenic flux, which results in
the self-assembled formation of droplets. During exposure to a reduced As
flux, such droplets locally etch the underlying $Al_{0.33}Ga_{0.67}As$ layer,
resulting in $\approx$$9\text{\,}\mathrm{nm}$-deep and
$\approx$$60\text{\,}\mathrm{nm}$ wide nanoholes on the surface. Then the
nanoholes are filled with GaAs by depositing
$\approx$$1.1\text{\,}\mathrm{nm}$ of GaAs on the surface, followed by an
annealing step of $45\text{\,}\mathrm{s}$. The temperature used for the
etching of the nanoholes was $600\text{\,}\mathrm{\SIUnitSymbolCelsius}$. The
droplet self-assembly process results in QDs with random position and a
surface density of about $2\times 10^{7}${\mathrm{cm}}^{-2}$$, suitable for
single QD spectroscopy.

##  Appendix B stiX characterization and optimization

To confirm that the collected photons indeed originate from stiX, several
characterisation experiments are done, as presented in the following.

Initially, we vary the TPE power, observing Rabi rotations to determine the
TPE $\pi$ power. While maintaining the TPE pulses at this power, we introduce
the stim. pulse tuned to the XX energy and scan their relative time delay (see
Methods in the main manuscript) from negative (i.e., stim. pulse arrives
before the TPE pulse) to positive while recording the generated X photons.
Optimum stiX is achieved when the stim. pulse arrives just after the TPE pulse
has maximized the XX state occupation. Arriving earlier will not stimulate the
de-excitation and a stim. pulse that arrives too late will allow the XX to
already decay spontaneously. This explains the observed dependence of the X
emission on the temporal delay between TPE and stim. pulses, a sharp increase
for short delays and an exponential reduction of the stimulation effect that
follows the XX lifetime. In Ref. 50 the optimum delay is about
$0.03\times\tau_{XX}\approx 5\,\text{ps}$ which matches our observations (Fig.
5d). The red (stim. pulse has H polarization), green (stim pulse has V
polarization) and blue (no stim. pulse) dots represent the different
polarization cases respectively. At time delay
$\approx$$7\text{\,}\mathrm{ps}$ we observe nearly two times the photon counts
compared to the reX case for an H-polarized stim. pulse, confirming the
successful stimulation. Note that, when we stimulate the V-polarization
cascade, we are effectively suppressing the collected decay pathway, which
shows as a drop in photon counts (see Fig.S5d, green dots).

As the stim. pulse removes the timing jitter from the exciton emission caused
by the biexciton decay, one can also observe the effect in the X photon
arrival time distribution (Fig. 5f). The standard reX exciton emission (blue
line) exhibits an exponential rise time that corresponds to the XX decay time
characteristic of the cascaded emission, before decaying exponentially. When
adding the stim. pulse (red line) the timing jitter due to the random
biexciton decay is removed and the rise time of the exciton vanishes. The
exciton now decays right after the arrival of the stim. pulse.

Once the time delay of the stim. pulse is optimized, one can turn to the
polarization control. To achieve maximum enhancement, the polarization of the
stimulation pulse that controls the emitted X polarization must match the
polarization that is collected, while the orthogonal polarization leads to
suppression. This is confirmed by rotating the stim. pulse polarization at
optimum delay and power by the means of an HWP (figure 5b). In reX, the
polarization of the excitation pulse does not have an effect on the X emission
polarization, as any linear polarization would generate the H and V cascades
to the ground state. However, with the stim. pulse polarization, one can
direct the emission cascade to arbitrary ratios of $X_{H}$ or $X_{V}$. We
observe an oscillatory trend with respect to the HWP orientation, signifying
that the $X_{H}$ intensity can be controlled via stiX polarization. Note that,
while the enhancement increases the emission by a factor of 2, the suppression
does not reduce the counts completely. This is also observed in other works on
stiX [52] and can be attributed to an imperfect preparation fidelity of the
reX, non-ideal stim. pulse polarization control as well as anisotropies in the
quantum dot that cause the polarization axes to not be fully orthogonal, such
that the stim. pulse also couples with the other polarization axis [61, 62].

Next, we investigate the power dependence of the stim. pulse by setting the
TPE pulse to $\pi$-power and sweeping the stim. pulse power (figure 5e). This
is again done for the stim. pulse with horizontal (enhancement, red dots) and
vertical (suppression, green dots) polarization. The coherent excitation and
de-excitation of the quantum dot states is evidenced by the presence of Rabi
rotations as a function of stim. pulse power consistent with previous works
[50, 52, 51]. While adding the stim. pulse depopulates the biexciton to
enhance or suppress the exciton emission in one polarization, higher TPE
powers can re-populate the biexciton state which compensates the effect of the
stim. pulse. Thus, the emission also oscillates as a function of stim. pulse
pump area.

Finally, we can quantify the photon enhancement at this optimal stiX
condition. At first, we switch off the stim. pulse, and record the X photon
counts upon TPE pulse power sweep, to observe the reX Rabi oscillations (Fig.
5c, blue dots). At $\pi$-power we observe $\approx$2600 counts. We now
introduce the stim. pulse, with its polarization set to H, and perform the TPE
power sweep. The red dots in Fig. 5c represent the recorded photon counts
which, at $\pi$-power reach $\approx$5200 counts, clearly demonstrating the
expected two-fold enhancement under optimal stiX conditions. Furthermore, we
observe that once we switch to orthogonal polarization of the stim. pulse, the
recorded photon counts are minimal, which represent the residual photon
emission via the V-cascade (Fig. 5c, green dots).

Finally, the hallmark and main motivation of the stiX scheme is the improved
indistinguishability that was already shown in the main text. In summary, all
these observations consistently confirm the successful realization of the stiX
scheme with two pulses spectrally cut from the same ps-laser.

Figure 5: stiX characterization: (a) Pulse shaping and detuning during the
experiment. TPE (light orange) and stim. (dark orange) pulses are spectrally
cut from an initial picosecond pulse (dashed orange line). (b) Integrated
$X_{H}$ counts as a function of the stimulation pulse polarization tuned via
an HWP (at optimal delay). (c) Measured two-fold $X_{H}$ photon enhancement
(red dots) under optimal stiX conditions of time delay, power and
polarization. The green dots represent the case where we suppress the
collected polarisation. Blue dots represent the reX case. (d) Integrated
$X_{H}$ photon counts as a function of the time separation of stim. pulse,
following the TPE pulse (red dots). Blue dots represent the $X_{H}$ photon
counts under reX. Green dots represent the drop in photon counts for a
vertically polarized stim. pulse. (e) Integrated $X_{H}$ counts for varying
stim. pulse power while keeping the TPE pulse at $\pi$ power. Red dots
represent the H cascade, which is enhanced, while green dots represent the
V-polarized cascade, which is suppressed. (f) Emission decay dynamics under
stiX (red) compared to reX (blue) in a lifetime measurement.

##  Appendix C Photon quality

In Table 2 we summarize the single-photon purities and indistinguishabilities
measured under various excitation conditions of reX and stiX methods. To
compute the $g^{(2)}(0)$, we first fit the recorded photon coincidences for a
time interval of $8\text{\,}\mathrm{ns}$ around zero delay with a Gaussian
function, and also the four side peaks (corresponding to time delays
$\pm$12.5\text{\,}\mathrm{ns}$$ and $\pm$25\text{\,}\mathrm{ns}$$) to then
calculate the ratio of the extracted areas. For calculating HOM visibility, we
again fit the recorded photon coincidences for a time interval of
$6\text{\,}\mathrm{ns}$ at zero delay for parallel and orthogonal
configurations and compute the ratio of the extracted areas.

Table 2: Single-photon purity, measured as $g^{(2)}(0)$, and indistinguishability (HOM Visibility $\mathrm{V_{HOM}}$) under various excitation pulse powers of reX, stiX and s-shell excitation. Uncertainties are extracted from the fitting procedure. Protocol | TPE power | stim. power | $g^{(2)}(0)$ | $\mathrm{V_{HOM}}$
---|---|---|---|---
reX | $\pi$ | - | 0.0004(1) | 0.58(3)
stiX | 0.25$\pi$ | $\pi$ | - | 0.85(3)
stiX | 0.5$\pi$ | $\pi$ | - | 0.73(5)
stiX | $\pi$ | $\pi$ | 0.0009(1) | 0.95(6)
stiX | 2$\pi$ | $\pi$ | 0.0003(1) | -
s-shell | - | - | 0.04(1) | 0.88(3)

##  Appendix D Quantum Efficiency

The blinking behavior of the quantum dot varies depending on the type of
excitation. In Fig. 6(a)-(c) we present the behaviour under resonant
excitation, reX and stiX of the exciton state, on the same quantum dot, as
measured under a long timescale $g^{(2)}(0)$, denoted here as
$g^{(2)}_{\text{LT}}(0)$ [63, 64]. We fit a symmetric exponential function
$g^{(2)}(\tau)=\text{A}*\exp(-|\tau/\tau_{\text{blinking}}|)+\text{B}$ (black
solid lines) to extract a blinking timescale $\tau_{\text{blinking}}$ and
$g^{(2)}_{\text{LT}}(0)=\text{A}+\text{B}$ . The extracted parameters are
listed in Table 3.

$\text{QE}=\frac{t_{\mathrm{on}}}{t_{\mathrm{off}}+t_{\mathrm{on}}}=\frac{1}{g^{(2)}_{\text{LT}}(0)}$
(21)

Figure 6: Quantum efficiency: Blinking behaviour of the quantum dot under different excitation schemes, as measured by long timescale $g^{(2)}(\tau)$. (a): resonant s-shell excitation, (b): stiX, and (c): reX . Black curve denote the fit. Table 3: Blinking behaviour analysis for different excitation techniques. Uncertainties are extracted from the fitting procedure. Protocol |  $\tau_{\text{blinking}}$ / $\mathrm{ms}$ | $g^{(2)}_{\text{LT}}(0)$ | QE
---|---|---|---
s-shell | 1.601(6) | 2.865(5) | 0.349(1)
reX | 0.0350(3) | 2.247(7) | 0.445(1)
stiX | 0.0326(3) | 2.184(7) | 0.458(1)

We calculate that under resonant s-shell excitation, the quantum efficiency of
the quantum dots is $\approx 0.35$. By introducing reX and stiX, we are able
to achieve higher quantum efficiencies of $\approx 0.45$ and $\approx 0.46$,
respectively.

##  Appendix E Time control of PNC

Here, we investigate the control of PNC further using the stiX scheme
parameters. Initially, we scan the time delay of the stim. pulse and record
the $X_{\text{H}}$ counts for various TPE pulse powers. The resulting two-
dimensional map of the recorded photon counts of $X_{H}$ photons as a function
of TPE power and stim. pulse time delay is displayed in Figure 7b. We observe
that the photon counts reach the expected two-fold enhancement (compared to
reX case, c.f. Figure 5c) at a time delay $\approx$$7\text{\,}\mathrm{ps}$
when TPE pulse power reaches $\approx$1$\pi$. Subsequently, we perform the PNC
experiment (see Methods) at every time delay and TPE power, and extract the
visibilities (same as in Figure 4 in the main manuscript). We then compute the
PNC (note that $V_{\text{HOM}}$ is assumed constant for all time delays) using
the same procedure as explained in Section IV.3 and F. The resulting two-
dimensional map of the computed PNC values for various time delays and TPE
powers is displayed in Fig. 7d.

To support our observations, we perform additional simulations on the mean
expected photons per excitation cycle ($X_{H}$) and the PNC. For computational
simplicity, omit the two photon modes and the coupling to them, such that the
Hamiltonian (4) reduces to
$\hat{H}=\hat{H}^{\text{QD}}+\hat{H}^{\text{TPE}}+\hat{H}^{\text{stim}}+\hat{H}^{\text{phonon}}$
. This is motivated by the agreement between the PNC and the quantum dot
coherence (see Fig. 2b and Fig. 3e) by time-integrating the respective
elements and normalizing by the decay rate $\gamma_{X}$, i.e.:

$\begin{split}X_{H}&\approx\gamma_{X}\int|\rho_{X_{H},X_{H}}|dt\\\
\text{PNC}&\approx\frac{\gamma_{X}}{2}\int|\rho_{g,X_{H}}|dt.\end{split}$ (22)

The results are displayed in Fig. 7a and 7c, and show an excellent match with
our experimental data.

Figure 7: stiX and PNC temporal behaviour: (a) Theory: Mean photon number per
excitation cycle depending on TPE pulse power and time seperation between TPE
and stim pulses ($\Delta t$). Every column represents a TPE pulse power sweep
and each row the corresponding stiX time delay scan, with stiX power set to
$\pi$. (b)Experiment: Integrated X photon counts for the same parameters as in
(a). (c) Theory: Simulated PNC of stiX. (d) Experiment: Qualitative (e.g.
$V_{HOM}$ is constant for all $\Delta t$) extraction of PNC from the
measurement.

##  Appendix F Estimation of $\lambda$ for PNC extraction from visibility

We compute PNC based on Equation (20), which depends on $\lambda$, a quantity
that signifies the purity of the state. Following the method in Ref. 28 we
extract this parameter by fitting the measured visibility $v$ (c.f. Figure 4
middle panel), as

$v\approx\lambda^{2}\rho_{0,0}\sqrt{V_{\text{HOM}}}+v_{0}.$ (23)

Here $V_{\text{HOM}}$ is the measured single-photon indistinguishability,
$v_{0}$ a residual visibility at $\pi$ power and $\rho_{0,0}$ is approximated
by $\rho_{0,0}\approx 1-\text{N}$ with N being the normalized photon counts
measured while sweeping the TPE pulse power. This approximation is valid since
the system does not include decay channels that do not end in the ground state
and we treat the photon density matrix in a two-dimensional subspace. Fitting
equation (23) we obtain $\lambda_{\text{stiX}}=0.73(3)$ for stiX (displayed as
red circles and squares, respectively indicating the measured visibilities
from two detectors) and $\lambda_{\text{reX}}=0.28(2)$ for reX. The results
are displayed in Fig. 8.

Figure 8: Computing $\lambda$: Squares (circles) represent the measured
visibilities $v_{1(2)}$ measured at APD1(2) respectively. Red and blue denote
stiX and reX processes. Dashed lines are linear fits according to equation 23.