SReferences

††thanks: Present address: Chimie ParisTech, Université PSL, CNRS, Institut de
Recherche de Chimie Paris, 75005 Paris, France††thanks: Email to:
<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Violation of Bell inequality by photon scattering on a two-level emitter

Shikai Liu  Center for Hybrid Quantum Networks (Hy-Q), The Niels Bohr
Institute, University of Copenhagen, DK-2100 Copenhagen Ø, Denmark  Oliver
August Dall’Alba Sandberg  Center for Hybrid Quantum Networks (Hy-Q), The
Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen Ø, Denmark
Ming Lai Chan  Center for Hybrid Quantum Networks (Hy-Q), The Niels Bohr
Institute, University of Copenhagen, DK-2100 Copenhagen Ø, Denmark  Björn
Schrinski  Center for Hybrid Quantum Networks (Hy-Q), The Niels Bohr
Institute, University of Copenhagen, DK-2100 Copenhagen Ø, Denmark  Yiouli
Anyfantaki  Center for Hybrid Quantum Networks (Hy-Q), The Niels Bohr
Institute, University of Copenhagen, DK-2100 Copenhagen Ø, Denmark  Rasmus
Bruhn Nielsen  Center for Hybrid Quantum Networks (Hy-Q), The Niels Bohr
Institute, University of Copenhagen, DK-2100 Copenhagen Ø, Denmark  Robert
Garbecht Larsen  Center for Hybrid Quantum Networks (Hy-Q), The Niels Bohr
Institute, University of Copenhagen, DK-2100 Copenhagen Ø, Denmark  Andrei
Skalkin  Center for Hybrid Quantum Networks (Hy-Q), The Niels Bohr Institute,
University of Copenhagen, DK-2100 Copenhagen Ø, Denmark  Ying Wang  Center for
Hybrid Quantum Networks (Hy-Q), The Niels Bohr Institute, University of
Copenhagen, DK-2100 Copenhagen Ø, Denmark  Leonardo Midolo  Center for Hybrid
Quantum Networks (Hy-Q), The Niels Bohr Institute, University of Copenhagen,
DK-2100 Copenhagen Ø, Denmark  Sven Scholz  Lehrstuhl für Angewandte
Festörperphysik, Ruhr-Universität Bochum, Universitätsstraße 150, 44801
Bochum, Germany  Andreas Dirk Wieck  Lehrstuhl für Angewandte Festörperphysik,
Ruhr-Universität Bochum, Universitätsstraße 150, 44801 Bochum, Germany  Arne
Ludwig  Lehrstuhl für Angewandte Festörperphysik, Ruhr-Universität Bochum,
Universitätsstraße 150, 44801 Bochum, Germany  Anders Søndberg Sørensen
Center for Hybrid Quantum Networks (Hy-Q), The Niels Bohr Institute,
University of Copenhagen, DK-2100 Copenhagen Ø, Denmark  Alexey Tiranov†
Center for Hybrid Quantum Networks (Hy-Q), The Niels Bohr Institute,
University of Copenhagen, DK-2100 Copenhagen Ø, Denmark  Peter Lodahl  Center
for Hybrid Quantum Networks (Hy-Q), The Niels Bohr Institute, University of
Copenhagen, DK-2100 Copenhagen Ø, Denmark

###### Abstract

Entanglement, the non-local correlations present in multipartite quantum
systems, is a curious feature of quantum mechanics and the fuel of quantum
technology. It is therefore a major priority to develop energy-conserving and
simple methods for generating high-fidelity entangled states. In the case of
light, entanglement can be realized by interactions with matter, although the
required nonlinear interaction is typically weak, thereby limiting its
applicability. Here, we show how a single two-level emitter deterministically
coupled to light in a nanophotonic waveguide is used to realize genuine
photonic quantum entanglement for excitation at the single photon level. By
virtue of the efficient optical coupling, two-photon interactions are strongly
mediated by the emitter realizing a giant nonlinearity that leads to
entanglement. We experimentally generate and verify energy-time entanglement
by violating a Bell inequality (Clauder-Horne-Shimony-Holt Bell parameter of
$S=2.67(16)>2$) in an interferometric measurement of the two-photon scattering
response. As an attractive feature of this approach, the two-level emitter
acts as a passive scatterer initially prepared in the ground state, i.e., no
advanced spin control is required. This experiment is a fundamental
advancement that may pave a new route for ultra-low energy-consuming synthesis
of photonic entangled states for quantum simulators or metrology.

Figure 1:  (color online) Two-photon energy-time entanglement induced by
coherent interaction of two photons with a quantum dot (QD) integrated into a
photonic crystal waveguide (PhC WG). (a) Operational principle of the photon
scattering and entanglement processes. A single-photon wave packet is
predominantly reflected by elastic scattering on a two-level emitter, while
the two-photon wave packet can be inelastically scattered in the forward
direction, thereby generating energy-time entanglement. The entanglement is
probed using two unbalanced Mach–Zehnder interferometers (UMZIs), see Figure
2. Each UMZI is used to realize time projections onto the superposition state
$\frac{\ket{s}+e^{i\phi}\ket{l}}{\sqrt{2}}$, as illustrated on a Bloch sphere,
where $\ket{s}$ ($\ket{l}$) corresponds to a photon taking the short (long)
path, and $\phi$ is the phase setting of the UMZI. $\phi=0$ (blue vector) and
$\phi=\pi$ (red vector) refer to the two settings for data in (b). (b)
Transmission intensity measurements through the PhC WG and one UMZI versus QD
detuning. The blue curve indicates the transmission dip by resonant scattering
of a weak coherent state $\ket{\alpha}$. Suppression of the elastically
scattered laser photons by a destructive interference phase ($\phi=\pi$)
reveals inelastically scattered photons $\ket{I}$ (red). (c) Measured
normalized second-order correlation function $g^{(2)}(\tau)$ of the light
transmitted through the PhC WG on resonance with the QD, reaching values above
200, a signature of strong bunching induced by the scattering. (d) Schematic
of the QD-embedded PhC WG structure with two mode adaptors, including shallow-
etched gratings (SEGs) and nanobeam access waveguides to the photonic crystal
section. (e) Calculated normalized joint spectral intensity for a laser
linewidth of $\Gamma_{L}/2\pi=100$ kHz, a Purcell enhanced QD linewidth of
$\Gamma/2\pi=2.3$ GHz, and assuming ideal coupling of the QD to the PhC WG.
$\Delta_{a(b)}$ is the frequency difference between the output photon
($a\text{ or }b$) and the input laser:
$\Delta_{a(b)}=\omega_{a(b)}-\omega_{p}$ . The width of the biphoton spectrum
is determined by the laser linewidth $\Gamma_{L}$, while each photon is
broadened by the QD linewidth $\Gamma$. Upper-right insert: the corresponding
spectra of the input coherent state $\ket{\alpha}$ (blue) and the output
biphoton state $\ket{2_{I}}$ (red), respectively. Bottom-left insert: enlarged
joint spectral intensity spanning a range of 1 MHz. (f) The corresponding
normalized joint temporal intensity. The biphoton correlation time is
determined by the QD lifetime $\tau_{QD}$, cf. Supplementary Note 5 for
characterization.

A quantum pulse of light interacting with a two-level emitter, see Figure
1(a,d), constitutes a new experimental paradigm in quantum optics [1, 2].
Despite its conceptual simplicity, significant quantum complexity can be
encoded in the system since a quantum pulse represents an infinitely large
(continuous) Hilbert space. From an experimental point of view, this is an
attractive setting since a two-level emitter can implement a highly nonlinear
operation on the incoming pulse without the need for demanding and error-
susceptible emitter preparation schemes. To enhance this photon-photon
nonlinearity, the main experimental challenge is to promote the radiative
coupling of the emitter such that it dominates deteriorating decoherence
processes – such advancements have been made in the past decades using
semiconductor quantum dots (QDs) in photonic crystal waveguides (PhC WGs) and
cavities [3]; see illustration of a PhC WG device in Figure 1(d).

It is an exciting ongoing research endeavor to exploit the properties and
applicability of the quantum nonlinear response of a two-level emitter. The
essential nonlinear operation has been explored using various quantum emitters
such as QDs [4], color centers in diamond [5], atoms [6], and single molecules
[7]. Further experimental advancements include quadrature squeezing of light
[8, 9], two-photon correlation dynamics and photonic bound states [10, 11].
Theoretical proposals include photon sorters for deterministic Bell state
analyzers [12], quantum logic gates [13, 14], and single-photon transistors
[15]. Furthermore, many-body waveguide quantum electrodynamics may be pushed
to new realms of strongly correlated light and matter [16, 17]. While it was
theoretically predicted that the two-level nonlinear response can induce
photon-photon correlations [1], whether this nonlinearity can lead to photon
entanglement has never been truly verified experimentally.

Here, we demonstrate that a two-level quantum emitter coherently coupled to a
PhC WG can induce strong energy-time entanglement between two scattered
photons, see Figure 1(a), sufficient for violating a Bell inequality and thus
local realism under the fair sampling assumption. The experiment couples a
continuous wave (incoming light) and a discrete quantum system (emitter),
offering a route to non-Gaussian photonic operations – a type of operation
vigorously searched for in continuous-variable quantum computing architectures
[18]. Previous research on entanglement generation with quantum emitters
exploited the strong excitation (Mollow) regime in bulk samples [19] or the QD
biexciton radiative cascade [20, 21, 22]. To the best of our knowledge, no
such studies have yet answered the above open question and proved
experimentally that passive scattering of weak fields from a two-level QD in a
PhC WG can induce genuine entanglement. This work opens a conceptually new and
advantageous route to energy-time entanglement generation that may be an
attractive alternative to four-wave mixing sources [23] since it operates at
the ultra-low energy consumption level of single photons and no complex and
decoherence-sensitive pumping schemes are required.

Figure 2:  (color online) Experimental setup and characterization of two-
photon energy-time entanglement. (a) Time correlation histograms of
coincidence counts for constructive (blue, $\phi_{a}+\phi_{b}=0$) and
destructive (red, $\phi_{a}+\phi_{b}=\pi$) interference between two photons
traversing the short ($s$) and long ($l$) arms of the UMZIs, respectively. (b)
Experimental setup including the PhC WG chip (light green area), spectral
filter (light orange area), and the Franson interferometer (light blue areas).
The two Bloch spheres illustrate the two independently controlled phases
$\phi_{a}$ and $\phi_{b}$. LP: linear polarizer; HWP: half-wave plate; QWP:
quarter-wave plate; SNSPD: superconducting nanowire single-photon detector;
FC: fiber collimator; BS: beam splitter; PBS: polarizing beam splitter; PC,
polarization controller; FBS, fiber beam splitter; PD: photodiode. In order to
control the phase difference between the two interferometer arms, we actively
stabilize the UMZIs with a PID module locked by the same laser that excites
the QD. (c) 2D correlation histogram of coincidence counts versus phase and
time delay.

We consider a single two-level emitter deterministically coupled to a single
propagating spatial mode in a PhC WG, cf. Figure 1(d). A weak coherent input
field is launched into the PhC WG and interacts with the emitter of coupling
efficiency $\beta=\frac{\gamma}{\Gamma}$ governed by the ratio between the
radiative decay rate into the waveguide mode $\gamma$ and the QD total decay
rate $\Gamma$ [24]. For $\beta=1$ with no decoherence processes, the single-
photon component is elastically reflected via interaction with the emitter,
while the two-photon component can be inelastically scattered into the forward
(transmission) mode, leading to energy exchange and photon bunching in time
[1, 15], as shown in Figure 1(a). The latter process is analogous to
degenerate four-wave mixing with a Kerr nonlinearity [23]. Experimentally, the
two-photon scattering process is studied in a Franson interferometer [25] with
time-resolved photon correlation measurements, see Figure 2. In this way, a
Clauder-Horn-Shimony-Holt (CHSH) Bell inequality entanglement criterion can be
tested where a Bell parameter of $S=2$ constitutes the locality bound [26].
Various experimental imperfections influence $S$, including the finite photon-
emitter coupling efficiency ($\beta$-factor), pure dephasing rate
($\gamma_{d}$ relative to the emitter linewidth $\Gamma$), and the strength of
the incoming light (mean photon number within the emitter lifetime $n$). These
imperfections result in a single-photon component (elastic scattering) that is
not fully reflected thereby reducing $S$. We found that $S$ is sensitive to
$\gamma_{d}$, $\Gamma$ and $n$ to first-order but is remarkably robust to
coupling loss, with a quartic dependence in the limit of $\beta\to 1$:

$\displaystyle S(\beta)\approx 2\sqrt{2}\left[1-(1-\beta)^{4}\right].$ (1)

The complete theory is presented in Supplementary Note 7, where the
experimental requirements for violating the Bell inequality are also
benchmarked in detail (Supplementary Figure S6).

FIGs. 1(a) and (d) show the conceptual scheme of the experiment. We use a
narrow linewidth continuous-wave laser as a weak coherent input state
$\ket{\alpha}$. The light is coupled in and out of the PhC WG via two mode
adaptors with the sample mounted in a cryostat operating at 4 K. Due to high
coupling efficiency of the photon-emitter interface, the photon state is
strongly modified by the nonlinear interaction, which induces energy-time
entanglement involving a continuum of optical modes in multidimensional
Hilbert space. The photon pairs generated in different temporal modes can
subsequently be analyzed in a Franson interferometer [25]. The present
experiment is conducted in the regime of weak resonant excitation, i.e., far
below saturation threshold of the emitter.

Figure 3:  (color online) Two-photon Franson interference measurements and
observation of a violation of the CHSH inequality. (a) Interference curves as
a function of $\phi_{b}$ with $\phi_{a}$ fixed at 0 (red) and $\pi/4$ (blue).
The data are modeled with a sinusoidal model whereby a visibility of 95(4)% is
extracted. (b) Measured correlation functions from which $S=2.67(16)$ is
recorded. (c) $S$ parameter versus $n$ (bottom x-axis) or the corresponding
pump power in the PhC WG (top x-axis). $n$ and pump powers are calibrated by
fitting the full set of transmission intensity data (Supplementary Note 2).
The solid curve is the theoretical model (Supplementary Note 7.5) with
parameters taken from the filtered power saturation $g^{(2)}$ measurements in
Supplementary Note 3, i.e., no additional fitting was performed. The black
dashed line represents the locality bound. The data in (a) and (b) are
recorded for the lowest $n$ (0.0024) or pump power (7.2 pW) in (c). The error
bars show the standard deviations assuming Poissonian distribution for the
photon statistics.

The quantum correlations induced by the nonlinear scattering are illustrated
by the two-photon joint spectral and temporal intensity distributions in FIGs.
1(e)-(f). The input weak coherent state $\ket{\alpha}$ resembles a Dirac delta
function in frequency, while the output entangled photon pair $\ket{2_{I}}$ is
Lorentzian broadened by the QD linewidth $\Gamma$. It can be expressed as
$\ket{2_{I}}=\frac{1}{2}\int{d\Delta}\mathcal{T}_{\Delta,-\Delta}\ket{1_{\Delta}}\ket{1_{-\Delta}}$
(Supplementary Note 6), where $\Delta=\Delta_{a}=-\Delta_{b}$ is the frequency
detuning of each outgoing photon relative to the pump frequency, and
$\mathcal{T}_{\Delta,-\Delta}=-4\beta^{2}/[\pi\Gamma(1+4\frac{\Delta^{2}}{\Gamma^{2}})]$
is the two-photon Lorentzian spectrum [27]. Energy conservation demands
$2\omega_{p}=\omega_{a}+\omega_{b}$, which introduces anti-correlation in the
two-photon joint spectral density, cf. Figure 1(e). The time uncertainty of
the generated photon pair is determined by the pump laser coherence time
$\tau_{L}>1~{}\mu$s (inversely proportional to the laser linewidth
$\Gamma_{L}/2\pi\approx 100$ kHz), which is much longer than the Purcell
enhanced QD lifetime $\tau_{QD}\approx$ 69 ps (Figure 1 (f)).

Figure 1(b) measures the transmission intensity of scattered photons versus QD
detuning using one of the two unbalanced Mach-Zehnder interferometers (UMZIs)
at different phases $\phi$ (see UMZI setup in Figure 2(b)). This allows
separate measurements of either the extinction of elastically scattered
photons from the weak coherent state (blue, $\phi=0$) or the inelastically
scattered photons (red, $\phi=\pi$). On resonance, the single-photon component
is primarily reflected due to destructive interference in the PhC WG, while
the transmitted mode consists of residual coherent photons from the laser and
inelastically scattered photons. For $\phi=0$, elastic scattering dominates as
the laser photons traversing the short and long paths constructively interfere
at the beamsplitter of one UMZI, thereby revealing a transmission dip (blue
data in Figure 1(b)). Conversely, for $\phi=\pi$, these laser photons
destructively interfere, allowing direct observation of the inelastically
scattered photons (red data in Figure 1(b)).

The pronounced extinction of the transmission intensity is indicative of
efficient radiative coupling to the PhC WG. By modelling the experimental data
sets, we extract $\beta=92\%$ and a Purcell enhancement from slow light in the
PhC WG of $F_{P}\approx 15.9$, which increases the QD decay rate to
$\Gamma/2\pi=2.3$ GHz (compared to $\approx 0.14$ GHz for QDs in bulk [28]).
In the entanglement characterization discussed below, a narrow bandwidth notch
spectral filter is implemented to suppress residual laser leakage due to
experimental imperfections (non-unity $\beta$-factor, residual slow spectral
diffusion, etc), see Supplementary Note 1 and Figure S7 for further details of
the filter and its impact on various parameter estimates of the QD.

The successful preparation of a two-photon component is quantified by second-
order photon correlation measurements. Here we observed a pronounced photon
bunching of $g^{(2)}(0)\approx 210$ after applying the spectral filter, see
Figure 1(c). This explicitly demonstrates that the incoming Poissonian photon
distribution is significantly altered by strong nonlinear interaction with the
QD [27, 15]. While our theoretical model does not take the filter into
account, it accurately describes the experimental data of Figure 1(c), by
adjusting the input model parameters, see Supplementary Notes 3 and 7.5 for
the details and justification. We modelled both the unfiltered and filtered
sets of second-order correlation ($g^{(2)}$) data versus the mean number of
pump photons $n$, see Supplementary Note 3.

Figure 2(b) illustrates the experimental setup. The scattered light from the
QD PhC WG is spectrally filtered and directed by a nonpolarizing fiber
beamsplitter to two identical UMZIs for entanglement analysis. The time
difference between the two arms of each UMZI is set to $\tau_{I}=3.6$ ns,
which is shorter than $\tau_{L}$ and longer than the two-photon correlation
time set by $\tau_{QD}$. We implement two-photon Franson interference
measurements by recording time-resolved correlations between photon pairs
while controlling the interferometric phases ($\phi_{a}$ and $\phi_{b}$).
Figure 2(a) reveals three distinct correlation peaks, corresponding to every
possible path that the two photons can take separately: long-short $\ket{l,s}$
(left), short-short $\ket{s,s}$ or long-long $\ket{l,l}$ (center), and short-
long $\ket{s,l}$ (right). Since the two paths $\ket{s,s}$ and $\ket{l,l}$
cannot be distinguished in time, the central peak corresponds to projection
onto the entangled state $\ket{s,s}+e^{i(\phi_{a}+\phi_{b})}\ket{l,l}$. By
tuning the interferometer phases such that $\phi_{a}+\phi_{b}=0~{}(\pi)$, we
observe constructive (destructive) interference of the central peak (see
Figure 2(a)), stemming from the two-photon energy-time entanglement. By
further measuring a 2D histogram shown in Figure 2(c), almost background-free
quantum interference is observed as a testimony of the highly efficient
spectral selection of the two-photon scattering component, see Supplementary
Note 4 for background noise comparison between filtered and unfiltered data.

In Figure 3(a), we scan the phase $\phi_{b}$ for two different phase settings
of interferometer $a$ ($\phi_{a}=0,\pi/4$). The Franson interference
visibility is defined as
$V=(R_{\text{max}}-R_{\text{min}})/(R_{\text{max}}+R_{\text{min}})$, where
$R_{\text{min}}$ ($R_{\text{max}}$) is the coincidence rate of the central
peak at the minimum (maximum) of the interference curve. To obtain higher
count rates with smaller fluctuations (error bars), the coincidence time
window is increased to 0.512 ns compared with its counterpart (0.064 ns) in
FIGs. 1-2. Fitting the data with a sinusoidal, we extract an interference
visibility of $V=95(4)\%$, which indicates the presence of entanglement [29].

The energy-time entangled photon pair induced by the nonlinear interaction is
thoroughly certified with a CHSH Bell inequality test [26]. The CHSH $S$
parameter is defined as
$S=\left|E(\phi_{a},\phi_{b})+E(\phi_{a},\phi_{b^{\prime}})-E(\phi_{a^{\prime}},\phi_{b})+E(\phi_{a^{\prime}},\phi_{b^{\prime}})\right|$,
where $E(\phi_{a},\phi_{b})$ denotes the correlation function required for the
CHSH inequality, which is a combination of four unnormalized $g^{(2)}$ after
the UMZIs at different phase settings (Supplementary Note 7.4). Figure 3(b)
shows the strongest correlations measured at the lowest value of $n$ in Figure
3(c), which corresponds to a pump power of 7.2 pW at a single photon level. We
record a pronounced violation of the CHSH Bell inequality $S=2.67(16)>2$ by
more than four standard deviations. This validates that non-local quantum
correlations can be induced by two-photon inelastic scattering of a
deterministically coupled two-level emitter. Figure 3(c) explores the power
dependence of the $S$ parameter and the experimental data agree well with the
theoretical model detailed in Supplementary Notes 7.5. The entanglement
quality is primarily limited by photon distinguishability contributions from
pure dephasing, as well as multi-photon scattering processes from finite $n$.

We have experimentally demonstrated the violation of the CHSH Bell inequality
by weak scattering of a single two-level emitter deterministically coupled to
light in a PhC WG. Our scheme operates at a picowatt pump power (single-photon
level) by exploiting the giant nonlinearity of the emitter, laying the
foundation for ultimately energy-efficient and integrated sources of entangled
photons. The source is easy to operate experimentally since no elaborate
excitation schemes or active spin control is required, which could constitute
significant overhead for future up-scaling of entanglement generation schemes.
Future experiments could exploit the creation of high-dimensional entanglement
[30] and the synthesis of photonic quantum states useful for quantum optics
neural network [31]. Another promising direction is to engineer the inelastic
scattering processes by many-body subradiant states using coupled QDs [32,
33]. Waveguide-mediated quantum nonlinear interactions will prove essential to
applications within photonic quantum computing [34], quantum communication
[35] and quantum sensing [36, 37].

Data availability
The data that support the figures in this manuscript are available from the
corresponding authors upon request.

Acknowledgement
We gratefully acknowledge financial support from Danmarks Grundforskningsfond
(DNRF 139, Hy-Q Center for Hybrid Quantum Networks), the Novo Nordisk
Foundation (Challenge project ”Solid-Q”). Furthermore, this project has
received funding from the European Union’s Horizon 2020 research and
innovation programmes under Grant Agreements No. 824140 (TOCHA,
H2020-FETPROACT-01-2018). B.S. acknowledges financial support from Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation), grant no. 449674892.
O.A.D.S. acknowledges funding from the European Union’s Horizon 2020 research
and innovation program under the Marie Skłodowska-Curie grant agreement no.
801199. M.L.C. acknowledges funding from the European Union’s Horizon 2020
Research and Innovation programme under grant agreement No. 861097 (project
name QUDOT-TECH).

Author contributions
S.L. and A.T. performed the experiments based on interferometers built by
R.B.N., R.G.L. and Y.A. S.L. and A.T. carried out the measurements, and
analyzed the data with help from M.L.C. S.L. prepared the figures with input
from A.T., A.S.S. and P.L. O.A.D.S. developed the theory with input from B.S
and A.S.S. S.L., A.T., O.A.D.S, and P.L. wrote the manuscript with input from
all the authors. S.S., A.D.W., and A.L. prepared the wafer. Y.W. and L.M.
fabricated the chip. A.T. and P.L. conceived the idea and supervised the
project.

Competing interests
P.L. is the founder of the company Sparrow Quantum, which commercializes
single-photon sources. The authors declare no other competing interests.

## References

  * Shen and Fan [2007] J.-T. Shen and S. Fan, Phys. Rev. Lett. 98, 153003 (2007).
  * Kiilerich and Mølmer [2019] A. H. Kiilerich and K. Mølmer, Phys. Rev. Lett. 123, 123604 (2019).
  * Lodahl  _et al._ [2022] P. Lodahl, A. Ludwig, and R. J. Warburton, Physics Today 75, 44 (2022).
  * Javadi  _et al._ [2015] A. Javadi, I. Söllner, M. Arcari, S. L. Hansen, L. Midolo, S. Mahmoodian, G. Kiršanskė, T. Pregnolato, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, Nature Communications 6, 8655 (2015).
  * Sipahigil  _et al._ [2016] A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Burek, J. Borregaard, M. K. Bhaskar, C. T. Nguyen, J. L. Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, and M. D. Lukin, Science 354, 847 (2016).
  * Hamsen  _et al._ [2017] C. Hamsen, K. N. Tolazzi, T. Wilk, and G. Rempe, Phys. Rev. Lett. 118, 133604 (2017).
  * Pscherer  _et al._ [2021] A. Pscherer, M. Meierhofer, D. Wang, H. Kelkar, D. Martín-Cano, T. Utikal, S. Götzinger, and V. Sandoghdar, Phys. Rev. Lett. 127, 133603 (2021).
  * Schulte  _et al._ [2015] C. H. H. Schulte, J. Hansom, A. E. Jones, C. Matthiesen, C. Le Gall, and M. Atatüre, Nature 525, 222 (2015).
  * Hinney  _et al._ [2021] J. Hinney, A. S. Prasad, S. Mahmoodian, K. Hammerer, A. Rauschenbeutel, P. Schneeweiss, J. Volz, and M. Schemmer, Phys. Rev. Lett. 127, 123602 (2021).
  * Jeannic  _et al._ [2022] H. L. Jeannic, A. Tiranov, J. Carolan, T. Ramos, Y. Wang, M. H. Appel, S. Scholz, A. D. Wieck, A. Ludwig, N. Rotenberg, L. Midolo, J. J. García-Ripoll, A. S. Sørensen, and P. Lodahl, Nature Physics 18, 1191 (2022).
  * Tomm  _et al._ [2023] N. Tomm, S. Mahmoodian, N. O. Antoniadis, R. Schott, S. R. Valentin, A. D. Wieck, A. Ludwig, A. Javadi, and R. J. Warburton, Nature Physics (2023), 10.1038/s41567-023-01997-6.
  * Witthaut  _et al._ [2012] D. Witthaut, M. D. Lukin, and A. S. Sørensen, Europhysics Letters 97, 50007 (2012).
  * Konyk and Gea-Banacloche [2019] W. Konyk and J. Gea-Banacloche, Physical Review A 99, 010301 (2019).
  * Krastanov  _et al._ [2022] S. Krastanov, K. Jacobs, G. Gilbert, D. R. Englund, and M. Heuck, npj Quantum Information 8, 103 (2022).
  * Chang  _et al._ [2007] D. E. Chang, A. S. Sørensen, E. A. Demler, and M. D. Lukin, Nature Physics 3, 807 (2007).
  * Fayard  _et al._ [2021] N. Fayard, L. Henriet, A. Asenjo-Garcia, and D. E. Chang, Physical Review Research 3, 033233 (2021).
  * Bello  _et al._ [2022] M. Bello, G. Platero, and A. González-Tudela, PRX Quantum 3, 010336 (2022).
  * Bourassa  _et al._ [2021] J. E. Bourassa, R. N. Alexander, M. Vasmer, A. Patil, I. Tzitrin, T. Matsuura, D. Su, B. Q. Baragiola, S. Guha, G. Dauphinais, K. K. Sabapathy, N. C. Menicucci, and I. Dhand, Quantum 5, 392 (2021), arxiv:2010.02905 [quant-ph] .
  * Peiris  _et al._ [2017] M. Peiris, K. Konthasinghe, and A. Muller, Physical Review Letters 118, 030501 (2017).
  * Muller  _et al._ [2008] A. Muller, W. Fang, J. Lawall, and G. S. Solomon, Physical Review Letters 101, 027401 (2008).
  * Jayakumar  _et al._ [2013] H. Jayakumar, A. Predojević, T. Huber, T. Kauten, G. S. Solomon, and G. Weihs, Physical Review Letters 110, 135505 (2013).
  * Hohn  _et al._ [2023] M. Hohn, K. Barkemeyer, M. von Helversen, L. Bremer, M. Gschrey, J. H. Schulze, A. Strittmatter, A. Carmele, S. Rodt, S. Bounouar, and S. Reitzenstein, arXiv: 2301.05697 (2023), 10.48550/ARXIV.2301.05697.
  * Glorieux  _et al._ [2010] Q. Glorieux, R. Dubessy, S. Guibal, L. Guidoni, J.-P. Likforman, T. Coudreau, and E. Arimondo, Physical Review A 82, 033819 (2010).
  * Lodahl  _et al._ [2015] P. Lodahl, S. Mahmoodian, and S. Stobbe, Reviews of Modern Physics 87, 347 (2015).
  * Franson [1989] J. D. Franson, Phys. Rev. Lett. 62, 2205 (1989).
  * Clauser  _et al._ [1969] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. Rev. Lett. 23, 880 (1969).
  * Le Jeannic  _et al._ [2021] H. Le Jeannic, T. Ramos, S. F. Simonsen, T. Pregnolato, Z. Liu, R. Schott, A. D. Wieck, A. Ludwig, N. Rotenberg, J. J. García-Ripoll, and P. Lodahl, Phys. Rev. Lett. 126, 023603 (2021).
  * Arcari  _et al._ [2014] M. Arcari, I. Söllner, A. Javadi, S. Lindskov Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, Phys. Rev. Lett. 113, 093603 (2014).
  * Kwiat  _et al._ [1993] P. G. Kwiat, A. M. Steinberg, and R. Y. Chiao, Physical Review A 47, R2472 (1993).
  * Martin  _et al._ [2017] A. Martin, T. Guerreiro, A. Tiranov, S. Designolle, F. Fröwis, N. Brunner, M. Huber, and N. Gisin, Phys. Rev. Lett. 118, 110501 (2017).
  * Steinbrecher  _et al._ [2019] G. R. Steinbrecher, J. P. Olson, D. Englund, and J. Carolan, npj Quantum Information 5 (2019), 10.1038/s41534-019-0174-7.
  * Ke  _et al._ [2019] Y. Ke, A. V. Poshakinskiy, C. Lee, Y. S. Kivshar, and A. N. Poddubny, Physical Review Letters 123, 253601 (2019).
  * Tiranov  _et al._ [2023] A. Tiranov, V. Angelopoulou, C. J. van Diepen, B. Schrinski, O. A. D. Sandberg, Y. Wang, L. Midolo, S. Scholz, A. D. Wieck, A. Ludwig, A. S. Sørensen, and P. Lodahl, Science 379, 389 (2023).
  * Alexander  _et al._ [2016] R. N. Alexander, P. Wang, N. Sridhar, M. Chen, O. Pfister, and N. C. Menicucci, Phys. Rev. A 94, 032327 (2016).
  * Zhong  _et al._ [2015] T. Zhong, H. Zhou, R. D. Horansky, C. Lee, V. B. Verma, A. E. Lita, A. Restelli, J. C. Bienfang, R. P. Mirin, T. Gerrits, S. W. Nam, F. Marsili, M. D. Shaw, Z. Zhang, L. Wang, D. Englund, G. W. Wornell, J. H. Shapiro, and F. N. C. Wong, New Journal of Physics 17, 022002 (2015).
  * Sánchez Muñoz  _et al._ [2021] C. Sánchez Muñoz, G. Frascella, and F. Schlawin, Phys. Rev. Research 3, 033250 (2021).
  * Kaiser  _et al._ [2018] F. Kaiser, P. Vergyris, D. Aktas, C. Babin, L. Labonté, and S. Tanzilli, Light: Science & Applications 7, 17163 (2018).
  * Zhou  _et al._ [2018] X. Zhou, I. Kulkova, T. Lund-Hansen, S. L. Hansen, P. Lodahl, and L. Midolo, Applied Physics Letters 113, 251103 (2018).
  * Uppu  _et al._ [2020] R. Uppu, F. T. Pedersen, Y. Wang, C. T. Olesen, C. Papon, X. Zhou, L. Midolo, S. Scholz, A. D. Wieck, A. Ludwig, and P. Lodahl, Science Advances 6 (2020), 10.1126/sciadv.abc8268.
  * Fan  _et al._ [2010] S. Fan, i. m. c. E. Kocabaş, and J.-T. Shen, Phys. Rev. A 82, 063821 (2010).
  * Chan  _et al._ [2023] M. L. Chan, A. Tiranov, M. H. Appel, Y. Wang, L. Midolo, S. Scholz, A. D. Wieck, A. Ludwig, A. S. Sørensen, and P. Lodahl, npj Quantum Information 9, 49 (2023).
  * Manz  _et al._ [2007] S. Manz, T. Fernholz, J. Schmiedmayer, and J.-W. Pan, Physical Review A 75, 040101 (2007).
  * Wesenberg and Mølmer [2004] J. H. Wesenberg and K. Mølmer, Physical Review Letters 93, 143903 (2004).

Supplementary Notes - Violation of Bell inequality by photon scattering on a
two-level emitter

###### Contents

  1. 1 Spectral filtering of quantum dot emission
  2. 2 Characterization of the QD transmission intensity
  3. 3 Two-photon correlation measurements
  4. 4 Additional Franson interference data
  5. 5 Optical spectroscopy
  6. 6 Theoretical model of two-photon scattering
  7. 7 Quantum dot dynamics and second-order correlation functions after the interferometer
    1. 7.1 Quantum dot dynamics
    2. 7.2 Interferometer transformation
    3. 7.3 Spectral diffusion and detector time jitter
    4. 7.4 CHSH Bell inequality equation as a function of $n$, $\beta$ and $\gamma_{d}$
    5. 7.5 Fitting the theoretical model to the data

##  1 Spectral filtering of quantum dot emission

In the experiment, spectral filtering is implemented in order to purify the
entanglement quality. The filter consists of two parts: a narrow-band
Fabry–Perot (FP) cavity for suppressing residual laser leakage transmitted
through the QD and a volume phase holographic (VPH) transmission grating for
suppressing the QD phonon sidebands, see Figure S1. The QD-emission is
prepared in the horizontal polarization with a polarization controller (PC).
After passing through a Faraday rotator (FR) and a half-wave plate (HWP) at
22.5∘, the polarization remains horizontal. Subsequently, a lens is used to
focus the collimated light to mode match the FP cavity. The FP cavity
resonantly transmits a narrow frequency window in a linewidth of 21 MHz so
that residual laser leakage (100 kHz linewidth) can be transmitted through the
FP cavity. This is ensured by locking the FP cavity to the laser frequency
using a PID servo-loop using a photodiode (PD) and a piezo-mounted cavity
mirror. In this way, the light reflected from the FP cavity primarily consists
of components inelastically scattered by the QD, while the laser component is
effectively filtered. The polarization of the reflected light is rotated
vertically by the magneto-optic effect in the FR and therefore reflected from
the polarizing beam splitter (PBS) to obtain the output beam. After the VPH
transmission grating, the first diffraction order is collected by the fiber
collimator (FC). The FP cavity is operated on resonance with the stabilization
laser at low input power, and the excitation laser frequency is scanned. In
this way, the reflection spectrum is recorded, see the inset of Figure S1. By
fitting the data to a Lorenzian function, we obtain an extinction of 85% with
a cavity linewidth of 21 MHz, in which the extinction is limited by minor
fluctuations in the FP stabilization during the long data acquisition time,
and mode matching between the input mode and intrinsic mode of the FP cavity.

The stabilization laser is tuned 60 GHz away from the excitation laser, and
the residual leakage is spectrally filtered using the VPH transmission grating
with 16 GHz linewidth (Figure S1). Both lasers are stabilized using the
wavemeter up to 5 MHz precision, which limits the extinction of the FP cavity
at the end.

Figure S1:  (color online) Illustration of the spectral filter implemented on
the QD emission. Details of the implementation are described in the text.

##  2 Characterization of the QD transmission intensity

The nonlinear characteristics of the QD response can be probed by recording
the transmission intensity versus excitation power using one single photon
detector. Figure S2 shows the measured data versus pump power and QD detuning.
The transmission data are normalized to the value obtained off-resonance, i.e.
for a large QD detuning. The incident laser power ($P$) is measured in the
optical setup before the cryostat and subsequently converted in order to
obtain the Rabi frequency $\Omega=2\sqrt{\eta P}$ or the mean photon number
$n=\frac{\Omega^{2}}{2\beta\Gamma^{2}}$ in the PhC WG within a QD lifetime
$\tau_{QD}$. $\eta$ is the proportionality (loss) factor connecting the
measured power to the Rabi frequency. The average pump power
($P_{\text{pump}}$) in the PhC WG is estimated by
$P_{\text{pump}}=nh\nu/\tau_{QD}$, where $h$ is the Planck’s constant and
$h\nu$ is the energy carried by a single photon at $\nu=318.6702$ THz.

Figure S2 (b) shows the theoretical fit of the experimental data from Figure
S2 (a). For better visualization, the power-dependent resonance transmission
is also shown in Figure S2 (c), where the blue dots and curve show the minimum
transmission intensity extracted from Figure S2 (a) and (b), respectively. We
extract $\eta=0.0012\text{ ns}^{-2}\mu\text{W}^{-1}$ and this value is used in
the power-dependent measurements of the CHSH inequality (Figure 2(a)) and
second-order correlations (Figure S3). In Figure S2 (c), the conversion
between the pump power $P_{\text{pump}}$ (bottom black x-axis) to $\Omega$ and
$n$ (top green x-axis) is explicitly shown. For simplicity, we use only $n$ to
describe the power dependence in the subsequent measurements. We find the
normalized transmission intensity ($T$) measurement to be almost insensitive
to the FP filter. We attribute this to the unnormalized transmitted intensity
(I) being dominated by laser leakage, i.e.,
$\text{I}^{\text{Res}}_{\text{laser}}\gg\text{I}^{\text{Res}}_{\text{inelastic}}$,
which is suppressed by the same extinction ratio $\eta_{\text{filter}}$ for
both resonant (with superscript: “Res”) and off-resonant (“Off”) data, and
this mostly cancels out the effect of the filter:

$\displaystyle
T_{\text{filtered}}=\frac{\text{I}^{\text{Res}}_{\text{inelastic}}+\text{I}^{\text{Res}}_{\text{laser}}\eta_{\text{filter}}}{\text{I}^{\text{Off}}_{\text{laser}}\eta_{\text{filter}}}\approx\frac{\text{I}^{\text{Res}}_{\text{laser}}}{\text{I}^{\text{Off}}_{\text{laser}}}\approx\frac{\text{I}^{\text{Res}}_{\text{inelastic}}+\text{I}^{\text{Res}}_{\text{laser}}}{\text{I}^{\text{Off}}_{\text{laser}}}=T_{\text{unfiltered}}.$
(S1)

Figure S2:  (color online) The power-dependent QD transmission intensity.
(a)-(b) the measured (left) and fitted (right) transmission intensity as a
function of pump power in the PhC WG and QD detuning. Based on the DC stark
effect, the QD detuning relative to the laser frequency is controlled by
tuning the QD bias voltage. (c). Power-dependent resonance transmission. The
blue dots and curve represent the minimum transmission intensity (resonance
transmission dip) extracted from (a) and (b), respectively. The two additional
top x-axes represent the Rabi frequency $\Omega/2\pi$ and mean photon number
$n$ within the QD lifetime, and the bottom axis shows the pump power
$P_{\text{pump}}$ in the PhC WG. The extracted modelling parameters are:
$\Gamma/2\pi=2.3$ GHz, $\gamma_{d}/2\pi=0.01$ GHz, $\sigma_{sd}/2\pi=0.39$
GHz, $\beta=0.92$, see Note 7 for the definition of the parameters.

##  3 Two-photon correlation measurements

The two-photon correlations are measured in a Hanbury-Brown-Twiss (HBT)
experiment by recording the normalized intensity correlation function
$g^{(2)}(\tau)$. It is recorded using a fiber beam splitter connected directly
to the QD source before the interferometer. The power dependent
$g^{(2)}(\tau)$ is recorded, see Figure S3 (a) and (b), and comparing the two
different cases with and without the FP cavity filter. Pronounced bunching
$(g^{(2)}(0)>1)$ is observed in both cases, signifying the two-photon
character of the light scattered by the QD. With the FP spectral notch filter,
a significantly enhanced bunching is found, leading to $g^{(2)}(0)\approx
210$, i.e., the filter successfully suppresses residual laser leakage which
contains uncorrelated photons. $g^{(2)}(0)$ decreases gradually with
increasing pump power (quantified by $n$), which is the experimental signature
of nonlinear saturation of the QD. The experimental data are fitted to the
same theoretical model giving the following parameters for the unfiltered
(filtered) cases: spectral diffusion of 0.52 (0.16) GHz, pure dephasing of
0.01 (0.01) GHz, coupling strength $\beta$ of 0.92 (0.96), with a time jitter
of the single-photon detectors of 100 ps width and the natural linewidth of
$\Gamma/2\pi=2.3$ GHz. The parameters are described in detail in section 7,
while the theoretical model of $g^{(2)}(\tau)$ can be found in Ref. [27]. The
narrow spectral filtering improves the effective performance of the photon-
emitter coupling of the QD to the photonic crystal waveguide (PhC WG).

Figure S3:  (color online) Second-order correlation data as a function of time
delay $\tau$ and number of incident photons $n$. (a)-(b) $g^{(2)}(\tau)$
measurements without and with the narrow FP spectral filter. All data are
normalized to the data recorded at long time delays $\tau$.  Figure S4:
(color online) Franson interference data with and without spectral filtering.
The first (second) column is the unfiltered (filtered) cases. (a)-(b) 2D
coincidence interference histograms versus detector delay $\tau$ and
interferometer phase $\phi_{b}$. (c)-(d). Line cuts (blue and red data)
extracted from (a) and (b) at the positions of the dashed line with the
corresponding color. The red (blue) data correspond to destructive
(constructive) interference. (e)-(f) Two-photon interference data (green and
orange data) extracted from the center peak in (a) and (b) and the
corresponding background (grey data) taken at $\tau=6$ ns , i.e. away from the
region of the three peaks. The applied time-bin window of the coincidence data
is 0.064 ns in (a)-(b) and 0.512 ns in (c)-(f). All the data from (a) to (f)
is measured using the same low power, i.e., $n=0.0024$.

##  4 Additional Franson interference data

The Franson interference data are recorded both with and without the narrow FP
filter and pronounced quantum correlations are recorded in both cases, see
Figure S4. Note that here the interferometric phase $\phi_{a}$ is fixed to 0
during the measurement. Figure S4 (a) and (b) show the 2D coincidence
interference histograms versus the time delay $\tau$ between two detectors and
the interferometric phase $\phi_{b}$. Figure S4 (c) and (d) are the
corresponding projections versus time delay comparing the two cases of
constructive (blue data, $\phi_{b}=0$) and destructive interference (red data,
$\phi_{b}=\pi$). Figure S4 (e) and (f) display the interference data of the
center peaks versus $\phi_{b}$ together with the phase-dependent background
level. For the unfiltered data in the left column, the phase-dependent
background is due to a small amount of laser leakage with narrow linewidth,
namely the elastic photon scattering component, which can bring unwanted local
classical interference when using one UMZI, and unwanted coincidence
interference when using both UMZIs. Since two laser photons are always
uncorrelated in time, this coincidence interference noise is time delay
independent, which can be seen from the 2D interference histogram in Figure S4
(a).

##  5 Optical spectroscopy

The QD lifetime is an essential parameter entering in the theoretical
description of the experiment. It is extracted experimentally by exciting the
QD through a higher-order transition (the p-shell) using a Ti:Sapphire mode-
locked pulsed laser (pulse duration of 3 ps). The emitted fluorescence from
the QD is sent into a grating filter with 25 GHZ FWHM bandwidth to remove the
laser component and then detected by a single photon detector, see Figure S5.
In order to record the time-resolved emission spectra, both the detector
signal and the periodic sync signal from the pulsed laser are connected to a
PicoHarp time-correlator device. This allows to reconstruct the emission time
of the photons giving a coincidence correlation histogram, which is determined
by the convolution of a single exponential emission decay and the instrument
response function of the detector. By fitting the data, the relaxation time of
the QD dipole used in this work is found to be $\tau_{QD}=\Gamma^{-1}=69$ ps,
while the lifetime of the second dipole is 2.6 ns. Since the fine structure
splitting of this QD is much larger than the linewidth of the optical
transition $\Gamma/2\pi$ we can effectively excite only a single dipole, such
that the two-level approximation holds.

The sample is cooled down to 4 K temperature using a closed-cycle cryostat
with a confocal microscope with a 0.8 numerical aperture objective. This was
used to excite the QD from free space during the characterisation, and couple
light into the PhC WG. Shallow etched gratings [38] allowed to couple linearly
polarized light into the PhC WG within a 50 nm (FWHM) bandwidth. Details about
the sample growth and fabrication can be found in [39].

The spectroscopy of the sample containing the QD and PhC WG used in this study
is described in [33], where it is referred as QD1.

Figure S5:  (color online) Lifetime measurement of the QD using p-shell
excitation. Decays of two neutral excitons are visible which follow the sum of
two exponentials with lifetimes of 69 ps and 2.6 ns taking into account the
instrument response function.

##  6 Theoretical model of two-photon scattering

Here, we will briefly discuss the theoretical model used to describe two-
photon scattering. The experiment relies on two-photon correlation
measurements and the detection of a single photon in one detector
conditionally prepares a quantum state within the QD lifetime $\tau_{QD}$. The
energy conservation $2\omega_{p}=\omega_{a}+\omega_{b}$ implies an uncertainty
on the energy of each scattered photon, which is proportional to $\Gamma$.
This is analogous to the original Franson experiment where the decay of a
three-level atom is considered [25]. We consider the resonant transmission of
a weak coherent state through the 1D PhC WG. The result for the two-photon
scattering can be written as [40, 27]:

$\displaystyle\ket{2_{I}}=\int
d\omega^{\text{in}}_{1}d\omega^{\text{in}}_{2}f(\omega^{\text{in}}_{1})f(\omega^{\text{in}}_{2})\int{d\omega_{a}d\omega_{b}}[\chi(\omega^{\text{in}}_{1})\chi(\omega^{\text{in}}_{2})\delta(\omega_{a}-\omega^{\text{in}}_{1})\delta(\omega_{b}-\omega^{\text{in}}_{2})+$
$\displaystyle+\frac{1}{2}\mathcal{T}_{\omega^{\text{in}}_{1}\omega^{\text{in}}_{2}\omega_{a}\omega_{b}}\delta(\omega_{a}+\omega_{b}-\omega^{\text{in}}_{1}-\omega^{\text{in}}_{2})]\ket{1_{\omega_{a}}}\ket{1_{\omega_{b}}},$
(S2)

where $\chi(\omega)$ are the single-photon scattering corresponding to the
transmission ($\chi(\omega)=t(\omega)$) and reflection
($\chi(\omega)=r(\omega)$) coefficients and
$\mathcal{T}_{\omega^{\text{in}}_{1}\omega^{\text{in}}_{2}\omega_{a}\omega_{b}}$
is the intrinsic two-photon correlation term.

In the case of a narrow band laser
$\omega^{\text{in}}_{1}=\omega^{\text{in}}_{2}=\omega_{p}$ and
$f(\omega)=\delta(\omega-\omega_{p})$ the state in the transmission mode of
the PhC WG can be written

$\displaystyle\ket{2_{I}}=\frac{1}{2}\int{d\omega_{a}}\mathcal{T}_{\omega_{p}\omega_{p}\omega_{a}2\omega_{p}-\omega_{a}}\ket{1_{\omega_{a}}}\ket{1_{2\omega_{p}-\omega_{a}}}.$
(S3)

The term $\mathcal{T}_{\omega_{p}\omega_{p}\omega_{a}\omega_{b}}$ describes
the correlated scattering, which includes the spectral entanglement. For a
two-level system in a non-chiral/mirror symmetric 1D PhC WG, it can be
expressed

$\displaystyle\mathcal{T}_{\omega_{p}\omega_{p}\omega_{a}2\omega_{p}-\omega_{a}}=\frac{4}{\pi\beta\Gamma}r(\omega_{p})r(\omega_{a})r(2\omega_{p}-\omega_{a}).$
(S4)

Using the reflection (transmission) scattering coefficients
$r(\omega)=-\frac{\beta\Gamma}{\Gamma-2i(\omega-\omega_{0})}$
($t(\omega)=1+r(\omega)$) and resonant condition $\omega_{p}=\omega_{0}$, we
can further write

$\displaystyle\mathcal{T}_{\omega_{p}\omega_{p}\omega_{a}2\omega_{p}-\omega_{a}}=\mathcal{T}_{\Delta}$
$\displaystyle=-\frac{4}{\pi\Gamma}r(\omega_{a})r(2\omega_{p}-\omega_{a})$
$\displaystyle=-\frac{4\beta^{2}}{\pi\Gamma(1-2i\Delta/\Gamma)(1+2i\Delta/\Gamma)}$
$\displaystyle=-\frac{4\beta^{2}}{\pi\Gamma(1+4\frac{\Delta^{2}}{\Gamma^{2}})}$
(S5)

where $\Delta=\omega_{a}-\omega_{p}$ is the detuning between the generated
photon and the pump laser and $\mathcal{T}_{\Delta}$ is the two-photon
spectrum given by the Lorentzian profile. Finally, the two-photon state can be
written as

$\displaystyle\ket{2_{I}}=\frac{1}{2}\int{d\Delta}\mathcal{T}_{\Delta}\ket{1_{a,\Delta}}\ket{1_{b,-\Delta}},$
(S6)

which clearly shows the anticorrelation in energy between the two photons due
to energy conservation.

The joint time distribution of two-photon fields can be described in terms of
the two-photon amplitude

$\displaystyle
A(t,t^{\prime})=\bra{0,0}E^{(+)}_{a}(0,t)E^{(+)}_{b}(0,t^{\prime})\ket{2_{I}}=\int{d\omega_{a}a_{\omega_{a}}e^{-i\omega_{a}t}}\int{d\omega_{b}a_{\omega_{b}}e^{-i\omega_{b}t^{\prime}}}\int{d\Delta\mathcal{T}_{\Delta}a^{\dagger}_{\omega_{p}+\Delta}a^{\dagger}_{\omega_{p}-\Delta}\ket{0}}$
$\displaystyle=\int{d\Delta
e^{-i(\omega_{p}+\Delta)t}e^{-i(\omega_{p}-\Delta)t^{\prime}}}\mathcal{T}_{\Delta}=e^{-i\omega_{p}t}e^{-i\omega_{p}t^{\prime}}\int{d\Delta
e^{i\Delta(t-t^{\prime})}\mathcal{T}_{\Delta}},$ (S7)

where the last integral represents the Fourier transform of the two-photon
spectral amplitude. Since each of the photons has a Lorentzian profile, the
Fourier transform of their multiplication corresponds to the convolution of
their individual Fourier transformations. The two-photon correlation will look
as the double-exponential decay with the decay time $\Gamma^{-1}$, defining
the duration of the two-photon correlation.

##  7 Quantum dot dynamics and second-order correlation functions after the
interferometer

In this section, we will describe the fields and compute the density matrix
used to evaluate the second-order correlation function. We also describe how
the interferometer transforms the fields such that they can be used to violate
a CHSH inequality. We first write the master equation for the system, compute
the steady state density matrix $\rho_{ss}$ and then apply the quantum
regression theorem to the light fields and $\rho_{ss}$ to obtain the second-
order correlation function $G^{(2)}(\tau)$.

###  7.1 Quantum dot dynamics

The QD is described by the Lindblad master equation for the system which reads
($\hbar=1$)

$\displaystyle\dot{\rho}=-i\left[H,\rho\right]+\mathcal{L}_{\text{decay}}[\rho]+\mathcal{L}_{\text{deph}}[\rho].$
(S8)

Here, the unitary part of the time evolution is given by the Hamiltonian

$\displaystyle H=\frac{\Omega}{2}\sigma_{x}+\Delta\sigma_{ee},$ (S9)

which accounts for the driving of the QD with Rabi frequency $\Omega$ and
detuning $\Delta$. The Rabi frequency $\Omega=\Omega(t)$ may in general be
time-dependent, $\sigma_{x}$ is the usual Pauli matrix and
$\sigma_{ij}=\ket{i}\bra{j}$. The decay Liouvillian

$\displaystyle\mathcal{L}_{\text{decay}}[\rho]=\Gamma\left[\sigma_{-}\rho\sigma_{+}-\frac{1}{2}\left\\{\sigma_{+}\sigma_{-},\rho\right\\}\right]$
(S10)

models the total decay of the emitter $\Gamma=\gamma+\gamma_{s}$ into both the
waveguide mode $\gamma$ and other modes $\gamma_{s}$ (referred to as side
modes). We additionally include a Liouvillian modelling dephasing

$\displaystyle\mathcal{L}_{\text{deph}}[\rho]=\frac{\gamma_{d}}{2}\left[\sigma_{z}\rho\sigma_{z}-\rho\right],$
(S11)

where $\gamma_{d}$ is the dephasing rate and $\sigma_{z}$, $\sigma_{+}$ and
$\sigma_{-}$ are the Pauli operators. The ratio of the waveguide decay to the
total decay defines the $\beta$-factor
$\beta=\frac{\gamma}{\Gamma}=\frac{\gamma}{\gamma+\gamma_{s}}$. The
transmitted field operator entering the $G^{(2)}$ calculations is
$a=a_{in}\mathbb{I}-\sqrt{\beta\Gamma/2}\sigma_{ge}$, where the input field
$\hat{a}_{in}$ is a coherent state of amplitude $\alpha(t)$, and mean photon
flux $|\alpha(t)|^{2}$. The coherent field is related to the Rabi frequency by
$\Omega/2=\sqrt{\beta\Gamma/2}\alpha(t)$ and thus the mean photon number
within lifetime is
$n=\frac{|\alpha|^{2}}{\Gamma}=\frac{\Omega^{2}}{2\beta\Gamma^{2}}$. In matrix
form, the master equation reads

$\displaystyle\dot{\rho}=\left(\begin{array}[]{cc}-\Gamma\rho_{\text{ee}}+\frac{1}{2}i\Omega(\rho_{\text{eg}}-\rho_{\text{ge}})&\frac{1}{2}i\Omega(\rho_{\text{ee}}-\rho_{\text{gg}})-\frac{1}{2}\rho_{\text{eg}}(\Gamma+2i\Delta+2\gamma_{d})\\\
-\frac{1}{2}\rho_{\text{ge}}(\Gamma-2i\Delta+2\gamma_{d})-\frac{1}{2}i\Omega(\rho_{\text{ee}}-\rho_{\text{gg}})&\Gamma\rho_{\text{ee}}-\frac{1}{2}i\Omega(\rho_{\text{eg}}-\rho_{\text{ge}})\\\
\end{array}\right)$ (S14)

The QD is continuously driven by a laser field, eventually reaching an
equilibrium state described by the steady state density matrix $\rho_{ss}$. To
find $\rho_{ss}$, we write the master equation in the form

$\displaystyle\dot{\boldsymbol{\rho}}=\mathbf{M}\boldsymbol{\rho},$ (S15)

where

$\displaystyle\mathbf{M}=\left(\begin{array}[]{cccc}-\Gamma&\frac{i\Omega}{2}&-\frac{i\Omega}{2}&0\\\
\frac{i\Omega}{2}&-\frac{\Gamma}{2}-i\Delta-\gamma_{d}&0&-\frac{i\Omega}{2}\\\
-\frac{i\Omega}{2}&0&-\frac{\Gamma}{2}+i\Delta-\gamma_{d}&\frac{i\Omega}{2}\\\
\Gamma&-\frac{i\Omega}{2}&\frac{i\Omega}{2}&0\\\ \end{array}\right)$ (S20)

is the coefficient matrix and

$\displaystyle\boldsymbol{\rho}=(\rho_{\text{ee}},\rho_{\text{eg}},\rho_{\text{ge}},\rho_{\text{gg}})^{T}.$
(S21)

The master equation has a general time-dependent solution in terms of the
matrix exponential

$\displaystyle\boldsymbol{\rho}(t)=\exp{\left(\mathbf{M}t\right)}\boldsymbol{\rho}(0).$
(S22)

Taking $t\to\infty$ leads to an expression for $\rho_{ss}$. Alternatively, we
can find the steady state (obtained when $\dot{\boldsymbol{\rho}}=0$) by
taking the nullspace of $\mathbf{M}$ and normalizing it so that it has unit
trace, which gives

$\displaystyle\rho_{ss}=\left(\begin{array}[]{cc}\frac{\Omega^{2}(\Gamma+2\gamma_{d})}{\Gamma\left(4\Delta^{2}+(\Gamma+2\gamma_{d}){}^{2}\right)+2\Omega^{2}(\Gamma+2\gamma_{d})}&-\frac{i\Gamma\Omega(\Gamma-2i\Delta+2\gamma_{d})}{\Gamma\left(4\Delta^{2}+(\Gamma+2\gamma_{d}){}^{2}\right)+2\Omega^{2}(\Gamma+2\gamma_{d})}\\\
\frac{i\Gamma\Omega(\Gamma+2i\Delta+2\gamma_{d})}{\Gamma\left(4\Delta^{2}+(\Gamma+2\gamma_{d}){}^{2}\right)+2\Omega^{2}(\Gamma+2\gamma_{d})}&\frac{1}{\frac{\Omega^{2}(\Gamma+2\gamma_{d})}{\Gamma\left(4\Delta^{2}+(\Gamma+2\gamma_{d}){}^{2}\right)+\Omega^{2}(\Gamma+2\gamma_{d})}+1}\\\
\end{array}\right).$ (S25)

###  7.2 Interferometer transformation

Using an input-output formalism, we find the field that comes out of the
interferometers $a$ and $b$ to be

$\displaystyle\hat{a}_{out}^{(a)}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(a_{in}(t)+a_{in}(t-L/c)e^{-i\phi_{a}}\right),$
(S26) $\displaystyle\hat{a}_{out}^{(b)}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(a_{in}(t)+a_{in}(t-L/c)e^{-i\phi_{b}}\right),$
(S27)

where a phase $\phi_{a,b}$ and a time delay $L/c$ is picked up when traveling
through the long arm of the interferometer with length difference $L$ and
speed of light $c$. Here $a_{in}$ is the state that enters the interferometer,
i.e., the field from the QD. The (unnormalized) second-order correlation
function after the interferometer

$\displaystyle G^{(2)}(\tau)$
$\displaystyle=\text{Tr}\left(\hat{a}^{(a)}(t+\tau)\hat{a}^{(b)}(t)\hat{\rho}\hat{a}^{(b)}(t)^{{\dagger}}\hat{a}^{(a)}(t+\tau)^{{\dagger}}\right),$
(S28)

can be calculated using the quantum regression theorem, so that
$G^{(2)}(\tau)$ inherits its functional dependence on $\Delta,\Gamma$, etc
from the density matrix, $\rho$. We assume that $t,\tau,\frac{1}{\Gamma}\ll
L/c$, so that correlations between time bins vanish, implying that $G^{(2)}$
can be factored into traces containing each separate time bin. For the centre
peak (where $t^{\prime}=t+\tau$), the factored $G^{(2)}$ is

$\displaystyle G^{(2)}(\tau)$
$\displaystyle=2\text{Tr}\left(a\left(t^{\prime}\right)a(t)\rho
a(t)^{\dagger}a\left(t^{\prime}\right)^{\dagger}\right)+2\text{Tr}\left(a\rho_{ss}a^{\dagger}\right)^{2}$
$\displaystyle+e^{-i(\phi_{a}+\phi_{b})}\text{Tr}\left(a\left(t^{\prime}\right)a(t)\rho\right)\text{Tr}\left(\rho
a(t)^{\dagger}a\left(t^{\prime}\right)^{\dagger}\right)$
$\displaystyle+e^{-i(\phi_{a}-\phi_{b})}\text{Tr}\left(a\left(t^{\prime}\right)\rho
a(t)^{\dagger}\right)\text{Tr}\left(a(t)\rho
a\left(t^{\prime}\right)^{\dagger}\right)$
$\displaystyle+e^{-i\phi_{a}}\left[\text{Tr}\left(a\rho_{ss}\right)\text{Tr}\left(a(t)\rho
a(t)^{\dagger}a\left(t^{\prime}\right)^{\dagger}\right)+\text{Tr}\left(a\left(t^{\prime}\right)a(t)\rho
a(t)^{\dagger}\right)\text{Tr}\left(\rho_{ss}a^{\dagger}\right)\right]$
$\displaystyle+e^{i(\phi_{a}-\phi_{b})}\text{Tr}\left(a(t^{\prime})\rho
a(t)^{\dagger}\right)\text{Tr}\left(a(t)\rho a(t^{\prime})^{\dagger}\right)$
$\displaystyle+e^{-i\phi_{b}}\left[\text{Tr}\left(a(t^{\prime})\rho
a(t)^{\dagger}a(t^{\prime})^{\dagger}\right)\text{Tr}\left(a\rho_{ss}\right)+\text{Tr}\left(a(t^{\prime})a(t)\rho
a(t)^{\dagger}\right)\text{Tr}\left(\rho_{ss}a^{\dagger}\right)\right]$
$\displaystyle+e^{i(\phi_{1}+\phi_{b})}\text{Tr}\left(a(t^{\prime})a(t)\rho\right)\text{Tr}\left(\rho
a(t)^{\dagger}a(t^{\prime})^{\dagger}\right)$
$\displaystyle+e^{i\phi_{b}}\left[\text{Tr}\left(a(t^{\prime})a(t)\rho
a(t^{\prime})^{\dagger}\right)\text{Tr}\left(\rho_{ss}a^{\dagger}\right)+\text{Tr}\left(a(t^{\prime})\rho
a(t)^{\dagger}a(t^{\prime})^{\dagger}\right)\text{Tr}\left(a\rho_{ss}\right)\right]$
$\displaystyle+e^{i\phi_{a}}\left[\text{Tr}\left(a(t^{\prime})a(t)\rho
a(t)^{\dagger}\right)\text{Tr}\left(\rho_{ss}a^{\dagger}\right)+\text{Tr}\left(a(t)\rho
a(t)^{\dagger}a(t^{\prime})^{\dagger}\right)\text{Tr}\left(a\rho_{ss}\right)\right].$
(S29)

When $\tau=\pm L/c+\epsilon$ we obtain non-central peaks. We define
$t^{\prime}=t+\epsilon$. The factored $G^{(2)}(\tau)$ for the non-central
peaks is then

$\displaystyle G^{(2)}(\tau)$
$\displaystyle=\text{Tr}\left(a\left(t^{\prime}\right)a(t)\rho
a(t)^{\dagger}a\left(t^{\prime}\right)^{\dagger}\right)+3\text{Tr}\left(a\rho_{ss}a^{\dagger}\right)^{2}$
$\displaystyle+e^{i(\phi_{a}-\phi_{b})}\text{Tr}\left(a\rho_{ss}\right)^{2}\text{Tr}\left(\rho
a(t)^{\dagger}a(t^{\prime})^{\dagger}\right)$
$\displaystyle+e^{i(\phi_{a}+\phi_{b})}\text{Tr}\left(a\rho_{ss}\right)\text{Tr}\left(a(t)\rho
a(t^{\prime})^{\dagger}\right)\text{Tr}\left(\rho_{ss}a^{\dagger}\right)$
$\displaystyle+e^{i\phi_{a}}\left[\text{Tr}\left(a(t^{\prime})\rho\right)\text{Tr}\left(a(t)\rho
a(t)^{\dagger}a(t^{\prime})^{\dagger}\right)+\text{Tr}\left(a\rho_{ss}\right)\text{Tr}\left(\rho_{ss}a\right)\text{Tr}\left(a(t)\rho
a(t)^{\dagger}\right)\right]$
$\displaystyle+e^{-i(\phi_{a}+\phi_{b})}\text{Tr}\left(a(t^{\prime})\rho
a(t)^{\dagger}\right)\text{Tr}\left(a(t)\rho_{ss}\right)\text{Tr}\left(\rho_{ss}a^{\dagger}\right)$
$\displaystyle+e^{-i\phi_{b}}\left[\text{Tr}\left(a(t^{\prime})\rho
a(t)^{\dagger}a(t^{\prime})^{\dagger}\right)\text{Tr}\left(a(t)\rho\right)+\text{Tr}\left(a(t^{\prime})\rho
a(t^{\prime})\right)\text{Tr}\left(a\rho_{ss}\right)\text{Tr}\left(\rho_{ss}a^{\dagger}\right)\right]$
$\displaystyle+e^{-i(\phi_{a}-\phi_{b})}\text{Tr}\left(a(t^{\prime})a(t)\rho\right)\text{Tr}\left(\rho_{ss}a^{\dagger}\right)^{2}$
$\displaystyle+e^{i\phi_{b}}\left[\text{Tr}\left(a(t^{\prime})a(t)\rho
a(t^{\prime})^{\dagger}\right)\text{Tr}\left(\rho_{ss}a^{\dagger}\right)+\text{Tr}\left(a(t^{\prime})\rho
a(t^{\prime})^{\dagger}\right)\text{Tr}\left(a\rho_{ss}\right)\text{Tr}\left(\rho_{ss}a^{\dagger}\right)\right]$
$\displaystyle+e^{-i\phi_{a}}\left[\text{Tr}\left(a(t^{\prime})a(t)\rho
a(t)^{\dagger}\right)\text{Tr}\left(\rho_{ss}a^{\dagger}\right)+\text{Tr}\left(a\rho_{ss}\right)\text{Tr}\left(a(t)\rho
a(t)^{\dagger}\right)\text{Tr}\left(\rho_{ss}a^{\dagger}\right)\right].$ (S30)

###  7.3 Spectral diffusion and detector time jitter

Below, we consider two important effects relevant to modelling experimental
imperfections of QD spectral diffusion and time jitter of the single-photon
detector. Spectral diffusion corresponds to a slow (compared to the QD
lifetime) frequency drift of the QD that can be modelled as a normal
distribution of QD detunings:

$\displaystyle
P_{SD}(\Delta,\sigma_{SD})=\frac{1}{\sqrt{2\pi\sigma_{SD}^{2}}}\exp\left(-\frac{\Delta^{2}}{2\sigma_{SD}^{2}}\right),$
(S31)

where $\sigma_{SD}$ is the standard deviation of the spectral diffusion. We
compute the effect of spectral diffusion on $G^{(2)}$ by taking the integral
over all possible detunings

$\displaystyle G^{(2)}(\sigma_{SD},t,\Gamma,n,\beta,\gamma_{d})=\int
P_{SD}(\Delta,\sigma_{SD})G^{(2)}(\Delta,t,\Gamma,n,\beta,\gamma_{d})d\Delta.$
(S32)

Similarly, we consider the instrument response function (IRF) due to the time
jitter of the single-photon detector, which is modelled as a Gaussian
distribution:

$\displaystyle
P_{IRF}(t,\sigma_{IRF})=\frac{1}{\sqrt{2\pi\sigma_{IRF}^{2}}}\exp\left(-\frac{t^{2}}{2\sigma_{IRF}^{2}}\right),$
(S33)

that influences the $G^{(2)}$ according to

$\displaystyle
G^{(2)}(\sigma_{SD},\tau,\sigma_{IRF},\Gamma,n,\beta,\gamma_{d})=\int
P_{IFR}(t-\tau,\sigma_{IRF})G^{(2)}(\sigma_{SD},t,\Gamma,n,\beta,\gamma_{d})dt$
(S34)

In the next section, the above $G^{(2)}$ expressions from the center peak will
be used to construct the coincidence rates entering the CHSH inequality
parameter $S$.

###  7.4 CHSH Bell inequality equation as a function of $n$, $\beta$ and
$\gamma_{d}$

In the experiment, two separate, unbalanced Mach-Zehnder interferometers
(labelled $a$ and $b$) are implemented to test the CHSH inequality using the
standard phase settings $\phi_{a}=0,\pi/2,\pi,3\pi/2$ and
$\phi_{b}=\pi/4,3\pi/4,5\pi/4,7\pi/4$ as settings for the interferometers. We
note that here two phase settings are used to mimic different output ports of
each UMZI, which ideally should have separate detectors. In the experiments,
however, we only use a single detector and hence we perform measurements with
angles differing by $\pi$. On the other hand, the first two angles for
$\phi_{a}$ and $\phi_{b}$ could in principle be varied at will to optimize the
Bell inequality violation, but for simplicity, we restrict ourselves to these
angles. Experimentally, we use polarization elements (HWP and LP) to control
the interferometer phase via the polarization angle (the polarization angle
maps to half of the setting of the phase). Since each interferometer is
applied with four different phase settings, there are 16 possible phase
combinations for coincidence measurements in the central peak correlation
function $G^{(2)}(\phi_{a},\phi_{b})$ (for brevity we here focus on the phase
and suppress the dependency of $G^{(2)}$ on other variables). We then use the
set of $G^{(2)}(\phi_{a},\phi_{b})$ as theoretical coincidence rates to
calculate each correlation function $E(\phi_{a},\phi_{b})$ required in the
CHSH inequality, written as:

$\displaystyle
E(\phi_{a},\phi_{b})=\frac{G^{(2)}(\phi_{a},\phi_{b})+G^{(2)}(\phi_{a\bot},\phi_{b\bot})-G^{(2)}(\phi_{a},\phi_{b\bot})-G^{(2)}(\phi_{a\bot},\phi_{b})}{G^{(2)}(\phi_{a},\phi_{b})+G^{(2)}(\phi_{a\bot},\phi_{b\bot})+G^{(2)}(\phi_{a},\phi_{b\bot})+G^{(2)}(\phi_{a\bot},\phi_{b})}$
(S35)

The CHSH parameter can then be determined from

$\displaystyle
S=\left|E(\phi_{a},\phi_{b})+E(\phi_{a},\phi_{b^{\prime}})-E(\phi_{a^{\prime}},\phi_{b})+E(\phi_{a^{\prime}},\phi_{b^{\prime}})\right|$
(S36)

where $\phi_{a\bot}$ and $\phi_{a^{\prime}}$ represent the orthogonal
(differing by $\pi$) and diagonal phase (differing by $\pi/2$) basis relative
to $\phi_{a}$.

To give concise analytical expressions for $S$, we will neglect spectral
diffusion and detector time jitter, and assume on-resonance weak excitation of
the QD ($\Delta=0$). In this case, a complete analytical expression of $S$ can
be derived but is not shown explicitly here since it is rather lengthy. For
simplicity, we write the first-order Maclaurin series in $n,\gamma_{d}$ and in
$1-\beta$

$\displaystyle S(\Gamma,n,\beta,\gamma_{d})$ $\displaystyle\approx
2\sqrt{2}+32\sqrt{2}(\beta-2)n+\gamma_{d}\left(\frac{224\sqrt{2}(4-3\beta)n}{\Gamma}-\frac{16\sqrt{2}(2\beta+3)}{\Gamma}\right)$
(S37)

At the limiting values $\gamma_{d}=0$, $\beta=1$ and $n\to 0$, we obtain the
result $2\sqrt{2}$, i.e. a maximum Bell inequality violation. We give also the
exact expression for various limiting cases below. If we assume $\gamma_{d}=0$
and $\beta=1$, we have

$\displaystyle S(n)=\frac{2\sqrt{2}(1-4n)^{2}}{1+8n+32n^{2}}\approx
2\sqrt{2}\left[1-16n+O(n^{2})\right]$ (S38)

where we have performed the lowest order Taylor series expansion about $n=0$.
Assuming $\gamma_{d}=0$ and $n\to 0$, we can write an exact $S$ as well as the
lowest order Taylor series expansion around $\beta=1$. We obtain

$\displaystyle
S(\beta)=\frac{2\sqrt{2}(1-2\beta)^{2}}{2-8\beta+10\beta^{2}-4\beta^{3}+\beta^{4}}\approx
2\sqrt{2}\left[1-(1-\beta)^{4}+O(1-\beta)^{5}\right]$ (S39)

Finally, assuming $\beta=1$, $n=0$ and performing a Taylor series expansion
around $\gamma_{d}=0$, we obtain

$\displaystyle
S(\gamma_{d})=\frac{2\sqrt{2}(\Gamma-2\gamma_{d}){}^{2}}{\Gamma^{2}+8\gamma_{d}{}^{2}+4\Gamma\gamma_{d}}\approx
2\sqrt{2}\left[1-\frac{16\sqrt{2}}{\Gamma}\gamma_{d}+O(\gamma_{d}^{2})\right]$
(S40)

By evaluating the lower-order Taylor series expansions in the above equations,
we can see how these parameters influence the $S$ parameter distinctly. For
instance, $S(\beta)$ scales quartically with photon loss $(1-\beta)$, which
makes the entanglement robust against a finite coupling efficiency. Figure S6
plots how $S$ depends on these imperfections based on the full theoretical
model without considering spectral diffusion and time jitter.

Table 1: Parameters extracted from fitting the experimental data. Parameter | With FP | Without FP
---|---|---
$\Gamma/2\pi$, GHz | $2.3$ | 2.3
$\beta$ | $0.96$ | 0.92
$\gamma_{d}/2\pi$, GHz |  $\leq$ 0.01 |  $\leq$ 0.01
$\sigma_{SD}/2\pi$, GHz | $0.16$ | 0.39-0.52
$\sigma_{IRF}$, ps | 100 | 100
Figure S6:  (color online) $S$ parameter and $g^{(2)}(0)$ for various
experimental imperfections. (a) CHSH $S$ parameter as a function of
$\beta$-factor for various values of dephasing rate $\gamma_{d}$ and in the
low power limit ($n=0.01$). (b) CHSH $S$ parameter as a function of $n$ for
various values of dephasing rate $\gamma_{d}$ and for $\beta=1$. (c) Second-
order correlation function $g^{(2)}(0)$ as a function of $\beta$-factor for
various values of the dephasing rate $\gamma_{d}$ and in the low power limit
($n=0.01$). (d) Second-order correlation function $g^{(2)}(0)$ versus $n$ for
various values of the dephasing rate $\gamma_{d}$ and for $\beta=1$. The
y-axis in (a)-(b) has a linear scale while the y-axis in (c)-(d) has a
logarithmic scale for better visualization.

###  7.5 Fitting the theoretical model to the data

. After the inclusion of spectral diffusion and time jitter, the observables
are non-analytic functions of several parameters inherited from the density
matrix and the input fields e.g.,
$S(\sigma_{SD},\sigma_{IRF},\Gamma,n,\beta,\gamma_{d})$. To simplify the
results, the delay is set to zero ($\tau=0$) in the theory, since we are
interested in the coincidence rate at the center of the interference
histogram. In practice, the experimental coincidence time is not infinitely
narrow and the data is binned with a width of 0.512 ns for the CHSH violation.
We do not consider this explicitly in the theoretical model, but we verified
that artificially increasing the time jitter to take this into account did not
significantly change the result.

Our fitting procedure is conducted as follows. First, we extract $\Gamma$ by
fitting data in section 5 and fix the obtained value for the remaining fits.
An estimate of the pure dephasing rate $\gamma_{d}\lesssim 0.01$ GHz is
obtained from a previous characterization of a similar sample [41]. Based on
this we fix $\gamma_{d}/2\pi=0.01$ GHz for compatibility with the measurement
of the Bell parameter $S$ (see below). Next, we fit the power saturation
transmission data in section 2 to extract the value of the fitting parameter
$\eta$, in good agreement with the power saturation $G^{(2)}$ measurement in
section 3. The remaining parameters $\sigma_{SD}$ and $\beta$ are then
collectively fitted with the two aforementioned sets of power saturation data.
To produce the power-dependent CHSH curve in Figure 3 (c), the fitting
parameters from the filtered power saturation $G^{(2)}$ data are plugged
directly into our CHSH model with no additional fitting. The $S$ parameter is
much more sensitive to the dephasing rate $\gamma_{d}$ than any of the other
measurements, c.f. Figure S6. Any value larger than the chosen value
$\gamma_{d}/2\pi=0.01$ GHz is incompatible with the observed value of $S$ at
low $n$. Lower values of $\gamma_{d}$ would, however also be compatible with
our measurements and the value used ($\gamma_{d}/2\pi=0.01$ GHz) thus provides
an upper bound.

We collect in Table I the parameters that are extracted from fitting the
experimental data comparing with and without the FP filter. The FP filter
effectively reduces leakage from the laser, which we model by varying the QD
coupling parameters, in particular a modification of the effective spectral
diffusion and an effective enhancement of $\beta$, whilst we keep a fixed pure
dephasing rate. This can be justified by considering the illustrative spectra
of the outgoing light presented in Figure S7. As seen in the figure, the
outgoing spectrum consists of a narrow elastic peak and a broad inelastic
emission. The narrow peak comes from laser leakage and vanishes in the ideal
limit of vanishing imperfections ($\beta=1$, $\gamma_{d}=\sigma_{SD}=0$ and
$n\to 0$). Under non-perfect conditions, the central narrow peak appears to
quadratic order in the imperfection parameters ($1-\beta$,
$\sigma_{SD}/\Gamma$, $n$ and $\gamma_{d}/\Gamma$) or through products of
them. Since for typical parameters, $n$ and $\gamma_{d}/\Gamma$ ($\sim 0.01$)
are much smaller than $1-\beta$ and $\sigma_{SD}/\Gamma$ ($\sim 0.1$), the
latter have a much stronger influence on the coherent peak. In contrast to the
narrow peak, the pure dephasing induces (via the quantum jump operator
$c=\sqrt{\gamma_{d}/2}\sigma_{z}$) a spectrally broad inelastic emission
linear in $\gamma_{d}/\Gamma$ (with width $\sim\Gamma+\gamma_{d}$) from the
QD’s excited state [42, 43] and so does inelastic multi-photon scattering
controlled by $n$. Upon application of the spectral filter, the narrow peak
from the laser light is strongly affected, but the broad peak produced by pure
dephasing and multi-photon scattering is largely unchanged. The influence of
the filter can thus be effectively captured by adjusting $\beta$ and the
spectral diffusion $\sigma_{SD}$. To illustrate this, spectra where the
parameters in question are changed individually are given in Figure S7. The
parameters recorded with the FP filter are obtained from fitting the data in
Figure 1 (c), Figure 3 (c) and Figure S3 (b). The parameters recorded without
the FP filter are obtained from the transmission data in Figure 1 (b), Figure
S2 (a) and Figure S3 (a). It is noted that the spectral diffusion also depends
on the acquisition time. Since the data in Figure 1 (b) were collected with a
longer integration time than the data in Figure S2 (a), a relatively larger
spectral diffusion linewidth of 0.52 GHz is found compared to the other case
of 0.39 GHz.

Figure S7:  (color online) Illustrative spectra
$|a(\omega)a^{\dagger}(\omega^{\prime})|$ with various imperfections. Except
when otherwise stated, we use the values $n=0.0024$, $\beta=1$,
$\sigma_{SD}=0$ and $\gamma_{d}=0$. (a) Spectra where the spectral diffusion
$\sigma_{SD}$ is gradually increased. The narrow peak at $\omega=0$ is
strongly affected, whilst the broad peak remains unchanged (b) Spectra for
various values of $n$. Both the central narrow peak and the broader peak are
scaled up with increasing $n$. (c) Spectra with decreasing $\beta$. As with
spectral diffusion, the dominant contribution is to increase the height of the
central narrow peak whilst the broader peak is relatively unchanged. (d)
Spectra with gradually increasing dephasing rate $\gamma_{d}$. The dominant
effect is to increase and broaden the wide peak. The central narrow peak
remains relatively unaffected. Note that in all plots, the width of the
central peak has been scaled by a factor of 100 whilst conserving the total
area so that the effect can be seen more clearly.

apsrev4-1 reflist_SM


*labelnamepunct,

# Higgs cross-section and properties at ATLAS and CMS

Sébastien Rettie  on behalf of the ATLAS and CMS Collaborations  Department of
Physics & Astronomy, University College London,
Gower St, Bloomsbury, London WC1E 6BT, UK

###### Abstract

Recent measurements of Higgs boson cross-section and properties are presented
using up to 139 $\text{fb}^{-1}$ of proton-proton collision data delivered by
the Large Hadron Collider at $\sqrt{s}\leavevmode\nobreak\
=\leavevmode\nobreak\ 13\leavevmode\nobreak\ \text{Te\kern-1.00006ptV}$ and
recorded by the ATLAS and CMS detectors. Three measurements are discussed. The
first is the measurement of Higgs boson production with sizeable transverse
momentum decaying to a $b\bar{b}$ pair. The remaining two measurements exploit
the $H\rightarrow WW^{*}$ decay channel in various production modes: gluon
fusion, vector boson fusion, and production in association with a $W$ or $Z$
boson. The results presented are compatible with Standard Model predictions.

##  1 Introduction

The dataset collected at the Large Hadron Collider (LHC) [1] during 2015-2018
(Run 2) provides an exquisite opportunity to explore the rich phenomenology of
the Higgs boson, and to test the validity of the Standard Model (SM) of
particle physics. So far, measurements carried out by the ATLAS [2] and CMS
[3] experiments are consistent with the SM Higgs boson. The next logical steps
are to increase the precision of these measurements, and carry out
differential measurements. In particular, the simplified template cross-
section (STXS) [4] framework allows for the categorization of events into
Higgs production modes by focusing on key quantities such as the transverse
momentum of the Higgs boson ($p_{\text{T}}^{H}$) and the number of jets in the
final state. The main production modes are gluon fusion ($ggH$), vector boson
fusion ($VBF$), associated production with a $W$ or $Z$ boson ($VH$, $V=W,Z$),
and associated production with top quarks ($t\bar{t}H$) [5]. In this article,
the most recent results by both experiments are presented: Section 2 describes
the analysis of $H\rightarrow b\bar{b}$ decays in the boosted regime by the
ATLAS experiment, and Section 3 presents results by both ATLAS and CMS in the
$H\rightarrow WW^{*}$ decay channel.

##  2 Boosted $H\rightarrow b\bar{b}$ Decays with ATLAS

While unexplored areas where new physics discoveries could be hiding are
numerous, one of the regions of phase space for the Higgs decay with a large
sensitivity to new physics is the so-called boosted regime, where the Higgs
boson has sizable transverse momentum. This regime is difficult to explore due
to the challenging nature of reconstructing the Higgs boson decay products. In
this analysis, the Higgs boson candidate is reconstructed as a single large-
radius jet containing two b-tagged track-jets. A second jet is required to
ensure a di-jet topology. Either the leading or sub-leading large-radius jet
can be considered as the Higgs candidate. The dominant background process for
this measurement are multi-jet events, which are modeled by an analytic
function. The measurement is inclusive in production modes; Table 1 gives the
relative fraction of the four main Higgs production modes present in the
fiducial measurements carried out in this analysis.

Table 1: Signal acceptance times efficiency for the signal regions in the fiducial measurements [6]. Process | $p_{\text{T}}^{H}>450\leavevmode\nobreak\ \text{Ge\kern-1.00006ptV}$ | $p_{\text{T}}^{H}>1\leavevmode\nobreak\ \text{Te\kern-1.00006ptV}$
---|---|---
All | 0.25 | 0.18
$ggH$ | 0.26 | 0.22
$VH$ | 0.27 | 0.19
$VBF$ | 0.22 | 0.15
$t\bar{t}H$ | 0.20 | 0.16

Inclusive, fiducial, and differential measurements are performed via a binned
maximum-likelihood fit to the Higgs candidate jet mass distribution. The event
yields measured in this analysis correspond to Higgs boson production cross-
section values in the fiducial region of

$\begin{array}[]{rcl}\sigma_{H}(p_{\text{T}}^{H}>450\leavevmode\nobreak\
\text{Ge\kern-1.00006ptV})&=&13\pm 57\text{ (stat.)}\pm 22\text{ (syst.)}\pm
3\text{ (theo.})\leavevmode\nobreak\ \text{fb},\\\ &&\\\
\sigma_{H}(p_{\text{T}}^{H}>650\leavevmode\nobreak\
\text{Ge\kern-1.00006ptV})&=&13\pm 16\text{ (stat.)}\pm 7\text{ (syst.)}\pm
3\text{ (theo.})\leavevmode\nobreak\ \text{fb},\\\ &&\\\
\sigma_{H}(p_{\text{T}}^{H}>1\leavevmode\nobreak\
\text{Te\kern-1.00006ptV})&=&3.4\pm 3.9\text{ (stat.)}\pm 1.0\text{
(syst.)}\pm 0.8\text{ (theo.})\leavevmode\nobreak\ \text{fb}.\\\ &&\\\
\end{array}$

These measurements are currently statistically dominated, and provide a first
measurement in the phase space $p_{\text{T}}^{H}>1\leavevmode\nobreak\
\text{Te\kern-1.00006ptV}$.

##  3 $H\rightarrow WW^{*}$ Decays

In the following, Section 3.1 presents the latest result from the ATLAS
Collaboration, focusing on the $ggH$ and $VBF$ production modes, and Section
3.2 describes the CMS analysis measuring the $VH$ production mode.

###  3.1 $ggH$ and $VBF$ ATLAS Measurements

This analysis measures the $ggH$ and $VBF$ production modes of the Higgs
boson, and its subsequent decay into a pair of $W$ bosons, with a final state
consisting of an electron, a muon, and missing transverse momentum. Events are
required to contain an opposite-sign muon-electron pair and missing transverse
momentum. Either a single-lepton or electron-muon trigger is required to have
fired. Most background processes are simulated with Monte Carlo methods,
however the contribution from non-prompt or fake leptons is derived from data.
Compared to the previous Run 2 results from ATLAS, several improvements to the
analysis have been incorporated in addition to the increase in data statistics
from 36 $\text{fb}^{-1}$ to 139 $\text{fb}^{-1}$, most notably a measurement
of the $ggH$ production mode in the final state with two or more reconstructed
jets and measurements of cross-sections in kinematic fiducial regions defined
in the STXS framework. A maximum likelihood fit to the discriminating variable
distributions (either the dilepton transverse mass for the $ggH$ production
mode or the output of a deep neural network for the $VBF$ production mode) is
performed in order to extract the parameters of interest. STXS measurements
are provided in a total of 11 categories, as shown in Figure 1.

Figure 1: Best-fit values and uncertainties for the cross-sections measured in
each of the STXS categories, normalised to the corresponding SM predictions.
The black error bars, green boxes and tan boxes show the total, systematic,
and statistical uncertainties in the measurements, respectively. The grey band
represents the theory uncertainty on the signal production corresponding to
the STXS category [7].

###  3.2 $VH\rightarrow WW^{*}$ CMS Measurements

This result presents measurements of $VH$ production in the $H\rightarrow
WW^{*}$ decay channel, where at least one $W$ boson decays leptonically. The
$VH$ production mode provides a direct probe of the Higgs boson coupling to
vector bosons, and $VH$ kinematics are sensitive to the effects of new physics
beyond the SM. The analysis selects events with final states containing two to
four leptons, and simulates the majority of the background processes with
Monte Carlo methods, with the exception of the non-prompt or fake lepton
backgrounds, which are derived from data. A maximum likelihood fit to the
discriminating variable distributions, either the transverse mass or the
output of a boosted decision tree, is performed in order to obtain the final
measurements. The signal strength, defined as the ratio between the measured
signal cross-section and the SM expectation, provides insight on the
compatibility between the measurements and the SM. The measurements are
carried out in 4 STXS bins, shown in Figure 2, where $p_{\text{T}}^{V}$
corresponds to the transverse momentum of the associated vector boson $V$. The
$p_{\text{T}}^{V}<150\leavevmode\nobreak\ \text{Ge\kern-1.00006ptV}$ phase
space has been measured in this channel for the first time.

Figure 2: Comparison between the combined signal strength, determined by
applying a single signal strength to both $p_{\text{T}}^{V}$ bins and all $VH$
production modes in the STXS fit, and the signal strengths for each
$p_{\text{T}}^{V}$ bin and production mode ($WH$, $ZH$, and $VH$ inclusive)
[8].

##  4 Conclusion

Tremendous progress has been made in measuring properties and cross-sections
of the Higgs boson at the LHC since its discovery 9 years ago. The field is
making big improvements in the precision of a large number of Higgs
measurements, both from using the full Run 2 dataset, and from continually
improving analysis methods. The ATLAS and CMS Collaborations are still only at
the early stages of exploring the Higgs sector; with the start of Run 3
approaching, new data and more analysis improvements will provide
unprecedented levels of precision, and potentially hints of new physics beyond
the SM.

## Acknowledgments

The author would like to thank the organizers for making the
55${}^{\text{th}}$ Rencontres de Moriond a success, and acknowledges the
support of the Natural Sciences and Engineering Research Council of Canada
(NSERC).

## References

  * [1] Lyndon Evans and Philip Bryant  “LHC Machine”  In _JINST_ 3.08 IOP Publishing, 2008, pp. S08001  DOI: 10.1088/1748-0221/3/08/s08001
  * [2] ATLAS Collaboration  “The ATLAS Experiment at the CERN Large Hadron Collider”  In _JINST_ 3.08 IOP Publishing, 2008, pp. S08003  DOI: 10.1088/1748-0221/3/08/s08003
  * [3] CMS Collaboration  “The CMS Experiment at the CERN LHC”  In _JINST_ 3, 2008, pp. S08004  DOI: 10.1088/1748-0221/3/08/S08004
  * [4] Nicolas Berger et al.  “Simplified Template Cross Sections - Stage 1.1”, 2019  URL: https://arxiv.org/abs/1906.02754
  * [5] LHC Higgs Cross Section Working Group  “Handbook of LHC Higgs Cross Sections: 3. Higgs Properties: Report of the LHC Higgs Cross Section Working Group” Comments: 404 pages, 139 figures, to be submitted to CERN Report. Working Group web page: https://twiki.cern.ch/twiki/bin/view/LHCPhysics/CrossSections, CERN Yellow Reports: Monographs CERN-2013-004  Geneva: CERN, 2013  URL: https://cds.cern.ch/record/1559921
  * [6] ATLAS Collaboration  “Study of Higgs-boson production with large transverse momentum using the $H\rightarrow b\bar{b}$ decay with the ATLAS detector”, 2021  URL: http://cds.cern.ch/record/2759284
  * [7] ATLAS Collaboration  “Measurements of gluon fusion and vector-boson-fusion production of the Higgs boson in $H\rightarrow WW^{*}\rightarrow e\nu\mu\nu$ decays using $pp$ collisions at $\sqrt{s}=13$ TeV with the ATLAS detector”, 2021  URL: http://cds.cern.ch/record/2759651
  * [8] CMS Collaboration  “Measurement of Higgs boson production in association with a W or Z boson in the H $\rightarrow$ WW decay channel”, 2021  URL: https://cds.cern.ch/record/2758367

Copyright 2021 CERN for the benefit of the ATLAS and CMS Collaborations.
Reproduction of this article or parts of it is allowed as specified in the CC-
BY-4.0 license.