# Bright electrically contacted circular Bragg grating resonators with
deterministically integrated quantum dots

Setthanat Wijitpatima  [  Normen Auler  [  Priyabata Mudi  Timon Funk  Avijit
Barua  [  Binamra Shrestha  [  Imad Limame  Sven Rodt  [  Dirk Reuter  [
Stephan Reitzenstein<EMAIL_ADDRESS>[

###### Abstract

Cavity-enhanced emission of electrically controlled semiconductor quantum dots
is essential in developing bright quantum devices for real-world quantum
photonic applications. Combining the circular Bragg grating (CBG) approach
with a PIN-diode structure, we propose and implement an innovative concept for
ridge-based electrically-contacted CBG resonators. Through fine-tuning of
device parameters in numerical simulations and deterministic nanoprocessing,
we produced electrically controlled single quantum dot CBG resonators with
excellent electro-optical emission properties. These include multiple
wavelength-tunable emission lines and a photon extraction efficiency (PEE) of
up to (30.4±3.4)%, where refined numerical optimization based on experimental
findings suggests a substantial improvement, promising PEE >50%. Additionally,
the developed quantum light sources yield single-photon purity reaching
(98.8±0.2)% [post-selected: (99.5±0.3)%] and a photon indistinguishability of
(25.8±2.1)% [post-selected: (92.8±4.8)%]. Our results pave the way for high-
performance quantum devices with combined cavity enhancement and deterministic
charge-environment controls, advancing the development of photonic quantum
information systems such as complex quantum repeater networks.

TU Berlin] Institute of Solid State Physics, Technische Universität Berlin,
Hardenbergstraße 36, Berlin 10623, Germany Paderborn] Department of Physics,
Paderborn University, Warburger Str. 100, 33098 Paderborn, Germany Berlin]
Institute of Solid State Physics, Technische Universität Berlin,
Hardenbergstraße 36, Berlin 10623, Germany Paderborn] Department of Physics,
Paderborn University, Warburger Str. 100, 33098 Paderborn, Germany TU Berlin]
Institute of Solid State Physics, Technische Universität Berlin,
Hardenbergstraße 36, Berlin 10623, Germany Paderborn] Department of Physics,
Paderborn University, Warburger Str. 100, 33098 Paderborn, Germany TU Berlin]
Institute of Solid State Physics, Technische Universität Berlin,
Hardenbergstraße 36, Berlin 10623, Germany

##  1 Introduction

In the field of photonic quanatum information processing the concept of
quantum repeaters 1 has emerged as a cornerstone of advanced quantum
communication, serving as a key element to extend the communication range
limited in simple point-to-point concepts by absorption to distances below
about 100 km, and thereby enabling high bit-rate long-range transmission 2.
Implementing quantum repeater networks requires on-demand single-photon
sources with high single-photon purity and indistinguishability, as well as
wavelength tunability and electrically controllable spin-photon interfaces 3,
4. These stringent criteria have driven extensive research efforts,
encompassing the exploration of suitable material systems and tailoring
practical device designs for real-world applications 5. In this context,
semiconductor quantum dots (QDs) have been the most outstanding candidates for
single-photon sources in quantum information technology scenarios 6. Not only
do they exhibit excellent quantum optical properties, but based on
semiconductor materials, they also allow for device integration using advanced
deterministic nanofabrication techniques, thus opening up possibilities to
maximize their optical performance 7.

For applications in photonic quantum information technology, external control
over the electronic states of QDs is of great importance, for instance, to
bring QDs in remote quantum light sources into spectral resonance which is
needed to enable entanglement distribution via Bell-state measurements in
quantum repeater networks 8, 9. In this aspect, researchers have reported
results on electrical charge control of QDs embedded within field-effect
structures such as PIN diodes 10, 11, 12, 13, and applied quantum-confined
Stark effect 14 for spectral fine-tuning 15, 16, 17.

Parallel studies have focused on enhancing the photon extraction efficiency
(PEE) of QD quantum light sources, for instance, via nanophotonic cavities.
Among these cavities, circular Bragg grating (CBG) resonators have gained
prominence due to their broadband emission enhancement in combination with
pronounced light-matter interaction in the Purcell regime of cavity quantum
electrodynamics 18, 19, 4. A CBG resonator is typically fabricated by etching
a series of circular trenches around a central disk with the embedded targeted
QD, creating a high refractive index contrast to realize tight lateral
confinement of the light field 20, 21. When a CBG resonator is paired with a
back-side mirror, such as gold or a distributed Bragg reflector (DBR),
redirecting the emitted photons upward, the vertical collection efficiency can
be significantly boosted to PEE values exceeding 60% in experiments 19, 4.
However, due to the described geometry of a CBG resonator, a QD integrated
inside such a structure is fundamentally electrically isolated. As a
consequence, while the CBG-QD devices have performed well as on-demand devices
under optical pumping 19, 4, the experimental demonstration of electrically-
contacted CBG resonators has remained elusive 22. Notably, electrically-
contacted QD molecule devices with a PIN-diode design and surface CBG
resonators were demonstrated, achieving experimental PEEs up to 24%, limited
by only a partial utilization of the CBG concept 13.

To solve the issue of implementing high brightness CBG single-photon sources
with electrical control of integrated QDs, recent design efforts have explored
ridge-based CBG approaches 23, 24. In these methods, instead of etching
complete circular trenches around QDs as done on optically pumped structures,
narrow ridges are unetched to retain the connection of doped layers in the
central mesa to the area outside of the resonator where metallic pads can be
flexibly prepared for external electrical control (see schematic drawings in
Figure 1). However, to date, only numerical designs and theoretical
discussions of the optical performance of such modified CBGs have been
reported 25, 22, 26.

Therefore, taking an important step forward, we present the first proof-of-
concept experimental results on electrically-controlled QDs in CBG resonators.
We first discuss aspects of designing an electrically-contacted CBG (eCBG)
along with numerical simulations to maximize its PEE, then discuss the device
fabrication and the optical characterization. The design utilizes a DBR as the
back-side mirror to simplify the device fabrication process, instead of using
a gold mirror, which requires a post-growth flip-chip process 19, 4. In this
scenario, the heterostructure including DBR, QDs, and the PIN diode can be
grown epitaxially in a single process using metal-organic chemical vapor
deposition or molecular beam epitaxy. The deterministic nanofabrication
process utilizes marker-based electron-beam lithography (EBL) in combination
with low-temperature cathodoluminescence (CL) mapping to determine the
positions and spectral features of suitable QDs before device integration. To
investigate the electro-optical performance of the fabricated eCBG devices and
assess the quantum optical properties of the deterministically integrated QDs,
bias-voltage dependent micro-photoluminescence ($\mu$PL) measurements and
time-resolved photon correlation measurements are performed.

##  2 Results and discussion

###  2.1 Device Design

Figure 1: Design schematics of an electrically tunable single-photon source
based on a QD integrated into an eCBG in (a) direct-ridge and (b) mazy-ridge
configuration. (c) The vertical cross-section of the device illustrates a QD
embedded in a PIN diode structure with back-side DBR consisting of 10
$\lambda/4$-thick mirror pairs. The device parameters denoted in the
schematics include central mesa radius ($r_{m}$), ring/gap widths
($w_{r1,r2/g1,g2}$), ridge width ($w_{econ}$), etch depth ($t_{etch}$) and QD
top/bottom spacer ($t_{top/bot}$).

Designing a nanophotonic device involves careful consideration of the
interplay between the geometry of individual structural elements in the
proximity of the photon source (in this case, a QD) and their respective
refractive indices. When modifying the well-known optically pumped CBG
resonator concept into a ridge-based eCBG, the introduction of contact ridges
which can lead to scattering and lateral photon losses is the primary factor
influencing, and potentially reducing, the PEE. In this context, two aspects
related to the electrical contacts are considered in particular: the
electrical contact path length and the width of the ridge.

From the electrical perspective, the ridge should ideally have the shortest
path length connecting the central mesa to the external areas to minimize the
electrical resistance. This condition constitutes our first proposed design
featuring direct-ridge electrical connections, illustrated in Figure 1a. In
this design, when considering the optical perspective, such a straight ridge
could provide an undesired optical loss channel for emitted photons from the
central mesa via waveguiding, thereby deteriorating the lateral confinement.
In fact, as a countermeasure, labyrinth-like structures have been proposed,
where the connections between rings were sectioned and rotated 25. As a
comparative experimental case study, another design featuring mazy-ridge
electrical connections to reduce the effect of lateral waveguides is also
proposed here and depicted in Figure 1b. To keep the electrical path as short
as possible, both designs feature only two rings, and in the mazy-ridge
design, only the connections inside the first ring are rotated.

A similar trade-off scenario applies to the width of the ridge as well. For
electrical purposes, the ridge width should ideally be as large as possible to
minimize the resistance, but as the ridge becomes wider, it could support the
propagation of waveguiding modes 27, 28, degrading the optical performance of
the device. While the optical performance could be predicted and optimized via
numerical simulations, the sensible prediction for the electrical behavior of
nanoscale structures requires experimental insights, for instance, on the
width of depleted layers at the etched surfaces. In fact, our eCBGs are
patterned and fabricated utilizing inductively coupled plasma reactive-ion
etching (ICP-RIE), which usually results in residual surface defects with
deteriorated electrical conductivity 29. As a consequence, a functional ridge
must be wide enough to accommodate an electrically active region, sided by
defective surfaces. For this, the minimum ridge width was initially selected
as 100 nm, and the maximum ridge width was intuitively limited by the
effective wavelength of typical InGaAs QD emissions (>900 nm) in GaAs medium,
which is about 250 nm.

Based on the considerations mentioned above, numerical optimizations were
performed to identify the optimal device parameters depicted in Figure 1c, by
applying a Bayesian optimization algorithm to maximize the PEE at the
numerical aperture (NA) of 0.81, which is the NA of the optics later used in
the experiments. The field distributions and PEE were calculated by a three-
dimensional (3D) finite-element method (FEM) in JCMsuite 30 based on the
direct-ridge configuration, exploiting two vertical mirror planes (one along
the main ridge axis and one perpendicular to it) to reduce the demanding
computational time and resources for full 3D simulations. To sensibly limit
the computation time for optimizations, the overall size of the simulated
structures was bound by setting search ranges for the mesa radius ($r_{m}$),
ring widths ($w_{r1,2}$), gap widths ($w_{g1,2}$), and top/bottom-spacer
($t_{top/bot}$) within 100 and 500 nm. The QD was modeled using two point-like
dipoles oriented along the x- and y-axis emitting at the wavelength of 930 nm,
and the absorption effect of dopants in p- and n-doped GaAs was neglected in
the simulation at this wavelength.

Table 1: Optimized device parameters and simulated PEEs obtained from the optimizations with the minimum $w_{econ}$ of 100 nm and other parameter search ranges limited to the maximum of 500 nm (except for $t_{etch}$, which was limited to the maximum of 800 nm). The PEEs (NA = 0.81) were calculated using two perpendicular in-plane (x and y) dipoles oscillating at the wavelength of 930 nm. Structure | Device parameters [nm] | Sim. PEE [%]
---|---|---
$w_{econ}$ | $r_{m}$ | $w_{g1,2}$ | $w_{r1,2}$ | $t_{etch}$
Direct-ridge | 100 | 243 | 268, 369 | 120, 327 | 630 | 43.2
Mazy-ridge | 100 | 243 | 268, 369 | 120, 327 | 630 | 42.1

The device parameters obtained from the numerical optimization and the
calculated PEEs for both designs are listed in Table 1. In this device design,
the top spacer ($t_{top}$) and bottom spacer ($t_{bot}$), which distance the
QDs from the surface and the DBR, were determined to be 333 nm and 119 nm,
respectively. The uppermost part of the top spacer was chosen as a 50 nm
p-doped layer, while the lowermost part of the bottom spacer was chosen as a
30 nm n-doped layer. It is important to point out that the optimal $w_{econ}$
value for the highest PEE was found at the lower boundary of the defined
search domain, which was 100 nm, implying a narrow ridge width was favored.
This numerical study, which predicts PEEs of 43.2% and 42.1% for the two
considered eCBG configurations, confirmed the assumptions mentioned earlier
regarding the ridge width and implied that the main limiting criterion for a
functional ridge width is the electrical conductivity which needed to be
investigated experimentally.

###  2.2 Deterministic Fabrication

Following the aforementioned optimized design parameters, the heterostructure
was epitaxially grown using MBE as follows. Firstly, at a substrate
temperature of 605 °C, a 100 nm GaAs buffer layer was deposited on an undoped
GaAs (100) wafer, followed by the growth of a DBR consisting of ten pairs of
AlAs and GaAs layers with nominal thicknesses of 79 nm and 67 nm,
respectively. Subsequently, a 30 nm Si-doped GaAs layer with a doping
concentration of $2\times 10^{18}$ cm-3 was grown at 555 °C. Then, the
substrate temperature was returned to 605 °C, and an 86 nm GaAs layer was
deposited. Afterward, a 3-minute pause was introduced during which the
substrate temperature was lowered from 605 °C to 505 °C, and the arsenic
pressure was gradually reduced from $2.25\times 10^{-5}$ mbar to $1.50\times
10^{-5}$ mbar. After a 2-minute temperature stabilization period, 3 nm of GaAs
were deposited. Indium was then deposited in a pulsed mode (4 s deposition, 4
s pause) over 20.5 deposition cycles. During the initial 10.5 cycles, the
substrate rotation was halted to create an indium gradient along the [110]
direction. The rotation was resumed for the remaining 10 cycles at a speed of
10 rpm. The QDs were then partially capped with 2.6 nm of GaAs at 485 °C.
Following this, the temperature was rapidly increased to 60 5°C, and a 280 nm
GaAs layer was grown. Lastly, a top gate structure consisting of 10 nm and 40
nm C-doped GaAs layers, with doping concentrations of $1\times 10^{18}$ cm-3
and $1\times 10^{19}$ cm-3, respectively, was deposited at 555 °C. All
temperatures mentioned were determined using band-edge thermometry.

Following the epitaxial growth, a sample piece of (5$\times$5) mm2 with low QD
density (about $2.5\times 10^{6}$ cm-2) was selected, on which electrical
contacts and Au markers were prepared via four EBL steps. The first
lithography was performed to pattern areas at the sample corners, which were
then etched by 410 nm using ICP-RIE to remove the top p-doped layer and
prepare for n-pads. Then, within the etched areas, n-contact pads were
patterned and deposited with thicknesses of 10 nm Ni, 50 nm Au0.88Ge0.12, and
40 nm Au. After that, the sample was rapidly annealed at 420 °C for 90 s with
a 500 K/min ramping rate under N2 atmosphere. For the third EBL, p-contacts
and markers were patterned, and 10 nm Ti and 90 nm Au were deposited. Lastly,
all p- and n-contacts were deposited with an additional 250 nm Au, to achieve
the total pad thickness of 350 nm to ensure the durability of the pads during
the wire bonding procedure.

Figure 2: (a) A CL map overlaid on the SEM picture of a marker field, showing
two spatially localized QDs (labeled QD1 and QD2) in a planar structure. (b) A
CL map of the same marker field after using marker-based EBL to pattern eCBGs
on the QDs deterministically with the main ridge axis aligned along
$[1\bar{1}0]$ orientation. QD1 was integrated into a direct-ridge eCBG,
whereas QD2 was integrated into a mazy-ridge eCBG. (c-d) SEM images of the
fully processed eCBG structures (e) CL spectra of QD1 and QD2 before and after
the integration into eCBGs.

Suitable QDs were identified based on their emission wavelength and separation
from other QDs on the sample with pre-patterned markers by scanning each
marker field using CL mapping at 20 K without applying any bias voltage (open-
circuit). A map consisting of an array of CL spectra and a scanning electron
microscope (SEM) image, as illustrated using an exemplary field in Figure 2a,
provides spectral and spatial information about suitable QDs, which can be
pre-selected for device fabrication. After identifying the QDs, marker-based
EBL 31 and ICP-RIE were performed to pattern eCBGs at the determined QD
positions. As the definitive minimum ridge width for an electrically active
eCBG was unknown, the ridge width was intentionally varied from 100 to 130 nm
on different patterned structures without changing any other parameters.

The enhancement from the eCBG integration on the QDs could be observed via a
one-to-one comparison by performing another CL mapping on the same marker
field after the integration, as depicted in Figure 2b, showing that devices
were patterned on the determined QDs. Note that the QD emission spectra could
be observed on all 14 patterned devices regardless of their patterned ridge
widths as they were visible in open-circuit states. In the marker field, two
QDs (namely QD1 and QD2) were integrated into a direct-ridge structure (eCBG1)
and a mazy-ridge structure (eCBG2) whose SEM images are illustrated in Figure
2c and d, respectively. The images indicate smooth etched surfaces and well-
defined fabrication conditions. Figure 2e shows the CL spectra of both QDs
before and after device integration using the same excitation and collection
parameters. It is important to note that the intensity between QDs could not
be compared fairly as the CL mapping was performed using a parabolic mirror,
which resulted in a non-uniform intensity profile throughout the map.
Qualitatively, the post-integration spectra of both QDs clearly indicate
strongly enhanced extraction efficiency and improved signal-to-noise ratio.

###  2.3 Electric-field Dependent Photoluminescence Measurements

Figure 3: Contour plot of bias-voltage dependent $\mu$PL spectra of the
eCBG2-QD recorded at 4 K showing several emission lines originating from
excitonic ($\mathrm{X}$), biexcitonic ($\mathrm{XX}$), singly-charged
($\mathrm{X}^{+/-}$, $\mathrm{XX}^{+/-}$), and doubly charged
($\mathrm{X}^{2+/2-}$) states observed at different bias voltages. The lines
observed at bias voltages under 1.4 V exhibit spectral shifts due to the
quantum-confined Stark effect, allowing for emission wavelength tuning via
external electrical controls.

After the deterministic fabrication, electrical connections between an
external voltage source and the diode structures of the fabricated devices
were made through wire bonding between a chip carrier and deposited metallic
contact pads. The chip carrier was then installed in the optical setup
described in Section 4.1 for bias-voltage dependent $\mu$PL measurements. The
eCBGs were investigated under pulsed (80 MHz) ps-mode excitation via wetting
layer at 865 nm using a Ti:Sapphire laser. By varying the external bias
voltage ($V_{ex}$) between 0 and 3 V, we observed that the fabricated direct-
ridge eCBGs whose ridges were narrower than (110$\pm$5) nm exhibited no
voltage dependency on their optical spectra, as well as the mazy-ridge eCBGs
whose ridges were narrower than (120$\pm$5) nm. Both eCBG1 and eCBG2, as well
as other eCBGs that consisted of wider ridges than these values, showed clear
and similar bias-voltage dependency, as illustrated using eCBG2 as an example
in Figure 3.

The $\mu$PL contour plot displays different QD emission lines observed at
different $V_{ex}$. Most lines, which appeared below about 1.4 V, featured the
quantum-confined Stark effect by which the emission wavelengths shifted
gradually over the $V_{ex}$ change, whereas the emission lines appearing above
1.4 V remained energetically constant. This behavior can be explained by the
electronic band-bending of the PIN diode structure, which was initially caused
by the built-in electric field ($E_{in}$) 14. At zero bias, the electronic
bands were strongly bent by the field which promotes the quantum tunneling
rate of electrons from the wetting layer to GaAs, reducing carrier capturing
into the QD, hence, the low PL emission rates were observed at 0 V 32, 33. As
the $V_{ex}$ increased, the band-bending and electron tunneling weakened,
enabling the carrier capturing in the QD and subsequent carrier recombination,
hence the emission lines became visible while shifting energetically 33, 15.
Notably, bias-induced energetic shifts of up to 0.7 meV were observed in the
studied devices. When the externally applied electric field became as equally
strong as the $E_{in}$, which was at around $V_{ex}$ = 1.4 V), flat-band
conditions were achieved, and no quantum confined Stark tuning could be
observed anymore.

The nature of observed emission lines was determined by performing power- and
polarization-dependent $\mu$PL measurements at the corresponding $V_{ex}$. For
the power-dependent measurements, the $\mu$PL spectra were recorded at
different optical excitation power. By selectively fitting the area intensity
of each emission line and plotting them against the excitation power in a log-
log scale, a power exponent ($m$) can be extracted from the slope of the
linear section in the plot. Figure 4a shows an example of a power-dependent
$\mu$PL measurement at 1.3 V where an emission line (later determined as the
$\mathrm{X}^{-}$ emission line) was selectively fitted. The same process was
done for other lines and the findings are listed in Table 2. With extracted
$m$’s, excitonic and biexcitonic lines were categorized by their linear
($m\sim 1$) and superlinear ($m$>1) power dependencies, respectively 34.
Complementarily, the information on polarization properties of these emission
lines was necessary to determine their origins unambiguously, caused by the
spin-related fine structures 35, 36, 37. For this, the polarization-dependent
measurements were performed by inserting a rotatable $\lambda/2$-plate and a
linear polarizer in the collection path of the $\mu$PL setup. By changing the
$\lambda/2$-plate angle, different polarization components of each emission
line were recorded. Figure 4b shows the two polarization components of each
emission line along $[1\bar{1}0]$ and [110] orientations of the sample. For
each line, the peak energy difference between the two polarization angles and
the linewidth (full-width half-maximum measured at a certain polarization
angle where the narrowest emission was observed) were determined and are also
listed in Table 2. Combining the findings from these two measurements, the
emission lines were determined as follows.

Figure 4: (a) Power-dependent $\mu$PL spectra from eCGB2 biased at 1.3 V, and the area intensity of the $\mathrm{X}^{-}$ emission line plotted against excitation power in a log-log scale. (b) Polarization-dependent $\mu$PL spectra from eCBG2 biased at different bias voltages, showing multiple emission lines. The two perpendicular polarization angles plotted correspond to the $[1\bar{1}0]$ (blue) and [110] (purple) orientations of the sample. Table 2: Emission lines observed from eCBG2 at different bias voltages and their relevant parameters, including emission wavelengths ($\lambda$) and central energies (E), power exponents ($m$) from power-dependent $\mu$PL measurements, fitted peak energy deviations ($\Delta$E) and minimum emission linewidths (FWHM) from polarization-dependent $\mu$PL measurements. Emission | Voltage [V] |  $\lambda$ [nm] | E [meV] | $m$ |  $\Delta$E [$\mu$eV] | FWHM [$\mu$eV]
---|---|---|---|---|---|---
$\mathrm{X}^{2+}$ | 0.6 | 909.11 | 1366.58 | 1.51±0.01 | 9.0±1.8 | 31.5±2.9
$\mathrm{X}^{+}$ | 0.8 | 909.14 | 1366.53 | 0.85±0.01 | None | 23.7±2.5
$\mathrm{XX}^{+}_{1}$ | 0.8 | 910.73 | 1364.15 | 1.50±0.02 | None | 24.8±4.6
$\mathrm{XX}^{+}_{2}$ | 0.8 | 910.80 | 1364.05 | 1.60±0.03 | None | 29.7±3.1
X | 1.2 | 908.92 | 1366.87 | 0.82±0.01 | 37.2±2.6 | 24.6±4.7
XX | 1.2 | 910.96 | 1363.81 | 1.28±0.01 | 34.4±2.4 | 27.2±2.4
$\mathrm{XX}^{-}_{1}$ | 1.3 | 911.79 | 1362.57 | 1.69±0.04 | None | 28.9±3.1
$\mathrm{XX}^{-}_{2}$ | 1.3 | 911.87 | 1362.44 | 1.78±0.02 | None | 29.4±1.2
$\mathrm{X}^{-}$ | 1.3 | 912.44 | 1361.60 | 0.97±0.01 | None | 16.6±1.1
$\mathrm{X}^{2-}$ | 1.4 | 912.59 | 1361.37 | 1.29±0.01 | 7.7±1.4 | 46.0±1.2

Firstly, the most fundamental states of QD, exciton-biexciton (X-XX) emissions
were observed at around 1.2 V exhibiting a strong polarization dependency with
the opposite energetic deviation direction. The energy deviations indicate the
X-XX fine-structure splittings of (37.2$\pm$2.6) for X and (34.4$\pm$2.4)
$\mu$eV for XX, respectively, which are typical values for an InGaAs QD 38.
Secondly, when varying $V_{ex}$ slightly to a lower value (0.8 V), three
prominent lines were observed without any polarization-dependent energetic
variation, consisting of a single line with $m\sim 1$ and double lines with
$m>1$. This suggests that they originate from singly-charged excitonic and
biexcitonic states. Moreover, the double lines exhibited clear polarization-
dependent intensities, where each line was more prominent than the other at
different polarization angles. This strongly supports that they originate from
the relaxation of the charged biexcitonic state into the two triplet states of
excited-trion, which typically generate elliptically-polarized photons 35, 39.
Thirdly, when decreasing $V_{ex}$ even further (<0.8 V), a new prominent line
appeared having $m>1$ and showing a polarization-dependent emission. The
emission was observed as a single line at a certain polarization angle, but
the line broadened with a lower peak intensity when observed at different
polarization angles. This suggests that the emission consists of two different
lines, one without polarization dependency and another with polarization-
dependent energetic variation, which could be attributed to the two triplet
transitions from doubly-charged excitonic states 40, 36.

Another similar set of lines was observed as well when varying the voltage to
the higher values ($V_{ex}$>1.2 V), indicating that they originate from the
analogous spin states with the opposite charge sign. Note that, at lower
voltages ($V_{ex}$<1.2 V) where the band-bending exists, the tunneling rate of
electrons is more significantly promoted than that of holes, due to the
smaller effective mass, suggesting the hole is more effectively captured into
the QD and the prominent emission lines are expected to originate from the
positively charged states 32, 33. On the other hand, at higher voltages
($V_{ex}$>1.2 V) where the tunneling weakens, the electron capturing in the QD
becomes effective, suggesting that the emission lines in this regime originate
from negatively charged states instead. Additionally, the redshifts of the
higher negatively-charged emission lines also support the assumptions, in
agreement with previously reported studies 40, 41, 42. Consequently, all the
prominent emission lines in the voltage-dependent $\mu$PL spectra were
assigned, ranging from the positive doubly-charged line ($\mathrm{X}^{2+}$) at
low voltages to the negative doubly-charged line ($\mathrm{X}^{2-}$) at high
voltages, as shown in Figure 3.

Understanding the spin-related origins of the emission lines is crucial for
application in, for instance, quantum repeater networks, which require well-
defined light-matter interactions between the flying and stationary qubits 6.
The results presented in this section demonstrate the determination of the
origins of multiple QD emission lines and confirm the capability of the
designed eCBG devices, allowing for deterministic controls of electronic
states of QDs in the cavity. This feature can significantly enhance the
performance of spin-photon interfaces, crucial for mediating photon
entanglements in quantum repeater networks 43. The emission wavelength
tunability of the QD in the cavity enabled by this design can also benefit the
applications with quantum memories 44. Moreover, the design can be directly
applied to QD molecules, which have promising applications in photonic
cluster-state generation 45, 13, 46, 47.

###  2.4 Optical Enhancements

Table 3: Structural parameters of fabricated device obtained from SEM imaging (with measurement errors $\leq$ 5 nm) and the simulated and measured PEEs (NA = 0.81). The simulations were performed using two perpendicular in-plane (x and y) dipoles oscillating at the corresponding wavelengths to the measured emissions. For planar QDs, the PEE was simulated for the wavelength range of 900 – 930 nm, and measured from nine different QDs. Structure | Device parameters [nm] | PEE [%] | $\lambda$ [nm]
---|---|---|---
$w_{econ}$ | $r_{m}$ | $w_{g1,2}$ | $w_{r1,2}$ | $t_{etch}$ | Simulated | Measured
Planar | - | - | - | - | 0 | 2.02$\pm$0.19 | 1.69$\pm$0.59 | 900 – 930
eCBG1 | 122 | 240 | 250, 368 | 137, 324 | 633 | 24.53 | 12.6$\pm$1.4 | 918.1
eCBG2 | 132 | 245 | 240, 368 | 137, 328 | 633 | 31.93 | 30.4$\pm$3.4 | 912.7

As mentioned in the previous section, the electrically functioning eCBGs were
experimentally found to require wider ridges than the originally optimized
value (100 nm). For a fair comparison between the simulation and experimental
results, the actual geometry of the fabricated structures (eCBG1 and eCBG2)
was measured using SEM, and the PEEs were re-simulated based on the measured
structural parameters which are listed in Table 3. Note that the experimental
PEE can deviate from the simulated value due to the possible mismatch of the
QD position and the integrated eCBG 48, which could not be determined in this
work, limited by the small diameter of the central mesa.

To obtain experimental PEEs, the $\mu$PL setup efficiency was measured using a
continuous-wave laser at the same wavelength as the QD emission, yielding the
value of (7.3$\pm$0.8)%. Then, nine different QDs in the planar structure and
the studied devices (eCBG1 and eCBG2) were electrically biased at 1.5 V to
achieve the flat-band conditions for maximum electric-field dependent PL
emissions 32, and the emission spectrum from each QD was recorded under pulsed
(80 MHz) excitation at 800 nm using a Ti:Sapphire laser with the corresponding
saturation optical pump power. The PEE was calculated using the formula:

$\mathrm{PEE}=\frac{n}{\eta_{setup}*f}$

where $n$ is the single-photon emission rate, $\eta_{setup}$ is the setup
efficiency, and $f$ is the excitation frequency. For eCBG1 and eCBG2, single-
photon emission rates were measured to be (732$\pm$18) kHz and (1.77$\pm$0.34)
MHz, equivalent to PEEs of (12.6$\pm$1.4)%. and (30.4$\pm$3.4)%, respectively.
For planar QDs, the average (and standard deviation) emission rate was found
to be (98.6$\pm$26.1) kHz, which led to an average PEE of (1.69$\pm$0.59)%.

The comparison between the simulated and experimental PEE for eCBG2 shows
impressive conformity, indicating accurate numerical modeling and good
fabrication quality. The reliability of this comparison can be supported by
the results from planar QDs, which also show good agreement between the
simulation and the measured value. Deviated from the simulation, eCBG1 yielded
noticeably lower efficiency, similar to the other two fabricated direct-ridge
eCBGs with the same nominal parameters. The cause for the larger deviation of
the experimental PEE from the simulated one for eCBG1 than that of eCBG2 may
be multi-factored. Firstly, although all eCBGs were fabricated under identical
conditions, since the actual eCBG-to-QD position offsets could not be
determined, the effect of alignment mismatch cannot be completely ruled out.
Secondly, the simulations were performed with a QD modeled as two equally
contributing perpendicular point-like dipoles, which might not represent the
actual QD emission well enough in the presence of ridge-like structures
breaking the cylindrical symmetry. As a result, the simulation may have
underestimated the undesired direct-ridge effect, causing a larger difference
between simulated and experimental PEEs only for direct-ridge eCBGs. An
additional finding described as follows supports this assumption further.

Figure 5: (a) Simulated PEEs (NA = 0.81) of eCBG1 and eCBG2 using a single
point-like in-plane dipole source oriented at different angles related to the
main ridge axis. (b) Normalized intensities of the $\mathrm{X}^{-}$ emission
lines from eCBG1 and eCBG2 in polarization-dependent PL measurements, where
the 0° and 90° correspond to the linear polarization components along $[110]$
and $[1\bar{1}0]$ orientations, respectively.

Due to the presence of the ridges in eCBGs, the polarization-dependent optical
performance of the fabricated devices was intuitively anticipated. On the
simulation side, the effect was predicted by calculating the PEE using a
single in-plane dipole oriented at different angles related to the main ridge
axis. The simulation results for eCBG1 and eCBG2, plotted in Figure 5a,
illustrate good enhancements for the dipoles oscillating perpendicularly to
the innermost ridge axis, whereas the enhancements deteriorated the most when
the dipole was placed in parallel instead. The simulated PEEs could be
normalized and calculated into a degree of linear polarization:
$DLP=(I_{max}-I_{min})/(I_{max}+I_{min})$, where $I_{max/min}$ are maximum and
minimum intensity (equivalent to PEE). Interestingly, both eCBG1 and eCBG2
yielded the same simulated DLP of 17%, regardless of the difference in the
ridge continuity. On the experimental side, the effect could be observed in
polarization-dependent $\mu$PL measurements mentioned in the previous section.
To exemplify the effect, the $\mathrm{X}^{-}$ emission line (912.44 nm) which
is intrinsically circularly polarized was selected, and the fitted
polarization-dependent intensity plots are shown in Figure 5b. Note that the
polarization angle of 0° corresponded to the polarization components along the
$[1\bar{1}0]$ orientation of the sample, to which the main ridge axes of the
fabricated devices were aligned. Qualitatively, the experimental plots closely
match the simulation results well, clearly confirming the polarization-
dependency of PEEs. The experimental DLPs were calculated in the same fashion
as earlier, yielding (36$\pm$1)% and (17$\pm$1)% for eCBG1 and eCBG2,
respectively. For eCBG1, the experimental result implies poorer enhancement of
the emission along the main ridge axis than the simulation predicted, which
was another piece of evidence signifying the underestimation of the direct-
ridge effect on the optical enhancement of the eCBG. On the other hand,
another perfect match between simulation and experimental results on eCBG2
could be observed for DLP values. Thus, we conclude that this polarization-
dependency of PEEs is mostly affected by the innermost ridges, which suggests
that the outer ridges can be modified (e.g. for larger widths) to support
electrical conductivity without significantly degrading the optical property.

Table 4: Optimized device parameters and simulated PEEs obtained from the optimizations with the minimum $w_{econ}$ of 120 nm and other parameter search ranges up to the maximum of 1000 nm. The PEEs (NA = 0.81) were calculated using two perpendicular in-plane (x and y) dipoles oscillating at the wavelength of 930 nm. Structure | Device parameters [nm] | Sim. PEE [%]
---|---|---
$w_{econ}$ | $r_{m}$ | $w_{g1,2}$ | $w_{r1,2}$ | $t_{etch}$
Direct-ridge | 120 | 256 | 960, 607 | 257, 628 | 630 | 55.65
Mazy-ridge | 120 | 256 | 960, 607 | 257, 628 | 630 | 56.54

To demonstrate the capability of improving the current designs, another
numerical optimization was performed by setting $w_{econ}$ to 120 nm and
expanding the parameter search ranges for ring and gap widths up to 1000 nm.
The newly optimized parameters yielded the PEE > 50%, listed in Table 4,
promising a substantial improvement without changing the heterostructure
parameters. With numerical refinement, the proposed design can be adapted with
more pairs of back-side DBR to boost the PEE even further.

###  2.5 Single-photon Characteristics

Figure 6: (a) Time-resolved $\mu$PL measurements of the $\mathrm{X}^{-}$
emission line from eCBG2 biased at 1.3 V and measured at 4 K under 1, 2, and 4
$\mu$W excitation powers, which correspond to the emissions with $\sim$
20$\%$, 40$\%$, and 70$\%$ of the saturation intensity, respectively. The
respective fitted profiles (red lines) yield decay lifetimes of 0.78, 0.83,
and 0.89 ns (with fitting errors < 0.01 ns). (b) The Hanbury Brown and Twiss
correlation statistics $g^{(2)}(\tau)$ of the same emission line under 4
$\mu$W excitation power. The histogram shows excellent multi-photon
suppression $g^{(2)}(0)$ of (1.2±0.2)%, calculated from the area under the
fitted profile (red line), and a post-selected $g^{(2)}(0)$ of (0.5±0.3)%.

In addition to the optical performance of the fabricated devices, the quantum
optical properties were also assessed by performing time-resolved $\mu$PL
measurements. For this, the $\mathrm{X}^{-}$ emission line (912.44 nm) from
eCBG2 was selected, as it appeared near the flat-band condition, resulting in
a bright emission and a resolution-limited linewidth of (16.6$\pm$1.1)
$\mu$eV. The experiment was conducted under pulsed (80 MHz) ps-mode excitation
at 865 nm using a Ti:Sapphire laser, and the detection path included a
monochromator as a spectral filter and a photon correlation setup equipped
with superconducting nanowire single-photon detectors (SNSPDs). Firstly, the
time-resolved $\mu$PL measurements were performed under different excitation
powers, and the results are illustrated in Figure 6. Note that excitation
powers of 1, 2, and 4 $\mu$W corresponded to the emissions with $\sim$ 20$\%$,
40$\%$, and 70$\%$ of the saturation intensity, respectively, as previously
shown in Figure 4a. The emission lifetimes were found to increase gradually
from (0.78$\pm$0.01) to (0.89$\pm$0.01) ns, as the power increased from 1 to 4
$\mu$W. For further measurements, the excitation power of 4 $\mu$W was applied
to maximize the emission rate.

Following the lifetime measurements, a Hanbury Brown and Twiss (HBT) setup 49
was used to assess the single-photon purity. The recorded photon correlation
histogram, plotted in Figure 6, shows a very low coincidence rate observed
near the zero-time delay, which can be expected from carrier-recapturing
processes due to the non-resonant excitation scheme. Nevertheless, the
emission features an excellent multi-photon suppression rate with a
$g^{(2)}(0)$ of (1.2$\pm$0.2)% and a post-selected $g^{(2)}(0)$ of
(0.5$\pm$0.3)%, which are equivalent to the non-post-selected and post-
selected single-photon purities of (98.8$\pm$0.2)% and (99.5$\pm$0.3)%,
respectively. Notably, the multi-photon suppression rate can be further
improved by either reducing the excitation power, at the cost of a lower
emission rate or by operating the device under resonant excitation 50.

Figure 7: The Hong-Ou-Mandel correlation statistics $g^{(2)}(\tau)$ of photons
emitted from the $\mathrm{X}^{-}$ emission line of eCBG2 biased at 1.3 V and
measured at 4 K, for parallel (a) and orthogonal (b) polarization. The non-
post-selected visibility ($V_{raw}$) of (25.8±2.1)% and the post-selected
visibility ($V_{post}$) of (92.8±4.8)% were calculated.

After determining the single-photon purity, the photon indistinguishability
was assessed using a Hong-Ou-Mandel (HOM) correlation setup 51. In this case,
a sequence of double excitation pulses with a 4 ns pulse separation was used,
which was matched with a 4 ns fiber delay line in one interferometer arm in
the HOM setup. The HOM interference histograms from parallel and orthogonal
polarization configurations are plotted in Figure 7. The non-ideal raw
visibility was obtained as $V_{raw}$ = (25.8$\pm$2.1)%, evidenced by observed
coincidences near the zero-time delay. These coincidences were observed,
potentially because the QD had a non-enhanced recombination lifetime allowing
the carriers to interact with the solid-state environment, and due to spectral
diffusion induced by charge fluctuations at the etched surface near the QD 52.
However, prominent indistinguishability could be observed at zero-time delay
in the parallel polarization configuration, leading to the post-selected
visibility $V_{post}$ = (92.8$\pm$4.8)%. This indicates that QD could
intrinsically emit indistinguishable photons, suggesting that optimizing the
design toward Purcell enhancement, as well as operating the device under
resonant excitation and applying surface passivation, could improve the raw
visibility of the device further.

##  3 Conclusion

In this report, we proposed and implemented an innovative concept for high-
performance electrically controlled QD-CBG single-photon sources. By going
beyond simple optically controlled structures, our work can To maximize the
brightness of the quantum devices we explored two different ridge-based eCBG
designs, based on the trade-off between electrical and optical performance.
The sensitivity of the electrical path length and the ridge width to the
performance prompted careful optimizations between numerical and experimental
results. A direct-ridge configuration featured better electrical connection,
requiring a slightly narrower minimum ridge width for an electrically
functioning device, while optically performing poorer, especially for the
polarization in parallel to the ridge axis. On the other hand, a mazy-ridge
configuration featured a slightly longer electrical path length, requiring a
wider minimum ridge width to function, but yielded superior optical
enhancement. The determinsitically fabricated mazy-ridge eCBG-QD achieved a
PEE of (30.4±3.4)%, in quantitative agreement with numerical simulations which
promise substantial improvement with PEE >50 % when using re-optimized
parameters based on the present experiment findings. The electrical operation
of the eCBG-QD yields precise charge control and a quantum-confined Stark
effect, providing robust capability to fine-tune the emission wavelength up to
0.7 nm and selectively enhance emission lines from specific QD states. The
single-photon emission characteristics show impressive a single-photon purity
of (98.8±0.2)% and post-selected visibility of (92.8±4.8)%. The results
underscore the high potential of our advanced quantum device concept, which
can pave the way for many applications of photonic quantum information
technology requiring not only excellent optical and quantum optical properties
but also tight electrical control of the integrated quantum emitters.

##  4 Experimental methods

###  4.1 Optical Setups

The $\mu$PL measurements were performed on the sample cooled to 4 K in a
closed-cycle cryostat using a Ti:Sapphire laser operating in pulsed ps-mode
(80 MHz) at tunable wavelengths (800-865 nm). The emitted photons were
collected with a cold aspheric objective lens (NA = 0.81). With a beam
splitter, which separates excitation and collection paths, the collected beam
was directed through a 900 nm long-pass filter to a monochromator with a
1500-line-per-nm grating and detected using a CCD camera. For time-resolved
measurements, two different correlation setups, equipped with SNSPDs, were
used: (1) HBT setup, where the spectrally filtered photons were sent through a
1:1 single-mode fiber beam splitter and detected with two SNSPDs were
recorded, and (2) Hong-Ou-Mandel setup, where the spectrally filtered photons
were sent through a polarization-maintaining beam splitter in combination with
polarization-control paths (with $\lambda/2-$ and $\lambda/4-$plates) and a
$\sim$4 ns time delay on one arm, before sent back into a 1:1 beam splitter
for two-photon interference effect and detected with two SNSPDs. In these
correlation setups, the time correlation between two detected photons in each
detector was recorded.

This work was financially supported by the German Federal Ministry of
Education and Research (BMBF) through the project “QR.X –
Quantenrepeater.Link” with funding IDs: 16KISQ012 and 16KISQ014. The authors
further thank Johannes Schall, Ching-Wen Shih, Chirag Palekar, Lucas Rickert,
Niels Heermeier, and Aris Koulas-Simos for technical support and guidance.

The data that support the findings of this study are available from the
corresponding author upon a reasonable request.

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# Some remarks on graphs without rainbow triangles

Peter Frankl  Alfréd Rényi Institute of Mathematics, Hungarian Academy of
Sciences.  Ervin Győri  Alfréd Rényi Institute of Mathematics, Hungarian
Academy of Sciences.  Zhen He  Alfréd Rényi Institute of Mathematics,
Hungarian Academy of Sciences.  Department of Mathematical Sciences, Tsinghua
University.  Zequn Lv  Alfréd Rényi Institute of Mathematics, Hungarian
Academy of Sciences.  Department of Mathematical Sciences, Tsinghua
University.  Nika Salia  Alfréd Rényi Institute of Mathematics, Hungarian
Academy of Sciences.  Asian University for Women.  Casey Tompkins  Alfréd
Rényi Institute of Mathematics, Hungarian Academy of Sciences.  Kitti Varga
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences.
Department of Computer Science and Information Theory, Budapest University of
Technology and Economics.  Xiutao Zhu  Alfréd Rényi Institute of Mathematics,
Hungarian Academy of Sciences.  Department of Mathematics, Nanjing University.

###### Abstract

Given graphs $G_{1}$, $G_{2}$ and $G_{3}$ on a common vertex set of size $n$,
a rainbow triangle is a triangle consisting of one edge from each $G_{i}$. In
this note we provide a counterexample to a conjecture of Frankl on the maximum
product of the sizes of the edge sets of three graphs avoiding a rainbow
triangle. Moreover we propose an alternative conjecture and prove it in the
case when every pair is an edge in at least one of the graphs.

The classical theorem of Mantel [7] asserts that the maximum number of edges
in an $n$-vertex graph containing no triangle is
$\left\lfloor{n^{2}/4}\right\rfloor$. This problem was generalized by Turán
[8] who showed that the maximum number of edges in an $n$-vertex graph with no
complete graph $K_{r}$ as a subgraph is obtained by taking a complete
$(r-1)$-partite graph with parts of size $\left\lfloor{n/(r-1)}\right\rfloor$
or $\left\lceil{n/(r-1)}\right\rceil$.

Many natural generalizations of these theorems have been considered. For a
graph $G$, let $E(G)$ denote the edge set of $G$ and $e(G)=|E(G)|$. Of
particular importance to the present note is an extremal problem due to
Keevash, Saks, Sudakov and Verstraëte [6]. They considered (among other
problems) the maximum of $e(G_{1})+e(G_{2})+\cdots+e(G_{k})$ across $k$ graphs
on a common vertex set of size $n$ with the property that there is no $K_{r}$
with each of its edges coming from a distinct $E(G_{i})$. Such a $K_{r}$ is
referred to as a _rainbow_ $K_{r}$. For this problem there are two natural
constructions. On the one hand, if $k\geq\frac{r^{2}-1}{2}$, then we take $k$
identical copies of the Turán graph. On the other hand, when $\binom{r}{2}\leq
k<\frac{r^{2}-1}{2}$ it is better to take $\binom{r}{2}-1$ copies of the
complete graph and let the remaining graphs have empty edge sets (of course
for $k<\binom{r}{2}$ it is optimal to take all $k$ graphs to be complete).
Keevash, Saks, Sudakov and Verstraëte [6] also proved that for $3$-chromatic
color-critical graphs and $n$ sufficiently large an analogous result holds in
which either a construction consisting of $k$ copies of a Turán graph, or a
construction consisting of complete graphs and graphs with no edges is
optimal. Chakraborti, Kim, Lee, Liu and Seo [2] showed the same holds for
$4$-chromatic color-critical graphs and almost all color-critical graphs of
chromatic number at least $5$, partially verifying a conjecture from [6]. In
the context of extremal set theory, rainbow extremal problems have also been
considered earlier, for example by Hilton [5].

After one has bounds on $e(G_{1})+e(G_{2})+\cdots+e(G_{k})$, it is natural to
consider maximizing other objective functions over $e(G_{1})$, $e(G_{2})$,…,
$e(G_{k})$. The problem of maximizing
$\min\left(e(G_{1}),e(G_{2}),e(G_{3})\right)$ while avoiding a rainbow
triangle was considered by Aharoni, DeVos, de la Maza, Montejano and Šámalin
[1], answering a question of Diwan and Mubayi [3]. For the problem of
maximizing the product $e(G_{1})e(G_{2})e(G_{3})$, Frankl [4] gave the
following conjecture.

######  Conjecture 1 (Frankl).

Let $G_{1},G_{2},G_{3}$ be graphs with a common vertex set on $n$ vertices
with no rainbow triangle. Then

$e(G_{1})e(G_{2})e(G_{3})\leq\left\lfloor{\frac{n^{2}}{4}}\right\rfloor^{3}.$

Taking $G_{1},G_{2},G_{3}$ to be $3$ copies of the complete bipartite graph
with almost equal parts attains this bound. Frankl proved that under the
additional assumption $E(G_{1})\subseteq E(G_{2})$ and $E(G_{1})\subseteq
E(G_{3})$, Conjecture 1 holds. We show that Frankl’s conjecture does not hold
in the general case.

Let $\gamma$ be the maximum of

$\frac{x^{2}}{2}\left(\frac{x^{2}}{2}+\frac{(1-x)^{2}}{2}\right)\left(x(1-x)+\frac{(1-x)^{2}}{2}\right)$
(1)

on $[0,1]$, and assume $\gamma$ is attained at $x=x_{0}$. Note that
$\frac{1}{52}<\gamma<\frac{1}{51}$ (and $x_{0}\approx.729$). We have the
following.

######  Theorem 1.

There exist graphs $G_{1},G_{2},G_{3}$ with a common $n$-vertex set and no
rainbow triangle such that

$e(G_{1})e(G_{2})e(G_{3})\geq\gamma n^{6}(1-o(1)).$

###### Proof.

Let $[n]=X\cup Y$ be a partition of $[n]$ with $X$ of size approximately
$x_{0}n$ and $Y$ is of size approximately $(1-x_{0})n$. Let $G_{1}$ consist of
a complete graph on $X$, and $G_{2}$ consist of the union of a complete graph
on $X$ and a complete graph on $Y$ and let $G_{3}$ consist of a complete graph
on $Y$ as well as all edges between $X$ and $Y$. Observe that the product
$e(G_{1})e(G_{2})e(G_{3})$ is asymptotically $\gamma n^{6}$. ∎

Moreover, we believe that the expression in Theorem 1 should be best possible.

######  Conjecture 2.

For three graphs $G_{1},G_{2},G_{3}$ on a common vertex set of size $n$ with
no rainbow triangle we have

$e(G_{1})e(G_{2})e(G_{3})\leq\gamma n^{6}(1+o(1)).$

We prove Conjecture 2 under an additional assumption that every pair is an
edge in at least one of the $E(G_{i})$. For convenience we say a pair of
vertices (or an edge) in the $n$-vertex ground set is colored if it belongs to
at least one of the sets $E(G_{1})$, $E(G_{2})$, $E(G_{3})$. An edge is
$t$-colored if it belongs to exactly $t$ of the sets $E(G_{1})$, $E(G_{2})$,
$E(G_{3})$.

######  Theorem 2.

Let $G_{1},G_{2},G_{3}$ be graphs with a common vertex set on $n$ vertices
with no rainbow triangle. If $n$ is sufficiently large and every pair of
vertices on the ground set is colored, then the construction described in
Theorem 1 maximizes $e(G_{1})e(G_{2})e(G_{3})$.

###### Proof.

Let $n$ be sufficiently large, and let $G_{1},G_{2},G_{3}$ be graphs with a
common set of $n$ vertices with no rainbow triangle and assume every edge is
colored and $e(G_{1})e(G_{2})e(G_{3})$ is maximal. For a vertex $v$, let
$N_{i}(v)$ be the set of neighbors of $v$ in $G_{i}$ which are not neighbors
of $v$ in $G_{l}$ for all $l\neq i$. Let $\\{i,j,l\\}=\\{1,2,3\\}$, then we
denote the set of neighbors of the vertex $v$ in $G_{i}$ and $G_{j}$ but not
in $G_{l}$ by $N_{i,j}(v)$.

Assume that $e(G_{1})e(G_{2})e(G_{3})\geq\gamma n^{6}(1-o(1)).$ Then we have
that $e(G_{1})+e(G_{2})+e(G_{3})\geq
3\sqrt[3]{e(G_{1})e(G_{2})e(G_{3})}>0.8n^{2}+\frac{n}{2}$, for sufficiently
large $n$. Since three-colored edges cannot be adjacent to any edge with at
least two colors, the number of three-colored edges is at most $n/2$. Hence
the number of two-colored edges is at least $0.3n^{2}$.

######  Claim 3.

The graph containing all edges with at least two colors is the union of vertex
disjoint cliques, such that every edge of each clique has the same coloring.

###### Proof.

Let $e=uv$ be a two-colored edge, and suppose $e$ has colors $1$ and $2$. If
$w\in N_{3}(v)$, then $uw$ is one-colored with color $3$. That is
$N_{3}(v)=N_{3}(u)$ and

$N_{1,3}(u)=N_{2,3}(u)=\emptyset.$ (2)

Let $w\in N_{1,2}(v)$, then $uw$ is not colored with color $3$. Assume by the
way of contradiction that the edge $uw$ is a one-colored edge. Without loss of
generality we may assume $uw$ is of color $1$. Then by the maximality of the
coloring, there is a vertex $w^{\prime}$ such that
$w^{\prime}\notin\\{v,u,w\\}$ and edges $uw^{\prime}$ and $ww^{\prime}$
colored with colors $1$ and $3$ in any order. This is a contradiction since
for any coloring of the edge $w^{\prime}v$ either the triangle $w^{\prime}vu$
or the triangle $w^{\prime}vw$ is a rainbow triangle. Hence $w\in N_{1,2}(u)$,
thus

$N_{1,2}(u)=N_{1,2}(v).$ (3)

The claim follows from (2) and (3). ∎

Let $A$ be a clique of maximum size in the graph consisting of edges of at
least two colors. Let $a$ be the size of $A$. Then
$\frac{(a-1)n}{2}\geq.3n^{2}$ since the maximum degree in this graph is $a-1$
and we observed earlier that there are at least $.3n^{2}$ two-colored edges.
Hence we have $a\geq 0.6n$. Since $a>\frac{n}{2}$ and there are at least
$.3n^{2}$ two colored edges, it follows that $\binom{a}{2}+\binom{n-a}{2}\geq
0.3n^{2}$. Thus we have $a\geq 0.723n$.

Assume that the edges of $A$ are colored with $1,2$, then we have $e(A)\geq
0.26n^{2}$ and it follows that $e(G_{3})\leq 0.24n^{2}$. From the maximality
of the product $e(G_{1})e(G_{2})e(G_{3})$ and Claim 3, we have that all one-
colored edges are of color $3$. Indeed, otherwise we could change all of the
one-colored edges to color $3$ and since $e(G_{3})<e(G_{1})$ and
$e(G_{3})<e(G_{2})$, this would increase the product. Moreover, for any
maximal clique $B$ in the graph different from $A$ and consisting of edges of
at least two colors, one of those colors must be color $3$ (for otherwise,
changing one of the colors in $B$ to $3$ would increase the product).

By maximality, it is easy to observe that there is at most one clique colored
with colors $i$ and $j$ for all $1\leq i<j\leq 3$. Let $d$ be the number of
three-colored edges. Then $G_{1}$ consists of a complete graph of size $a$, a
complete graph of size $b$ and a matching of size $d$. $G_{2}$ consists of
complete graphs of size $a$ and $c$ and a matching of size $d$. $G_{3}$
consists of all edges not in $A$. Without loss of generality we may assume
$c\geq b$. We may also assume that we do not have the case $b=c=0$ for
otherwise the product is only $O(n^{5})$. Then the following holds:

$\left(\binom{a}{2}+\binom{b}{2}+d\right)\left(\binom{a}{2}+\binom{c}{2}+d\right)\leq\left(\binom{a}{2}+\binom{b}{2}\right)\left(\binom{a}{2}+\binom{c+2d}{2}\right).$

Therefore $d=0$, otherwise by changing the coloring, we could increase the
product. Since $a+b+c=n$ and $a\geq 0.72n$, we have

$\left(\binom{a}{2}+\binom{b}{2}\right)\left(\binom{a}{2}+\binom{c}{2}\right)\leq\left(\binom{a}{2}+\binom{b+c}{2}\right)\binom{a}{2}.$

Therefore $b=0$, otherwise by changing the coloring, we could increase the
product. Thus, we have obtained that the $G_{1}$, $G_{2}$ and $G_{3}$ have the
form of the construction in the proof of Theorem 1. That is, we have a
partition of the ground set into two parts $X$ and $Y$, and $X$ is colored
with colors $1$ and $2$, $Y$ is colored with colors $2$ and $3$, and all edges
between $X$ and $Y$ are colored with $3$. The maximum product of the number of
edges among such constructions is again given asymptotically by maximizing the
expression (1), thus yielding an upper bound of the form $\gamma
n^{6}(1+o(1))$. ∎

## Acknowledgments

The research of Győri and Salia authors was partially supported by the
National Research, Development and Innovation Office NKFIH, grants K132696,
SNN-135643 and K126853. The research of Tompkins was supported by NKFIH grant
K135800.

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