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

# Compact Chirped Fiber Bragg Gratings for Single-Photon Generation from
Quantum Dots

Vikas Remesh  Institut für Experimentalphysik, Universität Innsbruck, 6020
Innsbruck, Austria  Ria G. Krämer  Institute of Applied Physics, Abbe Center
of Photonics, Friedrich Schiller University Jena, 07745 Jena, Germany  René
Schwarz  Institut für Experimentalphysik, Universität Innsbruck, 6020
Innsbruck, Austria  Florian Kappe  Institut für Experimentalphysik,
Universität Innsbruck, 6020 Innsbruck, Austria  Yusuf Karli  Institut für
Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria  Malte Per
Siems  Institute of Applied Physics, Abbe Center of Photonics, Friedrich
Schiller University Jena, 07745 Jena, Germany  Thomas K. Bracht  Institut für
Festkörpertheorie, Universität Münster, 48149 Münster, Germany  Condensed
Matter Theory, Department of Physics, TU Dortmund, 44221 Dortmund, Germany
Saimon Filipe Covre da Silva  Institute of Semiconductor and Solid State
Physics, Johannes Kepler University Linz, 4040 Linz, Austria  Armando Rastelli
Institute of Semiconductor and Solid State Physics, Johannes Kepler University
Linz, 4040 Linz, Austria  Doris E. Reiter  Condensed Matter Theory, Department
of Physics, TU Dortmund, 44221 Dortmund, Germany  Daniel Richter  Institute of
Applied Physics, Abbe Center of Photonics, Friedrich Schiller University Jena,
07745 Jena, Germany  Stefan Nolte  Institute of Applied Physics, Abbe Center
of Photonics, Friedrich Schiller University Jena, 07745 Jena, Germany
Fraunhofer Institute for Applied Optics and Precision Engineering IOF, Center
of Excellence in Photonics, 07745 Jena, Germany  Gregor Weihs  Institut für
Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria

(Date: August 27, 2024
XXX E-mail<EMAIL_ADDRESS>
###### Abstract

A scalable source of single photons is a key constituent of an efficient
quantum photonic architecture. To realize this, it is beneficial to have an
ensemble of quantum emitters that can be collectively excited with high
efficiency. Semiconductor quantum dots hold great potential in this context,
due to their excellent photophysical properties. Spectral variability of
quantum dots is commonly regarded as a drawback introduced by the fabrication
method. However, this is beneficial to realize a frequency-multiplexed single-
photon platform. Chirped pulse excitation, relying on the so-called adiabatic
rapid passage, is the most efficient scheme to excite a quantum dot ensemble
due to its immunity to individual quantum dot parameters. Yet, the existing
methods of generating chirped laser pulses to excite a quantum emitter are
bulky, lossy, and mechanically unstable, which severely hampers the prospects
of a quantum dot photon source. Here, we present a compact, robust, and high-
efficiency alternative for chirped pulse excitation of solid-state quantum
emitters. Our simple plug-and-play module consists of chirped fiber Bragg
gratings (CFBGs), fabricated via femtosecond inscription, to provide high
values of dispersion in the near-infrared spectral range, where the quantum
dots emit. We characterize and benchmark the performance of our method via
chirped excitation of a GaAs quantum dot, establishing high-fidelity single-
photon generation. Our highly versatile chirping module coupled to a photon
source is a significant milestone toward realizing practical quantum photonic
devices.

††preprint: APS/123-QED

## Introduction

Future quantum communication architectures rely on bright and efficient
sources of high-purity single-photons and entangled photon pairs with superior
photostability and scalability. Semiconductor quantum dots are the most
important candidates for this role, due to their high brightness [1, 2], high
degrees of entanglement [3, 4], low multiphoton rate [5] and near-
deterministic operating nature. An important feature of quantum dots is the
possibility to control the emission wavelength of an ensemble by properly
choosing materials, growth methods, and fabrication parameters. Further post-
growth tuning is offered by electric [6], magnetic[7], optical [8] or strain
field tuning [9]. However, these methods increase the complexity of the photon
source. The natural spread in the dots’ physical properties like size,
composition, and strain is commonly perceived as an obstacle to achieving
scalable quantum hardware. However, the resulting distribution in emission
wavelengths offers a convenient way to produce frequency-multiplexed single-
photons and entangled photon pairs. Therefore, it is important to develop a
simple, efficient, and long-term stable optical excitation method that
delivers high-quality photons from the source.

Several resonant and off-resonant optical excitation schemes have been
developed to generate single-photons or entangled photon pairs from quantum
dots [10, 11, 12, 13, 14, 15, 16, 17]. In particular, chirped excitation
relying on Adiabatic Rapid Passage (ARP) is a robust scheme for quantum dot
excitation as demonstrated before [18, 19, 20, 21, 22, 23]. The ARP excitation
is immune to fluctuations in the laser power and spectral properties of
quantum dots, which facilitates a collective driving to produce single-photons
[24, 23] or entangled photon pairs [25]. It is also known that in ARP, the
excitation efficiency depends on the sign of the chirp, i.e., positive chirp
achieves near-unity excitation efficiency, while for negative chirp,
preparation efficiencies are lower due to phonon influences [26].

Figure 1: Plug-and-play chirping module for quantum dots: (a) Working
principle of a CFBG for pulse chirping. The direction of laser pulse
propagation through the CFBG dictates the sign of the group delay dispersion
(GDD): entry and reflection from the left side results in negatively chirped
pulses and vice versa. The relation between the GDD and wavelength-dependent
time delay ($T$) is also illustrated. (b) Single-photon generation:
$2\text{\,}\mathrm{ps}$-long laser pulses, centred at
$794.5\text{\,}\mathrm{nm}$ are coupled to the CFBG to impart
$45\text{\,}{\mathrm{ps}}^{2}$ chirp. A fiber-optic circulator (FOC) helps
separate incident (unchirped) and reflecting (chirped) laser pulses. The
chirped laser pulses are directed to an optical cryostat where the quantum dot
is mounted. A polarizing beamsplitter (PBS) enables cross-polarization
rejection of scattered laser light. The generated single photons at exciton
(X) and biexciton (XX) energies are collected via the same path, and filtered
and detected using superconducting nanowire single photon detectors (SNSPD).

In most of the practical implementations, chirped pulses are prepared by
introducing the Group Delay Dispersion (GDD) in the laser spectral phase [27].
Examples include diffraction-grating stretchers [28, 29, 30, 31, 22, 32, 23]
and programmable spatial light modulators (SLM) [33, 34, 35, 21, 15]. While
the SLM-based techniques are user-friendly and enable arbitrary spectral phase
manipulation, they are pixellated devices with limited resolution and response
times and bear phase wrapping issues. Diffraction-grating stretchers, on the
other hand, can be designed to provide extremely high dispersion [36]. Yet
both of these approaches are bulky, beam-distorting, lossy, and inherit
spatiotemporal coupling artifacts and higher-order dispersions [37, 38]. To
compensate for optical aberrations and higher-order dispersions, one requires
rigorous alignment procedures which restrict practical applications. For
instance, to accomplish a robust, and optomechanically stable single-mode
laser coupling to a quantum dot source, we must ensure a fast and effective
approach without spatiotemporal artifacts and aberrations.

In this Letter, we demonstrate a simple, compact, plug-and-play, and
alignment-free approach for chirped pulse excitation of quantum emitters,
which circumvents the obstacles in other existing methods. Our design is based
on a chirped fiber Bragg gratings (CFBG), fabricated via a femtosecond phase
mask inscription, to provide tailored values of GDD around
$800\text{\,}\mathrm{n}\mathrm{m}$ with large spectral bandwidth and high
efficiency. Our results pave the way for realizing compact solid-state single-
photon sources in an advanced quantum photonic architecture.

## Chirped fiber Bragg gratings for quantum dot excitation

A fiber Bragg grating consists of an optical fiber with a periodic variation
of the refractive index in the core, providing a narrowband in-fiber
reflection. The reflecting wavelength $\lambda_{\text{B}}$ depends on the
grating period $\Lambda$ and the effective refractive index $n_{\text{eff}}$
of the guided mode in the fiber core. In a CFBG, however, the grating period
is not constant but is varied along the propagation axis $z$. This results in
the reflection of different wavelengths at different spatial positions of
propagation:

$\lambda_{\text{B}}(z)=2n_{\text{eff}}\Lambda(z).$ (1)

Hence, different parts of the laser spectrum that is reflected at the CFBG
acquire different time delays, i.e., the laser pulse gets chirped in time. In
Figure 1a we summarize the working principle of a CFBG (see also SI). Note
that by adjusting the grating period along the propagation axis, the
dispersive response of the CFBG is defined: a linear variation of the grating
period introduces second-order dispersion, a quadratic variation introduces
third-order dispersion, and so on [39]. Initially proposed for dispersion
cancellation in optical telecommunication waveguides [40, 41], CFBGs have been
employed for chirped pulse amplification [42, 43], pulse shaping [44, 45, 46,
39] and high time resolution spectroscopy [47, 48], among many others. Yet
despite the progress in this technology, CFBGs have not been explored for
chirped excitation of quantum emitters around $800\text{\,}\mathrm{nm}$.

Figure 2: Performance of the CFBG: (a) Simulated reflection spectrum (blue
curve) of the CFBG designed for $\pm$45\text{\,}{\mathrm{ps}}^{2}$$ chirp,
centered at $794.5\text{\,}\mathrm{nm}$ and simulated time delay ($T$, red
curve) of different wavelengths. The negative slope of the red curve
determines the dispersion parameter $D_{1\lambda}$, from which the GDD is
computed. (b) Measured reflection spectra of CFBGs along the
+$45\text{\,}{\mathrm{ps}}^{2}$ (magenta curve) and
$-45\text{\,}{\mathrm{ps}}^{2}$ (green curve) propagation directions (c) CFBG
dispersion measurement: measured time delays ($T$) of different wavelengths
via a time-of-arrival technique. Red dots denote the peak wavelengths recorded
in the +$45\text{\,}{\mathrm{ps}}^{2}$ direction and the black line denotes
the fit. The slope of the solid black line provides the dispersion parameter
$D_{1\lambda}$. Grey shade represents the excitation laser spectrum.

For single-photon generation from quantum dots, CFBGs have a huge potential,
as illustrated in Figure 1b. To start with, CFBGs are an important
breakthrough towards realizing a direct fiber-coupled, alignment-free, high-
efficiency, and scalable quantum dot photon source [49, 50, 51, 52, 53]. They
can be tailored to provide arbitrary values of GDD, with a spectral bandwidth
of $10\text{\,}\mathrm{nm}$ or more, coincident with the typical spectral
variability of a quantum dot ensemble [54, 55]. More importantly, CFBG-based
dispersion is robust against mechanical instabilities and vibrations. As it
does not rely on angular dispersion, CFBGs eliminate optical aberrations and
spatiotemporal couplings, to provide excellent single-mode coupling to the
quantum dot source. Furthermore, a diffraction-grating stretcher [29, 30, 56]
designed for $45\text{\,}{\mathrm{ps}}^{2}$ chirp is bulky (several meters
long, see SI for details). In comparison, a CFBG for the same GDD is only a
few millimeters long. The versatility of a CFBG is that it allows inverting
the GDD by switching the propagation direction, while in a grating stretcher,
adding the option to invert the GDD increases the footprint of the setup even
further. Interestingly, with CFBGs we can realize additive combinations [45],
i.e., one can cascade several CFBGs to obtain a range of GDDs, without
increasing the complexity of the setup.

## Experimental implementation

To inscribe the CFBGs we employ a phase mask scanning technique using
femtosecond laser pulses [57] with $800\text{\,}\mathrm{nm}$ wavelength,
$100\text{\,}\mathrm{fs}$ pulse duration at $200\text{\,}\mathrm{Hz}$
repetition rate (Spectra-Physics Spitfire Ace). The fiber core refractive
index modification is induced by multiphoton absorption of the incident laser
irradiation. The non-periodic modulation of the refractive index modification
is generated by a phase mask (i.e., a transmission grating, manufactured in-
house at the University of Jena), which defines the resulting distribution of
the grating period along the fiber axis. To spectrally characterize the
fabricated CFBGs in reflection and transmission, we use linearly polarized
light from a supercontinuum source (NKT, SuperK EXR-15) and an optical
spectrum analyzer (Yokogawa, OSA AQ6375). A fiber-optic circulator (PMOptics)
helps extract the reflected signal of the CFBG. From the transmitted signal,
we determine the reflection strength as well as any broadband losses
introduced by the inscription process. For dispersion characterization of the
CFBG, we rely on a time-of-arrival technique.

For quantum dot excitation, we tune the laser source to
$794.5\text{\,}\mathrm{n}\mathrm{m}$, such that a two-photon resonant
excitation to the biexciton state is enabled [3, 12, 22, 23]. The laser beam
is directly coupled to a CFBG and the resulting, dispersed pulses are directed
to the quantum dot sample (see SI and Ref. [23] for details) mounted in a
closed-cycle cryostat with a base temperature of $1.5\text{\,}\mathrm{K}$
(ICEOxford). The quantum dot emission is collected via the same path, and
exciton (X) and biexciton (XX) emission photons are filtered using notch
filters (BNF-805-OD3, FWHM $0.3\text{\,}\mathrm{nm}$, Optigrate) and directed
to superconducting nanowire single-photon detectors (SNSPD, Eos, Single
Quantum). In the setup we employ cross-polarization filtering for efficient
laser scattering rejection.

## Results

###  .1 CFBG results

For the experiments described in this paper, we realize a set of three CFBGs
with different GDDs ($10\text{\,}{\mathrm{ps}}^{2}$,
$22.5\text{\,}{\mathrm{ps}}^{2}$, and $45\text{\,}{\mathrm{ps}}^{2}$
respectively). The phase masks employed have their center period around
$1090\text{\,}\mathrm{nm}$, resulting in CFBG center period of
$545\text{\,}\mathrm{nm}$, which corresponds to a second-order reflection at
$794.5\text{\,}\mathrm{nm}$ (see SI for details). To calculate the required
linear period variation, i.e., the chirp rate of the CFBG, we use the transfer
matrix algorithm as described in Ref. [58]. The achievable bandwidth of the
CFBG is limited by its maximum length, which is $60\text{\,}\mathrm{mm}$ in
our setup. In Figure 2a we present the simulated reflection spectrum (blue
curve) and the wavelength-dependent time delay ($T$, red curve) of the
$45\text{\,}{\mathrm{ps}}^{2}$ CFBG. From the slope of the red curve, we
compute the dispersion parameter $D_{1\lambda}$, from which we obtain the GDD
(see SI for details).

In Figure 2b we present the measured reflection spectra for the
$\pm$45\text{\,}{\mathrm{ps}}^{2}$$ CFBG (magenta and green curves
respectively). We observe a reflectivity $>$90\text{\,}\mathrm{\char
37\relax}$$ from $792\text{\,}\mathrm{nm}$ to $796\text{\,}\mathrm{nm}$,
showing an excellent agreement with the simulation in Figure 2a. The average
sideband suppression is $\approx$25\text{\,}\mathrm{d}\mathrm{B}$$. Note that,
for the $+$45\text{\,}{\mathrm{ps}}^{2}$$ direction, lower wavelengths
experience more losses due to the cladding mode coupling [57] which arises
from the chosen inscription pattern.

Figure 3: Chirped excitation of a quantum dot using CFBG: (a) the recorded
exciton (X) photon counts (blue dots) as a function of pulse area, for the
resonant Rabi rotation (i.e., $0\text{\,}{\mathrm{ps}}^{2}$), (b) positively-
chirped excitation, and (c) negatively-chirped excitation. Chirp values are
displayed along with. Red curves denote the simulated photon counts for the
respective cases.

We attribute a nominal 10% loss to the absorption and scattering of the guided
light in the grating structure. Two other major factors for the loss are the
circulator and the connector losses, due to the mismatch of different core
diameters (for more details see SI). The latter can be avoided by directly
splicing the CFBG to the circulator. Losses due to the splicing of the bare
fiber with the inscribed CFBG and connectorized fibers are negligible.

To characterize the GDD response of CFBGs we perform a time-of-arrival
measurement which directly measures the wavelength-dependent time delay (see
SI for details). In Figure 2c we display the result for
+$45\text{\,}{\mathrm{ps}}^{2}$ CFBG. Based on the variation of the fitted
peak wavelengths with respect to their arrival times recorded with the SNSPD,
we obtain a
$D_{1\lambda}=$-136.77\text{\,}\mathrm{ps}\text{\,}{\mathrm{nm}}^{-1}$$, which
corresponds to $\text{GDD}=$45.9\text{\,}{\mathrm{ps}}^{2}$$, showing an
excellent match with the simulation in Figure 2a.

###  .2 QD experiment results

We then proceed to investigate the single-photon generation from quantum dots.
We select a quantum dot resonant within the CFBG bandwidth and characterize it
under two-photon excitation (see SI for details). The exciton (X) wavelength
is found to be $793.6\text{\,}\mathrm{nm}$ and the biexciton (XX) wavelength
is $795.4\text{\,}\mathrm{nm}$. To confirm the resonance condition, we perform
a Rabi rotation experiment upon the sweep of the laser power, while recording
the corresponding emission spectra. The integrated intensities corresponding
to the X emission (for simplicity, XX photon counts are not displayed) are
displayed in Figure 3a, clearly demonstrating the Rabi oscillations.

We then investigate the utility of CFBG for chirped excitation of the quantum
dot i.e., ARP experiment. To this end, we perform the laser power sweep
experiments on the same quantum dot using the three CFBGs
($10\text{\,}{\mathrm{ps}}^{2}$, $22.5\text{\,}{\mathrm{ps}}^{2}$ and
$45\text{\,}{\mathrm{ps}}^{2}$ respectively). The results are displayed in
Figure 3b, with respective chirp values labeled alongside. At
$10\text{\,}{\mathrm{ps}}^{2}$, we observe that the oscillatory behavior in
the recorded X photon counts diminishes. The same experiment at
$22.5\text{\,}{\mathrm{ps}}^{2}$ shows that the Rabi oscillations have nearly
vanished, and a plateau is attained for higher pulse power, signifying that
the process is closer to ARP condition. Finally, with the
$45\text{\,}{\mathrm{ps}}^{2}$ CFBG, we observe a clear plateau in the X
photon counts for pulse powers >2$\pi$, demonstrating that the ARP process is
induced, confirming the previous observations [19, 21, 22, 32, 24, 23].

To further validate the robustness of ARP excitation, we investigate the
excited state preparation efficiency under both conditions via a biexciton-to-
exciton cross-correlation experiment, as in Ref. [59]. We observe that the
preparation efficiencies are 93.5% in the case of Rabi rotation (largest
value, i.e., at $\pi$ power) and >94.4% for ARP condition (for all powers
>$2.5\text{\,}\pi$), in agreement with the previous reports [23, 4] on a
similar quantum dot system (for details see SI). Note that, the preparation
efficiency under Rabi rotation is extremely sensitive to the pulse power [23],
while under chirped pulse excitation, the preparation efficiency is
insensitive to power variations.

To demonstrate the versatility of CFBGs for chirped pulse excitation, we
switch the pulse propagation direction (i.e., to obtain a negatively chirped
pulse) and repeat the experiments, and the results are displayed in Figure 3c.
We observe that, as expected [19, 60], the quantum dot excitation efficiency
does not reach unity, and the X photon counts drop to 30% at higher pulse
areas, in agreement with earlier experimental observations [21, 22]. This
validates the fact, for chirped pulse excitation of quantum dots, the sign of
the chirp determines the efficiency, due to phonon influences [19, 60].

Finally, we also confirm the nature of the photon emission from the quantum
dot under the resonant Rabi and the chirped excitations. To this end, we
perform Hanbury-Brown and Twiss ($g^{(2)}(0)$) measurements and observe that
the computed $g^{(2)}(0)\approx 0.005$ (see SI), validating the single-photon
nature of the emission.

## Conclusions

To summarize, we demonstrated a systematic study of designing and fabricating
compact, plug-and-play chirped fiber Bragg gratings tailored to a GaAs quantum
dot resonance. Our high-efficiency femtosecond phase mask inscription method
of CFBGs enabled precise control over the temporal characteristics of the
excitation pulses. We established the versatility of the CFBGs by performing
the two-photon chirped pulse excitation of a quantum dot under positive and
negative chirps. Under positively chirped excitation, we observed a high-
efficiency generation of single photons at exciton and biexciton energies,
relying on the adiabatic rapid passage, while for negative chirps, we observed
lower efficiency state preparation, in accordance with previous reports. In
comparison with the prevailing methods for chirped pulse excitation, our
method is robust, highly efficient, and optomechanically stable. Our results
are in excellent agreement with theoretical simulations. In conclusion, these
open an exciting avenue toward realizing an all-fiber-coupled, and alignment-
free solid-state source of frequency-multiplexed single and entangled photon
pairs. The versatility of CFBGs also allows for the development of efficient
and compact pulse-shaping methods tailored to control various quantum states
in any solid-state platform.

## Acknowledgements

The authors gratefully acknowledge insightful discussions with Timothy Imogore
(Jena), Thorsten Goebel (Jena), Dmitry Pestov (IPG Photonics), Ron Stepanek
(Meadowlark Optics), and Thomas Niedereichholz (Hamamatsu). VR, RS, FK, YK and
GW acknowledge financial support through the Austrian Science Fund FWF
projects TAI-556N (DarkEneT), W1259 (DK-ALM Atoms, Light, and Molecules), FG5,
and I4380 (AEQuDot). TKB and DER acknowledge financial support from the German
Research Foundation DFG through project 428026575 (AEQuDot). AR and SFCdS
acknowledge the FWF projects FG 5, P30459, I4320, the Linz Institute of
Technology (LIT), the European Union’s Horizon 2020 research, and innovation
program under Grant Agreement Nos. 899814 (Qurope), 871130 (ASCENT+), and the
QuantERA II Program (project QD-E-QKD, FFG Grant No. 891366). RGK acknowledges
financial support from the German Federal Ministry of Education and Research
through project 13N16028 (MHLASQU) and the German Research Foundation DFG
(455425131, OH-SUPER). DR acknowledges financial support from the German
Research Foundation DFG through project 448663633 (Fs2CVBG).

## Author contributions

VR, GW and SN conceived and supervised the project. RGK, DR and MPS designed
and fabricated the chirped fiber Bragg gratings. SFCdS and AR provided the
quantum dot sample. RS, FK, YK, RGK, and VR built the setups and performed the
experiments. VR and RGK wrote the first draft of the manuscript with input
from other authors. FK, YK and RGK performed the data analysis and numerical
simulations. All the authors discussed the results and were involved in
writing the manuscript.

##  I Supplementary Information

###  I.1 Chirped laser pulses

Chirped pulses possess time-varying frequency. In the frequency domain, this
is described by

$E(\omega)=E_{0}\exp\left[-\frac{\left(\omega-\omega_{c}\right)^{2}}{\Delta\omega^{2}}\right]\exp\left[i\frac{\phi_{2}}{2}\left(\omega-\omega_{c}\right)^{2}\right],$
(2)

where $E_{0}$ is the amplitude of the Gaussian frequency envelope centered at
$\omega_{c}$ with a frequency bandwidth of $\Delta\omega$. The constant
$\phi_{2}$ describes the group delay dispersion, GDD, or linear chirp.

By definition, the GDD is the second derivative of the spectral phase with
respect to the angular frequency $\omega$, or the first derivative of group
delay ($T$) with respect to frequency. i.e.,

$\phi_{2}=\text{GDD}=\derivative[2]{\phi}{\omega}=\frac{\partial
T}{\partial\omega}=D_{2}(\omega).$ (3)

In a CFBG, the time delay of wavelengths is varied by controlling the Bragg
reflection resonances at various positions along the propagation direction in
the fiber core. Converting everything to the wavelength domain, we can relate
the dispersion parameter $D_{1\lambda}$ (specified as
$\mathrm{ps}\text{\,}{\mathrm{nm}}^{-1}$) to GDD (specified as
${\mathrm{ps}}^{2}$) through

$D_{1\lambda}=\frac{\partial T}{\partial\lambda}=-\frac{2\pi
c}{\lambda^{2}}\cdot\derivative[2]{\phi}{\omega}=-\frac{2\pi
c}{\lambda^{2}}\cdot\text{GDD}$ (4)

It is now clear that a positive GDD implies a shorter time delay for the red
part of the laser spectrum and vice versa for negative GDD.

###  I.2 CFBG details

The CFBGs employed in this work are fabricated via femtosecond phase mask
inscription technique [61, 57] The dispersion parameter $D_{1\lambda}$ can be
determined from the time delay between the reflections of various wavelength
components as

$D_{1\lambda}=\frac{\Delta t}{\Delta\lambda}.$ (5)

In the CFBG with the grating length $L$, the effective refractive index of the
guided light $n_{\text{eff}}$ and $c$ as the speed of light in vacuum, the
delay time is given by

$\Delta t=\frac{2Ln_{\text{eff}}}{c}.$ (6)

Note that each wavelength component travels twice the distance within the
CFBG, as it functions in reflection. Then

$D_{1\lambda}=\frac{2Ln}{c\Delta\lambda}$ (7)

which leads to

$L=\frac{c\Delta\lambda}{2n}D_{1\lambda}$ (8)

and in terms of GDD, this will be

$L=\frac{\pi c^{2}\Delta\lambda}{n\lambda^{2}}\text{GDD},$ (9)

which provides an estimate for the required grating length to provide a given
GDD.

In Figure 4 we present the results of the further reflectivity measurements on
CFBGs. Note that due to the chosen inscription pattern, for the positive GDD
input side, lower wavelengths experience more losses due to the cladding mode
coupling [57].

Figure 4: The measured reflectivities of $10\text{\,}{\mathrm{ps}}^{2}$ and
$22.5\text{\,}{\mathrm{ps}}^{2}$ CFBGs.

In the time-of-arrival (ToA) method of characterizing dispersion, we select
several narrow spectra from the reference laser spectrum using a 4$f$ pulse
shaper equipped with a motorized slit on its Fourier plane and propagate
through the CFBG. The resulting, dispersed laser pulses are simultaneously
recorded with a spectrometer and SNSPD (time resolution
$\approx$20\text{\,}\mathrm{ps}$$) using 10:90 fiber beam-splitter. While with
the spectrometer we obtain the wavelength component, with the SNSPD we record
its arrival time with respect to the reference laser spectrum. To extract the
wavelength component precisely, we employ a Gaussian fit to the measured laser
spectra, to obtain a plot of time delay as a function of central wavelength.
The slope extracted from this plot provides the dispersion parameter
$D_{1\lambda}$, from which the GDD can be obtained. In Figure 5 we display the
result of ToA measurement for $\pm$45\text{\,}{\mathrm{ps}}^{2}$$ and
$-22\text{\,}{\mathrm{ps}}^{2}$.

Figure 5: Characterization of CFBG dispersions via time-of-arrival (ToA)
measurement: (a) Recorded laser spectra for various slit positions in the 4$f$
shaper (b) Results of the dispersion characterization via the ToA measurement.
Grey-shaded area denotes the laser spectrum.

In Table 1 we summarize the parameters of all the CFBGs and their measured
performance parameters.

CFBG | 45 | 22 | 10
---|---|---|---
GDD${}_{\text{design}}$ ($\text{\,}{\mathrm{ps}}^{2}$) | $45\text{\,}$ | $22\text{\,}$ | $10\text{\,}$
Chirp rate ($\text{\,}\mathrm{nm}\text{\,}{\mathrm{cm}}^{-1}$) | $1.04\text{\,}$ | $1.54\text{\,}$ | $4.62\text{\,}$
Length ($\text{\,}\mathrm{mm}$) | $60\text{\,}$ | $60\text{\,}$ | $50\text{\,}$
Center wavelength ($\text{\,}\mathrm{nm}$) | $793.8\text{\,}$ | $795.6\text{\,}$ | $797.0\text{\,}$
FWHM ($\text{\,}\mathrm{nm}$) | $4.1\text{\,}$ | $8.1\text{\,}$ | $15.2\text{\,}$
D1λ ($\text{\,}\mathrm{ps}\text{\,}{\mathrm{nm}}^{-1}$) |  $-136.77\text{\,}$ ($135.56\text{\,}$) | $58.62\text{\,}$ | –
GDD${}_{\text{meas}}$ ($\text{\,}{\mathrm{ps}}^{2}$) |  $45.9\text{\,}$ ($-45.5\text{\,}$) | $-19.7\text{\,}$ | –
Table 1: Overview of CFBG parameters: The chirp rate, length, and measured
full-width at half maximum (FWHM) bandwidth of the fabricated CFBGs labeled
their GDD values (See SI for details).

###  I.3 Multiple order CFBG resonances

The change of the refractive index during the CFBG inscription process is a
result of multiphoton absorption in the fiber core due to the femtosecond
laser irradiation [57]. Due to the nature of this process, the underlying
refractive index change is nonlinear, i.e., the modulation of the refractive
index is non-sinusoidal. The pattern along the propagation axis $\Delta n(z)$
can be expressed as a sum of its Fourier terms $\Delta n_{m}$:

$\Delta n(z)=\frac{\Delta n_{0}}{2}+\sum_{m=1}^{\infty}\Delta
n_{m}\cos\left(\frac{2\pi m}{\Lambda}z+\Phi(z)\right)$ (10)

with the Fourier order $m$, the nominal grating period $\Lambda$ and the
grating chirp described by $\Phi(z)$. Consequently, the CFBG reflects at
multiple wavelengths, corresponding to each of the Fourier terms,

$\lambda_{m}(z)=2n_{\text{eff}}\frac{\Lambda(z)}{m}.$ (11)

Therefore, a CFBG with a center period of $545\text{\,}\mathrm{nm}$ will
reflect in first order at $\lambda_{1}$ = $1570\text{\,}\mathrm{nm}$, in
second order at $\lambda_{2}$ = $795\text{\,}\mathrm{nm}$, in third order at
$\lambda_{3}$ = $530\text{\,}\mathrm{nm}$ and so on.

###  I.4 QD simulations

Simulations of the pulse area-dependent biexciton population (see Figure 3 in
the main manuscript) are performed using a numerically complete path-integral
method described in detail in Ref. [62]. For this we set up a Hamiltonian
consisting of the quantum dot system $\hat{H}^{\text{QD}}$, the coupling to
the laser field $\hat{H}^{\text{Laser}}$ as well as the coupling to phonons
$\hat{H}^{\text{Phonon}}$

$\hat{H}=\hat{H}^{\text{QD}}+\hat{H}^{\text{Laser}}+\hat{H}^{\text{Phonon}}.$
(12)

We model the quantum dot as a four-level system consisting of the ground state
$|g\rangle$ two exciton states of orthogonal polarisation, $|x_{H}\rangle$ and
$|x_{V}\rangle$, and the biexciton state $|xx\rangle$. The ground state energy
is set to zero, while the two exciton states are assumed to be of the same
energy $\hbar\omega_{x}$, i.e., no fine structure splitting is present. The
biexciton state has a binding energy $E_{B}$ such that
$\hbar\omega_{xx}=2\hbar\omega_{x}-E_{B}$.

$\begin{split}\hat{H}^{\text{QD}}=&\hbar\omega_{x}\left(|x_{H}\rangle\langle
x_{H}|+|x_{V}\rangle\langle x_{V}|\right)+\hbar\omega_{xx}|xx\rangle\langle
xx|\end{split}$ (13)

Coupling to the laser field is modeled with a classical laser field

$\begin{split}\hat{H}^{\text{Laser}}(t)=-\frac{\hbar}{2}\Omega(t)&(|g\rangle\langle
x_{H}|+|g\rangle\langle x_{V}|\\\ &+|x_{H}\rangle\langle
xx|+|x_{V}\rangle\langle xx|+h.c.).\end{split}$ (14)

where $\Omega(t)$ is the Fourier transform of Equation 2. In Figure 3 we scale
the x-axis of the simulation to be proportional to
$\int_{-\infty}^{+\infty}|\Omega(t)|^{2}dt$, i.e., the pulse power.

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

$\hat{H}^{\text{Phonon}}=\hbar\sum_{\mathbf{k}}\omega_{\mathbf{k}}\hat{b}^{\dagger}_{\mathbf{k}}\hat{b}_{\mathbf{k}}+\hbar\sum_{{\mathbf{k}},S}n_{S}\left(\gamma_{\mathbf{k}}^{S}\hat{b}_{\mathbf{k}}^{\dagger}+\gamma_{\mathbf{k}}^{S^{*}}\hat{b}_{\mathbf{k}}\right)|S\rangle\langle
S|,$ (15)

coupling each mode $\mathbf{k}$ to the quantum dot state $|S\rangle$, where
$S\in\\{x_{H},x_{V},xx\\}$ and $n_{S}$ is the number of excitations, i.e., 1
for $x_{H}$ and $x_{H}$ and 2 for $xx$. The coupling constant
$\gamma_{\mathbf{k}}^{S}$ and the material parameters are taken to be the same
as in Ref. [63] and further simulation parameters are shown in Table 2.

frequency bandwidth | $\Delta\omega$ | 1.410 ${\mathrm{ps}}^{-1}$
---|---|---
Binding energy | $E_{B}$ | 4 meV
QD size | $a$ | 5 nm
temperature | $T$ | 1.5 K
Table 2: Overview of the parameters used for quantum dot simulation. Material
parameters are taken as in Ref. [63].

###  I.5 Comparison of various chirping methods

In this section, we compare various chirping methods. Grating stretcher: For
imparting $45\text{\,}{\mathrm{ps}}^{2}$ we designed and constructed a grating
stretcher in folded geometry (details in Ref.[23]). Here grating period
$d=$1200\text{\,}{\mathrm{mm}}^{-1}$$ at
$\lambda_{0}=$795\text{\,}\mathrm{nm}$$ and distance
$s=$21\text{\,}\mathrm{cm}$$ from the lens ($f=$75\text{\,}\mathrm{cm}$$) with
angle of incidence ($\theta_{i}=$2\text{\,}\mathrm{\SIUnitSymbolDegree}$$) and
grating efficiency is 70%. Note that in the grating stretcher, the inherited
third and fourth-order dispersions can be compensated in principle, yet, it
increases the experimental complexity. The overall efficiency strongly depends
on the efficiency of a single grating, i.e.,
$\eta\textsuperscript{4}_{\text{grating}}$. Additionally, optical aberrations
also contribute to lower fiber-coupling efficiencies. To illustrate this, in
Figure 6 we describe the variation in single-mode fiber coupling efficiency as
a function of grating position in our grating stretcher, as obtained by ray-
tracing simulations.

Figure 6: Variation in single-mode fiber-coupling efficiency in a grating
stretcher

In an SLM-equipped 4$f$ shapers, the difference from the grating stretcher is
that a programmable SLM is mounted at the Fourier plane. The straightforward
improvement is the deterministic, adaptive, and programmable spectral phase
manipulation, including higher-order components. Imparting negative dispersion
does not enhance the footprint, as in grating stretchers. There are two
classes of SLMs: dual-mask, transmissive SLMs [21, 35, 24], and two-
dimensional, reflective SLMs [64]. The transmissive SLMs operate by
controlling the phase between frequency components, depending on the voltage
applied, in a 4$f$ shaper [27]. They suffer from transmission losses of liquid
crystal layers. To perform simultaneous phase-amplitude shaping, one requires
dual mask SLMs, which when operated in a folded, double-pass configuration,
leads to a doubling of the transmission loss per liquid crystal layer.
Therefore, the total efficiency is
$\eta\textsuperscript{4}_{\text{LC}}\cdot\eta\textsuperscript{2}_{\text{grating}}$.
The reflective SLM also operates in a 4$f$ shaper, however, it functions based
on diffractive pulse shaping [64]. Here, the total efficiency is
$\eta_{\text{SLM(ref)}}\cdot\eta_{\text{SLM(diff)}}\cdot\eta\textsuperscript{2}_{\text{grating}}$,
where the first two terms refer to the reflection and diffraction efficiencies
respectively. In a reflective SLM, typically, the total efficiency is higher
than its transmissive counterpart.

In the case of CFBG, the maximum efficiency is
$\eta\textsuperscript{2}_{\text{FOC}}\cdot\eta_{\text{CFBG}}$, where
$\eta_{\text{FOC}}$ is the efficiency of the fiber-optic circulator. The
characterized efficiencies are $\eta\textsuperscript{2}_{\text{FOC}}=77\%$,
and $\eta\textsuperscript{2}_{\text{CFBG}}=90\%$. In our experiments, we
observed a total efficiency of 40%. We attribute the remaining losses to fiber
mating inefficiencies. This could be mitigated by splicing the CFBG ends to
the FOC.

Generally speaking, grating stretchers (or SLM-equipped 4$f$ shapers) can be
operated in a wide spectral range. Yet they have a fundamental issue: the GDD
is dictated by wavelength-dependent optical path delay resulting from angular
dispersion. This imposes significantly large spatiotemporal coupling. However,
the CFBG achieves spectral dispersion without angular dispersion. Thus, our
CFBGs eliminate angular distortions that affect the excitation laser mode
coupling to an emitter.

###  I.6 QD sample

Our sample consists of GaAs/AlGaAs quantum dots with exciton emission centered
around $793\text{\,}\mathrm{nm}$ grown by the Al-droplet etching method [65,
54]. The dots are embedded in the center of a lambda-cavity placed between a
bottom (top) distributed Bragg reflector consisting of 9 (2) pairs of
$\lambda/4$ thick Al0.95Ga0.05As/Al0.2Ga0.8As layers. In Figure 7 we present
the characterization of the quantum dot under two-photon excitation.

Figure 7: Two-photon excitation and characterization: (a) (a) Recorded
emission spectrum of the quantum dot under two-photon excitation. The exciton
(X) and biexciton (XX) wavelengths are identified as
$793.6\text{\,}\mathrm{nm}$ and $795.4\text{\,}\mathrm{nm}$ respectively. (b)
The recorded X and XX lifetimes under resonant Rabi and chirped excitations
($\text{GDD}=$45\text{\,}{\mathrm{ps}}^{2}$$) (c) Computed efficiencies of the
biexciton state preparation under resonant Rabi and chirped excitations based
on XX-X crosscorrelation measurements (d) The single-photon purity computed
from the $g^{(2)}(0)$ measurements.

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