High-throughput quantum photonic devices emitting indistinguishable photons in the telecom C-band

Single indistinguishable photons at telecom C-band wavelengths are essential for quantum networks and the future quantum internet. However, high-throughput technology for single-photon generation at 1550 nm remained a missing building block to overcome present limitations in quantum communication and information technologies. Here, we demonstrate the high-throughput fabrication of quantum-photonic integrated devices operating at C-band wavelengths based on epitaxial semiconductor quantum dots. Our technique enables the deterministic integration of single pre-selected quantum emitters into microcavities based on circular Bragg gratings. Respective devices feature the triggered generation of single photons with ultra-high purity and record-high photon indistinguishability. Further improvements in yield and coherence properties will pave the way for implementing single-photon non-linear devices and advanced quantum networks at telecom wavelengths.


INTRODUCTION
A quantum network 1 based on remote nodes interconnected via fiber-optical links and capable of transferring quantum information using flying qubits will provide the backbone for the implementation of protocols for secure communication 2,3 and distributed quantum computing 4 .Notably, the network can rely on the existing silica-fiber-based infrastructure, utilizing a low-loss channel for the transmission of photons with a wavelength in the telecom C-band around 1550 nm 5 .These quantum network architectures can benefit from existing components and classical signal management protocols, hence making it feasible to transfer quantum information over large distances 5 .
In recent years, the technology for the epitaxial growth of self-assembled quantum dots (QDs) has rapidly advanced, resulting in the demonstration of QD-based single-photon sources (SPSs) with excellent characteristics.These include high photon extraction efficiencies (∼ 79 %) 6 , high single photon generation rates (∼ 1 GHz) 7 , and near unity photon indistinguishability (> 96 %) 7,8 , however, all achieved outside the telecom-relevant C-band.Besides the extraordinary material quality, these characteristics are achieved owing to efficient light-matter coupling between a QD and a suitable photonic element.For efficient coupling, spectral and spatial matching is required between the quantum emitter and the engineered photonic mode, which is challenging due to the spatial and spectral distribution of epitaxial QDs.Until now, the QD coupling to photonic cavities operating around 1550 nm has only been realized using non-deterministic fabrication processes, limiting device yield and scalability 9 .
In this article, we report on the high-throughput fabrication of nanophotonic elements around pre-selected individual QDs emitting single and indistinguishable photons in the telecom C-band.For this purpose, we develop a near-infrared (NIR) imaging technique for self-assembled InAs/InP QDs utilizing a hybrid sample geometry with enhanced out-ofplane emission from single QDs 10 and a thermo-electrically cooled InGaAs camera in a wide-field imaging configuration.In combination with two electron-beam lithography (EBL) steps, our method enables an overall positioning accuracy of 90 nm of the QD with respect to the photonic element and allows for rapid data collection as compared to competing techniques based on scanning in-situ imaging 11 .We apply our technique for the deterministic integration of pre-selected QDs into circular Bragg grating (CBG) cavities.The proposed technological workflow allows us to greatly enhance the device fabrication yield reaching ∼ 30 %, which is a significant improvement compared to < 1 % that would typically be achieved with a random placement approach.The QD-CBG coupling is evidenced by a Purcell factor ∼ 5, and our devices demonstrate a state-of-the-art photon extraction efficiency of η = (16.6±2.7)% into the first lens with a numerical aperture (NA) of 0.4, a high single-photon purity associated with g (2) (0) = (3.2± 0.6) × 10 −3 , and a record-high photonindistinguishability of V = (19.3± 2.6) % for QD-based SPSs at C-band wavelengths.

RESULTS
Design of a QD structure for wide-field imaging Imaging at very low light levels at wavelengths >1 µm is challenging due to the high level of electronic noise of respective camera systems.Although cameras based on InGaAs achieve quantum efficiencies > 80%, they are characterized by a factor of > 10 4 higher dark currents compared to Si-based devices.The photon emission rate from the sample is therefore of key importance for the ability to image and localize single QDs.Following our previous work 10 , we have designed a planar sample geometry that significantly enhances the photon extraction efficiency, allowing to localize single QDs and the subsequent fabrication of photonic elements.
The planar QD structure consists of an epitaxially grown 312 nm-thick InP slab containing a single layer of InAs QDs.The InP slab is atop a 359 nm-thick SiO 2 layer with a bottom Al mirror bonded to a Si wafer carrier (Fig. 1a, see Methods).Overall, this geometry enhances the QD emission in the outof-plane direction by a factor > 7 as compared to bulk InP samples, reaching a total photon extraction efficiency of > 10 % from a single QD for NA = 0.4 10 .In fact, this design turned out to be crucial for the imaging step, as the SNR of the QD emission was insufficient for investigated structures without a backside mirror (see Supplementary Note 3).For QD localization later in the experiment, we structure the top InP layer in a mesh with fields of size (50 × 50) µm 2 separated by 10 µm-wide ridges (see Methods), where the field edges are used as alignment marks (AMs) for imaging.The fields are furthermore organized in blocks accompanied by InP crosses that allow us to align the electron beam to specific target QDs during the EBL process (see Supplementary Note 2 for the optical microscope image of the sample surface with fabricated cavities).
For the self-assembled Stranski-Krastanov QD epitaxy, we employed the near-critical growth regime in metalorganic vapor-phase epitaxy (MOVPE) 12 (see Methods), and obtained a QD surface density of 3.1 × 10 8 /cm 2 corresponding to an average QD separation of 1.5 µm.Since QDs exhibit a size, shape, and strain distribution, only a fraction of the QDs have their ground-state optical transition in the C-band.With a (1550 ± 8) nm bandpass filter, we find on average N F = 10 QDs per field, which translates to an effective QD density of 4 × 10 5 /cm 2 and an average QD separation of ∼ 16 µm.

Design of circular Bragg grating cavities
The CBG geometry is optimized using the modal method (see Supplementary Note 1) to enhance the cavity figures of merit at 1550 nm, namely the collection efficiency at the first lens and the Purcell factor (F P ).As opposed to other implementations 9 , we consider a simplified CBG geometry consisting of a central mesa and only four external rings.According to our calculations, this number is sufficient for high η and F P , providing a smaller footprint and less complexity in the fabrication process.The in-plane cavity dimensions include the central mesa radius of R 0 = 648 nm, the grating period of 747 nm, and the separation between InP rings (air gap) of 346 nm.For these geometrical parameters, Fig. 1b shows the calculated broadband η that amounts to nearly 62 % and 82 % at 1550 nm for a NA of 0.4 and 0.65, respectively, which is similar to other state-of-the-art CBG designs [13][14][15] .The wavelength dependence of the Purcell factor, presented in Fig. 1b, mimics the CBG cavity mode centered at 1550 nm and reaches a maximum value F P = 18.1 with a quality factor of 110.The influence of the cavity geometry on the dispersion of η and F P suggests that the cavity grating together with the Al mirror creates a photonic bandgap that governs the η-dependence and enhances the emission directionality, while the InP membrane thickness and central mesa diameter crucially affect the center wavelength of the F P -dependence.A scanning electron microscopy (SEM) image of a fabricated CBG cavity is shown in Fig. 1c.

Optical imaging and QD localization
The NIR imaging setup utilizes a wide-field bright microscope configuration as shown in Fig. 2a.The structure with QDs above the Al reflector is mounted in an optical cryostat at T = 4.2 K movable by an x-y-z stage for targeting fabricated fields that are imaged consecutively.For sample illumination, we use a 660 nm continuous-wave (CW) semiconductor laser diode, spatially shaped with a beam expander, and focused on the backside of a commercially-available microscope objective (NA = 0.65) with 50× magnification, 57 % transmission in the NIR, and 10 mm working distance.This configuration provides nearly homogeneous surface illumination across a (50 × 50) µm 2 field and high photon collection efficiency.The spatially-distributed QD microphotoluminescence (µPL) and scattered light from the field edges (here used as AMs) are collected by the same objective and pass through a cold mirror cutting off the laser light.Finally, the emission is projected onto a thermo-electrically cooled InGaAs-based camera with a (12.8 × 10.24) mm 2 chip and a pixel size of (20 × 20) µm 2 .With the 4× magnification lens in front of the camera, the setup has a 200× magnification, enabling the optimal filling of the entire camera chip with a single field (Fig. 2b).The 1550 nm band-pass filter with 8 nm full-width half-maximum (FWHM) placed in front of the imaging lens selects QDs with emission in the C-band.
Figure 2b shows a representative image of a field recorded with a camera integration time of 2.5 s.The QDs can clearly be recognized as individual bright spots with FWHM ≈ 1.6 µm (see Fig. 2h and Supplementary Note 3) and Airy rings around.The square-shaped outline of the field scatters light and is used as AM for QD localization.
The localization of QDs is performed by taking vertical and horizontal cross-sections both crossing at a QD emission spot in the µPL intensity map.Each cross-section thus contains the position of the target QD relative to two AMs (Figs. 2f-2g).
Gaussian profiles fitted to the QD and the AM signals are subsequently used to determine the QD peak position relative to the AMs.The average signal-to-noise ratio for QD emission spots is 10.6, emphasizing the importance of the 7× emission enhancement in the planar structure as compared to bulk InP.We find that for the brightest 10 % of all QDs with SNR > 15.5, the position is fitted with an uncertainty of < 54 nm in 1D and with an uncertainty of the AM position of < 36 nm, resulting in a total uncertainty of QD position in 1D of < 62 nm.In 2D, this translates to 80 nm accuracy for the QD localization.Finally, taking into account the EBL alignment accuracy of 40 nm as measured in our previous work 16 , we estimate the overall accuracy of 2D CBG placement to ∆R = 90 nm.The histograms of QD localization accuracy and overall cavity placement accuracy for all detected spots are shown in Fig. 2f and Fig. 2g respectively, and the 80 nm and 90 nm levels are marked for reference.Medians for the distributions are slightly larger, 127 nm and 133 nm, due to processing of all detected spots, irrespective of their brightness and expected cavity fabrication precision.Details on the localization algorithm, derivation and discussion of the uncertainties, and data on the accuracy of cavity positioning are given in the Supplementary Note 3. Following the localization of suitable QDs, CBGs are fabricated using EBL with proximity error correction and an optimized inductively coupled plasma-reactive ion etching (ICP-RIE) process (see Methods).The SEM image presented in Fig. 2c is taken from the same field after fabricating the CBG cavities around pre-selected QDs indicated by the circles in Fig. 2b.

Deterministic process yield
We use a µPL setup to quantify the process yield that we define as the ratio between the number of QD-CBG devices with QD emission spectra matching the CBG mode and the number of all CBGs investigated, and we obtain Y = 30 %.This value should be compared with the yield that would be obtained in a statistical QD-CBG fabrication process.As we estimate the average number of QDs per field of size F = 50 µm to N F = 10, the probability of covering one of them with the central mesa of diameter 2R 0 is Y random ∼ N F × (2R 0 /F) 2 = 0.67 % ≪ Y .Some of the QD (inside a CBG) emission spectra are significantly broadened (median linewidth of 0.76 nm, see Supplementary Note 6) as compared to the narrowest recorded linewidth of 0.14 nm (identical to the spectrometer resolution).We attribute the broadening to the impact of surface states and point defects caused by the cavity fabrication, effectively resulting in the spectral wandering of the QD emission line 17 .Such defects can also introduce non-radiative recombination centers in the close vicinity of or even into the QD, quench the optical emission, and effectively reduce the process yield.Using the temperature-dependent µPL studies, we make sure that even the broadened emission lines follow the expected Varshni trend, ensuring that these spectral lines can indeed be associated with the QD emission as the temperature dependence of the cavity mode energy is much weaker.

Purcell enhancement
In the following, we discuss the optical properties of two exemplary devices, QD-CBG #1 and QD-CBG #2, each containing a single pre-selected QD coupled to the CBG cavity mode (see Supplementary Note 6 for the properties of a third device QD-CBG #3). Figure 3a shows the narrow QD emission lines overlaid on the cavity mode with Q = 194, the latter obtained under high power cavity excitation, evidencing good spectral overlap between the cavity mode and the QD emission.We interpret the dominant QD emission lines in both devices as trions (CX), due to their linear intensity dependence on excitation power, and the lack of fine-structure splitting.This is in line with typical spectra for our InAs/InP QDs with preferential CX recombination where the average CX binding energy was measured to be 4.7 meV 10 .
The coupling between the QD and the CBG cavity is evidenced by the observation of a reduced emission decay time as compared to the decay of QDs in the planar reference structure.For the CX line in QD-CBG #1 and #2 we record decay times of τ #1 = (0.40 ± 0.01) ns and τ #2 = (0.53 ± 0.01) ns, respectively (Fig. 3b).To take statistical QD-to-QD fluctuations for the reference decay time into account, we estimate the average decay time of 8 QD CX lines observed from dots located outside of the cavities, i.e. without Purcell-induced modification of the radiative lifetime, and obtain τ ref = (1.99 ± 0.16) ns, while the single reference shown in Fig. 3b has a decay time of (2.01 ± 0.02) ns.Therefore, the measured Purcell factor for QD-CBG #1 is F P = (5.0 ± 0.4) and F P = (3.8± 0.3) for QD-CBG #2.Although the obtained Purcell factors are comparable with F P = 3 obtained in the non-deterministic fabrication approach 9 , we expect it to be much higher if the QD would perfectly match the cavity mode both spectrally and spatially.However, the expected Purcell factor decays rapidly with the dipole displacement r 0 from the cavity center (decreases by half for |r 0 | = 100 nm, see Supplementary Note 1), as the fabricated CBG is optimized for a higher-order mode that exhibits electric field minima along the cavity radial direction (see Supplementary Note 1).Hence, the relatively large total positioning uncertainty for QD-CBGs #1-#3 (∆R ∼ 140-150 nm) and the non-ideal spectral emitter-cavity overlap explain the reduced F P as compared to the model.

Photon extraction efficiency
We evaluate the photon extraction efficiency by recording the power-dependent µPL signal with a superconducting nanowire single-photon detector (SNSPD) in a calibrated optical setup (see Fig. 3c).The setup has a total transmission of (1.1 ± 0.2) % (see Supplementary Note 5).The measured η values are corrected by the factor 1 − g (2) (0), to account for the detection of secondary photons due to the refilling of QD states 18,19 .Here, the g (2) (0) value is obtained un- der the excitation power P sat corresponding to saturation of the CX line.Evaluating the CX emission, we obtain an extraction efficiency η #1 = (16.6 ± 2.7) % for QD-CBG #1 and η #2 = (13.3± 2.2) % for QD-CBG #2 using an objective with NA = 0.4.

Single-photon emission purity
The photon statistics of a quantum light source is of fundamental importance for applications in photonic quantum technologies.In the following, we investigate the single-photon purity of the emission from QD-CBG #2 by analyzing the photon autocorrelation function g (2) (τ) (cf.Supplementary Note 7 for details on the data analysis and complementary g (2) (τ) measurements).
Under weak quasi-resonant excitation at P ≪ P sat , the probability for charge-carrier recapture is strongly reduced, resulting in almost negligible background contributions (B = 0) (Fig. 4b) and a fitted value of g (2) (0) fit = (4.7 ± 2.6) × 10 −3 at P = 0.04 × P sat .Additionally, we evaluated the raw antibunching value by integrating the raw coincidences around τ = 0 over a full repetition period normalized by the Poisson level of the side peaks.This results in g (2) (0) raw = (3.2± 0.6) × 10 −3 , with the error deduced from the standard deviation of the distribution of counts in the side peaks.As discussed later, this result compares favorably with previous reports on non-deterministically fabricated QD-CBGs.

Photon indistinguishability
Finally, we explore the photon indistinguishability of QD-CBG #2 by Hong-Ou-Mandel (HOM)-type two-photon interference (TPI) experiments 20 (see Methods and Supplementary Note 7 for details on the experimental setup, data analysis, and complementary TPI measurements).The HOM histograms recorded for co-and cross-polarized measurement configurations are presented in Fig. 4c, and were obtained under pulsed quasi-resonant excitation with identical experimental conditions as the g (2) (τ) measurement presented in the previous section (at 0.04 × P sat ).The data shown in Fig. 4c is not corrected for multi-photon events or contributions from residual laser light.
The HOM histograms feature a characteristic pattern that we analyze following the methodology described in Ref. 21.The reduced area of the central peak in the co-polarized measurement, as compared to the maximally distinguishable crosspolarized measurement, is a distinct signature of the twophoton coalescence due to a significant degree of photon indistinguishability.The visibility of the TPI is calculated from the ratio of the fitted central peak area in the co-and crosspolarized measurements as V = 1 − A Co /A Cross .We obtain a TPI visibility of V = (19.3± 2.6) % with the accuracy being the propagated fitting errors reflecting the statistics of the experimental data.The temporally post-selected visibility at zero delay time (τ = 0) is V PS = 99.8 +0.2 −2.6 %, limited only by the system temporal response.

Photon coherence time
From the width of the central dip, we extract a photon coherence time of T 2 = (103 ± 13) ps, while the highest extracted coherence time is T 2 = (176 ± 9) ps, measured for the lowest excitation power (see Supplementary Note 7).Given the Purcell-reduced lifetime of T 1 = 400 ps, this results in a T 2 /T 1ratio of 0.44, comparing favorably with previous reports for QDs emitting in the telecom C-band 22,23 .The still relatively short coherence time observed in our work is mainly attributed to fluctuating charges in the QD environment, suggested by the observed time-dependent spectral diffusion of emission lines, which most probably limits the observed indistinguishability.The coherence properties may be further improved by implementing electrical charge stabilization via electric gates or using droplet epitaxy as an alternative growth technique 22 .
To gain further insights into the coherence properties, we performed direct measurements of the T 2 time using an all-fiberbased Michelson interferometer (MI; see Methods).We extract a coherence time of up to (62 ± 3) ps for the QD-CBG #2 under weak CW above-band excitation (see Supplementary Note 7).
Note that while this value is lower than the T 2 time extracted from the dip in the HOM experiment in Fig. 4c, a direct comparison is not possible due to the different excitation schemes applied.An analysis of MI data from a total of three different CBG devices yields T 2 values between 18 ps and 60 ps.These numbers compare favorably with MI-measured coherence times of 6-30 ps obtained for three different QDs in planar regions on the same sample.The observed coherence times are comparable with reports in the literature for SK InAs/InP QDs 22 .While the direct comparison of the T 2 values measured for QDs with and without CBG should be treated with care due to the relatively low statistics, these results indicate that the microcavity integration does not degrade the optical coherence of the emitted photons.Future work in this direction may include a more elaborate study allowing for deeper insights into the limiting dephasing mechanisms and their timescales, e.g., by applying photon-correlation Fourier spectroscopy 24 .

DISCUSSION
The scalable fabrication of active quantum photonic devices operating in the telecom C-band has been a long-standing challenge.This is mainly due to the random size and strain distribution of epitaxially grown QDs causing an inhomogeneous broadening of the emission and difficulties in localizing suitable QDs due to the high electronic noise level of detector arrays sensitive around 1.55 µm wavelength.In this work, we present a solution to this problem based on a hybrid sample design.We fabricate an InP layer containing epitaxial QDs on top of a Al reflector placed on a Si wafer carrier.This geometry significantly enhances the photon extraction efficiency from the QDs by a factor > 7 enabling the localization of single QDs in a wide-field imaging setup with a thermo-electrically cooled InGaAs camera.For the 10 % brightest QDs our setup achieves an imaging SNR > 15.5 and a localization uncertainty of ∼ 80 nm with respect to alignment marks.After final EBL processing, we achieve an overall uncertainty of ∼ 90 nm for fabricating a nano-photonic device around pre-selected QDs.The localization accuracy of our setup is comparable to setups operating in the 770-950 nm range (48 nm 25 , 30 nm 26 ) where Si-based sensors with four orders of magnitude lower electronic noise can be used.The accuracy in our setup can be further improved by using higher-NA objectives inside the cryostat together with an overall increased microscope magnification, which was reported to reduce the localization accuracy down to 5 nm 27 .Alternatively, in-situ EBL 28 or photolithography 29 scanning techniques provide a similar accuracy down to ∼ 35 nm, but are comparably slow, require cathodoluminescence signal for QD localization or, in case of photolithography, are not suitable to define reliably sub-µm features.
To exemplify our approach, we fabricate CBG cavities with a resonance wavelength of 1.55 µm around some of the preselected QDs.The low QD density of 3.1 × 10 8 /cm 2 guarantees their spatial isolation sufficient for the imaging procedure while maximizing the number of devices that can be fabricated per field.The QD-to-CBG coupling is evidenced by a Purcell factor of F P = (5.0 ± 0.4), further increasing the singlephoton emission rate and final source brightness.Using our approach, we obtain a total process yield of 30 % for finding a pre-selected QD spectrally matching the CBG cavity, which is a significant improvement compared to the yield achievable with a random placement approach (∼ 0.7 %).For our QD-CBG device, we measure a photon extraction efficiency of (16.6 ± 2.7) % with a NA = 0.4 objective, which is comparable to previously reported devices fabricated probabilistically and operating at C-band wavelengths 9 , as well as deterministically fabricated CBGs at O-band with In(Ga)As/GaAs QDs 30 .However, no cavity positioning accuracy, fabrication yield, and HOM visibility data were provided in Ref. 30.The discrepancy between the simulated and measured photon extraction efficiency is attributed to the residual defects in the epitaxial material and possible material damage due to the dry etching.The fabrication can be optimized to eliminate both effects.
Our QD-CBG devices feature excellent single-photon emission purities with raw values down to g (2) (0) = (3.2± 0.6) × 10 −3 , beating previous records for non-deterministically 9 and deterministically 30 fabricated QD-CBGs, as well as most QDs operating in the C-band 31,32 , while being not yet competitive with the state-of-the-art 33 .
Importantly, we report triggered TPI experiments for InPbased cavity-coupled QDs with emission wavelengths in the telecom C-band, which is crucial for applications in quantum information processing (QIP).We generate indistinguishable photons with a TPI visibility up to V = (19.3± 2.6) % and a post-selected value of V PS = 99.8 +0.2 −2.6 % at zero time delay.Previous reports on C-band QD-SPSs by other groups were based either on droplet epitaxy InAs/InP QDs in planar structures 22,34 or InAs QDs grown on GaAs followed by an InGaAs metamorphic buffer, also located in planar structures 23,35 or embedded in randomly placed CBGs 9 .The photon coherence time measured using a Michelson interferometer is up to (62 ± 3) ps for the QD-CBG #2 under weak CW above-band excitation.
Further improvement in the photon indistinguishability is of utmost importance for applications in QIP.This is challenged by the strong coupling of QDs to their semiconductor environment via charge and spin noise, both causing QD decoherence 21,36 .It is thus important to stabilize the QD environment by removing the excess charge carriers from the vicinity of QDs, e. g., by integrating them into a p-i-n junction, which is expected to increase the photon coherence and indistinguishability substantially.On the other hand, tuning the QD emission energy using strain 37,38 , or quantum-confined Stark effect 39,40 , would address the challenge of QD ensemble inhomogeneous broadening by fine-tuning the QD energy to match the cavity mode.The QD tuning is feasible using the reported approach but requires a different cavity design 41,42 .Implementing coherent optical pumping schemes, such as two-photon resonant excitation 43 , also for scalably fabricated devices, while avoiding the excess charge carriers that could originate, e.g., from the electrical QD excitation, is a crucial next step to further improve the photon coherence time and hence indistinguishability 44 .
Moreover, the InP material system used in our work appears to be advantageous for QD-based quantum photonic devices operating in the C-band and compared to GaAs-based devices.Despite the careful strain engineering involved in the epitaxy of QDs on GaAs 45 , the metamorphic buffer complicates the device engineering and QD growth.In contrast, an unstrained InP system is free from threading dislocations that would be a source of dangling bonds causing non-radiative recombination, thus lowering the efficiency 46 .
In conclusion, our work opens the route for the highthroughput fabrication of telecom C-band wavelength quantum photonic devices with QDs delivering flying qubits, i.e., single or entangled photons 47 , or acting as a non-linear element for QIP 48 .Improvements in our optical imaging setup will further increase the device yield and positioning accuracy, while the electric control and coherent excitation of QD emitters will further push the achievable photon indistinguishability.

Methods
Epitaxial growth and fabrication of planar structure with QDs-The structures were grown on epi-ready (001)-oriented InP substrates by the low-pressure metalorganic vapor-phase epitaxy (MOVPE) TurboDisc® reactor using arsine (AsH 3 ), phosphine (PH 3 ), tertiarybutylphosphine (TBP), trimethylgalium (TMG) and trimethylindium (TMIn) precursors with H 2 as a carrier gas.We grow the 0.5 µm-thick InP buffer followed by 200 nm-thick In 0.53 Ga 0.47 As lattice-matched to InP etch-stop layer and a 156 nm-thick InP layer at 610 • C.Then, the temperature is decreased to 493 • C, stabilized under TBP for 180 s and AsH 3 for 27 s.The nucleation of QDs occurs in the near-critical regime of Stranski-Krastanov growth mode after deposition of nominally 1.22 ML-thick InAs at growth rate 0.53 ML/s under TMIn and AsH 3 flow rates of 11.8 µmol/min and 590 µmol/min, respectively (V/III ratio of 50).Nucleated QDs are annealed for 3.5 s at the growth temperature in AsH 3 ambient before the deposition of a 156 nm-thick InP capping layer (12 nm at 493 • C, and the remaining 144 nm after increasing the temperature up to 610 • C) what finishes the growth sequence.
Fabrication of the sample for µPL imaging-After the QD epitaxy, SiO 2 is deposited in plasma-enhanced chemical vapor deposition (PECVD).This layer is intended to be 358.6 nm-thick, and it covered with a 120 nm-thick Al layer deposited via electron-beam evaporation.The flipped structure is bonded to the Si chip carrier utilizing spin-coated AP3000 adhesion promoter and benzocyclobutene (BCB) on Si and AP3000 on the InP wafer.The bonding is done by applying the force of ∼ 2 kN in vacuum at 250 • C. The substrate removal step is done by ∼ 60 min dip in HCl and the InGaAs etch-stop layer is subsequently removed in H 2 SO 4 :H 2 O 2 :H 2 O=1:8:80 mixture.Next, by employing electron-beam lithography (EBL) followed by inductively coupled plasma-reactive ion etching (ICP-RIE) to etch InP down to SiO 2 , the square imaging fields were fabricated with 50 µm side-length and AMs dedicated to EBL outside the fields.Therefore, there are different AMs for optical imaging (edges of the imaging fields) and for EBL alignment marks detection (InP crosses).This approach is justified by the simplification of the fabrication flow by avoiding the deposition of metallic AMs, relying instead on the outline of the field visible due to the µPL signal scattering from its edges.The material contrast between InP and SiO 2 /Al regions is sufficient for the AM detection during the EBL alignment step.
Modeling of the CBG-The QD is modeled as a classical dipole 49 , and the numerical simulations of the CBG geometry are performed using a modal method employing a true open boundary condition 50 .See the Supplementary Note 1 for further details.
Deterministic fabrication of QD-CBG devices-After determining the positions of QDs with respect to the AMs, the CBG pattern is defined in the CSAR e-beam resist using e-beam lithography using high-precision alignment based on the InP mark detection in JEOL JBX-9500FSZ e-beam writer.The pattern is transferred into the PECVD-deposited 110 nm-thick SiN x layer using ICP-RIE with SF 6 -based etch recipe.Residual CSAR is stripped in Remover 1165 followed by 10 min descum in the barrel-type plasma asher.Subsequently, the pattern is transferred into the InP layer in ICP-RIE by HBr-based etch.The calculated design is first scaled and fabricated using a nominally identical heterostructure to investigate the mode energy vs. size dependence and to account for the fabrication imperfections.Additionally, we experimentally determine the ∼15 nm temperature-induced blueshift of the mode energy between a room and low temperature resulting from the contraction of the structure (introducing size and strain changes), as well as from the change of the refractive indices of the layers.
Optical characterization of devices-The structure with QD-CBG devices is held in a helium-flow cryostat allowing for control of the sample temperature in the range of 4.2-300 K.For our standard µPL studies, the structures are optically excited through a microscope objective with NA = 0.4 or 0.65 and 20× magnification using 660 nm or 805 nm light generated with semiconductor laser diodes.The same objective is used to collect the µPL signal and to direct it for spectral analysis into a 1 m-focal-length monochromator equipped with a liquid-nitrogen-cooled InGaAs multichannel array detector, providing spatial and spectral resolution of ∼2 µm and ∼25 µeV, respectively.
The photon extraction efficiency and time-resolved µPL are measured in the same setup.Here, QDs are excited by ∼50 ps-long pulses with a repetition rate of 80 MHz and a central wavelength of 805 nm.At the same time, the second monochromator output port is equipped with the fiber coupling system, transmitting the signal to an NbN-based SNSPD (Scontel) with ∼87 % quantum efficiency in the range 1.5-1.6 µm and ∼ 200 dark counts per second.A multichannel picosecond event timer (PicoHarp 300 by PicoQuant GmbH) analyzes the single photon counts as a time-to-amplitude converter.The overall time resolution of the setup is ∼80 ps.Experimental setups are shown in Supplementary Note 4, and data on the setup transmission efficiency used for determining the photon extraction efficiency is given in Supplementary Note 5.
Photon autocorrelation measurements-For the photon-autocorrelation measurements, QD-CBG devices were optically excited using a Ti:Sapphire (Ti:Sa) laser (Coherent Mira-HP) or a widely tunable ps-pulsed laser system based on an optical parametric oscillator (OPO) (picoEmerald by APE GmbH) with repetition rates of 76 MHz and 80 MHz, respectively.We use a fiber-coupled bandpass filter (FWHM = ∼0.4nm) for spectrally selecting the QD emission, followed by a 50:50 fiber beam splitter.For the off-resonant excitation, we use a microscope objective with NA = 0.7 and 100× magnification and excite the QD emission with ∼2 ps-long pulses at 830 nm from the Ti:Sa.The signal is detected with a pair of SNSPDs with ∼87 % and ∼92 % quantum efficiency at 1550 nm.For the quasi-resonant excitation, we use an aspheric lens (NA = 0.6) mounted inside a low-vibration closed-cycle cryostat (attoDRY800 by Attocube Systems AG) cooled to 4.5 K. Here, the OPO-laser is used and adjusted to a pulse length of 5 ps.Single photons are detected via SNSPDs with 80 % detection efficiency at 1550 nm and 57 ps timing jitter (complete system temporal response).The excitation energy was determined in photoluminescence excitation experiments to be 0.835 37 eV (37.57meV above the QD emission energy, cf.Supplementary Note 7), which was also used for following TPI experiments.
Photon-indistinguishability measurements-In the TPI experiments, an additional 4 ns delay was introduced between consecutive laser pulses by adding an imbalanced free-space Mach-Zehnder interferometer (MZI) in the excitation path, which was compensated in the HOM setup on the detection side.Hence, the excitation sequence is composed of pairs of pulses separated by 4 ns, every 12.5 ns corresponding to 80 MHz laser repetition rate.Free-space waveplates were used to match the polarization for the TPI inside the fiber beam splitter.The exact polarization was set by using a polarimeter at the beam splitter output in combination with a laser tuned to the QD emission wavelength.Fine-tuning the relative delay between both MZI arms was used to match precisely the detection and excitation delay, respectively.The contrast of classical Michelson interference of the laser with itself was used for optimization.See Supplementary Note 7 for the details of the HOM data analysis and the scheme of the experimental setup.
Coherence measurements-For measurements of the T 2 time, an all-fiber-based Michelson interferometer (MI) was implemented 22 , consisting of a 2 × 2-port 50 : 50 fiber beam splitter with both exit ports terminated by a Faraday mirror, reflecting the light with 90 • polarization rotation.The necessary coarse and fine temporal delay is controlled by a variable optical delay stage and a piezo-driven fiber stretcher in the two MI arms, respectively.The single-photon signal is coupled to one input port of the MI and detected at the second input port using a SNSPD.The MI setup in its configuration features 80 % overall transmission (excluding the BS) and allows for the measurement of coherence times of up to 1 ns.The maximally achievable interference contrast was measured with a CW laser at 1550 nm to be 98 %, limited only by the slight intensity mismatch due to the reduced transmission through the optical delay line.For each temporal delay adjusted via the coarse variable delay line, a fine temporal scan is performed via the fiber stretcher, resulting in interference fringes with an amplitude depending on the overall delay.The interference fringes are evaluated by subtracting a constant amount of dark counts and evaluating the interference contrast via v = (I max − I min )/(I max + I min ).Finally, the T 2 time is extracted by fitting a two-sided exponential decay to the interference visibility v data as a function of the coarse delay set in the MI with the uncertainty representing the fit accuracy.Supplementary Fig. 5. Histogram of signal-to-noise ratio (SNR) for the recorded QD spots.

B. Localization algorithm
Our algorithm calculates the scaling factor P [px/µm] to change the map unit from pixels (px) to µm.For a given µPL map, we identify n QD spots and take 2n cross-sections of the map (each QD is sectioned in two directions).The width of each cross-section is taken to equal a few pixels centered at the QD spot maximum to increase the signal-to-noise ratio (SNR).We found the best SNR for 10 pixels and used this value consistently.
For the i-th cross-section, we determine the positions M i,l [px], M i,u [px]1 of the field boundaries (acting as reference/alignment marks, AMs; indices l and u stand for lower and upper field boundaries, the same indices are applied also for left and right field edges) and Q i [px] for the QD spot position.All values M i,l , M i,u , and Q i are the centers of the Gaussian peaks, fitted to respective maxima on the µPL map cross-section.Additionally, we assume that each field is a square of size F = 50 µm.
P is calculated by averaging over all cross-section-related coefficients P i recorded for a given field (P = Pi ) to ensure its highest accuracy, according to the formula Calculating P separately for each map accounts for possible slight changes in the magnification due to defocusing of the sample surface during the cryostat translation, however, we find very low dispersion of P coefficients for different µPL maps (see the following section and Supplementary Fig. 6).
Then, the i-th QD position Q i [µm] (vertical or horizontal) is calculated as C. Uncertainty of scaling the µPL maps The accuracy of the scaling factor ∆P i for a single (i-th) µPL map cross-section can be calculated by propagating the uncertainties in Supplementary Eq. ( 8): The uncertainty of the field size ∆F has two contributions, the uncertainty of the electron beam lithography alignment, estimated to ∆C = 40 nm [8] and the over-etching ∆x during the ICP-RIE step (estimated to be up to ∆x = 40 nm), potentially influencing F by 2∆x = 80 nm.∆M i,u and ∆M i,l are standard errors of the numerical fitting.
However, to increase the accuracy of the determination of the scaling factor, we average the P i values for all cross-sections taken in a particular µPL map, as can be seen in Supplementary Eq. ( 8), so as the uncertainty ∆P we adopt the estimator of the standard error of mean σ Pi for the sample defined as a set of all 2n factors P i = |M i,u − M i,l |/F determined for a given µPL map: where σ P i is the sample standard deviation.
According to calculated statistics for m = 84 exemplary µPL maps, we can compare the estimators of the standard error of mean σ Pi determined separately for each µPL map and reach the following conclusions: 1. We find that the coefficients P i are very close to each other, evidencing no perceptible elongation or distortion of the registered images.This can be evaluated by the analysis of the set of calculated (for all m maps) uncertainties ∆P = σ Pi scaled by the determined P factors: δ P = ∆P/P.We plot the histogram of δ P in Supplementary Fig. 6 and find the median value of δ P-distribution of 0.06 %.In other words, for half of the analyzed maps, the distribution of P i factors is sufficiently narrow to determine the P factor with relative uncertainty δ P < 0.06 %.This value includes the potential elongation of the image (difference in vertically and horizontally determined P i factors).
2. The stability of the setup and overall repetitivity of the imaging process is high as the image magnification varies very little between maps.This can be evaluated by the estimator of the standard error of mean σ P calculated not for different cross-sections [averaging P i values for a given map, as in Supplementary Eq. ( 8)] but for different maps (averaging P values for all maps).We obtain σ P/ P = 0.013 %.We take this value as the estimation of variation of the setup magnification during the imaging process.Then, the accuracy of cavity positioning ∆R depends primarily on the accuracy of the QD localization (∆Q) and EBL alignment uncertainty ∆C.The ∆Q itself depends on the fit standard errors (∆M, ∆Q), and scaling factor uncertainty ∆P.

D. Image rotation ϕ
Although we carefully align the image of the field with the horizontal and vertical axes of the detector array, we assume that there can be some indiscernible rotation of the image ϕ on the level of up to a few degrees.Even if this is the case, the rotation results first in the larger separation between AM peaks in the cross-sections (|M i,u − M i,l |/ cos ϕ instead of |M i,u − M i,l |) and this is translated to scaled P: P → P/ cos ϕ, according to Supplementary Eq. (8).The distance between lower AM and QD is however also elongated, According to Supplementary Eq. ( 9), the new QD position is: so that the slight image rotation has no influence on the determination of QD position Q i in our approach.
(PSF) of the setup, which for the approximation of 2D paraxial wide-field fluorescence microscope forms an Airy disc (first-order Bessel function of the first kind) [9].Our QD localization algorithm uses the approximation of the Bessel function by the Gaussian profile, which we use for fitting the QD-and AM-related signal in the cross-sections of µPL maps.
First, we calculate the standard deviation σ diff of the Gaussian curve that best approximates the PSF of our setup according to the formula [9]: λ QD NA = 0.501 µm (14)   with employed NA = 0.65 and λ QD = 1.55 µm.Then, we translate σ diff to full width at half-maximum (FWHM) as We take the calculated FWHM diff = 1.18 µm (see previous section) as the diffraction-limited spot size and compare it with the histogram of all FWHM values for registered QD spots in Supplementary Fig. 7a.The medians for the registered FWHM values are similar for horizontal and vertical cross-sections, 1.62 µm and 1.58 µm, respectively, which is ∼35 % more than the calculated FWHM diff .Almost all registered FWHM values are lower than 2×FWHM diff .Therefore, our imaging setup operates close to the diffraction limit with residual broadening originating most probably from the cryostat window between the QDs and microscope objective [10].The similarity between horizontal and vertical spot widths is evidenced by the histograms of ratios We record the temperature-dependent µPL signal for QD-CBG #2 in the temperature range of T = 4.2 K to 70 K under the continuous-wave (CW) excitation and present the stacked spectra in Supplementary Fig. 11a.Three emission lines, marked with rings, can be identified and their emission intensity and energy tracked as the temperature is increased.We plot also the mode profile for reference.Note the ∼ 2 nm redshift of the most intense line from the central wavelength of the cavity.
The µPL intensities change differently for lines with emission energy lower vs. higher compared to the mode profile.The short-wavelength lines, marked with red and blue rings, are tuned across the mode profile as the temperature raises.Their µPL intensity quench is greatly suppressed, in fact, the intensity of the line marked with a red circle has a maximum at T = 50 K.The behavior for the long-wavelength line, marked with a green ring, is opposite and its intensity quenches fast.
The temperature-dependent µPL intensity for the most intense line marked with the green ring is fitted with a standard formula assuming two activation processes [11]: where I 0 is the µPL intensity for T → 0, E a,1 and E a,2 are activation energies, and B 1 and B 2 are relative rates corresponding to the efficiency of involved processes.We achieve the activation energies of E a,1 = (6.9± 0.8) meV and E a,2 = (27.9± 3.1) meV.
We note that E a,2 is similar to the activation energy E Ref = (23.6 ± 0.8) meV found for a CX line from analogous QDs [7].Accordingly to Ref. [12], this activation energy can be attributed to the charge transfer to higher orbital states, based on the band structure calculations within the 8-band k • p framework.Hence, another mechanism is responsible for the identification of E a,1 energy which is absent in the case of high-energy lines (blue and red circles, see Supplementary Fig. 11b).This can potentially be associated with the enhancement of the QD emission rate when the overlap between the line and the mode energy is maximized.
In Supplementary Figure 11c we plot the temperature dependence of the emission energy for lines analyzed in Supplementary Fig. 11b and, additionally, for another investigated QD-CBG device, where the observation of the crossing of the temperature-tuned QD line and the cavity mode under high power off-resonant CW excitation was possible due to the QD energy being higher than the mode energy at T = 4.2 K.The temperature-induced evolution of cavity mode energy (black open squares) can be compared with the QD lines, evidencing a weaker temperature dependence, as expected for the cavity mode.Supplementary Fig. 17.Comparison of Hong-Ou-Mandel (HOM) histograms taken at a, 0.02 × P sat and b, P sat excitation power.HOM measurement recorded for 0.02 × P sat shows higher indistinguishability and longer coherence time than the one taken at P sat , see Supplementary Table 7.
As mentioned at the beginning of this section, in addition to comparing the co-and cross-polarized central peak areas, the indistinguishability can also be extracted by comparing the areas of the central to the outer peaks for only the co-polarized data.This analysis of the low-power HOM histogram is presented in Supplementary Fig. 18.Evaluating the visibility from the extracted center A 3 and side peak areas A 2 , A 4 , and calculating the visibility as V side peaks = 1 − 2A 3 /(A 2 + A 4 ) leads to (15.9 ± 9.9) % for the low excitation power (0.02 × P sat ), which agrees with the visibility result reported above obtained from the co-cross-comparison.The uncertainty is however larger, as the fitting relies on successfully separating the overlapping peak contributions.
Importantly, the obtained visibilities and the post-selected values compare favorably with the reports for GaAs-based QDs emitting at C-band [20-22].For planar QDs, a visibility of (14.4 ± 1.5) % has been reported under pulsed resonant excitation [21]  Supplementary Table 7. Extracted fitting parameters for recorded Hong-Ou-Mandel (HOM) histograms.MFR -mean fit residuals, TPItwo-photon interference, pol -polarized.while the raw visibility value of (71 ± 15) % was obtained under two-photon-resonant CW excitation [20].Values obtained under CW excitation can be related to the post-selected values determined from pulsed excitation.For QDs placed non-deterministically in CBGs, the only reported visibility so far is (8.1 ± 3.4) % and the post-selected on the order of 60 % [22].

Quasi-resonant excitation power
(a) (b) Supplementary Fig. 18.Data analysis of the low-power Hong-Ou-Mandel (HOM) histogram.a, Individual peak contributions extracted as part of the total fit according to Supplementary Eq. ( 23) for the lowest power HOM measurement (0.02 × P sat ).b, Close-up of the central peak.

E. Coherence measurements
To access the coherence time T 2 directly, we performed measurements using a Michelson interferometer (MI) under abovebarrier CW excitation (980 nm) of QDs integrated into CBGs and planar sample regions.To do so, we employed an all-fiber MI shown in Supplementary Fig. 19a, where the inset shows a typical interference scan.While the resulting T 2 values are not directly comparable to the values extracted from the HOM experiments, they provide lower bounds on the coherence time (as above-barrier CW excitation results typically in substantially enhanced decoherence effects), and can be used to compare different QD devices.The highest coherence time observed in the MI measurements was obtained from the QD-CBG #2 from the main text to be (62 ± 3) ps (c.f.Supplementary Fig. 19b), extracted from double-exponential fits.Note the artifacts, which are caused by temporary spectral jumps (telegraphic noise), as also discussed in the main text as one of the factors limiting the indistinguishability.The extracted T 2 value is in line with coherence times of ∼ 50 ps reported in the literature for InAs/InP QDs grown in the Stranski-Krastanov mode [23].Supplementary Fig. 19.a, Schematic of the all-fiber Michelson interferometer for 1550 nm.The single photon signal is split in a fiber beam splitter (BS) and back-reflected at two Faraday mirrors (FM), while the coarse delay on one arm is controlled with a variable optical delay line (VDL) and the fine scan of the relative path difference is done via a piezo fiber stretcher (FS).The counts after the interference are detected on a superconducting nanowire single photon detector (SNSPD).Inset: Example of the FS scan for 0 ps delay indicating constructive and destructive interference.b, Extracted visibility for different delay positions allows the determination of the T 2 time using an exponential fit.This is data under 980 nm CW excitation for the QD-CBG #2 from the main text.
To gain further insight, we have measured the T 2 time for the QD-CBG #2 from the manuscript for different excitation powers (Supplementary Fig. 20a) as well as for different temperatures (Supplementary Fig. 20b).As expected, the T 2 time drops when the temperature or the excitation power is increased.The slight increase in coherence when temperature increases might be explained by the fact that the trion transition occasionally showed a random telegraphic noise.Here, the different states may be associated with different coherence times, as confirmed by repeated measurements under the same conditions.The sudden spectral jumps during a Michelson scan yield sudden changes in signal rate, which leads to outliers for the extracted interference contrast (c.f., four data points in Supplementary Fig. 20b).
While the strong dependence of the coherence on excitation power makes a direct comparison between QDs in the planar state and in CBGs difficult, we chose a compromise of evaluating the coherence for 3 exemplary (one high coherence, one medium, and one low) planar and CBG-integrated QDs, respectively.The results are compared in Supplementary Fig. 21.As the planar QDs do not couple to the excitation laser efficiently, more power was required to get a sufficient signal, yielding an overall reduced coherence.Also, due to the smaller count rate, the maximum interference contrast is not well resolved.We do find, however, a spread of measured maximum coherence times between 6-30 ps for the set of planar QDs investigated (Supplementary Fig. 21a).
For the QDs deterministically integrated into CBG structures, a lower excitation power is possible.Even though only a small number of QDs were investigated, the spread of coherence times from 18-62 ps indicates at least no deterioration of the coherence by integrating it into CBG structures.If the integration yields a significant Purcell enhancement, thus reducing the T 1 time, while not strongly reducing the T 2 time, as it seems to be the case here, one can conclude that the CBG cavity brings the emission closer to the Fourier limit of T 1 = 2 × T 2 .

FIG. 2 .
FIG.2.Microphotoluminescence imaging of quantum dots (QDs).a, The optical setup used for imaging, BP -bandpass filter, NAnumerical aperture, b, a microphotoluminescence map of a (50 × 50) µm2  InP field containing seven localized InAs QDs emitting at C-band, c, scanning electron microscope image of CBGs fabricated atop the localized and preselected QDs, d, e, histograms of QD localization accuracy (d) and overall cavity placement accuracy (e) for all detected spots, with markers for 10 th distribution percentiles, f, g, exemplary microphotoluminescence map cross-sections showing the signal of the QD labeled with the orange circle in b and alignment marks together with Gaussian fits used for the QD localization, h, close-up of the QD signal from e, centered to 0 µm.The full width at half maximum of the fitted Gaussian profiles is ∼1.6 µm and the signal-to-noise ratio (SNR) is 12.
b e r o f Q D s S i g n a l -t o -n o i s e r a t i o C u t o f f f o r 1 0 % b r i g h t e s t Q D s : S N R = 1 5 .5

. 6 .
The histogram of relative microphotoluminescence map scaling uncertainty δ P = ∆P/P for the set of analyzed m = 84 exemplary microphotoluminescence maps.

J.Supplementary Fig. 7 .
Statistics on QD spot size and the accuracy of QD positioning Parameters of fitting the microphotoluminescence map cross-sections, shown separately for horizontal and vertical directions.a, the histogram of full width at half maximum (FWHM) values for the fits used for the determination of quantum dot (QD) position Q compared with the calculated diffraction-limited value FWHM diff = 1.18 µm [Supplementary Eq. (15)] and fitted with the normal distribution curves, b, histogram of the ratio between the widths of the fitted horizontal and vertical cross-sections of the QD Gaussian profiles, based on the panel (a), c, QD position fit uncertainty ∆Q/P as a function of the QD fit area, d, histogram of the QD position fit uncertainty ∆Q/P, e, alignment mark (AM) position fit uncertainty ∆M/P.Histograms in panels (d) and (e) are fitted with log-normal distributions.

10 .Supplementary Fig. 11 .
Additional microphotoluminescence (µPL) data.a, Histogram of recorded decay times for reference QD trion (CX) lines; arrow points to the quantum dot (QD) shown in Fig.3bin the article and in panel (c), b, µPL spectrum for QD-CBG #3, c, time-resolved µPL time traces for QD-CBG #3 and the reference QD.Analysis of the temperature-dependent microphotoluminescence spectra for quantum dot-circular Bragg grating (QD-CBG) device #2.a, Temperature-dependent µPL spectra for QD-CBG #2, overlapped with the cavity mode.b, Arrhenius plot for the integrated µPL intensity for three QD lines, indicated with the same symbols in panel (a).Inset: close-up of the marked region.c, µPL emission energy for the same QD lines.Black symbols stand for another QD-CBG device, with QD and cavity mode energies tracked up to T = 70 K.The mode crossing characteristic for the weak QD-cavity coupling regime is visible.

Coincidences
Supplementary Fig.16.Expected Hong-Ou-Mandel (HOM) histograms of coincidences with individual peak contributions to the five peak patterns formed by 4 ns delay and laser repetition of τ 0 = 12.5 ns, a, for perfectly indistinguishable photons, and b, for maximally distinguishable photons.The black lines represent the fits according to Supplementary Eq. (23)hom-pulsed-Cross that include all contributions, plotted for τ 1 = 550 ps lifetime.

SupplementaryFig. 21 .
Comparison of coherence measurements results.a, Data for the selection of planar QDs, b, QDs deterministically integrated into CBG structures.While the exact coherence time depends on power, the general trend is that the T 2 time is, on average, not reduced when integrating the QD into a photonic structure.