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Abstract
Efficient generation of entangled photon pairs at telecom wavelengths is a key ingredient for long-range quantum networks. While embedding semiconductor quantum dots into hybrid circular Bragg gratings has proven effective, it conflicts with p-i-n diode heterostructures which offer superior coherence. We propose and analyze hybrid circular photonic crystal gratings, incorporating air holes to facilitate charge carrier transport without compromising optical properties. Through numerical simulations, a broad cavity mode with a Purcell factor of 23 enhancing both exciton and biexciton transitions, and exceptional collection efficiency of 92.4% into an objective with numerical aperture of 0.7 are achieved. Furthermore, our design demonstrates direct coupling efficiency over 90.5% into a single-mode fiber over the entire telecom C-band. The hybrid circular photonic crystal grating thereby emerges as a promising solution for the efficient generation of highly coherent, polarization-entangled photon pairs.
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1. Introduction
Entangled photon pairs are essential building blocks for a quantum internet, enabling connections between distant nodes [1] through entanglement-based quantum key distribution [2,3] and quantum repeater schemes [4]. Although spontaneous parametric down-conversion (SPDC) sources have been the main workhorse for testing these scenarios, they face limitations in achieving high brightness without compromising the single-photon purity [5]. In contrast, epitaxial semiconductor quantum dots (QDs) offer on-demand single-photon generation with low multi-photon probability and high brightness simultaneously. Notably, photons from QDs demonstrate high indistinguishability comparable with that generated from SPDC sources, particularly under resonance fluorescence (RF) and stimulated resonant two-photon excitation (TPE) schemes [6–8]. Polarization-entangled photon pairs with fidelity up to 0.987 have been demonstrated via TPE of the biexciton state (XX) followed by a cascade decay via the exciton state (X) [3]. Recent progress has been achieved in improving the quality of the photons emitted by QDs directly in the telecom C-band [9–13]. This advancement paves the way for utilizing optical fiber networks with minimal transmission loss and therefore increasing the achievable distances in quantum communication applications [14]. However, due to the high refractive index of the semiconductor host matrix, total internal reflection limits the fraction of photons extracted from QDs, necessitating their integration into photonic nanostructures such as weak-coupling optical microcavities [6,15–19] or photonic crystal waveguides [20]. By harnessing the Purcell effect, the emitted photons can be efficiently funneled through the desired optical mode, resulting in high collection efficiency. In addition, the Purcell effect can enhance radiative decay rates of the excitonic transitions to improve the photon indistinguishability [21,22]. Among various photonic nanostructures, hybrid circular Bragg gratings (CBGs) have demonstrated superior suitability for the generation of polarization-entangled photon pairs [23–25]. The broad cavity mode enhances the radiative decay rates of X and XX asymmetrically, thereby increasing their lifetime ratio and improving the photon indistinguishability [26,27]. Its broadband collection efficiency enhancement facilitates more relaxed requirements on the QD emission wavelength. Moreover, its planar structure allows for convenient integration onto micromachined piezoelectric actuators for the efficient transfer of anisotropic strain [28,29]. This configuration serves as a powerful tool for the simultaneous tuning of emission wavelength and maximization of the degree of entanglement by eliminating the exciton fine structure [25,30].
Another challenge for QDs as quantum light sources, being embedded in a solid-state environment, is decoherence processes due to phonons, charge fluctuations, and spin fluctuations [31]. In addition to wavelength tuning via the Stark effect, highly coherent and blinking-free photon emission in the near-infrared range has been demonstrated by embedding QDs into a $p-i-n$ diode heterostructure to stabilize the charge environment via Coulomb blockade [32,33]. However, applying this charge-tuning technique necessitates electrical contacts on both top and bottom doping layers for vertical biasing. Conventional CBGs featuring fully etched trenches are incompatible with this requirement. An apparent solution involves incorporating straight bridges for electrical connection [34,35]. However, this introduces a waveguiding effect, reducing optical confinement and, consequently, the Purcell factor $F_\mathrm {P}$. Moreover, the discontinuous grating leads to a significant deviation from the desired Gaussian pattern in the far field, impairing prospective coupling efficiency into single-mode fibers (SMFs). These effects can be mitigated by narrowing the bridge such that the propagation mode is precluded. Nevertheless, the effectiveness of charge control would be undermined. These issues have been addressed by intentionally shifting the position of the bridges between consecutive rings [36]. Although this labyrinth design partially restores the optical confinement and the quasi-Gaussian far-field pattern, it remains deficient in collecting photons. Recently, by replacing trenches with air holes, the hole-CBG structure has demonstrated single-photon emission with high $F_\mathrm {P}$ close to the telecom O-band [37]. The possibility to optimize the structure also in the azimuthal direction suppresses higher-order diffraction, leading to high collection efficiency, especially into objectives with small numerical aperture (NA) and SMFs. In addition, its continuous surface allows for implementing electrical connections. While promising as a single-photon source, it still has limitations for generating polarization-entangled photon pairs. The limitation arises from the subtle lifting of the degeneracy between two orthogonal fundamental cavity modes due to the non-stringent rotational symmetry, reducing the theoretical maximum entanglement fidelity [38].
In the first decade of this century, sunflower-type circular photonic crystal (CPC) microcavities were extensively studied, showcasing isotropic photonic band gap and high quality factor $Q$ [39–41]. Here, we revitalize this concept by implementing its symmetric design into the hole-CBG. We conduct numerical investigations of the InP-based hybrid circular photonic crystal grating (CPCG) structure while maintaining a broadband reflector comprising a SiO2 layer and a gold mirror underneath. The azimuthal gaps between air holes serve as channels for charge carriers, rendering CPCG compatible with $p$-$i$-$n$ diode heterostructures. Design principles and optical properties are explored and assessed via finite-difference time-domain (FDTD) simulations. The optimized design yields a Purcell factor of 23 and collection efficiency of 92.4% into an objective with NA of 0.7. Additionally, we examine the impacts of QD displacement and orientation, demonstrating the robustness against such fabrication imperfections and the highly approximated circular symmetry of the CPCG. Finally, the maximum coupling efficiency of 95% is achieved for direct photon collection into a commercial SMF.
2. Simulation method
The optical properties of the CPCG are simulated using Ansys Lumerical FDTD 3D solutions. The thickness and the distance to the object of the perfectly matched layer boundaries are each at least half a wavelength in the corresponding medium to ensure sufficient absorption of incident light with minimal reflection. Because the optical properties are predominantly influenced by the CPCG region in the InP layer, its mesh size is set as 20 nm, equivalent to over 22 grids per wavelength, to better mimic the circular shapes. For the remaining simulation region, we employ automatic non-uniform meshes to alleviate the demands on computational resources. To precisely simulate the device performance at cryogenic temperatures, which are crucial for minimizing the phonon thermal occupation and hence improving emission properties [42], refractive indices in the telecom C-band at 4 K ($n_{\mathrm {InP}}=3.135$, $n_\mathrm {SiO_2}=1.443$) are calculated or extracted from the literature, respectively [43,44]. The emission from a QD is represented by an electric dipole oriented along the $x$ axis and positioned at the center of the InP layer, as pointed out by the red arrow in Fig. 1(b). The Purcell factor $F_\mathrm {P}$ is calculated as the ratio of the actual emitted power by the dipole in the given environment to the power emitted by the dipole in a homogeneous environment. We define the extraction efficiency $\eta _{\mathrm {ext}}$ as the portion of the total dipole power entering the upper hemisphere, and the collection efficiency $\eta _{\mathrm {coll}}$ as the ratio of the integrated far-field power within the objective aperture angle to the total dipole power. The collection efficiency $\eta _{\mathrm {coll}}$ is calculated for NA = 0.7 throughout the paper unless specified otherwise.
Fig. 1. Sketches of the hybrid CPCG device. (a) 3D illustration showing the InP layer with fully etched air holes in a sunflower-type geometry, on top of a SiO2 layer and a gold mirror, including electrical contacts on both $p$- and $n$-layers. (b) Cross-sectional schematic with the relevant structure parameters, where $R$, $\Lambda$, and $D$ denote the central disk radius, the grating period, and the hole diameter, respectively.
3. Results
3.1 Device design and parametric analysis
The hybrid CPCG device illustrated in Fig. 1 incorporates an InP layer hosting InAs QDs at the center, situated atop a SiO2 layer and a gold mirror. The InP layer thickness $t_\mathrm {InP}$ is set intentionally to 300 nm, which is thicker than the critical thickness that only supports the fundamental transverse electric mode. This prevents degradation of QD emission properties caused by the $p$-type dopant diffusion [34,45]. The risk of coupling to the second-order mode is mitigated since the QD position corresponds to its node, rendering it unexcitable. The initial SiO2 layer thickness $t_\mathrm {SiO_2}$ is chosen to be approximately half the wavelength in the medium, facilitating constructive interference between the upward emission and the reflected downward emission by the 100 nm-thick gold mirror. Surrounding the dipole source, the sunflower-type CPCG consists of air holes etched through the InP layer, with their positions in the $xy$ plane given by [39]
To understand how structure parameters affect the device performance, we perform several groups of simulations where only one parameter is changed each time. In principle, the photonic crystal grating diffracts a portion of the emitted photons into a unidirectional vertical emission and reflects the remainder to establish the optical confinement. Therefore, the cavity mode characteristics are governed by the size of the central disk and the reflectivity of the grating. The radius of the disk $R$ is the most influential parameter affecting the cavity resonance wavelength $\lambda _\mathrm {cav}$, inducing a redshift when it increases, as shown in Fig. 2(a). Concurrently, $F_\mathrm {P}$ and $\eta _{\mathrm {coll}}$ increase due to the proximity of the redshifted cavity mode to the upper band edge of the photonic crystal grating, where the reflectivity is slightly higher [24]. This enhances the optical confinement and the diffraction efficiency. Conversely, Fig. 2(b) shows that an increase in the grating period $\Lambda$ weakens and broadens the cavity mode and flattens the $\eta _{\mathrm {coll}}$ curve. We attribute this to the reduced optical confinement for a fixed wavelength when the grating reflection spectrum redshifts. The hole diameter $D$ also plays an important role in controlling the device performance. When $D$ is larger, the effective refractive index of the disk is smaller, resulting in a blueshift of the cavity mode, as shown in Fig. 2(c). Additionally, a larger $D$ intensifies the perturbation to the cavity mode, reducing the $Q$ factor by enhancing the radiation loss, which translates to favorable higher $\eta _{\mathrm {ext}}$ [47]. However, $\eta _{\mathrm {coll}}$ is slightly reduced due to an enlarged emission half-angle, an effect that becomes more pronounced for smaller NA.
Fig. 2. Impacts of structure parameter variations on the optical properties of the CPCG. (a)-(c) Purcell factor $F_\mathrm {P}$ (solid) and collection efficiency $\eta _{\mathrm {coll}}$ (dashed) as a function of emission wavelength for different (a) $R$, (b) $\Lambda$, and (c) $D$. (d) Top panel: $F_\mathrm {P}$ and cavity spectral width $W_{F_\mathrm {P}>10}$ as a function of $t_\mathrm {SiO_2}$, showing that an increase in $F_\mathrm {P}$ is consistently accompanied with a reduction in $W_{F_\mathrm {P}>10}$, and vice versa. Bottom panel: $\eta _{\mathrm {coll}}$ and cavity resonance wavelength $\lambda _\mathrm {cav}$ as a function of $t_\mathrm {SiO_2}$, revealing their weak dependence on $t_\mathrm {SiO_2}$.
The SiO2 layer thickness determines the optical path difference between the original upward emission and the reflected downward emission by the gold mirror. This parameter offers a convenient means to fine-tune the device performance with minimal impact on $\lambda _\mathrm {cav}$, as suggested by Fig. 2(d). The maximum $F_\mathrm {P}$ of 303 is attained when $t_\mathrm {SiO_2} = 210\;\textrm{nm}$, albeit at the expense of greatly suppressed $\eta _{\mathrm {coll}}$. One plausible explanation for this phenomenon is that, in this configuration, the optical path difference is approximately equal to $\lambda _\mathrm {cav}$. Accounting for the 180° phase change upon reflection by the gold mirror, the reflected light destructively interferes with the original emission, impeding vertical emission from the QD. This can be equivalently regarded as an enhancement of the vertical contribution to the $Q$ factor, thereby leading to the observed maximum $F_\mathrm {P}$ [48]. This hypothesis is also supported by the fact that the cavity spectral width reaches its minimum at the same time. In order to achieve efficient photon collection, constructive interference needs to be satisfied. High $\eta _{\mathrm {coll}}$ exceeding 90% can be achieved over a broad range from 290 nm 630 nm, relaxing the constraints on $t_\mathrm {SiO_2}$. This flexibility also enables the replacement of the SiO2 layer with a spin-coated indium tin oxide layer, which can serve as the bottom $p$-contact [49]. Consequently, the need for precise control of the dry etching depth to expose the $p$-layer is alleviated, potentially enhancing the fabrication yield, especially for thin-film structures. A trade-off between $F_\mathrm {P}$ and $\eta _{\mathrm {coll}}$ can be tailored for various applications. For instance, $F_\mathrm {P}$ approaches 130 while $\eta _{\mathrm {coll}}$ remains above 90% when $t_\mathrm {SiO_2}={290}\;\textrm{nm}$, which is appealing for single-photon generation. As we focus on the generation of polarization-entangled photon pairs, a broad cavity mode is crucial for the simultaneous Purcell enhancement of both X and XX transitions and easier QD-cavity spectral coupling, improving optical properties and scalability. Therefore, we designate the optimal $t_\mathrm {SiO_2}$ as 610 nm, achieving the broadest cavity mode without compromising $\eta _{\mathrm {coll}}$.
3.2 Optimal device performance
The preceding section highlights the ability to optimize the device performance by adjusting structure parameters. The final parameter set is chosen as $R = {760}\;\textrm{nm}$, $\Lambda = {630}\;\textrm{nm}$, $D = {200}\;\textrm{nm}$, and $t_{\mathrm {SiO_2}} = {610}\;\textrm{nm}$. Due to the smaller refractive index contrast in the photonic crystal grating, more periods ($N=10$) are necessary for sufficient optical confinement and efficient light extraction (see Fig. S1 of Supplement 1). This optimized device exhibits a broad cavity mode with a $F_\mathrm {P}$ of 23, a $Q$ factor around 215, and a mode volume $V$ around 0.71 ($\lambda$/$n$)$^3$. For QDs emitting photons in the telecom C-band, the typical spectral separation between X and XX is around 6 nm [9,10,13]. In our simulation, when XX is in resonance with the cavity, X experiences a $F_\mathrm {P}$ of roughly 6.8. Applying such asymmetric Purcell enhancement to the state-of-the-art lifetimes measured under TPE predicts a theoretical photon indistinguishability of 92.9% [13,27]. In addition, considering the measured statistics of lifetimes and coherence times of X and XX of InAs/InP QDs grown in the droplet epitaxy mode, $F_\mathrm {P}>10$ is vital to realize transform-limited emission from both transitions [50]. The corresponding cavity spectral width $W_{F_\mathrm {P}>10}$ is 8.22 nm, which is broad enough to enhance the X and XX transitions simultaneously. Thanks to the broadband nature of the grating structure, maximum $\eta _{\mathrm {coll}}$ of 92.4% is achieved while exceeding 80% over a spectral range of 60 nm from 1513 nm 1573 nm, as shown in Fig. 3(a). This experimentally advantageous feature ensures efficient photon collection even when QDs are not in resonance with the cavity.
Fig. 3. Optical properties of the optimized CPCG structure. (a) $F_\mathrm {P}$ and $\eta _{\mathrm {coll}}$ as a function of emission wavelength. (b) NA-dependent $\eta _{\mathrm {coll}}$ at $\lambda _\mathrm {cav}$ showing efficient photon collection. The inset shows the highly converged far-field pattern, calculated by combining the far-field patterns of two orthogonal dipoles. The concentric circles and the boundary represent the acceptance angle for NA = 0.2, 0.4, 0.65, and 0.7, respectively. (c) Cross-sectional electric field intensity distribution in logarithmic scale showing the unidirectional emission. (d) In-plane intensity distribution of the cavity mode, with air holes represented by white circles.
It is noteworthy that the effective refractive index of the air-hole ring gradually decreases, from the innermost to the outermost ring. This characteristic suppresses higher-order diffraction, contributing to a highly converged unidirectional far-field pattern, evident in the inset of Fig. 3(b) and the cross-sectional electric field intensity distribution in Fig. 3(c). As a result, high $\eta _{\mathrm {coll}}$ persists for objectives with small NA, achieving 75.4% for NA = 0.4, as plotted in Fig. 3(b). This feature makes CPCGs attractive for employing cost-effective objectives or direct fiber coupling. The in-plane intensity distribution of the cavity mode, illustrated in Fig. 3(d), reveals the interaction between the QD and the CPCG. The residual field intensity upon encountering the first ring clarifies the strong correlation between $\lambda _\mathrm {cav}$ and $R$. As discussed earlier, substituting the complete trench with air holes reduces the interaction area, resulting in a gentler change of the electric field at the cavity edges and, consequently, a slightly higher $Q$ factor in relative to CBGs [47]. Superior results may be achieved using advanced optimization algorithms [51,52], including the incorporation of additional structure parameters such as chirping the grating period or apodizing the hole diameter, which remain interesting for future studies.
3.3 Robustness against fabrication imperfections
QDs are randomly distributed across the wafer due to the inherent randomness of the growth mechanism, posing a challenge to their integration into photonic nanostructures, as the performance relies heavily on the spatial coupling between the QD and the cavity. While fabricating arrays of multiple devices on high-density QD samples is an approach, it may compromise the single-photon purity due to nonresonant feeding if more than one QD is present [53]. Therefore, deterministic fabrication of photonic nanostructures on low-density QD samples emerges as the most promising solution. Optical positioning techniques, especially the wide-field photoluminescence imaging for semiconductor QDs in the near-infrared region, have enabled deterministic device fabrication with positioning accuracy mainly limited by the electron beam lithography (EBL) alignment uncertainty [54]. However, the accuracy diminishes in the telecom range due to the dimer emission of telecom QDs and the inefficiency of InGaAs image sensors used in telecom cameras. To our best knowledge, the cutting-edge deterministic device fabrication in the telecom C-band achieves median overall cavity placement accuracy of approximately 130 nm [55]. Thus, it is crucial to investigate how the optical properties of CPCGs change when the QD is displaced from the cavity center with a similar level of mismatch.
To quantify the degradation of device performance caused by QD displacement, we move the dipole along $x$ and $y$ axes in steps of 20 nm. When the dipole is moved along the $x$ axis, aligned with its orientation, the most obvious characteristic is the preserved maximum $\eta _{\mathrm {coll}}$. At the largest displacement of 140 nm, $F_\mathrm {P}$ decreases by approximately 45%, as shown in Fig. 4(a), (b). On the other hand, the cavity mode diminishes rapidly when the dipole is displaced along the $y$ axis, while the reduction in $\eta _{\mathrm {coll}}$ is only 3%. Compared with CBGs with shorter wavelengths, the relaxed influence of dipole displacement on the Purcell factor can be attributed to the larger effective wavelength in the material and reduced InP refractive index in the telecom C-band. These findings suggest that when a QD is not at the center of the CPCG, its horizontal and vertical linearly polarized dipoles undergo different levels of Purcell enhancement. The difference in radiative decay rates unbalances the branching ratio of the XX-X cascade, leading to a weakened entanglement [38].
Fig. 4. Impacts of fabrication imperfections on the optical properties of the CPCG. (a), (b) Colormaps of $F_\mathrm {P}$ and $\eta _{\mathrm {coll}}$ values for different dipole displacements at cavity resonance, showing stronger dependence along the $y$ axis for an $x$-oriented dipole. (c), (d) $\lambda _\mathrm {cav}$, and corresponding $F_\mathrm {P}$, $\eta _{\mathrm {coll}}$, $Q$, and $V$ as functions of (c) the degree of tilted sidewall $\Delta D$ and (d) ellipticity, where the H and V modes are represented by solid and dotted lines, respectively.
In the nanofabrication process of InP-based PC cavities, typical imperfections are tilted sidewalls and air holes deformation into elliptical shapes. A thorough examination of the robustness of the CPCG performance against them is necessary. Tilted sidewalls occur when the dry etching process is isotropic, while the opening of the air holes stays unaffected since it is defined by EBL, which is sketched in the inset of Fig. 4(c). It effectively causes an enlargement of $R$ and a reduction of $D$, yielding a redshifted cavity mode and reduced $\eta _{\mathrm {coll}}$. Interestingly, the $Q$ factor increases significantly, which is counter-empirical [56]. This phenomenon is attributed to changes in the vertical confinement provided by the bottom reflector. As $\Delta D$ increases, emission becomes less directional, and consequently, the reflected light starts to interfere with the upward emission destructively. Simulations with only an InP slab support this explanation, showing an anticipated reduction in the $Q$ factor (see Fig. S4 of Supplement 1). Notably, $\Delta D$ of 100 nm corresponds to a tilt angle of 80.5°. The observed strong dependence justifies the importance of achieving vertical sidewalls. Additionally, we investigate the impact of air hole deformation. We define the hole ellipticity $e = D_{y}/D_{x}$, where $D_{x}$ remains constant [57]. As a result, the circular symmetry is broken, lifting the degeneracy between H and V modes, where H mode shows stronger dependence in terms of $\lambda _\mathrm {cav}$ and $Q$ factor, as shown in Fig. 4(d).
3.4 Direct coupling into single-mode fibers
For real-world applications, portable integrated quantum light sources are highly desirable. Direct coupling into optical fibers is particularly attractive as it eliminates the need for bulky optics, thereby allowing for significant miniaturization of the footprint. Moreover, while an objective is limited to collecting signals from a finite field of view, direct fiber coupling enables scaling up to access multiple emitters on the same chip. Two commercial SMFs commonly deployed for telecom applications are considered, namely 980-HP and UHNA1 with mode field diameters (MFDs) of 6.8 $\mathrm{\mu}$m and 4.8 $\mathrm{\mu}$m, respectively. The MFD represents the diameter where the mode intensity decreases to $1/e^2$ of its peak value. The normalized fundamental fiber mode amplitude $E_{\mathrm {SMF}}$ can be approximated by a Gaussian function [58]:
where $r_{\mathrm {SMF}}=\sqrt {x^2+y^2}$ is the radial distance from the beam center, and the beam radius is denoted by $w=\mathrm {MFD}/2$. We calculate the near-field amplitude of circularly polarized photons emitted from the XX-X cascade by combining the near-field patterns of two orthogonal dipoles, denoted by $E_{\mathrm {near}}$, to mimic the actual photon emission. To evaluate the compatibility of the CPCG with direct fiber coupling applications, we calculate the mode coupling efficiency $\eta _{\mathrm {MC}}$ as the fraction of the dipole power that enters the fundamental mode of an SMF using the overlap integral method, as described by [58]Fig. 5. Direct coupling of the emission from CPCGs into SMFs. (a) Cross-sectional schematic illustrating the direct fiber coupling configuration, with the MFD of the far-field pattern denoted by the red line. (b) $\eta _{\mathrm {MC}}$ of the optimized CPCG at $\lambda _\mathrm {cav}$ when coupling into 980-HP and UHNA1 fibers as a function of $h$. (c) Intensity profile lineshapes showing good agreement between the near-field pattern of the optimized CPCG and the Gaussian function of the propagation mode of a UHNA1 fiber on the fiber facet at $h={1.2}\;\mathrm{\mu}\textrm {m}$. (d) $\eta _{\mathrm {MC}}$ when coupling into 980-HP and UHNA1 fibers at their respective optimal $h$ as a function of emission wavelength.
4. Conclusion
In summary, we performed FDTD simulations to numerically study a CPCG design tailored for charge-tunable semiconductor QD-based single-photon or entangled photon-pair sources. By replacing the fully etched trenches in the CBG with air holes, the CPCG allows for electrical access to QDs without deteriorating the device performance. It outperforms alternative proposals such as the labyrinth geometry by achieving collection efficiency of 92.4% into an objective with NA = 0.7 and a more Gaussian-like far-field pattern. The optimized cavity mode, featuring a Purcell factor of 23, is sufficiently broad to offer improved photon indistinguishability via the asymmetric Purcell enhancement of exciton and biexciton transitions. We conclude that the performance of the CPCG remains robust against lateral QD displacement, tilted sidewall, and elliptical hole shape in single-photon generation scenarios. Furthermore, direct fiber coupling can be achieved with near-unity efficiency. The hybrid design allows for integration onto piezoelectric platforms, enabling dynamic strain-tuning of the emitter-cavity system for wavelength tuning and the elimination of exciton fine structure. The proposed design therefore enables a combination of charge-tuning and strain-tuning toward the development of application-ready quantum light sources emitting bright and indistinguishable photons with high entanglement fidelity. This capability is instrumental for global quantum communication through existing optical fiber networks.
Funding
European Research Council (GA101043851, GA715770); Bundesministerium für Bildung und Forschung (03ZU1209DD, 13N16291, 16KISQ015, 16KISQ117); Deutsche Forschungsgemeinschaft (390837967, EXC-2123, GZ: INST 187/880-1 AOBJ: 683478); Niedersächsisches Ministerium für Wissenschaft und Kultur (76251-1009/2021).
Acknowledgments
The authors gratefully acknowledge the German Federal Ministry of Education and Research (BMBF) within the projects QR.X (16KISQ015), SemIQON (13N16291), SQuaD (16KISQ117), and QVLS-iLabs: Dip-QT (03ZU1209DD), the European Research Council (QD-NOMS - No. GA715770, MiNet – No. GA101043851), MWK Niedersachsen (QuanTec - 76251-1009/2021), and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) within the project InterSync (GZ: INST 187/880-1 AOBJ: 683478), and under Germany’s Excellence Strategy (EXC-2123) Quantum Frontiers (390837967). P.L. acknowledges the China Scholarship Council (CSC201807040076). C.M. is grateful to Peter Lodahl and the Rosenfeld Foundation for supporting the research visit at the Niels Bohr Institute, University of Copenhagen. We would like to thank Leonardo Midolo and Frederik Benthin for fruitful discussions.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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