Inverse-designed photon extractors for optically addressable defect qubits

Optically addressable defect qubits in materials like diamond [1, 2] and silicon carbide [3] are promising for the realization of a wide range of quantum technologies including single-photon generation, quantum metrology, and quantum information protocols [4, 5]. In all of these applications, photon collection efficiency is a key figure of merit. The photoluminescence (PL) detection efficiency is intrinsically limited by the non-directional emission of PL and total internal reflection between the high-index host material and its low-index environment. To enable high PL collection efficiency, photonic structures such as solid-immersion lenses [6], waveguides [7, 8] and microcavities [9, 10, 11, 12] have been utilized. Nanophotonic devices are particularly attractive for their small footprint and potential for scalable integration. However, optimizing nanophotonic structures for efficient defect integration is often nontrivial and bespoke due to the unique constraints imposed by the targeted application for a specific defect system. For high-sensitivity quantum metrology with nitrogen vacancy (NV) centers in diamond, photonic structures should be optimized for efficient broadband extraction of PL from NVs located a few nm from the sensing surface. In contrast, for quantum information applications, photonic coupling to the sharp zero-phonon line (ZPL) of deep NV centers (> 100 nm from the surface) and operation under the high-cooperativity (Purcell-enhanced) regime is preferable.

Here we present a flexible inverse-design optimization framework that can reconcile a wide range of design constraints and generate planar dielectric gallium phosphide(GaP)-on-diamond photonic structures. We selected GaP due to its high refractive index ($n=3.31$) and low loss at the NV emission energy. The structural patterning is constrained to the GaP layer in order to minimize perturbations of the defect environment. For a dipole located 100 nm from the surface and oriented perpendicular to the NV axis, an optimized 1.5 µm $\times$ 1.5 µm device is calculated to provide a 15.7-fold PL enhancement of the free-space PL collection. The 100 nm dipole depth corresponds to the surface-defect separation at which NV centers synthesised by implantation and annealing have been shown to exhibit high optical coherence[13]. Orders of magnitude larger enhancement factors are found for devices designed for defects positioned closer to the surface. Recent work[14, 15] on fundamental bounds to related design goals such as maximizing scattering cross-section or near field radiation extraction suggest that inverse-designed devices can have performance approaching the absolute theoretical limit.

Experimentally, we fabricate the optimized GaP-on-diamond photon extractors and observe efficient PL collection from implanted single NV centers. These versatile devices exhibit a broadband PL enhancement for wavelengths in the measured range of 575 nm to 750 nm with up to a 14-fold zero-phonon-line enhancement for single NV centers. Extensive optical characterization of the NV centers performed both before and after fabrication provides insight into changes in the local NV environment. Post-fabrication sample treatment substantially improves device NV optical stability and points to important fabrication/design considerations for future defect qubit-photonics integration.

The photon extractors are designed for robust enhancement of ZPL photon collection from near-surface NV centers, and modeled via a 3D finite-difference frequency-domain (FDFD) solver, with the frequency set at the negatively-charged NV^- ZPL of 637 nm. Topology optimization [16] was used to design the device within a design region situated directly on top of the diamond substrate with dimensions 1.5 µm $\times$ 1.5 µm $\times$ 0.25 µm. A harmonic dipole source representing the NV center is situated 100 nm below the diamond-GaP interface.

The optimization objective for any given dipole polarization is the net Poynting flux through a square collection surface $\partial F$ of sidelength 1.5 µm, situated 0.4 µm above the top surface of the device. To make the device robust with respect to uncertainty in the NV center polarization state, a minimax formulation was used so as to maximize the minimum Poynting flux contribution from dipoles with $\hat{\mathbf{x}}$, $\hat{\mathbf{y}}$, and $\hat{\mathbf{z}}$ polarizations. The optimization problem is formulated as follows:

A schematic representation and $xy$ cross-section of the fabricated design are shown in Fig. 1a with the fabricated devices shown in Fig. 1b. Fig. 1c shows the LDOS and collection flux enhancement spectra of the device for different dipole polarizations, with $\hat{x}$ and $\hat{y}$ polarizations producing nearly the same spectrum due to the resulting near mirror device symmetry. A few salient features of the optimized design are worth commenting upon. To begin with, the extractor yields orders of magnitude larger flux enhancements for $\hat{z}$ compared to in-plane polarized dipoles. Such a vast difference in relative improvement follows from the fact that the bulk of the radiation from a $\hat{z}$ polarized dipole underneath a bare diamond-air interface undergoes total internal reflection, suppressing the collection efficiency by more than a order of magnitude compared to that of an in-plane polarized dipole; the main role of the extractor is therefore to out-couple such previously trapped radiation. Further, device performance is robust against spatial displacement of the emitter; the enhancement factor decreases by less than 50% for spatial displacements smaller than 70 nm from the original location of the NV center (see Fig. SI.1, Supplement 1).

Simulating an experimental setup with microscope NA of 0.7, the emitter being an NV center in (100) terminated diamond at a depth of 100 nm coupled to the extractor patterned in GaP (including the resist mask), the fabricated design (Fig. 1a) achieves a total collection flux enhancement of 15.7 relative to a bare interface. This is attributed to a 1.4 times local density of states (LDOS) enhancement and a 11 times collection efficiency enhancement (Fig. 1c). For comparison, a full GaP slab with the same thickness as the device achieves a 1.2 times LDOS enhancement but actually reduces collection efficiency by 15% leading to roughly the same collected flux as that of the bare interface. With regards to the absolute collection efficiency, 1.8% of total NV radiation can be collected from a bare interface, 1.6% collected with a full GaP slab, and 20% collected with the optimized design (Section SI.2, Supplement 1 for simulation details and NV flux enhancement spectrum). For calculations of the far field radiation pattern of the extractor device see Fig. SI.3, Supplement 1.

While the extractor leads to large enhancements in the total collected flux, it does not significantly increase the net radiation from the NV center, quantified by the LDOS; hence, most of the observed improvement comes from a higher collection efficiency as opposed to fluorescence rate enhancement via the Purcell effect. Fig. 1d, e explores this trade-off for a $\hat{z}$-polarized dipole. At each targeted defect depth, we use TO to design two devices, one with LDOS enhancement as the maximization objective and the other with total collected flux enhancement as the maximization objective. For shallow dipoles, optimizing for larger flux (square data points) results in a substantial contribution due to LDOS enhancement. However, the divergence between these two design objectives is observed to grow rapidly with increasing dipole depth: beyond a depth of 75 nm, devices designed to maximize the collected flux maintain a flux enhancement ratio of about 100, while their LDOS enhancement is close to one. Notably, beyond a dipole depth of 150 nm, even LDOS optimized devices are only able to achieve LDOS enhancements of roughly two, as the result of the rapidly decaying access of the device to the dipole’s near field. Devices that combine the benefits of Purcell enhancement with increased collection efficiency through interference are therefore only needed for NV centers sufficiently close ($\lesssim$ 100 nm) to the interface.

Finally, with regards to the device footprint and material choice, we note that for the target emitter depth of 100 nm for the fabricated design, the device footprint of $(1.5\,$µm)² covers a large effective NA (relative to the dipole), and TO attempts with larger device footprints show diminishing increase in device performance. As GaP is a strong dielectric with a high susceptibility, there are limited gains to switching to a higher susceptibility material (Fig. SI.4, Supplement 1), and the low loss of GaP among high susceptibility materials in this frequency range make it particularly attractive for integrated photonics applications [12].

We fabricate photon extractors (Fig. 1b) on two samples, one for ensemble-averaged measurements (Sample A) and the other for single-emitter characterization (Sample B). Sample A is a high pressure high temperature synthesised diamond (Element Six, N $<$ 200 ppm, B $<$ 0.1 ppm) implanted with ¹⁴N accelerated to 20 keV and vacuum annealed at 800 ^∘C. Here, annealing forms NVs primarily by vacancy recombination with native nitrogen. The resulting NV distribution is dictated by the vacancy diffusion profile [20], yielding a dense layer ($\approx$100 NVs per 800 nm excitation spot diameter) of NV centers $\lesssim$100 nm below the diamond surface. Sample B is a chemical vapor deposition diamond (Element Six, electronic grade, N < 5 ppb, B < 1 ppb) implanted with ¹⁵N accelerated to 85 keV and vacuum annealed at 1200 ^∘C (see Methods). During annealing, NV centers are formed primarily by vacancy recombination with the implanted nitrogen, yielding a thin layer ($\approx$ 3 NVs per excitation spot) of single NV centers 100 nm$~{}\pm~{}$20 nm from the surface.

After NV-formation, a 250 nm thick gallium phosphide (GaP) membrane is transferred to each sample via a wet lift-off process [21, 11, 12]. Electron beam lithography and subsequent plasma RIE etching of the GaP layer forms the 1.5 µm $\times$ 1.5 µm photon extractors. The small footprint was chosen for compatibility with on-chip electrodes enabling optical frequency tuning [22, 12]. Over 100 000 devices are fabricated in multiple arrays on the 2 mm $\times$ 2 mm diamond substrate. An array of fabricated devices and a false color SEM image highlighting the material stack is shown in Fig. 1b. On average, each feature is measured to be within 10% of the design, with near vertical ($\theta=$ 88^∘) GaP sidewalls. See Section SI.5, Supplement 1 for more details on fabrication.

This work utilizes inverse-design techniques to optimize the photon collection efficiency from single NV centers under three conditions chosen for future scalability: (1) the NV center is created via implantation, 100 nm from the surface (2) the diamond defect qubit host is not etched and (3) the device lateral dimensions are 1.5  µm. Despite these constraints, the resulting 14-fold enhancement performs similar to related broadband devices in the literature (6 to 20-fold enhancement) such as solid immersion lenses[37, 6], diamond nanowires[38], bullseye gratings[39], diamond metalenses [40] and parabolic reflectors[41], all of which required directly etching the diamond via harsh processes that degrade near-surface NV optical properties [42]. Other designs based on hybrid metal-dielectric gratings / plasmonic resonances may achieve Purcell enhancement along with high directionality (up to 15-fold enhancement), but they require the NV to be either very close (few nm) to the surface[43] or embedded in the device within a nanodiamond[44, 45, 46], both extremely challenging environments for the realization of high charge stability and spectral purity required for quantum information applications. We focused on 100 nm-deep centers due to the ability create optically coherent centers at this depth [13]; however due to the (relatively) large depth, the resulting design predominantly works as a photon extractor with little emission (LDOS) enhancement. As near-surface defect qubit engineering advances, our inverse-design platform can readily be used for even higher enhancement of photon emission from shallower defects . Moreover, the platform provides the design flexibility for integration with emerging technologies such as stamp-transferred devices [36] that can eliminate NV exposure to nearly all fabrication processes to further preserve defect qubit coherence.

During the preparation of this manuscript we were made aware of a contemporaneous theoretical proposal by Wambold et al.[47]. We see the two schemes as complementary, both showcasing the ability of nanophotonic inverse design for handling non-trivial design objectives and constraints. Our design is geared towards quantum information applications, while Wambold et al. is focused on quantum metrology.

This material is based upon work supported by the National Science Foundation under Grants EFMA-1640986 (photonic design and fabrication) and ECCS-1807566 (NV synthesis and characterization). The photonic devices were fabricated at the Washington Nanofabrication Facility, a National Nanotechnology Coordinated Infrastructure (NNCI) site at the University of Washington which is supported in part by funds from the National Science Foundation (awards NNCI-1542101, 1337840 and 0335765). We thank E.R. Schmidgall for help with fabrication process development and pre-implantation etching of our diamond samples.

The authors declare no conflicts of interest.