Quantum networks based on color centers in diamond

With the ability to transfer and process quantum information, large-scale quantum networks will enable a suite of fundamentally new applications, from quantum communications to distributed sensing, metrology, and computing. This perspective reviews requirements for quantum network nodes and color centers in diamond as suitable node candidates. We give a brief overview of state-of-the-art quantum network experiments employing color centers in diamond, and discuss future research directions, focusing in particular on the control and coherence of qubits that distribute and store entangled states, and on efficient spin-photon interfaces. We discuss a route towards large-scale integrated devices combining color centers in diamond with other photonic materials and give an outlook towards realistic future quantum network protocol implementations and applications.

With the Panama-Pacific International Exposition of 1914 fast approaching, AT&T's leadership grew increasingly alarmed: it was still impossible to hold a coast-to-coast telephone call, despite the company touting it for years. The essential problem "was a satisfactory telephone repeater," recalled an AT&T senior manager from conversations with the company's chief engineer John J. Carty 1 . Whereas mechanical repeaters adequately boosted voice signals on the metroscale, they added so much noise that signals quickly became unintelligible over longer distances. A fundamentally new repeater technology was needed. Tasked with this urgent mission, Bell Laboratories developed the "audion" trion vacuum tube, just in time for the 1914 Exposition. Over the ensuing decades, the triode made way to the transistor, copper wires yielded to optical fiber, and binary digits ("bits") became the universal language of information, transcending physical modality. Today, as the authors edit these sentences over a globe-spanning video conference, the world is at the cusp of the next information revolution, as quantum bits ("qubits") have become the universal units fueling a new generation of 'quantum information technologies'. And once again, an essential challenge is to develop a "satisfactory repeater" -this time, capable of relaying (but not amplifying! 2 ) quantum information signals. This article reviews the progress towards one such "quantum repeater" as well as other quantum network technology using color centers in diamond to connect quantum information among spins and photons. a) Electronic mail: englund@mit.edu b) Electronic mail: r.hanson@tudelft.nl

I. A BRIEF INTRODUCTION TO QUANTUM NETWORKS
In a future quantum network (see Fig. 1), remote parties are connected by sharing long-lived entangled states 3,4 . Arguably, the most promising way of linking distant nodes is to employ fiber-or free-space photonic communication channels to establish entanglement. While all photon-based schemes are associated with losses that scale with distance 5 , motivating the need for quantum repeaters 6 , heralding entanglement generation on successful photon transmission events maps these losses into reduced entanglement generation rates without lowering entanglement fidelities [7][8][9] .
Apart from fundamental tests of physics 20 , small-scale quantum networks have been used to demonstrate key network protocols such as non-local quantum gates 17 , entanglement distillation 36 , and very recently entanglement swapping 37 . These networks are currently limited to a few nodes 37 , distances of up to one kilometer 20 , and entanglement generation rates in the Hz to kHz regime [10][11][12][13][14][15][16]19,20 . A major challenge for the coming decade is to transition from the current proof-of-principle experiments to large-scale quantum networks for use in fields such as distributed quantum computation 38,39 , quantum enhanced sensing 40,41 , and quantum secure communication 42 .
FIG. 1. Schematic overview of a future large-scale quantum network, consisting of nodes containing optically interfaced qubits (purple) with long coherence times. Photons routed via optical fibers or free-space channels serve as mediators to create entanglement (blue wiggled lines). Local area and trunk backbone quantum repeaters (dark and light red circles, respectively) are used to enable a high entanglement generation rate over large distances, overcoming photon transmission loss. The entanglement generation is heralded, meaning that detection of certain photon statistics signals the successful generation of an entangled state that is available to a network user for further processing and applications such as networked quantum computation, quantum secured communication, quantum enhanced sensing, and potential not yet discovered tasks.
In this perspective article, we focus on color centers in diamond as potential building blocks for large-scale quantum networks. We identify their strengths and challenges, current research trends, and open questions, with a focus on qubit control, coherence, and efficient spin-photon interfaces. We lay out a path towards large-scale, integrated devices, and discuss possible protocols and functionalities such devices could enable.
This article is organized as follows: Sec. II introduces the requirements for the nodes of a quantum network and gives a high-level overview of why color centers in diamond are promising candidates to meet these demands. We then give a brief overview of state-of-the-art quantum network experiments employing color centers in diamond in Sec. III, to give the reader a sense of the current research status. In Sec. IV, we introduce the physics behind color centers in diamond, to understand their strengths and challenges for quantum network applications in the remainder of this article. Sec. V discusses open questions regarding the coherence and control of nuclear spin memory qubits surrounding color centers in diamond that can be used as additional quantum resources. In Sec. VI, we outline the need and strategies for an enhancement of the spinphoton interface of diamond color centers and discuss promising future research in this direction. Sec. VII outlines a path towards large-scale, integrated diamond devices that may enable future quantum network experiments with high rates. Finally, Sec. VIII concludes by illuminating possible applications for future quantum networks.

II. REQUIREMENTS FOR A QUANTUM NETWORK NODE
To make a material platform suitable for a node in a quantum network in which entanglement is mediated by photons, it has to fulfill three main requirements 4 . First, the capability to interface at least one qubit efficiently with optical photons (at telecommunication wavelengths for fiber-based systems), to establish remote entanglement at high rates. Second, the ability to store quantum states during entanglement generation; in particular, this requires qubit coherence times under full network activity to be longer than the time it takes to generate entanglement between nodes. Third, the capability to store several entangled states per node with a capability for high-fidelity operations between them, to enable multi-qubit protocols such as error correction.
Color centers in diamond satisfy most of these requirements, and have enabled some of the most advanced experimental demonstrations of quantum network protocols to date. We here give a concise, high level overview of the achieved and projected key capabilities of color centers in diamond [see Fig. 2(a)], motivating a detailed treatment (and also including the relevant literature) in the remains of this article: 1. The color center contains an individually controllable, optically active spin (communication qubit), with access to several long-lived nuclear spins (memory qubits) in its surrounding. These memory qubits can be manipulated with high fidelity to free up the communication qubit and enable multi-qubit protocols. They have a long coherence time enabling robust state storage during subsequent network activity.
2. The internal level structure is suited to generate remote entanglement. In particular, color centers feature spinstate-selective optical transitions that can entangle the color center's spin state with a photonic state, e.g. in the number / polarization / time basis (see Fig. 2 3. Optical emission of color centers is bright and can be collected with high efficiency, enabling high entanglement generation rates. This emission is in the visible to near-IR spectrum, and thus at wavelengths that are associated with higher fiber transmission losses than for photons in the telecom band. However, it is possible Two recent experiments showcasing the potential of color centers in diamond as quantum network node candidates can be seen in Fig. 3, together defining the state of the art in quantum networks. Fig. 3(a) shows a laboratory-scale quantum network based on nitrogen-vacancy (NV) centers in diamond 37 . Three NV centers in separate cryostats, linked via optical fiber channels in a line configuration, are operated as independent quantum network nodes. The centers are set to a common optical emission frequency via electric field tuning (DC Stark shift) 19 . The network is used to demonstrate distribution of three-partite Greenberger-Horne-Zeilinger (GHZ) entangled states across the nodes, as well as entanglement swapping to achieve any-to-any connectivity in the network 37 . This network sets the state of the art for entanglement-based quantum networks.
Entanglement between the nodes is generated using a photon-emission-based scheme, in which spin-photon entanglement is locally generated at each NV node using microwave and laser pulses. The photonic modes are then interfered on a beam splitter, thereby erasing the which-path information. Subsequent photon detection heralds the generation of a spin-spin entangled state between two NV nodes 19,43 . Early experiments 19 had employed two-photon entangling schemes reaching fidelities ∼ 90% 20 at ∼ mHz rates. More recent experiments (including the three-node network discussed here) used a single-photon scheme 43 yielding fidelities ∼ 82% at rates of ∼ 10 Hz 37 .
Importantly, these schemes generate entanglement in a heralded fashion, making the fidelity robust to photon loss and the generated entangled states available for further use. In the vicinity of the NV center, a 13 C nuclear spin is used as a memory qubit at the middle node, coupled to the NV center electron spin via the hyperfine interaction. The nuclear spin is controlled via tailored microwave pulses applied to the NV center spin 44 that simultaneously decouple the NV electron spin from the other nuclear spins. Characterization of the nuclear spin environment of an NV center has enabled communication qubit coherence times above one second 45 . In addition, entangled states have been stored in 13 C memory qubits for up to ∼ 500 subsequent entanglement generation attempts 37,46,47 . Combined with the demonstrated capability of entanglement distillation (generating one higher fidelity state from two lower fidelity states) 36 and deterministic entanglement delivery (by generating entanglement between two nodes faster than it is lost, albeit not under full network activity) 43 , these experiments highlight the potential of the NV center as a quantum network node. As we discuss in detail below, a main challenge for the NV center is the low fraction of emitted and detected coherent photons, limiting the entanglement rates.
Recently, a new class of defect centers in diamond has also gathered the attention of the community: the family of group-IV color centers in diamond, of which the negatively charged silicon-vacancy (SiV) center is the most studied member. Fig. 3(b) shows a schematic of a SiV center embedded in a photonic crystal cavity, which increases the interaction of light with the communication qubit as compared to emission in a bulk diamond material (discussed in Sec. VI below). Spinphoton entanglement with fidelity ≥ 94 % has been generated in this system using cavity quantum electrodynamics (cQED) reflection-based schemes, in which photons impinging on the cavity containing the communication qubit are reflected or transmitted depending on the spin-state of the communica- tion qubit 48 . While entanglement between distant SiV centers has not yet been demonstrated, strong collective interactions between two SiV centers 49 in the same cavity and probabilistic entanglement generation between two SiVs 50,51 in one waveguide have been observed. In addition, photon outcoupling efficiencies from the emitter-cavity system into fiber > 0.9 have enabled the demonstration of memory-enhanced quantum communication 23 . This experiment sets the state of the art for quantum repeater experiments.
Additionally, recent experiments have demonstrated key quantum network node capabilities, including dynamical decoupling of the SiV spin from the bath with coherence times above 1 ms 22,52 , 13 C nuclear spin initialization, control, and readout via the central SiV electron spin 53,54 , and frequency control of SiV centers via strain tuning 51,55,56 . Combined with recent efforts to fabricate SiV centers in nanophotonic waveguides at a large scale and their integration in photonic circuits 56 , these results show the potential of silicon-vacancy FIG. 4. Schematic atomic structure of NV (a) and group-IV (b) color centers in diamond and ground state level structure of group-IV color centers (c). The inversion symmetry of group-IV color centers leads to a vanishing permanent electric dipole moment, making those defect centers suitable for integration in nanophotonic structures. This structure also results in orbital and spin double degeneracy in the ground and excited state (c, only ground state shown for simplicity) that is lifted by (mostly) spin-orbit coupling (∆ gs ) and by application of an external magnetic field (B), respectively. Phonon-induced transitions (γ + / γ − ) between orbital states limit (communication) qubit coherence times (see text).
centers and other closely related group-IV color centers as nodes for a quantum network (see Sec. VI).

IV. OPTICAL AND SPIN PROPERTIES OF COLOR CENTERS IN DIAMOND
Nitrogen-vacancy and group-IV color centers in diamond feature different structural symmetries, which result in different properties of their spin-photon interface. Here, we briefly introduce the most important differences between those color centers in diamond, to the point necessary to understand current limitations and research directions described in the remains of this article; we refer the reader to the many excellent articles that cover the physics of these color centers in great detail [57][58][59] . In this article, we consider all color centers to be in the negative charge state, except where indicated differently; this negative charge state is the one most commonly studied and used in experiments geared towards entanglement generation. Fig. 4 shows the difference in atomic structure of nitrogenvacancy (NV) centers and group-IV color centers (consisting of silicon-vacancy (SiV) 54,65,66 , germanium-vacancy (GeV) 70 , tin-vacancy (SnV) 24,71 and lead-vacancy (PbV) 75 centers). An NV center is formed by a nitrogen atom replacing a carbon atom of the diamond lattice, next to a lattice vacancy, giving rise to a C 3v symmetry [ Fig. 4(a)]. The NV center possesses a permanent electric dipole moment, making the optical transition frequencies sensitive to charge fluctuations in its environment 57 . This sensitivity to electric fields has been used as a resource to tune different NV centers onto resonance 80 , counteracting local strain inhomogeneities in the host diamond crystal which shift the transition frequencies of color centers. However, it has also hindered the nanophotonic integration of NV centers due to degradation of their optical properties near surfaces 81,82 .
The NV center features an outstanding spin energy relax- Overview of optical and orbital properties of group-IV (SiV, GeV, SnV, PbV) and NV defect centers in diamond. All listed numbers are approximate values. We consider all defects to be in the negative charge state. The Debye-Waller Factor characterizes the fraction of radiative emission into the ZPL, while we define the quantum efficiency as the ratio of radiative to all decay. Ground state splitting denotes splitting between orbital states in the optical ground state of group-IV color centers (see Fig. 4(c) and text). T oper is the calculated operation temperature below which the expected orbital lifetime is above 100 ms. We estimate this temperature by calculating the phonon transition rate where ∆ gs is the orbital ground state splitting, T is the temperature, and k B the Boltzmann constant (as γ − > γ + , γ + determines the qubit coherence times), as in Refs. 24,71 . We then find the proportionality factor to map this to a concrete temperature by using the measured values of Ref. 66 . ation time of over 8 h 83 . Quantum states have been stored in the NV center's electronic spin for over one second by decoupling the electron from unwanted interactions in its environment using tailored microwave pulse sequences 45 . Fidelities in quantum networks experiments have reached > 0.99% for single qubit gates and ∼ 98% for two-qubit gates (between NV electron and 13 C nuclear spins) 37 .
For group-IV color centers, the group-IV atom takes an interstitial lattice site between two lattice vacancies. The resulting inversion-symmetric D 3 d structure of the defect [ Fig. 4(b)] results in no permanent electric dipole and thus a first-order insensitivity to electric field fluctuations. As a consequence, tuning optical transition frequencies of group-IV color centers by electric fields is relatively inefficient 84,85 , potentially necessitating other techniques (discussed in Sec. VI below). On the other hand, the vanishing permanent electric dipole makes these color centers excellent candidates for integration into photonic nanostructures (also discussed in Sec. VI below).
Group-IV emitters also feature a different level structure than the NV center: both optical ground and excited states are formed by doubly-degenerate orbital states, which are each split into pairs by spin-orbit interaction and the Jahn-Teller effect. Under an external magnetic field, the double spin degeneracy in each pair (S = 1/2) is lifted 58,65,86 . The qubit subspace can then for example be defined over the two opposing spin sublevels of the optical ground state's lower orbital branch [see Fig. 4(c)]. This, however, means that direct microwave driving of the qubit levels is forbidden in first order, as one would need to flip both spin and orbital quantum numbers. Typically, this complication is overcome by working with emitters with transversal strain ∼ ∆ gs , which mixes the orbital levels, such that the qubit state can be effectively described by the spin only and thus be driven with higher efficiency 54 . Coherent microwave control of group-IV color centers has recently been demonstrated for SiV centers 52,54 , with infidelities on the percent level 52 . For other group-IV color centers, only incoherent microwave driving of spin transitions has been achieved so far 24,70 . The reported infideli-ties for coherent SiV control are a result of qubit dephasing during microwave pulses, as relatively low Rabi frequencies (∼ 10 MHz) are used to limit heating-induced decoherence. It is expected that the use of superconducting waveguides 87 will further improve microwave pulse fidelities (as a result of potentially significantly reduced ohmic heating) for all color centers.
Spin dephasing in the group-IV qubit subspace is dominated by phonon-assisted transitions to the upper ground state orbital branch, which leads to the acquisition of a random phase: while the phonon transitions are in principle spinconserving, the detuning between the spin states varies depending on which orbital state is occupied, see Fig. 4(c) 88,89 . In Tab. I, we map reported ground state splittings into a concrete performance metric by extrapolating the temperatures T oper below which we expect an orbital coherence time > 100 ms due to low phonon occupation. From this, we see that a larger ground state splitting (associated with stronger spinorbit coupling strengths for heavier ions) is beneficial for maintaining long qubit coherence. However, this requires higher microwave driving powers for similar off-axis strain values, as the strain-induced orbital mixing is related to the spin-orbit coupling strength, and thus the overlap between orbital states is reduced 54 . All-optical spin control schemes represent an alternative to microwave driving, as demonstrated e.g. for SiV 64,90,91 and GeV 70 centers. For SnV centers, spin initialization by optical pumping has been observed 24 . Another way of defining and driving a qubit could be to mix the (mostly) orthogonal spin components via an off-axis magnetic field, and use surface acoustic waves to drive phonon transitions between the orbital levels, with demonstrated Rabi frequencies ∼ 30 MHz 92 . Such an off-axis magnetic field, however, is at odds with achieving the high spin cyclicity needed for single shot spin state readout and high fidelity entanglement generation, necessitating a careful tradeoff 54 . For magnetic fields aligned with the color center's symmetry axis, readout fidelities above 99.99% have been achieved 48,52 .
Tab. I also shows a comparison of the most important optical properties for both group-IV color centers and nitrogen-vacancy centers in diamond. To generate entanglement, only the fraction of coherent photons emitted in the zero-phononline (ZPL, for which no lattice vibrations assist the transition 57 ) can be used 19 . Importantly, all quantified zerophonon-line emission fractions for group-IV color centers are significantly larger than for the NV center, as the inversion symmetry of these emitters causes their excited states to have greater overlap with the optical ground state 86 , resulting in a higher fraction of direct photon transitions 93,94 . While most recent quantum memory experiments with SiV centers have operated in dilution refrigerators (see below) in order to suppress thermal phonon population 52,91 , the SnV center with relatively high expected operating temperature requirements might be particularly promising for use as quantum network node. We note that the neutral charge state variants of group-IV color centers are also promising, as they potentially combine the favorable spin-photon interface of negative group-IV color centers with the excellent spin coherence times and easy qubit manipulation of negative nitrogen vacancy centers 86,95-98 . We do not discuss it extensively here due to its early stage of study.

V. ENHANCING THE MEMORY QUBITS
We now discuss the state of the art and future research directions regarding the memory qubits of color centers in diamond. The most common isotope of carbon is 12 C. However, the 13 C isotope with natural abundance of about 1.1% carries a nuclear spin of 1/2. In the past decade, techniques have been developed (mostly on NV-based systems) to control these nuclear spins via the position-dependent hyperfine coupling. Universal nuclear spin control using electron decoupling sequences that are on resonance with a single 13 C spin has been demonstrated 44 . This has enabled the demonstration of coherence times above 10 seconds, and electronnuclear gate fidelities ∼ 98% for individual 13 C spins in NVbased systems 36,99 . Additionally, up to one memory qubit per communication node has been used in an NV-based quantum network setting to date 36,37 . For the case of SiV centers (and group-IV color centers in general), on the other hand, the spinhalf nature of the system leads to a vanishing first-order sensitivity of decoupling sequences to individual 13 C hyperfine parameters, thus requiring long decoupling times (and / or potentially off-axis magnetic fields that are at odds with a high spin cyclicity of optical transitions) to isolate out single nuclear spins. So far, electron-nuclear gate fidelities have been limited to ∼ 59% for SiV centers 48 ; 13 C nuclear spins close to group-IV color centers have not yet been used in quantum networks experiments.
A first area of research is thus to extend the number of available nuclear spin qubits per network node, as well as their control speeds and fidelities. Recently developed gate schemes based on interleaving radiofrequency (RF) driving of 13 C nuclear spins (to control previously unaddressable qubits) with dynamical decoupling pulses (to decouple from the spin bath) have allowed the creation of a local quantum register of an NV communication qubit and up to 9 surrounding nuclear  Fig. 5(bottom)], creation of a local 7-qubit GHZ state, as well as memory coherence times of single and two qubit states of over ten seconds 99 . This result (achieved on a similar device as used for distant entanglement generation 19,36,37 ) shows that it is realistic to expect diamond-based quantum nodes containing many qubits in the near-term future. Such registers could enable scalable, modular quantum computation 38 and universal, fault-tolerant error correction 44,100-102 . Additionally, we expect that RF driving of nuclear spins will overcome current limitations in manipulating nuclear spins for group-IV color centers, as the nuclear spin transition frequencies depend on the hyperfine parameters on first order in this case 54 . Overcoming issues related to sample heating and background amplifier noise could then also enable to address multiple nuclear spins simultaneously, thus reducing decoherence associated with long gate times 99 . Additionally, cross-talk between nuclear spins, as well as unwanted coupling to other spins could be reduced when using the full information of the environment of a color center 103 (potentially acquired while involving automated techniques 104 ) to simulate and tailor gate sequences for a specific spin environment, and computationally optimizing them for overall protocol fidelity. Another way of extending the num-ber of controllable nuclear spin qubits per node is to also employ the nuclear spin of a nuclear spin containing color center atom isotope, e.g. of 14 N (nuclear spin of 1) 105 or 29 Si (nuclear spin of 1/2) 66 .
A second research direction is to increase the coherence time per available memory qubit under full network activity (in particular entanglement generation). The "always-on" nature of the hyperfine interaction of 13 C nuclear spin memory qubits and the NV center has limited memory coherence in quantum networks to ∼ 500 entanglement generation attempts 37,47 . Uncontrolled electron dynamics, which result for instance from control infidelities and stochastic electron spin initialization during an entangling sequence, cause dephasing of the nuclear spin memory qubits 47 . Techniques such as higher magnetic fields at the color center location can speed up gate times and shorten the entanglement generation element, thus reducing the time over which random phases can be picked up 37,47 . Additionally, decoupling pulses on the memory qubits that suppress quasi-static noise in the environment have shown initial promise to prolong nuclear memory qubit coherence under network activity for NV centers 47 . Other promising routes to extend the nuclear spin memory coherence time involve reducing the color center's state-dependent coupling strength (the main dephasing channel), e.g. by employing decoherence-protected subspaces (formed from two or more individual spins, or pairs of strongly-coupled spins that mostly cancel the state-dependent hyperfine interaction term) 45,46,106 , and using isotopically purified samples for which weakly coupled 13 C nuclear spin qubits can be controlled 107,108 ). Other methods are to engineer systems of coupled defects (e.g. involving a 13 C nuclear spin qubit coupled to a P1 center in the vicinity of an NV center 109 , or to use the nitrogen nuclear spin of a second NV center (whose nitrogen nuclear spin is used as a memory) in proximity to the communicator NV center 110,111 .

VI. ENHANCING THE SPIN-PHOTON INTERFACE
Optically-mediated entanglement generation with high rates requires quantum network nodes with efficient spinphoton interfaces. However, the relatively high refractive index of diamond (∼ 2.4) leads to significant total internal reflection at the diamond-air interface, which limits photon collection efficiencies. Entanglement rates are further reduced by the finite fraction of photons emitted into the zero-phonon line (ZPL) that can be used for entanglement generation (see Sec. IV above). Thus, techniques that can increase the fraction of collected photons at ZPL wavelengths, and / or increase the interaction of light with the emitter are required. Here, we discuss both strategies, as well as their suitability for enhancing the spin-photon interface of different diamond color centers, and the state of the art and future avenues for this research. We note that strong light-matter interactions also open the door to entangling schemes alternative from the photonemission-based protocols discussed above 7,8 , e.g. based on spin-dependent cavity reflections 9,23,112-114 .
A. Strategies for enhancing the spin-photon interface The two most common strategies to enhance the spinphoton interface are by increasing the photon flux at the detector through improvements in photon collection, and through cavity or Purcell enhancement, which increases light-matter interaction and spontaneous emission rates (see Fig. 6). Here, we only give a brief overview of these methods, and refer to the many extensive diamond-color-center specific review articles that cover them in detail, see e.g. Refs. 59,[115][116][117] .

Collection efficiency enhancement
A common method to enhance collection efficiency from color centers in diamond is to fabricate dome-shaped solid immersion lenses around the emitters [ Fig. 6(a)], which leads to a higher fraction of light being collected, as total internal reflection is avoided by the light striking the diamond surface at a perpendicular angle. These devices have been used to increase emission from NV 118-122 , SiV 64,65 , and GeV 70,123 centers; all diamond-based remote entanglement generation experiments reported to date have used these devices, with detection efficiencies of emitted photons of up to ∼ 10% 120 .
A second class of collection efficiency enhancement methods is based on nanostructures that modify the far-field emission of a dipole emitter. These structures comprise different dimensions, designs, and supported wavelength ranges, and include parabolic reflectors 124,125 [Fig. 6(b) Fig. 6(c)]. While they offer largescale fabrication and higher collection efficiencies than solid immersion lenses (reported dipole-waveguide coupling efficiencies above 55% 56,125 ), the demands in nano-fabrication and color center placement are more stringent than for solidimmersion lenses, and emitters have to be brought in close proximity to nano-fabricated surfaces. This proximity to surfaces is believed to be the cause of optical instability for some diamond color centers, as discussed in detail below. Such instabilities make it challenging to produce indistinguishable photons from remote centers as required for entanglement generation 138,139 .

Purcell enhancement
Another way of enhancing the spin-photon interface is to embed the dipole emitter inside an optical cavity, making use of the Purcell effect, described by a Purcell factor F P . This factor scales with the ratio of the cavity quality factor, Q (inversely proportional to the cavity's energy decay rate), to the cavity mode volume, V (describing the cavity light field confinement) 113 . Briefly, a cavity enhances the density of states around its resonance frequency, such that the spontaneous emission of an embedded dipole emitter is increased when on resonance with the cavity following Fermi's golden rule. Additionally, the cavity funnels emitted light into a well-defined mode that can be readily matched e.g. to that of optical fibers. A commonly used parameter to quantify the radiative emission enhancement by the cavity (defined as the ratio of the dipole-cavity emission, γ cav , compared to the emission in a homogeneous medium, γ hom = γ rad + γ nonrad ) is the cooperativity, C, defined as where g is the dipole-cavity coupling rate, κ is the energy decay rate from the cavity, γ rad is the free-space radiative emitter decay rate, γ nonrad is the non-radiative emitter decay rate, β 0 is the fraction of photon emission from the emitter into the zero-phonon line (given by the Debye-Waller factor), and η is the quantum efficiency (defined as the ratio of radiative to all decay). While this quantifies the efficiency of the spin-photon interface, it does not take into account the coherence of the emitter: in practice, solid state emitters often experience dephasing due to unwanted interactions with the environment of the solid state host, characterized by a pure dephasing rate γ dep , which manifests in broadening of the optical transitions above the lifetime limited value 48,78,137,[142][143][144] . Therefore, it is useful to introduce the coherent cooperativity, C coh , which quantifies the ratio of coherent decay into the cavity mode to undesired decay, and is given as 117,145 C coh = 4g 2 κ(γ hom + γ dep ) = C γ hom γ hom + γ dep . (2) C coh can be interpreted as the probability of coherent atomphoton interaction, and plays an important role in the fidelity and efficiency of many near-deterministic quantum protocols (which require C coh 1) 145 .

B. Looking forward: key challenges and potential solutions
Having introduced different theoretical concepts to increase the spin-photon interface of color centers in diamond, we now discuss the experimental state of the art of such enhancement, highlighting key challenges and potential research directions.

NV centers
While the nitrogen-vacancy (NV) center has been a workhorse of quantum network demonstrations with color centers in diamond, its low fraction of optical emission at the zero-phonon line (ZPL) frequency of 3%, as well as limited collection efficiency from solid immersion lenses currently restrict local entanglement generation rates to <100 Hz 37,43 . Due to their permanent electric dipole moment, NV centers experience large spectral shifts on short timescales caused by charge fluctuations in the environment when close to surfaces (∼ microns) 82 . As a result, despite reported Purcell factors of up to 70 in small mode-volume photonic crystal cavities 81 , the regime of C coh ≥ 1 has remained out of reach. We note that the origins of the surface noise are still a subject of active research, and that recent experiments geared at understanding the effect and origins of surface noise on the NV center spin 173 , and the local electrostatic environment of NV centers 174 show promise towards understanding the causes of surface charge noise. Combined with proposals for active optical driving to reduce spectral diffusion effects 175 , this could revive the field of NV centers in nanophotonic structures in the future.
The challenge imposed by nearby surfaces can be circumvented by embedding microns-thin diamond membranes between two highly reflective mirrors 176 . A recently reported fabrication technique has formed NV centers with bulk-like optical properties in µm-thin diamond membranes that can be embedded in such cavities via a combination of electron irradiation, high temperature annealing, and a tailored diamond etching sequence 82 [ Fig. 7(a)]. This has allowed for the first demonstration of resonant excitation and detection of optically coherent NV centers, Purcell enhanced by an optical cavity 79 , as would be required for cavity-enhanced entanglement generation. However, the Purcell enhancement in this experiment was limited to ∼ 3, (mostly) due to cavity-length fluctuations ∼ 100 pm induced by a closed-cycle cryocooler. Previously, it has been shown (using off-resonant excitation) that Purcell enhancement factors of up to 30 (corresponding to 46% of light emitted in the ZPL) can be achieved for such open micro-cavities in a liquid helium bath cryostat (featuring lower cavity length fluctuations) 78 [ Fig. 7(b)]. Combined, these experiments demonstrate that there is a near-term path from the current C coh ∼ 0.1 towards C coh ∼ 1 and coherent photon collection efficiencies ∼ 10%. In addition, recent results suggest that it should be possible to reach the required pm-scale cavity length fluctuations even in closedcycle cryostats 30,177 , which would remove the need for a liquid helium infrastructure at each quantum network node.
Another active area of research is to create shallow, stable NV centers [178][179][180] at controlled locations; both the electron irradiation technique employed in Ref. 79 , as well as recently introduced laser-writing techniques 181 can form optically coherent NV centers, but rely on native nitrogen in the sample to recombine with introduced lattice vacancies and thus miss precise control of the site of NV center formation. However, for maximal Purcell enhancement, NV centers should be positioned at an antinode of the cavity field. While such precision can be achieved using ion implantation, recent research has shown that NV centers created via this technique suffer from increased optical linewidths compared to NV centers formed during growth 182 , even after extended high temperature treatments to restore the diamond lattice 183 .
While it will remain challenging to achieve coherent cooperativites 1 (and thereby enter the near-deterministic spinphoton interface regime 145 ), the open cavity approach is projected to speed up current emission-based entanglement generation schemes by ∼ 2 orders of magnitude 79 . This could allow for continuous deterministic entanglement generation (generating high fidelity entanglement faster than it is lost), and enable experiments such as the formation of a quantum repeater surpassing direct transmission 184 , and deviceindependent quantum key distribution 185 using NV centers.

Group-IV color centers
The first-order electric field insensitivity of group-IV color centers in diamond 24,67,73,75,86,137,[186][187][188] has enabled the demonstration of close-to-lifetime limited optical transitions even in heavily fabricated nanophotonic structures 21,50,54,56,137,187 . To date, photonic crystal cavity coupling of group-IV color centers in diamond at low temperatures has been demonstrated for SiV 23,[48][49][50]54 and SnV centers 74 , with demonstrated C coh > 100 for the case of the SiV center 23 . This system allows for spin-state dependent photon reflection, enabling high fidelity single shot spin state readout with demonstrated fidelities > 99.9% [see Fig. 7(c,d)]. This has recently enabled the first demonstration of memoryenhanced quantum communication 23 .
A key requirement for many remote entanglement generation schemes is the ability to tune two group-IV color centers located on separate chips to a common resonance frequency. So far, tuning the emission frequency of group-IV emitters has been demonstrated using strain 51,55,56,89,189 , electric fields 84,85 , and Raman-type 50,190 schemes (although the latter is only compatible with emission-based entanglement generation schemes), but only for single emitters, or several emitters in one structure. The first two tuning techniques deform the orbitals of the group-IV color centers, and thus the color's inversion symmetry is broken, leading to an observed increase in transition linewidths and spectral diffusion under applied external strain / electric field (potentially due to an increase in sensitivity to charge noise in the environment). We note that recent experiments indicate a larger tuning range of transitions for strain tuning (as compared to electric field tuning) for the same induced line broadening 56,84,85 .
While close-to-lifetime limited linewidths of group-IV color centers in nanophotonic structures have been observed, experiments still routinely show spectral shifts and charge instabilities, leading to broadening ∼ several lifetime-limited linewidths 49,54,56,137 , as well as large local strain fields. While dynamic strain tuning can be used to suppress slow spectral diffusion (seconds timescale) 51 , it is challenging to improve the homogeneous linewidth (sub-microsecond timescale) that enters the coherent cooperativity using this approach. Typically, emitters are created by high-energy implantation, followed by a high temperature annealing step to form group-IV vacancy centers and to reduce the effects of lattice damage from the implantation process 183 . However, there is evidence that even such high temperature treatments can not fully recover the original diamond lattice 182 . A recently developed promising method to overcome this limitation employs low energy shallow ion implantation, combined with overgrowth of diamond material 191 . Another strategy could be to combine low-density ion implantation with electron irradiation, to reduce the amount of damage created in the lattice via the (heavy) ion implantation 192 . Combined with controlled engineering of the diamond Fermi level 96,193,194 , these techniques could increase the quality and stability of group-IV color center optical transitions 49,54,56,73,75,137 .
Another area of active development involves the design and fabrication of the nanophotonic structures. Despite re- cent progress 195,196 , growing high quality, thin film diamond on large scales is challenging, and (to the best of our knowledge) there is no wet processing technique that can etch bulk single-crystal diamond along its crystal planes, thus requiring either laborious thinning of diamond on a low index material, or sophisticated techniques to produce suspended, wavelength-thick diamond devices. While initial research focused on creating sub-µm-thin diamond films via selectively wet-etching a localized graphitized diamond layer [197][198][199][200][201][202] or etching down a super-polished thin diamond membrane 61,81,163,164,203 , these methods typically feature low device yield (as a consequence of challenges in device handling and initial material thickness variations). Thus, recent research has focused on fabricating structures directly in commercially available bulk diamond material via a combination of a hard mask and an angled diamond etch (leading to a triangular device cross-section) 50,54,132,140,[204][205][206] , or a quasiisotropic (dry) undercut etch, which selectively etches along certain diamond crystal planes (leading to rectangular device cross-section) 56,168,[207][208][209][210][211] . Known limitations to these processes include mixing of TE-and TM-like modes that will ultimately limit device quality factors for triangular devices, as well as a relatively high bottom surface roughness for quasiisotropic etched devices (although it is worthwhile to note that the fabrication parameter space is somewhat less explored for the latter technique).
Currently, all listed fabrication methods routinely achieve photonic crystal cavity quality factors ∼ 10 4 , about two orders of magnitude lower than simulated values 209 . These deviations are caused by a combination of surface roughness 54 , non-uniform hole sizes 74 , and deviations from the expected device cross-section 210 ; we expect that an order-of-magnitude improvement in device quality factors is within reach upon further optimization. Furthermore, although photonic crystal nanocavities can be designed and fabricated to resonate at the ZPL frequency, they are especially sensitive to process variations, leading to a resonance wavelength spread of ∼ 5 nm across devices 209 . Nonetheless, this spread in frequency may be overcome by cavity tuning methods 50,81,212 , which are in any case needed for the precise overlapping of the ZPL and cavity frequency.
As nanofabrication methods are constantly refined, it is also likely that current methods of optimally aligning color centers within nanophotonic structures (to guarantee maximal overlap between the optical mode and the dipole emitter) can be further improved. Towards this end, various methods have already been demonstrated, including the targeted fabrication around pre-located centers 124 , or ion implantation into devices using either masks 54 or focused implantation 50,56 . Compared to the SiV center for which C coh ∼ 100 has been demonstrated, GeV and SnV centers have intrinsically higher radiative efficiencies and thus potentially higher cooperativities, which may increase cavity-QED-based protocol fidelities (see discussion below).
We note that group-IV color centers in diamond are also suitable for integration into tuneable, open micro-cavities, as recently demonstrated for GeV 68 and SiV 161,162 centers at room temperature. It should be possible to achieve C coh 1 for these systems, potentially providing a viable alternative to the more fabrication-intensive nanostructures.

VII. TOWARDS LARGE-SCALE QUANTUM NETWORKS
Having discussed the core quantum network components based on diamond color centers, we now turn to what is required to build these into a network capable of distributing entanglement over large distances at high rates.
First, a future quantum internet will likely make use of existing fiber infrastructure. This necessitates the matching of the photon wavelength from the network node to the telecommunication bands using quantum frequency conversion. Recently, it has been demonstrated that spin-photon entanglement can be preserved after frequency down-conversion of a ∼ 637 nm photon entangled with the spin-state of the NV center to telecommunication wavelengths (∼ 1588 nm) 215 . Similar conversion techniques can be applied to other color centers and quantum systems [216][217][218][219][220] . Future work will focus on increasing the system efficiency and potentially integrating onchip conversion (see below).
Second, future quantum networks covering large distances will require many quantum repeater stations --each with a large number of qubits --for multiplexing, purification and error-correction. For example, 5-10 repeater stations with kilohertz entangled bit (ebit) rates need a total of O(10 8 ) data qubits to reach mega-ebits per second communication 221 . Although color center devices are already produced using standard nanofabrication techniques, the needed scale for highrate, high-fidelity networks would require large-scale manufacturing of quantum nodes. This task not only entails qubit and device production but also the packaging of efficient optical and microwave signals to and from many independent color centers at once.
Optical technologies such as photonic integrated circuits (PICs) may play an important role in addressing these challenges. Their programmability 222 and access to a large number of spatial modes 223 are especially of interest to quantum network applications. Similar to their bulk optics counterparts, PICs comprise of low-loss on-chip components, such as waveguides, filters and switches. References 222-225 review the device concepts and state of the art of PIC technologies that may be relevant to quantum network applications. Driven by foundry adoption, as well as new frontiers in data communication 226,227 , photonic processors 228,229 and optical quantum computing 230 , integrated photonic circuits have advanced dramatically in manufacturability and complexity over the last decade. The first notable feature of PICs for quantum technologies is their compact footprint, which not only promotes dense integration, but also reduces phase errors in quantum interference of photons 222 . Next, phase modulators in PICs can implement on-chip switches for routing photons within an optical network 222 , and material nonlinearities can be used for efficient frequency conversion between visible and telecommunication photons 231 . Finally, multi-channel optical access can be accomplished using standard fiber arrays, and electrical packaging for large-scale control of color centers can be achieved using potential metal layers in a PIC stack.
Photonic circuits in diamond have been previously demonstrated for opto-mechanics 169 and nonlinear optics 232 . Gallium phosphide-on-diamond photonics have also been used to route the emission from NV centers 154 . However, the nondeterministic creation and integration of color centers in devices, as well as the absence of single-crystal diamond wafers have limited the scale of diamond integrated photonics (see Sec. VI). One way to combine the functionalities and performance of industry-leading photonic circuits with diamond is through heterogeneous integration of diamond with other material systems. Also known as "hybrid photonics" 233,234 , this approach is akin to modern integrated circuits in that discrete chips are separately optimized and fabricated and then populated into a larger circuit board. Examples of heterogeneous integration include modulators and lasers in silicon photonics 235 . In the context of quantum photonics, recent successes include quantum emitters 233,234 and single-photon detectors 236 , which are otherwise difficult to achieve in a single material platform with high performance.
Heterogeneous integration of diamond color centers with PICs circumvents the yield issues associated with all-diamond architectures, thereby allowing potentially many addressable qubits within a single chip. A recent result showing the large-scale integration of color centers in diamond with hybrid photonic integrated circuits 56 is shown in Fig. 8(a-c). Diamond quantum micro-chiplets, each consisting of 8 diamond nanophotonic waveguides with at least one addressable group-IV color center, are integrated with PICs based on aluminum nitride [ Fig. 8(c)]. After fabrication using the quasi-isotropic undercut technique (see Sec. VI), a total of 16 chiplets numbering to 128 waveguides were positioned with sub-microns accuracy on the PIC [ Fig. 8(a)] using a pick-and-place technique [ Fig. 8(b)]. The coupling of the color center to the waveguide can be as high as 55 %, and the diamond-PIC coupling and PIC-to-optical fiber coupling are reported to be 34 % and 11 %, respectively. In addition, electrodes in this hybrid PIC enabled the in-situ tuning of optical transition frequencies within this integrated device architecture 56 . The availability of multiple color centers per waveguide potentially allows for spectral multiplexing 237,238 , which is a hardware-efficient path to multiplying the total number of qubits to N s × N f , where N s and N f are the number of spatial and frequency channels, respectively.
With further improvements in waveguide-emitter coupling as well as diamond-nanocavity integration with PICs, such a hybrid architecture could become an important building block of future quantum network nodes. Looking forward,  213 . This allows concurrent communication links with only local information used in each node (f, data and communication qubits depicted in light and dark red, respectively). Figure (e) adapted with permission from Ref. 213 . Figure (f) reprinted with permission from Ref. 214 , in accordance with creativecommons.org/licenses/by/4.0/legalcode. we expect future developments to also focus on integrating chip components for color center technologies, such as CMOS-integrated microwave electronics for spin control 239 , on-chip beamsplitters for photon quantum interference 230 , optical switches for channel connectivity 222 , single-photon detectors 236,240-242 for heralded entanglement, and quantum frequency conversion 243,244 or frequency tuning 245 on the same microphotonic platform. While these functionalities have been realized in other PIC platforms, a key challenge is to bring together various technologies and materials for implementing a chip-based node. Here, the advances in heterogeneous integration will be critical for controlling and deploying a large number of such chips.
As an example of such a chip, Fig. 8(d) shows a modular architecture with many unit cells per network node 39 . Each unit cell consists of atomic memories, photonic switches, and single-photon detectors. The atomic memories include communication qubits for inter-cell entanglement and memory qubits for long-term storage and intracell information process-ing. The switching network selects one of the adjacent cells, and the photo-detectors herald successful entanglement events between cells. Such a photonic architecture could allow for entanglement routing within a chip-based node, potentially boosting the communication rate 9,213 . By establishing connectivity between unit cells using optical switches, the resulting network is scalable because adding nodes does not require any modification in the existing network.
Once multiple long-lived memory qubits that can be controlled with high fidelity are available per quantum network node, the entanglement fidelity can be greatly improved through entanglement distillation (purification), where low fidelity entangled pairs are employed as a resource and transformed to a smaller number of high fidelity pairs after local operations and classical communication 246,247 . Such a scheme has recently been demonstrated in a proof-ofprinciple experiment between two distant NV centers that each have access to an additional 13 C memory 36 . While such distillation requires additional time resources as a result from two-way-classical-communication, quantum error correction 248 can achieve a more favorable key rate per qubit scaling 221 . Such error correction can be performed if the operational error is smaller than the threshold of a given code (e.g. ≈ 0.189 for the Calderbank-Shor-Stean (CSS) code 249 ). If the photon-loss of a link is sufficiently small, heralded entanglement generation can also be replaced with error correction, allowing to move beyond two-way signalling associated with the heralding 250 . In addition, multi-qubit network nodes can be operated in a way that outperforms more simple linear repeater schemes. For example, Fig. 8(f) shows a 2D network with nearest-neighbor connectivity that allows multi-path entanglement routing 214 . Crucially, the algorithm only requires local knowledge about which of the entanglement generation attempts with its nearest neighbor succeeded while still achieving faster key distribution than a linear repeater chain 214 .

VIII. CONCLUSIONS
We have provided a perspective on the emerging field of quantum networks based on diamond color centers. Diamond color centers already define the state of the art in multi-node entanglement-based networks 37 and in memoryenhanced quantum key distribution 23 . We expect the next years to see rapid progress on photonic interfaces and integration of color centers, paving the way for first experiments on long-distance quantum links. From the basic building blocks, larger-scale devices will be designed and constructed. Control layers of higher abstraction -akin to the current Internet -are currently being developed 251,252 .
A future functional quantum network will support many interesting applications, such as distributed quantum computing 253 , accessing a quantum server in the cloud with full privacy 254 and stabilizing quantum clocks 41,255 . Color centers in diamond may play an essential role in these networks, as the "satisfactory repeater", or perhaps the "transistor of the quantum age." support from Brookhaven National Laboratory, which is supported by the U.S. Department of Energy