# Quantum networks based on color centers in diamond

Maximilian Ruf  QuTech, Delft University of Technology, PO Box 5046, 2600 GA
Delft, The Netherlands  Kavli Institute of Nanoscience Delft, Delft University
of Technology, PO Box 5046, 2600 GA Delft, The Netherlands  Noel H. Wan
Department of Electrical Engineering and Computer Science, Massachusetts
Institute of Technology, Cambridge, Massachusetts 02139, USA  Research
Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge,
Massachusetts 02139, USA  Hyeongrak Choi  Department of Electrical Engineering
and Computer Science, Massachusetts Institute of Technology, Cambridge,
Massachusetts 02139, USA  Research Laboratory of Electronics, Massachusetts
Institute of Technology, Cambridge, Massachusetts 02139, USA  Dirk Englund
<EMAIL_ADDRESS>Department of Electrical Engineering and Computer Science,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
Research Laboratory of Electronics, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139, USA  Brookhaven National Laboratory, Upton,
New York 11973, USA  Ronald Hanson<EMAIL_ADDRESS>QuTech, Delft
University of Technology, PO Box 5046, 2600 GA Delft, The Netherlands  Kavli
Institute of Nanoscience Delft, Delft University of Technology, PO Box 5046,
2600 GA Delft, The Netherlands

###### Abstract

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.

††preprint: AIP/123-QED Figure 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.

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 Gertner (2012). Whereas mechanical repeaters adequately boosted voice
signals on the metro-scale, 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! Park (1970)) 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.

##  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 Kimble (2008); Wehner, Elkouss, and Hanson
(2018). 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 Pirandola  _et al._ (2017), motivating the need for
quantum repeaters Briegel  _et al._ (1998), heralding entanglement generation
on successful photon transmission events maps these losses into reduced
entanglement generation rates without lowering entanglement fidelities
Cabrillo  _et al._ (1999); Barrett and Kok (2005); Duan and Kimble (2004).

Optically-mediated remote entanglement of individually controllable qubits has
been generated for different materials platforms, including quantum dots
Delteil  _et al._ (2016); Stockill  _et al._ (2017), trapped ions Moehring
_et al._ (2007); Hucul  _et al._ (2015); Stephenson  _et al._ (2020), neutral
atoms Ritter  _et al._ (2012); Hofmann  _et al._ (2012); Daiss  _et al._
(2021); Langenfeld  _et al._ (2021), and nitrogen-vacancy centers in diamond
Bernien  _et al._ (2013); Hensen  _et al._ (2015). Other promising systems,
including so-called group-IV defects in diamond Bhaskar  _et al._ (2017);
Nguyen  _et al._ (2019a); Bhaskar  _et al._ (2020); Trusheim  _et al._ (2020),
defects in SiC Nagy  _et al._ (2019); Bourassa  _et al._ (2020); Lukin,
Guidry, and Vučković (2020), and rare-earth ions in solid-state hosts Raha
_et al._ (2020); Kindem  _et al._ (2020); Merkel, Ulanowski, and Reiserer
(2020), show great potential for quantum network applications, although remote
entanglement has not yet been generated. Another less explored approach is to
link distant superconducting quantum processors using coherent conversion of
microwave photons to telecom frequencies Chu and Gröblacher (2020);
Mirhosseini  _et al._ (2020); Forsch  _et al._ (2020); Hease  _et al._ (2020);
Krastanov  _et al._ (2020).

Apart from fundamental tests of physics Hensen  _et al._ (2015), small-scale
quantum networks have been used to demonstrate key network protocols such as
non-local quantum gates Daiss  _et al._ (2021), entanglement distillation Kalb
_et al._ (2017), and very recently entanglement swapping Pompili  _et al._
(2021). These networks are currently limited to a few nodes Pompili  _et al._
(2021), distances of up to one kilometer Hensen  _et al._ (2015), and
entanglement generation rates in the Hz to kHz regime Delteil  _et al._
(2016); Stockill  _et al._ (2017); Moehring  _et al._ (2007); Hucul  _et al._
(2015); Stephenson  _et al._ (2020); Ritter  _et al._ (2012); Hofmann  _et
al._ (2012); Bernien  _et al._ (2013); Hensen  _et al._ (2015). 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 Nickerson, Fitzsimons, and Benjamin (2014);
Choi  _et al._ (2019), quantum enhanced sensing Gottesman, Jennewein, and
Croke (2012); Kómár  _et al._ (2014), and quantum secure communication Ekert
and Renner (2014).

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 spin-photon 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

Figure 2: Overview of a quantum network node based on color centers in diamond
and simplified optical level scheme. (a) Schematic of a quantum network node
in diamond, consisting of an optically-active (communication) qubit (purple)
embedded between two highly reflective mirrors (an optical cavity) to enhance
the interaction strength of photons with the qubit. The state of the color
center qubit can be swapped onto long-lived nuclear spin (memory) qubits in
the surroundings (orange) using high fidelity gates that employ microwave or
RF pulses delivered via dedicated lines (gray). Tuning of network nodes to an
optimal frequency operation point is e.g. enabled by applying a static
electric field via electrodes (yellow). Frequency-conversion (black box) can
be used to down-convert photons entangled with the state of the communication
qubit to telecommunication wavelengths, for which photon transmission losses
are low. The color center features spin-resolved optical transitions between
an optical ground (GS) and excited state (ES), enabling optically-mediated
remote entanglement generation (b).

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
Wehner, Elkouss, and Hanson (2018). 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. 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. 2.

The internal level structure is suited to generate remote entanglement. In
particular, color centers feature spin-state-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(b).)

  3. 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 to efficiently convert this emission to the telecom
band while maintaining quantum correlations.

  4. 4.

Diamond is a solid-state material, potentially enabling nanophotonic device
fabrication at a large scale. Such devices can also be integrated with
photonic circuits, and efficient interfacing of color centers embedded in such
systems has been demonstrated.

##  III State of the art of diamond-based quantum networks

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 Pompili  _et
al._ (2021). 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) Bernien  _et al._ (2013). 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 Pompili  _et al._ (2021). 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 Bernien  _et al._ (2013); Humphreys  _et al._ (2018). Early experiments
Bernien  _et al._ (2013) had employed two-photon entangling schemes reaching
fidelities $\sim 90\%$ Hensen  _et al._ (2015) at $\sim$ mHz rates. More
recent experiments (including the three-node network discussed here) used a
single-photon scheme Humphreys  _et al._ (2018) yielding fidelities $\sim
82\%$ at rates of $\sim 10$ HzPompili  _et al._ (2021).

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 13C 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 Taminiau  _et al._
(2014) 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
Abobeih  _et al._ (2018). In addition, entangled states have been stored in
13C memory qubits for up to $\sim 500$ subsequent entanglement generation
attempts Reiserer  _et al._ (2016); Kalb  _et al._ (2018); Pompili  _et al._
(2021). Combined with the demonstrated capability of entanglement distillation
(generating one higher fidelity state from two lower fidelity states) Kalb
_et al._ (2017) and deterministic entanglement delivery (by generating
entanglement between two nodes faster than it is lost, albeit not under full
network activity) Humphreys  _et al._ (2018), 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.

Figure 3: State-of-the-art experiments showing the potential of color centers
in diamond as quantum network nodes. (a) Schematic of a 3-node quantum network
based on nitrogen-vacancy (NV) centers in diamond, linked with optical fibers.
One of the network nodes features a 13C nuclear spin that is used as a memory
qubit to store (part of) an entangled state and free up the communication
qubit, while subsequent entanglement is created using this communication qubit
a second time. Figure from Pompili  _et al._ (2021), reprinted with permission
from AAAS. (b) Schematic of a silicon-vacancy (SiV) center (blue circle)
embedded in a fiber-coupled (dark grey) all-diamond photonic crystal cavity
(light grey area with holes). Coherent emitter-cavity cooperativities $\sim
100$ have enabled high-fidelity reflection-based spin-photon entanglement
generation Bhaskar  _et al._ (2020), and control of individual 13C nuclear
spins via microwave pulses applied to the electron spin of the silicon vacancy
spin qubit (yellow electrodes) has been demonstrated Nguyen  _et al._ (2019b).
Figure adapted with permission from Ref. Nguyen  _et al._ (2019b). Copyright
(2019) by the American Physical Society.

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). Spin-photon entanglement with fidelity $\geq$ 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 communication qubit Nguyen  _et al._ (2019b). While
entanglement between distant SiV centers has not yet been demonstrated, strong
collective interactions between two SiV centers Evans  _et al._ (2018) in the
same cavity and probabilistic entanglement generation between two SiVs
Sipahigil  _et al._ (2016); Machielse  _et al._ (2019) 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 Bhaskar  _et al._ (2020). 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 Sukachev  _et al._ (2017); Nguyen  _et al._
(2019a), 13C nuclear spin initialization, control, and readout via the central
SiV electron spin Metsch  _et al._ (2019); Nguyen  _et al._ (2019c), and
frequency control of SiV centers via strain tuning Sohn  _et al._ (2018);
Machielse  _et al._ (2019); Wan  _et al._ (2020). Combined with recent efforts
to fabricate SiV centers in nanophotonic waveguides at a large scale and their
integration in photonic circuits Wan  _et al._ (2020), these results show the
potential of silicon-vacancy 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

Figure 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 ($\Delta_{gs}$)
and by application of an external magnetic field ($B$), respectively. Phonon-
induced transitions ($\gamma_{+}$ / $\gamma_{-}$) between orbital states limit
(communication) qubit coherence times (see text).

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 Doherty  _et al._ (2013); Hepp
(2014); Bradac  _et al._ (2019). 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.

Vacancy | ZPL | Quantum | Debye-Waller | Radiative | Ground state splitting | $T_{\textrm{oper}}$
---|---|---|---|---|---|---
| wavelength [nm] | efficiency [%] | factor | Lifetime [ns] | [GHz] | [K]
SiV | 737 Bhaskar  _et al._ (2020) |  $(1-10)$ Neu, Agio, and Becher (2012); Sipahigil  _et al._ (2016) / $60$ Riedrich-Möller  _et al._ (2014) |  $(0.65-0.9)$ Neu  _et al._ (2011); Dietrich  _et al._ (2014) |  $1.7$ Rogers  _et al._ (2014); Sipahigil  _et al._ (2016) |  $50$ Hepp  _et al._ (2014); Rogers  _et al._ (2014); Pingault  _et al._ (2017) | 0.1
GeV |  $602$ Iwasaki  _et al._ (2015) |  $(12-25)$ Høy Jensen  _et al._ (2020); Nguyen  _et al._ (2019d) |  $\sim 0.6$ Siyushev  _et al._ (2017) |  $(1.4-6)$ Iwasaki  _et al._ (2015); Bhaskar  _et al._ (2017) |  $170$ Siyushev  _et al._ (2017) | 0.4
SnV |  $620$ Iwasaki  _et al._ (2017); Trusheim  _et al._ (2020) |  $\sim 80$ Iwasaki  _et al._ (2017) |  $0.57$ Görlitz  _et al._ (2020) |  $(4.5-7)$ Tchernij  _et al._ (2017); Trusheim  _et al._ (2020); Rugar  _et al._ (2021) |  $850$ Iwasaki  _et al._ (2017); Trusheim  _et al._ (2020) | 1.8
PbV |  $520$ Trusheim  _et al._ (2019) | not known | not known |  $>3$ Trusheim  _et al._ (2019) | 5700 Trusheim  _et al._ (2019) | 9.8
NV |  $637$ Doherty  _et al._ (2013) |  $>85$ Radko  _et al._ (2016) |  $0.03$ Faraon  _et al._ (2011); Riedel  _et al._ (2017) |  $(11-13)$ Faraon  _et al._ (2011); Kalb  _et al._ (2018); Ruf  _et al._ (2021) | not applicable | not applicable
Table 1: 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).
$T_{\textrm{oper}}$ 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
$\gamma_{+}\propto\Delta_{gs}^{3}/(e^{(\Delta_{gs/k_{b}T)}}-1)$ [see Fig.
4(c)], where $\Delta_{gs}$ is the orbital ground state splitting, $T$ is the
temperature, and $k_{B}$ the Boltzmann constant (as $\gamma_{-}$ >
$\gamma_{+}$, $\gamma_{+}$ determines the qubit coherence times), as in Refs.
Iwasaki  _et al._ (2017); Trusheim  _et al._ (2020). We then find the
proportionality factor to map this to a concrete temperature by using the
measured values of Ref. Pingault  _et al._ (2017).

Fig. 4 shows the difference in atomic structure of nitrogen-vacancy (NV)
centers and group-IV color centers (consisting of silicon-vacancy (SiV) Hepp
_et al._ (2014); Pingault  _et al._ (2017); Nguyen  _et al._ (2019c),
germanium-vacancy (GeV) Siyushev  _et al._ (2017), tin-vacancy (SnV) Iwasaki
_et al._ (2017); Trusheim  _et al._ (2020) and lead-vacancy (PbV) Trusheim
_et al._ (2019) 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${}_{3\textrm{v}}$ 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 Doherty  _et al._ (2013).
This sensitivity to electric fields has been used as a resource to tune
different NV centers onto resonance Bernien  _et al._ (2012), 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 Faraon  _et al._ (2012); Ruf  _et al._ (2019).

The NV center features an outstanding spin energy relaxation time of over 8 h
Astner  _et al._ (2018). 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
Abobeih  _et al._ (2018). Fidelities in quantum networks experiments have
reached $>0.99\%$ for single qubit gates and $\sim 98\%$ for two-qubit gates
(between NV electron and 13C nuclear spins) Pompili  _et al._ (2021).

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 De Santis
_et al._ (2021); Aghaeimeibodi  _et al._ (2021), 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 Hepp  _et al._ (2014); Hepp
(2014); Thiering and Gali (2018). 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
$\sim\Delta_{gs}$, 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 Nguyen  _et al._ (2019c). Coherent microwave control of group-IV
color centers has recently been demonstrated for SiV centers Sukachev  _et
al._ (2017); Nguyen  _et al._ (2019c), with infidelities on the percent level
Sukachev  _et al._ (2017). For other group-IV color centers, only incoherent
microwave driving of spin transitions has been achieved so far Siyushev  _et
al._ (2017); Trusheim  _et al._ (2020). The reported infidelities for coherent
SiV control are a result of qubit dephasing during microwave pulses, as
relatively low Rabi frequencies ($\sim$ 10 MHz) are used to limit heating-
induced decoherence. It is expected that the use of superconducting waveguides
Sigillito  _et al._ (2014) 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
spin-conserving, the detuning between the spin states varies depending on
which orbital state is occupied, see Fig. 4(c) Jahnke  _et al._ (2015);
Meesala  _et al._ (2018). In Tab. 1, we map reported ground state splittings
into a concrete performance metric by extrapolating the temperatures
$T_{\textrm{oper}}$ 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 spin-orbit 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 Nguyen  _et
al._ (2019c). All-optical spin control schemes represent an alternative to
microwave driving, as demonstrated e.g. for SiV Rogers  _et al._ (2014);
Pingault  _et al._ (2014); Becker  _et al._ (2018) and GeV Siyushev  _et al._
(2017) centers. For SnV centers, spin initialization by optical pumping has
been observed Trusheim  _et al._ (2020). 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 $\sim$ 30 MHz
Maity  _et al._ (2020). 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 Nguyen  _et al._ (2019c). For magnetic fields aligned with the color
center’s symmetry axis, readout fidelities above 99.99% have been achieved
Sukachev  _et al._ (2017); Nguyen  _et al._ (2019b).

Tab. 1 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-phonon-line (ZPL, for which no lattice vibrations assist the transition
Doherty  _et al._ (2013)) can be used Bernien  _et al._ (2013). Importantly,
all quantified zero-phonon-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 Thiering and Gali (2018), resulting in a higher fraction
of direct photon transitions Franck and Dymond (1926); Condon (1926). While
most recent quantum memory experiments with SiV centers have operated in
dilution refrigerators (see below) in order to suppress thermal phonon
population Sukachev  _et al._ (2017); Becker  _et al._ (2018), 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 Green  _et al._ (2017); Rose
_et al._ (2018); Thiering and Gali (2018); Green  _et al._ (2019); Zhang  _et
al._ (2020). 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 12C. However, the 13C 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 13C
spin has been demonstrated Taminiau  _et al._ (2014). This has enabled the
demonstration of coherence times above 10 seconds, and electron-nuclear gate
fidelities $\sim 98\%$ for individual 13C spins in NV-based systems Kalb  _et
al._ (2017); Bradley  _et al._ (2019). Additionally, up to one memory qubit
per communication node has been used in an NV-based quantum network setting to
date Kalb  _et al._ (2017); Pompili  _et al._ (2021). For the case of SiV
centers (and group-IV color centers in general), on the other hand, the spin-
half nature of the system leads to a vanishing first-order sensitivity of
decoupling sequences to individual 13C 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 $\sim 59\%$ for SiV centers Nguyen  _et al._ (2019b); 13C nuclear
spins close to group-IV color centers have not yet been used in quantum
networks experiments.

Figure 5: State-of-the-art research for increasing memory qubit numbers and
control for color centers in diamond. (a) Schematic overview of a local
10-qubit quantum register in diamond, formed from a nitrogen-vacancy center
(purple), a native 14N nuclear spin, and 8 surrounding 13C nuclear spins. (b)
Connection map of the spin register in (a), demonstrating entanglement
generation between all pairs of qubits in the register. Figure adapted with
permission from Ref. Bradley  _et al._ (2019), in accordance with
creativecommons.org/licenses/by/4.0/legalcode.

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 13C 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 spin qubits [see Fig. 5(top)] that allow entangled state
generation between all pairs of qubits [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 Bradley  _et al._ (2019). This result (achieved on
a similar device as used for distant entanglement generation Bernien  _et al._
(2013); Kalb  _et al._ (2017); Pompili  _et al._ (2021)) 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 Nickerson, Fitzsimons, and Benjamin (2014) and universal, fault-
tolerant error correction Taminiau  _et al._ (2014); Waldherr  _et al._
(2014); Cramer  _et al._ (2016); Abobeih (2021). 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
Nguyen  _et al._ (2019c). 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 Bradley  _et al._ (2019). 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 Abobeih  _et al._
(2019) (potentially acquired while involving automated techniques Jung  _et
al._ (2021)) to simulate and tailor gate sequences for a specific spin
environment, and computationally optimizing them for overall protocol
fidelity. Another way of extending the number 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 14N (nuclear spin of 1) Pfaff
_et al._ (2014) or 29Si (nuclear spin of 1/2) Pingault  _et al._ (2017).

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 13C
nuclear spin memory qubits and the NV center has limited memory coherence in
quantum networks to $\sim$ 500 entanglement generation attempts Kalb  _et al._
(2018); Pompili  _et al._ (2021). 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 Kalb  _et al._ (2018). 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 Kalb  _et al._ (2018); Pompili  _et al._ (2021).
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 Kalb  _et al._
(2018). 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) Reiserer  _et al._ (2016); Abobeih  _et al._ (2018); Bartling  _et al._
(2021), and using isotopically purified samples for which weakly coupled 13C
nuclear spin qubits can be controlled Pfender  _et al._ (2017, 2019)). Other
methods are to engineer systems of coupled defects (e.g. involving a 13C
nuclear spin qubit coupled to a P1 center in the vicinity of an NV center
Degen  _et al._ (2020), 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 Yamamoto  _et al._ (2013); Degen (2021).

##  VI Enhancing the spin-photon interface

Optically-mediated entanglement generation with high rates requires quantum
network nodes with efficient spin-photon interfaces. However, the relatively
high refractive index of diamond ($\sim 2.4$) 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
photon-emission-based protocols discussed above Cabrillo  _et al._ (1999);
Barrett and Kok (2005), e.g. based on spin-dependent cavity reflections
Sørensen and Mølmer (2003); Duan and Kimble (2004); Reiserer and Rempe (2015);
Bhaskar  _et al._ (2020); Chen, Bersin, and Englund (2021).

###  VI.1 Strategies for enhancing the spin-photon interface

The two most common strategies to enhance the spin-photon 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. Schröder  _et al._ (2016); Johnson, Dolan, and Smith (2017); Bradac
_et al._ (2019); Janitz, Bhaskar, and Childress (2020).

####  VI.1.1 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
Hadden  _et al._ (2010); Marseglia  _et al._ (2011); Robledo  _et al._ (2011);
Jamali  _et al._ (2014); Riedel  _et al._ (2014), SiV Hepp  _et al._ (2014);
Rogers  _et al._ (2014), and GeV Siyushev  _et al._ (2017); Chen  _et al._
(2019a) centers; all diamond-based remote entanglement generation experiments
reported to date have used these devices, with detection efficiencies of
emitted photons of up to $\sim$ 10% Robledo  _et al._ (2011).

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 Wan  _et al._ (2018); Hedrich  _et
al._ (2020) [Fig. 6(b)], nanopillars Maletinsky  _et al._ (2012), nanowires
Hausmann  _et al._ (2010); Babinec  _et al._ (2010), nanocones Jeon  _et al._
(2020), gratings Choy  _et al._ (2013); Li  _et al._ (2015a) and diamond
waveguides Burek  _et al._ (2012); Momenzadeh  _et al._ (2015); Shields  _et
al._ (2015); Mouradian  _et al._ (2015); Patel  _et al._ (2016); Sipahigil
_et al._ (2016); Bhaskar  _et al._ (2017); Rugar  _et al._ (2020a); Wan  _et
al._ (2020) [Fig. 6(c)]. While they offer large-scale fabrication and higher
collection efficiencies than solid immersion lenses (reported dipole-waveguide
coupling efficiencies above 55% Hedrich  _et al._ (2020); Wan  _et al._
(2020)), the demands in nano-fabrication and color center placement are more
stringent than for solid-immersion 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 Legero  _et al._ (2003, 2006).

Figure 6: Devices used to enhance the spin-photon interface of color centers
in diamond. Solid immersion lenses Robledo  _et al._ (2011) (a), parabolic
reflectors Wan  _et al._ (2018) (b, adapted with permission from Ref. Wan  _et
al._ (2018). Copyright (2017) American Chemical Society.), and waveguide (c,
SEM image of device from the same chip used in Ref. Rugar  _et al._ (2020a))
allow to enhance photon collection efficiencies from embedded color centers,
compared to emission in a bulk host material. Photonic crystal Burek  _et al._
(2014) [d, Figure adapted with permission from Ref. Burek  _et al._ (2016),
copyright by The Optical Society (OSA)] and open, tunable micro-cavities Ruf
_et al._ (2021) (e, Figure adapted with permission from Ref. Ruf  _et al._
(2021), in accordance with creativecommons.org/licenses/by/4.0/legalcode)
enhance collection efficiency, while modifying the nanophotonic environment at
the same time. This allows to additionally enhance the number of photons
emitted at a certain wavelength via the Purcell effect, compared to collection
enhancement tools.

####  VI.1.2 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) Reiserer and Rempe (2015). 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,
$\gamma_{\textrm{cav}}$, compared to the emission in a homogeneous medium,
$\gamma_{\textrm{hom}}=\gamma_{\textrm{rad}}+\gamma_{\textrm{nonrad}}$) is the
cooperativity, $C$, defined as

$C=\frac{4g^{2}}{\kappa\gamma_{\textrm{hom}}}=\frac{\gamma_{cav}}{\gamma_{hom}}=F_{P}\beta_{0}\eta,$
(1)

where $g$ is the dipole-cavity coupling rate, $\kappa$ is the energy decay
rate from the cavity, $\gamma_{\textrm{rad}}$ is the free-space radiative
emitter decay rate, $\gamma_{\textrm{nonrad}}$ is the non-radiative emitter
decay rate, $\beta_{0}$ is the fraction of photon emission from the emitter
into the zero-phonon line (given by the Debye-Waller factor), and $\eta$ 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
$\gamma_{\textrm{dep}}$, which manifests in broadening of the optical
transitions above the lifetime limited value Robledo  _et al._ (2010); Santori
_et al._ (2010); Riedel  _et al._ (2017); Jantzen  _et al._ (2016); Nguyen
_et al._ (2019b); Rugar  _et al._ (2020a). Therefore, it is useful to
introduce the coherent cooperativity, $C_{\textrm{coh}}$, which quantifies the
ratio of coherent decay into the cavity mode to undesired decay, and is given
as Borregaard, Sørensen, and Lodahl (2019); Janitz, Bhaskar, and Childress
(2020)

$C_{\textrm{coh}}=\frac{4g^{2}}{\kappa(\gamma_{\textrm{hom}}+\gamma_{\textrm{dep}})}=C\frac{\gamma_{\textrm{hom}}}{\gamma_{\textrm{hom}}+\gamma_{\textrm{dep}}}.$
(2)

$C_{\textrm{coh}}$ can be interpreted as the probability of coherent atom-
photon interaction, and plays an important role in the fidelity and efficiency
of many near-deterministic quantum protocols (which require
$C_{\textrm{coh}}\gg 1$) Borregaard, Sørensen, and Lodahl (2019).

To date, cavity enhancement of diamond color centers has been demonstrated for
different implementations, including diamond-on-insulator whispering gallery
mode resonators Faraon  _et al._ (2011); Hausmann  _et al._ (2012) and
nanodiamond plasmonic antennas Bogdanov  _et al._ (2020). Another class of
devices are based on the coupling of non-diamond resonators to color centers
in nanodiamonds Wolters  _et al._ (2010); Van Der Sar  _et al._ (2011); Fehler
_et al._ (2019, 2020) or diamond films Englund  _et al._ (2010); Barclay  _et
al._ (2011); Gould  _et al._ (2016). Open and tunable micro-cavities
geometries have also been demonstrated for embedded nanodiamonds Albrecht  _et
al._ (2013); Kaupp  _et al._ (2013); Albrecht  _et al._ (2014); Johnson  _et
al._ (2015); Kaupp  _et al._ (2016); Benedikter  _et al._ (2017) and thin
diamond membranes Riedel  _et al._ (2017); Høy Jensen  _et al._ (2020);
Häußler  _et al._ (2019); Salz  _et al._ (2020); Ruf  _et al._ (2021) [Fig.
6(d)]. Steady progress in diamond nanofabrication has also led to all-diamond
photonic crystal cavities Faraon  _et al._ (2012); Hausmann  _et al._ (2013);
Li  _et al._ (2015b); Riedrich-Möller  _et al._ (2015); Sipahigil  _et al._
(2016); Schröder  _et al._ (2017); Evans  _et al._ (2018); Jung  _et al._
(2019); Nguyen  _et al._ (2019b, c); Bhaskar  _et al._ (2020)[Fig. 6(e)], with
demonstrated coherent cooperativities exceeding 100 for the case of the SiV
center Bhaskar  _et al._ (2020).

Quantum network applications also require the efficient coupling of light from
these nanostructures to optical fibers. Different methods of coupling to
fiber-based communication channels have been realized to date, including
optimized free-space couplers with efficiencies up to $\sim 25\%$ Dory  _et
al._ (2019); Rugar  _et al._ (2020a), notch couplers with efficiencies $\sim
1\%$ Sipahigil  _et al._ (2016), grating couplers with efficiencies $\sim
30\%$ Hausmann  _et al._ (2012); Rath  _et al._ (2013), and double- Khanaliloo
_et al._ (2015a); Patel  _et al._ (2016); Mitchell, Lake, and Barclay (2019)
and single- Burek  _et al._ (2017); Nguyen  _et al._ (2019c); Evans  _et al._
(2018); Bhaskar  _et al._ (2020) sided fiber-tapers with efficiencies of up to
$\sim 90\%$.

###  VI.2 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.

####  VI.2.1 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 Humphreys  _et al._ (2018); Pompili
_et al._ (2021). 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 ($\sim$ microns) Ruf
_et al._ (2019). As a result, despite reported Purcell factors of up to 70 in
small mode-volume photonic crystal cavities Faraon  _et al._ (2012), the
regime of $C_{\textrm{coh}}\geq 1$ 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 Bluvstein, Zhang, and Jayich (2019), and the local
electrostatic environment of NV centers Sangtawesin  _et al._ (2019) show
promise towards understanding the causes of surface charge noise. Combined
with proposals for active optical driving to reduce spectral diffusion effects
Fotso  _et al._ (2016), this could revive the field of NV centers in
nanophotonic structures in the future.

Figure 7: State-of-the-art research for enhancing the spin-photon interface of
color centers in diamond for quantum networks applications. (a) Confocal
microscope image of a typical $\upmu\mathrm{m}$-thin etched diamond sample
containing NV centers with bulk-like optical properties, bonded via Van-der-
Waals forces to a mirror containing microwave striplines. Location and
direction of a sample height trace measurement is indicated with a red arrow.
Such samples have recently enabled the first demonstration of resonant
excitation and detection of coherent Purcell enhanced NV centers Ruf  _et al._
(2021). Figure adapted with permission from Ref. Ruf  _et al._ (2019), further
permissions related to the material excerpted should be directed to the
American Chemical Society. (b) NV center excited state decay in the ZPL for
different emitter-cavity detunings for an NV center in an open micro-cavity
under pulsed off-resonant excitation. Inset shows normalized decay curves.
Figure adapted with permission from Ref. Riedel  _et al._ (2017), in
accordance with creativecommons.org/licenses/by/4.0/legalcode. (c) Spin-state
dependent reflection spectrum of a SiV center in a critically coupled photonic
crystal cavity with $C_{\textrm{coh}}\sim 100$. The spin-state dependent
photon reflection with high collection efficiency enables spin-state readout
with fidelity $>$ 99.9 % in a single shot (d), and has recently enabled the
first demonstration of memory-enhanced quantum communication Bhaskar  _et al._
(2020). Figures (c) and (d) are adapted with permission from Ref. Bhaskar  _et
al._ (2020).

The challenge imposed by nearby surfaces can be circumvented by embedding
microns-thin diamond membranes between two highly reflective mirrors Janitz
_et al._ (2015). A recently reported fabrication technique has formed NV
centers with bulk-like optical properties in $\upmu\mathrm{m}$-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 Ruf  _et al._ (2019) [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 Ruf  _et al._ (2021), as would
be required for cavity-enhanced entanglement generation. However, the Purcell
enhancement in this experiment was limited to $\sim 3$, (mostly) due to
cavity-length fluctuations $\sim$ 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) Riedel  _et al._ (2017)
[Fig. 7(b)]. Combined, these experiments demonstrate that there is a near-term
path from the current $C_{\textrm{coh}}\sim 0.1$ towards $C_{\textrm{coh}}\sim
1$ and coherent photon collection efficiencies $\sim 10\%$. In addition,
recent results suggest that it should be possible to reach the required pm-
scale cavity length fluctuations even in closed-cycle cryostats Merkel,
Ulanowski, and Reiserer (2020); Fontana  _et al._ (2021), 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
Lekavicius, Oo, and Wang (2019); Kasperczyk  _et al._ (2020); Yurgens  _et
al._ (2021) at controlled locations; both the electron irradiation technique
employed in Ref. Ruf  _et al._ (2021), as well as recently introduced laser-
writing techniques Chen  _et al._ (2019b) 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 van Dam  _et al._ (2019), even after
extended high temperature treatments to restore the diamond lattice Chu  _et
al._ (2014).

While it will remain challenging to achieve coherent cooperativites $\gg 1$
(and thereby enter the near-deterministic spin-photon interface regime
Borregaard, Sørensen, and Lodahl (2019)), the open cavity approach is
projected to speed up current emission-based entanglement generation schemes
by $\sim$ 2 orders of magnitude Ruf  _et al._ (2021). 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 Rozpȩdek  _et
al._ (2019), and device-independent quantum key distribution Murta  _et al._
(2019) using NV centers.

####  VI.2.2 Group-IV color centers

The first-order electric field insensitivity of group-IV color centers in
diamond Sipahigil  _et al._ (2014); Evans  _et al._ (2016); Thiering and Gali
(2018); Müller  _et al._ (2014); Iwasaki  _et al._ (2015); Tchernij  _et al._
(2017); Trusheim  _et al._ (2020); Rugar  _et al._ (2020a); Trusheim  _et al._
(2019) has enabled the demonstration of close-to-lifetime limited optical
transitions even in heavily fabricated nanophotonic structures Evans  _et al._
(2016); Sipahigil  _et al._ (2016); Bhaskar  _et al._ (2017); Nguyen  _et al._
(2019c); Rugar  _et al._ (2020a); Wan  _et al._ (2020). To date, photonic
crystal cavity coupling of group-IV color centers in diamond at low
temperatures has been demonstrated for SiV Sipahigil  _et al._ (2016); Evans
_et al._ (2018); Nguyen  _et al._ (2019b, c); Bhaskar  _et al._ (2020) and SnV
centers Rugar  _et al._ (2021), with demonstrated $C_{\textrm{coh}}>100$ for
the case of the SiV center Bhaskar  _et al._ (2020). 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 memory-enhanced quantum
communication Bhaskar  _et al._ (2020).

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 Meesala  _et al._ (2018); Sohn
_et al._ (2018); Maity  _et al._ (2018); Machielse  _et al._ (2019); Wan  _et
al._ (2020), electric fields De Santis  _et al._ (2021); Aghaeimeibodi  _et
al._ (2021), and Raman-type Sipahigil  _et al._ (2016); Sun  _et al._ (2018)
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 Wan  _et al._ (2020); De Santis  _et al._ (2021); Aghaeimeibodi
_et al._ (2021).

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 $\sim$ several
lifetime-limited linewidths Evans  _et al._ (2018); Nguyen  _et al._ (2019c);
Rugar  _et al._ (2020a); Wan  _et al._ (2020), as well as large local strain
fields. While dynamic strain tuning can be used to suppress slow spectral
diffusion (seconds timescale) Machielse  _et al._ (2019), 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 Chu  _et al._ (2014). However, there is evidence that
even such high temperature treatments can not fully recover the original
diamond lattice van Dam  _et al._ (2019). A recently developed promising
method to overcome this limitation employs low energy shallow ion
implantation, combined with overgrowth of diamond material Rugar  _et al._
(2020b). 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 McLellan  _et al._ (2016). Combined
with controlled engineering of the diamond Fermi level Collins (2002); Rose
_et al._ (2018); Murai  _et al._ (2018), these techniques could increase the
quality and stability of group-IV color center optical transitions Tchernij
_et al._ (2017); Evans  _et al._ (2018); Nguyen  _et al._ (2019c); Trusheim
_et al._ (2019); Rugar  _et al._ (2020a); Wan  _et al._ (2020).

Another area of active development involves the design and fabrication of the
nanophotonic structures. Despite recent progress Schreck  _et al._ (2017);
Nelz  _et al._ (2019), 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-$\upmu\mathrm{m}$-thin
diamond films via selectively wet-etching a localized graphitized diamond
layer Parikh  _et al._ (1992); Fairchild  _et al._ (2008); Gaathon  _et al._
(2013); Lee  _et al._ (2014); Piracha  _et al._ (2016a, b) or etching down a
super-polished thin diamond membrane Faraon  _et al._ (2012); Hausmann  _et
al._ (2013); Riedrich-Möller  _et al._ (2014); Li  _et al._ (2015b); Cady  _et
al._ (2019), 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) Burek  _et al._ (2012); Bayn  _et al._ (2014); Burek  _et al._
(2014); Sipahigil  _et al._ (2016); Atikian  _et al._ (2017); Zhang  _et al._
(2018); Nguyen  _et al._ (2019c), or a quasi-isotropic (dry) undercut etch,
which selectively etches along certain diamond crystal planes (leading to
rectangular device cross-section) Khanaliloo  _et al._ (2015b, c); Mouradian
_et al._ (2017); Wan, Mouradian, and Englund (2018); Dory  _et al._ (2019);
Zheng  _et al._ (2019); Wan  _et al._ (2020). 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 quasi-isotropic 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 $\sim 10^{4}$, about two orders of magnitude lower than
simulated values Mouradian  _et al._ (2017). These deviations are caused by a
combination of surface roughness Nguyen  _et al._ (2019c), non-uniform hole
sizes Rugar  _et al._ (2021), and deviations from the expected device cross-
section Wan, Mouradian, and Englund (2018); 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
$\sim$ 5 nm across devices Mouradian  _et al._ (2017). Nonetheless, this
spread in frequency may be overcome by cavity tuning methods Mosor  _et al._
(2005); Faraon  _et al._ (2012); Sipahigil  _et al._ (2016), 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 Wan  _et al._ (2018), or ion implantation into devices using
either masks Nguyen  _et al._ (2019c) or focused implantation Sipahigil  _et
al._ (2016); Wan  _et al._ (2020). Compared to the SiV center for which
$C_{\textrm{coh}}\sim 100$ 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 Høy Jensen  _et al._ (2020) and SiV Häußler  _et al._ (2019); Salz  _et
al._ (2020) centers at room temperature. It should be possible to achieve
$C_{\textrm{coh}}\gg 1$ for these systems, potentially providing a viable
alternative to the more fabrication-intensive nanostructures.

##  VII Towards large-scale quantum networks

Figure 8: Schematic of realized and envisioned architectures for large-scale
hybrid integrated circuits enabling entanglement routing. (a) False-color SEM
image of a AlN-on-sapphire photonic integrated circuit (PIC), containing a
diamond quantum microchiplet [QMC, red dashed area, enlarged view in (c)].
These microchiplets with embedded color centers are pre-fabricated and
-characterized in the diamond host material, and transferred onto the PIC
using a pick-and-place technique (b), allowing to optimize PICs and
microchiplets separately. This technique has been used to generate a defect-
free array of 128 nanophotonic waveguides containing individually addressable
quantum emitters, coupled to the PIC Wan  _et al._ (2020). Images (a-c)
adapted with permission from Ref. Wan  _et al._ (2020). (d) Integrated multi-
quantum memory node in a modular architecture Choi  _et al._ (2019). Each unit
cell (black dashed line) contains a diamond-based atomic memory (red), a fast,
reconfigurable switching network allowing all-to-all connectivity (realized
e.g. via Mach-Zehnder interferometers), and photo-detectors. The atomic memory
includes one communication and several memory qubits, with the communication
qubit coupled to a diamond nanophotonic structure. Figure reprinted with
permission from Ref. Choi  _et al._ (2019), in accordance with
creativecommons.org/licenses/by/4.0/legalcode. (e) Entanglement routing in a
quantum network. A multi-path routing algorithm (realized e.g. via re-
configurable PICs, dashed black box) can improve the entanglement generation
rate (wiggled blue lines) between distant nodes (white circles), compared to a
linear repeater topology Lee  _et al._ (2020). 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. Lee  _et al._ (2020). Figure (f) reprinted
with permission from Ref. Pant  _et al._ (2019), in accordance with
creativecommons.org/licenses/by/4.0/legalcode.

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 $\sim 637$ nm photon
entangled with the spin-state of the NV center to telecommunication
wavelengths ($\sim 1588$ nm) Tchebotareva  _et al._ (2019). Similar conversion
techniques can be applied to other color centers and quantum systems
Krutyanskiy  _et al._ (2019); Bock  _et al._ (2018); Ikuta  _et al._ (2018);
Yu  _et al._ (2020); De Greve  _et al._ (2012). Future work will focus on
increasing the system efficiency and potentially integrating on-chip
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 Muralidharan  _et
al._ (2016). Although color center devices are already produced using standard
nanofabrication techniques, the needed scale for high-rate, 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 Bogaerts
_et al._ (2020) and access to a large number of spatial modes Harris  _et al._
(2018) 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 Bogaerts  _et al._
(2020); Blumenthal  _et al._ (2018); Harris  _et al._ (2018); Zhu  _et al._
(2021) 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 Sun  _et al._ (2015);
Atabaki  _et al._ (2018), photonic processors Wetzstein  _et al._ (2020);
Shastri  _et al._ (2021) and optical quantum computing Wang  _et al._ (2020),
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
Bogaerts  _et al._ (2020). Next, phase modulators in PICs can implement on-
chip switches for routing photons within an optical network Bogaerts  _et al._
(2020), and material nonlinearities can be used for efficient frequency
conversion between visible and telecommunication photons Wang  _et al._
(2018). 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 Rath  _et al._ (2013) and nonlinear optics Hausmann  _et al._
(2014). Gallium phosphide-on-diamond photonics have also been used to route
the emission from NV centers Gould  _et al._ (2016). However, the non-
deterministic 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" Kim  _et al._ (2020); Elshaari  _et al._ (2020), 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 Komljenovic  _et al._ (2018). In the context of quantum
photonics, recent successes include quantum emitters Kim  _et al._ (2020);
Elshaari  _et al._ (2020) and single-photon detectors Najafi  _et al._ (2015),
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 Wan  _et al._ (2020) 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 Wan  _et al._ (2020). The availability of multiple color centers
per waveguide potentially allows for spectral multiplexing Bersin  _et al._
(2019); Chen  _et al._ (2020), which is a hardware-efficient path to
multiplying the total number of qubits to $N_{s}\times N_{f}$, 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, we
expect future developments to also focus on integrating chip components for
color center technologies, such as CMOS-integrated microwave electronics for
spin control Kim  _et al._ (2019), on-chip beamsplitters for photon quantum
interference Wang  _et al._ (2020), optical switches for channel connectivity
Bogaerts  _et al._ (2020), single-photon detectors Reithmaier  _et al._
(2015); Najafi  _et al._ (2015); Schwartz  _et al._ (2018); Gyger  _et al._
(2021) for heralded entanglement, and quantum frequency conversion Guo  _et
al._ (2016); Jankowski  _et al._ (2020) or frequency tuning Hu  _et al._
(2020) 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 Choi  _et al._ (2019). 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 processing. 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 Duan and Kimble (2004); Lee  _et
al._ (2020). 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 Bennett  _et al._ (1996); Deutsch  _et al._ (1996). Such a
scheme has recently been demonstrated in a proof-of-principle experiment
between two distant NV centers that each have access to an additional 13C
memory Kalb  _et al._ (2017). While such distillation requires additional time
resources as a result from two-way-classical-communication, quantum error
correction Jiang  _et al._ (2009) can achieve a more favorable key rate per
qubit scalingMuralidharan  _et al._ (2016). Such error correction can be
performed if the operational error is smaller than the threshold of a given
code (e.g. $\approx 0.189$ for the Calderbank-Shor-Stean (CSS) code Poulin
(2006)). 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 Borregaard
_et al._ (2020). 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 Pant  _et al._ (2019). 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 Pant  _et al._ (2019).

##  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 Pompili  _et al._ (2021) and
in memory-enhanced quantum key distribution Bhaskar  _et al._ (2020). 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 Dahlberg  _et al._
(2019); Pirker and Dür (2019).

A future functional quantum network will support many interesting
applications, such as distributed quantum computing Cuomo, Caleffi, and
Cacciapuoti (2020), accessing a quantum server in the cloud with full privacy
Broadbent, Fitzsimons, and Kashefi (2009) and stabilizing quantum clocks
Hodges  _et al._ (2013); Kómár  _et al._ (2014). Color centers in diamond may
play an essential role in these networks, as the “satisfactory repeater”, or
perhaps the “transistor of the quantum age.”

## Acknowledgements

We thank Ben Dixon, Mihir Bhaskar, Eric Bersin, Johannes Borregaard, Tim
Taminiau, Matteo Pompili, and Matteo Pasini for feedback on the manuscript,
Hans Beukers for the calculations of operation temperatures of color centers,
and Matteo Pompili, Mihir Bhaskar, Christian Nguyen, Daniel Riedel, Alison
Rugar, Shahriar Aghaeimeibodi, Conor Bradley, Mihir Pant, Yuan Lee, Michael
Burek and Cleaven Chia for providing originals of figures. D.E. acknowledges
support from Brookhaven National Laboratory, which is supported by the U.S.
Department of Energy, Office of Basic Energy Sciences, under Contract No. DE-
SC0012704. N.W. acknowledges support from the MITRE Quantum Moonshot
initiative. H.C. acknowledges the Claude E. Shannon Fellowship and Samsung
Scholarship. R.H. and M.R. acknowledge financial support from the EU Flagship
on Quantum Technologies through the project Quantum Internet Alliance, from
the Netherlands Organisation for Scientific Research (NWO) through a VICI
grant and the Zwaartekracht program Quantum Software Consortium, and the
European Research Council (ERC) through a Consolidator Grant.

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