# Coherent microwave, optical, and mechanical quantum control of spin qubits
in diamond

Laura Orphal-Kobin  Department of Physics & IRIS Adlershof, Humboldt-
Universität zu Berlin, Newtonstr. 15, 12489 Berlin, Germany  Cem Güney Torun
Department of Physics & IRIS Adlershof, Humboldt-Universität zu Berlin,
Newtonstr. 15, 12489 Berlin, Germany  Julian M. Bopp  Department of Physics &
IRIS Adlershof, Humboldt-Universität zu Berlin, Newtonstr. 15, 12489 Berlin,
Germany  Ferdinand-Braun-Institut, Gustav-Kirchhoff-Str. 4, 12489 Berlin,
Germany  Gregor Pieplow  Department of Physics & IRIS Adlershof, Humboldt-
Universität zu Berlin, Newtonstr. 15, 12489 Berlin, Germany  Tim Schröder
<EMAIL_ADDRESS>Department of Physics & IRIS Adlershof,
Humboldt-Universität zu Berlin, Newtonstr. 15, 12489 Berlin, Germany
Ferdinand-Braun-Institut, Gustav-Kirchhoff-Str. 4, 12489 Berlin, Germany

###### Abstract

Diamond has emerged as a highly promising platform for quantum network
applications. Color centers in diamond fulfill the fundamental requirements
for quantum nodes: they constitute optically accessible quantum systems with
long-lived spin qubits. Furthermore, they provide access to a quantum register
of electronic and nuclear spin qubits and they mediate entanglement between
spins and photons. All these operations require coherent control of the color
center’s spin state. This review provides a comprehensive overview of the
state-of-the-art, challenges, and prospects of such schemes, including, high
fidelity initialization, coherent manipulation, and readout of spin states.
Established microwave and optical control techniques are reviewed, and
moreover, emerging methods such as cavity-mediated spin-photon interactions
and mechanical control based on spin-phonon interactions are summarized. For
different types of color centers, namely, nitrogen-vacancy and group-IV color
centers, distinct challenges persist that are subject of ongoing research.
Beyond fundamental coherent spin qubit control techniques, advanced
demonstrations in quantum network applications are outlined, for example, the
integration of individual color centers for accessing (nuclear) multi-qubit
registers. Finally, we describe the role of diamond spin qubits in the
realization of future quantum information applications.

##  I Introduction

Quantum information processing applications are expected to have a big impact
on industry and society. For example, long distance quantum networks [1] would
enable secure quantum communication,[2, 3] distributed quantum computing,[4,
5, 6] and enhanced quantum sensing.[7] A quantum node in such a network
requires an optically accessible, sufficiently coherent quantum system
composed of qubits that can be controlled with high fidelity. Wide-band gap
solid-state materials stand out, since they can host a large range of color
centers capable of hosting long-lived, optically active spin qubits,[8, 9] and
moreover, provide compatibility with photonic integration and on-chip
scalability.[10, 11, 12, 13, 14, 15, 16, 17]

Color centers in diamond have been investigated extensively regarding quantum
applications since the 1990s, when the nitrogen-vacancy color center’s (NV)
ground state spin-triplet structure and spin coherence have been optically
detected in landmark experiments.[18, 19] While the NV became initially
attractive as an excellent quantum nanoscale sensor, both the NV and group IV
vacancy color centers (G4Vs) continue developing as promising resources for
quantum information processing.[20, 21, 22] The NV is well-known for its long
spin coherence times of more than one second,[23] but limited in optical
coherence due to its susceptibility to local electric field fluctuations
causing spectral diffusion. However, recently, spectrally stable photon
emission in nanostructures has been demonstrated under resonant
excitation.[24] G4Vs exhibit due to their inversion symmetry high optical
coherence. Extensive research from 2011 on,[25, 26] exploring different types
of G4Vs and experimental methods, also lead to an increase of spin-coherence
times from sub-microseconds up to the order of tens of milliseconds.[27, 28]
Moreover, color centers in diamond provide access to a long lived register of
electronic and nuclear spin qubits.[29, 30, 31]

Figure 1: Illustration of the different ingredients of coherent quantum
control. Color centers in diamond constitute spin qubits. Initialization,
manipulation, and readout of the spin states can be achieved by coherent
microwave, optical, and mechanical control methods. Quantum information can be
transmitted in flying qubits after spin-photon entanglement generation, e.g.,
to a distant quantum node in a quantum network.

This review is centered on the stack of control operations that are required
for any quantum node.[32] This stack is composed of qubit initialization, its
coherent control and readout as well as its ability to mediate entanglement
between spins and photons. So far, continuous development enabled spin qubit
initialization fidelities of more than $99\%$,[33] and averaged spin qubit
state readout fidelities of more than $97\%$.[34] Moreover, color centers in
diamond have been used to demonstrate, employing advanced control protocols,
spin-photon [35] and spin-spin entanglement,[36, 34, 37] the implementation of
a three-node quantum network [33] as well as qubit teleportation.[38] For
entanglement generation, time-bin encoding, to date, has turned out to be the
most efficient, however, the NV also allows for polarization encoding.[35]
Despite these milestone achievements, there are still persisting challenges
that need to be tackled for the implementation of large-scale quantum networks
based on color centers in diamond, which will be discussed in the scope of
this work as well.

In this review, we aim to provide a concise overview of the current and
emerging methods for (coherent) control of spin qubits in diamond towards
quantum information processing applications, as indicated in the illustration
in Figure 1. We do not discuss the role of color centers in diamond for
sensing and secure communication. Moreover, for information on other pre-
requisites for large scale quantum network applications, such as diamond
nanofabrication,[16] hybrid photonic architectures,[39, 40, 41, 42] and
frequency conversion to the telecom band [43, 44] we refer to more specialized
works and reviews.

This review is organized as follows: In Section II some fundamental properties
of color centers in diamond are introduced. The following sections then
provide an overview of coherent control techniques. We begin with microwave
control in Section III and optical control in Section IV, which are the most
established and are a foundation for most advanced control techniques. We then
review more recent methods for realizing single- and two-qubit gates. In
particular, we present spin qubit-photon gates mediated by enhanced light-
matter interaction in nanocavities (Sections IV.4 and IV.5). In Section V we
provide an overview of single qubit gates enabled by mechanical spin control,
which is mediated by application of dynamical strain. Beyond fundamental
control principles, we briefly outline advanced demonstrations in quantum
network applications as well as potential future directions of quantum
information processing and quantum computing based on color centers in diamond
in Section VI.

##  II Color centers in diamond

In diamond, more than 500 optically active defects exist [45]. For quantum
applications color centers with single photon emission and optically
accessible spin ground states are most relevant. Here, we focus on the
nitrogen-vacancy color center (NV) and the negatively charged group IV vacancy
color centers (G4V). There are many more defect centers that have been
investigated. Their application range, however, to date has been limited and
for a detailed introduction we point at the respective references. Examples
are color centers based on magnesium,[46] chromium,[47] xenon,[48] and the
nickel-nitrogen complex NE8,[49] as well as the nickel-vacancy center with
near infrared optical activity.[50, 51]

###  II.1 Material considerations

Diamond is a wide-bandgap semicondutor with a bandgap of about 5.5 eV. Its
optical transparency and capability to host a large variety of color centers
make it an appealing platform for quantum optics. The color centers discussed
in this review are formed from substitutional atoms (impurities) in
combination with single lattice vacancies. Within the bandgap, they give rise
to a localized atom-like system that can, depending on its charge state, host
a long lived spin-qubit, which is amenable to microwave, optical, and
mechanical manipulation, and, moreover, optical initialization and readout. In
diamond, long spin coherence times can be maintained due to low phonon
interaction (high Debye temperature) and low magnetic noise from nearby spins,
since it is primarily composed of spin-0 12C atoms (around 99$\%$).[45, 52]

For quantum operations based on interference of indistinguishable photons,
photons emitted into the spectrally narrow zero-phonon line (ZPL) are used.
The Debye-Waller factor is defined by the ratio of photons emitted into the
ZPL, compared to the total emission including the ZPL as well as the
spectrally broad and red-detuned phonon-sideband. It varies for different
color centers from 3$\%$ for the NV up to 60-80$\%$ for the centrosymmetric
G4Vs.[26, 53] The high refractive index of diamond $n=2.41$ leads to a
critical angle of $24.5^{\circ}$ of the total internal reflection at the
diamond-air interface, confining most of the emitted photons in the material.
Fabrication of efficient diamond nanostructures enable high photon collection
efficiencies, for example, by collecting photons through an objective (free-
space) or by directly coupling to an integrated fiber.[16, 54, 55, 56]
Nanostructures can also be used to increase the spin-photon coupling, which
leads to the enhancement of emission into a well defined optical mode (Purcell
enhancement, see also Section IV.4), which also improves collection
efficiencies. Finally, as a solid-state material, diamond facilitates photonic
integration, an essential aspect for scalability, which is a critical factor
in the development of quantum technology applications.[13]

###  II.2 Color center formation

Typically, artificial diamond is used for quantum applications, due to high
purity and because it allows for the control of various growth parameters,
such as the density of a desired defect species. Diamond can be produced
through two primary methods: chemical vapor deposition (CVD) or the high-
pressure high-temperature (HPHT) technique. The CVD setup is rather simple and
the method allows for flexible control over impurity components and
concentrations.[57] Electronic grade diamond crystals are formed by
isotopically purified 12C, which is obtained from a plasma composed of methane
and hydrogen. The 12C are then deposited onto a substrate, which seeds the
growth of the diamond crystal. Defects are introduced by adding a certain
concentration of e.g., nitrogen or silicon, into the growth chamber. The
addition of impurity components for a well defined time, can even create
"delta-doped" layers with precise control of defect center depth at the
nanometer scale.[58, 59] To enhance the formation yield of atom-vacancy color
centers, the diamond is exposed to electron irradiation,[60] creating lattice
damage. While smaller atoms can be incorporated during growth and still
maintain good diamond crystal properties, for larger atoms, such as tin and
lead, ion implantation into pure electronic grade diamond substrates with
subsequent high-temperature (and possibly high-pressure) annealing is
applied.[53, 61, 62] The annealing causes an atom migration within the diamond
lattice, leading to the formation of atom-vacancy complexes and a recovery of
the diamond lattice.

In comparison to defects generated through ion implantation, native (in-grown)
single defect centers typically exhibit superior optical properties due to
minimal damage to the local lattice environment.[63] For large atoms, shallow
ion implantation with subsequent overgrowth is a promising approach.[64]

Optimising defect formation is still a subject of active research. Co-
implantation of atoms and Coulomb-driven defect engineering can lead to
improved properties regarding charge-state, spin and optical coherence, as
well as increased color center formation yield.[65, 66, 67] A review on color
center generation in diamond can be found in ref. [68].

###  II.3 Nitrogen-vacancy color center

The NV is the most explored color center in diamond. It first made its strides
as a platform for quantum sensing of magnetic fields and temperature [60, 69,
70, 71, 72] and later emerged as a candidate for quantum information
applications.

The NV is a point defect, that consists of a nitrogen atom and a vacancy,
substituting a nearest-neighbor pair of carbon atoms in the diamond lattice.
It exhibits C3v symmetry along the $\left\langle 111\right\rangle$ directions,
i.e., it is non-inversion symmetric leading to a permanent electric dipole
moment which makes the NV energy levels sensitive to electric fields. In its
neutral charge state (NV0) with one unpaired electron, it forms a spin $S=1/2$
quantum system. In the negative charge state (NV-), an extra electron is
present resulting together with the unbound vacancy electron in a $S=1$
system. From here on in NV refers to the NV-, which, so far, is the most
relevant charge state for quantum applications.

The schematic energy level structure of the NV- is shown in Figure 2(a).
Unperturbed, the energy levels of the NV- involve a ground state
(${}^{3}\textrm{A}_{2}$), an excited state spin triplet (${}^{3}\textrm{E}$)
[73, 74, 75] and two singlets (${}^{1}\rm A_{1}$, ${}^{1}\rm E$). The ground
state spin triplet ${}^{3}\textrm{A}_{2}$ shows a zero field splitting of
$\sim$2.87 GHz between the $m_{\rm s}=0$ and $m_{\rm s}=\pm 1$ sublevels.
Analogous to the zero field splitting of the ground states the excited state
triplet ${}^{3}\textrm{E}$ shows a splitting of $\sim$1.42 GHz.[76] By
applying an external magnetic field the energy degeneracy of the $m_{s}=\pm 1$
spin sublevels of the ground state spin triplet can be lifted due to the
Zeemann effect. A subset of states of the ground state triplet such as for
example $m_{\rm s}=0$ and $m_{\rm s}=+1$ levels is then typically used as a
spin qubit.

Figure 2: Schematic electronic level structure. (a) Negatively charged
nitrogen-vacancy color center ($I=1$). (b) Negatively charged group-IV vacancy
color center ($I=1/2$).

The excited state and ground state manifolds are optically coupled through
spin preserving transitions, leading to a total of six ZPL transitions at
$\sim 637$ nm. In addition, the two intermediate singlet states ${}^{1}\rm
A_{1}$ and ${}^{1}\rm E$ are connected through an optical transition with a
wavelength of 1042 nm.[77, 78] The two singlet states form another (spin-
dependent) decay channel from the excited to the ground states. An electronic
decay (intersystem crossing) via these meta-stable states happens
predominantly for $m_{s}=\pm 1$ spin-states and are not spin conserving which
enables, e.g., spin state initialization through non-resonant optical
pumping.[79, 80]

NVs formed from naturally the abundant 14N isotope, which has a nuclear spin
$I=1$ (natural abundance of 99.63$\%$), ion implantation typically relies on
15N isotopes with nuclear spin $I=1/2$. Both isotopes naturally lead to two
distinct hyperfine structures. The Debye-Waller factor represents the ratio of
photon emission into the ZPL. For the NV only 3$\%$ of the emitted photons are
in the ZPL at 637 nm and can be potentially considered to be optically
coherent and useful for quantum operations, while the large amount is emitted
in the phonon sideband (640-800 nm).

###  II.4 Group-IV color centers

Although the NV has shown exceptional performance in terms of spin coherence
properties (see Section III), its limitations in terms of optical properties
(see Section IV), have led to the investigation of another family of color
centers, known as the G4Vs. These color centers derive their name from the
specific group of atoms in the periodic table from which they are composed.
G4Vs are formed by an atom, such as silicon,[81, 82, 26] germanium,[83, 84]
tin,[85, 86, 53] and lead,[61, 87] which is positioned at an interstitial site
between two vacancy defects in the crystal lattice.[88, 89] The D3d point
group inversion symmetric arrangement prevents the formation of a permanent
electric dipole and therefore inhibits first-order DC Stark Shifts.[90, 91]

Completely unperturbed, the electronic structure of the G4V is composed of an
$E_{g}$ ground state level and an $E_{u}$ excited state level.[92] Their
degeneracy is lifted by the spin orbit and Jann-Teller interaction, leading to
four observable optical transitions [93, 94] as depicted in Figure 2(b).
Because G4Vs are effective spin 1/2 systems, the levels are two fold
degenerate, and can be split through the Zeemann interaction in the presence
of a magnetic field. A suitable spin qubit can be chosen, for example, between
the two electronic states of the lower ground state branch. The optical
selection rules for the four states in the ground state manifold and the four
states in the excited state manifold strongly depend on the direction of the
magnetic field, and the strength of the spin-orbit interaction. Under the
right conditions, the spectral fine structure of G4Vs results in 16 possible
optical transitions.[92]

For G4Vs, more than 50$\%$ of the emission spectrum consists of coherent
photons in the ZPL. The Debye-Waller factor increases inversely proportional
with the dopant atom size,[95] which reduces distortions of the lattice. A
comprehensive review of G4V and their application to quantum technologies can
be found in ref. [25].

Interestingly, the ground and excited state branches are coupled by strain,
and therefore to the lattice phonon field. Depending on the temperature,
phononic processes [96, 97] can lead to the decoherence of the two lowest
branch spin states due to thermal occupation of the phonoic modes. The
temperature for which these processes become the leading decoherence mechanism
is determined by the splitting of the ground state branches. Depending on the
G4V, phonons are frozen out at a few hundred mK (SiV) or at cryogenic
temperatures in the range of a few Kelvin (SnV, PbV). The heavier G4Vs, such
as the SnV and PbV are much more desirable, due to their longer coherence
times at higher temperatures.

Also for G4V different dopant isotopes can be implanted, leading to a coupling
to different nuclear spins.[31, 98]

In Table 1, some fundamental properties of different types of color centers in
the negative charge state in diamond are summarized.

Table 1: Summary of fundamental diamond color center properties in their negative charge state. Color center | ZPL resonance | Debye-Waller factor | Lifetime (natural linewidth) | Ground state splitting
---|---|---|---|---
NV [99, 75] | 637 nm | $3\%$ | 12 ns (13 MHz) | 2.87 GHz
SiV [100, 101, 102, 82] | 737 nm | $\sim 70\%$ |  $1.2-1.7$ ns (110 MHz) |  $\sim 50$ GHz
GeV [84, 103, 104] | 602 nm | $\sim 70\%$ |  $1.1-6$ ns (45 MHz) |  $150-170$ GHz
SnV [53, 85, 86, 105, 106, 62] | 619 nm | $\sim 60\%$ |  $4-8$ ns (27 MHz) |  $850$ GHz
PbV [61, 107, 95, 87] |  $550$ nm | $\sim 30\%$ | 4.4 ns (36 MHz) |  $3.9-5.7$ THz

##  III Microwave control

Since the first demonstrations of optically detected magnetic resonance (ODMR)
with NVs in the 1990s,[18, 19] color centers in diamond became a well
established platform for (nano-)sensors of magnetic fields and temperature [7,
70]. They have since then also proved themselves as promising candidates for
quantum network applications.[33, 108] The spin state splitting frequencies
(under an applied magnetic field) are determined in ODMR experiments.[109] The
interaction potential that couples a microwave field to a color center’s
electrons is typically described by [110]

$H_{\rm B}=\frac{\mu_{B}}{\hbar}\sum_{i}(\bf l_{i}+g_{e}\bf s_{i})\bf B\,,$
(1)

where $\mu_{B}$ the Bohr magneton, $\hbar$ the reduced Planck constant,
$g_{e}$ the electron g-factor, $\bf l_{i}=\bf r_{i}\times\bf p_{i}$ is the
orbital magnetic moment of the $i$th electron, $\bf p_{i}$ its momentum, $\bf
s_{i}$ its spin, and $\bf B$ the magnetic field used to drive the center. The
effective Hamiltonians describing the interaction of a magnetic field with the
ground state manifold are slightly distinct for the NV and the G4Vs because
they have a different spin. For the NV, the effective Hamiltonian can be
written in a fairly compact form:

$H_{\rm B}^{\rm NV}=\frac{\mu_{B}}{\hbar}\bf S\cdot\hat{g}\cdot\bf B\,,$ (2)

where $\bf S$ is the total spin-1 operator and $\hat{g}={\rm
diag}\\{g_{\perp},g_{\perp},g_{\parallel}\\}$. $g_{\perp},g_{\parallel}$ are
the nonaxial and axial effective g-factor components.

From [101] the effective Hamiltonian for G4Vs becomes

$\displaystyle\begin{aligned} H_{\rm
B}^{G4V}=&\frac{f\gamma_{L}}{\hbar}B_{z}\begin{pmatrix}0&i\\\
-i&0\end{pmatrix}\otimes\begin{pmatrix}1&0\\\ 0&1\\\
\end{pmatrix}+\frac{\gamma_{S}}{\hbar}\begin{pmatrix}1&0\\\ 0&1\\\
\end{pmatrix}\\\ &\otimes\begin{pmatrix}B_{z}&B_{x}-{\rm i}B_{y}\\\ B_{x}+{\rm
i}B_{y}&-B_{z}\\\ \end{pmatrix}~{},\end{aligned}$ (3)

which is written in the basis of the Hilbert space
$\mathcal{H}=\mathcal{E}_{g}\otimes\mathcal{S}_{\rm 1/2}$, where
$\mathcal{E}_{g}$ is spanned by $|e_{gx}\rangle,|e_{gy}\rangle$ and
$\mathcal{S}_{1/2}$ by $|\downarrow\rangle,|\uparrow\rangle$. In Equation (3),
$\gamma_{L}=\gamma_{s}/2$, where $\gamma_{s}$ is the gyromagnetic ratio, $f$
is a quenching factor, and $\bf B=(B_{x},B_{y},B_{z})$. Both these
Hamiltonians are the starting point for modelling the interaction of a
magnetic field with the vacancy’s spin.

Coupling to a resonant microwave driving field induces rotations on the Bloch
sphere, known as Rabi oscillations. Microwave pulses with well-defined
duration, power, and phase – for example a $\pi$-pulse around a certain axis –
can generate controlled spin state population flipping or rotations in the
equator plane of the Bloch sphere. Full quantum control can be achieved by
control of either the microwave polarization or phase, resulting in the
implementation of a set of universal single-qubit gates. By coherent microwave
manipulation of the electronic spin and by leveraging the Zeeman interaction,
a complete set of single- and two-qubit gates can be engineered operating on
two electronic spins. Such control procedures were used to produce
entanglement between two electronic spins with high fidelity.[111]

###  III.1 Electron spin coherence

To quantify the coherence times of a qubit system, the electron spin-lattice
relaxation time $T_{1}$ (longitudinal relaxation) and spin coherence time
$T_{2}$ (transverse relaxation) have been investigated using microwave
control. The $T_{1}$ time captures the population decay between the two qubit
states and sets the fundamental limit for spin coherence, $T_{2}\leq
2\,T_{1}$. It is determined by phononic processes, such as thermal relaxation
by emission, absorption, or scattering of phonons in the diamond lattice.
Moreover, depending on the environment also magnetic and electric charge noise
can cause random spin-flips. The spin coherence time determines, how long
phase information can be stored in a spin qubit, the memory time of the
quantum system. The $T_{2}^{\star}$ time, representing the overall coherence
between two states, corresponds to inhomogeneous dephasing processes and is
recorded in a Ramsey experiment (free-induction decay).[109] When the electron
spin undergoes precession in the equator plane after being prepared in an
equal superposition state ($\pi/2$-pulse), an additional refocusing
$\pi$-pulse can decouple the quantum system from quasi-static magnetic fields,
protecting the coherence of the spin state. Such a Hahn echo measurement
yields the homogeneous $T_{2}$ time. For comparison, an overview of spin
properties in different solid-state materials can be found in ref. [9].

Decoherence of the electron spin is mainly caused by the interaction with
neighboring spins in the environment, which is a common mechanism for solid-
state systems. In diamond, the spin density is relatively low, however, a
residual paramagnetic bath is comprised of 13C spins, P1 centres (nitrogen
impurities), and others. The dephasing can be considered as a random magnetic
bath field which is directed along the color center’s quantization axis. To
suppress the interaction of the electron spin with the surrounding spin bath
and to prevent decoherence of quantum states, dynamical decoupling (DD)
schemes are used. The spin echo pulse sequence can be extended to a train of
periodic $\pi$-pulses seperated by a defined delay time $\tau$,
$(\tau-\pi-\tau)^{N}$, called Carr-Purcell-Meiboom-Gill (CPMG) sequence.[112]
While CPMG is a single-axis decoupling protocol, decoupling sequences
constructed of rotations over two axes preserve robustness against pulse
errors and maintain coherence of arbitrary quantum states $\ket{x}$ and
$\ket{y}$.[113, 114] E.g., the XY4 and XY8 protocols are commonly used in
which the phase of the pulses is alternated. The coherence time scales with
the number of applied pulses $N$.

Since G4Vs are spin 1/2 systems, they are not only subject to magnetic field
fluctuations – the predominant dephasing mechanism for the NV. Depending on
the environment properties, resonant dipolar coupling with the spin bath (e.g.
P1 centers) additionally contributes significantly to decoherence. Such a
Markovian coupling cannot be mitigated by DD pulses.[115]

###  III.2 Noise spectroscopy and coupling to nuclear spin registers

While for the electron spin coherence, nuclear spins in the environment seem
to potentially cause detrimental magnetic fluctuations, they can also be used
as a powerful resource. Nuclear spins are significantly less affected by
magnetic fields in comparison to electron spins due to the difference in the
nuclear and Bohr magneton. By coupling a single electron spin to multiple
nuclear spins in a controlled fashion, multi-qubit registers can be formed to
store quantum information for long times, and be used for quantum
algorithms.[116] Here, the color center acts as an optically active interface
qubit and the nearby spins form weakly interacting ancilla qubits, e.g. 13C
and 14N nuclei for the NV. In DD spectroscopy, the electron spin operates as a
quantum sensor to resolve the nuclear-spin environment.[117, 118, 119]
Coherent coupling between spins can be mediated through microwave pulse
sequences tuned to the respective hyperfine coupling strengths [120, 121].
Then DD sequences can be applied in order to decouple the target spins from
the remaining spin bath. Despite increased quantum memory times and
facilitating quantum operations in advanced quantum protocols,[29, 33, 38]
controllable coupling to nuclear ancilla qubits is used for nuclear-assisted
high-fidelity electronic spin readout [122, 123, 124] and leads to enhancement
in sensing protocols.[125, 126]

Recently, a method based on time-asymmetric Hahn echo sequences has been
proposed for noise spectroscopy with reduced experimental times and, in turn,
more robustness against drifts and accumulation of pulse imperfections
compared to DD schemes.[127]

###  III.3 State-of-the-art microwave control experiments

The NV is famous for its long spin coherence times. Even at room temperature,
spin coherence in isotopically purified diamond of $>$1 ms for the electron
spin [128] and $>$1 s for the 13C nuclear spin [129] could be preserved. At
cryogenic temperatures of 3.7 K, electron spin relaxation times $T_{1}$ of one
hour and spin coherence times $T_{2}>1$ s using tailored DD sequences [23]
have been demonstrated. For NVs coupled to nanocavities, spin coherence times
exceeding 200 µs have been shown.[130, 131] For NVs, microwave $\pi$-pulse
fidelities of more than $\geq 99\%$ are reported.[33]

Based on magnetic dipolar coupling of NV electron spins, two qubit NV-NV
entanglement has been demonstrated at room temperature.[132, 111] Moreover, in
a ten-qubit register consisting of one electron spin and nine nuclear spins, a
single-qubit state was preserved for more than 75 s by using DD sequences on
the electron spin and selective phase-controlled driving of the nuclear spins
[29] (see Figure 3(a)-(c)). Also entanglement between 13C spin-pairs
controlled by an nearby NV has been demonstrated, emphasizing the potential of
nuclear spins as qubits.[133]

Figure 3:  Micrwoave control and coherence of spin qubits. Top row: Ten-qubit
spin register with an NV as central qubit. (a) Illustration of a multi-qubit
register constituted by an NV electron and 14N nuclear spin coupled to eight
13C nuclear spins by two-qubit gates. (b) In nuclear spin spectroscopy, the NV
electron spin is used as a quantum sensor of its nuclear environment. Here,
the electron spin is prepared in a superposition state and a DDrf gate
(controlled $\pm\pi/2$-rotation) is applied for different frequencies. If
resonant with a nearby nuclear spin, the selective interaction becomes
apparent as a dip in the coherence of the electron spin. (c) To confirm that
arbitrary quantum states can be protected, the six cardinal states are
prepared and the average state fidelity under DD is recorded. Here the decay
curves for the 14N nuclear spin are shown for varying number of DD pulses
$\alpha$. With 256 pulses, a state fidelity exceeding the classical memory
bound of 2/3 at a time of 75.3 s is measured. Bottom row: Control of an SnV
spin qubit. (d) Schematics of the experimental setup using confocal microscopy
at 1.7 K. An SnV spin qubit is incorporated in a diamond nanopillar. Pulsed
microwave control is achieved via a wire bond across the sample 60 µm apart
the qubit. The $\braket{100}$ diamond is oriented along the z-axis of the
vector magnet. (e) Gate characterization for quantification of spin control
fidelities is performed by applying $N$ gates to the qubit, denoted as unitary
rotation $U_{i}$ involving only $\pi$-pulses or a random Clifford gate. Data
from the first 30 gates is fitted to the function $\alpha F^{N}$. A
$\pi$-pulse fidelity of $F=99.51\%$ with 54 ns pulses, and an average random
Clifford fidelity of $F=95.04\%$ with 52 ns pulses have been extracted. (f)
The spin coherence time is measured using different DD sequences. Here a
$T_{2}$ time of 650 µs could be preserved by applying a XY16 protocol (top).
Considering a XY protocol, the coherence time scales with the number of
applied decoupling pulses according to $aN^{\chi}$ with $\chi=0.505$ (bottom).
Figures (a)-(c) are rearranged and adapted from Bradley et al. [29] under the
terms of the Creative Commons Attribution 4.0 International (CC BY 4.0)
license. Copyright 2019, American Physical Society. Figures (d)-(f) are
rearranged and adapted from Rosenthal et al. [134] under the terms of the
Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright
2023, American Physical Society.

The first demonstrations of microwave control of G4V color center states have
been achieved with the ‘lighter’ defect elements, the SiV and GeV color
centers, at a few K temperatures.[135, 103] In this temperature regime,
phononic interactions limit the coherence time. Therefore, further
investigations were performed at 100-400 mK temperatures and moreover DD
sequences (CPMG) have been applied to extend $T_{2}$ times to 13 ms for an
SiV.[28] Here, magnetic field noise from the nuclear spin environment was
identified as the main decoherence channel. A recent work involving the GeV at
300 mK benchmarks the state-of-the-art G4V spin control performance using
driving $\pi$-pulses with a duration of 77 ns, which yields T2,CPMG of 24
ms.[27] In particular in dilution refrigerators with limited cooling power,
drive-induced heat load needs to be considered carefully. When sequences with
fast and large amplitude $\pi$-pulses are applied, coherence times may be
limited.[136]

Table 2: Selection of reported electronic spin qubit coherence times of color centers in diamond in their negative charge state. In the referenced examples, for the NV, the spin qubit is encoded in the $\ket{m_{s}=0}$ and $\ket{m_{s}=-1}$ states and for the G4Vs in the lower branch ground states $\ket{1,\downarrow}$ and $\ket{2,\uparrow}$. Color center | Temperature | $T_{2}^{\star}$ | $T_{2,\rm Hahn}$ | $T_{2,\rm CPMG(N)}$ | $T_{2,\rm XY(N)}$
---|---|---|---|---|---
NV[137] | 100 mK | 160 µs | 1 ms | 0.56 s (16k) | 1.58 s (10k) @ 3.7 K [23]
SiV |  |  |  |  |
Bulk [135] | 3.6 K | 115 ns | – | – | –
Bulk [28] | 100 mK | 1.5-13 µs | – | 13 ms (32) | –
GeV |  |  |  |  |
Solid immersion lens [27] | 300 mK | 1.43 µs | 440 µs | 24 ms (8) | 18 ms (8)
SnV |  |  |  |  |
Nanopillar [134] | 1.7 K | 396.6 ns | 170 µs | – | 650 µs (16)
Membrane [138] | 1.7 K | 2.5 µs | 100 µs | 1.57 ms (128) | 223 µs (18) @ 4 K

Coherent microwave control of the SnV spin qubit encoded in the lower ground
state branch was also demonstrated (see Figure 3(d)-(f)) with $\pi$-pulse
fidelities of 99.51% [134] and 99.36%.[138] As an important pre-requisite for
high-fidelity microwave operations for the G4Vs based on heavier group IV
defects, i.e. tin and lead, applying strain to the emitters is necessary.
Strain introduces orbital mixing and suppresses dephasing processes at
temperatures $>4$ K. The interplay of strain and spin-orbit coupling is a
topic of ongoing investigations. Recent theory works identify regimes to
suppress phonon-mediated decoherence by changing magnetic-field and strain
bias to allow higher temperature operation,[97] and there are indications that
for certain magnetic field configurations, strain is not only unnecessary but
can limit the efficiency of the coherent control.[139] Finally, the larger
dopants are promising candidates for increased coherence times at temperatures
in the K regime. For example, the PbV center is predicted to exhibit a spin
coherence time on the order of a millisecond at 9 K.[61]

While the NV’s ($S=1$) resonance condition is in first order sensitive to the
particular coupling terms between two spins, G4Vs ($S=1/2$) couple only weakly
to nuclear spins. However, initialization and read-out of a 13C nuclear spin
was demonstrated for the first time by using the SiV.[140, 141] In addition,
nuclear spins of G4V dopant isotopes can be accessed (even at zero-magnetic
fields).[31, 98]

The reported microwave spin coherence times for the different color centers
are listed in Table 2. For the NV, the spin qubit encoding is usually
implemented in the ground state spin states $\ket{m_{s}=0}$ and
$\ket{m_{s}=\pm 1}$. For the G4V, in the referenced examples, the lower branch
ground states $\ket{1,\downarrow}$ and $\ket{2,\uparrow}$ have been used.

##  IV Optical control

Qubit state initialization via optical pumping and optical single-shot readout
are key operations in quantum control protocols, such as spin-photon
entanglement generation. In fundamental research, optical control is performed
in a confocal microscopy setup. At low temperatures, the spin-selective
optical transitions can be spectrally resolved and addressed. While off-
resonant excitation or optical pumping resonant to the color centers’ neutral
charge state can be used for re-initialization in their negative charge
state,[142, 143] here we present resonant excitation and Raman schemes for
optical control between ground and excited states for deterministic photon
retrieval, and for optical control of the ground state spin.

For initialization and single-shot readout of spin qubits, resonant excitation
schemes are applied and the state-dependent fluorescence is recorded. In
particular, for entanglement generation the coherent photons emitted into the
ZPL are collected. Separation techniques of excitation laser light and single
signal photons are cross-polarization, temporal filtering, and spatial
filtering.

###  IV.1 Optical coherence and resonant control schemes

By the application of laser pulses resonant to the ZPL transitions (orbital
transitions), Rabi oscillations are driven through the interaction with the
electric field

$H_{\rm El.}=-\textbf{d}\cdot\textbf{E}\,,$ (4)

leading to the generalized Rabi-frequency

$\Omega^{\prime}=\sqrt{\Omega^{2}+\Delta^{2}}\,,$ (5)

where $\Omega=\textbf{d}\cdot\textbf{E}/\hbar$ is the resonant Rabi frequency,
d corresponds to the transition dipole moment, E to the electric field vector,
and $\Delta$ to the detuning from the transition resonance.[144] From a Rabi
measurement, the corresponding $\pi$-pulse parameters, i.e. excitation power
and pulse duration, can be directly extracted. The pulse area is defined by
$\Theta=\int\Omega(t)\,$d$t$.

In general, the optical coherence is limited by spontaneous emission, phonon-
induced relaxation processes, spin-mixing and spectral diffusion. Depending on
the color center and type of sample, the impact of these processes will be
different. The decoherence mechanisms prevent high initialization and qubit
gate fidelities.

Figure 4:  Optical control. (a) Simplified level scheme of the NV color center
showing the optical $E_{x}$ and $A_{1}$ transitions. Single shot readout is
performed by resonant excitation of the $E_{x}$ transition (left solid red
line) which results with large probability in photon emission (left dashed red
line). With low probability, spin-mixing occurs (light yellow dashed line).
For charge state initialization in the ground ${m_{s}=0}$ spin state, the
$A_{1}$ transition is optically pumped. Non-spin conserving transitions via
the metastable singlet states (infared, not shown here) enable spin state
initialization with high fidelities ${>99.7\%}$. (b) top: Pulse sequence for
electronic spin readout. After charge state initializtaion (not shown) either
the $A_{1}$ or $E_{x}$ transition is optically pumped for initialization in
the ${m_{s}=0}$ or ${m_{s}=\pm 1}$ spin state. After optional microwace
manipulation, the spin state is readout by a laser pulse resonant to the
$E_{x}$ transition. bottom: Photon count statistics integrated over 10.000
measurements during an electronic spin readout pulse with 100 µs duration
after initialization in the ${m_{s}=\pm 1}$ (red) or ${m_{s}=0}$ (grey, inset)
spin state. (c) Optical quantum control of the spin levels through a lambda
configuration. When ${\Delta_{\rm two-photon}=0}$, i.e. two-photon resonance
is achieved, a coherent population transfer via the excited state without ever
occupying it (Raman scheme) can be carried out. When ${\Delta_{\rm Raman}=0}$,
the coherent population trapping (CPT) effect occurs involving the ground
state (spin) levels and inhibits excitation processes. The linewidth of the
two-photon resonance as well as the efficacy of the population trapping
(fluorescence contrast) can be analyzed to acquire the rates of the incoherent
processes that affect the driven levels. (d) An example CPT measurement with a
SiV color center. One of the lasers is stabilized on the B4 transition and a
second laser frequency scans around the B3 transition. When both lasers are in
resonance, the fluorescence significantly reduces. A characteristic CPT dip is
obtained from this measurement. By varying the laser powers, it is possible to
estimate the dip’s linewidth at zero power, which corresponds to the dephasing
rate of the investigated levels. The transition labels in the inset are
adapted to the level scheme in (c). Figure (b) is adapted and reprinted from
Robledo et al. [145] with permission. Copyright 2011, Nature Publishing Group.
Figure (d) is reprinted from Rogers et al. [146] under the terms of the
Creative Commons Attribution 3.0 International (CC BY 3.0) license. Copyright
2014, American Physical Society.

For the NV’s $E_{x}$ transition connecting the ground and excited $m_{s}=0$
states, more than 10 oscillation cycles have been resolved at an excitation
power of 38 µW with Rabi frequencies of $\Omega=2\pi\times 410$ MHz
corresponding to a $\pi$-pulse duration of 1.2 ns duration.[147] In early
works, paving the way for the first remote spin-spin entanglement
demonstration based on NVs,[36] high fidelity spin qubit state preparation,
manipulation, and readout have been demonstrated.[145] State initialization in
the $m_{s}=0$ ($m_{s}=\pm 1$) spin ground state can be achieved by resonant
optical pumping of the $A_{1}$ ($E_{x}$) transition between the ground and
excited $m_{s}=\pm 1$ ($m_{s}=0$) spin states with a high (low) probability of
relaxation via the non-spin conserving metastable infrared singlet state
(spin-mixing in the excited states), as shown in Figure 4(a). Here, a
preparation error of the $m_{s}=0$ ground state of $(0.3\pm 0.1)\%$ was
extracted. The performance of resonant single-shot readout is characterized
after spin state initialization by resonant excitation of the $E_{x}$
transition. While for initialization in the $m_{s}=\pm 1$ spin state,
negligible numbers of detected photons are expected, for the initial $m_{s}=0$
state on average six photons were detected during the readout window with
optimal duration of 40 µs, resulting in an average readout fidelity of $F_{\rm
avg}=(F_{m_{\rm s}=0}+F_{m_{\rm s}=\pm 1})/2=(93.2\pm 0.5)\%$ (see example
photon count histograms with 100 µs readout window in Figure 4(b)). For
entanglement generation, usually optical $\pi$-pulses resonant to the $E_{x}$
transition with a duration of about 2 ns are used. In advanced protocols after
delivery of entanglement, the NV state is readout in a particular basis with
fidelities of $>95\%$ for the bright $m_{\rm s}=0$ ground state and with
$>99.5\%$ for the $m_{\rm s}=\pm 1$ dark state employing methods as described
above.[34] In a low-strain NV environment, spin-mixing and phonon-induced
transitions within the excited states while driving the $E_{x}$ transition are
significantly reduced. A completely different approach for spin-state readout
is based on photoelectric readout by spin-to-charge conversion.[148, 149] This
method allows for suppression of spin-flip errors and support fully integrated
quantum devices. Single-shot averaged readout fidelities $>95\%$ with a 10 µs
spin-to-charge conversion duration have been demonstrated for an NV electron
spin in the presence of high strain and fast spin-flip process.[148]

Also for G4V color centers, optical spin state initialization and readout have
been demonstrated. To access the electronic spin optically, an external
magnetic field with slight misalignment with respect to the color center’s
symmetry axis is applied, inducing spin state mixing.[150] At low temperatures
on the order of hundreds of mK, the lower orbital branch of the ground state
is effectively decoupled from the phonon bath, i.e. thermalization is
inhibited, resulting in an orbital polarization in the lower branch.
Initialization can be performed by spin pumping: by selective excitation of
one of the spin states targeting any of the excited states will populate the
other spin state due to radiative decay from the excited state to the other
spin state when a misaligned magnetic field is applied. If the highest lying
excited state branch is targeted, fast phononic decay processes result in a
similar decay chain and therefore also allow initialization. When the spin is
fully initialized, the excitation field cannot drive transition cycles anymore
and the color center is in a dark state. It has been demonstrated that, e.g.
by driving the D1 transition, a G4V color center can be initialized into the
spin-up ground state $\ket{2,\uparrow}$ [135] (see level scheme in Figure
2(c)).

The initialization fidelity can be extracted by recording the time-resolved
fluorescence during the excitation pulse. The spin polarization is determined
by the ratio of the optical pumping rate to the spin relaxation rate.[103] By
applying a second laser pulse resonant to the first initialization pulse and
by measuring the recovery of the fluorescence, the spin can be read out. A
comparison between the fluorescence peak areas of the initialization and
readout pulses yields the spin polarization.[135] High fidelity spin state
single-shot readout can be achieved when high, spin-flip free, cyclicity
($\propto\langle i|{\bf d}\cdot{\bf E}|j\rangle$ with $|i\rangle$, $|j\rangle$
being the involved levels) of the spin conserving transition is maintained,
e.g. driving transition $\sim 10^{5}$ times. The required cyclicity is
obtained by aligning the angle of the applied magnetic field with the G4Vs
symmetry axis.[28] However, even for large magnetic field misalignment of
$54.7\degree$ between the magnetic field and the symmetry axis of the defect,
a reasonable cyclicity of spin-conserving transitions could be achieved,
allowing for a readout fidelity of $74\%$, which surpasses the threshold of
$50\%$ for meaningful readout implementations.[151] Here, the spin was
initialized via the spin-conserving $B2$ transition and for readout a laser
pulse resonant to the $A1$ transition was applied.

Rabi oscillations and $\pi$-pulse excitation with G4Vs have been
demonstrated.[152, 153] At large driving powers $P/P_{\rm sat}=364$, an
optical $\pi$-rotation was performed by a $\pi$-pulse of 1.71 ns duration
(much faster than $T_{1}$) with $77.1\%$ fidelity.[153] However, the spin
state degeneracy was not lifted by applying a magnetic field.

In Table 3 selected reported spin-state initialization and readout fidelities
for the SiV, GeV, and SnV are summarized.

In addition to the presented resonant methods, a novel coherent excitation
scheme, called Swing-UP of the quantum EmitteR population (SUPER), [154] has
been developed. By using high intensity, two-color, ultrashort, and non-
resonant pulses, it is possible to rotate the qubit vector around two
different Bloch sphere axes simultaneously such that their combined effect
results in a full inversion. This scheme has been demonstrated with quantum
dots [155, 156] and recently, for the first time with color centers in diamond
[157]. The non-resonant nature of the scheme enables spectral filtering of the
excitation field and detection of the coherent single photons in the ZPL.

Table 3: Selection of reported spin-state initialization and readout fidelities for G4Vs in their negative charge state based on optical control methods. The magnetic field angle describes the angle between the applied magnetic field direction and the G4V symmetry axis. The labeling of the involved transitions matches the labeling in Figure 2 and Figure 4.  | SiV | GeV | SnV
---|---|---|---
| [135] | [28] | [103] | [27] | [151] | [138]
Settings |  |  |  |  |  |
Temperature | 3.6 K | 100 mK | 2 K | 300 mK | 2 K | 1.7 K
Magnetic field (strength, angle) | $109.5\degree$ | 2.7 kG, $<0.5\degree$ | 0.3 T, $54.7\degree$ | 0.1 T, $0\degree$ | 0.2 T, $54.7\degree$ | 81.5 mT, $0\degree$
Initialization |  |  |  |  |  |
Transition | $D1$ |  $A1$, $B2$ | – | $B2$ | $B2$ | $A1$
Pulse duration | 500 ns | 30 ms | – | 1 ms | 200 µs | 24.2 µs
Spin polarization | $85\%$ | – | $59\%$ | $98\%$ | $98.9\%$ | $98.82\%$
Readout/gate operation |  |  |  |  |  |
Transition | $D1$ | $A1$ | – | – | $A1$ | –
Pulse duration, readout window | 500 ns, 30 ns | 150 ms, 20 ms | – | – | 200 µs | –
Fidelity (average) | – | $89\%$ | – | – | $74\%$ | –

###  IV.2 Coherent Raman control

While microwave control methods are effective for high-fidelity quantum
operations, they face challenges such as limited scalability,[158]
heating,[159] and a limited frequency bandwidth (1 GHz - 300 GHz).[160] This
has led to exploring optical control methods as an alternative for coherently
controlling quantum states. Since the energy difference in a spin qubit
typically doesn’t fall within the optical spectrum, optical control must be
engineered using so called Raman control techniques, which are sometimes
summarized under the process of stimulated Raman adiabatic passage
(STIRAP).[161] These schemes can be used for systems that have a lambda level
structure, where two chosen long-lived quantum states are optically coupled to
an energetically higher excited state with a comparatively short lifetime as
shown in Figure 4(c). In reality, states and couplings outside the lambda
level configuration can interfere with the desired Raman control operation.
Because any other unwanted optical interaction may introduce errors in the
quantum gate operations, they have to be carefully accounted for when
designing such a control method.[162, 163]

So called coherent population trapping (CPT) schemes [164] are a subset of
STIRAP. In CPT, two optical transitions within a lambda system are driven
simultaneously by two lasers, fulfilling the two-photon resonance condition
$\Delta_{\rm two-photon}=0$. Through a quantum interference effect, one of the
adiabatic eigenstates of the driven system no longer contains the short-lived
excited state. This allows for the optical control of the spin qubit without
any involvement of the short-lived excited state, which would otherwise reduce
the fidelity of the desired operation.

In the rotating wave approximation, the lambda system’s Hamiltonian is
typically written as (using for example only the states
$|1,\downarrow\rangle$, $|2,\uparrow\rangle$, $|A,\downarrow\rangle$ as a
basis, (see Figure 4(c)) [161]:

$H(t)/\hbar=\begin{pmatrix}0&0&\Omega_{1}(t)/2\\\ 0&\Delta_{\rm two-
photon}&\Omega_{2}(t)/2\\\ \Omega_{1}(t)/2&\Omega_{2}(t)/2&\Delta_{\rm
Raman}\end{pmatrix}\,,$ (6)

where $\Omega_{1,2}$ is given by Equation (4), $d\rightarrow d_{1,2}$ are the
respective dipole transition operators (assumed to be real-valued), $\bf
E\rightarrow\bf E_{1,2}(t)$, which are the time-dependent amplitudes of two
lasers with central frequencies $\omega_{1,2}$. The two-photon detuning is
given by $\Delta_{\rm two-photon}=\omega_{1}-\omega_{2}$ and the single photon
detuning is $\Delta_{\rm Raman}=\omega_{\rm A1}-\omega_{1}$ with $\omega_{\rm
A1}$ as the optical transition frequency. The authors of [162] used the
Hamiltonian in Equation (6) as a basis for theoretically designing fast Raman
gates which act on the ground state qubit of the SiV and SnV. They simulated
gates exceeding a fidelity of 99$\%$ for pulses with $20$ ps duration. The
method was further refined in [163] addressing leakage errors. Another
theoretical proposal for a double resonant fractional STIRAP was proposed in
[165] which employs a complex sequence of pulses required to remove the state-
dependent outcome of the single resonant pulse sequence.

For a large single photon detuning $\Delta_{\rm Raman}$ and $\Delta_{\rm two-
photon}=0$, Equation (6) can be approximated by adiabatic elimination of the
excited state to become [161]

$H_{\rm eff}(t)/\hbar=\begin{pmatrix}0&\Omega_{\rm eff}(t)/2\\\ \Omega_{\rm
eff}(t)/2&\Delta_{\rm eff}(t)\end{pmatrix}\,,$ (7)

where

$\displaystyle\begin{aligned} &\Omega_{\text{eff
}}(t)=-\frac{\Omega_{1}(t)\Omega_{2}(t)}{\Delta_{\rm Raman}}\,,\\\
&\Delta_{\text{eff
}}(t)=\frac{\Omega_{2}(t)^{2}-\Omega_{1}(t)^{2}}{2\Delta_{\rm
Raman}}\,.\end{aligned}$ (8)

The time-dependent effective Hamiltonian $H_{\rm eff}(t)$ generates rotations
of the qubit on the Bloch sphere. By extending Equation (6) to include the
relative phase of the two Raman lasers, rotations around the $x$ and $y$-axis
of the Bloch sphere can be engineered and therefore arbitrary SU(2)
operations.[166]

CPT may also be employed spectroscopically to investigate the decoherence time
of a qubit. For this purpose, one laser is set to match one of the transitions
($\Delta_{\rm Raman}=0$) and then $\Delta_{\rm two-photon}$ is varied by
tuning the other laser’s frequency while the spontaneous emission
(fluorescence) of the system is monitored. When $\Delta_{\rm two-photon}=0$
(two-photon resonance), a noticeable decrease or ‘dip’ in the fluorescence
signal occurs, because the system is driven into the dark state, where the
population is coherently trapped in the ground states. The linewidth and
contrast of the fluorescence dip depends on the coherence of the probed
levels, which can be determined by analyzing the dip’s width and depth.

The initial implementation of CPT for analyzing a diamond spin qubit,
specifically an NV color center, was reported in [99]. The authors found an
effective spin qubit ground state decoherence time of 833 ns in a temperature
range between 2 and 10 K. In later experiments with G4V color centers,
T${}_{2}^{*}$ dephasing times were found to be 45 ns at 4 K [150] for the
lower branch $\ket{1}$-$\ket{2}$ and 35 ns at 5 K [146] for the upper branch
$\ket{3}$-$\ket{4}$ SiV spin levels (Figure 4(d)), 19 ns at 2 K for the lower
branch GeV spin levels [103] and 5 µs at 1.7 K for lower branch SnV spin
levels.[151] One similar application of dephasing measurements with CPT showed
that applying strain can extend coherence times from 40 ns to 250 ns for a
lower branch SiV spin.[167] Other applications of this effect include
acquiring the CPT dip as a two-photon resonance identification method with
SnVs as an initial step for optical control.[168]

The first experimental demonstration of an optical STIRAP scheme with a color
center was performed with an NV [169] and involved controlling the m${}_{\rm
s}=\pm 1$ levels. This was a remarkable result as the splitting of these
levels is usually below 500 MHz under the typical magnetic fields used in
experiments, and are inaccessible by microwave generators. The required two
pulses were generated using acousto-optic modulators (AOM). Employing $\sim$
600 ns $\pi$-pulses, a T${}_{2}^{*}$ time of 1 µs at 5 K was measured. In
addition, adiabatic passage by engineering pulses with varying rise times was
also investigated. The fluorescence responses with relative pulse delays were
measured and found to be in agreement with simulated values. In [170] another
method was developed, which involved a single detuned pulse generated from an
electro-optic modulator (EOM). The pulse covered both frequency components due
to its broadband nature, enabling faster gates with a duration of 25 ns and a
fidelity of $>$85% at 7 K.

The STIRAP method was also applied to G4Vs. For example, the SiV ground state
orbital levels were controlled with an ultrashort 1 ps single broadband pulse
[152] with 30% fidelity at 5 K. The fidelity was limited by the experimental
constraints as the control pulse had to be applied after the SiV had already
been partially thermalized. In a later work,[115] spin control was
demonstrated at 40 mK by freezing out phonons. Two required frequency
components were obtained by sideband generation from an EOM. By cascading with
an AOM, 8 ns $\pi$-pulses were generated and T${}_{2}^{*}=29$ ns, T${}_{\rm
2,Hahn}=138$ ns were obtained. In the latest G4V optical control work,[168] an
SnV was controlled similarly employing EOM sideband generation. A 139 ns
$\pi/2$-pulse with 90.9% fidelity was generated. Higher rotations were not
achievable due to laser power limitations. Decreasing the detuning was not
possible either as both the gate fidelities and coherence times were limited
by the optical scattering from the excited state. A T${}_{2}^{*}=1.3$ µs,
T${}_{\rm 2,Hahn}=28.3$ µs, and T${}_{\rm 2,CPMG4}=300$ µs were obtained by
applying optical refocusing pulses.

An early review of optical control methods for the SiV can be found in ref.
[171].

###  IV.3 Spectral diffusion, coherent photons, and spectral control

For high-fidelity resonant optical initialization and readout necessary for
spin-photon entanglement, as well as to maintain optical coherence of the
emitted single photons, spectral stability is a crucial prerequisite. Optical
pulses with high energy and large power not only excite or re-initialize the
color center, but also lead to ionization of other defect states, i.e., nearby
defects in the bulk diamond or defect states on the surface of diamond
nanostructures (see Figure 5(a)). Random changes in the electrostatic
environment over time affect the ZPL transition energy via the DC Stark shift,
which in turn leads to spectral diffusion. Depending on the color center’s
crystallographic geometry and defect environment, an inhomogeneous broadening
of the ZPL resonance far exceeding the lifetime-limited linewidth can be
observed.

The DC Stark shift contributes directly to the Hamiltonian. It is defined as

$\Delta E_{\rm Stark}=-\Delta\textbf{d}(\textbf{F})\cdot\textbf{F}=-\Delta
dF-\frac{1}{2}\Delta\alpha F^{2}\,,$ (9)

where F is the electric field, and $\Delta d$ and $\Delta\alpha$ represent the
field-induced difference of the permanent electric dipole moment and the
polarizability between the ground and excited orbital states. The effect of
spectral diffusion is in particular detrimental for single photon emission
from solid-state quantum systems.

The NV exhibits a permanent electric dipole moment, which makes it in first-
order sensitive to external electric fields and susceptible to spectral
diffusion. Detailed theoretical work on the effect of electric fields on the
excited state manifold can be found in [74, 172]. In nanostructures,
inhomogeneous linewidths up to several GHz are observed.[173] However, the use
of native NVs in a nitrogen-rich substrate, advanced fabrication methods, and
specific control schemes based on resonant excitation enabled maintenance of
spectral stability over minutes and the emission of optically coherent photons
on a timescale of one second for NVs in nanostructures (NV-to-surface distance
of less than 125 nm) [24] (see Figure 5(b)).

Figure 5:  Spectral diffusion and spectral control. (a) Schematics,
illustrating the process of spectral diffusion of a spin qubit in a diamond
nanopillar. At rough surfaces, the density of defects (charge traps and
electron donors) is very large. Green laser irradiation leads to
photoionization of defects and moving charges, in turn causing fluctuations of
the electrostatic environment and spectral diffusion of the ZPL resonances (DC
Stark shift). (b) In photoluminescence excitation (PLE) measurements, the
spectral dynamics of the ZPL resonance is measured. While additional dim green
laser irradiation causes large spectral jumps of several GHz, red laser
irradiation only induces slow drifts of the ZPL resonance and spectral
stability can be preserved over minutes (no initialization pulses applied).
(c) Active tuning of the NV ZPL transition frequencies can be achieved by
applying an external voltage. Here, PLE spectra as a function of applied
voltage are shown. (d) The ZPL transition frequencies can be modelled with
good agreement. Figure (b) is adapted from Orphal-Kobin et al. [24] under the
terms of the Creative Commons Attribution 4.0 International (CC BY 4.0)
license. Copyright 2023, American Physical Society. Figures (c) and (d) are
adapted from Acosta et al. [174] under the terms of the Creative Commons
Attribution 3.0 International (CC BY 3.0) license. Copyright 2012, American
Physical Society.

G4V are inversion-symmetric and are therefore insensitive to first-order DC
Stark shifts. Moreover, they even show a suppressed response to second-order
effects.[90] (Nearly) transform-limited linewidths were demonstrated for
different color centers in several works,[102, 175, 53, 176] also in
nanodiamonds and nanostructures.[177, 178, 179, 104, 106, 180, 153] In a
recent work, even at temperatures above 10 K, transform-limited photon
emission was shown for the lead-vacancy color center. A model for decoherence
processes due to phonon-induced processes for different G4V color centers was
derived.[107]

Especially for quantum networks, indistinguishable photons are required for
quantum interference, and in turn, for spin-spin entanglement generation.
Usually, in a Hong-Ou-Mandel-type (HOM) setup,[181] two-photon interference on
a beam splitter, considered as a linear optical gate, is used for Bell-state
measurements. Photonic indistinguishability describing photons with an equal
quantum state relates to all optical degrees of freedom: frequency,
polarization, temporal mode, and spatial mode. The HOM visibility, often
quantified by the coherence time of the emitters, is a direct measure of
photon indistinguishability and determines the entanglement generation
fidelity. Theoretical predictions for the visibility and the output Bell-state
fidelity set by the single-photon characteristics of different emitters are
analyzed in detail in ref. [182]. In two-photon quantum interference
experiments with color centers in diamond, often the visibility is determined
by measuring the zero-delay second-order intensity correlation function for
aligned $g_{\parallel}^{(2)}(0)$ and orthogonal $g_{\perp}^{(2)}(0)$
polarization of the input modes

$V=1-\frac{g_{\parallel}^{(2)}(0)}{g_{\perp}^{(2)}(0)}\,.$ (10)

For NVs in microstructures for the first time in 2012, a time-resolved two-
photon interference contrast of $66\%$ has been demonstrated[183] and later on
even a visibility of up to $90\%$, applying a pulse train of 10 optical
$\pi$-pulses that excite both emitters.[34] For the SiV in bulk, a visibility
of $72\%$ [175] and for the SnV in a single-mode waveguide a visibility of
$63\%$ [153] was reported. Whereas in experiment, the visibility is not only
limited by non-perfect indistinguishability due to spectral broadening, but
additionally by contributions from a finite detector time resolution and
background events.

Measures that are taken to reduce spectral diffusion, are the improvement of
color center formation methods,[63, 64] i.e. using preferentially in-grown
defects or only shallow ion implantation and over-growth, as well as high-
pressure high-temperature annealing to reduce the number of vacancies in the
diamond lattice (see Section II.2). Moreover, micro- and nanofabrication
methods are optimized to reduce the number of surface defects.[184, 185]
Certain surface terminations preserve the spin coherence for shallow color
centers [186] and might also improve optical coherence. Active approaches to
mitigate spectral diffusion are adjusted excitation schemes [24, 151] and
active feedback schemes.[174] Furthermore, methods based on pulsed optical
control protocols have been proposed for suppressing spectral diffusion [187]
and even demonstrated for spectral shaping, for example, using SiC.[188]

The sensitivity of color centers to external electric fields [174, 91] and to
strain [189, 190, 191] can be used for active tuning of the emitter’s
resonances up to several GHz (see Figure 5(c) and (d), Section V).

###  IV.4 Optical micro- and nanostructures for controlled light-matter
interaction

In the past years, a multitude of diamond nanostructures evolved.[16] Often,
these structures aim for better photon collection efficiencies, i.e. lower
photon losses, which are currently an obstacle not only for fiber-based
quantum communication networks.[20] Amongst others, bullseye gratings,[192]
parabolic reflectors [54] as well as inverted nanocones [56] have been
investigated.

Nanocavities denote a special class of diamond nanostructures since they do
not only affect the embedded color center’s spatial emission characteristics,
but also the interaction strength between photons and the color centers
themselves.[193, 194] In general, the interaction cross section between light
and single (artificial) atoms is weak. It can be increased significantly
employing an optical cavity which surrounds the atom.[195] Thus, embedding a
diamond color center into a photonic crystal (PhC) cavity etched into the
diamond substrate, enhances on the one hand single-photon generation processes
and allows for optical nonlinearities (refer to Section IV.5) on the other
hand.

In the simplest case, an atom with a ground state $\ket{g}$ and an excited
state $\ket{e}$ is coupled to a cavity with a quality factor
$Q=\omega_{\mathrm{c}}/\left(2\kappa\right)$ derived from the cavity’s
resonance frequency $\omega_{\mathrm{c}}$ and its decay rate $\kappa$.
Furthermore, the cavity possesses a mode volume

$\displaystyle V=\frac{\bigintss_{\bf r}\epsilon\\!\left(\bf r\right)\left|\bf
E\\!\left(\bf r\right)\right|^{2}\mathrm{d}^{3}\bf
r}{\max\\!\left[\epsilon\\!\left(\bf r\right)\left|\bf E\\!\left(\bf
r\right)\right|^{2}\right]}$ (11)

defined as an integral over the cavity mode’s spatial electric field intensity
distribution $\left|\bf E\\!\left(\bf r\right)\right|^{2}$ multiplied by the
relative permittivity $\epsilon$ at position $\bf r$ and normalized to the
maximal field energy. Altogether with the vacuum permittivity $\epsilon_{0}$,
the reduced Planck constant $\hbar$, and the respective electric transition
dipole matrix element $\mu_{\mathrm{ge}}$, the single-photon coupling constant
becomes [196, 195]

$\displaystyle
g=\sqrt{\frac{\mu_{\mathrm{ge}}^{2}\omega_{\mathrm{c}}}{2\hbar\epsilon_{0}V}}\,.$
(12)

With the atom’s decay rate $\gamma$, the emitter-cavity cooperativity is
$C=g^{2}/\left(2\kappa\gamma\right)$. If $C\gg 1$, the single cavity-embedded
atom is able to interact nonlinearly with light [195]. Two regimes fulfill the
condition $C\gg 1$. Firstly, the weakly-coupled Purcell regime identified by
$\kappa/2\gg g\gg\gamma$ leads to Purcell enhancement: the cavity-incorporated
emitter’s emission rate is risen by the Purcell factor $F_{\mathrm{P}}$
compared to its bulk emission rate.[197] This also increases the emitter’s
homogeneous linewidth, which is advantageous to approach a spectral overlap
between multiple distinct emitters (see Section IV.3). Likewise, the emitter’s
Purcell factor-dependent Debye-Waller factor $DW$ rises. Secondly, the
strongly-coupled regime characterized by $g\gg\gamma,\kappa$ leads to new
effects facilitating optical spin control and readout as well as single-photon
nonlinear processes as detailed in the next section.

Nowadays, there are several realizations of diamond color centers coupled to
PhC cavities: NV centers have been coupled to two-dimensional [198, 199] and
one-dimensional nanobeam [130, 200] cavities with rectangular cross-sections.
For SiV centers, a two-dimensional cavity [201] and one-dimensional cavities
with a triangular cross-section [202, 141, 203] exist. Recently, one-
dimensional nanobeam cavities with a rectangular cross-section containing SnV
centers have been fabricated.[204, 205]

All of the aforementioned cavity designs rely on holes or elliptic features.
To extract part of the light $\kappa_{\mathrm{out}}$ stored in a PhC cavity
without introducing scattering losses $\kappa_{\mathrm{loss}}$, tapered
cavity-waveguide interfaces with slowly vanishing cavity features are employed
[203] (see Figure 6(a)). Such interfaces might suppress scattering losses even
better maintaining the cavity performance common for hole-based designs by
replacing the holes with corrugation-based ’Sawfish’ designs.[206, 207]
Considering the cavity’s total decay rate
$\kappa=\kappa_{\mathrm{out}}+\kappa_{\mathrm{loss}}$, the cavity-waveguide
coupling efficiency $\beta_{\mathrm{WG}}=\kappa_{\mathrm{out}}/\kappa$, and
the probability to emit a photon into the cavity mode
$\beta_{\mathrm{C}}=2C/\left(1+2C\right)$, the overall probability $\eta$ to
firstly emit a ZPL photon into the cavity and to secondly transfer it to the
attached waveguide becomes
$\eta=DW\times\beta_{\mathrm{C}}\beta_{\mathrm{WG}}$.[195]

Figure 6:  Jaynes–Cummings model and spin-dependent photon reflection. (a)
Model of a single (artificial) atom (orange sphere) coupled to a PhC cavity
(gray corrugated structure) leading to the Jaynes–Cummings Hamiltonian in
cavity quantum electrodynamics. The embedded atom possesses a ground state
$\ket{g}$ and an excited state $\ket{e}$ with a decay rate $\gamma$. It
couples to the (asymmetric) cavity with a rate determined by the single-photon
coupling constant $g$. The total cavity decay rate $\kappa$ consists of the
left and right (incoupling) cavity mirror’s respective decay rates
$\kappa_{\mathrm{l}}$ and $\kappa_{\mathrm{r}}$ as well as of the cavity
scattering losses $\kappa_{\mathrm{loss}}$. Thus, it becomes
$\kappa=\kappa_{\mathrm{l}}+\kappa_{\mathrm{r}}+\kappa_{\mathrm{loss}}$. (b)
Under an atom-cavity detuning $\Delta$, the SiV’s
$737\text{\,}\mathrm{nm}\text{/}$ optical transition (red) couples to a PhC
cavity. An external magnetic field $B_{\mathrm{ext}}$ splits the SiV’s spin-
conserving transitions (orange and blue) allowing for spin-selective
reflection at the cavity input interface: photons at a fixed frequency are
reflected (transmitted) if the SiV is in spin state $\ket{\uparrow}$
$\left(\ket{\downarrow}\right)$. (c) Cavity reflectivity
$R\\!\left(\Delta_{\mathrm{c}}\right)$ depending on the embedded SiV’s spin
state for a large (small) SiV-cavity detuning depicted in the left (right)
panel. With a large detuning $\Delta$, the transitions
$f_{\mathrm{\uparrow\uparrow^{{}^{\prime}}}}$ and
$f_{\mathrm{\downarrow\downarrow^{{}^{\prime}}}}$ can be clearly
distinguished. However, a small detuning $\Delta$ enables the highest contrast
(red shaded areas) in the spin-dependent cavity reflectivity. Figure (b) is
reprinted with permission from Nguyen et al.[141] Copyright 2019 American
Physical Society. Figure (c) is reprinted with permission from Nguyen et al.
[208] Copyright 2019 American Physical Society.

To eventually interface diamond waveguides with optical fibers by evanescent
coupling, a tapered fiber tip is brought in close contact with a diamond
waveguide which narrows down adiabatically.[209] Recent works apply these
fiber-optical interfaces due to their near-unity coupling efficiencies.[104,
141, 208, 203, 31] Similar adiabatic photonic interfaces between diamond and
silicon or aluminium nitride waveguides allow for hybrid integration of
diamond waveguides [17] and microchiplets [10] by pick-and-place. Hybrid
integration methods, more scalable than pick-and-place, comprise wafer bonding
and transfer printing.[39, 40]

###  IV.5 Cavity-mediated spin-photon interaction

(Artificial) atoms with an optical transition of frequency
$\omega_{\mathrm{a}}$ between a ground $\ket{g}$ and an excited $\ket{e}$
state coupled to (PhC) cavities do not only offer increased emission rates and
an improved photon collection efficiency. Under the term ‘cavity quantum
electrodynamics’, a multitude of effects occurring for cavity-coupled atoms
has been studied.[210] These effects involve the ability of a cavity to
reflect single photons impinging on its mirrors depending on whether an atom
is coupled to the cavity or whether the cavity is empty. This effect can be a
fundamental ingredient for spin-photon entanglement creation.

Off-resonance, an empty optical cavity reflects photons with unity
probability. However, if such a cavity is tuned to resonance, a Lorentzian dip
with a full width at half maximum of $2\kappa$ appears in its reflectivity.
With an atom coupled to the cavity, the reflectivity changes dramatically –
even in the Purcell regime. Applying input-output theory to the
Jaynes–Cummings Hamiltonian allows to derive an expression for the frequency-
dependent amplitude reflection coefficient $r\\!\left(\omega\right)$ of a
coupled atom-cavity system.[195, 211] This coefficient relates to the
reflectivity via
$R\\!\left(\omega\right)=\left|r\\!\left(\omega\right)\right|^{2}$. With the
cavity (atomic) detuning
$\Delta_{\mathrm{c}\left(\mathrm{a}\right)}\equiv\omega-\omega_{\mathrm{c}\left(\mathrm{a}\right)}$
from the probe frequency $\omega$, the cavity’s strong mirror decay rate
$\kappa_{\mathrm{l}}$, its weak (incoupling) mirror decay rate
$\kappa_{\mathrm{r}}$, and the atom’s excited state decay rate $\gamma$ (see
Figure 6(a)), it is

$\displaystyle
r\\!\left(\omega\right)=1-\frac{2\kappa_{\mathrm{r}}\left(\mathrm{i}\Delta_{\mathrm{a}}+\gamma\right)}{\left(\mathrm{i}\Delta_{\mathrm{c}}+\kappa\right)\left(\mathrm{i}\Delta_{\mathrm{a}}+\gamma\right)+g^{2}}\,.$
(13)

For an atom coupled to a cavity under zero atom-cavity detuning
$\Delta\equiv\left|\omega_{\mathrm{c}}-\omega_{\mathrm{a}}\right|=0$, the
Lorentzian dip in the cavity’s reflectivity with a minimum at
$\Delta_{\mathrm{c}}=0$ becomes replaced by two dips offset from the cavity
resonance by the coupling constant $\pm g$.[195] If only an optical transition
related to one of the cavity-embedded atom’s spin states couples to the
cavity, this effect enables mapping time-bin photonic qubits to the atom’s
spin degree of freedom by rendering the atom-cavity coupling spin-dependent
[208] (see Figure 6(b)). In contrast to empty cavities ($g=0$), the phase
$\phi\\!\left(\omega\right)=\arg\\!\left[r\\!\left(\omega\right)\right]$ of
the light reflected off the atom-coupled cavity does not undergo a phase shift
around the cavity resonance $\Delta_{\mathrm{c}}=0$. The phase difference
between photons reflected off an empty and an atom-coupled cavity can be
engineered to be $\mathrm{\pi}$.[195, 211] Particularly for a spin-dependent
atom-cavity coupling, this enables a further polarization-based route towards
entanglement generation.[212]

At a small atom-cavity detuning $\Delta<\kappa$, $R\\!\left(\omega\right)$
describes a Fano line shape, which yields the highest contrast in the cavity
reflectivity depending on the atom-cavity coupling. As soon as the detuning
rises further ($\Delta>\kappa$), the reflectivity contrast decreases, but the
atom-cavity system’s resonances become narrower. For a cavity-coupled SiV,
this regime leads to highly spin-conserving optical transitions, and thus
allows for efficient single-shot readout of the SiV’s spin configuration [208]
(see Figure 6(c)).

Employing diamond color centers as artificial atoms integrated into diamond
nanocavities and utilizing their spin-dependent optical transitions to spin-
selectively couple to the cavity, a couple of pioneering studies establish
cavity-mediated spin readout and control schemes. These schemes are exploited
amongst others for spin-photon entanglement generation. In 2018, two SiV
centers were coupled to a diamond nanocavity and the cavity’s reflectivity was
measured dependent on the spin state of one of the two SiV centers.[213]
Furthermore, controlling the SiV spin states, photon-mediated interactions
between the two SiV centers could be deterministically switched on and off. In
a follow-up work, a cavity-embedded SiV center was used to generate Bell
states between the SiV spin and a photon being reflected off the cavity, again
depending on the SiV spin state.[208] Also, the usability of such a system as
a node within a quantum network to implement a quantum repeater was
examined.[141] Rising the same system’s cooperativity to $C=$105(11)$$
recently allowed the demonstration of non-destructive single-shot cavity
reflection-based readout of the SiV spin state with a fidelity of about
$$99.98$\,\%$.[136] To map polarization-encoded photonic qubits to the spin
degree of freedom of a solid-state artificial atom, the PEPSI scheme was
proposed.[212] In a first implementation of a single-photon nonlinearity based
on a waveguide-coupled SnV, a spin-gated $$11(1)$\,\%$ decrease in the
reflectivity was demonstrated.[31] Moreover, recently, a two-node quantum
network in the Boston urban area could be realized, facilitated by reflection-
based spin-photon gates employing cavity-embedded SiV centers and time-bin
encoding.[37]

##  V Mechanical control

The interplay between spin and the crystal strain offers another avenue for
coherent control of spin qubits in diamond. The key mechanism behind the spin-
phonon interaction is the coupling of the color center to strain,[214, 215] or
in other words: to a displacement of the carbon atoms in the color center’s
environment. The displacement directly interacts with the electronic degree of
freedom through a distortion of the orbitals that are populated by the color
center’s electrons. In the presence of a magnetic field, dynamic strain caused
by either bulk oscillations, i.e. phonons, or macroscopic oscillations of the
diamond system, can then directly influence the electronic spin degree of
freedom as shown in Figure 7.

Figure 7:  Overview of the diamond-based surface accoustic wave (SAW) device
used in [215]. (a) A microwave signal, when applied to one of the interdigital
transducers (IDTs), leads to the generation of acoustic waves, due to the
piezoelectric effect in aluminum nitride (AlN). A focused laser beam is used
to interrogate silicon-vacancies (SiVs). (Inset) The scanning electron
microscope (SEM) image shows a pair of microwave transducers. The scale bar
shows 20 µm. (b) The electronic structure of the SiV when subjected to an
external magnetic field. The solid red arrow points to the optical transition
used for the spin’s initialization and reading, while the dashed red arrows
show other optical transitions. The blue arrow marks the acoustic transition
between the spin qubit’s two levels. (c) A pulse sequence for coherently
driving the SiV involving an acoustic pulse at 3.43 GHz, which matches the SiV
spin transition frequency. The SAW pulse is produced with a peak microwave
input of 2 mW. (d) Observation of the normalized population in the
$|1\downarrow\rangle$ state as the duration of the acoustic pulse is varied. A
theoretical model fit is depicted in red. (Inset) The Rabi frequency as a
function of the square root of the peak microwave input power shows the
anticipated linear trend. The Rabi frequency errors are around 0.1 MHz.
Figures (a)-(d) are rearranged and adapted from Maity et al. [215] under the
terms of the Creative Commons Attribution 4.0 International (CC BY 4.0)
license. Copyright 2020, Nature Publishing Group.

To leverage the interaction with the mechanical degree of freedom many
approaches have been explored. One such avenue involves the harnessing of
surface acoustic waves (SAWs),[216, 217, 215] which interact with shallow
implanted color centers. An example of a SAW device for control of an SiV is
shown in Figure 7a. The utilization of mechanical oscillators, such as bulk
mechanical oscillators,[218, 219] an oscillating cantilever,[214, 220, 221]
and diamond microdisk resonators [222] have also been implemented. A more
extensive description of mechanical spin interfaces and their properties can
be found in the topical reviews.[223, 224] Mechanical control has also been
achieved by magnetically linking the oscillation of a mechanical resonator to
NVs.[225, 226] For this review, we categorize it under microwave control since
the resonator’s motion mainly exposes the color center to a time-dependent
magnetic field.

The interest for exploring mechanical interactions with an optically active
spin system are wide ranging. It starts with understanding the fundamentals of
the spin-phonon coupling mechanism with the electronic spin of a color center,
which has direct implications for its coherence properties,[96, 97] their
coherent optical efficiency,[227, 228] and ensemble properties that can be
greatly affected by a dynamic strain environment.[229, 230] Furthermore, the
mechanical interaction can be used to couple a macroscopic mechanical degree
of freedom to a quantum degree of freedom, allowing for the study of quantum
mechanical oscillators and for creating non-classical states of these hybrid
systems.[231] Another interesting application for spin-mechanical hybrid
systems lies in the transduction of microwaves to the optical regime for
coupling superconducting qubits in a quantum network with the goal of scaling
quantum computers.[232, 233, 234] Phononic interactions can also be used to
create phononic quantum networks connecting spins.[235, 236] In recent
research,[221] a device named "Strain-Transduction by Resonantly Actuated
Integrated Nanoelectromechanical Systems" (STRAINEMS) was introduced. Its
primary purpose was to show how strain can be adjusted and controlled at the
chip level for color centers. This work serves as a foundational step towards
establishing that mechanical control can be scaled up for quantum information
processors.

The application that will take center stage here is the coherent control of
spin defects in diamond.[218, 215] Establishing such control will have larger
implications, due to the spin hyperfine interaction used for controlling e.g.
13C spins in the environment of the electronic spin.[237]

###  V.1 Mechanical control of G4Vs

Because the spin-strain coupling strength is quite large for G4Vs ($1.5$
PHz/strain, SiV [215]) compared to the NV ($5.46-19.63$ GHz/strain [214]),
mechanical control is an interesting avenue for efficient control of the G4V,
reducing for example heating during control sequences. The mechanical control
is facilitated through the interaction of a color center with strain. In [215]
the authors outline the coupling of a spin to a dynamical strain field to the
SiV, which can be generalized to other G4Vs. The Hamiltonian that best
captures the spin-strain interaction is

$H_{\rm G4V}=H_{\rm SO}+H_{\rm Z}+H_{\rm strain}\,,$ (14)

where $H_{\rm SO}$ is the spin-orbit interaction, $H_{\rm strain}$ is the
strain interaction, and $H_{\rm Z}$ the axial Zeeman Hamiltonian which
describes the interaction with a magnetic field. In the basis $|E_{\rm
gx},\downarrow\rangle,|E_{\rm gy},\downarrow\rangle,|E_{\rm
gx},\uparrow\rangle,|E_{\rm gy},\uparrow\rangle$ – a direct product of the
orbital degrees of freedom ($E_{\rm gx},E_{\rm gy}$) and the spin degrees of
freedom ($\downarrow,\uparrow$) – the spin-orbit and Zeeman term are given by
[101]

$\displaystyle H_{\rm SO}$
$\displaystyle=\frac{\lambda}{2}\sigma_{y}\otimes\sigma_{z}~{},$ (15)
$\displaystyle H_{\rm Z}$
$\displaystyle=\frac{\gamma_{s}}{2}B_{z}\mathbbm{1}\otimes\sigma_{z}+\frac{\gamma_{s}}{2}(B_{x}\mathbbm{1}\otimes\sigma_{x}+B_{y}\mathbbm{1}\otimes\sigma_{y})\,,$
(16)

where $\lambda$ is the the magnitude of the spin-orbit coupling, $B_{i}$ are
the components of the magnetic field in the internal reference frame of the
G4V,[101] $\sigma_{i}$ are the Pauli matrices, and $\gamma_{s}$ is the
gyromagnetic ratio. For the Zeeman term, the coupling between different
orbitals is neglected because of the quenching of the orbital interaction. The
electronic level structure of the SiV is shown in Fig. 7b, with the respective
optically and acoustically coupled levels.

The unperturbed G4Vs are part of the D3d symmetry group, coupling to phonons
with A1g and Eg symmetry:

$H_{\rm strain}=H_{\rm strain}^{\rm A_{\rm 1g}}+H_{\rm strain}^{\rm E_{\rm
g}}\,.$ (17)

The A1g representation of the strain tensor [101, 215] is given by

$H^{\rm A_{\rm 1g}}_{\rm strain}=\alpha\mathbbm{1}\otimes\mathbbm{1}\,,$ (18)

where
$\alpha=t_{\perp}(\epsilon_{xx}+\epsilon_{yy})+t_{\parallel}\epsilon_{yy}$,
$\epsilon_{ik}$ is a component of the strain tensor $\epsilon$ and
$t_{\perp/\parallel}$ can be determined experimentally. $H^{\rm A_{1g}}_{\rm
strain}$ induces a global shift of the energy values and do not couple the
spins. The Eg components, however, can induce spin mixing, in the presence of
an off-axis magnetic field. The representation of the Eg component of the
strain is

$H^{\rm E_{\rm g}}_{\rm
strain}=(-\beta\sigma_{z}+\gamma\sigma_{x})\otimes\mathbbm{1}\,,$ (19)

where $\beta=d(\epsilon_{xx}-\epsilon_{yy})+f\epsilon_{zx}$ and
$\gamma=-2d\epsilon_{xy}+f\epsilon_{yz}$. Again, $d,f$ are scaling factors,
which can be determined empirically. The magnitude of the spin-strain response
for an slightly off-axis magnetic field can be determined from $\Omega=\langle
E_{g+},\downarrow^{\prime}|H_{\rm strain}|E_{g-},\uparrow^{\prime}\rangle$,
where $|E_{g+},\downarrow^{\prime}\rangle$ and
$|E_{g-},\uparrow^{\prime}\rangle$ are the two lowest energy eigenstates of
$H_{\rm SO}+H_{\rm Z}$. It is assumed that $|B_{x}|,|B_{y}|\ll|B_{z}|$ acts as
a perturbation. In [215] the authors find

$\displaystyle|E_{g+},\downarrow^{\prime}\rangle\approx|E_{g+},\downarrow\rangle+\frac{\gamma_{s}}{2\lambda}(B_{x}+{\rm
i}B_{y})|E_{g+},\uparrow\rangle$ (20)
$\displaystyle|E_{g-},\uparrow^{\prime}\rangle\approx|E_{g-},\uparrow\rangle+\frac{\gamma_{s}}{2\lambda}(B_{x}-{\rm
i}B_{y})|E_{g-},\downarrow\rangle\,,$ (21)

where $|E_{g\pm}\rangle=|E_{gx}\rangle\pm{\rm i}|E_{gy}\rangle$ so that

$\Omega=-\frac{\gamma_{s}}{\lambda}(\beta+{\rm i}\gamma)(B_{x}-{\rm
i}B_{y})\,.$ (22)

The spin-strain Hamiltonian in the new basis
$|E_{g+},\downarrow^{\prime}\rangle,|E_{g-},\uparrow^{\prime}\rangle$ then is

$H=-\frac{\epsilon}{2}\sigma_{z}+{\rm Re}[\Omega]\sigma_{x}-{\rm
Im}[\Omega]\sigma_{y}\,,$ (23)

where $\epsilon=\gamma_{s}B_{z}/2$ are the eigenenergies of the non-
interacting system. A time-dependent strain field with $\beta(t)$ and
$\gamma(t)$ at the position of the color center, for example caused by surface
acoustic waves can then be used to induce rotations on the spin qubit’s Bloch
sphere, which are mediated by the generalized Rabi frequency $\Omega(t)$. In
summary, mechanical control of the spin qubit of a G4V in the presence of a
slightly off-axis magnetic field is possible, if a time-dependent strain field
interacts with the G4V. In Fig. 7c a pulse sequence is shown for the
initialization of the spin state followed by an acoustic control and readout
pulse. The obtained Rabi oscillations as a function of the acoustic pulse
duration can be seen in Fig. 7d.

The hyperfine interaction of the electronic spin with the nuclear spins of 13C
in the environment of the G4V, can be leveraged to extend the mechanical
control to the nuclear spins.[237]

###  V.2 Mechanical control of the NV

Similar to the G4Vs, the negatively charged NV spin can interact with a strain
field. In contrast to the G4V, the NV is a spin-1 system and has C3v symmetry,
which changes the structure of the interaction and therefore the details of
the mechanical control:[214]

$H/h=\underbrace{D_{0}S_{z}^{2}}_{H_{0}}+\underbrace{\gamma_{\mathrm{NV}}\bf
S\cdot\bf B}_{H_{\text{Zeeman
}}}+\underbrace{d_{\|}\epsilon_{z}S_{z}^{2}-d_{\perp}/2\left[\epsilon_{+}S_{+}^{2}+\epsilon_{-}S_{-}^{2}\right]}_{H_{\text{strain
}}}\,,$ (24)

where $D_{0}=2.87$ GHz is the zero-field splitting, $\gamma_{\rm NV}=2.8$
MHz/G the NVs gyromagnetic ratio, $h$ Planck’s constant, $\bf B$ the external
magnetic field, and $\bf S$ is the NV electron spin operator with components
$S_{x}$, $S_{y}$ and $S_{z}$, and $S_{\pm}=S_{x}\pm{\rm i}S_{y}$. The
Hamiltonian $H_{\rm strain}$ describes the coupling of the NV spin to lattice
strain $\epsilon_{j}$ along coordinate $j\in\\{[100],[010],[001]\\}$, where
$[ijk]$ are the lattice directions of the fcc-diamond lattice and
$\epsilon_{\pm}=-\epsilon_{y}\pm{\rm i}\epsilon_{x}$. The strain coupling
constants corresponding to strain longitudinal and transverse to the NV axis
are denoted by $d_{\parallel}=5.46$ GHz/strain and $d_{\perp}=19.63$
GHz/strain.

The influence of $\epsilon_{z}$ is a modification of $D_{0}$ and thus, only
alters the energy difference between the states $|\pm 1\rangle$ and
$|0\rangle$ (eigenstates of $\bf S$). On the contrary, with
$\epsilon_{x,y}\neq 0$, the states $|+1\rangle$ and $|-1\rangle$ can mix and
are therefore amenable to mechanical control.

An oscillating $B_{\perp}$ field can drive transitions between $|0\rangle$ and
$|\pm 1\rangle$, whereas $d_{\perp}$ can directly couple $|+1\rangle$ and
$|-1\rangle$, which are not directly coupled by $B_{\perp}$. This allows for
establishing direct transitions between all levels of the NV’s ground state
manifold.

##  VI Perspectives

High fidelity coherent quantum control is a necessary prerequisite for almost
all quantum applications, which can be roughly divided into two broad
categories: applications that do not require entanglement and those that do.
Sensing, for example, does not necessarily require entanglement, but can
greatly benefit from it.[7] For most future quantum information applications,
however, such as wide area quantum networks and quantum computing,
entanglement is the key enabling quantum phenomenon. Here we seek to provide a
non-extensive overview of some of the potential applications for which color
centers may produce the required entanglement either by exploiting
interactions with nuclear spins, electronic spins, phonons as well as photons.

As mentioned in previous sections, color centers such as the NV [33] and SiV
[136] have already made an impact in the realm of quantum networks, where one
of the key challenges is the distribution of entanglement across various nodes
in the network. An implementation of a network with diamond spins and the NV
is shown in Figure 8 which was adapted from [238]. Quantum networks have wide
ranging applications, some of them being the well known quantum key
distribution, blind quantum computing, worldwide timekeeping, navigation and
sensing, fundamental test of physics as well as the scaling of quantum
computers (see e.g. [239] and references therein).

Figure 8:  Distributed quantum information processing with diamond-based
spins: (a) Formation of multi-qubit nodes utilizing electron spins for
’communication’ (depicted in purple) and nuclear spins for ’data’ (shown in
yellow). These allow for complete control, with the electron spins’ ability to
coherently interact with photons facilitating the heralded creation of
photonic entanglement across distant nodes. The nuclear spins contribute to
the prolonged preservation and manipulation of the quantum information. (b)
Interaction between electron and nuclear spins: The nuclear spins’ rotation at
the Larmor frequency around an external magnetic field ${\rm\bf B}$ is
influenced by the electron spin’s presence. Variations in the rotation
frequency and axis occur based on the charge and spin states of the NV, which
include NV- (in purple, $m_{s}=\\{-1,0,+1\\})$ and NV0 (in orange,
$m_{s}=\\{-1/2,+1/2\\}$). (c) NV centre’s dynamical decoupling spectroscopy: A
specific resonance at 44.832 $\mu$s (highlighted in red) indicates a solitary
13C spin, characterized by hyperfine parameters $A_{\parallel}=2\pi\times
80(1)$ Hz and $A_{\perp}=2\pi\times 271(4)$ Hz. Another resonance at 44.794
$\mu$s (in black) relates to the collective 13C spin bath. The solid lines
represent a theoretical model (details in Supplementary Note of [238]). (d)
Nuclear spin’s intrinsic decoherence timescales for various electron states:
Solid lines indicate model fits, while dashed lines offer a visual guide. ’N’
signifies the count of spin-echo pulses. Error bars represent one standard
deviation. Figure and caption adapted from Bradley et al. [238] under the
terms of the Creative Commons Attribution 4.0 International (CC BY 4.0)
license. Copyright 2022, Nature Publishing Group.

For quantum network applications, spin-photon entanglement [35] is often the
most intense focus of research efforts: The stationary electronic spin hosted
by a color center either directly acts as a memory or as a quantum bus between
a longer lived memories made up of nuclear spins.[33] An excellent overview of
methods for establishing entanglement between stationary qubits and photons
can, for example, be found in the tutorial [240]. Here, the authors provide a
high level overview over various building blocks for entanglement protocols.

Once a flying qubit is entangled with the spin-qubit, spin-spin entanglement
is the next logical step, which was demonstrated in a range of seminal works
[36, 241, 242, 34, 33] with NVs at the core of the spin-photon interface.
Similarly, the SiV has made great strides towards quantum networks, from the
first demonstrations of a heralded entanglement scheme,[208] memory-enhanced
quantum communication [136] to demonstrating spin-spin entanglement in a
telecommunication network.[37] In the context of quantum networks, color
centers have also been proposed as transducers for scaling quantum computers
based on, for example, superconducting qubits.[234]

Color centers are of course not only a potential building block to scale
quantum hardware based on already established systems, but they are a viable
resource for quantum applications in their own right. The electronic spin
provides optical access to a register of nuclear spins, which makes the
electronic spin a quantum bus providing access to a small scale processing
node based on nuclear [29] or electronic spins.[243] Most recently a spin-
network of 50 nuclear spin has been mapped by the authors of [30] by using
spin-echo double resonance (SEDOR). Similarly SEDOR was used to map electronic
spins [244] in an NV’s environment. Both experiments show that the building
blocks for small to intermediate scale quantum processors based on color
centers in diamond are actively being developed.

Color centers are also viable for generating resource states [245, 246, 247,
248] for second and third generation quantum repeater architectures, which
make use of so called photonic graph states.[249, 20, 250] In the same vein
they can be used to generate the discrete variable cluster states, that have
been proposed as a resource for measurement based fault tolerant quantum
computing.[251, 252]

Long-term coherence maintenance in quantum memories or registers is another
goal, enabling permanent quantum state storage. This advancement will boost
quantum memories’ applications in quantum technologies, such as secure quantum
tokens [253] and quantum money,[254] which are more reliant on long coherence
times. Recent demonstrations of minute-long storage times [29] suggest that
permanent quantum state storage is becoming more feasible.

In conclusion, despite ongoing challenges, diamond color centers hold great
promise for quantum applications. This review has outlined various quantum
control approaches and examined their state-of-the-art performance. Coherent
manipulation of spin qubits in diamond enabled entanglement generation and
fundamental quantum network applications. Advancing these applications
requires integrated photonics to allow for device scaling while, at the same
time, preserving the optical and spin coherence properties. A major challenge
is the charge and spin noise affecting shallow color centers near surfaces,
causing spectral diffusion and spin dephasing. Another challenge is
experimental complexity; for instance, the well-established SiV center
requires operation at millikelvin temperatures in dilution refrigerators.
Heavier color centers like SnV and PbV could operate at cryogenic
temperatures, but their ability to generate spin-photon entanglement is not
yet proven. In summary, diamond color centers offer potential for both local
and remote quantum operations, leveraging both spin and photonic states. With
continued rapid progress, the implementation of large-scale quantum networks
and quantum information processing using color centers is a realistic goal.

Acknowledgements

The authors acknowledge funding by the European Research Council (ERC,
Starting Grant project QUREP, No. 851810), the German Federal Ministry of
Education and Research (BMBF, project DiNOQuant, No. 13N14921; project QPIS,
No. 16KISQ032K; project QPIC-1, No. 13N15858; project QR.X, No. KIS6QK4001),
and the Einstein Foundation Berlin (Einstein Research Unit on Quantum
Devices).

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