# Rational Design of Efficient Defect-Based Quantum Emitters

Mark E. Turiansky<EMAIL_ADDRESS>Materials Department, University of
California, Santa Barbara, CA 93106-5050, U.S.A.  Kamyar Parto  Department of
Electrical and Computer Engineering, University of California, Santa Barbara,
CA 93106-5080, U.S.A.  Galan Moody  Department of Electrical and Computer
Engineering, University of California, Santa Barbara, CA 93106-5080, U.S.A.
Chris G. Van de Walle<EMAIL_ADDRESS>Materials Department,
University of California, Santa Barbara, CA 93106-5050, U.S.A.

###### Abstract

Single-photon emitters are an essential component of quantum networks, and
defects or impurities in semiconductors are a promising platform to realize
such quantum emitters. Here we present a model that encapsulates the essential
physics of coupling to phonons, which governs the behavior of real single-
photon emitters, and critically evaluate several approximations that are
commonly utilized. Emission in the telecom wavelength range is highly
desirable, but our model shows that nonradiative processes are greatly
enhanced at these low photon energies, leading to a decrease in efficiency.
Our results suggest that reducing the phonon frequency is a fruitful avenue to
enhance the efficiency.

##  I Introduction

The generation of single photons is essential to power the second quantum
revolution and to realize the promise of the quantum internet [1, 2, 3].
Photons are natural carriers of quantum information. They have very weak
interactions with the environment and, thanks to decades of development in
fiber-optic technologies, can be transmitted over long distances with minimal
loss. An ideal single-photon emitter acts as a “photon gun”, producing single
photons on demand. Three properties should be optimized [4]: (i) brightness,
as quantified through the intensity and efficiency, (ii) photon purity, as
quantified through the photon autocorrelation function measured in a Hanbury
Brown-Twiss experiment, and (iii) indistinguishability, as quantified through
the ability of photons to interfere in a Hong-Ou-Mandel experiment. In
addition, if long-range transmission is required, the photon energies should
lie within telecom wavelengths, 1260 nm to 1675 nm (between 0.74 and 0.98 eV),
to take advantage of fiber optics [5].

Single photons in a well-defined quantum state can be produced in a variety of
ways; point defects embedded in a semiconductor or insulator are a
particularly promising platform. (We use the term “point defects” to refer to
both native defects, which are intrinsic to the lattice, as well as extrinsic
impurities or a complex of the two.) Point defects can be used as qubits,
quantum memories, or single-photon emitters, all essential components of
quantum networks, and have been demonstrated to operate even at room
temperature [6, 7, 8, 9].

Long-range networking using defect-based single-photon emitters has been
demonstrated [7, 10, 11], mainly relying on the nitrogen-vacancy (NV) center
in diamond [12, 13]. However it is known that the optical interface of the NV
center is not ideal. In particular, less than 3% of the emitted photons are in
the zero-phonon line (ZPL)—in other words, useful for quantum information—due
to coupling to phonons. This is a result of the interaction between the
electronic states of the defect with the diamond host lattice, referred to as
electron-phonon coupling. Electron-phonon coupling also broadens the ZPL,
leading to dephasing which reduces indistinguishability. Alternatives to the
NV center that have weaker electron-phonon coupling and stronger emission into
the ZPL, such as the silicon-vacancy (SiV) center, are being pursued [10, 13,
7, 9].

In addition to its impact on the ZPL, electron-phonon coupling has a second
effect, namely the introduction of nonradiative decay. After a photon is
absorbed at a quantum defect, the system is in an excited state. Ideally, it
will decay radiatively by emitting a photon, but it may also decay through
some other mechanism; nonradiative decay mediated by electron-phonon coupling
can lead to an alternative recombination channel that may dominate over the
radiative process.

The NV center produces photons in the visible spectrum; for many applications,
emission at lower energies is desirable. As noted above, telecom-wavelength
emitters are desirable for long-range networking. In addition, cavity coupling
is often required to enhance brightness [14], and high-quality cavities are
easier to fabricate at longer wavelengths. Numerous luminescent centers in
diamond have been observed and characterized [15], yet centers that produce
telecom-wavelength photons are rare. More generally, reports of longer-
wavelength single-photon emitters in any material are scarce, which is
surprising given the ubiquity of defects. In diamond, it has been previously
suggested that this may be due to strong electron-phonon coupling, which
causes nonradiative processes to dominate [16]. To our knowledge, no rigorous
investigation of this proposal or its extension to other materials exists.

Here we develop a model that captures the essential role that electron-phonon
coupling plays in point-defect-based single-photon emitters. Our results show
that nonradiative processes dominate at smaller transition energies and
indicate that obtaining a high-efficiency single-photon emitter at longer
wavelengths is indeed difficult. However, our model allows us to suggest
productive avenues for improving the efficiency; one example is by reducing
the average phonon frequency. We also assess several approximations commonly
employed in the literature surrounding the evaluation of emission rates. The
perspective provided by our model sheds new light on results in the existing
literature.

##  II Basic Properties

We consider point defects embedded in a semiconductor or insulator. Defects
often give rise to states that lie well within the band gap [6, 7]. For a
given charge state of the defect, these states are occupied with electrons,
and in some cases an electron can be excited from an occupied to an unoccupied
state of the defect, thus defining the ground state ($g$) and excited state
($e$) of a two-level system, separated by an energy $\Delta E$ [Fig. 1(a)].

Figure 1:  Energy level diagram for a transition between intra-defect states.
The ground $g$ and excited $e$ states are labeled. Orange corresponds to the
conduction band and blue to the valence band.

The transition energy $\Delta E$ of the emitter is an important factor in the
long-range transmission efficiency. A single-photon emitter that produces
photons within the telecom wavelength range of 1260–1675 nm ($\Delta E$
between 0.74 and 0.98 eV) could take advantage of fiber optics [5], with loss
values below 0.2 dB/km. Free-space communication is another option [17].
Several low-loss atmospheric windows exist; one commonly utilized window
covers wavelengths of 780–850 nm ($\Delta E$ between 1.46 and 1.59 eV).

An important issue for an ideal single-photon emitter is stability. For an
emitter based on point defects, one important form of stability is charge-
state stability: the ideal emitter avoids exchanging electrons with the
valence or conduction bands. Processes that result in an exchange of electrons
with the bulk bands are referred to as charge dynamics. Charge dynamics can
occur nonradiatively (through thermal ionization or capture processes) or
radiatively (through the absorption or emission of a photon). To avoid charge
dynamics, the deep defect states should be far from the band edges, for both
the ground and excited states. Such a system can be operated at elevated
temperatures, a feature highly desirable for quantum applications.

It is worth noting that a single-photon emitter based on point defects can
also be realized with involvement of a carrier in the valence or conduction
band, bound to the defect in an excitonic state [18, 19, 20], i.e., the
excited state involves an electron or hole in a hydrogenic wavefunction.
Alternatively, a bulk exciton (in which the electron and hole reside in band
states) can be bound to a point defect [18, 21, 22, 23]. Such systems are
inherently susceptible to charge dynamics since they arise from interactions
with the bulk states of the host material. The binding energy will determine
the overall stability, but such emitters typically need be operated at low
temperature.

Spin dynamics occur when the total spin of the defect changes, which is
another form of stability that needs to be addressed. Due to the weak nature
of the interactions involved in changing the spin of the defect, spin dynamics
can lead to the defect being in a dark, non-emissive state for long periods of
time. Thus spin dynamics can be detrimental to the efficiency of a single-
photon emitter and should be avoided, for instance by selecting a defect with
a level structure that precludes changes of spin. We note that spin dynamics
is not always harmful; there can be advantages to having a single-photon
emitter with a ground-state spin that can be manipulated through spin
dynamics. Indeed, this is an essential feature of the NV center: a transition
from the triplet to the singlet manifold enables optical manipulation of the
ground-state spin. In this paper, we focus on designing a highly efficient
emitter, and for that purpose we will assume spin dynamics are negligible.

Given the above definitions, we can address two of the three parameters for an
optimal single-photon emitter, namely purity and indistinguishability [4].
Photon purity is evaluated in a Hanbury Brown-Twiss experiment, in which the
photon autocorrelation function $g^{(2)}(\tau)$ is measured as a function of
the time delay $\tau$ between the two light beams [24]. At short time delays,
a single-photon emitter exhibits an “antibunching dip” with $1-g^{(2)}(0)$
quantifying the photon purity, which should be unity for a perfect single-
photon emitter. Our idealized defect with only two levels will exhibit unity
purity, assuming a single defect can be isolated.

Indistinguishability is the probability of two-photon interference in a Hong-
Ou-Mandel experiment [25]. Indistinguishability can be affected by spectral
diffusion, which causes the transition energy to vary due to fluctuating
charges in the vicinity of the emitter. Resonant or quasi-resonant optical
pumping is usually effective for reducing or eliminating spectral diffusion.
Here we assume our model defect to be an idealized two-level emitter that has
a fixed transition energy, and thus unity indistinguishability in the absence
of pure dephasing mechanisms. Pure dephasing occurs when the quantum state
picks up a phase from coupling to the local environment. A dominant source of
pure dephasing in solid-state systems is elastic scattering with acoustic
phonons [26, 27], which becomes negligible at low temperatures. Throughout
this work, we will assume low temperature, allowing us to assume unity
indistinguishability for our model system.

The remaining parameter, brightness, is the main focus of this paper and is
addressed in Sec. III.

##  III Model

###  III.1 Radiative Properties

The rate at which a transition from the excited state to the ground state
produces photons via the electric-dipole interaction is given by [28, 29]

$\Gamma_{\rm R}^{(0)}={\left(\frac{\mathcal{E}_{\rm
eff}}{\mathcal{E}_{0}}\right)}^{2}\frac{n_{r}\mu^{2}(\Delta
E)^{3}}{3\pi\epsilon_{0}c^{3}\hbar^{4}}\;,$ (1)

where $n_{r}$ is the index of refraction, which we will take to be 2.4, the
value for diamond [30], which is also close to the value for boron nitride,
silicon nitride, silicon carbide, and other insulator host materials. $\mu$ is
the transition dipole moment, which defines the strength of the transition.
Local-field effects describe the fact that the electric field at the defect
may be different from that in bulk, due to scattering of light at the defect
[28]. This is captured in the prefactor $\mathcal{E}_{\rm
eff}/\mathcal{E}_{0}$, which is the ratio of the effective electric field at
the defect to the bulk value. Various models [31, 28] have been proposed to
estimate this ratio; they produce a value larger than—but close to—one. We
will set $\mathcal{E}_{\rm eff}/\mathcal{E}_{0}$=1, a common assumption in the
literature [29].

The transition dipole moment $\mu$ plays a key role in determining the overall
radiative rate. The lower limit on $\mu$ (and therefore $\Gamma_{\rm
R}^{(0)}$) is zero, for a forbidden transition where the dipole moment is zero
by symmetry. An upper limit on $\mu$ can be estimated by introducing the
concept of the oscillator strength $f$, given by

$f=\frac{2m_{e}(\Delta E)}{e^{2}\hbar^{2}}\mu^{2}\;.$ (2)

The oscillator strengths for transitions to all possible final states for a
given initial state in the system must sum up to one (the Thomas-Kuhn sum rule
[28]). If we assume a single transition dominates the sum, then $f\approx 1$
and for $\Delta E=1$ eV, we find an upper limit on $\mu$ of $\approx$1.95 eÅ.
In practice, some of the largest transition dipole moments are found for
transitions between orbitals of $s$ and $p$ character. Such orbitals are
common in $sp$-bonded covalent materials and have a transition dipole moment
on the order of 1 eÅ. For example, the transition dipole moment for the carbon
dimer in hexagonal boron nitride (h-BN) is 1.06 eÅ [32]. The blue line in Fig.
2 depicts Eq. (1) evaluated for $\mu=1$ eÅ as a function of $\Delta E$.

Figure 2:  The radiative rate without phonons $\Gamma_{\rm R}^{(0)}$ (solid)
and with phonons $\Gamma_{\rm R}$ (dotted) for a transition dipole moment
$\mu=1$ eÅ (blue), 0.1 eÅ (orange), and 0.01 eÅ (green). The nonradiative rate
$\Gamma_{\rm NR}$ is shown by the dashed lines for different values of the
Huang-Rhys factor $S$ given by the color bar. For $\Gamma_{\rm R}$, the Huang-
Rhys factor is assumed to be $S=3$. Energies that fall within the range of
telecom wavelengths are shaded in pink, and energies that fall within the
free-space communication window are shaded in grey.

When symmetry or conservation rules lead to a forbidden electric-dipole
transition, magnetic-dipole transitions may be observed, as is commonly the
case for rare-earth impurities [33, 34]. While our focus is on electric-dipole
transitions here, magnetic-dipole transitions have a similar cubic dependence
on the energy [33], and many of our arguments can be applied to such
transitions with the magnitude of the radiative rate rescaled.

If this electronic system were isolated, its description would be complete.
However, the presence of the semiconducting or insulating host lattice
provides a phonon bath for the electronic states to couple to. Electron-phonon
coupling has two main effects on our system: (i) it gives rise to the phonon
sideband in the luminescence spectrum, and (ii) it provides nonradiative
pathways for energy to be dissipated.

###  III.2 Electron-Phonon Coupling

In a three-dimensional solid with $N$ atoms, there are 3$N$ vibrational modes
to couple to. Here we will study the interaction with a single, dispersionless
phonon mode with energy $\hbar\Omega$. A single-mode approximation has been
effectively used as the basis of first-principles calculations of luminescence
lineshapes [35] and radiative [36] and nonradiative capture rates [37, 38];
such calculations have achieved both a qualitative and quantitative
description of experimental observations [35, 39, 40]. A single-mode
approximation can thus be considered a realistic approximation to the
electron-phonon coupling problem. The single mode in this approximation is
known as the accepting mode [41, 42]. It is a local vibrational mode that is
not necessarily a normal mode of the system and is dominated by the motion of
the defect and its nearest neighbor atoms. Such a mode is thus highly
sensitive to the chemical nature of these atoms. Unless otherwise stated, we
will assume a phonon energy of $\hbar\Omega=100$ meV, which is common for the
first-row elements that are often present in materials that are used as hosts
for quantum defects.

####  III.2.1 Phonon Sideband

The effect of this phonon mode on the electronic system, within the single-
mode approximation, is shown in the configuration-coordinate diagram in Fig.
3(a). One consequence of the coupling between the electronic and vibronic
degrees of freedom is that the ground and excited states may not share the
same equilibrium geometry. We define the mass-weighted difference in atomic
geometries $\Delta Q$ as

${(\Delta Q)}^{2}=\sum_{I}M_{I}{\lvert{\bf R}_{I,g}-{\bf
R}_{I,e}\rvert}^{2}\;,$ (3)

where $I$ labels the atomic sites, $M_{I}$ is the $I$th atomic mass, and ${\bf
R}_{I,g/e}$ are the coordinates of the $I$th site in the ground ($g$) or
excited ($e$) state.

Figure 3:  (a) A configuration-coordinate diagram for the model system and (b)
a schematic photoluminescence lineshape function $L(E)$. Blue indicates the
excited state, and red indicates the ground state. We assume both states share
the same phonon frequency $\Omega$. The vibronic wavefunctions are shown
schematically. The ZPL energy $E_{\rm ZPL}$ and the average emission energy
$E_{\rm em}$, as defined in the text, are labeled in black. The Franck-Condon
energy $E_{\rm FC}$ describes the average energy emitted as phonons and is
written in terms of the Huang-Rhys factor $S$. We can see that the Huang-Rhys
factor is the average number of phonons emitted during the radiative process.

Energy must be conserved during the radiative transition. When all of the
energy is dissipated in the form of a photon (without phonons), the photon is
said to be emitted into the ZPL. $E_{\rm ZPL}$ is the energy of the emitted
photons and is given by

$E_{\rm ZPL}=\Delta
E+\frac{1}{2}\left(\hbar\Omega_{e}-\hbar\Omega_{g}\right)\;,$ (4)

where $\Omega_{g/e}$ are the vibrational frequencies of the ground ($g$) and
excited ($e$) states. When $\Omega_{g}=\Omega_{e}$, as we will assume here,
$E_{\rm ZPL}$ is identical to the energy separation of the ground and excited
states $\Delta E$.

A non-zero $\Delta Q$ means that phonons may be emitted during the radiative
emission process. We assume low temperature, such that only the vibrational
ground state is occupied in the initial state of the system. Since the phonons
have energy and energy must be conserved, the emitted photons occur at an
energy lower than $E_{\rm ZPL}$. The vibronic wavefunctions $\chi$ of the
ground and excited state are depicted in Fig. 3(a). The emission of phonons
during the radiative transition gives rise to the phonon sideband observed in
luminescence [schematically shown in Fig. 3(b)]. The average emitted photon
energy (including photons emitted into both the ZPL and the phonon sideband)
is given by

$\displaystyle E_{\rm em}$ $\displaystyle=\sum_{n=0}^{\infty}(\Delta
E-n\hbar\Omega){\lvert\braket{\chi_{e0}}{\chi_{gn}}\rvert}^{2}$ (5)
$\displaystyle=\Delta E-S\,\hbar\Omega\;.$ (6)

The second equality in Eq. (6) follows from the assumption that the phonon
frequencies are the same in the ground and excited state (see Appendix A). The
dimensionless parameter $S$ is the Huang-Rhys factor, defined as [37, 28]:

$S=\frac{1}{2\hbar}{(\Delta Q)}^{2}\Omega\;.$ (7)

The Huang-Rhys factor quantifies the strength of electron-phonon coupling and
can be interpreted as the average number of phonons emitted during the
radiative emission process. From Eq. (6) we can see that $E_{\rm em}$ must be
smaller than or equal to $\Delta E$, as expected. The total radiative emission
rate, including photons emitted into the phonon sideband, is given by (see
Appendix B):

$\Gamma_{\rm R}={\left(\frac{\mathcal{E}_{\rm
eff}}{\mathcal{E}_{0}}\right)}^{2}\frac{n_{r}\mu^{2}(\Delta E)^{2}E_{\rm
em}}{3\pi\epsilon_{0}c^{3}\hbar^{4}}\;,$ (8)

which is reduced from $\Gamma_{\rm R}^{(0)}$ [Eq. (1), the rate in the absence
of electron-phonon coupling] by a factor of $E_{\rm em}/\Delta E$.

$\Gamma_{\rm R}$ is shown in Fig. 2 for the case of $S=3$. For $\Delta E>1$
eV, $\Gamma_{\rm R}$ is close to $\Gamma_{\rm R}^{(0)}$, i.e., the impact of
phonons on the radiative rate is quite modest. At lower energies, however,
coupling to phonons severely reduces the radiative rate, and at $\Delta
E=S\,\hbar\Omega$, $\Gamma_{\rm R}$ drops to zero. This indicates the
transition from the Marcus inverted region ($\Delta E>S\,\hbar\Omega$) to the
Marcus region ($\Delta E<S\,\hbar\Omega$) [41], see Fig. 4. The existence of
these two distinct regimes was an important prediction of the seminal theory
of Marcus [43]. In the Marcus region, the overlaps between the vibronic
wavefunctions decrease, and luminescence is suppressed (as shown in Fig. 4).
The strong radiative transitions that we are interested in thus occur all in
the Marcus inverted region; this is indeed the most common scenario for
transitions at defects in the solid state (as opposed to electron transfer in
chemistry, which is what Marcus [43] focused on).

Figure 4:  A configuration coordinate diagram in the Marcus (a) inverted and
(b) normal regions. Blue indicates the excited state, and red indicates the
ground state. We assume both states share the same phonon frequency $\Omega$.
The vibronic wavefunctions are shown schematically. In the normal region,
luminescence is suppressed since the overlap between vibronic wavefunctions of
the ground and excited states is negligible.

It is common practice to approximate $\Gamma_{\rm R}$ by $\Gamma_{\rm
R}^{(0)}$: as we can see, this approximation is valid for $\Delta E\gg
S\,\hbar\Omega$, when the system is well within the Marcus inverted region.
This approximation also benefits from a fortuitous cancellation. As previously
mentioned, local-field effects are commonly ignored and tend to increase the
rate, while electron-phonon coupling reduces the rate. Some cancellation may
thus occur between these two effects, improving the agreement between
experiment and theory when both are neglected.

Photons that are emitted without emitting phonons correspond to the ZPL. These
are the photons that are in a well-defined quantum state useful for quantum
information applications. The fraction of photons emitted into the ZPL is
given by the Debye-Waller factor $e^{-S}$ [28], and the overall rate of
emission into the ZPL is

$\Gamma_{\rm ZPL}=e^{-S}\Gamma_{\rm R}^{(0)}\;.$ (9)

Equation (9) corresponds to the $n=0$ term in Eq. (21). The Huang-Rhys factor
$S$ should therefore be as small as possible to efficiently emit photons in a
well-defined quantum state.

####  III.2.2 Nonradiative Decay

The second consequence of electron-phonon coupling is the introduction of
alternative decay mechanisms. Electron-phonon coupling enables nonradiative
relaxation via multiphonon emission [37, 28]. A semi-classical picture of the
multiphonon process is shown in Fig. 5. The system is initially in the
excited-state configuration with the vibronic states occupied based on thermal
equilibrium. If the system has enough energy, it can surmount the barrier
defined by the crossing point between the ground- and excited-state potential
energy surfaces. An electronic transition occurs, and multiple phonons are
emitted in the process of relaxing down to the equilibrium configuration of
the ground state.

Semiclassically, surmounting the barrier is the rate-limiting step, and this
would be a thermally activated process. However, the actual process is
quantum-mechanical, and even at very low temperatures the nonradiative process
still occurs due to tunneling through the barrier. We will show that this
nonradiative mechanism can be dominant at small $\Delta E$ even at low
temperature. Going to higher temperatures would increase the nonradiative rate
even more; for the purposes of our discussion we focus on the low-temperature
case.

Figure 5:  Semiclassical view of the nonradiative process illustrated with a
configuration-coordinate diagram for the model system. Blue indicates the
excited state, and red indicates the ground state. We assume both states share
the same phonon frequency $\Omega$. The vibronic wavefunctions are shown
schematically, and the nonradiative process is indicated by the black arrows.

Since quantum mechanics governs the behavior of the system the nonradiative
transition rate of the multiphonon emission process can be written using
Fermi’s golden rule [37, 28]:

$\Gamma_{\rm
NR}=\frac{2\pi}{\hbar}W_{eg}^{2}\sum_{n}{\lvert\braket{\chi_{e0}\lvert\hat{Q}-Q_{0}\rvert\chi_{gn}}\rvert}^{2}\\\
\times\delta(\Delta E-n\hbar\Omega)\;,$ (10)

where $Q_{0}$ is the geometry for the perturbative expansion of the electron-
phonon coupling to linear order. In this expression we assumed that only the
vibrational ground state is occupied in the initial state of the system (as
also assumed in Sec. III.2.1). $W_{eg}$ is the electron-phonon coupling matrix
element and is system dependent; we will use a representative value of 0.1
eV/(amu1/2 Å), which is comparable to the value for telecom-wavelength
emitters in c-BN [44], and evaluate the nonradiative transition rate using the
implementation in the Nonrad code [38].

As seen in Fig. 2, the nonradiative transition rate $\Gamma_{\rm NR}$
increases exponentially as $\Delta E$ decreases. This poses a particular
problem for obtaining efficient emission at long wavelengths: even for a very
strong electric-dipole transition ($\mu=1$ eÅ), the nonradiative rate
dominates over the radiative rate unless the Huang-Rhys factor is smaller than
$S\approx 1$. Suppressing the nonradiative rate is thus of utmost important
for obtaining efficient emitters at energies below 1.5 eV, placing severe
constraints on candidate defect centers.

For completeness, we mention that other nonradiative decay mechanisms are
possible; e.g., an Auger-Meitner process could be important in some systems
[45, 41, 46]. In the Auger-Meitner process, energy is dissipated by exciting a
free carrier to higher energies through the Coulomb interaction. Since this
mechanism is active only in the presence of free carriers, the Auger-Meitner
process can be suppressed in samples with low carrier concentrations.
Furthermore, it was found that the process depends weakly on the transition
energy [46], in contrast to the exponential dependence of the multiphonon
process. Therefore we expect the Auger-Meitner process not to be the dominant
decay mechanism at longer wavelengths.

###  III.3 Quantum Efficiency

The most common measure of efficiency is the internal quantum efficiency
(IQE):

$\eta_{\rm IQE}=\frac{\Gamma_{\rm R}}{\Gamma_{\rm R}+\Gamma_{\rm NR}}\;,$ (11)

which quantifies the probability of producing a photon (with any energy) per
given excitation. However, as previously noted, only photons in the ZPL are
useful for quantum information applications. A more useful measure is
therefore the efficiency of ZPL emission,

$\eta_{\rm ZPL}=\frac{\Gamma_{\rm ZPL}}{\Gamma_{\rm R}+\Gamma_{\rm
NR}}=\frac{e^{-S}\,\Gamma_{\rm R}^{(0)}}{\Gamma_{\rm R}+\Gamma_{\rm NR}}\;.$
(12)

The common practice of replacing $\Gamma_{\rm R}$ with $\Gamma_{\rm R}^{(0)}$
means that $\eta_{\rm ZPL}\approx e^{-S}\eta_{\rm IQE}$. In the literature, it
is common to report the Debye-Waller factor and internal quantum efficiency
for a given defect. These values can then be used to estimate $\eta_{\rm
ZPL}$, but care should be taken to make sure the limits of the approximation
($S$ is not too large) are not overstepped.

The $\eta_{\rm ZPL}$ as a function of $\Delta E$ for a strong radiative
transition ($\mu=1$ eÅ) is shown in Fig. 6. Larger Huang-Rhys factors clearly
suppress $\eta_{\rm ZPL}$, predominantly because of the larger nonradiative
recombination rate $\Gamma_{\rm NR}$. The results confirm the conclusions of
Fig. 2: even a relatively small Huang-Rhys factor ($S\approx 1$) will suppress
the quantum efficiency below $10^{-3}$ in the telecom-wavelength range.

Figure 6:  Quantum efficiency of ZPL emission $\eta_{\rm ZPL}$ for a strong
electric-dipole transition $\mu=1$ eÅ. Different colors indicate different
values of the Huang-Rhys factor $S$ given by the color bar. Energies that fall
within the range of telecom wavelengths are shaded in pink, and energies that
fall within the free-space communication window are shaded in grey.

We now discuss approaches to improve the efficiency. So far, we have kept the
phonon energy $\hbar\Omega$ fixed to a value of 100 meV, representative of a
case (such as diamond) where high-frequency vibrations dominate the electron-
phonon coupling. In Fig. 7, we investigate the impact of the phonon energy
$\hbar\Omega$ on the efficiency $\eta_{\rm ZPL}$, for the case where the
emission energy is kept fixed at $\Delta E=0.80$ eV (in the telecom C-band).
Larger phonon frequencies are clearly detrimental for quantum efficiency.

Phonon frequencies are determined by atomic masses and force constants. The
representative phonon energy is determined by both the host lattice and the
defect. Light host atoms (particularly first-row elements) will inevitably
lead to higher frequencies because of their small masses and short bond
lengths (which lead to larger force constants). A host material with heavier
atoms would therefore be preferred. However, if the defect involves a light
impurity, local vibrational modes with larger phonon energies are to be
expected; hydrogen would be particularly problematic in this respect. We also
note that heavier lattices may increase the Huang-Rhys factor, which would
negate any improvements in efficiency from lower phonon energies. This is
particularly problematic in soft lattices (e.g., in the halide perovskites
[47, 48]). Still, we suggest that careful exploration of the impact of phonon
frequencies is a promising route for improving quantum efficiencies.

Figure 7:  The quantum efficiency $\eta_{\rm ZPL}$ as a function of phonon
energy $\hbar\Omega$ for $\Delta E=0.80$ eV. Different colors indicate
different values of the Huang-Rhys factor $S$ given by the color bar.

##  IV Discussion

###  IV.1 Tolerating Inefficiency

Given the inherent difficulty in obtaining a telecom-wavelength transition, or
longer wavelengths in general, with high quantum efficiency, a natural
question arises: how much inefficiency can be tolerated? One method to
overcome low quantum efficiency is through coupling to a photonic cavity [13,
49, 50].

We will consider a cavity with frequency $\omega_{c}$ tuned to be on resonance
with the emitter (i.e., $\hbar\omega_{c}=E_{\rm ZPL}$). The strength of
coupling between the cavity and emitter is given by [50]

$g=\sqrt{\frac{\xi\,\omega_{c}}{2\hbar\epsilon_{0}V_{c}}}\lvert\mu\rvert\cos\theta_{d}\;,$
(13)

where $\theta_{d}$ is the angle between the polarization of the defect and the
cavity. We will assume perfect alignment and take $\cos\theta_{d}=1$.
$\xi=e^{-S}$ accounts for the fact that only emission into the ZPL will couple
to the cavity [51, 49]. $V_{c}$ is the cavity mode volume, which can be
designed to give a certain coupling strength.

The cavity is defined by several decay rates. $\Gamma_{p}=\omega_{c}/(2Q)$ is
the total decay rate of a cavity with total quality factor $Q$.
$\Gamma_{p}=\Gamma_{c}+\kappa$, where $\Gamma_{c}=\omega_{c}/(2Q_{i})$ is the
intrinsic decay rate with intrinsic quality factor $Q_{i}$. $\kappa$ is the
rate of the decay into the output mode from the cavity, which is the useful
light that can be extracted. For such a cavity, the efficiency of light
extraction is given by [25, 50]

$\eta_{\rm
cav}=\frac{\kappa}{(\Gamma_{p}+\Gamma_{e})(1+\frac{\Gamma_{p}\Gamma_{e}}{4g^{2}})}\;,$
(14)

where $\Gamma_{e}=\Gamma_{\rm R}+\Gamma_{\rm NR}$ is the total decay rate of
the emitter.

In the Purcell regime, the cavity enhances the photon density of states
leading to an enhancement of the spontaneous emission rate [52]. To achieve
this, the strength of coupling between the cavity and emitter should be weak
($2g<\Gamma_{p}+\Gamma_{e}$). Thus Rabi oscillations between the emitter and
cavity are avoided. Furthermore, the cavity should decay faster than the
emitter produces a photon ($\Gamma_{p}>\Gamma_{e}$). This avoids the produced
photon from staying in the cavity long enough to interact with emitter again,
in other words avoiding high cooperativity [13]. The $\kappa$ that optimizes
the efficiency of light extraction $\eta_{\rm cav}$ can be obtained by taking
a derivative of Eq. (14) with respect to $\kappa$ (recalling that
$\Gamma_{p}=\Gamma_{c}+\kappa$) and setting it equal to zero. The optimal
value is given by [50]

$\kappa_{\rm
opt}=\Gamma_{c}\sqrt{\left(1+\frac{\Gamma_{e}}{\Gamma_{c}}\right)\left(1+\frac{4g^{2}}{\Gamma_{e}\Gamma_{c}}\right)}\;.$
(15)

For a given emitter and the above definitions of the cavity, the remaining
parameters to be determined are $V_{c}$ and $Q_{i}$. These parameters are
influenced by materials choice, in particular the ability to fabricate
photonic structures with that material. Here we will assume
$V_{c}=0.1\lambda_{c}^{3}$ where $\lambda_{c}=cn_{r}/\omega_{c}$. This value
should be achievable for emitters operating at visible and longer wavelengths
[53, 50, 49]. We note that constructing photonic cavities is easier at
wavelengths longer than visible; e.g., values as low as
$V_{c}=10^{-4}\lambda_{c}^{3}$ have been demonstrated [54, 55]. While
decreasing $V_{c}$ increases $\kappa$ and therefore $\eta_{\rm cav}$, it also
increases $g$, which could potentially push the system out of the Purcell
regime. Thus the choice of $V_{c}=0.1\lambda_{c}^{3}$ is realistic while also
avoiding strong coupling.

We then determine $Q_{i}$ by maximizing $\eta_{\rm cav}$ subject to the
constraint that the system is within the Purcell regime. Furthermore, we
restrict $Q_{i}$ to be less than $10^{8}$, which is a value used for visible
wavelength emitters [49]. While increasing $Q_{i}$ increases $\eta_{\rm cav}$,
the constraint that $\Gamma_{p}>\Gamma_{e}$ may no longer be satisfied. Indeed
the main effect of the optimizing $Q_{i}$ is to maintain
$\Gamma_{p}>\Gamma_{e}$: At longer wavelengths, $\Gamma_{e}$ increases due to
the increase in $\Gamma_{\rm NR}$, and therefore, $Q_{i}$ must decrease to
increase $\Gamma_{p}$ accordingly.

The resulting values of $\eta_{\rm cav}$ are shown in Fig. 8. (Optimized
values of $Q_{i}$ and the resulting Purcell enhancement are shown in Appendix
C.) When the transition energy is greater than 1.5 eV, unity efficiency can be
obtained even for $S=3$. For comparison, without the cavity $\eta_{\rm ZPL}$
was no larger than 6% when $S=3$ (yellow line in Fig. 6). At telecom
wavelengths, unity efficiency can be obtained for an emitter with $S<0.31$,
which was only possible for $S<0.01$ in the absence of a cavity (Fig. 6).

Figure 8:  Quantum efficiency of emission from a cavity $\eta_{\rm cav}$ for a
strong electric-dipole transition $\mu=1$ eÅ and with the cavity parameters
specified in the text. Different colors indicate different values of the
Huang-Rhys factor $S$ given by the color bar. Energies that fall within the
range of telecom wavelengths are shaded in pink, and energies that fall within
the free-space communication window are shaded in grey.

Our results have shown that obtaining an efficient single-photon emitter is
much easier at larger transition energies. Quantum frequency conversion [56,
57] could therefore be used to convert a higher energy photon down to telecom
wavelengths for transmission. The efficiency of conversion can in principle
approach unity, and efficiencies exceeding 50% have been demonstrated [56].
However, the conversion efficiency strongly depends on the specifics of the
system, especially with regard to the introduction of noise channels:
conversion from visible or ultraviolet wavelengths to telecom wavelengths is
particularly challenging [56]. Quantum frequency conversion from near-infrared
(795 nm) to O-band telecom (1342 nm) wavelengths with an efficiency of 33% is
one of the highest values achieved [58].

###  IV.2 Examples

A number of defects that act as single-photon emitters have been observed
experimentally; in addition, many have been predicted theoretically. Some of
these defects and their properties, such as emission energy and efficiency,
are summarized in Table 1. From Table 1, we can see that the essential physics
captured by our model is confirmed: smaller transition energies ($\Delta E$)
tend to have lower quantum efficiencies $\eta$ due to enhanced nonradiative
processes. As previously mentioned, the NV center and SiV center in diamond
have been used for a variety of networking demonstrations [7, 10, 11].
$\eta_{\rm ZPL}$ takes a value of 2.3% for the NV center and 7.5% for the SiV
center, which are relatively high compared to that of the telecom-wavelength
emitters discussed below. The relatively high $\eta_{\rm ZPL}$ values are
likely due to the large transition energy $\Delta E$ of 1.95 eV for the NV
center and 1.68 eV for the SiV center.

Table 1:  Selected host materials and defects that could serve as single-photon emitters. Data is shown for NV [13, 12], SiV [13], and SiV2:H [59] in diamond; $V_{\rm B}$-CB and $V_{\rm B}$-SiB in c-BN [44]; CB-CN [32], 2-eV emitters [60, 61, 62], and the B dangling bond (DB) [63, 64] in h-BN; excitonic emitters in WSe2 [65, 66]; and Er in MgO [34]. We assume $\eta_{\rm ZPL}\approx e^{-S}\eta_{\rm IQE}$ here, as described in the text. When a range of values is given in the literature, we use the most favorable values to estimate $\eta_{\rm ZPL}$. The final column indicates if quantum frequency conversion (QFC) is necessary to transmit the photons over long distances using fiber optics (i.e., the photons need to be converted to telecom wavelengths).  Host | Defect |  $\Delta E$ [eV] | $S$ |  $\Gamma_{\rm ZPL}$ [MHz] | $\eta_{\rm IQE}$ | $\eta_{\rm ZPL}$ | Needs QFC?
---|---|---|---|---|---|---|---
Diamond | NV- | 1.95 | 3.5 | 2.1 | 0.76 | $2.3\times 10^{-2}$ | Yes
SiV- | 1.68 | 0.13–0.29 | 47 | 0.10 | $7.5\times 10^{-2}$ | Yes
(SiV2:H$)^{-}$111Tentative attribution from Ref. 59. | 1.02 | 0.72 | 3.6 | $2.0\times 10^{-3}$ | $1.0\times 10^{-3}$ | No
c-BN |  $(V_{\rm B}$-C${}_{\rm B})^{0}$ | 0.95 | 1.5 | 1.2 | $5.4\times 10^{-4}$ | $1.2\times 10^{-4}$ | No
$(V_{\rm B}$-Si${}_{\rm B})^{0}$ | 0.89 | 1.5 | 1.6 | $3.2\times 10^{-4}$ | $7.1\times 10^{-5}$ | No
h-BN | (CB-C${}_{\rm N})^{0}$ | 4.31 | 2.0 | 120 | 1.0 | $1.4\times 10^{-1}$ | Yes
“2-eV emitters” | 1.6–2.2 | 2–3 | 120 | 0.06–0.87 | $1.2\times 10^{-1}$ | Yes
B DB | 2.06 | 2.3 | 2.6 | 0.06–0.12 | $1.2\times 10^{-2}$ | Yes
WSe2 | Unknown | 1.6–1.7 |  $<0.1$ 222No phonon sideband observed in experiments. | 13 | 0.05 | $5.0\times 10^{-2}$ | Yes
MgO | Er | 0.80 |  $<0.1$ 22footnotemark: 2 | $5.0\times 10^{-5}$ | 1.0 | $1.0\times 10^{0}$ | No

The $\eta_{\rm IQE}$ values of the NV and SiV centers are surprisingly low;
one might expect that should be closer to unity. The SiV center has a Huang-
Rhys factor $S$ no larger than 0.29, for which even a telecom-wavelength
transition should have a high efficiency. While the NV center has a larger
Huang-Rhys factor ($S$=3.5), one would still expect a high $\eta_{\rm IQE}$
within our model given the high transition energy. Based on our model (Fig. 6)
we can conclude that a direct nonradiative process via multiphonon emission is
not a limiting factor in the efficiency of the NV and SiV centers.

We suggest that the low $\eta_{\rm IQE}$ values are due to the presence of
spin dynamics. While we do not quantitatively evaluate the rates here, one can
qualitatively see from the schematic in Fig. 9 that the effect of spin
dynamics is to drain away some population from the excited state, leading to a
lower overall efficiency. Indeed, a key component of the NV center—which is
integral to its widespread adoption as the prototype quantum defect—is the
ability to optically control the ground-state spin, which is enabled by an
intersystem crossing between the triplet and singlet manifolds of the center
[12]. While spin dynamics can influence the efficiency, it is a second-order
concern and can provide an overall benefit in some cases, as exemplified by
the NV center.

Figure 9:  Schematic illustration of spin dynamics using an energy level
diagram that show the influence of the intersystem crossing on the system. The
ground $g$ and excited $e$ states are labeled, as well as the metastable
excited state $m$ that enables the intersystem crossing. Radiative transitions
are indicated by solid arrows and nonradiative transitions by wiggly arrows.
The effect of the metastable state is to draw population away from the optical
excited state $e$, thereby lowering the efficiency.

Similar to spin dynamics, charge dynamics can be understood by including
additional states in the model. While spin dynamics involves transitions
within the defect, charge dynamics involves exchanging charge with the local
environment (e.g., the bulk bands or nearby traps). It is generally desirable
to avoid charge dynamics completely. Both charge and spin dynamics can be
observed through the photon autocorrelation function, which can be measured
spectroscopically [24, 67].

We recently predicted $V_{\rm B}$-CB and $V_{\rm B}$-SiB in c-BN as potential
NV-center analogues that emit in the telecom range [44]. These centers have
quantum efficiencies of $\sim$10-4. Comparing with a recently observed O-band
emitter in diamond [59] is informative. This emitter has an inferred quantum
efficiency of $\sim$10-3, and was tentatively attributed to (SiV2:H$)^{-}$.
The larger quantum efficiency for the diamond emitter could be partly due to
the fact that its effective phonon frequency (68.4 meV) is lower that the
values for the defects in c-BN ($\sim$100 meV).

In Table 1, there is a notable exception to the low quantum efficiencies
observed at telecom wavelengths, namely Er in MgO. The radiative transition of
Er in MgO arises from a transition between $f$ orbitals. Symmetry dictates
that the electric-dipole transition is forbidden, and the emission arises from
a magnetic-dipole transition. No discernible phonon sideband is observed in
experiments [34], indicating that the Er ion is decoupled from the lattice
(small $S$). As a result the coupling to phonons is so weak that the
nonradiative process is suppressed, giving high quantum efficiency even though
$\Gamma_{\rm ZPL}$ is low. This result is in line with the predictions of our
model if we assume $S\sim 10^{-2}$. While this might be appealing for having
an efficient single-photon emitter in the telecom, there is an obvious
drawback: due to the weakness of the magnetic-dipole transition, the overall
rate of single-photon emission is only $\sim$50 Hz [34]. With cavity coupling,
Purcell-enhanced rates up to 21 kHz have been observed [68], which are
promising but also still orders of magnitude slower than visible and near-IR
emitters.

From our survey, some of the most efficient emitters are found in h-BN. The
single-photon emission at ultraviolet energies has been attributed to the
carbon dimer (CB-CN) [32]. At these energies, the nonradiative process via
multiphonon emission is negligible, giving a high internal quantum efficiency.
The overall efficiency of single-photon emission is then governed by the
Debye-Waller factor, which is high due to the relatively low Huang-Rhys
factor, resulting in a high overall efficiency of emission into the ZPL.

Another class of emitters in h-BN emit in the visible spectrum and are known
as the “2-eV emitters” [69, 70, 60]. They are notoriously heterogeneous, as
indicated by the spread of values in Table 1. We proposed the boron dangling
bond (DB) as the likely origin of this emission [63]; DBs can naturally
explain the heterogeneity due to their sensitivity to the local environment
[64]. Moreover, the spread in observed efficiencies has been linked to the
excitation power [71]; the spread can be explained by charge dynamics
(photoionization) within the B DB model [67]. The brightest 2-eV emitters are
competitive with the carbon dimer in terms of overall efficiency. However,
there is a clear advantage to the 2-eV emitters: some of the emitters produce
photons with a wavelength near 795 nm, which means the already established
quantum frequency conversion to telecom wavelengths with an efficiency of 33%
[58] could be applied to these emitters. More generally, two-dimensional
materials also benefit from improved light-extraction efficiency.

First-principles calculations can be a powerful tool to predict novel defect-
based single-photon emitters. Our work clearly shows the importance of
assessing nonradiative decay at longer wavelengths. Unfortunately, this issue
is often overlooked. In SiC, a near-infrared emitter OC-$V_{\rm Si}$ with ZPL
energy $\approx$1.2 and Huang-Rhys factor $S=2.01$ was recently predicted
[72], and also in SiC, the ClC-$V_{\rm Si}$ was predicted to be a telecom-
wavelength emitter with Huang-Rhys factor in excess of 3.5 [73]. While these
defects have lower effective phonon frequencies that are beneficial for
efficiency, the low energy of the transition and large Huang-Rhys factors are
concerning from the perspective of efficiency. Given the availability of open-
source codes such as Nonrad [38] to evaluate the nonradiative rate from first-
principles, we strongly recommend assessing the efficiency of predicted
single-photon emitters to determine their feasibility.

###  IV.3 Future Directions

In our opinion, the “Goldilocks” single-photon emitter has yet to be
uncovered. We are optimistic that efforts to reduce the phonon frequency, as
discussed in Sec. III.3, will prove fruitful. One route of exploration to
control the phonon frequency could be to focus on chemical trends. For
example, the group-III nitrides (BN, AlN, and GaN) are being explored as hosts
for single-photon emitters [74]. While these materials all contain N, the
effective phonon frequency within the one-dimensional model may be lower in
GaN than in BN if the phonon mode is dominated by the motion of the heavier Ga
atoms.

The observation that transitions between $f$ orbitals leads to small Huang-
Rhys factors may also be a fruitful line of exploration. It may be possible to
identify a rare-earth ion and host material combination for which the
electric-dipole transition is not forbidden and the radiative rate is sizable.
Along these lines, transition metals may provide a “middle ground” that
balances moderate emission rates with reduced electron-phonon coupling.

Based on the considerations of cavity coupling, we can suggest that the
“Goldilocks” emitter may likely have a transmission energy around 1.5 eV. At
this energy, the nonradiative processes are not so severe, and cavity coupling
can result in unity efficiency. Indeed defects whose Huang-Rhys factor is
$S$$\approx$3 could still achieve unity efficiency, which is far less
restrictive compared to telecom wavelengths. Moreover, the cavity parameters
necessary to realize unity efficiency should be achievable in experiments. An
emitter at 1.5 eV then naturally falls within the window of low atmospheric
loss for free-space communication [17]. In light of the much higher
efficiencies achievable at shorter wavelengths, we suggest that if telecom
wavelengths are required for transmission in optical fibers, quantum frequency
conversion should be considered alongside direct generation.

##  V Conclusion

In summary, we have investigated the impact of electron-phonon interactions on
the efficiency of defect-based single-photon emitters. We utilized a model
that captures the essential physics of coupling to phonons, based on the
formalism of fully first-principles calculations. Various approximations that
have been implicitly used in the literature were discussed, and their validity
addressed. We demonstrated that nonradiative transitions via multiphonon
emission become dominant when the transition energy is below 1.5 eV,
dramatically reducing the efficiency with which such defects can produce
single photons. The findings were discussed in the context of values reported
in the existing literature. We proposed engineering approaches to enhance the
efficiency, including reducing the phonon frequencies by utilizing chemical
trends. The search is still on to find the optimal single-photon emitter.

###### Acknowledgements.

M.E.T. was supported by the U.S. Department of Energy, Office of Science,
National Quantum Information Science Research Centers, Co-design Center for
Quantum Advantage (C2QA) under contract number DE-SC0012704. K.P., G. M., and
C.G.VdW. were supported by the National Science Foundation (NSF) through
Enabling Quantum Leap: Convergent Accelerated Discovery Foundries for Quantum
Materials Science, Engineering and Information (Q-AMASE-i) Award No.
DMR-1906325. The research used resources of the National Energy Research
Scientific Computing Center, a DOE Office of Science User Facility supported
by the Office of Science of the U.S. Department of Energy under Contract No.
DE-AC02-05CH11231 using NERSC award BES-ERCAP0021021.

## Data Availability

The data that supports the findings of this study can be obtained readily from
the figures or are available from the corresponding author upon reasonable
request.

## Author Contributions

Mark E. Turiansky: Writing - original draft (lead); Writing - review & editing
(equal); Formal analysis (lead); Conceptualization (equal); Visualization
(lead); Software (lead); Methodology (equal) Kamyar Parto: Writing - review &
editing (supporting); Conceptualization (supporting). Galan Moody: Supervision
(supporting); Writing - review & editing (supporting); Conceptualization
(supporting); Funding acquisition (supporting). Chris G. Van de Walle:
Supervision (lead); Writing - review & editing (equal); Conceptualization
(equal); Funding acquisition (lead); Methodology (equal).

##  Appendix A Derivation of the Average Emitted Photon Energy $E_{\rm em}$

Here we derive the average emitted photon energy $E_{\rm em}$ including
photons emitted into both the phonon sideband and the ZPL Eq. (6)]. Within the
harmonic approximation, the phonon overlap integrals
${\braket{\chi_{e0}}{\chi_{gn}}}$ can be expressed in terms of the Huang-Rhys
factor [28]:

${\lvert\braket{\chi_{e0}}{\chi_{gn}}\rvert}^{2}=e^{-S}\frac{S^{n}}{n!}\;.$
(16)

Therefore Eq. (5) can be written as

$E_{\rm em}=\Delta
E\,e^{-S}\sum_{n=0}^{\infty}\frac{S^{n}}{n!}-\hbar\Omega\,e^{-S}\sum_{n=0}^{\infty}\frac{nS^{n}}{n!}\;.$
(17)

The first summation is just the series expansion of the exponential,

$e^{S}=\sum_{n=0}^{\infty}\frac{S^{n}}{n!}\;.$ (18)

Furthermore, we can use

$Se^{\lambda S}=\frac{d}{d\lambda}e^{\lambda
S}=\frac{d}{d\lambda}\sum_{n=0}^{\infty}\frac{(\lambda
S)^{n}}{n!}=\sum_{n=0}^{\infty}\frac{n\lambda^{n-1}S^{n}}{n!}\;,$ (19)

and take $\lambda\rightarrow 1$ to address the second series. Plugging these
results into Eq. (17) gives Eq. (6).

##  Appendix B Derivation of the Total Radiative Emission Rate $\Gamma_{\rm
R}$

The luminescence intensity $I(\hbar\omega)$ is the number of photons emitted
per unit time per unit energy, for a given photon energy $\hbar\omega$ [36,
28]. Within the Condon approximation and at low temperature where only the
ground vibrational level of the excited state is occupied [75],

$I(\hbar\omega)={\left(\frac{\mathcal{E}_{\rm
eff}}{\mathcal{E}_{0}}\right)}^{2}\frac{n_{r}\mu^{2}(\Delta
E)^{2}}{3\pi\epsilon_{0}c^{3}\hbar^{4}}\\\
\times\sum_{n=0}^{\infty}\hbar\omega\,{\lvert\braket{\chi_{e0}}{\chi_{gn}}\rvert}^{2}\delta(\Delta
E-n\hbar\Omega-\hbar\omega)\;.$ (20)

The two factors of $\Delta E$ in the numerator come from the fact that the
intensity is derived from minimal coupling to the electric field [36, 28];
thus the coupling entails evaluating momentum matrix elements, which are
related to the transition matrix element by $p=i\,m_{e}(\Delta E/e\hbar)\mu$.
The total radiative rate is an integral of the luminescence intensity over all
photon energies:

$\displaystyle\Gamma_{\rm R}$ $\displaystyle=\int
d(\hbar\omega)\,I(\hbar\omega)$ $\displaystyle=\frac{n_{r}\mu^{2}(\Delta
E)^{2}}{3\pi\epsilon_{0}c^{3}\hbar^{4}}\sum_{n=0}^{\infty}(\Delta
E-n\hbar\Omega){\lvert\braket{\chi_{e0}}{\chi_{gn}}\rvert}^{2}\;.$ (21)

Using the definition of the average emission energy [Eq. (5)], we arrive at
Eq. (8).

##  Appendix C Optimized Cavity Quality Factor

At a given value of $V_{c}$, we optimize the intrinsic quality factor $Q_{i}$
to maximize the efficiency $\eta_{\rm cav}$ while satisfying three
constraints. We constrain the value to be $10^{2}<Q_{i}<10^{8}$. Furthermore,
the value must maintain the cavity in the Purcell region, where
$2g<\Gamma_{p}+\Gamma_{e}$ and $\Gamma_{p}>\Gamma_{e}$. For a given emission
energy $\Delta E$ and Huang-Rhys factor $S$, the emitter decay rate
$\Gamma_{e}$ is determined. For each $\Gamma_{e}$, we define a fine grid of
$Q_{i}$ values from $10^{2}$ to $10^{8}$ and evaluate $\eta_{\rm cav}$ using
Eqs. 14 and 15. We then select the value of $Q_{i}$ for which $\eta_{\rm cav}$
is maximal, while satisfying the constraints. The resulting values of $Q_{i}$
are shown in Fig. 10. When $\Gamma_{\rm NR}<\Gamma_{\rm R}$, the optimal value
of $Q_{i}$ is pinned at the maximal value ($10^{8}$), but when $\Gamma_{\rm
NR}>\Gamma_{\rm R}$, $Q_{i}$ must drop rapidly to maintain
$\Gamma_{p}>\Gamma_{e}$.

Figure 10:  The optimized intrinsic quality factor $Q_{i}$ plotted as a
function of the emission energy $\Delta E$ of the single-photon emitter.
Different colors indicate different values of the Huang-Rhys factor $S$ given
by the color bar.

From the optimized $Q_{i}$, we can calculate the Purcell enhancement $F_{p}$
from [50]

$F_{p}=\frac{3}{4\pi^{2}}Q\left(\frac{\lambda_{c}^{3}}{V_{c}}\right)\;,$ (22)

where we remind the reader that $Q$ here is the total quality factor [i.e.,
from $\Gamma_{p}=\omega_{c}/(2Q)=\Gamma_{c}+\kappa$, where
$\Gamma_{c}=\omega_{c}/(2Q_{i})$ and $\kappa$ depends on $\Gamma_{c}$]. Our
obtained values of $F_{p}$ are shown in Fig. 11. $F_{p}$ needs to be no larger
than $10^{4}$ to obtain the results shown in the main text. This is an
achievable value given that diamond nanophotonic resonators can theoretically
achieve values of $10^{5}$ [13].

Figure 11:  The Purcell enhancement $F_{p}$ resulting from the numerically
optimized values of $Q_{i}$ as a function of the emission energy $\Delta E$ of
the single-photon emitter. Different colors indicate different values of the
Huang-Rhys factor $S$ given by the color bar.

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