# Enhancing two-photon spontaneous emission in rare earths using graphene and
graphene nanoribbons

Colin Whisler  Gregory Holdman  D. D. Yavuz  Victor W. Brar  University of
Wisconsin-Madison, Department of Physics, 1150 University Ave., Madison,
Wisconsin 53706, USA<EMAIL_ADDRESS>
###### Abstract

The enhancement of two-photon spontaneous emission (2PSE) from trivalent and
divalent rare earth ions in proximity to graphene and graphene nanoribbons is
calculated for achievable experimental conditions using a combination of
finite difference time domain simulations and direct computation of transition
rates between energy levels in rare earths. For Er3+, we find that the 2PSE
rate is initially 8 orders lower than the single-photon spontaneous emission
rate but that, with enhancement, 2PSE can reach 2.5% of the overall decay.
When graphene nanoribbons are used, we also show that the emission of free-
space photon pairs from Er3+ at 3 - 3.2 $\mu$m via 2PSE can be increased by
$\sim 400$. Our calculations show significantly less relative graphene-
enhanced 2PSE than previous works, and we attribute this variation to
differences in emitter size and assumed graphene mobility. We also show that
the internal energy structure of the ion can have an impact on degree of 2PSE
enhancement achievable and find that divalent rare earths are more favorable.

##  I Introduction

Two photon spontaneous emission (2PSE) is a decay process whereby an
electronic transition occurs via emission of a photon pair whose combined
energy is equal to the difference between the excited and ground states
Göppert-Mayer (1931). Compared to single-transition decay, 2PSE usually occurs
at a much lower rate, as it requires a transition to an intermediate state. In
atomic systems these states are typically found at energies far from the
initial and final states, and they are only weakly connected through an
electric dipole perturbation. This leads to 2PSE decay rates that are commonly
$10^{8}$ lower than single-channel transition rates Breit and Teller (1940).
Despite this low efficiency, 2PSE has generated interest as a mechanism for
the creation of entangled photon pairs that would be useful in quantum
information systems Hayat, Ginzburg, and Orenstein (2007). In particular, the
2PSE process naturally generates photon pairs that are entangled polarization
Perrie  _et al._ (1985); Radtke, Surzhykov, and Fritzsche (2008) and frequency
states Hayat  _et al._ (2008). Moreover, the theoretical bounds on 2PSE permit
it to, in principle, generate photon pairs at higher rate than state-of-the-
art parametric down-conversion schemes, and over a wider range of frequencies.

Figure 1: Left: schematic of an emitter beneath a graphene nanoribbon, with
single-transition decay shown in blue and 2PSE shown in red. Right: energy
levels for the processes involved in two-photon spontaneous emission for Er3+.
An electron relaxes from the 4I13/2 state to the 4I15/2 state, but the matrix
elements that determine the process’s strength are defined by the electric
dipole transitions to the higher 4f105d and 4f106s states.

In order to enhance 2PSE to a level that is useful in real-world applications,
multiple research efforts have investigated solid state systems — including
quantum wells Hayat, Ginzburg, and Orenstein (2008) and quantum dots Ota  _et
al._ (2011) — that permit higher rates of 2PSE. In some experiments, the
intermediate state is engineered to exist between the initial and final
states, creating a resonant condition in the decay process that enhances the
2PSE transition rate. Other works have considered metallic nanostructures as a
method of enhancing 2PSE by supporting plasmonic resonances Poddubny  _et al._
(2012). More recently, it was theoretically predicted that 2D materials that
support highly confined optical modes — including surface plasmons in graphene
Muniz  _et al._ (2020); Rivera  _et al._ (2016) and surface polaritons in
hexagonal boron nitride Rivera  _et al._ (2017) — could dramatically enhance
2PSE from nearby emitters through the Purcell effect. The underlying premise
of those predictions is that the enhancement of single- and two-photon
processes scale with the third and sixth powers, respectively, of the
confinement factor of an optical mode Rivera  _et al._ (2016). In 2D
materials, this confinement factor — which can be approximated as the ratio
between the free-space and surface mode wavelengths — can be $\sim 10^{2}$,
large enough to allow 2PSE rates to approach those of single-transition decay.
Indeed, theoretical studies have shown that hydrogen atoms suspended above
graphene Rivera  _et al._ (2016), graphene nanodisks Muniz  _et al._ (2020),
and monolayer hexagonal boron nitride Rivera  _et al._ (2017) can exhibit 2PSE
rates that become comparable to or even exceed the corresponding single-
transition decay processes. However, despite numerous predictions, there has
been no experimental evidence of 2PSE occurring in graphene-coupled emitters,
even though many works have studied such systems in detail Tielrooij  _et al._
(2015); Cano  _et al._ (2020); Lin  _et al._ (2013); Federspiel  _et al._
(2015); Chen  _et al._ (2010); Yeltik  _et al._ (2013); Treossi  _et al._
(2009); Guzelturk and Demir (2016); Tisler  _et al._ (2013); Stöhr  _et al._
(2012); Yu  _et al._ (2015); Gaudreau  _et al._ (2013); Kim  _et al._ (2010).
Factors that make observations of 2PSE challenging include: (1) the strong
quenching effect of graphene on nearby emitters, driven by non-radiative
energy transfer as well as coupling to non-radiative surface modes; (2) the
high inefficiencies and low speeds of detectors in the mid- to far-IR, where
graphene-mediated 2PSE is likely to occur; (3) the competing processes that
drive emitter decay, which can also depend on doping and distance from
graphene. Even with these barriers, however, a strongly enhanced 2PSE rate
should perturb the overall observed lifetime of a graphene-coupled emitter,
and such a perturbation has not been revealed in detailed lifetime
measurements.

In this work we perform a comprehensive theoretical analysis of 2PSE from
fluorescent rare earth atoms that are optically coupled to graphene and
graphene nanoribbons. Our focus on rare earths reflects the fact that they can
exhibit high efficiency emission at sufficiently long wavelengths to couple
with graphene plasmonic modes which, for achievable carrier densities, are
heavily damped at wavelengths $<$1 $\mu$m. Infrared emissions from these ions
occur between their 4f electron states, which are located close to the
nucleus, causing the emission spectra to remain largely consistent across
different host crystals Liu and Jacquier (2005). For Er3+, photonic crystal
designs have been proven effective in enhancing emission sufficiently to
observe fluorescence of individual ions Dibos  _et al._ (2018). Rare earth
doped substrates can also be grown via ion implantation or molecular beam
epitaxy, where the placement of dopants can be controlled with nm-precision,
and multiple experiments have shown that graphene can affect the emission
properties of nearby rare earth ions Tielrooij  _et al._ (2015); Cano  _et
al._ (2020). Here we theoretically investigate both single-transition decay
and 2PSE from rare earths near graphene as a number of parameters are varied,
including emitter orientation, rare earth species, graphene nanostructure
geometry, and graphene doping levels.

A basic illustration of our model system is shown in Figure 1, where a rare
earth atom — in this case trivalent erbium (Er3+) — is placed near a graphene
sheet or nanoribbon. Upon excitation, the atom can decay non-radiatively or
radiatively via single-transition decay or 2PSE, emitting photons and/or
graphene surface plasmons in the process. All of these processes are affected
by the graphene, which provides pathways for non-radiative and radiative decay
that are dependent on the graphene carrier density, and also on the graphene
geometry.

In order to precisely quantify the graphene-rare earth interaction, we first
determine the relative rates of single-transition decay and 2PSE from the rare
earth atoms by numerically calculating the transition probabilities between
all intermediate states, which is detailed for Er3+ in Section II. In Section
III we use full field finite difference time domain (FDTD) simulations to
calculate the enhancement of both single-transition decay and 2PSE from Er3+
due to both radiative and non-radiative processes enabled by the presence of
graphene. Next, in Section IV we explore how graphene affects the 2PSE in
divalent samarium and thulium, which have energy levels that are more
favorable to 2PSE in comparison to Er3+. Finally, we discuss how our
calculations compare to previous works, and experimental configurations where
we predict graphene-enhanced 2PSE to be observable.

##  II Determination of 2PSE rates in rare earths

Contrary to single-photon emission, in which an ion transitions from an
excited state $\hbar\omega_{e}$ to the ground state $\hbar\omega_{g}$ by
emitting a single photon of energy
$\hbar\omega_{0}=\hbar(\omega_{e}-\omega_{g})$, 2PSE allows the ion to emit
two simultaneous photons, one of energy $\hbar\omega$ and the other
$\hbar(\omega_{0}-\omega)$. As a second-order process, 2PSE requires a sum
over all intermediate states $\hbar\omega_{n}$ that are connected to the
initial and final states by electric dipole selection rules. The total 2PSE
rate $\Gamma_{0}$ can be written in terms of the spectral 2PSE rate
$\gamma_{0}$, which takes the form Craig and Thirunamachandran (1984)

$\Gamma_{0}=\int_{0}^{\omega_{0}}\gamma_{0}(\omega)d\omega\\\
=\frac{4c}{3\pi}\alpha^{2}k^{5}\int_{0}^{1}y^{3}(1-y)^{3}\times\left|\sum_{n}r_{gn}r_{ne}\right.\\\
\left.\left(\frac{1}{(\omega_{n}-\omega_{e})/\omega_{0}+y}+\frac{1}{(\omega_{n}-\omega_{e})/\omega_{0}+1-y}\right)\right|^{2}dy$
(1)

where $r_{gn}$ and $r_{ne}$ represent the electric dipole matrix elements
connecting the intermediate energy level to the ground state and excited
state, respectively. The variable of integration $y$ represents the fraction
of the total emission energy accounted for by the first photon, with the
second accounting for $1-y$. A diagram of the intermediate transition
processes involved in 2PSE for Er3+ is shown in Figure 1, along with the
4I13/2 $\rightarrow$ 4I15/2 transition for single-channel decay.

Although the 4I13/2 $\rightarrow$ 4I15/2 transition is the dominant emission
process in Er3+ within a material host (e.g. Y2O3), it is classically electric
dipole forbidden due to both states having the 4f11 configuration, which has
odd parity. The strength of the transition arises from the mixing of these
states with higher-lying even parity states in the 4f105d and 4f106s
configurations, which occurs due to the host crystal structure breaking the
symmetry of the wavefunction Judd (1962); Ofelt (1962). Similarly, for 2PSE,
the matrix elements $r_{gn}$ and $r_{ne}$ in Eq. (1) that connect the 4I13/2
and 4I15/2 states to other 4f11 states will be nonzero; however, the dominant
contributions will originate from the electric dipole allowed even-parity
(4f10) states. To determine $r_{gn}$ and $r_{ne}$, we use Cowan’s atomic
structure codes to calculate the wavefunctions, diagonalize the matrix, and
calculate the resulting transition matrix elements Cowan (1981). This allows
us to numerically calculate the energy levels for all 4f105d and 4f106s states
along with the oscillator strengths for all electric dipole transitions
between them and the 4I13/2 and 4I15/2 states. The energy levels for the
4f105d and 4f106s levels used for $\omega_{n}$ in Eq. (1) are adjusted from
the outputs of Cowan’s code by adding a constant amount to each, setting the
lowest-energy dipole-allowed state to the published value of 73458 cm-1 above
the ground state Meftah  _et al._ (2016). Using these matrix elements in Eq.
(1) and a single-photon emission frequency of $\omega_{0}=6500$ cm-1 Sardar
_et al._ (2007), we compute a spontaneous two-photon emission rate for Er3+ of
$\Gamma_{0}=2.943\times 10^{-7}\,\mathrm{s}^{-1}.$ Comparing this value to the
experimentally determined single-photon spontaneous emission rate of 125 s-1
for Er3+ in Y2O3 Weber (1968), we show that the two processes differ by more
than eight orders of magnitude; this relative difference is comparable to what
has been predicted for the 2s $\rightarrow$ 1s two-photon transition in
hydrogen compared to the 2p $\rightarrow$ 1s single-photon transition Breit
and Teller (1940).

##  III Enhancement of 2PSE and single-transition decay in Er3+ by graphene
and graphene nanoribbons

While intrinsic 2PSE rates are substantially less than those of single-
transition decay, it has been predicted that they can be made comparable by
using Purcell enhancement, wherein the decay rate of an emitter is enhanced in
the presence of confined optical modes Rivera  _et al._ (2016). This process
can occur in any optical cavity, but it can be especially strong in graphene
and other 2D materials, which support optical surface waves with wavelengths
much shorter than free space. These modes — which include surface plasmons and
phonon polaritons — enhance single-photon emission rates by a factor that
scales with $\eta_{0}^{3}=(\lambda_{0}/\lambda_{p})^{3}$, where $\lambda_{0}$
is the free-space wavelength and $\lambda_{p}$ is the wavelength of the
surface mode. For two-photon processes, the enhancement scales as
$\eta_{0}^{6}$. While metals can support surface modes with $\eta_{0}\sim 10$,
graphene plasmons exhibit $\eta_{0}>100$ Jablan, Buljan, and Soljačić (2009),
leading to substantially more 2PSE enhancement over single-transition decay
processes. Moreover, when the graphene is patterned into a nanostructure that
supports resonant plasmonic modes, the emission enhancement is further
increased by the Q factor of the resonance, it can be made to selectively
occur at specific frequencies, and the emitted plasmonic modes can more
efficiently out-couple to free space.

In general, the modification of the differential two-photon spontaneous
emission rate in the presence of a surface is given by Muniz  _et al._ (2019)

$\frac{\gamma(\omega,\textbf{r})}{\gamma_{0}(\omega)}=\frac{\sum_{i,j}|D_{ij}(\omega,\omega_{0}-\omega)|^{2}P_{i}(\omega,\textbf{r})P_{j}(\omega_{0}-\omega,\textbf{r})}{\sum_{i^{\prime},j^{\prime}}|D_{i^{\prime}j^{\prime}}(\omega,\omega_{0}-\omega)|^{2}}.$
(2)

Here $P_{i}(\omega,\textbf{r})$ depends on the confinement factor $\eta_{0}$,
the wavenumber $k$, and the surface-emitter distance $z_{0}$ according to
$P_{i}(\omega,\textbf{r})\sim\eta_{0}^{3}\text{exp}(-2\eta_{0}kz_{0})$ Rivera
_et al._ (2016) and represents the Purcell enhancement of an electric dipole
at location r emitting at frequency $\omega$ and oriented along the axis
$i=[x,y,z]$. The matrix $D_{ij}$ depends on the electric dipole matrix
elements $\textbf{d}_{ne}$ and $\textbf{d}_{gn}$ connecting the intermediate
states $n$ to the excited and ground states, respectively, and is given by

$D(\omega_{1},\omega_{2})=\sum_{n}\left(\frac{\textbf{d}_{ne}\textbf{d}_{gn}}{\omega_{n}-\omega_{e}+\omega_{1}}+\frac{\textbf{d}_{gn}\textbf{d}_{ne}}{\omega_{n}-\omega_{e}+\omega_{2}}\right).$
(3)

For a hydrogenic ion whose excited and ground configurations are both s
orbitals, $D_{ij}$ will be diagonal, as the dipole-allowed intermediate p
states are aligned along a specific x-, y-, or z-axis and can only be reached
when both transitions have the same polarization. For more complicated ions
such as Er3+, however, the off-diagonal terms will in general not be zero, and
their values may be computed by using the Wigner-Eckart theorem to calculate
the irreducible tensor components for each transition Cowan (1981); Eckart
(1930); Wigner (1927). After minor simplifications, we find that the off-
diagonal components for Er3+ contribute 3/4 as strongly as the diagonal ones,
so the differential 2PSE enhancement is given by

$\frac{\gamma(\omega,\textbf{r})}{\gamma_{0}(\omega)}\approx\sum_{i,j}A_{i,j}P_{i}(\omega,\textbf{r})P_{j}(\omega_{0}-\omega,\textbf{r})$
(4)

with

$A=\begin{pmatrix}2/15&1/10&1/10\\\ 1/10&2/15&1/10\\\
1/10&1/10&2/15\end{pmatrix}.$ (5)

Further details of this calculation can be found in the Supplemental Material
for this paper.

Figure 2: Results for a single dipole orientation. All results consider an
Er3+ emitter 5 nm away from graphene structures, with dipole moment oriented
along the short dimension of the ribbons (or parallel to the sheet). (a)
Purcell enhancement for the emitter near 20 nm graphene ribbons. Horizontal
lines correspond to the Fermi energies plotted in (b), while vertical lines
show the doubly-resonant frequencies for photon emission. Electric field
profiles at $E_{F}=0.385$ eV are shown below, calculated at a lateral distance
of 7 nm along the ribbon with the field strength normalized to the value with
no graphene present. (b) Differential 2PSE enhancement $\gamma/\gamma_{0,max}$
for the emitter near graphene ribbons for doubly-resonant Fermi energies
(pink, dotted; orange, solid) and a non-resonant Fermi energy (blue, dashed).
(c) Integrated 2PSE enhancement $\Gamma/\Gamma_{0}$ for the emitter near 20 nm
wide graphene ribbons. (d) Differential 2PSE enhancement
$\gamma/\gamma_{0,max}$ at $E_{F}=0.565$ eV for the emitter with no graphene
present (pink, dotted), modified by graphene ribbons (blue, dashed), and
modified by a graphene sheet (orange, solid). (e, f) Differential 2PSE
enhancement $\gamma/\gamma_{0,max}$ for the emitter near graphene ribbons (e)
or a graphene sheet (f). The results in (b), (d), (e), and (f) are normalized
by the maximum value of free-space differential 2PSE.

Computational methods such as finite difference and finite element methods
provide a tool that can be used to analyze the modification of emission rates
in a variety of plasmonic environments Akselrod  _et al._ (2014). To calculate
the $P_{i}$ factors in Eq. (4), we use FDTD (Lumerical) simulations to monitor
the power emitted from an electric dipole source placed 5 nm below the
graphene as a function of frequency and graphene Fermi level ($E_{F}$). We
consider the effects of both a continuous graphene sheet as well as ribbons
designed in an array with a period equal to three times the ribbon width, with
the dipole source positioned beneath the center of the ribbon. In all
simulations, the graphene dielectric properties are modeled using a surface
conductivity formalism Hanson (2008), with a mobility set to 500
cm2/V$\cdot$s, a value consistent with graphene grown via chemical vapor
deposition (CVD) and fitted experimentally in previous measurements of
graphene plasmons Hong  _et al._ (2022); Jang  _et al._ (2014); Kim  _et al._
(2018); Kobayashi  _et al._ (2013); Pirkle  _et al._ (2011).

We note that interactions between an emitter and nearby graphene occur through
three main mechanisms depending on the Fermi energy to which the graphene is
tuned Cox  _et al._ (2012); Karanikolas, Marocico, and Bradley (2015);
Tielrooij  _et al._ (2015); Cano  _et al._ (2020); Koppens, Chang, and García
de Abajo (2011); Biehs and Agarwal (2013); Manjavacas  _et al._ (2012). When
the Fermi energy $E_{F}$ is less than half the emitter energy, that energy is
transferred to graphene by exciting an electron-hole pair. At higher values of
$E_{F}$, the emitter instead transfers most of its energy to free space in the
form of propagating photons, or it is damped by Ohmic loss in the graphene
sheet. At still higher values of Fermi energy, when $E_{F}$ exceeds
$\sim$70$\%$ of the emission energy, the emission couples to plasmonic modes
supported by the graphene. All three can, in principle, contribute to the
enhancement of the decay rate; however, the enhancement by plasmonic modes is
orders of magnitude larger than the other interaction mechanisms, and when two
photons/plasmons are considered, this difference is increased further, such
that decay pathways via two-plasmon transitions dominate. We therefore
identify the emission enhancement due to graphene as Purcell enhancement
throughout this text, although other non-radiative pathways make minor
contributions.

In Figure 2(a), we plot $P_{x}(\omega,\textbf{r})$ for the case of a dipole
emitter placed 5 nm beneath a 20 nm wide graphene nanoribbon, with the dipole
axis oriented along the short axis of the ribbon. The effects of changing the
emitter’s position beneath the ribbon, the emitter’s orientation, or the
ribbon’s width are all discussed in the Supplemental Material for this paper.
For a given value of $E_{F}$, it is observed that $P_{x}(\omega,\textbf{r})$
shows two maxima of magnitude $10^{5}$ \- $10^{6}$, which correspond to the
1st and 2nd order plasmonic resonances supported by the graphene nanoribbon,
consistent with previous works Koppens, Chang, and García de Abajo (2011).

For 2PSE, emission of one photon at frequency $\omega$ is accompanied by
another at $\omega_{0}-\omega$, and the total enhancement is determined by the
factor $P_{i}(\omega,\textbf{r})\times P_{j}(\omega_{0}-\omega,\textbf{r})$.
When both photons are matched to plasmonic resonances, the 2PSE shows strong
enhancement at particular energies; conversely, when one or more photons are
not matched to a plasmonic resonance, there is suppression of 2PSE. This
behavior is illustrated in Figure 2(b), where it is shown that for
$E_{F}=0.565$ eV there is a strong enhancement of the differential 2PSE near
0.4 eV (3.1 $\mu$m) due to the plasmonic resonance supported in the graphene
nanoribbon at that energy, which simultaneously enhances the emission of both
2PSE photons. Likewise, when $E_{F}$ is tuned to 0.385 eV, plasmonic
resonances are supported in the nanoribbon at both 0.32 and 0.49 eV. These
support simultaneous emission of photons at both energies which, combined,
equal the overall transition energy. This process leads to correspondingly
large differential 2PSE enhancements at those photon energies. For an $E_{F}$
of 0.385 eV, however, the nanoribbons support resonances at 0.34 and 0.53 eV,
and thus each resonant emission must be accompanied by a non-resonant process
with minimal enhancement, causing the differential 2PSE to be low at both
energies. This behavior is further illustrated in Figure 2(c), where the
wavelength-integrated 2PSE enhancement is plotted as a function of $E_{F}$,
showing maxima when the nanoribbons support plasmonic resonances at energies
that can be summed or doubled to equal the overall Er3+ transition energy.

The localized resonances in graphene nanoribbons provide energy selectivity to
the 2PSE process and also aid in free-space emission; however, their
relatively low quality factors ($Q\sim 10$) make it such that they do not
enhance 2PSE substantially above what is realized in unpatterned graphene,
which supports highly confined plasmonic modes over a large bandwidth. This is
illustrated in Figure 2(d,e,f), where the differential 2PSE enhancement is
compared between bare graphene and graphene nanoribbons over a range of
carrier densities. Both exhibit enhancements of 1010-1011, and while the
differential enhancement at $\frac{1}{2}\omega_{0}\sim 0.4$ eV can be greater
for tuned nanoribbons than unpatterned graphene, the total 2PSE enhancement is
comparable for both geometries.

The above analysis considers only emitters with dipole orientations along the
x-axis as defined in Figure 1, which most effectively couple to the plasmonic
resonances of the ribbon. Aligning the dipole along the y- and z-axes allows
for emission into the unbound plasmonic resonances that propagate along the
length of the ribbon and behave identically to the unpatterned graphene, and
into localized “dark” mode resonances that have no net dipole moment but can
still couple to nearby emitters Koppens, Chang, and García de Abajo (2011).
The overall enhancement factors for different dipole orientations do not vary
significantly; however, the “dark” resonances do occur at different
frequencies than the resonances observed for an x-polarized dipole (see
Supplemental Material). As a result, when all dipole orientations are averaged
together (including cross terms), the net effect of the nanoribbons is a Fermi
level-dependent 2PSE enhancement that has less well-defined maxima, but that
reaches an enhacement of $4\times 10^{10}$ and $5\times 10^{10}$ for the
nanoribbons and unpatterned graphene, respectively, as shown in Figure 4(a).
The lateral position of the emitter also affects the magnitude of the Purcell
enhancement (see Supporting Information). Emitters located in the middle of
the gap between two ribbons experience approximately an order of magnitude
less enhancement than those directly beneath the nanoribbons.

Figure 3: Enhancement of two-photon emission. All results consider an Er3+
emitter 5 nm away from graphene structures, with all dipole orientations
averaged according to Eq. (4). (a) Enhancement of purely radiative (photon-
photon) emission. (b) Enhancement of purely radiative (photon-photon) emission
in the 3 - 3.2 $\mu$m range.

Although 2PSE is dramatically enhanced in both patterned and unpatterned
graphene, the majority of the emission occurs in the form of confined
plasmonic modes, which mostly decay non-radiatively. However, some fraction of
the emitted plasmons weakly couple to free space as far-field photons. In
Figure 3 we calculate the total, Fermi energy-dependent, enhancement of 2PSE
from Er3+ that results in far-field photon pairs due to continuous graphene
and graphene nanoribbons. These results, which consider randomized dipole
orientations, were performed by monitoring far-field radiation from a dipole
with and without the presence of graphene. The resulting far-field enhancement
factors were then utilized in place of the $P_{i}(\omega,r)$ factors in Eq.
(4) to obtain the graphene nanoribbon enhancement of two-photon far-field
emission. We find only modest enhancements, with the continuous graphene sheet
providing no benefit, and the graphene nanoribbons enhancement reaching $\sim
35$. However, as shown in Figure 3(b), the differential enhancement exceeds
400 when emitting near $\frac{1}{2}\omega_{0}\sim 0.4$ eV (3.1 $\mu$m), where
the emitted photon pairs have similar energies. Our calculations show that 20
nm wide nanoribbons display substantially more enhancement than 14 nm
nanoribbons, despite the latter supporting larger confinement factors in the
resonant modes at $\sim 0.4$ eV (3.1 $\mu$m). We attribute this behavior to
the better free-space photon-plasmon wavelength matching of the 20 nm
nanoribbons. However, we note that even with enhancement factors of 400, the
free-space 2PSE is still six orders smaller than conventional single-
transition decay for Er3+. The degree of free-space emission enhancement can
reach $\sim 10^{6}$ when we instead consider emission in which one decay
radiates to the far-field and the other decays non-radiatively, creating a
non-entangled, down-converted photon; this behavior is demonstrated in the
Supplemental Material of this paper.

Figure 4: Comparison of single-transition decay and two-photon emission. All
results consider an Er3+ emitter 5 nm away from graphene structures, with all
dipole orientations averaged according to Eq. (4). (a) Enhancement
$\Gamma/\Gamma_{0}$ of total two-photon emission relative to a dipole with no
graphene present. (b) Enhancement of single-transition decay at 1538 nm
relative to a dipole with no graphene present. (c) Total single-transition
decay rate compared to total two-photon emission rate in the presence of 20 nm
ribbons and a continuous graphene sheet. (d) Fraction of total decay rate that
can be attributed to 2PSE for 20 nm graphene ribbons.

While we find the production of free-space photons to be inefficient via
graphene-enhanced 2PSE, it is interesting to investigate how it affects the
overall lifetime of the Er3+. In order to understand the relative contribution
of the 2PSE, it is necessary to also calculate the enhancement by graphene of
conventional single-transition decay for the Er3+, which is found by
calculating the Fermi level-dependent factor $P_{i}(\omega,\textbf{r})$ at
$\omega=0.8$ eV with the dipole 5 nm from the graphene surface, averaged over
all orientations. Figure 4(b) shows the results of those calculations for a
continuous sheet of graphene, as well as nanoribbbons of 14 nm and 20 nm
widths, revealing enhancement factors of $\sim 10^{4}$, consistent with
previously published predictions Koppens, Chang, and García de Abajo (2011).
The total emission rates of both single-transition decay and 2PSE can be
obtained by multiplying their respective enhancement factors by their free-
space rates, which is shown in Figure 4(c). These results show that while the
2PSE enhancement is six orders higher than single-transition enhancement for
Er3+, the overall 2PSE rate is still less than single-transition decay by more
than a factor of 10. These results are due to the large difference in initial
base rates between single-transition decay and 2PSE, which differ by more than
$10^{8}$. A similar result regarding 14 nm ribbons is shown in the
Supplemental Material of this paper. Finally, Figure 4(d) shows the Fermi
level-dependent fraction of 2PSE compared to the total decay rate of Er3+
modified by 20 nm graphene ribbons. It can be seen that the 2PSE can exceed
2.5% of the total decay at an optimal Fermi energy of $E_{F}\sim 0.5$ eV.

##  IV Enhancement of 2SPE in Sm2+ and Tm2+

The low relative amounts of 2PSE achievable in Er3+ via graphene plasmon
enhancement are due primarily to the low initial base rates of 2PSE, as well
as the large amount of enhancement in competing single-transition decay
processes. This motivates the search for other emitters where one or both of
those factors are mitigated. To this end, we investigate other rare earth ions
that are potentially better suited for realization of a 2PSE-dominant emission
source. In particular, one factor contributing to the low base 2PSE rates in
Er3+ is the large energy spacing between the low-lying 4f11 states and the
dipole-allowed 4f105d and 4f106s states. This causes the
$\omega_{n}-\omega_{e}$ term in the denominator of Eq. (1) to be much larger
than the single-photon emission frequency $\omega_{0}$, reducing the overall
2PSE rate. In divalent rare earth ions, the intermediate states have much
lower relative energies above the ground level when compared to their
trivalent counterparts Dorenbos (2003). We consider two such divalent ions,
Sm2+ and Tm2+. Both of these ions have been synthesized and measured in
photoluminescence experiments in fluorides as well as other compounds Suta and
Wickleder (2019). The first dipole-allowed excited states in these ions are at
approximately 24500 cm-1 and 25000 cm-1 relative to the ground state,
respectively, which are lower than the corresponding Er3+ level by a factor of
more than two. Experimental measurements of Sm2+ ions in SrF2 show single-
photon emission at 14350 cm-1 with a rate of 83.3 s-1 when temperature
conditions are optimal ($<$45 K) Alam and Di Bartolo (1967). For Tm2+ ions in
CaF2, the corresponding process occurs at 8966 cm-1 with a rate of 278 s-1 at
liquid nitrogen temperatures Kiss (1962). Applying Eq. (1) to these ions,
using oscillator strengths computed through Cowan’s code and energy levels
adjusted to match the experimentally measured first dipole-allowed level
Martin, Zalubas, and Hagan (1978), we find the free-space 2PSE rates for Sm2+
and Tm2+ to be 1.928$\times$10-6 s-1 and 2.386$\times$10-6 s-1, respectively.
These base rates for Sm2+ and Tm2+ are closer to the experimentally measured
single-transition decay rates by factors of 9.8 and 3.6, respectively, in
comparison to Er3+.

Next, we explore the relative enhancements by graphene plasmons of 2PSE and
single-transition decay in Sm2+ and Tm2+ using the same methodology described
in Section III. Both the single-transition decay and 2PSE rates for the
divalent emitters 5 nm beneath 20 nm graphene ribbons are shown in Figure 5,
computed here using the theoretical single-photon emission frequencies of
15223 cm-1 (1.89 eV) for Sm2+ and 8421 cm-1 (1.04 eV) for Tm2+ Cowan (1981).
These results show that for Sm2+, the 2PSE rate can equal the single-
transition decay rate, while for Tm2+ the 2PSE rate is able to reach nearly
20% of the single-transition rate. Both ions, however, require higher Fermi
levels to achieve a maximal 2PSE fraction. For Tm2+, the optimal Fermi energy
of 0.65 eV is beyond what is achievable via electrostatic gating, but can be
reached with ionic liquid gating schemes Fang  _et al._ (2014); Siegel  _et
al._ (2021); Tielrooij  _et al._ (2015); for Sm2+, however, Fermi energies
exceeding 1.1 eV are required, which is potentially realizable using chemical
doping techniques Khrapach  _et al._ (2012); Song  _et al._ (2014).

Figure 5: Comparison of total single-transition decay and 2PSE rates for (a)
Sm2+ and (b) Tm2+ when modified by 20 nm graphene ribbons. All results
consider an emitter 5 nm away from graphene structures, with all dipole
orientations averaged according to Eq. (4).

##  V Discussion

Our calculations predict 2PSE rates via graphene plasmon enhancement that are
significantly lower than previously published results, which predicted that
2PSE could easily exceed single-transition decay rates when enhanced with
surface modes in 2D materials Muniz  _et al._ (2020); Rivera  _et al._ (2016,
2017). There are two primary reasons for this difference. First, previous
works studied emitter systems with large transition moment lengths
($\textbf{d}=\bra{f}\textbf{x}\ket{i}$), such as the 4s $\rightarrow$ 3s
transition of hydrogen. Using the 5p state as the intermediate and
incorporating all three dipole polarizations, the hydrogen 4s $\rightarrow$ 3s
transition has an average transition moment length of $\sim 2.3$ Å between the
two transitions that occur. Such systems have intrinsically large 2PSE base
rates, since the 2PSE rate approximately scales as $\textbf{d}^{4}$ Rivera
_et al._ (2017). For the 4s $\rightarrow$ 3s transition, for example, the 2PSE
base rate is $\sim 10^{-2}\,\mathrm{s}^{-1}$. However, coupling an isolated
hydrogen atom to a graphene surface that is a few nanometers away is
experimentally challenging. In contrast, fluorescent solid state “color
centers” such as those we study here — which have larger atomic numbers — have
substantially smaller dipole lengths, leading to lower overall 2PSE base
rates; for Er3+, each intermediate has an average transition moment length of
around 0.4 Å or lower, and as shown in Section II, this emitter has a 2PSE
base rate of $2.943\times 10^{-7}\,\mathrm{s}^{-1}$. Thus, while it is
experimentally easier to couple graphene to solid state emitters, the 2PSE
enhancement must be much larger in order to be comparable to competing single-
transition processes. Second, our calculations use a graphene model that
considers a carrier mobility of 500 cm2/V$\cdot$s, which is consistent with
what is typical for graphene samples grown via chemical vapor deposition and
placed on oxide surfaces Hong  _et al._ (2022); Jang  _et al._ (2014); Kim
_et al._ (2018); Kobayashi  _et al._ (2013); Pirkle  _et al._ (2011) but is
lower than mobilities considered in previous theoretical works, which assumed
mobilities of 10000 cm2/V$\cdot$s Muniz  _et al._ (2020). The effect of the
lower mobility is to increase the single-channel decay rate enhancement, and —
for the graphene nanoribbons — to decrease the 2PSE enhancement. This leads to
a smaller fraction of ions exhibiting 2PSE in comparison to single-channel
processes.

Thus, a main conclusion of this work is that when achievable material systems
are considered, the prospects of realizing 2PSE as a dominant emission process
are reduced. However, our calculations also indicate the conditions where the
effects of 2PSE may be observable in experimentally testable systems. For
example, rare earth doped substrates prepared via ion implantation or
molecular beam epitaxy can be engineered to place ions at a fixed distance
from an overlying graphene sheet. Lifetime-dependent emission measurements of
such sample geometries have confirmed modification of the overall single-
transition rate of Er3+ by both interband and intraband processes in graphene
Tielrooij  _et al._ (2015); Cano  _et al._ (2020). Our calculations indicate
that similar experiments can be used to search for evidence of 2PSE by probing
the Fermi energy-dependent lifetime of Er3+ at 1538 nm, which should be
affected by 2PSE near $E_{F}\sim 0.5$ eV, as shown in Figure 4(d). The
presence of such a 2PSE contribution to overall decay rates would require
lifetime measurements with a time resolution of $\sim$ 1 ns. Moreover, it is
critical in such measurements that the Er3+ have a highly uniform distance
distribution from the graphene. Previous results were performed using
experimental configurations with $\sim$ 1 $\mu$s time resolution and with a
broad distribution of distances, such that the signal was dominated by ions
located further from the graphene, which are more likely to emit radiatively
than through a plasmonic surface mode Tielrooij  _et al._ (2015); Cano  _et
al._ (2020). We note that such measurements would also be aided by measuring
high mobility samples, where the substrate is treated or engineered to
minimize charge disorder. Larger mobilities would suppress the enhancement of
single-transition processes and allow 2PSE to play a larger relative role,
making detection easier.

Direct detection of photons emitted from rare earth ions via 2PSE is also
possible; however, only a small fraction ($\sim 10^{-9}$) of plasmon pairs
generated via 2PSE are emitted to free space as photon pairs. In the limit of
high excitation rates, the number of emitted photon pairs in the 3 - 3.2
$\mu$m wavelength range can be estimated to be 222 photon pairs per second.
This assumes a Y2O3 substrate with a doped plane of 0.2% Er3+ that is 5 nm
below an array of 20 nm graphene nanoribbons with $E_{F}=0.6$ eV and with
collection over a 10 $\mu$m $\times$ 10 $\mu$m area. These numbers can be
improved by using larger dopant concentrations and collection areas, and
larger graphene nanoribbons which have lower Purcell enhancements but more
effectively couple plasmonic modes to free space. However, when emitters are
considered that are placed at random lateral positions relative to the
graphene nanoribbons, the overall 2PSE will be reduced.

Another possible route to verify our theoretical results is by probing two-
photon absorption, the inverse process to 2PSE. In a two-photon absorption
(2PA) measurement, two photons with frequencies summing to $\omega_{0}$ are
incident on an atomic system simultaneously, elevating an electron to a higher
energy level from which it can decay Lin and Vučković (2010). The efficiency
of this process is also dependent on $P_{i}(\omega,\textbf{r})\times
P_{j}(\omega_{0}-\omega,\textbf{r})$, which, for single-frequency excitation,
must satisfy the condition $\omega=\omega_{0}/2$. Thus, the large Purcell
enhancements supported by graphene will also drive large 2PA, which can be
monitored by measuring fluorescence at $\omega_{0}$. For graphene systems
coupled to rare earth ions, it would be required to perform 2PA measurements
on nanoribbons or some other patterned nanostructure, as the plasmonic modes
in a continuous sheet would not couple to free-space excitation.

Our calculations also point to the importance of utilizing emitters with large
initial 2PSE base rates for experiments whose goal is to boost 2PSE via
Purcell enhancement to a significant fraction of overall emission. While
previous works showed that atomic hydrogen can achieve this condition, the
experimentally viability of such an approach is low. Nevertheless, other
emitter systems could be considered, including fluorescent dyes such as
ruthenium complexes and rhodamine 123, and fluorescent biomolecules, such as
green fluorescent protein chromophore. These emitters are expected to have
transition dipole moments corresponding to lengths of 1.1 - 1.3 Å Stark  _et
al._ (2019), 1.7 Å Chung, Tregidgo, and Suhling (2016), and 1.4 Å Bublitz,
King, and Boxer (1998), respectively. Alternatively, our results show that a
detailed consideration of the intermediate energy levels allows for
identification of emitters with larger 2PSE base rates. Specifically, we find
that emitters with intermediate energy levels that are near the excited state,
such as divalent rare earths, have larger 2PSE rates which, via Purcell
enhancement, can approach single-transition rates.

##  VI Conclusion

We have calculated the 2PSE base rates for Er3+, Sm2+, and Tm2+, and, using
full field simulations, determined the degree to which Purcell enhancement
from nearby graphene and graphene nanoribbons can enhance overall 2PSE. We
show that the overall 2PSE rate enhancement of Er3+ ions can exceed $4\times
10^{10}$ and that it is strongly dependent on the graphene Fermi level.
Moreover, we show that the production of entangled free-space photon pairs
near 3.1 $\mu$m via 2PSE can be enhanced by 400 by graphene nanoribbons. In
contrast to previous studies of hydrogen, we find that the 2SPE rate for Er3+
does not become a significant fraction of the overall decay, and we attribute
this discrepancy to the low initial 2PSE base rate for Er3+ and the lower
graphene mobility used in our calculations. These two factors, however,
represent realistic experimental conditions, and we outline ways in which 2PSE
could be observed in Er3+-doped substrates coated with graphene; such systems
have already been explored experimentally in regimes where single-transition
processes are affected by graphene. Finally, we find that controlling the
emitter species can affect the 2PSE contribution to overall emission.
Specifically, we show that divalent rare earths that have intermediate states
close in energy to the initial excited state have higher overall base rates of
2PSE, and we find that for Sm2+ and Tm2+, Purcell enhancement can be used to
make 2PSE a large overall fraction of the decay.

These findings provide specific guidance for future experiments that aim to
generate photon pairs via 2PSE using Purcell enhancement. Moreover, they can
be used to engineer next-generation devices that produced entangled photons at
high rates for quantum information science applications.

##  VII Acknowledgments

This material is based upon work supported by the U.S. Department of Energy
Office of Science National Quantum Information Science Research Centers as
part of the Q-NEXT center, which supported the work performed by C. Whisler
and V. W. Brar. G. Holdman was supported by a Defense Advanced Research
Projects Agency Young Faculty Award (YFA D18AP00043). D. D. Yavuz was
supported by the Vilas Associates Award of the UW-Madison.

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