  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 Er$^{3+}$, 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 Er$^{3+}$ 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.
