Theoretical study of the enhancement of spontaneous two-photon emission in Er³⁺ by graphene plasmons

Spontaneous two-photon emission (STPE) occurs when an excited emitter decays through the spontaneous production of two photons rather than one photon. This process provides an intriguing potential source of entangled photons with a broad distribution of frequencies. In most cases, the rate of STPE is dramatically slower than that of single-photon emission; however, it has recently been predicted that, for hydrogenic atoms, the optical surface waves in 2D materials could be used to significantly increase the STPE rate. Here we investigate the efficiency of this process for the ⁴I_13/2 → ⁴I_15/2 transition of trivalent erbium, which occurs around 1550 nm and is used in telecommunications. We perform a comprehensive calculation of all matrix elements involved in the STPE process and couple this with finite-difference time-domain simulations of the emission rate enhancement of an erbium emitter placed near a graphene sheet. We derive experimental conditions whereby graphene plasmons augment the STPE rates to become comparable to single-photon transition rates. Furthermore, we show that patterning the graphene into an array of nanoribbons allows the radiative two-photon emission to become strongly enhanced when the graphene’s Fermi energy is tuned to the proper resonance value.