Enhanced multipolar transitions near 2D material nanoislands

Generally, the spontaneous emission rate of an emitter is computed via classical electromagnetic simulations, modelling the dipolar source as a point. The quantum and classical simulations usually give identical results since the size of the quantum emitter (QE) is orders of magnitude smaller than the wavelength. However, close to nanophotonic structures sustaining strongly confined fields, such as graphene and other 2D materials, the approximation is not valid anymore: the dipolar transition is no longer dominant and can be surpassed by the rates of higher-order transitions such as quadrupolar, two-photons... [1,2]. Engineering such high-order transitions through the complex nanophotonic environment promises new infrared sources, entangled two-photon sources and other novel physical effects.
However, the determination of such rates requires the spatial variation of the Green's function along the spatial extent of the QE, which is not easily accessed by conventional numerical solvers. Therefore, the state of the art is restricted to nanostructures for which the Green's function is analytically known.
Here, we compute the high-order transition rates of a hydrogen-like atom close to graphene nanoislands (square, triangle and crescent geometries), by developing a method valid for any material of any shape. To this end, we combine the quantum-electrodynamic treatment of Rivera et al. [1] with the calculation of the Green's function through the eigenpermittivity expansion method developed by Chen et al. [3]. The formulation rewrites as an integral over the mode profiles (40 modes computed with COMSOL) and the wavefunction, which is faster to compute than 6-dimensional integral over the total Green's tensor. Figure 1 shows from left to right the dipolar (6p - 4s), quadrupolar (6d - 4s) and octupolar (6f - 4s) transition rates of an H-like quantum emitter as a function of its position 5 nm above a graphene equilateral triangle of 50 nm side length. The QE is x-oriented and the free-space wavelength of these transitions is 2.63 µm. In these results, the quadrupolar transitions rates maxima that are forbidden in vacuum are only 2 orders of magnitudes smaller than the dipolar transition rates (as expected in [1]). Furthermore, we show that the quadrupolar transitions rates at the corner of the triangle are 100 times stronger than the dipolar rates, demonstrating a local breakdown of the selection rules.