Quantization of graphene plasmons

In this article we perform the quantization of graphene plasmons using both a macroscopic approach, based on the classical expression for the average electromagnetic energy in a dielectric medium, and a quantum hydrodynamic model, in which graphene electrons are modeled as a charged fluid. Both models allow one to take into account the dispersion in the optical response, with the hydrodynamic model also allowing for the inclusion of the momentum dependence of the optical response (nonlocal effects). Using both methods, the electromagnetic field mode functions, and the respective frequencies, are determined for two different graphene structures. We show how to quantize graphene plasmons, considering that graphene is a dispersive medium, within the local and nonlocal descriptions. It is found that the dispersion of graphene's optical response leads to a nontrivial normalization condition for the mode functions. The obtained mode functions are then used to calculate the decay of an emitter, represented by a dipole, via the excitation of graphene surface plasmon polaritons. The obtained results are compared with the total spontaneous decay rate of the emitter and a near perfect match is found in the relevant spectral range. It is found that nonlocal effects in graphene's conductivity become relevant for the emission rate for small Fermi energies and small distances between the dipole and the graphene sheet.B.A.F. and N.M.R.P. acknowledge support from the European Commission through the project "Graphene-Driven Revolutions in ICT and Beyond" (Ref. No. 785219), and the Portuguese Foundation for Science and Technology (FCT) in the framework of the Strategic Financing UID/FIS/04650/2013. Additionally, N.M.R.P. acknowledges COMPETE2020, PORTUGAL2020, FEDER, and the Portuguese Foundation for Science and Technology (FCT) through Projects No. PTDC/FIS-NAN/3668/2013 and No. POCI-01-0145-FEDER-028114. B.A. acknowledges the hospitality of CeFEMA, where he was a visiting researcher during part of the time in which this work was developed, and financial support from Portuguese Foundation for Science and Technology (FCT) through Project No. CEECIND/02936/2017