Canonical quantization for quantum plasmonics with finite nanostructures

Abstract

The quantization of plasmons has been analyzed mostly under the assumption of an infinite-sized bulk medium interacting with the electromagnetic field. We reformulate it for finite-size media, such as metallic or dielectric nano-structures, highlighting sharp differences. By diagonalizing the Hamiltonian by means of a Lippmann-Schwinger equation, we show the contribution of two sets of bosonic operators, one stemming from medium fluctuations, and one from the electromagnetic field. The results apply to general models including dissipative and dispersive responses.

Critical review of quantum plasmonic models for finite-size media

“…* jauslin@u-bourgogne.fr Our interpretation of this discrepancy is that the former results [1][2][3][4][5][6][7][8][9][10][11] were obtained under the implicit hypothesis of an infinite bulk medium, although it has been extensively extrapolated to finite media in later works. Our results [17] have some conceptual consequences and may lead to quantitative divergences from past studies. The present article aims at addressing in detail the reasons that the formulas obtained for bulk systems cannot be applied to systems with a finite medium.…”

“…Their goal was to introduce dissipation in the quantization of light in a homogeneous (therefore infinite) medium. It was followed by many other works [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] which used two main approaches:…”

“…The main criterion for the choice of the microscopic model is that if one integrates the equations for the medium and one inserts the obtained currents into the microscopic Maxwell equations one should obtain the macroscopic Maxwell equations [10,11]. In a recent work [17] another quantization procedure, designed specifically to treat plasmonic systems with a finite-size medium, was formulated. Starting from the same classical model, this approach leads to results substantially different from the formulas derived and used in the past literature.…”

Fluctuation-dissipation theorem and fundamental photon commutation relations in lossy nanostructures using quasinormal modes

“…A few years after the introduction of the Green function quantization scheme, the method was further confirmed by approaches using a canonical quantization formalism in combination with a Fano-type diagonalization [26,27]. While it was confirmed that, for the case of absorptive bulk media, the theory is consistent with the fundamental axioms of quantum mechanics, e.g., the preservation of the fundamental commutation relations between the electromagnetic field operators [15,28], it has been debated recently, whether the theory can be applied to nonabsorbing media or finite-size absorptive media, e.g., dielectric nanostructures in a nonabsorbing background medium [29,30]. This is an important limit, in which results of the well-known quantization of lossless dielectrics as in the seminal approach from Ref.…”

“…We do this, since the QNMs are not a good representation of the fields outside the scattering geometry, and they need a regularization to prevent spatial divergence [49,50]. Here, we again assume that G ff (r, r , ω) is an accurate approximation to the full Green function inside the resonator geometry region, to obtain the Green function for r, r outside the scattering geometry as [49] G α (r, r ) = G FF,α (r, r ) + G others,α (r, r ) with the QNM contribution (29) and all other contributions associated to the background are summarized in G others,α (r, r ). The α-dependent fields F μ (r, ω; α) are regularized QNM functions,…”