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

Abstract

We provide theory and formal insight on the Green function quantization method for absorptive and dispersive spatial-inhomogeneous media in the context of dielectric media. We show that a fundamental Green function identity, which appears, e.g., in the fundamental commutation relation of the electromagnetic fields, is also valid in the limit of nonabsorbing media. We also demonstrate how the zero-point field fluctuations yields a nonvanishing surface term in configurations without absorption, when using a more formal procedure of the Green function quantization method. We then apply the presented method to a recently developed theory of photon quantization using quasinormal modes [Franke et al., Phys. Rev. Lett. 122, 213901 (2019)] for finite nanostructures embedded in a lossless background medium. We discuss the strict dielectric limit of the commutation relations of the quasinormal mode operators and present different methods to obtain them, connected to the radiative loss for nonabsorptive but open resonators. We show exemplary calculations of a fully three-dimensional photonic crystal beam cavity, including the lossless limit, which supports a single quasinormal mode and discuss the limits of the commutation relation for vanishing damping.

Quantized quasinormal-mode description of nonlinear cavity-QED effects from coupled resonators with a Fano-like resonance

“…accounts for the radiative loss, whereH μ (s, ω) = ∇ × F μ (s, ω)/(iωμ 0 ) is the regularized QNM magnetic field andn points outward of S. Importantly, we have radiative loss even if the system has no material loss, as explained in more detail in Ref. [57]. Numerically, we have verified that the radiative part of S has to be chosen in the far field (i.e., at least half a wavelength away from the resonator) in order to get a convergent value of S rad μμ .…”

“…where k 0 = ω/c, and (r, ω) = R (r, ω) + i I (r, ω) is the dielectric permittivity. The noise current densityĵ N (r, ω) = ω √h 0 I (r, ω)/π b(r, ω) counteracts the dissipation, such that the commutation relations between the electromagnetic field operators are spatially preserved for the dissipative materials [42,54,55] as well as nondissipative dielectrics [56,57]. A formal solution of Eq.…”

Quantized quasinormal mode theory of coupled lossy and amplifying resonators

“…In this part, we summarize the main results from the lossy QNM quantization. For the case of purely lossy media (which includes the limit of a lossless dielectrics [42]), the noise current density ĵN (r, ω) is equal to the lossinduced noise ĵL N (r, ω), extending over all space, so that…”

“…is a dissipation-induced QNM overlap matrix, which yields a positive definite form [20,42]. After applying the limit α → 0, S µη can be written as a sum S µη = S nrad µη + S rad µη , where [20,42]…”

“…However, the presence of a continuum of photon noise operators usually prevents the use of the theory for light-matter interaction beyond the weak coupling limit. For purely lossy media, this problem was recently tackled via a dissipative mode quantization scheme [20,42,43] based on a quasinormal mode [44][45][46][47][48] (QNM) expansion, which allowed for a construction of a discrete photon Fock basis for a lossy resonator problem.…”