Quantizing polaritons in inhomogeneous dissipative systems

Abstract

In this article we provide a a general analysis of canonical quantization for polaritons in dispersive and dissipative electromagnetic media. We compare several approaches based either on the Huttner Barnett model [B. Huttner, S. M. Barnett, Phys. Rev. A \textbf{46}, 4306 (1992)] or the Green function, Langevin noise method [T. Gruner, D.-G. Welsch, Phys. Rev. A \textbf{53}, 1818 (1996)] which includes only material oscillators as fundamental variables. We show in order to preserve unitarity, causality and time symmetry one must necessarily include with an equal footing both electromagnetic modes and material fluctuations on the evolution equations. This becomes particularly relevant for all nanophotonics and plasmonics problems involving spatially localized antennas or devices.Comment: This paper is submitted for publication (any comments are welcome

Equivalence between the Hamiltonian and Langevin noise descriptions of plasmon polaritons in a dispersive and lossy inhomogeneous medium

“…In this description f (0) ω and f †(0) ω are respectively lowering and rising bosonic vector field operators associated with the fluctuating bath of material oscillators, i.e., rigorously equivalent to those operators given in the standard DLN approach. Moreover, in [71][72][73] we showed that these noise operators are related to the total field operators at the initial time t 0 , i.e., f (0) ω (x, t) = f ω (x, t 0 )e −iω(t−t0) . This is essential since the choice of retarded causal Green functions involves necessarily a boundary condition in the remote past at t 0 < t. Therefore as discussed in [72] our formalism preserves time symmetry and allows other equivalent descriptions involving 'advanced' Green functions and boundary conditions at a future time t f > t. The present choice is of course dictated by physical considerations not part of QED but connected to thermodynamics and cosmology.…”

“…Moreover, in [71][72][73] we showed that these noise operators are related to the total field operators at the initial time t 0 , i.e., f (0) ω (x, t) = f ω (x, t 0 )e −iω(t−t0) . This is essential since the choice of retarded causal Green functions involves necessarily a boundary condition in the remote past at t 0 < t. Therefore as discussed in [72] our formalism preserves time symmetry and allows other equivalent descriptions involving 'advanced' Green functions and boundary conditions at a future time t f > t. The present choice is of course dictated by physical considerations not part of QED but connected to thermodynamics and cosmology. We also point out that in the general case, i.e., when external systems such as fluorescent molecules are coupled to the fields we have to add to P a contribution P (mol.)…”

“…We start with the canonical description given in [71,72] in the Heisenberg picture. It is based on a dual formalism involving an electric potential vector operator F(x, t) such that ∇ · F = 0 (dual Coulomb gauge) and D = E + P = ∇ × F, where D is the transverse displacement field, E the electric field, and P the total dipole density of the medium.…”

“…We also point out that in the general case, i.e., when external systems such as fluorescent molecules are coupled to the fields we have to add to P a contribution P (mol.) (x, t) [73] which we let here unspecified. Within this dual formalism we can show that the electromagnetic field operators satisfy Maxwell's equations and, since both D and the magnetic field B are transverse, it is not necessary to make the distinction between transverse quantized and longitudinal otherwise un-quantized fields [71].…”

“…Few years ago, it was proposed that the equivalence between the Hamiltonian and DLN approaches should finally be rigorous [64][65][66][67][68][69][70]. However, we recently showed [71][72][73] that a full Hamiltonian description, generalizing the Huttner-Barnett results [14][15][16][17][18][19][20] and valid for any inhomogeneous dielectric systems, must not only include the material oscillator degrees of freedom, i.e., like in the DLN method, but also add the previously omitted quantized photonic degrees of freedom associated with fluctuating optical waves coming from infinity and scattered by the inhomogeneities of the medium [72]. Furthermore, the inclusion of both photonic and material fluctuations on a equal footing appears necessary in order to preserve the full unitarity of the quantum evolution and to conserve time symmetry [56].…”