Dual-Lagrangian description adapted to quantum optics in dispersive and dissipative dielectric media

Abstract

We develop a dual description of quantum optics adapted to dielectric systems without magnetic property. Our formalism, which is shown to be equivalent to the standard one within some dipolar approximations discussed in the article, is applied to the description of polaritons in dielectric media. We show that the dual formalism leads to the Huttner-Barnett equations [B. Huttner, S. M. Barnett, Phys. Rev. A 46, 4306 (1992)] for QED in dielectric systems. More generally, we discuss the role of electromagnetic duality in the quantization procedure for optical systems and derive the structure of the dynamical laws in the various representations.

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

“…(x, t) corresponds to the total scattered field induced by P which depends on the Green dyadic propagator ∆ (v) ret. (τ, x, x ′ ) in vacuum [71,72]. We have explicitly…”

“…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.…”

“…(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].…”

“…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.…”

“…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].…”