Quantum electrodynamics of lossy samples in vacuum: modified Langevin noise formalism

Quantum behavior of the electromagnetic field in unbounded macroscopic media displaying absorption is properly described by the Langevin noise formalism (macroscopic quantum electrodynamics) where the field is assumed to be entirely produced by medium fluctuating sources via the dyadic Green's function. On the other hand, such formalism leads to inconsistencies when applied to finite-size lossy objects placed in vacuum since it does not take into account the scattering modes, whose homomeneous plane wave asymptotic behavior is ruled out in unbounded lossy media. Accordingly a modified Langevin noise formalism has been proposed to encompass the scattering modes and its consistency has been numerically validated in few specific geometries. In this paper we analytically derive the modified Langevin noise formalism from the enstablished canonical quantization of the electromagnetic field in macroscopic media, thus proving that it models any possible scenario involving linear, inhomegeneous and magnetodielectric samples. The derivation starts from quantum Maxwell equations in the Heisenberg picture together with their formal solution as the superposition of the medium assisted field and the scattering modes. We analytically prove that each of the two field parts can be expressed in term of particular bosonic operators, which in turn diagonalize the electromagnetic Hamiltonian and whose associated quasi-particles are medium assisted and scattering polaritons, respectively. The key ingredient underpinning our reasoning is a peculiar integral relation linking the far field amplitude of the dyadic Green's function and the scattering modes, relation we rigorously derive and physically explain by identifying the scattering modes as fields generated by infinitely far dipole point sources.