Description of spontaneous photon emission and local density of states in the presence of a lossy polaritonic inhomogeneous medium

We provide a description of spontaneous emission in a dispersive and dissipative linear inhomogeneous medium based on the generalized Huttner-Barnett model [Phys. Rev. A 46, 4306 (1992)]. Our discussion considers on an equal footing both the photonic and material fluctuations which are necessary to preserve unitarity of the quantum evolution. Within this approach we justify the results obtained in the past using the Langevin noise method that neglects the removal of photonic fluctuations. We finally discuss the concept of local density of states in a lossy and dispersive inhomogeneous environment that provides a basis for theoretical studies of fluorescent emitters near plasmonic and polaritonic antennas.

Figure 1

Sketch of the typical contributions to the electric field in the presence of a dielectric medium if there is no additional source ${\mathbf{P}}_{\mathrm{\Psi }}(\mathbf{x},t)$. The electric field ${\mathbf{E}}^{\left(0\right)}(\mathbf{x},t)$ is associated with incident photon modes scattered by the dielectric medium (in the case of a vacuum input field this corresponds to the quantum fluctuations). ${\mathbf{E}}_{\mathrm{eff}.}^{\left(s\right)}(\mathbf{x},t)$ denotes the electric induced by the dipole distribution ${\mathbf{P}}^{\left(0\right)}(\mathbf{x},t)$, i.e., to the material field ${\mathbf{X}}_{\omega }^{\left(0\right)}(\mathbf{x},t)$.

Figure 2

Sketch of the different contributions to the total electric field in presence of a radiating atomic dipole. In (a) the atom is in vacuum and the electric field is separated into a ${\mathbf{E}}^{\left(v\right)}(\mathbf{x},t)$ contribution and a radiation field ${\mathbf{E}}_{\sigma }^{\left(s\right)}(\mathbf{x},t)$ associated with the dipole. In (b) we have a contribution ${\mathbf{E}}^{\left(0\right)}(\mathbf{x},t)$, i.e., associated with scattered photon modes (see Fig. 1), and two contributions of the scattered fields, i.e., induced by the fluctuating current ${\mathbf{E}}_{\mathrm{eff}.,f}^{\left(s\right)}(\mathbf{x},t)$ and the atomic dipole ${\mathbf{E}}_{\mathrm{eff}.,\sigma }^{\left(s\right)}(\mathbf{x},t)$.