Quantum electrodynamics near a dispersive and absorbing dielectric

We build up a consistent theory of quantum electrodynamics in the presence of macroscopic polarizable media. We use the Huttner-Barnett model of a dispersive and absorbing dielectric medium and formulate the theory in terms of interacting quantum fields. We integrate out the damped polaritons by using diagrammatic techniques and find an exact expression for the displacement-field (photon) propagator in the presence of a dispersive and absorbing dielectric half-space. This offers a route to traceable perturbative calculations of the same kind as in free-space quantum electrodynamics. As a worked-through example, we consider the interaction of a neutral atom with a dispersive and absorbing dielectric half-space. For that, we use the multipolar coupling $\mu ·\mathbf{D}$ of the atomic dipole moment to the electromagnetic displacement field. We apply this formalism to calculate the one-loop correction to the atomic electron propagator and to find the energy-level shift and changes in the spontaneous decay rates for a neutral atom close to an absorptive dielectric mirror.

Figure 1

The atomic dipole is located a distance $\mathcal{Z}$ away from the dielectric half-space of complex and frequency-dependent permittivity $\epsilon \left(\omega \right)$. The transverse propagator $D_{ik}(\mathbf{r},{\mathbf{r}}^{\prime };\omega )$ of the dielectric displacement field in this geometry is given by Eq. (96).

Figure 2

Plot of the coefficients $c_i^{\sigma }\left(n\right)$ that enter Eq. (118) for different values of the static refractive index $n\equiv n\left(0\right)$.

Figure 3

Plot of the exact ground-state energy shift (contributions due to the perpendicular component of the atomic dipole moment) $\Delta E_g^{\parallel }$, Eq. (111), multiplied by ${\mathcal{Z}}^4{\omega }_{mg}$ as a function of $\mathcal{Z}{\omega }_{mg}$ for various values of the damping parameter $\gamma$. The solid line represents the energy shift caused by the nonabsorptive and nondispersive dielectric half-space with static refractive index $n\left(0\right)=\sqrt{2}$.

Figure 4

Normalized lifetime ${\tau }_{\parallel }$, Eq. (133), of the atomic state $|i\rangle$ plotted as a function of $|{\omega }_{mi}|/{\omega }_{\mathrm{T}}$. The sequence of graphs corresponds to various distances of the atom from the mirror $\mathcal{Z}{\omega }_{\mathrm{T}}$ as indicated. The three different line styles (colors) indicate distinct choices of the damping constant $\gamma$ in the dielectric: black solid line: $\gamma /{\omega }_{\mathrm{T}}=0.05$; blue dashed line: $\gamma /{\omega }_{\mathrm{T}}=0.5$; and red dot-dashed line: $\gamma /{\omega }_{\mathrm{T}}=5$. For sufficiently high frequencies $|{\omega }_{mi}|$, the dielectric becomes transparent. If the atom is close to the surface and absorption is small, the interaction is resonant at $|{\omega }_{mi}|/{\omega }_{\mathrm{T}}\approx 1$, i.e., when the frequency of the atomic transition coincides with the absorption line of the dielectric.