Duality, decay rates, and local-field models in macroscopic QED

Any treatment of magnetic interactions between atoms, molecules, and optical media must start at the form of the interaction energy. This forms the base on which predictions about any number of magnetic atom-light properties stands, spontaneous decay rates and forces included. As is well known, the Heaviside-Larmor duality symmetry of Maxwell's equations, where electric and magnetic quantities are exchanged, is broken by the usual form of the magnetic interaction energy. We argue that this symmetry can be restored by including general local-field effects and that local fields should be treated as a necessity for correctly translating between the microscopic world of the dipole and the macroscopic world of the measured fields. This may additionally aid in resolving a long-standing debate over the form of the force on a dipole in a medium. Finally, we compute the magnetic dipole decay rate in a magnetodielectric with local-field effects taken into account and show that macroscopic quantum electrodynamics can be made to be dual symmetric at an operator level, instead of only for expectation values.

Figure 1

(a) The magnetic induction $\widehat{\mathbf{B}}$ from a solenoid is unchanged by the presence of a paramagnetic medium. (b) The corresponding magnetic field $\widehat{\mathbf{H}}$, which is screened by the paramagnetic medium, is represented by a thinner line.

Figure 2

(a) Relative permittivity and permeability described by Eqs. (13) and (14). (b) Corresponding complex index of refraction $n\left(\omega \right)=\sqrt{\varepsilon \left(\omega \right)\mu \left(\omega \right)}=\eta \left(\omega \right)+i\kappa \left(\omega \right)$.

Figure 3

(a) Purcell factor ${\gamma }_H/{\gamma }_0$ as a function of dipole resonance ${\omega }_A$ in the example medium seen in Fig. 2. (b) Purcell factor ${\gamma }_B/{\gamma }_0$ as a function of dipole resonance ${\omega }_A$ in the example medium seen in Fig. 2. In both cases, we have chosen the spherical radius $R_{\text{sphere}}$ such that $R_{\text{sphere}}^3=3R^3/4\pi$. With reference to an electric resonance at ${\omega }_{\text{Te}}=2\pi c/100\text{nm}$, $R_{\text{sphere}}\simeq \{1.27,3.82,12.7\}$ Å, respectively.

Figure 4

(a) Purcell factor $\gamma /{\gamma }_0$ as a function of dipole resonance ${\omega }_A$ in the example medium seen in Fig. 2. As with Fig. 3, we have chosen the spherical radius $R_{\text{sphere}}$ such that $R_{\text{sphere}}^3=3R^3/4\pi$ and chosen $R_{\text{sphere}}\simeq \{1.27,3.82,12.7\}$ Å, respectively, with reference to an electric resonance at ${\omega }_{\text{Te}}=2\pi c/100\text{nm}$. (b) Comparison of Purcell factors ${\gamma }_H/{\gamma }_0$, ${\gamma }_B/{\gamma }_0$, and $\gamma /{\gamma }_0$ as a function of dipole resonance ${\omega }_A$ in the example medium seen in Fig. 2.