Constitutive relations and adiabatic invariants for electromagnetic waves in a dynamic Lorentz medium

Constitutive relations for Lorentz media with a time-varying density of oscillators and a time-varying intrinsic frequency of oscillators are derived ab initio. For an electromagnetic wave propagating in a dynamic Lorentz medium, combinations of the wave energy and frequency (adiabatic invariants) that are conserved during slow time variations of the medium are obtained. If the oscillator density varies, the invariants differ from the known ones for nondispersive dielectrics and plasmas and have a different form for increasing and decreasing oscillator densities. In both cases, however, they predict that the frequency shift is accompanied by a decrease of the wave energy. If the intrinsic frequency varies, the invariant coincides with the one for a nondispersive dielectric and predicts the wave amplification and attenuation with the increase and decrease of the frequency, respectively. It is also shown that a medium with a decreasing density of oscillating dipoles cannot be described by a time-varying dielectric permittivity $\ensuremath{\varepsilon}(t)$ even in the nondispersive limit.