Hawking effect in dielectric media and the Hopfield model

We consider the so-called Hopfield model for the electromagnetic field in a dielectric dispersive medium in a framework in which one allows a space-time dependence of microscopic parameters, aimed at a phenomenological description of a space-time varying dielectric perturbation induced by means of the Kerr effect. We discuss the analogue Hawking effect by introducing a simplified model which avoids some difficulties which characterize in the full Hopfield model, still keeping the same dispersion relation. Our main result is an analytical calculation of the spontaneous thermal emission in the single-branch case, which is provided nonperturbatively for the first time in the framework of dielectric black holes. A universal mechanism for thermality which is shared both by optical black holes and acoustic black holes is also pointed out.

Figure 1

Asymptotic dispersion relation for the Cauchy case in the comoving frame. The monotone branch is $G_{+}$, the nonmonotone one is $G_{-}$. The dashed line divides antiparticle states (below it) and particle ones (above it). Two lines at $\omega =\text{const}$ and at $-\omega =\mathrm{const}$ are also drawn, and relevant states introduced in the text are explicitly indicated.

Figure 2

Circuits in the complex $s$-plane defining the relevant modes as described in the text.

Figure 3

Asymptotic dispersion relation for the Sellmaier dispersion relation of a diamondlike material (qualitative plot). The lab frame is adopted. Lines of constant $\omega$ are represented by straight lines.