Quantum optical effective-medium theory for layered metamaterials

The quantum optics of metamaterials starts with the question whether the same effective-medium theories apply as in classical optics. In general the answer is negative. For active plasmonics but also for some passive metamaterials, we show that an additional effective-medium parameter is indispensable besides the effective index, namely the effective noise-photon distribution. Only with the extra parameter can one predict how well the quantumness of states of light is preserved in the metamaterial. The fact that the effective index alone is not always sufficient and that one additional effective parameter suffices in the quantum optics of metamaterials is both of fundamental and practical interest. Here from a Lagrangian description of the quantum electrodynamics of media with both linear gain and loss, we compute the effective noise-photon distribution for quantum light propagation in arbitrary directions in layered metamaterials, thereby detailing and generalizing our recent work [ E. Amooghorban et al., Phys. Rev. Lett. ${\bf 110}$, 153602 (2013)]. The effective index with its direction and polarization dependence is the same as in classical effective-medium theories. As our main result we derive both for passive and for active media how the value of the effective noise-photon distribution too depends on the polarization and propagation directions of the light. Interestingly, for TE-polarized light incident on passive metamaterials, the noise-photon distribution reduces to a thermal distribution, but for TM-polarized light it does not. We illustrate the robustness of our quantum optical effective-medium theory by accurate predictions both for power spectra and for balanced homodyne detection of output quantum states of the metamaterial.