  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.
