Quantum Optical Effective-Medium Theory for Loss-Compensated Metamaterials

Abstract

A central aim in metamaterial research is to engineer sub-wavelength unit cells that give rise to desired effective-medium properties and parameters, such as a negative refractive index. Ideally one can disregard the details of the unit cell and employ the effective description instead. A popular strategy to compensate for the inevitable losses in metallic components of metamaterials is to add optical gain material. Here we study the quantum optics of such loss-compensated metamaterials at frequencies for which effective parameters can be unambiguously determined. We demonstrate that the usual effective parameters are insufficient to describe the propagation of quantum states of light. Furthermore, we propose a quantum optical effective-medium theory instead and show that it correctly predicts the properties of the light emerging from loss-compensated metamaterials.PACS numbers: 42.50. Nn, 78.67.Pt, 78.20.Ci, 42.50.Ct Metamaterials are intensely studied, since they allow the propagation and control of light in new and often counterintuitive ways. These man-made structures are composed of strongly subwavelength unit cells, with effective dielectric parameters often not occurring in nature, such as a negative refractive index [1,2]. Unlike in classical optics, the possible benefits of metamaterials in quantum optics have not been explored so far, for example to manipulate single photons. More fundamentally, it is an important open question whether the same effectivemedium parameters suffice to describe the propagation of quantum states of light in metamaterials.The constituents and geometry of a unit cell can be complicated and interesting, but they are designed to allow the effective description of the metamaterial as a homogeneous medium. For subwavelength unit cells, unique effective dielectric parameters can be identified, independent of the method used to retrieve them. Our results go against the common belief that experiments at the operating frequency do not reveal information about the unit cell beyond the usual effective refractive index.Noble metals, an important ingredient of metamaterials, are inherently lossy. For many applications it is naturally desirable to have less loss. Complementary strategies are to replace the metals by other material [3] or to compensate for the metal loss [4][5][6][7][8]. As an important branch of active plasmonics [9], active loss compensation with the use of gain material has already proved experimentally successful, for example in surface plasmon polariton propagation [10,11] and in metamaterials [12].In recent years the quantum optics of attenuating [13][14][15][16][17][18][19] and amplifying [20][21][22][23][24] dielectric media was developed, where optical modes are described as open quantum systems. There are important similarities with classical optics, for example the classical Green function plays a central role also in quantum optics, but new is that both with loss and gain there is quantum noise associated. Evidently, effective-medium theories that neglect qu...

Quantum optical effective-medium theory and transformation quantum optics for metamaterials

“…13 We find that for s-polarized light incident on metamaterials without loss compensation, the effective noise photon density is a thermal distribution for all angles of incidence. This is a direct generalization of the result we found earlier for normally incident light.…”

“…The result of all this 14,25 is that the material polarization field (a quantum operator) is not described by P = ε 0 χE, but rather by…”

“…13 we studied these questions for the first time, limiting our studies to light propagation normal to the layers. We found that for passive multilayers systems as well as for binary multilayer metamaterials with gain in both types of layers, that the effective index actually describes the average amount of quantum noise well.…”

“…13 that the QOEI theory breaks down for the important class of loss-compensated metamaterials, because it underestimates the effective amount of quantum noise. This has observable consequences.…”

“…Our Quantum Optical Effective-Medium (QOEM) theory of Ref. 13 repairs this by not only determining an effective index but also an effective noise photon distribution N eff,σ (k, ω, T ), depending on the polarization state σ (s-polarized (TE) or p-polarized (TM)), on the wavevector parallel to the layers k, and on temperature T . For loss-compensated metamaterials the effective noise photon distribution is given by…”