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...

A path-integral approach to the quantization of the electromagnetic field in a linearly amplifying magnetodielectric medium is presented. Two continua of inverted harmonic oscillators are used to describe the polarizability and magnetizability of the amplifying medium. The causal susceptibilities of the amplifying medium, with negative imaginary parts in finite frequency intervals, are identified and their relation to microscopic coupling functions are determined. By carefully relating the twopoint functions of the field theory to the optical Green functions, we calculate the Casimir energy and Casimir forces for a multilayer magnetodielectric medium with both gain and loss. We point out the essential differences with a purely passive layered medium. For a single layer, we find different bounds on the Casimir force for fully amplifying and for lossy media. The force is attractive in both cases, also if the medium exhibits negative refraction. From our Lagrangian we also derive by canonical quantization the postulates of the phenomenological theory of amplifying magnetodielectrics.