# Quantum metamaterials: a brave new world

Metamaterials are artificial, composite structures that present effective properties not encountered in natural materials. This broad definition can apply to many materials and types of properties. Consider, for instance, fiber-reinforced cement, a composite medium, made of cement and metallic fibers, with mechanical properties that neither of its components possesses alone. This is a (mechanical) metamaterial. The kinds of properties that we will be interested in here relate not to mechanics, but to electromagnetic waves. That is, we consider metamaterials whose electromagnetic properties, permittivity and permeability, can be tailored. (There is a similar concept with phonons, i.e., phononic metamaterials.) The field of composite media in electromagnetics is a rather old one. However, it has greatly expanded in recent years, in particular through the concepts of negative index, cloaking, superlenses, and even plasmonics.

Various methods have been suggested to introduce quantum degrees of freedom. The use of metal in metamaterials and in plasmonics soon gave rise to the idea of inserting some gain inside these structures, which can be done by inserting quantum dots.^{1, 2} However, this approach uses quantum degrees of freedom not to modify the effective properties, but to compensate the losses. Somewhat differently, Plumridge and colleagues have shown the possibility of tailoring the anisotropic effective permittivity tensor of a layered medium doped by quantum wells.^{3} In recent work, Zagoskin and co-workers, who coined the term ‘quantum metamaterial,’ proposed using a set of charge qubits (an array of Josephson junctions).^{4} They showed that in such a system, a band gap can exist (depending on the microscopic quantum state of the qubits), resulting in a quantum photonic crystal. Another approach is to consider coupled-atom optical cavities.^{5} These metamaterials can be used to design a reconfigurable superlens possessing a negative index gradient for single-photon imaging.

Our approach to quantum metamaterials proceeds along two paths.^{6} The first kind of structure is a set of nanowires containing quantum dots, such as that depicted in Figure 1. Such a structure is interesting for several reasons: it is simple, it has many interesting properties (negative index, effective magnetism, all-dielectric cloaking), and it can be experimentally realized fairly easily. Furthermore, the quantum dots have a dipole with a controllable orientation. Each nanorod can be characterized by its scattering matrix, which relates the diffracted field to the incident field. For a wavelength that is sufficiently large compared to the diameter of the nanorod, the diffracted field can be characterized by an electric dipole and a magnetic dipole. We have shown that, for *H*_{||} polarized fields (i.e., magnetic fields parallel to the rods), the collective resonances of the microscopic magnetic dipoles induce an effective permeability (this remains true for dielectric spheres).^{7} We have also proved the existence of a negative permittivity and a negative permeability for *E*_{||} fields.^{8} Now, the quantum dots inserted inside the nanorods allow control of the transverse dipole (the magnetic one for *E*_{||} fields). By pumping the dots (see Figure 1), it is possible to switch this dipole on or off, opening or closing a conduction band (see Figure 2). This kind of structure thus possesses a quantum reconfigurable band structure.

The second type of quantum metamaterial we are studying is made of artificial crystals of ultracold atoms, arranged periodically in optical traps, in what is called a Mott insulator state. Such artificial crystals have been realized in several laboratories, and exhibit a much larger periodicity than common crystals (of the order of a fraction of a micron). These quantum systems, whose extreme control and tunability allow their use as quantum clocks,^{9} show interesting interplays between the geometry of the lattice, the internal atomic structure, and the motional degrees of freedom of the atoms in their local trapping potential. In particular, we have shown that such structures may possess a full photonic band gap, as is the case for a four-level atomic structure arranged in a diamond lattice,^{10} or a two-level atomic structure in a cubic lattice.^{11} However, a four-level atomic structure in a cubic lattice does not possess any full photonic band gap.^{11} In fact, we have shown that the opening or closing of a band depends on the interplay between the lattice geometry and the internal quantum structure. That a collective property such as the photonic band depends on a quantum number shows the merit of the concept of quantum metamaterials in this context of cold atoms. Remarkably, such artificial crystals have recently been used to produce a 1D photonic-crystal lasing medium, where multiple Bragg reflections and the presence of a cold atom cloud simultaneously provide gain and feedback.^{12}

In summary, we have proposed new ways to make metamaterials enter the quantum world. Inserting artificial atoms (quantum dots) into periodic nanostructures or arranging real atoms (ultracold gases) into artificial periodic lattices can open new possibilities for collective light-matter behavior. In the future, we will study these new structures both experimentally (lattices of doped dielectric nanorods) and theoretically. By investigating several fundamental aspects linked to quantum fluctuations, we aim to reveal the existence of non-trivial collective phenomena resulting from hybridization of internal and global states.

*Mauro Antezza acknowledges financial support from the Julian Schwinger Foundation.*

and

University of Montpellier 2

Montpellier, France

http://www.coulomb.univ-montp2.fr

Didier Felbacq is a member of the Institut Universitaire de France.

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