**2.8 Quantum and atomistic MMs**

The domains of quantum MMs studies in near IR or optical region are still shallow but promising as the quantum degrees of freedom are incorporated [90]. In the photonic structure, the quantum wells have been used to describe the permittivity influence over the structure behavior electromagnetically. Studying layered MMs supplied with two quantum wells of GaAs, showed an effective permittivity tensor resulting in a negative refraction [91]. Many proposals have been done to extend the quantum magnetism of the MMs via organic synthesis or molecular engineering. Theory showed that Cu-CoPc <sup>2</sup> (copper phthalocyanine and cobalt phthalocyanine chains) provided a relatively robust ferromagnetism [92].

A chain of studies has been implemented that can be described as a development of work. It has been discovered that a full quantum process happens between two level atoms and a quantized electromagnetic field [93]. Then Cavity Array MMs (CAM), where 2D network of coupled atom-optical cavities were scrutinized to analyze the model via 2D photonic crystal membrane [34]. For a reasonable hypothesis, Jaynes-Cummings-Hubbard Hamiltonian method can be used to depict a system that exhibits a quantum phase transition [94]. So, it is possible to work as a quantum simulator [95]. Also, negative refraction and cloaking phenomena were elucidated. Moreover, the polaritons hybrids that formed of atomic and photonic states are an exciting system.

The dielectric function εQD expresses the quantum dots [96]:

$$\mathbf{e}\_{\rm QD} \ (\mathbf{o}) = \mathbf{e}\_{\rm b} + \left( \mathbf{f}\_{\rm c}(\mathbf{E\_h}) \mathbf{-f}\_{\rm v}(\mathbf{E\_h}) \right) \ \frac{\mathbf{a}}{\mathbf{o}^2 - \mathbf{o}\_0^2 + 2\mathrm{i}\mathbf{o}\,\eta} \tag{18}$$

**Figure 7.** *A cavity array metamaterials [34].*

Where, (fc(Eh) –fv(Eh), is the difference between population levels. **Figure 7**, is showing cavity array metamaterials [34].
