**7. Metamaterials in SQUIDs**

The investigations on the metamaterials comprising SQUIDs (Superconducting quantum interference devices) are going on intensively from the last decade [252,253]. These are the artificial materials which exhibit special properties such as negative magnetic permeability [254], dynamic multi-stability and switching [252], broad band tunability [255], etc. As we know that simple version of SQUIDs contains superconducting ring interrupted by Josephson junction [256]. The two-dimensional arrays of SQUIDs in different lattice geometries establish nonlinear metasurfaces. This can project theoretically in both quantum [257] and classical regimes [258,259]. By implementing this various designs and structures have been investigated in one and two dimensions [260–263]. With the help of microwave transmission measurements recently reported the degree of spatiotemporal coherence of SQUID metamaterials [264]. Further, these SQUID metamaterials provide novel testbed for exploring complex spatiotemporal dynamics. Interestingly, it is proved that SQUID metamaterials support spatially inhomogeneous states like chimera states. These chimera states gained much attention for both experimental and theoretical viewpoints [265,266]. In various discrete systems it was identified that different dynamic states such as solitary state chimeras [267], spiral wave chimeras [268], and imperfect chimeras [269]. In addition, the chimera states in SQUID metasurfaces controls the speed of the propagat-

ing electromagnetic waves [270]. Lazarides et al. [271] demonstrated the generation of chimera states by SQUID metasurfaces which are subjected to dc-flux gradient and driven by ac-flux. Figure 5 shows the schematic representation of SQUID metamaterial (Figure from Ref. [271]). They reported that the flux gradient of dc-field and amplitude of the ac-flux controls desynchronized clusters of chimera state along with their location and size. Further, the chimera states are distinguished from non-chimera states by ac driving flux. Finally, the proposed SQUID oscillator is an example for system with inertia and driving which plays a prominent role in the advancement of metamaterials science. In addition, J. Hizanidis et al. [272] presented 2D SQUID lattice with nearest neighbour interactions which can form states with Turning-like pattern. In general, single SQUID undergoes complex bifurcations at low coupling limit, where as in 2D SQUID the Turning like pattern arises near synchronization to desynchronization transformation region. Further, owing to extreme multi-stability of single SQUID, 2D chimera states have formed at near resonant regime. The formed chimera states can be tuned by dc-flux also control their position and multiplicity. Finally, recent reports revealed that the imaging of the chimera states in SQUID metamaterials is prominent candidates for verifying the theoretical findings [273].

**Figure 5.** Schematic representation of SQUID metamaterial (Figure from Ref. [271]).
