Chimera States in Networks of Locally and Non-locally Coupled SQUIDs

Johanne Hizanidis,Nikos Lazarides,Giorgos P Tsironis

doi:10.3389/fams.2019.00033

Abstract

Planar and linear arrays of SQUIDs (superconducting quantum interference devices) operate as nonlinear magnetic metamaterials in microwaves. Such {\em SQUID metamaterials} are paradigmatic systems that serve as a test-bed for simulating several nonlinear dynamics phenomena. SQUIDs are highly nonlinear oscillators which are coupled together through magnetic dipole-dipole forces due to their mutual inductance; that coupling falls-off approximately as the inverse cube of their distance, i.~e., it is non-local. However, it can be approximated by a local (nearest-neighbour) coupling which in many cases suffices for capturing the essentials of the dynamics of SQUID metamaterials. For either type of coupling, it is numerically demonstrated that chimera states as well as other spatially non-uniform states can be generated in SQUID metamaterials under time-dependent applied magnetic flux for appropriately chosen initial conditions. The mechanism for the emergence of these states is discussed in terms of the multistability property of the individual SQUIDs around their resonance frequency and the attractor crowding effect in systems of coupled nonlinear oscillators. Interestingly, controlled generation of chimera states in SQUID metamaterials can be achieved in the presence of a constant (dc) flux gradient with the SQUID metamaterial initially at rest.

Highlights

The notion of metamaterials refers to artificially structured media designed to achieve properties not available in natural materials
The simplest version of a SQUID consists of a superconducting ring interrupted by a Josephson junction (JJ) (Figure 1A), which can be modeled by the equivalent electrical circuit of Figure 1B; according to that model, the SQUID features a self-inductance L, a capacitance C, a resistance R, and a critical current Ic which characterizes an ideal JJ
The emergence of chimera and multi-chimera states in a 1D SQUID metamaterial driven by an ac flux field is demonstrated numerically, using a well-established model that relies on equivalent electrical circuits

Summary

Introduction

The notion of metamaterials refers to artificially structured media designed to achieve properties not available in natural materials. They were comprising subwavelength resonant elements, such as the celebrated split-ring resonator (SRR). The latter, in its simplest version, is just a highly conducting metallic ring with a slit, that can be regarded as an effectively resistive–inductive–capacitive (RLC) electrical circuit. An important subclass of metamaterials is that of superconducting ones [17, 18], in which the elementary units (i.e., the SRRs) are made by a superconducting material, typically Niobium (Nb) [19] or Niobium Nitride (NbN) [20], as well as perovskite superconductors such as yttrium barium copper oxide (YBCO) [21].

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Conclusion