Superconducting Quantum Interference Device Metamaterials Research Articles

Tuning of strong nonlinearity in radio-frequency superconducting-quantum-interference-device meta-atoms.

Strong nonlinearity of a self-resonant radio-frequency (rf) superconducting-quantum-interference-device (SQUID) meta-atom is explored via intermodulation (IM) measurements. Previous work in zero dc magnetic flux showed a sharp onset of IM response as the frequency sweeps through the resonance. A second onset at higher frequency was also observed, creating a prominent gap in the IM response. By extending those measurements to nonzero dc flux, different dynamics are revealed, including dc flux tunability of the aforementioned gaps and enhanced IM response near geometric resonance of the rf SQUID. These features observed experimentally are understood and analyzed theoretically through a combination of a steady-state analytical modeling and a full numerical treatment of the rf SQUID dynamics. The latter in addition predicts the presence of chaos in narrow parameter regimes. The understanding of intermodulation in rf SQUID metamaterials is important for producing low-noise amplification of microwave signals and tunable filters.

Pattern formation and chimera states in 2D SQUID metamaterials.

The Superconducting QUantum Interference Device (SQUID) is a highly nonlinear oscillator with rich dynamical behavior, including chaos. When driven by a time-periodic magnetic flux, the SQUID exhibits extreme multistability at frequencies around the geometric resonance, which is manifested by a "snakelike" form of the resonance curve. Repeating motifs of SQUIDs form metamaterials, i.e., artificially structured media of weakly coupled discrete elements that exhibit extraordinary properties, e.g., negative diamagnetic permeability. We report on the emergent collective dynamics in two-dimensional lattices of coupled SQUID oscillators, which involves a rich menagerie of spatiotemporal dynamics, including Turing-like patterns and chimera states. Using Fourier analysis, we characterize these patterns and identify characteristic spatial and temporal periods. In the low coupling limit, the Turing-like patterns occur near the synchronization-desynchronization transition, which can be related to the bifurcation scenarios of the single SQUID. Chimeras emerge due to the multistability near the geometric resonance, and by varying the dc component of the external force, we can make them appear and reappear and, also, control their location. A detailed analysis of the parameter space reveals the coexistence of Turing-like patterns and chimera states in our model, as well as the ability to transform between these states by varying the system parameters.

Controlled generation of chimera states in SQUID metasurfaces using DC flux gradients

SQUID (Superconducting QUantum Interference Device) metamaterials, subject to a time-independent (dc) flux gradient and driven by a sinusoidal (ac) flux field, support chimera states that can be generated with zero initial conditions. The dc flux gradient and the amplitude of the ac flux can control the number of desynchronized clusters of such a generated chimera state (i.e., its `heads') as well as their location and size. The combination of three measures, i.e., the synchronization parameter averaged over the period of the driving flux, the incoherence index, and the chimera index, is used to predict the generation of a chimera state and its multiplicity on the parameter plane of the dc flux gradient and the ac flux amplitude. Moreover, the full-width half-maximum of the distribution of the values of the synchronization parameter averaged over the period of the ac driving flux, allows to distinguish chimera states from non-chimera, partially synchronized states.

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

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.

Coherent oscillations of driven rf SQUID metamaterials.

Through experiments and numerical simulations we explore the behavior of rf SQUID (radio frequency superconducting quantum interference device) metamaterials, which show extreme tunability and nonlinearity. The emergent electromagnetic properties of this metamaterial are sensitive to the degree of coherent response of the driven interacting SQUIDs. Coherence suffers in the presence of disorder, which is experimentally found to be mainly due to a dc flux gradient. We demonstrate methods to recover the coherence, specifically by varying the coupling between the SQUID meta-atoms and increasing the temperature or the amplitude of the applied rf flux.

Robust chimera states in SQUID metamaterials with local interactions.

We report on the emergence of robust multiclustered chimera states in a dissipative-driven system of symmetrically and locally coupled identical superconducting quantum interference device (SQUID) oscillators. The "snakelike" resonance curve of the single SQUID is the key to the formation of the chimera states and is responsible for the extreme multistability exhibited by the coupled system that leads to attractor crowding at the geometrical resonance (inductive-capacitive) frequency. Until now, chimera states were mostly believed to exist for nonlocal coupling. Our findings provide theoretical evidence that nearest-neighbor interactions are indeed capable of supporting such states in a wide parameter range. SQUID metamaterials are the subject of intense experimental investigations, and we are highly confident that the complex dynamics demonstrated in this paper can be confirmed in the laboratory.

Chimera states and synchronization in magnetically driven SQUID metamaterials

Superconducting QUantum Interference Device (SQUID) metamaterials are superconducting artificial media whose function relies both on their geometry and the extraordinary properties of superconductivity and the Josephson effect. Recent experiments on one- and two-dimensional radio-frequency (rf) SQUID metamaterials have revealed their wide-band tuneability, significantly reduced losses, dynamic multistability, and tunable broadband transparency. The simplest version of an rf SQUID involves a superconducting ring interrupted by a Josephson junction; this device is a highly nonlinear resonator with a strong response to applied magnetic fields. SQUID metamaterials exhibit peculiar magnetic properties such as negative diamagnetic permeability, predicted both for the quantum and the classical regime. The applied alternating fields induce (super)currents in the SQUID rings, which are therefore coupled through dipole-dipole magnetic forces. This interaction is weak due to its magnetic nature. However, it couples the SQUIDs non-locally since it falls-off as the inverse cube of their center-to-center distance.

Chimeras in SQUID metamaterials

Regular lattices comprising superconducting quantum interference devices (SQUIDs) form magnetic metamaterials exhibiting extraordinary properties, including tunability, dynamic multistability, and negative magnetic permeability. The SQUIDs in a metamaterial interact through nonlocal, magnetic dipole-dipole forces that makes it possible for counter-intuitive dynamic states referred to as chimera states to appear; the latter feature clusters of SQUIDs with synchronous dynamics which coexist with clusters exhibiting asynchronous behavior. The spontaneous appearance of chimera states is demonstrated numerically for one-dimensional SQUID metamaterials driven by an alternating magnetic field in which the fluxes threading the SQUID rings are randomly initialized; then, chimera states appear generically for sufficiently strong initial excitations, which exhibit relatively long lifetimes. The synchronization and metastability levels of the chimera states are discussed in terms of appropriate measures. Given that both one- and two-dimensional SQUID metamaterials have been already fabricated and investigated in the laboratory, the presence of a chimera state could in principle be detected with presently available experimental setups.

Wide-band tuneability, nonlinear transmission, and dynamic multistability in SQUID metamaterials

Superconducting metamaterials comprising rf SQUIDs (Superconducting QUantum Interference Devices) have been recently realized and investigated with respect to their tuneability, permeability and dynamic multistability properties. These properties are a consequence of intrinsic nonlinearities due to the sensitivity of the superconducting state to external stimuli. SQUIDs, made of a superconducting ring interrupted by a Josephson junction, possess yet another source of nonlinearity, which makes them widely tuneable with an applied dc dlux. A model SQUID metamaterial, based on electric equivalent circuits, is used in the weak coupling approximation to demonstrate the dc flux tuneability, dynamic multistability, and nonlinear transmission in SQUID metamaterials comprising non-hysteretic SQUIDs. The model equations reproduce the experimentally observed tuneability patterns, and predict tuneability with the power of an applied ac magnetic magnetic field. Moreover, the results indicate the opening of nonlinear frequency bands for energy transmission through SQUID metamaterials, for sufficiently strong ac fields.

Protecting SQUID metamaterials against stray magnetic fields

Using superconducting quantum interference devices (SQUIDs) as the basic, low-loss elements of thin-film metamaterials has one main advantage: their resonance frequency is easily tunable by applying a weak magnetic field. The downside, however, is a strong sensitivity to stray and inhomogeneous magnetic fields. In this work, we demonstrate that even small magnetic fields from electronic components destroy the collective, resonant behaviour of the SQUID metamaterial. We also show how the effect of these fields can be minimized. As a first step, magnetic shielding decreases any initially present fields, including the earth’s magnetic field. However, further measures such as improvements in the sample geometry have to be taken to avoid the trapping of Abrikosov vortices.

Multistability and self-organization in disordered SQUID metamaterials

Planar arrays of magnetoinductively coupled rf SQUIDs (Superconducting QUantum Interference Devices) belong to the emergent class of superconducting metamaterials that encompass the Josephson effect. These SQUID-based metamaterials acquire their electromagnetic properties from the resonant characteristics of their constitutive elements, i.e., the individual rf SQUIDs. In its simplest version, an rf SQUID consists of a superconducting ring interrupted by a Josephson junction. We investigate numerically the response of a two-dimensional rf SQUID metamaterial with respect to the driving frequency of an externally applied alternating magnetic field in the presence of disorder arising from critical current fluctuations of the Josephson elements; in effect, the resonance frequencies of individual SQUIDs are distributed randomly around a mean value. Bistability is observed in the current amplitude–frequency curves both in ordered and disordered SQUID metamaterials; moreover, bistability is favored by disorder through the improvement of synchronization between SQUID oscillators. Relatively weak disorder widens significantly the bistability region by helping the system to self-organize and leads to nearly homogeneous states that change smoothly with varying driving frequency. Also, the total current of the metamaterial is enhanced, compared with that of uncoupled SQUIDs, through the synergetic action of coupling and synchronization. The existence of simultaneously stable states that provide either high or low total current, allows the metamaterial to exhibit different magnetic responses that correspond to different values of the effective magnetic permeability. At low power of the incident field, high current amplitude states exhibit extreme diamagnetic properties corresponding to negative magnetic permeability in a narrow frequency interval.

Dissipative Breathers in rf SQUID Metamaterials

The existence and stability of dissipative breathers in rf SQUID (Superconducting Quantum Interference Device) arrays is investigated numerically. In such arrays, the nonlinearity which is intrinsic to each SQUID, along with the weak magnetic coupling of each SQUID to its nearest neighbors, result in the formation of discrete breathers. We analyze several discrete breather excitations in rf SQUID arrays driven by alternating flux sources in the presence of losses. The delicate balance between internal power losses and input power, results in the formation of dissipative discrete breather (DDB) structures up to relatively large coupling parameters. It is shown that DDBs may locally alter the magnetic response of an rf SQUID array from paramagnetic to diamagnetic or vice versa.