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Superconducting metamaterials comprising rf Superconducting QUantum Interference Devices (SQUIDs) 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 field. Moreover, the results indicate the opening of nonlinear frequency bands for energy transmission through SQUID metamaterials, for sufficiently strong ac fields. [ABSTRACT FROM AUTHOR]

A one dimensional, parity-time ( $\mathcal{PT}$)-symmetric magnetic metamaterial comprising split-ring resonators having both gain and loss is investigated. In the linear regime, the transition from the exact to the broken $\mathcal{PT}$-phase is determined through the calculation of the eigenfrequency spectrum for two different configurations; the one with equidistant split-rings and the other with the split-rings forming a binary pattern ( $\mathcal{PT}$ dimer chain). The latter system features a two-band, gapped spectrum with its shape determined by the gain/loss coefficient as well as the interelement coupling. In the presence of nonlinearity, the $\mathcal{PT}$ dimer chain configuration with balanced gain and loss supports nonlinear localized modes in the form of a novel type of discrete breathers below the lower branch of the linear spectrum. These breathers that can be excited from a weak applied magnetic field by frequency chirping, can be subsequently driven solely by the gain for very long times. The effect of a small imbalance between gain and loss is also considered. Fundamental gain-driven breathers occupy both sites of a dimer, while their energy is almost equally partitioned between the two split-rings, the one with gain and the other with loss. We also introduce a model equation for the investigation of classical $\mathcal{PT}$ symmetry in zero dimensions, realized by a simple harmonic oscillator with matched time-dependent gain and loss that exhibits a transition from oscillatory to diverging motion. This behavior is similar to a transition from the exact to the broken $\mathcal{PT}$ phase in higher-dimensional $\mathcal{PT}$-symmetric systems. A stability condition relating the parameters of the problem is obtained in the case of a piece-wise constant gain/loss function that allows the construction of a phase diagram with alternating stable and unstable regions. [ABSTRACT FROM AUTHOR]

We propose a superconducting metamaterial design consisting of meta-atoms which are each composed of a direct current superconducting quantum interference device and a superconducting rod. This design provides negative refraction index behavior for a wide range of structure parameters. [ABSTRACT FROM AUTHOR]

$${\mathcal {PT}}$$ -symmetric dimers with a time-periodic gain/loss function in a balanced configuration where the amount of gain equals that of loss are investigated analytically and numerically. Two prototypical dimers in the linear regime are investigated: a system of coupled classical oscillators, and a Schrödinger dimer representing the coupling of field amplitudes, each system representing a wide class of physical models. Through a thorough analysis of their stability behavior, we find that turning on the coupling parameter in the classical dimer system leads initially to decreased stability but then to re-entrant transitions from the exact to the broken $${\mathcal {PT}}$$ -phase and vice versa, as it is increased beyond a critical value. On the other hand, the Schrödinger dimer behaves more like a single oscillator with time-periodic gain/loss. In addition, we are able to identify the conditions under which the behavior of the two dimer systems coincides and/or reduces to that of a single oscillator. [ABSTRACT FROM AUTHOR]