PT-Symmetric Nonlinear Metamaterials and Zero-Dimensional Systems

Abstract

A one dimensional, parity-time ($\mathcal{P}\mathcal{T}$)-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 $\mathcal{P}\mathcal{T}$-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 ($\mathcal{P}\mathcal{T}$ dimer chain). The latter system features a two-band, gapped spectrum with its shape determined by the gain/loss coefficient as well as the inter-element coupling. In the presense of nonlinearity, the $\mathcal{P}\mathcal{T}$ dimer chain with balanced gain and loss supports nonlinear localized modes in the form of novel 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. Fundamendal 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 $\mathcal{P}\mathcal{T}$ 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 behaviour is similar to a transition from the exact to the broken $\mathcal{P}\mathcal{T}$ phase in higher-dimensional $\mathcal{P}\mathcal{T}-$ symmetric systems. A stability condition relating the parameters of the problem is obtained in the case of piecewise constant gain/loss function that allows for the construction of a phase diagram with alternating stable and unstable regions.

10 pages, 11 figs, subm. to Applied Physics A for the Proceedings of the META13 Conference

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Cited by in corpus (9)

• Nonlinear waves in $\mathcal{P}\mathcal{T}$-symmetric systems
• Experimental realization of Floquet PT-symmetric systems
• Photonic Floquet media with a complex time-periodic permittivity
• Flat bands and PT-symmetry in quasi-one-dimensional lattices
• $\mathcal{P}\mathcal{T}$ symmetry of the Su-Schrieffer-Heeger model with imaginary boundary potentials and next-nearest-neighboring coupling
• Compact Localized States in Engineered Flat-Band $\mathcal{P}\mathcal{T}$ Metamaterials
• $\mathcal{P}\mathcal{T}-$Symmetric Dimers with Time-Periodic Gain/Loss Function
• Topological split-ring resonator based metamaterials with $\mathcal{P}\mathcal{T}$ symmetry relying on gain and loss
• Resilience of $\mathcal{P}\mathcal{T}$ symmetry against stochasticity in a gain-loss balanced oscillator