Abstract
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.
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References
V. Achilleos, P.G. Kevrekidis, D.J. Frantzeskakis, R. Carretero-González, Dark solitons and vortices in \(\mathcal{PT}\)-symmetric nonlinear media: from spontaneous symmetry breaking to nonlinear pt phase transitions. Phys. Rev. A 86, 013808 (2012) (7pp)
N.V. Alexeeva, I.V. Barashenkov, A.A. Sukhorukov, Y.S. Kivshar, Optical solitons in pt-symmetric nonlinear couplers with gain and loss. Phys. Rev. A 85, 063837 (2012) (13pp)
S.M. Anlage, The physics and applications of superconducting metamaterials. J. Opt. 13, 024001 (2011) (10pp)
I.V. Barashenkov, S.V. Suchkov, A.A. Sukhorukov, S.V. Dmitriev, Y.S. Kivshar, Breathers in pt-symmetric optical couplers. Phys. Rev. A 86, 053809 (2012) (12pp)
A.D. Boardman, V.V. Grimalsky, Y.S. Kivshar, S.V. Koshevaya, M. Lapine, N.M. Litchinitser, V.N. Malnev, M. Noginov, Y.G. Rapoport, V.M. Shalaev, Active and tunable metamaterials. Laser Photonics Rev. 5(2), 287–307 (2010)
S.V. Dmitriev, A.A. Sukhorukov, Y.S. Kivshar, Binary parity-time-symmetric nonlinear lattices with balanced gain and loss. Opt. Lett. 35, 2976–2978 (2010)
R. El-Ganainy, K.G. Makris, D.N. Christodoulides, Z.H. Musslimani, Theory of coupled optical pt-symmetric structures. Opt. Lett. 32, 2632–2634 (2007)
L. Esaki, New phenomenon in narrow germanium p–n junctions. Phys. Rep. 109, 603–605 (1958)
N. Feth, M. König, M. Husnik, K. Stannigel, J. Niegemann, K. Busch, M. Wegener, S. Linden, Electromagnetic interaction of spit-ring resonators: the role of separation and relative orientation. Opt. Express 18, 6545–6554 (2010)
S. Flach, A.V. Gorbach, Discrete breathers—advances in theory and applications. Phys. Rep. 467, 1–116 (2008)
A. Guo, Observation of pt-symmetry breaking in complex optical potentials. Phys. Rev. Lett. 103, 093902 (2009)
F. Hesmer, E. Tatartschuk, O. Zhuromskyy, A.A. Radkovskaya, M. Shamonin, T. Hao, C.J. Stevens, G. Faulkner, D.J. Edwardds, E. Shamonina, Coupling mechanisms for split-ring resonators: theory and experiment. Phys. Status Solidi, B Basic Res. 244, 1170–1175 (2007)
D.W. Hook, Non-hermittian potentials and real eigenvalues. Ann. Phys. (Berlin) 524(6–7), A106 (2012)
T. Jiang, K. Chang, L.M. Si, L. Ran, H. Xin, Active microwave negative-index metamaterial transmission line with gain. Phys. Rev. Lett. 107, 205503 (2011)
N. Lazarides, M. Eleftheriou, G.P. Tsironis, Discrete breathers in nonlinear magnetic metamaterials. Phys. Rev. Lett. 97, 157406 (2006) (4pp)
N. Lazarides, M.I. Molina, G.P. Tsironis, Breather induction by modulational instability in binary metamaterials. Acta Phys. Pol. A 116(4), 635–637 (2009)
N. Lazarides, M.I. Molina, G.P. Tsironis, Breathers in one-dimensional binary metamaterial models. Physica B 405, 3007–3011 (2010)
N. Lazarides, V. Paltoglou, G.P. Tsironis, Nonlinear magnetoinductive transmission lines. Int. J. Bifurc. Chaos 21, 2147–2156 (2011)
N. Lazarides, G.P. Tsironis, Gain-driven discrete breathers in pt-symmetric nonlinear metamaterials. Phys. Rev. Lett. 110, 053901 (2013) (5pp)
K. Li, P.G. Kevrekidis, Pt-symmetric oligomers: analytical solutions, linear stability, and nonlinear dynamics. Phys. Rev. E 83, 066608 (2011) (7pp)
Z. Lin, H. Ramezani, T. Eichekraut, T. Kottos, H. Cao, D.N. Christodoulides, Unidirectional invisibility induced by pt-symmetric periodic structures. Phys. Rev. Lett. 106, 213901 (2011) (4pp)
K.G. Makris, R. El-Ganainy, D.N. Christodoulides, Z.H. Musslimani, Beam dynamics in pt-symmetric optical lattices. Phys. Rev. Lett. 100, 103904 (2008)
A.E. Miroshnichenko, B.A. Malomed, Y.S. Kivshar, Nonlinearly pt-symmetric systems: spontaneous symmetry breaking and transmission resonances. Phys. Rev. A 84, 012123 (2011) (4pp)
M.I. Molina, N. Lazarides, G.P. Tsironis, Bulk and surface magnetoinductive breathers in binary metamaterials. Phys. Rev. E 80, 046605 (2009)
H. Ramezani, D.N. Christodoulides, V. Kovanis, I. Vitebskiy, T. Kottos, Pt-symmetric Talbot effects. Phys. Rev. Lett. 109, 033902 (2012)
H. Ramezani, T. Kottos, R. El-Ganainy, D.N. Christodoulides, Unidirectional nonlinear pt-symmetric optical structures. Phys. Rev. A 82, 043803 (2010) (6pp)
N.N. Rosanov, N.V. Vysotina, A.N. Shatsev, I.V. Shadrivov, D.A. Powell, Y.S. Kivshar, Discrete dissipative localized modes in nonlinear magnetic metamaterials. Opt. Express 19, 26500 (2011)
C.E. Rüter, K.G. Makris, R. El-Ganainy, D.N. Christodoulides, M. Segev, D. Kip, Observation of parity–time symmetry in optics. Nat. Phys. 6, 192 (2010)
M. Sato, B.E. Hubbard, A.J. Sievers, B. Ilic, D.A. Czaplewski, H.G. Graighead, Observation of locked intrinsic localized vibrational modes in a micromechanical oscillator array. Phys. Rev. Lett. 90, 044102 (2003) (4pp)
J. Schindler, A. Li, M.C. Zheng, F.M. Ellis, T. Kottos, Experimental study of active LRC circuits with pt symmetries. Phys. Rev. A 84, 040101(R) (2011)
I. Sersić, M. Frimmer, E. Verhagen, A.F. Koenderink, Electric and magnetic dipole coupling in near-infrared split-ring metamaterial arrays. Phys. Rev. Lett. 103, 213902 (2009)
L.M. Si, T. Jiang, K. Chang, T.-C. Chen, X. Lv, L. Ran, H. Xin, Active microwave metamaterials incorporating ideal gain devices. Materials 4, 73–83 (2011)
S.V. Suchkov, B.A. Malomed, S.V. Dmitriev, Y.S. Kivshar, Solitons in a chain of parity-time invariant dimers. Phys. Rev. A 84, 046609 (2011)
O. Sydoruk, A. Radkovskaya, O. Zhuromskyy, E. Shamonina, M. Shamonin, C. Stevens, G. Faulkner, D. Edwards, L. Solymar, Tailoring the near-field guiding properties of magnetic metamaterials with two resonant elements per unit cell. Phys. Rev. B 73, 224406 (2006)
A. Szameit, M.C. Rechtsman, O. Bahat-Treidel, M. Segev, Pt-symmetry in honeycomb photonic lattices. Phys. Rev. A 84, 021806(R) (2011)
B. Wang, J. Zhou, T. Koschny, C.M. Soukoulis, Nonlinear properties of split-ring resonators. Opt. Express 16, 16058 (2008)
W. Xu, W.J. Padilla, S. Sonkusale, Loss compensation in metamaterials through embedding of active transistor based negative differential resistance circuits. Opt. Express 20, 22406 (2012)
D.A. Zezyulin, V.V. Konotop, Nonlinear modes in finite-dimensional pt-symmetric systems. Phys. Rev. Lett. 108, 213906 (2012) (5pp)
N.I. Zheludev, The road ahead for metamaterials. Science 328, 582–583 (2010)
Acknowledgement
This work was partially supported by the European Union’s Seventh Framework Programme (FP7-REGPOT-2012-2013-1) under grant agreement no 316165, and by the Thales Project ANEMOS, cofinanced by the European Union (European Social Fund—ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF)—Research Funding Program: THALES. Investing in knowledge society through the European Social Fund.
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Tsironis, G.P., Lazarides, N. \(\mathcal{PT}\)-symmetric nonlinear metamaterials and zero-dimensional systems. Appl. Phys. A 115, 449–458 (2014). https://doi.org/10.1007/s00339-013-8035-2
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DOI: https://doi.org/10.1007/s00339-013-8035-2