Experimental Demonstration of Controllable $\mathcal{PT}$ and Anti-$\mathcal{PT}$ Coupling in a Non-Hermitian Metamaterial

Experimental Demonstration of Controllable $PT$ and Anti- $PT$ Coupling in a Non-Hermitian Metamaterial

Chang Li, Ruisheng Yang, Xinchao Huang, Quanhong Fu, Yuancheng Fan, and Fuli Zhang

Phys. Rev. Lett. 132, 156601 – Published 8 April 2024

Abstract

Non-Hermiticity has recently emerged as a rapidly developing field due to its exotic characteristics related to open systems, where the dissipation plays a critical role. In the presence of balanced energy gain and loss with environment, the system exhibits parity-time ( $PT$ ) symmetry, meanwhile as the conjugate counterpart, anti- $PT$ symmetry can be achieved with dissipative coupling within the system. Here, we demonstrate the coherence of complex dissipative coupling can control the transition between $PT$ and anti- $PT$ symmetry in an electromagnetic metamaterial. Notably, the achievement of the anti- $PT$ symmetric phase is independent of variations in dissipation. Furthermore, we observe phase transitions as the system crosses exceptional points in both anti- $PT$ and $PT$ symmetric metamaterial configurations, achieved by manipulating the frequency and dissipation of resonators. This work provides a promising metamaterial design for broader exploration of non-Hermitian physics and practical application with a controllable Hamiltonian.

Received 15 August 2023
Accepted 11 March 2024

DOI:https://doi.org/10.1103/PhysRevLett.132.156601

Physics Subject Headings (PhySH)

Metamaterials

Non-Hermitian systems

P-symmetry T-symmetry

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Chang Li ^1,2,*, Ruisheng Yang^1,3,4, Xinchao Huang ^1,5, Quanhong Fu¹, Yuancheng Fan^1,†, and Fuli Zhang^1,‡

¹Key Laboratory of Light Field Manipulation and Information Acquisition Ministry of Industry and Information Technology and School of Physical Science and Technology Northwestern Polytechnical University, Xi’an 710129, China
²European Center for Quantum Sciences (CESQ-ISIS, UMR7006), University of Strasbourg and CNRS, Strasbourg, France
³Institute of Precision Optical Engineering, School of Physics Science and Engineering, Tongji University, Shanghai 200092, China
⁴Shanghai Frontiers Science Research Base of Digital Optics, Tongji University, Shanghai 200092, China
⁵European XFEL GmbH, Holzkoppel 4, 22869 Schenefeld, Germany

^*lichangphy@gmail.com
^†phyfan@nwpu.edu.cn
^‡fuli.zhang@nwpu.edu.cn

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Vol. 132, Iss. 15 — 12 April 2024

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Figure 1
Non-Hermiticity in metamaterials. (a) Diagram of two strongly coupled resonators. The eigenfrequencies and dissipation rates are denoted by $ω_{i}$ and $Γ_{i}$ ( $i = 1$ , 2). The coupling strength between them is $k$ with phase delay $ϕ$ . (b) Eigenenergy as a function of $ϕ$ and $Δ Γ$ in Eq. (3). The real and imaginary parts of $E_{\pm}$ are calculated with fixed $k = 1$ and $Γ_{1} = 1$ . The blue dots denote exceptional points where two eigenmodes degenerate with same real and imaginary parts. (c) Eigenenergy as a function of $Δ ω$ and $Δ Γ$ in Eq. (4) with fixed $k = 1$ and $Γ_{1} = 1$ . The blue dots are exceptional points. (d) Metamaterial configuration. The image to the left shows one sample used for experiments. All samples contain $7 \times 14$ units and each unit is $10 \times 20 {mm}^{2}$ . The sketch of one unit is shown on the right. The I-shaped cut wire with short arm length $L$ is at the center, and a couple of SRRs with rotation angle $θ$ are symmetrically located on both sides, with a distance $δ$ to the center. The probing EM field is alongside the $z$ axis and perpendicular to the sample, and the electric component is parallel to the long arm of cut wire alongside $x$ axis. (e) Image of the experiment setup. A linear-polarized microwave is generated from the first horn and collimated by a lens with a focus of 1.2 m. The sample mount is located at the focus, which is a flat microwave absorber with a window at center, and samples are stuck to the surface.
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Figure 2
Achieve anti- $PT$ symmetry with complex coupling. The transmission spectra are obtained both from experiment (left) and simulation (right) to find out the anti- $PT$ symmetry. Five groups of spectra correspond to the distance $δ = [4.5, 5.5, 6.0, 6.5, 7.5] mm$ . Other configuration parameters are fixed, $L = 2.5 mm$ and $θ = 0 °$ . The transparent window vanishes at $δ = 6 mm$ which indicates anti- $PT$ symmetry with coupling phase $ϕ = π / 2$ .
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Figure 3
Anti- $PT$ symmetry independent of dissipation variation. (a) The real part of $E_{\pm}$ in Eq. (4) as a function of $ϕ$ and $Γ_{2}$ . The parameters used for calculation are extracted from experimental spectra, and they are as follows: $ω_{1} = 0$ , $Γ_{1} = 0.34 GHz$ , $k = 30 MHz$ . $ω_{2} = - 40 MHz$ when $ϕ = π / 2$ , and it varies linearly $0.6 GHz / π$ with $ϕ$ . (b) Transmission spectra from experiment and simulation. Samples used here have fixed $L = 2.5 mm$ and $δ = 7 mm$ . $θ$ is varied from 30° to 90°, and the transmission peak vanishes at $θ = 60 °$ .
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Figure 4
Anti- $PT$ and $PT$ symmetry phase transition. (a) Transmission spectra of the anti- $PT$ symmetry phase transition. The spectra are obtained with $L = [2.30, 2.38, 2.46, 2.50, 2.54, 2.62, 2.70] mm$ , with fixed $δ = 6 mm$ and $θ = 0 °$ . (b) Eigenenergy of anti- $PT$ symmetry phase transition. The real (left) and imaginary (right) parts of the eigenenergy are extracted from the experimental transmission spectra. The purple and green dots represent $E_{\pm}$ , and red triangles are the resonant frequencies of the cut wire (CW). The coupling strength, $k$ , is estimated first with the frequencies of the eigenmodes, and then the real and imaginary parts of the eigenenergy are calculated afterwards as the blue and red curves showing here. (c) Sample for $PT$ symmetry phase transition. The size of each unit is still the same, but the SRRs are at the back surface with small center offset $δ = 0.2 mm$ from cut wire. We use $L = 2.8 mm$ to keep resonant frequencies of the cut wire and the SRRs are both at 10 GHz. (d) Transmission spectra of $PT$ symmetry phase transition. $θ$ is linearly varied from 70° to 90°. Two resonant modes coincide around $θ = 82 °$ . (e) Eigenenergy of $PT$ symmetry phase transition. The real (blue square) and imaginary (orange circle) parts of the eigenenergy are obtained from the experiment spectra. The spectra of real (blue curve) and imaginary (orange dash dot) parts are drawn after fitting the coupling strength as 151 MHz with obtained resonant frequencies. The pink (green) region represents the $PT$ symmetric (symmetry broken) phase.
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Physical Review Letters

Experimental Demonstration of Controllable PT and Anti-PT Coupling in a Non-Hermitian Metamaterial

Chang Li, Ruisheng Yang, Xinchao Huang, Quanhong Fu, Yuancheng Fan, and Fuli Zhang

Phys. Rev. Lett. 132, 156601 – Published 8 April 2024

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Experimental Demonstration of Controllable $PT$ and Anti- $PT$ Coupling in a Non-Hermitian Metamaterial