Manipulating spectral topology and exceptional points by nonlinearity in non-Hermitian polariton systems

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Manipulating spectral topology and exceptional points by nonlinearity in non-Hermitian polariton systems

Jan Wingenbach, Stefan Schumacher, and Xuekai Ma

Phys. Rev. Research 6, 013148 – Published 7 February 2024

Abstract

Exceptional points (EPs), with their intriguing spectral topology, have attracted considerable attention in a broad range of physical systems, with potential sensing applications driving much of the present research in this field. Here, we investigate spectral topology and EPs in systems with significant nonlinearity, exemplified by a nonequilibrium exciton-polariton condensate. With the possibility to control loss and gain and nonlinearity by optical means, this system allows for a comprehensive analysis of the interplay of nonlinearities (Kerr type and saturable gain) and non-Hermiticity. Not only do we find that EPs can be intentionally shifted in parameter space by the saturable gain, but we also observe intriguing rotations and intersections of Riemann surfaces and find nonlinearity-enhanced sensing capabilities. With this, our results illustrate the potential of tailoring spectral topology and related phenomena in non-Hermitian systems by nonlinearity.

Received 9 May 2023
Accepted 8 January 2024

DOI:https://doi.org/10.1103/PhysRevResearch.6.013148

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Exciton polariton Nonlinear optics Polaritons

Non-Hermitian systems

Nonlinear Dynamics

Authors & Affiliations

Jan Wingenbach ¹, Stefan Schumacher^1,2,3, and Xuekai Ma ¹

¹Department of Physics and Center for Optoelectronics and Photonics Paderborn, Paderborn University, Warburger Strasse 100, 33098 Paderborn, Germany
²Institute for Photonic Quantum Systems, Paderborn University, Warburger Straße 100, 33098 Paderborn, Germany
³Wyant College of Optical Sciences, University of Arizona, Tucson, Arizona 85721, USA

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Vol. 6, Iss. 1 — February - April 2024

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Images

Figure 1
System and excitation setup. (a) Sketch of the Riemann surface (real part only) of a two-level non-Hermitian system containing an EP for varying level separation $Δ E_{L} = E_{α} - E_{β}$ and varying relative level coupling $μ / Δ Γ$ . (b) Scheme for realization with polariton condensates, including the rectangular external potential $V (r)$ and excitation with two nonresonant pump beams, $P_{1} (r)$ and $P_{2} (r)$ . Polariton density (top row; in $µ m^{- 2}$ ) and phase (bottom row) of the (c) third and (d) fourth modes of the potential $V (r)$ that are chosen here as an example for the realization of an EP. (e) Vortex-antivortex mode at the EP. Further details are given in the text.
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Figure 2
Nonlinearity-induced enhancement of sensing at the EP. (a) Deviation of the eigenvalue splitting (real part) from the linear case at $μ / Δ Γ = 2$ , close to the EP as a function of the polariton-polariton interaction $g_{c}$ and for different polariton-reservoir interactions $g_{r}$ . The values of the splitting deviation and the corresponding increased or decreased EP sensitivity are given as a percentage of the energy splitting in the linear case. (b) Eigenvalues $E_{\pm}$ along the mode degeneracy line $Δ E = 0$ for different interaction strengths $g_{c}$ and $g_{r} = 0$ , illustrating the change in the energy splitting. Eigenvalues are renormalized to the total mode energy $E$ at the EP.
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Figure 3
Nonlinearity-induced rotation of Riemann surfaces. Riemann surfaces (real parts) resulting from the eigenvalue calculations, Eq. (4), for (a)–(c) $g_{r} = 0$ and (d)–(f) $g_{r} = 2 g_{c}$ for different polariton-polariton interactions: (a) and (d) $g_{c} = 0 µ eV µ m^{2}$ , (b) and (e) $g_{c} = 3 µ eV µ m^{2}$ and (c) and (f) $g_{c} = 6 µ eV µ m^{2}$ . The eigenvalues are depicted against the energy difference $Δ E_{L}$ and relative gain and loss difference $Δ Γ / μ$ . For $g_{c} \neq 0$ the nonlinear correction $Δ E_{NL}$ to the energy difference is plotted with $Δ Γ$ . Energies are given in $µ eV$ .
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Figure 4
Rotation of the Riemann surfaces with higher-order EPs. Riemann surfaces (real parts) containing two EPs (a) in the linear regime and (b) and (c) under the impact of the polariton-polariton interaction $g_{c} = 6 µ eV µ m^{2}$ and the saturable gain. In (b) the total densities of the modes forming the EPs are identical, whereas in (c) the total densities inside each EP are distinct. The eigenvalues are depicted against the eigenenergy difference $E_{L}$ and the gain and loss difference $Δ Γ$ . For $g_{c} \neq 0$ the nonlinear correction to the energy difference is plotted with $Δ Γ$ . The lines at the bottom of the surface plot illustrate the orientation of the $Δ E = 0$ trace to visualize the rotation of the Riemann surfaces. (d) Deflection angle of the two projected lines as a function of the total densities of the adjacent modes for a nonlinearity of $g_{c} = 6 µ eV µ m^{2}$ . Energies are given in $meV$ .
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Figure 5
Nonlinearity-induced shift of the required gain difference $Δ Γ$ to access the EP. Inverted plot to illustrate the EP-position ( $x$ -axis) shifting in parameter space under variation of the nonlinearity parameter $g_{c}$ .
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Figure 6
All-optical vortex-antivortex switching under EP encircling. Illustration of the (a) first and (b) second EP encirclings. The contours of the stationary states are displayed at the corners of the sketched trajectory. The initial and final states are highlighted, and the phase information of the vortex-antivortex pair at the EP is displayed at the bottom. After the first encircling the topological charge of the vortex pair remains unchanged (highlighted by gray and yellow boxes). After the second encircling the topological charge of the vortex pair is switched (highlighted by a red box).
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