🌀 nonlinear sciences

Re-entrant parity-time phase transitions in locally coupled ring resonators

This paper investigates two parity-time-symmetric ring resonators coupled via a finite-width super-Gaussian profile, demonstrating that local coupling induces re-entrant phase transitions and multiple exceptional points in the linear regime while enabling the formation of dynamically persistent nonlinear waveforms.

Original authors: Nguyen Duc Anh Quan, Le Xuan The Tai, Doan Quang Tri, Pawel S. Jung, Marek Trippenbach, Nguyen Viet Hung

Published 2026-08-11

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CC BY 4.0

Original authors: Nguyen Duc Anh Quan, Le Xuan The Tai, Doan Quang Tri, Pawel S. Jung, Marek Trippenbach, Nguyen Viet Hung

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where light doesn't just travel; it dances in a delicate balance between creation and destruction. In the realm of physics, there's a special kind of stage called a "Non-Hermitian system." Think of it like a musical duet where one singer is given a microphone that boosts their voice (gain) and the other has a microphone that muffles theirs (loss). Usually, this would make the song chaotic and out of control. But, if the two singers are perfectly matched and mirror each other, something magical happens: the chaos cancels out, and the music stays perfectly harmonious. This is called "Parity-Time (PT) symmetry." Scientists love studying this because it helps them understand how to control light in tiny, high-tech devices like lasers and sensors. The big question is: what happens if you change the rules of how these two singers interact? Do they stay in harmony, or does the music break into a noisy mess?

This paper explores that question using a very specific setup: two tiny, circular tracks for light, known as ring resonators. Usually, scientists imagine these rings are glued together all the way around, like two tires pressed against each other along their entire length. But in this study, the researchers asked a different question: What if the rings only touch in a small, specific spot? They simulated two rings that are linked only over a finite, narrow section, described by a smooth, bell-shaped curve (a super-Gaussian profile). They found that this "local" connection creates a surprisingly complex dance. Instead of a simple switch between harmony and chaos, the system can flip back and forth multiple times. As they tweaked the strength of the gain and loss, or the width of the touching spot, the light waves would enter a chaotic state, then suddenly snap back to harmony, and then go chaotic again. This "re-entrant" behavior means the system has multiple "tipping points" rather than just one.

The researchers also looked at what happens when the light gets really bright and starts interacting with itself (nonlinearity). They took the stable patterns they found in the linear (low-light) world and slowly turned up the volume. They discovered that some of these light patterns could survive the increase in brightness and keep dancing stably for a long time, while others would quickly fall apart. Interestingly, the patterns that started as excited, wiggly states were much harder to keep stable than the calm, ground-state patterns. They also tested a different type of light-matter interaction (saturable nonlinearity) and found it allowed the light to stay stable over a wider range of brightness.

In short, this paper shows that by simply changing where and how much two light rings touch, you can create a rich landscape of stability and chaos. It's like tuning a radio where, instead of just finding one clear station, you can find a station, lose it, find it again, lose it again, all by turning a single dial. This gives scientists a new geometric tool to control how light behaves in complex systems, potentially leading to better ways to manage light in future optical devices. The findings are based on detailed mathematical models and computer simulations, which act as a rigorous testbed for these ideas, showing that the "re-entrant" behavior is a real possibility in these specific ring setups.

Technical Summary: Re-entrant Parity–Time Phase Transitions in Locally Coupled Ring Resonators

Problem Statement
The paper investigates the spectral and dynamical properties of two identical ring resonators coupled over a finite angular region, a configuration that deviates from the standard assumption of spatially uniform coupling. While previous studies have established the behavior of Parity–Time (PT) symmetric dimers with uniform coupling or localized interactions in specific limits, a systematic characterization of how a continuously tunable, finite-width local coupling profile affects multimode PT transitions is lacking. Specifically, the authors address how local coupling lifts the degeneracy of counterpropagating modes and influences the topology of exceptional points (EPs) and phase transitions in a balanced gain-loss system.

Methodology
The study employs a theoretical framework based on coupled nonlinear Schrödinger equations for the slowly varying fields $\psi_1(x, t)$ and $\psi_2(x, t)$ in two rings. The system features balanced linear gain ( $+\gamma$ ) and loss ( $-\gamma$ ) and is coupled via a real, even, high-order super-Gaussian profile $\kappa(x)$ , which approximates a flat coupling window with steep edges.

Linear Regime: The authors analyze the linear eigenproblem $H\phi = \lambda\phi$ . They derive analytical spectra for two limiting cases:
- Homogeneous-coupling limit: Where the coupling width $W$ is much larger than the ring circumference.
- Fixed-amplitude narrow-contact limit: Where $W \to 0$ at fixed peak coupling $\kappa_0$ .
- Finite-width regime: Treated numerically using Fourier collocation to discretize the periodic domain. The authors track eigenvalues and eigenvectors to identify EPs (where eigenvalues and eigenvectors coalesce) and map the spectral growth rate $\Gamma_{\max}$ .
Nonlinear Regime: To study dynamical stability, the authors use an adiabatic continuation method. Selected linear eigenstates (ground and excited modes) in the unbroken-PT phase serve as seeds. The nonlinear coefficient $\sigma$ is ramped up from zero to a target value over a finite time, followed by propagation to observe if the resulting finite-amplitude waveforms remain dynamically persistent (bounded power and stationary intensity profile). Both Kerr ( $\beta=0$ ) and saturable ( $\beta=1$ ) nonlinearities are considered.

Key Contributions and Results

Lifting of Angular Degeneracy: Unlike uniform coupling, the finite-width local coupling introduces nonzero spatial Fourier components. These components mix different angular harmonics, breaking the continuous rotational symmetry responsible for the degeneracy of clockwise ( $+n$ ) and counterclockwise ( $-n$ ) modes. Consequently, each excited angular doublet ( $n \ge 1$ ) splits into four distinct branches classified by spatial parity (even and odd combinations of $\cos(nx)$ and $\sin(nx)$ ). The ground state ( $n=0$ ) retains only two branches.
Multiple Exceptional Points and Re-entrant Transitions: The collisions among these four branches generate multiple exceptional-point boundaries rather than a single threshold. The resulting phase diagrams in the $(W, \gamma)$ and $(W, \kappa_0)$ planes exhibit complex structures, including disconnected "islands" and "tongues" of broken-PT phases.
- Re-entrant Behavior: The paper demonstrates that varying the gain-loss strength ( $\gamma$ ), coupling width ( $W$ ), or peak coupling amplitude ( $\kappa_0$ ) can induce a sequence of unbroken–broken–unbroken transitions. A system can transition from a real spectrum to a complex-conjugate spectrum and back to a real spectrum as a single parameter is monotonically varied. This re-entry is mode-selective, affecting specific branch pairs while leaving others real.
Nonlinear Dynamics and Stability:
- Kerr Nonlinearity: Adiabatic propagation from linear seeds yields finite-amplitude Kerr waveforms that remain dynamically persistent over the simulated interval. The ground mode supports a significantly larger range of nonlinear strength ( $\sigma_{f,\max} \approx 6$ ) compared to the first excited mode ( $\sigma_{f,\max} \approx 2$ ). The narrower range for excited modes is attributed to their higher sensitivity to nonlinear detuning and mixing with nearby modes.
- Saturable Nonlinearity: Comparisons with a saturable response model show broader ranges of dynamical persistence for both ground and excited branches, consistent with the bounded nature of the saturable phase shift.

Significance and Claims
The authors claim that the spatial profile of inter-resonator coupling serves as a geometric control parameter for multimode PT systems. By tuning the coupling width, one can manipulate the location and multiplicity of exceptional points, thereby selecting specific PT-phase sequences (including re-entrant transitions) at fixed gain and loss levels.

The work establishes that the finite-width coupling regime introduces spectral behaviors absent in uniformly coupled dimers, specifically the resolution of degenerate modes into parity-dependent branches and the emergence of disconnected broken-PT domains. While the nonlinear results demonstrate dynamical persistence over the simulated observation interval, the authors modestly note that these findings do not constitute a complete linear stability analysis, which remains numerically challenging for locally coupled rings.

Ultimately, the paper concludes that engineering the spatial profile of coupling provides an effective means of controlling spectral structure, exceptional-point boundaries, and nonlinear wave dynamics in non-Hermitian ring-resonator systems.

Technical Summary: Re-entrant Parity–Time Phase Transitions in Locally Coupled Ring Resonators

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