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Higher-order exceptional points in a non-reciprocal waveguide beam splitter

This paper demonstrates that introducing tunable non-reciprocity into a generalized lossy waveguide beam splitter enables the realization of higher-order exceptional points and the analysis of NOON-state evolution, offering a promising route for advanced sensing in non-Hermitian quantum systems without requiring strict dissipation.

Original authors: Hamed Ghaemi-Dizicheh, Shahram Dehdashti, Andreas Hanke, Ahmed Touhami, Janis Nötzel

Published 2026-10-27

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CC BY 4.0

Original authors: Hamed Ghaemi-Dizicheh, Shahram Dehdashti, Andreas Hanke, Ahmed Touhami, Janis Nötzel

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

In the hidden world of quantum physics, scientists often study systems that do not behave like the solid, predictable objects we see in daily life. Instead, they look at systems where energy can be lost or gained, creating a landscape that is far more fluid and strange. Within these systems, there are special moments called exceptional points. Imagine a place on a map where two different paths suddenly merge into a single, indistinguishable trail. At these points, the fundamental properties of the system change in a dramatic way, becoming incredibly sensitive to even the tiniest nudge from the outside world. This sensitivity is a powerful tool; the more paths that merge at once, the sharper the system's reaction becomes, offering a potential key to building sensors that can detect changes far smaller than anything current technology can measure. However, reaching these high-level merging points has been difficult, usually requiring a perfect and fragile balance of energy loss that is hard to maintain in the real world.

A team of researchers has now found a way to reach these high-level merging points without needing such a delicate balance of energy loss. By studying a device that splits light into two paths, they discovered that simply making the light travel differently in one direction than the other is enough to create these powerful points. This device, known as a waveguide beam splitter, acts like a junction where light can flow between two channels. In a standard version of this device, light flows equally in both directions, like water in a calm river. The researchers introduced a new element: non-reciprocity. This means the device was engineered so that light moving forward behaves differently than light moving backward, effectively breaking the symmetry of the flow. They found that by tuning this one-way behavior, they could force the system's energy levels to merge into a single point, creating what is called a higher-order exceptional point, even if the system had no energy loss at all.

The scientists focused on a specific setup where they sent a group of identical photons, or particles of light, into the device. They looked at what happened when the system was perfectly balanced versus when it was made to be one-way. In the balanced state, the system only reached this special merging point if they carefully added a specific amount of energy loss, a process that is difficult to control precisely. But when they turned on the non-reciprocal feature, the rules changed. The researchers showed that as they increased the one-way nature of the light flow, the amount of energy loss needed to reach the merging point dropped. In the most extreme case, where the light flowed only in one direction, the merging point appeared even when there was absolutely no energy loss. This means the device could achieve its most sensitive state without the usual complications of managing dissipation, opening a new path for creating robust and precise measurement tools.

To understand how this works, the team tracked the behavior of the light as it moved through the device. They used a special type of light state, known as a NOON state, which is a highly entangled group of photons that acts as a single, unified quantum object. When they watched this state evolve through the device, they saw that the one-way flow dramatically altered how the light distributed itself. In a normal, balanced device, the light would oscillate back and forth between the two paths. But with the non-reciprocal setting, the light behaved differently, often settling into specific patterns that revealed the underlying merging of the system's properties. The researchers also compared this quantum behavior to what would happen with classical light, which does not have these quantum connections. They found that the unique, sensitive patterns only appeared with the quantum light, proving that the effect relies on the delicate nature of quantum mechanics.

The results of this study suggest that non-reciprocal devices offer a flexible way to engineer these sensitive points without the strict requirements of previous methods. By adjusting the directionality of the light flow, scientists can now lower the threshold for reaching these high-sensitivity states, making them easier to achieve in practical applications. The work highlights that the direction in which light travels is just as important as how much energy is lost, providing a new degree of freedom for designing advanced optical systems. This discovery could lead to the development of next-generation sensors capable of detecting minute changes in their environment, as well as new ways to manipulate quantum states for information processing. The findings confirm that by breaking the symmetry of light flow, we can unlock new regimes of behavior in quantum systems, turning a theoretical curiosity into a practical engineering tool.

Technical Summary: Higher-order exceptional points in a non-reciprocal waveguide beam splitter

Problem Statement
Non-Hermitian systems exhibit exceptional points (EPs), where eigenvalues and eigenstates coalesce, leading to enhanced sensitivity to external perturbations that scales with the order of the EP. While higher-order EPs offer significant potential for advanced sensing, their realization is practically challenging due to stringent symmetry requirements and the need for precise fine-tuning of system parameters. Previous approaches, such as those utilizing lossy waveguide beam splitters in the quantum domain, often require specific dissipation rates or post-selected photon subspaces to achieve these states. Furthermore, engineering specific dissipation without incurring undesirable losses or diminishing interference effects remains difficult in experimental systems.

Methodology
The authors investigate a generalized lossy waveguide beam splitter with asymmetric (non-reciprocal) coupling. The system is modeled using a second-quantized Hamiltonian for two coupled waveguides, where one waveguide is lossy. The coupling amplitudes are parameterized to introduce non-reciprocity ( $\nu \neq \nu'$ ), controlled by a parameter $\eta$ ranging from reciprocal ( $\eta=0$ ) to fully unidirectional ( $\eta=1$ ).

To analyze the spectrum, the authors employ the Schwinger representation to map the two-mode $N$ -photon subspace to an $SU(2)$ algebra. By utilizing the Baker–Campbell–Hausdorff formula and similarity transformations, they analytically derive the eigenvalues of the non-Hermitian Hamiltonian. The study focuses on the $N$ -photon subspace, where the system dimension is $N+1$ .

For dynamical analysis, the authors examine the evolution of NOON-states (maximally path-entangled states) and classical statistical mixtures. They solve the Schrödinger equation for the time-evolution operator using the Wei-Norman method, expressing the dynamics in terms of $SU(2)$ operators. The evolution is analyzed via the mode occupation function $P(m; |\psi(z)\rangle)$ , which tracks the probability distribution of photon numbers in the waveguides as a function of propagation distance (analogous to time).

Key Results

Analytical Spectrum and EP Conditions: The study derives an analytical expression for the eigenvalues, showing that the critical dissipation threshold ( $\Gamma_c$ ) required to reach an exceptional point of order $N+1$ is dependent on the non-reciprocity parameter. Specifically, $\Gamma_c = 2\nu_0\sqrt{1-\eta^2}$ .
- In the reciprocal limit ( $\eta=0$ ), the system requires a specific dissipation rate ( $\Gamma_c = 2\nu_0$ ) to reach the EP.
- As non-reciprocity increases, the required dissipation threshold decreases.
- In the extreme unidirectional limit ( $\eta=1$ ), the critical dissipation vanishes ( $\Gamma_c = 0$ ), meaning higher-order EPs emerge even in the absence of dissipation.
Eigenvalue Flow and Spin Projections: Numerical simulations for $N=4$ demonstrate that increasing $\eta$ lowers the dissipation value at which eigenvalues coalesce. The authors visualize this using spin-projection components ( $J_x, J_y, J_z$ ). In reciprocal systems, the transition to the EP is marked by a distinct change in spin projections as dissipation increases. In unidirectional systems, the system resides at the EP even at zero dissipation, and the spin projections remain robust against further increases in loss.
Dynamics of NOON-states: The evolution of NOON-states reveals distinct behaviors based on the regime:
- Below EP: In reciprocal systems, mode occupation oscillates. In non-reciprocal systems with low loss, this periodicity is disrupted, and photons preferentially occupy higher-order modes.
- At/Above EP: Once the system reaches or exceeds the critical dissipation (or enters the unidirectional regime), the mode occupation stabilizes, with energy accumulating in low-loss regions.
- Quantum vs. Classical: A comparison between quantum-coherent NOON-states and classical statistical mixtures in the unidirectional, lossless regime shows that quantum coherence is essential for observing direction-dependent asymmetries, interference fringes, and amplitude revivals. Classical mixtures lack these features.

Significance and Claims
The paper claims that introducing non-reciprocity as a tunable parameter provides a new pathway to realize higher-order exceptional points without the stringent dissipation requirements typically associated with such systems. The primary contribution is the demonstration that non-reciprocity effectively lowers the critical dissipation threshold for EP formation, eventually allowing for the emergence of higher-order EPs in lossless, unidirectional systems.

The authors assert that these findings open new pathways for realizing higher-order EPs in non-reciprocal open quantum systems. They highlight the potential for these systems in advanced sensing applications and precise measurement equipment, leveraging the enhanced sensitivity inherent to higher-order EPs. Additionally, the work underscores the importance of quantum coherence in non-reciprocal dynamics, suggesting that coherent quantum inputs offer operational advantages over classical counterparts in photonic non-reciprocal systems. The study concludes that non-reciprocal platforms are versatile tools for engineering higher-order EPs, with potential applications in interferometric systems and quantum state manipulation.

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