Non-Hermitian systems, characterized by balanced gain and loss distributions, exhibit unique physics including phase transitions and exceptional points (EPs). EPs are spectral singularities where eigenvalues and eigenmodes coalesce, reducing dimensionality. While previously demonstrated in meticulously fabricated nanostructures, this study aimed to replicate these phenomena in a simpler, more accessible system: a standard single-mode optical fiber. The research question was whether the precise control achievable through stimulated Brillouin scattering (SBS) in a standard fiber could enable the observation of anti-parity-time (PT) symmetry, phase transitions, and EP singularities. This is significant because it moves away from complex nanofabrication, making the study of EPs more accessible and potentially paving the way for integration into existing optical communication infrastructure. Existing studies mostly utilized integrated nanophotonic devices and nanostructures, which are compact but challenging to fabricate and tune precisely. The use of standard optical fibers with readily available components provides a significant advantage in terms of cost, accessibility, and scalability for future applications.

Two-level systems, often modeled with 2x2 Hermitian Hamiltonians, are prevalent in physics. However, systems with gain and/or loss necessitate non-Hermitian Hamiltonians, which generally have complex eigenvalues. PT symmetry, a balance of gain and loss, results in real eigenvalues; however, varying the non-Hermiticity parameter leads to a phase transition and complex eigenvalues at an EP. The extreme sensitivity of eigenvalues around the EP to parameter perturbations has spurred interest in diverse applications. Optical realizations often use coupled microresonators with precise control of parameters. This work contrasts this by using a readily available platform—standard telecommunication single-mode fiber—leveraging the precise control provided by Brillouin scattering processes. The Brillouin-enhanced four-wave mixing (BE-FWM) process, known for applications in optical phase conjugation and all-optical calculus, was utilized to generate and control the non-Hermitian interaction between two probe tones.

The experimental setup involved launching two strong continuous pump waves and two weak continuous probe waves into opposite ends of a standard single-mode fiber. The probe and pump frequencies were precisely controlled to achieve strong SBS interactions. The frequency separation between the pump and probe waves was matched to the Brillouin frequency shift (BFS) of the fiber. Backward stimulated Brillouin scattering (SBS) amplified the probe tones, while BE-FWM, controlled by the pump waves, enabled coupling between them. The Hamiltonian describing this anti-PT symmetric system was formulated and solved to predict the behavior of the eigenvalues as a function of system and environmental parameters.The experimental measurements used a single narrow-band laser diode, electro-optic modulators (EOMs) to generate the pump and probe waves, an erbium-doped fiber amplifier (EDFA) for the pump waves, and a coherent photoreceiver to measure the output probe waves. The complex amplitudes of the output probe waves were obtained via digital Fourier transformation of the acquired data. The transfer matrix was retrieved through least-squares fitting of experimental data. Eigenvalues were extracted from the diagonalized transfer matrix, accounting for the propagation length of the fiber. The experimental setup is shown in Fig. 6. Data analysis involved fitting the experimental data to the theoretical model to extract the eigenvalues and eigenmodes of the system.

The experiments successfully demonstrated the transition from anti-PT-symmetric to broken-symmetry regimes by tuning the pump power and frequency detuning between the probe waves, passing through an EP. The observed eigenvalue splitting matched theoretical predictions with high accuracy (discrepancies around 0.001 m⁻¹). The system exhibited drastically enhanced sensitivity to changes in frequency detuning and Brillouin frequency shift (BFS) near the EP. This enhanced sensitivity resulted from the square-root dependence of eigenvalue splitting on the deviation from the EP. Experimental measurements confirmed the enhanced sensitivity to BFS variations, corresponding to temperature changes as small as ±0.05 Kelvin. The study also explored the topological aspects of the EP, observing the exchange of eigenvalues when the parameters were adiabatically varied around the EP, illustrating the topological nature of non-Hermitian degeneracies. The eigenmode projections exhibited spectral narrowing near the EP, transitioning from a linewidth of 30 MHz to 400 kHz. This narrowing was accompanied by significant variation in projection amplitudes depending on the input phase. These findings validated the theoretical model and demonstrated a phase transition in both the eigenvalue and eigenmode spectra.

This work successfully demonstrates the observation of anti-PT-symmetric phase transitions and EP singularities in a standard single-mode optical fiber using SBS-induced coupling. The high precision of the measurements (uncertainty of 0.001 m⁻¹, ten orders of magnitude smaller than the optical wavenumber) allowed for detailed investigation of the EP dynamics. The enhanced sensitivity to BFS changes near the EP holds promise for high-precision Brillouin sensing applications, but further investigation is needed to analyze the noise contribution in the system. The observation of eigenvalue swapping around the EP highlights the topological features accessible in this platform, opening avenues for exploring phenomena like asymmetric energy transfer and mode switching. The simplicity and accessibility of the fiber-optic platform make it a powerful tool for studying non-Hermitian physics and potential applications. The work compares favorably to previous research using integrated devices which generally feature tighter control, but at the expense of considerable complexity in their design and fabrication. By implementing the desired phenomena in readily available components, this study showcases a valuable advancement in the field.

This study successfully demonstrated the realization of non-Hermitian physics, including phase transitions and EP singularities, in a standard single-mode optical fiber using stimulated Brillouin scattering. The high precision of the measurements allowed for a detailed characterization of the system's behavior around the EP, highlighting its potential for high-precision sensing applications. The inherent flexibility and accessibility of the fiber-optic platform offer a unique environment for exploring the exotic physics of non-Hermitian systems and their potential for technological advancements. Future work could explore higher-order EPs, examine noise contributions in detail, and investigate applications such as spatially resolved BFS measurements.

The two-level system model used in the study is a simplification that neglects the generation of higher-order sidebands in the BE-FWM process. While the impact of these sidebands was minor in the current experiment, a more comprehensive model including higher-order terms might offer improved accuracy. The enhanced sensitivity to BFS changes is only effective within a limited dynamic range, requiring a closed-loop control mechanism for practical sensing applications. Additionally, the impact of amplified spontaneous Brillouin scattering (ASBS) noise on the sensing performance warrants further investigation. Further studies are necessary to fully assess the advantages of this technique over traditional Brillouin sensing methods.