Observation of anti-parity-time-symmetry, phase transitions and exceptional points in an optical fibre

Two-level systems commonly model diverse physical phenomena and are typically governed by Hermitian Hamiltonians with real eigenvalues and unitary dynamics. Non-Hermitian systems with gain/loss, however, can host parity-time (PT) symmetry that yields real spectra up to a phase transition at exceptional points (EPs), where eigenvalues and eigenmodes coalesce. Prior optical implementations often rely on precisely engineered micro/nanophotonic resonators with tight gain/loss balance and coupling control, posing fabrication and tuning challenges. This work asks whether EP physics, including phase transitions, enhanced sensitivities, and topological features, can be realized and precisely tuned in a simple, standard optical fibre platform using off-the-shelf components. The authors demonstrate an anti-PT-symmetric two-level system formed by two co-propagating probe tones in a single-mode telecom fibre, coupled and amplified via backward stimulated Brillouin scattering (SBS) and Brillouin-enhanced four-wave mixing (BE-FWM). They precisely control the non-Hermitian parameters (frequency detuning and coupling via pump power), observe an EP-driven phase transition, showcase square-root response near the EP, and explore topological encircling of the EP. The platform also offers enhanced sensitivity to changes in the Brillouin frequency shift (BFS), relevant for temperature and strain sensing.

The paper positions its contribution within the broader study of non-Hermitian physics, PT and anti-PT symmetry, and EP phenomena observed across photonics, acoustics, atomic systems, and electronic circuits. It highlights that most optical realizations use high-Q microresonators or nanostructures with stringent fabrication/tuning demands and limited post-fabrication control. It references demonstrations of PT/anti-PT symmetry, EP-enhanced sensing, and topological effects (mode switching, energy transfer, geometric phases). It also surveys Brillouin interactions, BE-FWM, and their applications (phase conjugation, all-optical processing, memory, spectroscopy, and distributed sensing), motivating the choice of SBS in fibre to realize tunable non-Hermitian dynamics without resonators. The discussion acknowledges ongoing debates on EP-based sensing advantages versus noise penalties and notes related fibre-loop PT systems that still rely on resonant degrees of freedom, contrasting them with the present traveling-wave, in-situ amplification/coupling approach.

Principle and model: Two continuous probe tones (complex amplitudes A1, A2) counterpropagate relative to two strong, equal-power SBS pump tones in a standard single-mode telecom fibre (length L=10 m), with probe and pump pairs frequency-separated by Δν≈1 MHz and each pump-probe pair offset by the fibre BFS νB (~10.762 GHz). Backward SBS generates acoustic waves at νB−Δν, νB, νB+Δν (SBS linewidth Γ/2π≈30 MHz), mediating amplification and imaginary-valued coupling between probe tones via BE-FWM. The probe evolution follows i dA/dz = H A with an anti-PT-symmetric 2×2 non-Hermitian Hamiltonian H that includes: diagonal SBS gain terms ig, imaginary coupling ig/2, and an effective wavenumber mismatch Δk (optical mismatch plus contributions from off-resonant Brillouin gain). The eigenvalues σ± satisfy σ±= i g ± sqrt(Δk^2 − (g/2)^2), yielding an EP when |Δk|=g/2, which is tunable via Δν and pump power Pp (through g=γ Pp, with γ the SBS gain coefficient). Deviations from BFS by ΔνB perturb the Hamiltonian with a real-valued coupling term, modifying eigenvalue behaviors and enabling BFS sensing near the EP. Experimental setup: All optical tones derive from a single narrow-linewidth laser at ν0 (~1550 nm). Probes: A single-sideband suppressed-carrier EOM (SSB-SC) driven by an AWG generates two probe tones at ν0−fIF±Δν/2 with independently set amplitudes/phases (fIF=5.332 GHz). Each probe component power is 3 dBm; probes are launched into one fibre end with polarization aligned. Pumps: A double-sideband SC EOM driven at Δν/2 produces two tones at ν0±Δν/2, then an SSB-SC EOM driven at ν0−fIF upshifts to ν0+fIF±Δν/2, ensuring each pump minus corresponding probe equals νB. Pumps are amplified by EDFA and injected from the opposite fibre end via a circulator, with per-tone power Pp=27–28 dBm. Polarization is aligned and fibre length kept below the beat length to maintain alignment. Detection and retrieval: The output probe is coherently detected using an optical local oscillator (OLO) derived from the same laser, downshifted by fLO=5.181 GHz via a DSB-SC EOM, and mixed in a balanced coherent receiver (1.6 GHz bandwidth). The receiver output is digitized at 2 GS/s; FFT components at fIF−fLO±Δν/2 yield complex envelopes A1,2(L). All RF sources and scope are locked to a common 10 MHz reference with calibrated phases. For fixed {Δν, ΔνB, Pp}, the input probe vector A(0) is scanned over N=8–18 states (equal magnitudes, varying relative phase φm) using the AWG. A 2×2 transfer matrix M between z=0 and L is estimated by least squares from measured input-output pairs, then diagonalized to extract eigenvalues via relation eig(M)=exp(−i σ L), assuming a uniform fibre. Parameter sweeps over Δν, ΔνB and Pp map eigenvalue surfaces and EP conditions. Operating conditions: BFS νB≈10.762 GHz (measured), Δν scanned around EP in the 1–1.5 MHz range, Pp set to 27 dBm (EP at ΔνEP≈1.145 MHz) and 28 dBm (EP shifted to ΔνEP≈1.31 MHz). SBS gain on probes ~5–6 dB on average. The system largely adheres to a two-level model over the 10 m fibre, with higher-order sidebands minimal but discussed as a source of small deviations.

- EP and phase transition: Measured eigenvalues σ± vs Δν at Pp=27 dBm show coalescence at ΔνEP=1.145 MHz, transitioning from anti-PT-symmetric (equal Im parts, split Re parts) to broken symmetry (split Im parts). Square-root scaling of eigenvalue splitting ∝√|Δν−ΔνEP| is observed near the EP, crossing over to linear behavior far from the EP. Measurement uncertainty in eigenvalues is ~0.001 m−1. - Enhanced BFS sensitivity: Near the EP, small offsets ΔνB from BFS produce eigenvalue splitting with square-root dependence and scaling with g. Maximum sensitivity was obtained for νB=10.7583 GHz ±50 kHz and ΔνEP=1.31 ±0.01 MHz at Pp=28 dBm. The ±50 kHz νB uncertainty corresponds to ±0.05 K temperature change; further improvements are expected with thermal stabilization and gain stabilization (<0.1%). - Topological encircling: Calculated and measured Riemann surfaces of σ± as functions of 2Δk/g and ΔB (normalized BFS offset) exhibit intersection of Im parts at ΔB=0 and avoided crossing of Re parts. Experimental parameter loops around the EP demonstrate eigenvalue swapping after one encirclement, evidencing the EP’s topological nature. - Eigenmode projection narrowing: Projections of output probe states onto instantaneous eigenmodes show significant linewidth squeezing as the EP is approached: from the Brillouin linewidth (~30 MHz) to 4.4 MHz (anti-PT regime near EP) and down to ~400 kHz at the EP. Below the EP, asymmetry and discontinuities at BFS appear in the projections, consistent with branch cuts on the Riemann sheets. - Practical sensitivity estimates: Given BFS dependence of ~1 MHz/K and ~1 MHz per 20 ppm strain, the platform’s EP-enhanced response can, in principle, resolve ~0.1 K temperature changes or ~2 ppm axial strain under stable conditions.

The results confirm that a simple, standard single-mode telecom fibre can host precisely tunable non-Hermitian dynamics, including an anti-PT-symmetric phase transition and an EP with square-root branch-point behavior. By leveraging SBS/BE-FWM, the authors introduce in-situ traveling-wave amplification and imaginary coupling without requiring resonators or bespoke nanofabrication, enabling fine control of non-Hermitian parameters (detuning and gain) and high measurement precision. The observed enhanced eigenvalue response to Δν and ΔνB near the EP underpins potential sensing applications (temperature/strain via BFS), while the demonstrated topological encircling and eigenvalue swapping establish the platform’s capability to explore non-Hermitian topology, with prospects for asymmetric energy transfer and mode switching upon dynamic encircling. The eigenmode projection narrowing further evidences degeneracy-induced spectral features, though it also increases susceptibility to noise. Compared to resonator-based PT systems, this traveling-wave fibre approach offers simplicity, compatibility with existing fibre infrastructure, and broad tunability. The study also acknowledges and contextualizes the ongoing debate about noise and ultimate sensitivity advantages of EP-based sensing, suggesting the platform is well-suited for systematic investigations of these limits.

This work demonstrates, for the first time in a single standard optical fibre, anti-PT-symmetric dynamics, a clear phase transition through an exceptional point, enhanced spectral response near the EP, topological eigenvalue swapping upon encircling, and dramatic narrowing of eigenmode projections. The platform relies solely on off-the-shelf components and traveling-wave SBS/BE-FWM, enabling precise control and measurement of non-Hermitian parameters with eigenvalue uncertainties ≈0.001 m−1. These findings open practical avenues for fibre-based EP studies and potential sensing applications leveraging BFS sensitivity, as well as fundamental explorations of non-Hermitian topology. Future directions include: realizing higher-order EPs by adding probe tones; distributed BFS sensing using Brillouin protocols; implementing dynamic encircling via z-dependent pump profiles to observe handedness-dependent energy transfer; integrating the concept into on-chip Brillouin photonics; developing closed-loop stabilization for extended dynamic range; refining multi-level Hamiltonian models (including higher-order sidebands) for increased accuracy; and quantitatively assessing noise and fundamental sensitivity limits near EPs.

- Modeling: The primary analysis uses a two-level Hamiltonian, neglecting cascaded higher-order sidebands that can build up with SBS gain; this leads to small discrepancies (e.g., average gain underestimation, slight asymmetries for ±ΔνB). - Dynamic range: EP-enhanced sensitivity persists over a limited BFS variation range (<~1 MHz), necessitating closed-loop tracking for practical sensing. - Noise considerations: Potential noise amplification (e.g., amplified spontaneous Brillouin scattering) near EP conditions was not fully analyzed; EP-based sensitivity may be offset by increased noise, affecting SNR. - Experimental stability: Residual pump power imbalance, polarization drifts, temperature fluctuations, and open-loop gain stability limit precision (e.g., <0.1% gain stability desired to reduce ΔνEP uncertainty below 10 kHz). - Encircling protocol: The experiment retrieves eigenvalues across different static settings rather than continuously encircling the EP under a single excitation, so handedness-dependent energy transfer was not directly observed. - Fibre losses and approximations: Linear losses are neglected relative to Brillouin gain; slowly varying envelope and undepleted pump approximations are assumed, which may deviate at higher gains or longer fibres.