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Generalized Petermann factor of non-Hermitian systems at exceptional points

Julius Kullig and Jan Wiersig*

Henning Schomerus†

  • Institut für Physik, Otto-von-Guericke-Universität Magdeburg, Postfach 4120, D-39016 Magdeburg, Germany

  • *Contact author: jan.wiersig@ovgu.de
  • †Contact author: h.schomerus@lancaster.ac.uk
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Phys. Rev. Research 7, 043246 – Published 2 December, 2025

DOI: https://doi.org/10.1103/5sfv-54h2

Abstract

The nonorthogonality of modes in open systems significantly modifies their resonant response, resulting in quantitative and qualitative deviations from Breit-Wigner resonance relations. For isolated resonances with a Lorentzian lineshape, the deviations amount to a quantitative enhancement of the resonance linewidth, given by the Petermann factor, which is determined by the overlap of left and right eigenmodes of the underlying effectively non-Hermitian Hamiltonian. The Petermann factor diverges at exceptional points (EPs), where complex resonance frequencies become degenerate, and right and left eigenmodes are orthogonal to each other. This divergence signifies a qualitative departure from a Lorentzian lineshape, which has moved into the focus of recent attention. In this work, we develop the analog of the Petermann factor for this qualitatively modified response at an EP, and describe how this EP Petermann factor manifests in a variety of physical scenarios. First, we identify this analog in physical terms as an enhancement of the response of a system to external or parametric perturbations. Utilizing two natural orthogonally projected reference systems based on the right and left eigenvectors, we show that each choice carries a precise geometric interpretation that naturally extends the notion of the Petermann factor for isolated resonances to EPs. The two choices can be combined into an overall EP Petermann factor, which again can be expressed in purely geometric terms. Second, we illuminate the geometric mechanisms that determine the size of the EP Petermann factor, by considering the role of modes participating in the degeneracy and those that remain spectrally separated from it. This also leads to a systematic description to evaluate this factor. Third, we design a system that allows us to study the EP Petermann factor in a specific physical setup, consisting of two microrings coupled to a waveguide with embedded semitransparent mirrors. With this example, we demonstrate that our approach gives more accurate results for the spectral response strength than conventional truncation schemes. These results complete the description of systems operating at exceptional points in the same way as the original Petermann factor does for isolated resonances, and pave the way to the design of open systems with unconventional spectral response, be it in emission or in response to static and dynamic perturbations.

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References (59)

  1. R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, and D. N. Christodoulides, Non-Hermitian physics and PT symmetry, Nat. Phys. 14, 11 (2018).
  2. Ş. K. Özdemir, S. Rotter, F. Nori, and L. Yang, Parity–time symmetry and exceptional points in photonics, Nat. Mater. 18, 783 (2019).
  3. L. Huang, S. Huang, C. Shen, S. Yves, A. S. Pilipchuk, X. Ni, S. Kim, K. Y. Chiang, D. A. Powell, J. Zhu, Y. Cheng, Y. Li, A. F. Sadreev, A. Alù, and A. E. Miroshnichenko, Acoustic resonances in non-Hermitian open systems, Nat. Rev. Phys. 6, 11 (2024).
  4. E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Exceptional topology of non-Hermitian systems, Rev. Mod. Phys. 93, 015005 (2021).
  5. H. Feshbach, Unified theory of nuclear reactions, Ann. Phys. 5, 357 (1958).
  6. H. Feshbach, A unified theory of nuclear reactions. II, Ann. Phys. 19, 287 (1962).
  7. L. N. Trefethen and M. Embree, Spectra and Pseudospectra (Princeton University Press, Princeton, NJ, 2005).
  8. K. G. Makris, L. Ge, and H. E. Türeci, Anomalous transient amplification of waves in non-normal photonic media, Phys. Rev. X 4, 041044 (2014).
  9. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Beam dynamics in PT symmetric optical lattices, Phys. Rev. Lett. 100, 103904 (2008).
  10. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Observation of parity-time symmetry in optics, Nat. Phys. 6, 192 (2010).
  11. J. Wiersig, A. Eberspächer, J.-B. Shim, J.-W. Ryu, S. Shinohara, M. Hentschel, and H. Schomerus, Nonorthogonal pairs of copropagating optical modes in deformed microdisk cavities, Phys. Rev. A 84, 023845 (2011).
  12. J. Wiersig, Structure of whispering-gallery modes in optical microdisks perturbed by nanoparticles, Phys. Rev. A 84, 063828 (2011).
  13. Y. V. Fyodorov and D. V. Savin, Statistics of resonance width shifts as a signature of eigenfunction nonorthogonality, Phys. Rev. Lett. 108, 184101 (2012).
  14. M. Davy and A. Z. Genack, Selectively exciting quasi-normal modes in open disordered systems, Nat. Commun. 9, 4714 (2018).
  15. H. Schomerus, Nonreciprocal response theory of non-Hermitian mechanical metamaterials: Response phase transition from the skin effect of zero modes, Phys. Rev. Res. 2, 013058 (2020).
  16. T. Ozawa and H. Schomerus, Geometric contribution to adiabatic amplification in non-Hermitian systems, Phys. Rev. Res. 7, 013173 (2025).
  17. K. Petermann, Calculated spontaneous emission factor for double-heterostructure injection lasers with gain-induced waveguiding, IEEE J. Quantum Electron. 15, 566 (1979).
  18. A. E. Siegman, Lasers (University Science Books, Sausalito, CA, 1986).
  19. H. Schomerus, K. M. Frahm, M. Patra, and C. W. J. Beenakker, Quantum limit of the laser linewidth in chaotic cavities and statistics of residues of scattering matrix poles, Physica A 278, 469 (2000).
  20. H. Schomerus, Excess quantum noise due to mode nonorthogonality in dielectric microresonators, Phys. Rev. A 79, 061801(R) (2009).
  21. Y. D. Chong and A. D. Stone, General linewidth formula for steady-state multimode lasing in arbitrary cavities, Phys. Rev. Lett. 109, 063902 (2012).
  22. J. C. Pillay, Y. Natsume, A. D. Stone, and Y. D. Chong, Generalized sub-Schawlow-Townes laser linewidths via material dispersion, Phys. Rev. A 89, 033840 (2014).
  23. J. C. Budich and E. J. Bergholtz, Non-Hermitian topological sensors, Phys. Rev. Lett. 125, 180403 (2020).
  24. H. Wenzel, U. Bandelow, H.-J. Wünsche, and J. Rehberg, Mechanisms of fast self pulsations in two-section DFB lasers, IEEE J. Quantum Electron. 32, 69 (1996).
  25. M. V. Berry, Mode degeneracies and the Petermann excess-noise factor for unstable lasers, J. Mod. Opt. 50, 63 (2003).
  26. S.-Y. Lee, J.-W. Ryu, J.-B. Shim, S.-B. Lee, S. W. Kim, and K. An, Divergent Petermann factor of interacting resonances in a stadium-shaped microcavity, Phys. Rev. A 78, 015805 (2008).
  27. T. Kato, Perturbation Theory for Linear Operators (Springer, New York, NY, 1966).
  28. J. Wiersig, Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: Application to microcavity sensors for single-particle detection, Phys. Rev. Lett. 112, 203901 (2014).
  29. W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, Exceptional points enhance sensing in an optical microcavity, Nature (London) 548, 192 (2017).
  30. H. Hodaei, A. Hassan, S. Wittek, H. Carcia-Cracia, R. El-Ganainy, D. Christodoulides, and M. Khajavikhan, Enhanced sensitivity at higher-order exceptional points, Nature (London) 548, 187 (2017).
  31. Z. Xiao, H. Li, T. Kottos, and A. Alù, Enhanced sensing and nondegraded thermal noise performance based on PT-symmetric electronic circuits with a sixth-order exceptional point, Phys. Rev. Lett. 123, 213901 (2019).
  32. J. Wiersig, Prospects and fundamental limits in exceptional point-based sensing, Nat. Commun. 11, 2454 (2020).
  33. J. Wiersig, Review of exceptional point-based sensors, Photon. Res. 8, 1457 (2020).
  34. R. Kononchuk, J. Cai, F. Ellis, R. Thevamaran, and T. Kottos, Exceptional-point-based accelerometers with enhanced signal-to-noise ratio, Nature (London) 607, 697 (2022).
  35. F. Binkowski, J. Kullig, F. Betz, L. Zschiedrich, A. Walther, J. Wiersig, and S. Burger, Computing eigenfrequency sensitivities near exceptional points, Phys. Rev. Res. 6, 023148 (2024).
  36. J. Wiersig, Response strengths of open systems at exceptional points, Phys. Rev. Res. 4, 023121 (2022).
  37. J. T. Chalker and B. Mehlig, Eigenvector statistics in non-Hermitian random matrix ensembles, Phys. Rev. Lett. 81, 3367 (1998).
  38. J. Wiersig, Nonorthogonality constraints in open quantum and wave systems, Phys. Rev. Res. 1, 033182 (2019).
  39. S. A. van Langen, P. W. Brouwer, and C. W. J. Beenakker, Fluctuating phase rigidity for a quantum chaotic system with partially broken time-reversal symmetry, Phys. Rev. E 55, R1 (1997).
  40. J. Wiersig, Petermann factors and phase rigidities near exceptional points, Phys. Rev. Res. 5, 033042 (2023).
  41. H. Schomerus, Eigenvalue sensitivity from eigenstate geometry near and beyond arbitrary-order exceptional points, Phys. Rev. Res. 6, 013044 (2024).
  42. S. Bid and H. Schomerus, Uniform response theory of non-Hermitian systems: Non-Hermitian physics beyond the exceptional point, Phys. Rev. Res. 7, 023062 (2025).
  43. J. Wiersig, Moving along an exceptional surface towards a higher-order exceptional point, Phys. Rev. A 108, 033501 (2023).
  44. G. Yoo, H.-S. Sim, and H. Schomerus, Quantum noise and mode nonorthogonality in non-Hermitian PT-symmetric optical resonators, Phys. Rev. A 84, 063833 (2011).
  45. A. Pick, B. Zhen, O. D. Miller, C. W. Hsu, F. Herandez, A. W. Rodriguez, M. Soljačić, and S. G. Johnson, General theory of spontaneous emission near exceptional points, Opt. Express 25, 12325 (2017).
  46. A. Hashemi, K. Busch, D. N. Christodoulides, Ş. K. Özdemir, and R. El-Ganainy, Linear response theory of open systems with exceptional points, Nat. Commun. 13, 3281 (2022).
  47. S. Soleymani, Q. Zhong, M. Mokim, S. Rotter, R. El-Ganainy, and Ş. K. Özdemir, Chiral and degenerate perfect absorption on exceptional surfaces, Nat. Commun. 13, 599 (2022).
  48. Y.-Y. Zou, Y. Zhou, L.-M. Chen, and Y. Peng, Detecting bulk and edge exceptional points in non-Hermitian systems through generalized Petermann factors, Front. Phys. 19, 23201 (2024).
  49. J. Kullig, D. Grom, S. Klembt, and J. Wiersig, Higher-order exceptional points in waveguide-coupled microcavities: Perturbation induced frequency splittings and mode patterns, Photonics Res. 11, A54 (2023).
  50. Within our response framework, this Berry phase can be determined from the invariance of the resonant part (51) of the Green's function when one encircles the EP in parameter space, after which the eigenvalues are cyclically interchanged.
  51. Y.-J. Cheng, C. G. Fanning, and A. E. Siegman, Experimental observation of a large excess quantum noise factor in the linewidth of a laser oscillator having nonorthogonal modes, Phys. Rev. Lett. 77, 627 (1996).
  52. M. A. van Eijkelenborg, A. M. Lindberg, M. S. Thijssen, and J. P. Woerdman, Resonance of quantum noise in an unstable cavity laser, Phys. Rev. Lett. 77, 4314 (1996).
  53. M. Röntgen, N. E. Palaiodimopoulos, C. V. Morfonios, I. Brouzos, M. Pyzh, F. K. Diakonos, and P. Schmelcher, Designing pretty good state transfer via isospectral reductions, Phys. Rev. A 101, 042304 (2020).
  54. Q. Zhong, J. Ren, M. Khajavikhan, D. N. Christodoulides, Ş. K. Özdemir, and R. El-Ganainy, Sensing with exceptional surfaces in order to combine sensitivity with robustness, Phys. Rev. Lett. 122, 153902 (2019).
  55. J. Pomplun, S. Burger, L. Zschiedrich, and F. Schmidt, Adaptive finite element method for simulation of optical nano structures, Phys. Status Solidi B 244, 3419 (2007).
  56. J. Kullig and J. Wiersig, Calculating the spectral response strength of non-Hermitian systems with an exceptional point directly from wave simulations, Phys. Rev. Res. 7, 013223 (2025).
  57. X. Wang and G. Ma, Experimental measurement of non-Hermitian left eigenvectors, Front. Phys. 20, 054202 (2025).
  58. J. Kullig, J. Wiersig, and H. Schomerus, Source code for simulations: Generalized Petermann factor of non-Hermitian systems at exceptional points, Zenodo (2025), http://doi.org/10.5281/zenodo.15641127.
  59. J. Wiersig, Non-Hermitian effects due to asymmetric backscattering of light in whispering-gallery microcavities, in Parity-time Symmetry and Its Applications, edited by D. Christodoulides and J. Yang (Springer, Singapore, 2018), pp. 155–184.

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Phys. Rev. Research 7, 043246– Published 2 December, 2025
  • Received 12 June 2025
  • Accepted 13 November 2025
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