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Vector-light interference with a multimode N00N state

Elvis Pillinen*, Tero Setälä, and Andreas Norrman

  • *Contact author: elvis.pillinen@uef.fi

Phys. Rev. Research 8, 013184 – Published 19 February, 2026

DOI: https://doi.org/10.1103/zq23-n1ql

Abstract

Multimode N00N states stand out as an excellent platform to explore higher-order quantum interference phenomena, with benefits in quantum-enhanced estimation of multiple parameters beyond classical limits. Here, we investigate higher-order polarization properties, geometric phase features, and associated wave-particle duality structures in double-slit vector-light interference involving a multimode N00N state. We particularly show that the interferometric polarization modulation and the accompanied geometric phase accumulation are N times faster for the multimode N00N state than for one-photon or classical light. We also formulate Nth-order extensions of the three wave-particle duality relations established for a single vector-light photon: the two via distinguishabilities and visibilities [A. Norrman et al., Phys. Rev. Lett. 119, 040401 (2017)], and the one via the geometric phase and which-path information ratio [E. Pillinen et al., Phys. Rev. A 111, 063709 (2025)]. Our work contributes to the broader fundamental understanding of how vectorial quantum light acts in higher-order interference, with possible opportunities in precision metrology outside scalar-field regimes.

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References (60)

  1. Z.-Y. J. Ou, Multi-Photon Quantum Interference (Springer, New York, 2007).
  2. R.-B. Jin, Z.-Q. Zeng, C. You, and C. Yuan, Quantum interferometers: Principles and applications, Prog. Quantum Electron. 96, 100519 (2024).
  3. J. P. Dowling, Quantum optical metrology—The lowdown on high-N00N states, Contemp. Phys. 49, 125 (2008).
  4. V. Giovannetti, S. Lloyd, and L. Maccone, Advances in quantum metrology, Nat. Photon. 5, 222 (2011).
  5. E. Polino, M. Valeri, N. Spagnolo, and F. Sciarrino, Photonic quantum metrology, AVS Quantum Sci. 2, 024703 (2020).
  6. H. Shin, K. W. C. Chan, H. J. Chang, and R. W. Boyd, Quantum spatial superresolution by optical centroid measurements, Phys. Rev. Lett. 107, 083603 (2011).
  7. F. Wolfgramm, C. Vitelli, F. A. Beduini, N. Godbout, and M. W. Mitchell, Entanglement-enhanced probing of a delicate material system, Nat. Photon. 7, 28 (2013).
  8. Y. Israel, S. Rosen, and Y. Silberberg, Supersensitive polarization microscopy using NOON states of light, Phys. Rev. Lett. 112, 103604 (2014).
  9. S. Slussarenko, M. M. Weston, H. M. Chrzanowski, L. K. Shalm, V. B. Verma, S. Woo Nam, and G. J. Pryde, Unconditional violation of the shot-noise limit in photonic quantum metrology, Nat. Photon. 11, 700 (2017).
  10. M. Hiekkamäki, F. Bouchard, and R. Fickler, Photonic angular superresolution using twisted N00N states, Phys. Rev. Lett. 127, 263601 (2021).
  11. M. Hiekkamäki, R. F. Barros, M. Ornigotti, and R. Fickler, Observation of the quantum Gouy phase, Nat. Photon. 16, 828 (2022).
  12. P. C. Humphreys, M. Barbieri, A. Datta, and I. A. Walmsley, Quantum enhanced multiple phase estimation, Phys. Rev. Lett. 111, 070403 (2013).
  13. L. Zhang and K. W. C. Chan, Quantum multiparameter estimation with generalized balanced multimode NOON-like states, Phys. Rev. A 95, 032321 (2017).
  14. M. Hiekkamäki and R. Fickler, High-dimensional two-photon interference effects in spatial modes, Phys. Rev. Lett. 126, 123601 (2021).
  15. S. Hong, J. ur Rehman, Y.-S. Kim, Y.-W. Cho, S.-W. Lee, H. Jung, S. Moon, S.-W. Han, and H.-T. Lim, Quantum enhanced multiple-phase estimation with multi-mode N00N states, Nat. Commun. 12, 5211 (2021).
  16. J. Rehman, S. Hong, S.-W. Lee, Y.-S. Kim, Y.-W. Cho, H. Jung, S. Moon, H. Shin, S.-W. Han, and H.-T. Lim, Optimal strategy for multiple-phase estimation under practical measurement with multimode NOON states, Phys. Rev. A 106, 032612 (2022).
  17. C. Rosales-Guzmán, B. Ndagano, and A. Forbes, A review of complex vector light fields and their applications, J. Opt. 20, 123001 (2018).
  18. J. Wang, F. Castellucci, and S. Franke-Arnold, Vectorial light-matter interaction: Exploring spatially structured complex light fields, AVS Quantum Sci. 2, 031702 (2020).
  19. T. Setälä, J. Tervo, and A. T. Friberg, Stokes parameters and polarization contrasts in Young's interference experiment, Opt. Lett. 31, 2208 (2006).
  20. A. T. Friberg and T. Setälä, Electromagnetic theory of optical coherence, J. Opt. Soc. Am. A 33, 2431 (2016).
  21. A. Norrman, K. Blomstedt, T. Setälä, and A. T. Friberg, Complementarity and polarization modulation in photon interference, Phys. Rev. Lett. 119, 040401 (2017).
  22. A. Norrman, A. T. Friberg, and G. Leuchs, Vector-light quantum complementarity and the degree of polarization, Optica 7, 93 (2020).
  23. G. Jaeger, M. A. Horne, and A. Shimony, Complementarity of one-particle and two-particle interference, Phys. Rev. A 48, 1023 (1993).
  24. G. Jaeger, A. Shimony, and L. Vaidman, Two interferometric complementarities, Phys. Rev. A 51, 54 (1995).
  25. B.-G. Englert, Fringe visibility and which-way information: An inequality, Phys. Rev. Lett. 77, 2154 (1996).
  26. M. Jakob and J. A. Bergou, Complementarity and entanglement in bipartite qudit systems, Phys. Rev. A 76, 052107 (2007).
  27. M. Jakob and J. A. Bergou, Quantitative complementarity relations in bipartite systems: Entanglement as a physical reality, Opt. Commun. 283, 827 (2010).
  28. J.-H. Huang, S. Wölk, S.-Y. Zhu, and M. S. Zubairy, Higher-order wave-particle duality, Phys. Rev. A 87, 022107 (2013).
  29. S. Pancharatnam, Generalized theory of interference and its applications, Proc. Indian Acad. Sci. 44, 247 (1956).
  30. M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. Lond. A 392, 45 (1984).
  31. M. V. Berry, The adiabatic phase and Pancharatnam's phase for polarized light, J. Mod. Opt. 34, 1401 (1987).
  32. P. Hariharan, The geometric phase, Prog. Opt. 48, 149 (2005).
  33. E. Cohen, H. Larocque, F. Bouchard, F. Nejadsattari, Y. Gefen, and E. Karimi, Geometric phase from Aharonov–Bohm to Pancharatnam–Berry and beyond, Nat. Rev. Phys. 1, 437 (2019).
  34. K. Y. Bliokh, M. A. Alonso, and M. R. Dennis, Geometric phases in 2D and 3D polarized fields: Geometrical, dynamical, and topological aspects, Rep. Prog. Phys. 82, 122401 (2019).
  35. C. P. Jisha, S. Nolte, and A. Alberucci, Geometric phase in optics: From wavefront manipulation to waveguiding, Laser Photon. Rev. 15, 2100003 (2021).
  36. C. Cisowski, J. B. Götte, and S. Franke-Arnold, Geometric phases of light: Insights from fiber bundle theory, Rev. Mod. Phys. 94, 031001 (2022).
  37. A. Hannonen, H. Partanen, A. Leinonen, J. Heikkinen, T. K. Hakala, A. T. Friberg, and T. Setälä, Measurement of the Pancharatnam–Berry phase in two-beam interference, Optica 7, 1435 (2020).
  38. A. Leinonen, A. Hannonen, H. Partanen, J. Heikkinen, T. Setälä, A. T. Friberg, and T. K. Hakala, Noncyclic continuous Pancharatnam–Berry phase in dual-beam interference, Commun. Phys. 6, 132 (2023).
  39. E. Pillinen, A. Halder, A. T. Friberg, T. Setälä, and A. Norrman, Geometric phase and wave-particle duality of the photon, Phys. Rev. A 111, 063709 (2025).
  40. J. Brendel, W. Dultz, and W. Martienssen, Geometric phases in two-photon interference experiments, Phys. Rev. A 52, 2551 (1995).
  41. H. Kobayashi, Y. Ikeda, S. Tamate, T. Nakanishi, and M. Kitano, Nonlinear behavior of geometric phases induced by photon pairs, Phys. Rev. A 83, 063808 (2011).
  42. W.-R. Qi, R. Liu, L.-J. Kong, Z.-X. Wang, S.-Y. Huang, C. Tu, Y. Li, and H.-T. Wang, Pancharatnam–Berry geometric phase memory based on spontaneous parametric down-conversion, Opt. Lett. 45, 682 (2020).
  43. A. Hannonen, H. Partanen, J. Tervo, T. Setälä, and A. T. Friberg, Pancharatnam-Berry phase in electromagnetic double-pinhole interference, Phys. Rev. A 99, 053826 (2019).
  44. R. J. Glauber, The quantum theory of optical coherence, Phys. Rev. 130, 2529 (1963).
  45. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).
  46. M. Hanhisalo, M. S. Mirmoosa, T. Setälä, Ł. Rudnicki, and A. Norrman, Quantum uncertainty of optical coherence, Phys. Rev. A 110, 063702 (2024).
  47. J. Ellis and A. Dogariu, Complex degree of mutual polarization, Opt. Lett. 29, 536 (2004).
  48. O. Korotkova and E. Wolf, Generalized Stokes parameters of random electromagnetic beams, Opt. Lett. 30, 198 (2005).
  49. J. Tervo, T. Setälä, A. Roueff, P. Réfrégier, and A. T. Friberg, Two-point Stokes parameters: Interpretation and properties, Opt. Lett. 34, 3074 (2009).
  50. J. J. Gil and R. Ossikovski, Polarized Light and the Mueller Matrix Approach (CRC Press, Boca Raton, FL, 2022), 2nd ed.
  51. A. R. Brown and L. Susskind, The second law of quantum complexity, Phys. Rev. D 97, 086015 (2018).
  52. C. Gerry and P. Knight, Introductory Quantum Optics (Cambridge University Press, Cambridge, 2023), 2nd ed.
  53. See Supplemental Material at https://link.aps.org/supplemental/10.1103/zq23-n1ql for detailed derivations of the Nth-order Stokes parameters in Eq. (13), the total phase in Eq. (29), the dynamic phase in Eq. (30), and the geometric phase in Eq. (33), which includes Refs. [54, 55, 56].
  54. E. Sjöqvist, Geometric phases in quantum information, Int. J. Quantum Chem. 115, 1311 (2015).
  55. The equation commonly given in mathematical tables does not lead to a continuous solution. See, for example, Eq. (324) in D. Zwillinger, CRC Standard Mathematical Tables and Formulae (CRC Press, Boca Raton, FL, 2003), 31st ed.
  56. Continuous extension: J. Ming, Integrate ∫1a+cosxdx, mathematics stack exchange, https://math.stackexchange.com/questions/134577/integrate-int-frac1a-cos-x-dx (2019).
  57. J. Samuel and R. Bhandari, General setting for Berry's phase, Phys. Rev. Lett. 60, 2339 (1988).
  58. T. van Dijk, H. F. Schouten, W. Ubachs, and T. D. Visser, The Pancharatnam–Berry phase for non-cyclic polarization changes, Opt. Express 18, 10796 (2010).
  59. T. van Dijk, H. F. Schouten, and T. D. Visser, Geometric interpretation of the Pancharatnam connection and non-cyclic polarization changes, J. Opt. Soc. Am. A 27, 1972 (2010).
  60. Z. Zhou, Y. Margalit, S. Moukouri, Y. Meir, and R. Folman, An experimental test of the geodesic rule proposition for the noncyclic geometric phase, Sci. Adv. 6, eaay8345 (2020).

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