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Harnessing hybrid frequency-entangled qudits through quantum interference

Sheng-Hung Wang1,2, Po-Han Chen1,2, Cheng-Yu Yang3, Yen-Hung Chen2,3,4,*, and Pin-Ju Tsai1,2,3,†

  • *Contact author: yhchen@dop.ncu.edu.tw
  • †Contact author: tpinju@ncu.edu.tw
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Phys. Rev. Research 7, 043152 – Published 12 November, 2025

DOI: https://doi.org/10.1103/99qy-y833

Abstract

High-dimensional (HD) quantum entanglement expands the Hilbert space, offering a robust framework for quantum information processing with enhanced capacity and error resilience. In this work, we present an HD frequency-domain entangled state, the hybrid frequency-entangled qudit (HFEQ), generated via Hong-Ou-Mandel (HOM) interference, exhibiting both discrete-variable (DV) and continuous-variable (CV) characteristics in the frequency domain. By tuning HOM interference, we generate and control HFEQs with dimensions D=5,7,9, and 11, confirming their DV nature. Franson interferometry verifies global frequency correlations with a visibility of 97.6±0.6% and CV entanglement within individual frequency modes with a visibility exceeding 95%. Furthermore, based on the developed theoretical model, the generated HFEQs exhibit a Schmidt number that exceeds the upper bound D of conventional frequency-entangled qudits, certifying additional entanglement inherent to the hybrid structure. Our findings provide deeper insight into the physical nature of frequency-entangled qudits generated by quantum interference and introduce a resource for HD time-frequency quantum information processing.

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References (66)

  1. J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, Beating the channel capacity limit for linear photonic superdense coding, Nat. Phys. 4, 282 (2008).
  2. S. Ecker, F. Bouchard, L. Bulla, F. Brandt, O. Kohout, F. Steinlechner, R. Fickler, M. Malik, Y. Guryanova, R. Ursin et al., Overcoming noise in entanglement distribution, Phys. Rev. X 9, 041042 (2019).
  3. Y. I. Bogdanov, M. V. Chekhova, S. P. Kulik, G. A. Maslennikov, A. A. Zhukov, C. H. Oh, and M. K. Tey, Qutrit state engineering with biphotons, Phys. Rev. Lett. 93, 230503 (2004).
  4. S. Karan, R. Prasad, and A. K. Jha, Postselection-free controlled generation of a high-dimensional orbital-angular-momentum entangled state, Phys. Rev. Appl. 20, 054027 (2023).
  5. H. Cao, S.-C. Gao, C. Zhang, J. Wang, D.-Y. He, B.-H. Liu, Z.-W. Zhou, Y.-J. Chen, Z.-H. Li, S.-Y. Yu et al., Distribution of high-dimensional orbital angular momentum entanglement over a 1 km few-mode fiber, Optica 7, 232 (2020).
  6. M. Hiekkamäki and R. Fickler, High-dimensional two-photon interference effects in spatial modes, Phys. Rev. Lett. 126, 123601 (2021).
  7. M. P. Edgar, D. S. Tasca, F. Izdebski, R. E. Warburton, J. Leach, M. Agnew, G. S. Buller, R. W. Boyd, and M. J. Padgett, Imaging high-dimensional spatial entanglement with a camera, Nat. Commun. 3, 984 (2012).
  8. F. Devaux, A. Mosset, P.-A. Moreau, and E. Lantz, Imaging spatiotemporal Hong-Ou-Mandel interference of biphoton states of extremely high Schmidt number, Phys. Rev. X 10, 031031 (2020).
  9. F. Steinlechner, S. Ecker, M. Fink, B. Liu, J. Bavaresco, M. Huber, T. Scheidl, and R. Ursin, Distribution of high-dimensional entanglement via an intra-city free-space link, Nat. Commun. 8, 15971 (2017).
  10. B. Brecht, D. V. Reddy, C. Silberhorn, and M. G. Raymer, Photon temporal modes: A complete framework for quantum information science, Phys. Rev. X 5, 041017 (2015).
  11. I. Gianani, M. Sbroscia, and M. Barbieri, Measuring the time–frequency properties of photon pairs: A short review, AVS Quantum Sci. 2, 011701 (2020).
  12. K.-C. Chang, X. Cheng, M. C. Sarihan, and C. W. Wong, Recent advances in high-dimensional quantum frequency combs, Newton 1, 100024 (2025).
  13. H. Yu, A. O. Govorov, H.-Z. Song, and Z. Wang, Time-encoded photonic quantum states: Generation, processing, and applications, Appl. Phys. Rev. 11, 041318 (2024).
  14. H.-H. Lu, M. Liscidini, A. L. Gaeta, A. M. Weiner, and J. M. Lukens, Frequency-bin photonic quantum information, Optica 10, 1655 (2023).
  15. J. M. Lukens and P. Lougovski, Frequency-encoded photonic qubits for scalable quantum information processing, Optica 4, 8 (2017).
  16. T. Yamazaki, T. Arizono, T. Kobayashi, R. Ikuta, and T. Yamamoto, Linear optical quantum computation with frequency-comb qubits and passive devices, Phys. Rev. Lett. 130, 200602 (2023).
  17. C. Cui, K. P. Seshadreesan, S. Guha, and L. Fan, High-dimensional frequency-encoded quantum information processing with passive photonics and time-resolving detection, Phys. Rev. Lett. 124, 190502 (2020).
  18. H.-C. Weng and C.-S. Chuu, Implementation of Shor's algorithm with a single photon in 32 dimensions, Phys. Rev. Appl. 22, 034003 (2024).
  19. Z. Zhang, J. Mower, D. Englund, F. N. Wong, and J. H. Shapiro, Unconditional security of time-energy entanglement quantum key distribution using dual-basis interferometry, Phys. Rev. Lett. 112, 120506 (2014).
  20. J. Nunn, L. J. Wright, C. Söller, L. Zhang, I. A. Walmsley, and B. J. Smith, Large-alphabet time-frequency entangled quantum key distribution by means of time-to-frequency conversion, Opt. Express 21, 15959 (2013).
  21. T. Zhong, H. Zhou, R. D. Horansky, C. Lee, V. B. Verma, A. E. Lita, A. Restelli, J. C. Bienfang, R. P. Mirin, T. Gerrits et al., Photon-efficient quantum key distribution using time–energy entanglement with high-dimensional encoding, New J. Phys. 17, 022002 (2015).
  22. H. Yu, S. Sciara, M. Chemnitz, N. Montaut, B. Crockett, B. Fischer, R. Helsten, B. Wetzel, T. A. Goebel, R. G. Krämer et al., Quantum key distribution implemented with d-level time-bin entangled photons, Nat. Commun. 16, 171 (2025).
  23. K.-C. Chang, M. C. Sarihan, X. Cheng, Z. Zhang, and C. W. Wong, Large-alphabet time-bin quantum key distribution and Einstein–Podolsky–Rosen steering via dispersive optics, Quantum Sci. Technol. 9, 015018 (2024).
  24. F. Graffitti, P. Barrow, A. Pickston, A. M. Brańczyk, and A. Fedrizzi, Direct generation of tailored pulse-mode entanglement, Phys. Rev. Lett. 124, 053603 (2020).
  25. K. Niizeki, D. Yoshida, K. Ito, I. Nakamura, N. Takei, K. Okamura, M.-Y. Zheng, X.-P. Xie, and T. Horikiri, Two-photon comb with wavelength conversion and 20-km distribution for quantum communication, Commun. Phys. 3, 138 (2020).
  26. A. Khodadad Kashi and M. Kues, Frequency-bin-encoded entanglement-based quantum key distribution in a reconfigurable frequency-multiplexed network, Light Sci. Appl. 14, 49 (2025).
  27. É. Descamps, A. Keller, and P. Milman, Gottesman-Kitaev-Preskill encoding in continuous modal variables of single photons, Phys. Rev. Lett. 132, 170601 (2024).
  28. S. Merkouche, V. Thiel, A. O. Davis, and B. J. Smith, Heralding multiple photonic pulsed Bell pairs via frequency-resolved entanglement swapping, Phys. Rev. Lett. 128, 063602 (2022).
  29. S. Merkouche, V. Thiel, and B. J. Smith, Spectrally resolved four-photon interference of time-frequency-entangled photons, Phys. Rev. A 105, 023708 (2022).
  30. D. L. Vitullo, M. G. Raymer, B. J. Smith, M. Karpiński, L. Mejling, and K. Rottwitt, Entanglement swapping for generation of heralded time-frequency-entangled photon pairs, Phys. Rev. A 98, 023836 (2018).
  31. K.-C. Chang, X. Cheng, M. C. Sarihan, and C. W. Wong, Towards optimum Franson interference recurrence in mode-locked singly-filtered biphoton frequency combs, Photonics Res. 11, 1175 (2023).
  32. K.-C. Chang, X. Cheng, M. C. Sarihan, A. K. Vinod, Y. S. Lee, T. Zhong, Y.-X. Gong, Z. Xie, J. H. Shapiro, F. N. Wong et al., 648 Hilbert-space dimensionality in a biphoton frequency comb: Entanglement of formation and Schmidt mode decomposition, npj Quantum Inf. 7, 48 (2021).
  33. K.-C. Chang, X. Cheng, M. C. Sarihan, and C. W. Wong, Time-reversible and fully time-resolved ultra-narrowband biphoton frequency combs, APL Quantum 1, 016106 (2024).
  34. X. Cheng, K.-C. Chang, M. C. Sarihan, A. Mueller, M. Spiropulu, M. D. Shaw, B. Korzh, A. Faraon, F. N. Wong, J. H. Shapiro et al., High-dimensional time-frequency entanglement in a singly-filtered biphoton frequency comb, Commun. Phys. 6, 278 (2023).
  35. R. Ikuta, R. Tani, M. Ishizaki, S. Miki, M. Yabuno, H. Terai, N. Imoto, and T. Yamamoto, Frequency-multiplexed photon pairs over 1000 modes from a quadratic nonlinear optical waveguide resonator with a singly resonant configuration, Phys. Rev. Lett. 123, 193603 (2019).
  36. N. Fabre, G. Maltese, F. Appas, S. Felicetti, A. Ketterer, A. Keller, T. Coudreau, F. Baboux, M. I. Amanti, S. Ducci et al., Generation of a time-frequency grid state with integrated biphoton frequency combs, Phys. Rev. A 102, 012607 (2020).
  37. T. Yamazaki, R. Ikuta, T. Kobayashi, S. Miki, F. China, H. Terai, N. Imoto, and T. Yamamoto, Massive-mode polarization entangled biphoton frequency comb, Sci. Rep. 12, 8964 (2022).
  38. Z. Xie, T. Zhong, S. Shrestha, X. Xu, J. Liang, Y.-X. Gong, J. C. Bienfang, A. Restelli, J. H. Shapiro, F. N. Wong et al., Harnessing high-dimensional hyperentanglement through a biphoton frequency comb, Nat. Photon. 9, 536 (2015).
  39. H.-H. Lu, K. V. Myilswamy, R. S. Bennink, S. Seshadri, M. S. Alshaykh, J. Liu, T. J. Kippenberg, D. E. Leaird, A. M. Weiner, and J. M. Lukens, Bayesian tomography of high-dimensional on-chip biphoton frequency combs with randomized measurements, Nat. Commun. 13, 4338 (2022).
  40. E. Ortiz-Ricardo, C. Bertoni-Ocampo, M. Maldonado-Terrón, A. G. Zurita, R. Ramirez-Alarcon, H. C. Ramirez, R. Castro-Beltrán, and A. B. U’Ren, Submegahertz spectral width photon pair source based on fused silica microspheres, Photonics Res. 9, 2237 (2021).
  41. K. V. Myilswamy, S. Seshadri, H.-H. Lu, M. S. Alshaykh, J. Liu, T. J. Kippenberg, A. M. Weiner, and J. M. Lukens, Time-resolved Hanbury Brown–Twiss interferometry of on-chip biphoton frequency combs using Vernier phase modulation, Phys. Rev. Appl. 19, 034019 (2023).
  42. Z.-X. Yang, Z.-Q. Zeng, Y. Tian, S. Wang, R. Shimizu, H.-Y. Wu, S. Liu, and R.-B. Jin, Spatial–spectral mapping to prepare frequency entangled qudits, Opt. Lett. 48, 2361 (2023).
  43. A. Shukhin, I. Hurvitz, S. Trajtenberg-Mills, A. Arie, and H. Eisenberg, Two-dimensional control of a biphoton joint spectrum, Opt. Express 32, 10158 (2024).
  44. C. L. Morrison, F. Graffitti, P. Barrow, A. Pickston, J. Ho, and A. Fedrizzi, Frequency-bin entanglement from domain-engineered down-conversion, APL Photonics 7, 066102 (2022).
  45. R.-B. Jin, Z.-Q. Zeng, C. You, and C. Yuan, Quantum interferometers: Principles and applications, Prog. Quantum Electron. 96, 100519 (2024).
  46. N. B. Lingaraju, H.-H. Lu, S. Seshadri, P. Imany, D. E. Leaird, J. M. Lukens, and A. M. Weiner, Quantum frequency combs and Hong–Ou–Mandel interferometry: The role of spectral phase coherence, Opt. Express 27, 38683 (2019).
  47. R.-B. Jin, R. Shimizu, M. Fujiwara, M. Takeoka, R. Wakabayashi, T. Yamashita, S. Miki, H. Terai, T. Gerrits, and M. Sasaki, Simple method of generating and distributing frequency-entangled qudits, Quantum Sci. Technol. 1, 015004 (2016).
  48. Y. Chen, S. Ecker, L. Chen, F. Steinlechner, M. Huber, and R. Ursin, Temporal distinguishability in Hong-Ou-Mandel interference for harnessing high-dimensional frequency entanglement, npj Quantum Inf. 7, 167 (2021).
  49. B. Li, B. Yuan, C. Chen, X. Xiang, R. Quan, R. Dong, S. Zhang, and R.-B. Jin, Spectrally resolved two-photon interference in a modified Hong–Ou–Mandel interferometer, Opt. Laser Technol. 159, 109039 (2023).
  50. R.-B. Jin, Z.-Q. Zeng, D. Xu, C.-Z. Yuan, B.-H. Li, Y. Wang, R. Shimizu, M. Takeoka, M. Fujiwara, M. Sasaki et al., Spectrally resolved Franson interference, Sci. China Phys. Mech. Astron. 67, 250312 (2024).
  51. R.-B. Jin, R. Shimizu, T. Ono, M. Fujiwara, G.-W. Deng, Q. Zhou, M. Sasaki, and M. Takeoka, Spectrally resolved NOON state interference, arXiv:2104.01062.
  52. L. Hong, Y. Zhang, Y. Chen, and L. Chen, Fast quantifier of high-dimensional frequency entanglement through Hong–Ou–Mandel interference, Adv. Quantum Technol. 6, 2300012 (2023).
  53. N. Fabre, A. Keller, and P. Milman, Time and frequency as quantum continuous variables, Phys. Rev. A 105, 052429 (2022).
  54. J. Park, T. Jeong, H. Kim, and H. S. Moon, Time-energy entangled photon pairs from Doppler-broadened atomic ensemble via collective two-photon coherence, Phys. Rev. Lett. 121, 263601 (2018).
  55. A. O. Davis, P. M. Saulnier, M. Karpiński, and B. J. Smith, Pulsed single-photon spectrometer by frequency-to-time mapping using chirped fiber Bragg gratings, Opt. Express 25, 12804 (2017).
  56. C. Chen, C. Bo, M. Y. Niu, F. Xu, Z. Zhang, J. H. Shapiro, and F. N. Wong, Efficient generation and characterization of spectrally factorable biphotons, Opt. Express 25, 7300 (2017).
  57. R.-B. Jin, R. Shimizu, K. Wakui, H. Benichi, and M. Sasaki, Widely tunable single photon source with high purity at telecom wavelength, Opt. Express 21, 10659 (2013).
  58. J. D. Franson, Bell inequality for position and time, Phys. Rev. Lett. 62, 2205 (1989).
  59. H. Kim, S. M. Lee, O. Kwon, and H. S. Moon, Two-photon interference of polarization-entangled photons in a Franson interferometer, Sci. Rep. 7, 5772 (2017).
  60. B. Li, Y.-H. Li, Y. Cao, J. Yin, and C.-Z. Peng, Pure-state photon-pair source with a long coherence time for large-scale quantum information processing, Phys. Rev. Appl. 19, 064083 (2023).
  61. W. Tittel, J. Brendel, N. Gisin, and H. Zbinden, Long-distance Bell-type tests using energy-time entangled photons, Phys. Rev. A 59, 4150 (1999).
  62. O. Kovalenko, Y.-S. Ra, Y. Cai, V. C. Usenko, C. Fabre, N. Treps, and R. Filip, Frequency-multiplexed entanglement for continuous-variable quantum key distribution, Photonics Res. 9, 2351 (2021).
  63. F. R. Cardoso, D. Z. Rossatto, G. P. Fernandes, G. Higgins, and C. J. Villas-Boas, Superposition of two-mode squeezed states for quantum information processing and quantum sensing, Phys. Rev. A 103, 062405 (2021).
  64. I. Hurvitz, A. Shukhin, L. Vidro, H. Eisenberg, and A. Arie, Phase analysis of biphoton joint spectra by interference between different SPDC sources, Optica Quantum 2, 358 (2024).
  65. A. L. Grimsmo, J. Combes, and B. Q. Baragiola, Quantum computing with rotation-symmetric bosonic codes, Phys. Rev. X 10, 011058 (2020).
  66. I. Ali Khan and J. C. Howell, Experimental demonstration of high two-photon time-energy entanglement, Phys. Rev. A 73, 031801 (2006).

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Phys. Rev. Research 7, 043152– Published 12 November, 2025
  • Received 15 April 2025
  • Accepted 16 October 2025
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