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  • Open Access

Steady-state dynamics of quantum frequency combs in microring resonators

Patrick Tritschler*

Torsten Ohms

André Zimmermann

Peter Degenfeld-Schonburg

  • *Contact author: patrick.tritschler@de.bosch.com
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Phys. Rev. A 113, 053706 – Published 5 May, 2026

DOI: https://doi.org/10.1103/c8rj-c51q

Abstract

Optical frequency combs are utilized in a wide range of optical applications, including atomic clocks, interferometers, and various sensing technologies. They are often generated via four-wave mixing in chip-integrated microring resonators, a method that requires low optical input power due to the high-quality factor of the resonator, making it highly efficient. While the classical properties of optical frequency combs are well established, this work investigates the quantum-mechanical characteristics of the individual comb modes. We derive closed-form analytical expressions describing the squeezing, second-order correlation, and joint spectral intensity between the generated signal and idler modes. This comprehensive theoretical framework enables an intuitive understanding and optimization of the quantum features across the comb, revealing conditions for substantial squeezing and entanglement relevant for quantum information processing. Our findings highlight the profound impact of design and dispersion on these quantum properties and offer a foundational tool for chip-integrated quantum applications, including quantum sensing, computing, and communication.

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References (53)

  1. S. B. Papp, K. Beha, P. Del'Haye, F. Quinlan, H. Lee, K. J. Vahala, and S. A. Diddams, Microresonator frequency comb optical clock, Optica 1, 10 (2014).
  2. S. Y. Zhang, J. T. Wu, Y. L. Zhang, J. X. Leng, W. P. Yang, Z. G. Zhang, and J. Y. Zhao, Direct frequency comb optical frequency standard based on two-photon transitions of thermal atoms, Sci. Rep. 5, 15114 (2015).
  3. T. Fortier as a representative of the BACON collaboration, Frequency combs for precision synthesis and characterization of optical atomic standards, J. Phys.: Conf. Ser. 2889, 012021 (2024).
  4. A. Foltynowicz, T. Ban, P. Masłowski, F. Adler, and J. Ye, Quantum-noise-limited optical frequency comb spectroscopy, Phys. Rev. Lett. 107, 233002 (2011).
  5. N. Picqué and T. W. Hänsch, Frequency comb spectroscopy, Nat. Photon. 13, 146 (2019).
  6. N. Kuse and M. E. Fermann, Frequency-modulated comb LIDAR, APL Photon. 4, 106105 (2019).
  7. N. Horiuchi, Optical frequency comb for multi-sensors, Nat. Photon. 18, 648 (2024).
  8. P. Del'Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Optical frequency comb generation from a monolithic microresonator, Nature (London) 450, 1214 (2007).
  9. T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, Microresonator-based optical frequency combs, Science 332, 555 (2011).
  10. T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, Universal formation dynamics and noise of Kerr-frequency combs in microresonators, Nat. Photon. 6, 480 (2012).
  11. W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, Silicon microring resonators, Laser Photon. Rev. 6, 47 (2012).
  12. L. Chang, W. Xie, H. Shu, Q.-F. Yang, B. Shen, A. Boes, J. D. Peters, W. Jin, C. Xiang, S. Liu, G. Moille, S.-P. Yu, X. Wang, K. Srinivasan, S. B. Papp, K. Vahala, and J. E. Bowers, Ultra-efficient frequency comb generation in AlGaAs-on-insulator microresonators, Nat. Commun. 11, 1331 (2020).
  13. S. Fujii and T. Tanabe, Dispersion engineering and measurement of whispering gallery mode microresonator for Kerr frequency comb generation, Nanophotonics 9, 1087 (2020).
  14. T. Fortier and E. Baumann, 20 years of developments in optical frequency comb technology and applications, Commun. Phys. 2, 153 (2019).
  15. N. Liu, Y. Liu, J. Li, L. Yang, and X. Li, Generation of multi-mode squeezed vacuum using pulse pumped fiber optical parametric amplifiers, Opt. Express 24, 2125 (2016).
  16. Z. Yang, M. Jahanbozorgi, D. Jeong, S. Sun, O. Pfister, H. Lee, and X. Yi, A squeezed quantum microcomb on a chip, Nat. Commun. 12, 4781 (2021).
  17. M. A. Guidry, D. M. Lukin, K. Y. Yang, R. Trivedi, and J. Vučković, Quantum optics of soliton microcombs, Nat. Photon. 16, 52 (2022).
  18. D. V. Strekalov, C. Marquardt, A. B. Matsko, H. G. L. Schwefel, and G. Leuchs, Nonlinear and quantum optics with whispering gallery resonators, J. Opt. 18, 123002 (2016).
  19. S. Gerke, J. Sperling, W. Vogel, Y. Cai, J. Roslund, N. Treps, and C. Fabre, Full multipartite entanglement of frequency-comb Gaussian states, Phys. Rev. Lett. 114, 050501 (2015).
  20. O. Pinel, P. Jian, R. M. de Araújo, J. Feng, B. Chalopin, C. Fabre, and N. Treps, Generation and characterization of multimode quantum frequency combs, Phys. Rev. Lett. 108, 083601 (2012).
  21. N. C. Menicucci, S. T. Flammia, and O. Pfister, One-way quantum computing in the optical frequency comb, Phys. Rev. Lett. 101, 130501 (2008).
  22. L. Caspani, C. Xiong, B. J. Eggleton, D. Bajoni, M. Liscidini, M. Galli, R. Morandotti, and D. J. Moss, Integrated sources of photon quantum states based on nonlinear optics, Light: Sci. Appl. 6, e17100 (2017).
  23. X. Lu, Q. Li, D. A. Westly, G. Moille, A. Singh, V. Anant, and K. Srinivasan, Chip-integrated visible–telecom entangled photon pair source for quantum communication, Nat. Phys. 15, 373 (2019).
  24. X. Cheng, K.-C. Chang, M. C. Sarihan, A. Mueller, M. Spiropulu, M. D. Shaw, B. Korzh, A. Faraon, F. N. C. Wong, J. H. Shapiro, and C. W. Wong, High-dimensional time-frequency entanglement in a singly-filtered biphoton frequency comb, Commun. Phys. 6, 278 (2023).
  25. P. Tritschler, T. Ohms, C. Schweikert, O. Sözen, R. H. Klenk, S. Abdani, W. Vogel, G. Rademacher, A. Zimmermann, and P. Degenfeld-Schonburg, Chip-integrated single-mode coherent-squeezed light source using four-wave mixing in microresonators, arXiv:2502.16278.
  26. V. G. Lucivero, R. Jiménez-Martínez, J. Kong, and M. W. Mitchell, Squeezed-light spin noise spectroscopy, Phys. Rev. A 93, 053802 (2016).
  27. Y. Michael, L. Bello, M. Rosenbluh, and A. Pe'er, Squeezing-enhanced Raman spectroscopy, npj Quantum Inf. 5, 81 (2019).
  28. P. Tritschler, T. Ohms, A. Zimmermann, F. Zschocke, T. Strohm, and P. Degenfeld-Schonburg, Optical interferometer using two-mode squeezed light for enhanced chip-integrated quantum metrology, Phys. Rev. A 110, 012621 (2024).
  29. D. I. Herman, M. Walsh, M. K. Kreider, N. Lordi, E. J. Tsao, A. J. Lind, M. Heyrich, J. Combes, J. Genest, and S. A. Diddams, Squeezed dual-comb spectroscopy, Science 387, 653 (2025).
  30. K. Goda, O. Miyakawa, E. E. Mikhailov, S. Saraf, R. Adhikari, K. McKenzie, R. Ward, S. Vass, A. J. Weinstein, and N. Mavalvala, A quantum-enhanced prototype gravitational-wave detector, Nat. Phys. 4, 472 (2008).
  31. H. Vahlbruch, A. Khalaidovski, N. Lastzka, C. Gräf, K. Danzmann, and R. Schnabel, The GEO 600 squeezed light source, Class. Quantum Grav. 27, 084027 (2010).
  32. W. Jia, Squeezing the quantum noise of a gravitational-wave detector below the standard quantum limit, Science 385, 1318 (2024).
  33. J. E. Bourassa, R. N. Alexander, M. Vasmer, A. Patil, I. Tzitrin, T. Matsuura, D. Su, B. Q. Baragiola, S. Guha, G. Dauphinais, K. K. Sabapathy, N. C. Menicucci, and I. Dhand, Blueprint for a scalable photonic fault-tolerant quantum computer, Quantum 5, 392 (2021).
  34. K. Fukui, High-threshold fault-tolerant quantum computation with the Gottesman-Kitaev-Preskill qubit under noise in an optical setup, Phys. Rev. A 107, 052414 (2023).
  35. J. M. Arrazola, V. Bergholm, K. Brádler, T. R. Bromley, M. J. Collins, I. Dhand, A. Fumagalli, T. Gerrits, A. Goussev, L. G. Helt, J. Hundal, T. Isacsson, R. B. Israel, J. Izaac, S. Jahangiri, R. Janik, N. Killoran, S. P. Kumar, J. Lavoie, A. E. Lita, et al., Quantum circuits with many photons on a programmable nanophotonic chip, Nature (London) 591, 54 (2021).
  36. C. Reimer, M. Kues, P. Roztocki, B. Wetzel, F. Grazioso, B. E. Little, S. T. Chu, T. Johnston, Y. Bromberg, L. Caspani, D. J. Moss, and R. Morandotti, Generation of multiphoton entangled quantum states by means of integrated frequency combs, Science 351, 1176 (2016).
  37. Z. Vernon and J. E. Sipe, Strongly driven nonlinear quantum optics in microring resonators, Phys. Rev. A 92, 033840 (2015).
  38. S. Coen, H. G. Randle, T. Sylvestre, and M. Erkintalo, Modeling of octave-spanning Kerr frequency combs using a generalized mean-field Lugiato-Lefever model, Opt. Lett. 38, 37 (2013).
  39. L. A. Lugiato, F. Prati, M. L. Gorodetsky, and T. J. Kippenberg, From the Lugiato–Lefever equation to microresonator-based soliton Kerr frequency combs, Philos. Trans. R. Soc. A 376, 20180113 (2018).
  40. C. W. Gardiner and M. J. Collett, Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation, Phys. Rev. A 31, 3761 (1985).
  41. M. J. Collett and C. W. Gardiner, Squeezing of intracavity and traveling-wave light fields produced in parametric amplification, Phys. Rev. A 30, 1386 (1984).
  42. P. Tritschler, C. Schweikert, R. H. Klenk, S. Abdani, O. Sözen, W. Vogel, G. Rademacher, T. Ohms, A. Zimmermann, and P. Degenfeld-Schonburg, Nonlinear optical bistability in microring resonators for enhanced phase sensing, Phys. Rev. Lett. 134, 123802 (2025).
  43. P. Zoller and C. W. Gardiner, Quantum noise in quantum optics: The stochastic Schrödinger equation, arXiv:quant-ph/9702030.
  44. H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control (Cambridge University Press, Cambridge, 2009).
  45. D. F. Walls and G. J. Milburn, in Quantum Optics, edited by D. F. Walls and G. J. Milburn (Springer, Berlin, 2008), pp. 7–27.
  46. R. Kubo, Generalized cumulant expansion method, J. Phys. Soc. Jpn. 17, 1100 (1962).
  47. K. Zielnicki, K. Garay-Palmett, D. Cruz-Delgado, H. Cruz-Ramirez, M. F. O'Boyle, B. Fang, V. O. Lorenz, A. B. U'Ren, and P. G. Kwiat, Joint spectral characterization of photon-pair sources, J. Mod. Opt. 65, 1141 (2018).
  48. R. Simon, Peres-Horodecki separability criterion for continuous variable systems, Phys. Rev. Lett. 84, 2726 (2000).
  49. A. Peres, Separability criterion for density matrices, Phys. Rev. Lett. 77, 1413 (1996).
  50. P. Horodecki, Separability criterion and inseparable mixed states with positive partial transposition, Phys. Lett. A 232, 333 (1997).
  51. A. Serafini, F. Illuminati, and S. D. Siena, Symplectic invariants, entropic measures and correlations of Gaussian states, J. Phys. B 37, L21 (2004).
  52. C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Gaussian quantum information, Rev. Mod. Phys. 84, 621 (2012).
  53. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Quantum cryptography, Rev. Mod. Phys. 74, 145 (2002).

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Phys. Rev. A 113, 053706– Published 5 May, 2026
  • Received 13 October 2025
  • Accepted 20 April 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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