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Time-resolved second-order autocorrelation function of parametric down-conversion

Dmitri B. Horoshko1,2,*, Shivang Srivastava1,3, Filip Sośnicki4,5, Michał Mikołajczyk4, Michał Karpiński4, Benjamin Brecht5, and Mikhail I. Kolobov1

  • 1Univ. Lille, CNRS, UMR 8523 - PhLAM - Physique des Lasers Atomes et Molécules, F-59000 Lille, France
  • 2Institut für Quantenoptik, Universität Ulm, Ulm D-89081, Germany
  • 3Telecom Paris, Institut Polytechnique de Paris, 19 Place Marguerite Perey, 91120 Palaiseau, France
  • 4Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warszawa, Poland
  • 5Paderborn University, Department of Physics, Integrated Quantum Optics, Institute for Photonic Quantum Systems (PhoQS), Warburger Straße 100, 33098 Paderborn, Germany

  • *Contact author: dmitri.horoshko@uni-ulm.de
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Phys. Rev. A 112, 023703 – Published 7 August, 2025

DOI: https://doi.org/10.1103/7ckm-tm3r

Abstract

We study a possibility of measuring the time-resolved second-order autocorrelation function of one of two beams generated in type-II parametric down-conversion by means of temporal magnification of this beam, bringing its correlation time from the picosecond to the nanosecond scale, which can be resolved by modern photodetectors. We show that such a measurement enables one to infer directly the degree of global coherence of that beam, which is linked by a simple relation to the number of modes characterizing the entanglement between the two generated beams. We illustrate the proposed method by an example of photon pairs generated in a periodically poled potassium titanyl phosphate (KTP) crystal with a symmetric group velocity matching for various durations of the pump pulse, resulting in different numbers of modes. Our theoretical model also shows that the magnified double-heralded autocorrelation function of one beam exhibits a local maximum around zero delay time, corresponding to photon bunching at a short time scale.

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References (66)

  1. R. J. Glauber, The quantum theory of optical coherence, Phys. Rev. 130, 2529 (1963).
  2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, Cambridge, 1995).
  3. Y. Arakawa and M. J. Holmes, Progress in quantum-dot single photon sources for quantum information technologies: A broad spectrum overview, Appl. Phys. Rev. 7, 021309 (2020).
  4. C. Toninelli, I. Gerhardt, A. Clark, A. Reserbat-Plantey, S. Götzinger, Z. Ristanović, M. Colautti, P. Lombardi, K. Major, I. Deperasińska, W. H. Pernice, F. H. L. Koppens, B. Kozankiewicz, A. Gourdon, V. Sandoghdar, and M. Orrit, Single organic molecules for photonic quantum technologies, Nat. Mater. 20, 1615 (2021).
  5. M. W. Doherty, N. B. Manson, P. Delaney, F. Jelezko, J. Wrachtrup, and L. C. Hollenberg, The nitrogen-vacancy colour centre in diamond, Phys. Rep. 528, 1 (2013).
  6. A. Christ, K. Laiho, A. Eckstein, K. N. Cassemiro, and C. Silberhorn, Probing multimode squeezing with correlation functions, New J. Phys. 13, 033027 (2011).
  7. B. H. Kolner, Space-time duality and the theory of temporal imaging, IEEE J. Quantum Elect. 30, 1951 (1994).
  8. R. Salem, M. A. Foster, and A. L. Gaeta, Application of space–time duality to ultrahigh-speed optical signal processing, Adv. Opt. Photon. 5, 274 (2013).
  9. J. Lavoie, J. M. Donohue, L. G. Wright, A. Fedrizzi, and K. J. Resch, Spectral compression of single photons, Nat. Photon. 7, 363 (2013).
  10. M. Karpiński, M. Jachura, L. J. Wright, and B. J. Smith, Bandwidth manipulation of quantum light by an electro-optic time lens, Nat. Photon. 11, 53 (2017).
  11. S. Mittal, V. V. Orre, A. Restelli, R. Salem, E. A. Goldschmidt, and M. Hafezi, Temporal and spectral manipulations of correlated photons using a time lens, Phys. Rev. A 96, 043807 (2017).
  12. G. Patera, J. Shi, D. B. Horoshko, and M. I. Kolobov, Quantum temporal imaging: application of a time lens to quantum optics, J. Opt. 19, 054001 (2017).
  13. G. Patera, D. B. Horoshko, and M. I. Kolobov, Space-time duality and quantum temporal imaging, Phys. Rev. A 98, 053815 (2018).
  14. J. Shi, G. Patera, D. B. Horoshko, and M. I. Kolobov, Quantum temporal imaging of antibunching, J. Opt. Soc. Am. B 37, 3741 (2020).
  15. F. Sośnicki, M. Mikołajczyk, A. Golestani, and M. Karpiński, Aperiodic electro-optic time lens for spectral manipulation of single-photon pulses, Appl. Phys. Lett. 116, 234003 (2020).
  16. C. Joshi, B. M. Sparkes, A. Farsi, T. Gerrits, V. Verma, S. Ramelow, S. W. Nam, and A. L. Gaeta, Picosecond-resolution single-photon time lens for temporal mode quantum processing, Optica 9, 364 (2022).
  17. F. Sośnicki, M. Mikołajczyk, A. Golestani, and M. Karpiński, Interface between picosecond and nanosecond quantum light pulses, Nat. Photon. 17, 761 (2023).
  18. D. N. Klyshko, Photons and Nonlinear Optics (Gordon and Breach, New York, 1988).
  19. M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, Theory of two-photon entanglement in type-II optical parametric down-conversion, Phys. Rev. A 50, 5122 (1994).
  20. W. P. Grice and I. A. Walmsley, Spectral information and distinguishability in type-II down-conversion with a broadband pump, Phys. Rev. A 56, 1627 (1997).
  21. T. E. Keller and M. H. Rubin, Theory of two-photon entanglement for spontaneous parametric down-conversion driven by a narrow pump pulse, Phys. Rev. A 56, 1534 (1997).
  22. M. I. Kolobov, The spatial behavior of nonclassical light, Rev. Mod. Phys. 71, 1539 (1999).
  23. B. Huttner, S. Serulnik, and Y. Ben-Aryeh, Quantum analysis of light propagation in a parametric amplifier, Phys. Rev. A 42, 5594 (1990).
  24. D. B. Horoshko, Generator of spatial evolution of the electromagnetic field, Phys. Rev. A 105, 013708 (2022).
  25. Y. R. Shen, Quantum statistics of nonlinear optics, Phys. Rev. 155, 921 (1967).
  26. D. B. Horoshko, M. I. Kolobov, V. Parigi, and N. Treps, Few-mode squeezing in type-I parametric downconversion by complete group velocity matching, Opt. Lett. 49, 4078 (2024).
  27. R. W. Boyd, Nonlinear Optics (Academic, New York, 2020).
  28. D. B. Horoshko and M. I. Kolobov, Generation of monocycle squeezed light in chirped quasi-phase-matched nonlinear crystals, Phys. Rev. A 95, 033837 (2017).
  29. C. M. Caves and D. D. Crouch, Quantum wideband traveling-wave analysis of a degenerate parametric amplifier, J. Opt. Soc. Am. B 4, 1535 (1987).
  30. L. S. Telegin and A. S. Chirkin, Reversal and reconstruction of the profile of ultrashort light pulses, Sov. J. Quantum Electr. 15, 101 (1985).
  31. B. H. Kolner and M. Nazarathy, Temporal imaging with a time lens, Opt. Lett. 14, 630 (1989).
  32. L. Mouradian, F. Louradour, V. Messager, A. Barthelemy, and C. Froehly, Spectro-temporal imaging of femtosecond events, IEEE J. Quantum Electron. 36, 795 (2000).
  33. C. V. Bennett and B. H. Kolner, Principles of parametric temporal imaging. I. System configurations, IEEE J. Quantum Elect. 36, 430 (2000).
  34. C. V. Bennett and B. H. Kolner, Principles of parametric temporal imaging. II. System performance, IEEE J. Quantum Elect. 36, 649 (2000).
  35. M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, Silicon-chip-based ultrafast optical oscilloscope, Nature (London) 456, 81 (2008).
  36. S. Meir, Y. Tamir, H. Duadi, E. Cohen, and M. Fridman, Ultrafast temporal SU(1,1) interferometer, Phys. Rev. Lett. 130, 253601 (2023).
  37. M. Mazelanik, A. Leszczyński, M. Lipka, M. Parniak, and W. Wasilewski, Temporal imaging for ultra-narrowband few-photon states of light, Optica 7, 203 (2020).
  38. M. Mazelanik, A. Leszczyński, and M. Parniak, Optical-domain spectral super-resolution via a quantum-memory-based time-frequency processor, Nat. Commun. 13, 691 (2022).
  39. B. Niewelt, M. Jastrzebski, S. Kurzyna, J. Nowosielski, W. Wasilewski, M. Mazelanik, and M. Parniak, Experimental implementation of the optical fractional fourier transform in the time-frequency domain, Phys. Rev. Lett. 130, 240801 (2023).
  40. S. Srivastava, D. B. Horoshko, and M. I. Kolobov, Erecting time telescope for photonic quantum networks, Opt. Express 31, 38560 (2023).
  41. G. Patera, D. B. Horoshko, M. Allgaier, M. I. Kolobov, and C. Silberhorn, Modal approach to quantum temporal imaging, Phys. Rev. A 108, 043716 (2023).
  42. S. Srivastava, D. B. Horoshko, and M. I. Kolobov, Making entangled photons indistinguishable by a time lens, Phys. Rev. A 107, 033705 (2023).
  43. N. Quesada and J. E. Sipe, Time-ordering effects in the generation of entangled photons using nonlinear optical processes, Phys. Rev. Lett. 114, 093903 (2015).
  44. A. Christ, B. Brecht, W. Mauerer, and C. Silberhorn, Theory of quantum frequency conversion and type-II parametric down-conversion in the high-gain regime, New J. Phys. 15, 053038 (2013).
  45. T. Lipfert, D. B. Horoshko, G. Patera, and M. I. Kolobov, Bloch-Messiah decomposition and Magnus expansion for parametric down-conversion with monochromatic pump, Phys. Rev. A 98, 013815 (2018).
  46. W. P. Grice, A. B. U'Ren, and I. A. Walmsley, Eliminating frequency and space-time correlations in multiphoton states, Phys. Rev. A 64, 063815 (2001).
  47. D. B. Horoshko, L. La Volpe, F. Arzani, N. Treps, C. Fabre, and M. I. Kolobov, Bloch-Messiah reduction for twin beams of light, Phys. Rev. A 100, 013837 (2019).
  48. D. B. Horoshko and M. I. Kolobov, Interferometric sorting of temporal Hermite-Gauss modes via temporal Gouy phase, Phys. Rev. A 110, 033721 (2024).
  49. C. K. Law and J. H. Eberly, Analysis and interpretation of high transverse entanglement in optical parametric down conversion, Phys. Rev. Lett. 92, 127903 (2004).
  50. D. B. Horoshko, G. Patera, A. Gatti, and M. I. Kolobov, X-entangled biphotons: Schmidt number for 2D model, Eur. Phys. J. D 66, 239 (2012).
  51. A. Gatti, T. Corti, E. Brambilla, and D. B. Horoshko, Dimensionality of the spatiotemporal entanglement of parametric down-conversion photon pairs, Phys. Rev. A 86, 053803 (2012).
  52. P. R. Sharapova, G. Frascella, M. Riabinin, A. M. Pérez, O. V. Tikhonova, S. Lemieux, R. W. Boyd, G. Leuchs, and M. V. Chekhova, Properties of bright squeezed vacuum at increasing brightness, Phys. Rev. Res. 2, 013371 (2020).
  53. V. A. Averchenko, G. Frascella, M. Kalash, A. Cavanna, and M. V. Chekhova, Reconstructing two-dimensional spatial modes for classical and quantum light, Phys. Rev. A 102, 053725 (2020).
  54. D. A. Kopylov, T. Meier, and P. R. Sharapova, Theory of multimode squeezed light generation in lossy media, Quantum 9, 1621 (2025).
  55. I. B. Bobrov, S. S. Straupe, E. V. Kovlakov, and S. P. Kulik, Schmidt-like coherent mode decomposition and spatial intensity correlations of thermal light, New J. Phys. 15, 073016 (2013).
  56. V. Ansari, J. M. Donohue, B. Brecht, and C. Silberhorn, Tailoring nonlinear processes for quantum optics with pulsed temporal-mode encodings, Optica 5, 534 (2018).
  57. V. Giovannetti, L. Maccone, J. H. Shapiro, and F. N. C. Wong, Generating entangled two-photon states with coincident frequencies, Phys. Rev. Lett. 88, 183602 (2002).
  58. 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).
  59. F. König and F. N. C. Wong, Extended phase matching of second-harmonic generation in periodically poled KTiOPO4 with zero group-velocity mismatch, Appl. Phys. Lett. 84, 1644 (2004).
  60. P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U'Ren, C. Silberhorn, and I. A. Walmsley, Heralded generation of ultrafast single photons in pure quantum states, Phys. Rev. Lett. 100, 133601 (2008).
  61. P. Grangier, G. Roger, and A. Aspect, Experimental evidence for a photon anticorrelation effect on a beam splitter: A new light on single-photon interferences, Europhys. Lett. 1, 173 (1986).
  62. A. B. U'Ren, C. Silberhorn, J. L. Ball, K. Banaszek, and I. A. Walmsley, Characterization of the nonclassical nature of conditionally prepared single photons, Phys. Rev. A 72, 021802(R) (2005).
  63. E. Bocquillon, C. Couteau, M. Razavi, R. Laflamme, and G. Weihs, Coherence measures for heralded single-photon sources, Phys. Rev. A 79, 035801 (2009).
  64. S. Bettelli, Comment on “Coherence measures for heralded single-photon sources,” Phys. Rev. A 81, 037801 (2010).
  65. C. T. Lee, Theorem on nonclassical states, Phys. Rev. A 52, 3374 (1995).
  66. D. B. Horoshko, S. De Bièvre, G. Patera, and M. I. Kolobov, Thermal-difference states of light: Quantum states of heralded photons, Phys. Rev. A 100, 053831 (2019).

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Phys. Rev. A 112, 023703– Published 7 August, 2025
  • Received 27 February 2025
  • Accepted 10 July 2025
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