- Physical Review Journals
All JournalsPhysics Magazine

Physical Review A

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Phase-space open-system dynamics of second-order nonlinear interactions with pulsed quantum light

Emanuel Hubenschmid1,* and Victor Rueskov Christiansen2,†

  • *Contact author: emanuel.hubenschmid@uni-konstanz.de
  • †Contact author: victorrc@phys.au.dk
PDF

Phys. Rev. A 113, 063734 – Published 25 June, 2026

DOI: https://doi.org/10.1103/hjxr-f4kz

Abstract

The theoretical description of broadband, multimode quantum pulses undergoing a second-order χ(2)-nonlinear interaction can be quite intricate due to the large dimensionality of the underlying phase space. However, in many cases only a few broadband (temporal) modes are relevant before and after the nonlinear interaction. Here we present an efficient framework to calculate the relation between the quantum states at the input and output of a nonlinear element in their respective relevant modes. Since the number of relevant input and output modes may differ, resulting in an open quantum system, we introduce the generalized Bloch-Messiah decomposition (GBMD), reducing the description to an equal number of input and output modes. The GBMD enables us to calculate the multimode Wigner function of the output state by convolving the rescaled Wigner function of the reduced input quantum pulse with a multivariate Gaussian phase-space function. We expand on this result by considering two examples: a parametric amplification of a broadband Fock state in the infrared frequencies and a THz-to-optical frequency up-conversion of a pulsed two-mode squeezed vacuum state. We investigate the effect, the convolution and thermalization due to entanglement breakage have on the output Wigner function by calculating the von Neumann entropy of the output Wigner function. The methods presented here can be used to optimize the amplification or frequency conversion of broadband quantum states, opening an avenue to the generation and characterization of optical quantum states on ultrafast timescales.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (59)

  1. H. Vahlbruch, A. Khalaidovski, N. Lastzka, C. Gräf, K. Danzmann, and R. Schnabel, The GEO 600 squeezed light source, Classical Quantum Gravity 27, 084027 (2010).
  2. J. Aasi, J. Abadie, B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Abernathy, C. Adams, T. Adams, P. Addesso, R. X. Adhikari, et al., Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light, Nat. Photonics 7, 613 (2013).
  3. K. E. Dorfman, F. Schlawin, and S. Mukamel, Nonlinear optical signals and spectroscopy with quantum light, Rev. Mod. Phys. 88, 045008 (2016).
  4. S. Mukamel, M. Freyberger, W. Schleich, M. Bellini, A. Zavatta, G. Leuchs, C. Silberhorn, R. W. Boyd, L. L. Sánchez-Soto, A. Stefanov, et al., Roadmap on quantum light spectroscopy, J. Phys. B 53, 072002 (2020).
  5. A. Datta, Sensing with quantum light: A perspective, Nanophotonics 14, 1993 (2025).
  6. A. Khan, F. Albarelli, and A. Datta, A tensor network approach to sensing quantum light-matter interactions, PRX Quantum 6, 040343 (2025).
  7. B. Korzh, C. C. W. Lim, R. Houlmann, N. Gisin, M. J. Li, D. Nolan, B. Sanguinetti, R. Thew, and H. Zbinden, Provably secure and practical quantum key distribution over 307 km of optical fibre, Nat. Photonics 9, 163 (2015).
  8. H. J. Kimble, The quantum internet, Nature (London) 453, 1023 (2008).
  9. M. Walschaers, C. Fabre, V. Parigi, and N. Treps, Statistical signatures of multimode single-photon-added and -subtracted states of light, Phys. Rev. A 96, 053835 (2017).
  10. C. Riek, P. Sulzer, M. Seeger, A. S. Moskalenko, G. Burkard, D. V. Seletskiy, and A. Leitenstorfer, Subcycle quantum electrodynamics, Nature (London) 541, 376 (2017).
  11. T. L. M. Guedes, M. Kizmann, D. V. Seletskiy, A. Leitenstorfer, G. Burkard, and A. S. Moskalenko, Spectra of ultrabroadband squeezed pulses and the finite-time Unruh-Davies effect, Phys. Rev. Lett. 122, 053604 (2019).
  12. M. Kizmann, T. L. M. Guedes, D. V. Seletskiy, A. S. Moskalenko, A. Leitenstorfer, and G. Burkard, Subcycle squeezing of light from a time flow perspective, Nat. Phys. 15, 960 (2019).
  13. A. Kawasaki, R. Ide, H. Brunel, T. Suzuki, R. Nehra, K. Nakashima, T. Kashiwazaki, A. Inoue, T. Umeki, F. China, and M. Yabuno, et al., Broadband generation and tomography of non-Gaussian states for ultra-fast optical quantum processors, Nat. Commun. 15, 9075 (2024).
  14. K. Y. Spasibko, D. A. Kopylov, V. L. Krutyanskiy, T. V. Murzina, G. Leuchs, and M. V. Chekhova, Multiphoton effects enhanced due to ultrafast photon-number fluctuations, Phys. Rev. Lett. 119, 223603 (2017).
  15. C. Riek, D. V. Seletskiy, A. S. Moskalenko, J. F. Schmidt, P. Krauspe, S. Eckart, S. Eggert, G. Burkard, and A. Leitenstorfer, Direct sampling of electric-field vacuum fluctuations, Science 350, 420 (2015).
  16. A. S. Moskalenko, C. Riek, D. V. Seletskiy, G. Burkard, and A. Leitenstorfer, Paraxial theory of direct electro-optic sampling of the quantum vacuum, Phys. Rev. Lett. 115, 263601 (2015).
  17. I.-C. Benea-Chelmus, F. F. Settembrini, G. Scalari, and J. Faist, Electric field correlation measurements on the electromagnetic vacuum state, Nature (London) 568, 202 (2019).
  18. F. F. Settembrini, F. Lindel, A. M. Herter, S. Y. Buhmann, and J. Faist, Detection of quantum-vacuum field correlations outside the light cone, Nat. Commun. 13, 3383 (2022).
  19. F. Lindel, A. Herter, V. Gebhart, J. Faist, and S. Y. Buhmann, Entanglement harvesting from electromagnetic quantum fields, Phys. Rev. A 110, 022414 (2024).
  20. G. Adesso, S. Ragy, and A. R. Lee, Continuous variable quantum information: Gaussian states and beyond, Open Syst. Inf. Dyn. 21, 1440001 (2014).
  21. 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).
  22. O. Tziperman, V. R. Christiansen, I. Kaminer, and K. Mølmer, Parametric amplification of a quantum pulse, Phys. Rev. A 110, 053712 (2024).
  23. W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, Pulsed squeezed light: Simultaneous squeezing of multiple modes, Phys. Rev. A 73, 063819 (2006).
  24. E. Hubenschmid, T. L. M. Guedes, and G. Burkard, Complete positive operator-valued measure description of multichannel quantum electro-optic sampling with monochromatic field modes, Phys. Rev. A 106, 043713 (2022).
  25. U. Leonhardt and H. Paul, Realistic optical homodyne measurements and quasiprobability distributions, Phys. Rev. A 48, 4598 (1993).
  26. R. Yanagimoto, R. Nehra, R. Hamerly, E. Ng, A. Marandi, and H. Mabuchi, Quantum nondemolition measurements with optical parametric amplifiers for ultrafast universal quantum information processing, PRX Quantum 4, 010333 (2023).
  27. E. Sendonaris, J. Williams, R. Nehra, R. Gray, R. Sekine, L. Ledezma, and A. Marandi, Ultrafast single-photon detection using nanophotonic parametric amplifiers, Phys. Rev. Appl. 25, 044078 (2026).
  28. G. Gallot and D. Grischkowsky, Electro-optic detection of terahertz radiation, J. Opt. Soc. Am. B 16, 1204 (1999).
  29. S. Virally, P. Cusson, and D. V. Seletskiy, Enhanced electro-optic sampling with quantum probes, Phys. Rev. Lett. 127, 270504 (2021).
  30. I.-C. Benea-Chelmus, J. Faist, A. Leitenstorfer, A. S. Moskalenko, I. Pupeza, D. V. Seletskiy, and K. L. Vodopyanov, Electro-optic sampling of classical and quantum light, Optica 12, 546 (2025).
  31. D. Adamou, L. Hirsch, T. Shields, S. Yoon, A. C. Dada, J. M. R. Weaver, D. Faccio, M. Peccianti, L. Caspani, and M. Clerici, Quantum-enhanced time-domain spectroscopy, Sci. Adv. 11, eadt2187 (2025).
  32. H. Xu, An SVD-like matrix decomposition and its applications, Linear Algebra Appl. 368, 1 (2003).
  33. H. Xu, A numerical method for computing an SVD-like decomposition, SIAM J. Matrix Anal. Appl. 26, 1058 (2005).
  34. M. Houde, W. McCutcheon, and N. Quesada, Matrix decompositions in quantum optics: Takagi/Autonne, Bloch–Messiah/Euler, Iwasawa, and Williamson, Can. J. Phys. 102, 497 (2024).
  35. 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).
  36. V. Ansari, J. M. Donohue, M. Allgaier, L. Sansoni, B. Brecht, J. Roslund, N. Treps, G. Harder, and C. Silberhorn, Tomography and purification of the temporal-mode structure of quantum light, Phys. Rev. Lett. 120, 213601 (2018).
  37. M. Freyberger, K. Vogel, and W. P. Schleich, From photon counts to quantum phase, Phys. Lett. A 176, 41 (1993).
  38. M. Kalash and M. V. Chekhova, Wigner function tomography via optical parametric amplification, Optica 10, 1142 (2023).
  39. S. Onoe, T. L. M. Guedes, A. S. Moskalenko, A. Leitenstorfer, G. Burkard, and T. C. Ralph, Realizing a rapidly switched Unruh-DeWitt detector through electro-optic sampling of the electromagnetic vacuum, Phys. Rev. D 105, 056023 (2022).
  40. E. Hubenschmid, T. L. Guedes, and G. Burkard, Optical time-domain quantum state tomography on a subcycle scale, Phys. Rev. X 14, 041032 (2024).
  41. B. Brecht, D. V. Reddy, C. Silberhorn, and M. Raymer, Photon temporal modes: A complete framework for quantum information science, Phys. Rev. X 5, 041017 (2015).
  42. M. G. Raymer and I. A. Walmsley, Temporal modes in quantum optics: Then and now, Phys. Scr. 95, 064002 (2020).
  43. C. Bloch and A. Messiah, The canonical form of an antisymmetric tensor and its application to the theory of superconductivity, Nucl. Phys. 39, 95 (1962).
  44. S. L. Braunstein, Squeezing as an irreducible resource, Phys. Rev. A 71, 055801 (2005).
  45. C. K. Law, I. A. Walmsley, and J. H. Eberly, Continuous frequency entanglement: Effective finite Hilbert space and entropy control, Phys. Rev. Lett. 84, 5304 (2000).
  46. E. Hubenschmid and G. Burkard, Time-domain field correlation measurements enable tomography of highly multimode quantum states of light, Commun. Phys. 9, 89 (2026).
  47. E. Hubenschmid, Subcycleq (Version 3.0.0) [Software], Zenodo, 2026, https://doi.org/10.5281/zenodo.18888929.
  48. S. Namba, Electro-optical effect of zincblende, J. Opt. Soc. Am. 51, 76 (1961).
  49. A. Leitenstorfer, S. Hunsche, J. Shah, M. C. Nuss, and W. H. Knox, Detectors and sources for ultrabroadband electro-optic sampling: Experiment and theory, Appl. Phys. Lett. 74, 1516 (1999).
  50. N. Quesada and J. E. Sipe, Effects of time ordering in quantum nonlinear optics, Phys. Rev. A 90, 063840 (2014).
  51. 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).
  52. D. T. F. Marple, Refractive index of ZnSe, ZnTe, and CdTe, J. Appl. Phys. 35, 539 (1964).
  53. R. W. Boyd, Nonlinear Optics, 4th ed. (Academic Press, London, 2020).
  54. J. Tucker and D. F. Walls, Quantum theory of parametric frequency conversion, Ann. Phys. 52, 1 (1969).
  55. N. Quesada and J. E. Sipe, High efficiency in mode-selective frequency conversion, Opt. Lett. 41, 364 (2016).
  56. P. Sulzer, K. Oguchi, J. Huster, M. Kizmann, T. L. M. Guedes, A. Liehl, C. Beckh, A. S. Moskalenko, G. Burkard, D. V. Seletskiy, and A. Leitenstorfer, Determination of the electric field and its Hilbert transform in femtosecond electro-optic sampling, Phys. Rev. A 101, 033821 (2020).
  57. A. S. Holevo, M. Sohma, and O. Hirota, Capacity of quantum Gaussian channels, Phys. Rev. A 59, 1820 (1999).
  58. W. Magnus, On the exponential solution of differential equations for a linear operator, Commun. Pure Appl. Math. 7, 649 (1954).
  59. W. Vogel and D.-G. Welsch, Quantum Optics (Wiley-VCH Verlag, Weinheim, 2006).

Outline

Information

Phys. Rev. A 113, 063734– Published 25 June, 2026
  • Received 3 December 2025
  • Revised 6 March 2026
  • Accepted 29 May 2026
Crossmark - Check for updates button
Creative Commons Logo

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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation