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Experimental validation of a recursive density matrix expansion for four-wave mixing: Propagation and suppression effects in dense rubidium vapor

Wagner Albuquerque1,2,*, Larissa F. Gonçalves1, Ammis S. Alvarez1, Natalia R. de Melo1, and José Ferraz1,2,†

  • *Contact author: wagner.albuquerque@ufrpe.br
  • †Contact author: jose.ferraz@ufrpe.br
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Phys. Rev. A 114, 013522 – Published 29 July, 2026

DOI: https://doi.org/10.1103/yy3x-mbkq

Abstract

We present a combined experimental and theoretical study of degenerate four-wave mixing (FWM) in a thermal rubidium vapor on the D2 line. The experiment employs a forward phase-matched geometry with a single laser, where the atomic density—and thus the optical depth—is controlled via the cell temperature. We measure the generated FWM signal across a wide range of densities, observing a characteristic suppression and saturation at high optical depths. To explain these phenomena, we develop a theoretical model based on a semiclassical treatment of the atomic response that includes nonlinear propagation effects. The model's core is a recursive Liouville equation derived from a systematic expansion of the system's density matrix, which provides an analytical solution for the optical coherence and the generated field in a closed four-level excitation loop. This theoretical framework shows excellent agreement with the experimental data across all densities. By confronting theory and experiment, we demonstrate that our model simultaneously captures the roles of absorption, propagation, and interference in shaping the nonlinear optical response, showing that the observed suppression of the FWM signal at high atomic density arises from the combined action of these competing mechanisms.

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References (72)

  1. L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, New York, 1987).
  2. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, Electromagnetically induced transparency: Optics in coherent media, Rev. Mod. Phys. 77, 633 (2005).
  3. Q. Glorieux, T. Aladjidi, P. D. Lett, and R. Kaiser, Hot atomic vapors for nonlinear and quantum optics, New J. Phys. 25, 051201 (2023).
  4. H. Schmidt and A. Imamoglu, Giant Kerr nonlinearities obtained by electromagnetically induced transparency, Opt. Lett. 21, 1936 (1996).
  5. M. D. Lukin and A. Imamoglu, Nonlinear optics and quantum entanglement of ultraslow single photons, Phys. Rev. Lett. 84, 1419 (2000).
  6. S. L. McCall and E. L. Hahn, Self-induced transparency by pulsed coherent light, Phys. Rev. Lett. 18, 908 (1967).
  7. M. D. Crisp, Propagation of small-area pulses of coherent light through a resonant medium, Phys. Rev. A 1, 1604 (1970).
  8. J. E. Rothenberg, D. Grischkowsky, and A. C. Balant, Observation of the formation of the 0π pulse, Phys. Rev. Lett. 53, 552 (1984).
  9. Y. Silberberg and I. Bar-Joseph, Transient effects in degenerate four-wave mixing in saturable absorbers, IEEE J. Quantum Electron. 17, 1967 (1981).
  10. J. D. Briscoe, D. Pizzey, R. M. Potvliege, S. A. Wrathmall, and I. G. Hughes, Light propagation through an atomic vapor in the presence of a magnetic field of arbitrary direction: Taking the non-orthogonality of the electric field modes into account, Opt. Express 33, 32099 (2025).
  11. A. A. C. de Almeida, A. S. Alvarez, N. R. de Melo, M. R. L. da Motta, M. P. M. de Souza, and S. S. Vianna, Intensity correlations in transmission and four-wave-mixing channels in an atomic vapor, Phys. Rev. A 111, 043518 (2025).
  12. A. C. Tam and W. Happer, Long-range interactions between cw self-focused laser beams in an atomic vapor, Phys. Rev. Lett. 38, 278 (1977).
  13. A. W. McCord and R. J. Ballagh, Mutual beam reshaping by two interacting radiation modes, J. Opt. Soc. Am. B 7, 73 (1990).
  14. L. S. Costanzo, A. S. Coelho, D. Pellegrino, M. S. Mendes, L. Acioli, K. N. Cassemiro, D. Felinto, A. Zavatta, and M. Bellini, Zero-area single-photon pulses, Phys. Rev. Lett. 116, 023602 (2016).
  15. A. J. A. Carvalho, R. S. N. Moreira, J. Ferraz, S. S. Vianna, L. H. Acioli, and D. Felinto, Enhanced absorption of weak ultrashort light pulses by a narrowband atomic medium, Phys. Rev. A 101, 053426 (2020).
  16. Q. Fontaine, T. Bienaimé, S. Pigeon, E. Giacobino, A. Bramati, and Q. Glorieux, Observation of the Bogoliubov dispersion in a fluid of light, Phys. Rev. Lett. 121, 183604 (2018).
  17. M. E. Crenshaw, Quasiadiabatic approximation for a dense collection of two-level atoms, Phys. Rev. A 54, 3559 (1996).
  18. M. E. Crenshaw and C. M. Bowden, Effects of local fields on spontaneous emission in dielectric media, Phys. Rev. Lett. 85, 1851 (2000).
  19. P. Weisman, A. D. Wilson-Gordon, and H. Friedmann, Effect of propagation on pulsed four-wave mixing, Phys. Rev. A 61, 053816 (2000).
  20. V. S. Egorov, V. N. Lebedev, I. B. Mekhov, P. V. Moroshkin, I. A. Chekhonin, and S. N. Bagayev, Coherent interaction of laser pulses in a resonant optically dense extended medium under the regime of strong field-matter coupling, Phys. Rev. A 69, 033804 (2004).
  21. L. Zhang and J. Evers, Diffractionless image propagation and frequency conversion via four-wave mixing exploiting the thermal motion of atoms, Phys. Rev. A 89, 013817 (2014).
  22. J. Lang, D. Chang, and F. Piazza, Interaction-induced transparency for strong-coupling polaritons, Phys. Rev. Lett. 125, 133604 (2020).
  23. Z. Bai, C. S. Adams, G. Huang, and W. Li, Self-induced transparency in warm and strongly interacting Rydberg gases, Phys. Rev. Lett. 125, 263605 (2020).
  24. N. R. de Melo and S. S. Vianna, Two-photon resonant forward four-wave mixing in rubidium vapor involving Rydberg states, J. Opt. Soc. Am. B 31, 1735 (2014).
  25. V. V. Konotop, E. Lakshtanov, and B. Vainberg, Designing lasing and perfectly absorbing potentials, Phys. Rev. A 99, 043838 (2019).
  26. A. Vernier, S. Franke-Arnold, E. Riis, and A. S. Arnold, Enhanced frequency conversion in a warm atomic vapor using four-wave mixing, Opt. Express 18, 17020 (2010).
  27. A. Montaña Guerrero, R. L. Rincón Celis, P. Nussenzveig, M. Martinelli, A. M. Marino, and H. M. Florez, Continuous variable entanglement in an optical parametric oscillator based on a nondegenerate four wave mixing process in hot alkali atoms, Phys. Rev. Lett. 129, 163601 (2022).
  28. A. Gahl, J. Seipenbusch, A. Aumann, M. Möller, and W. Lange, Self-induced planar and cylindrical splitting of a laser beam in sodium vapor, Phys. Rev. A 50, R917 (1994).
  29. T. Ackemann, A. Heuer, Y. A. Logvin, and W. Lange, Light-shift-induced level crossing and resonatorless optical bistability in sodium vapor, Phys. Rev. A 56, 2321 (1997).
  30. C. F. McCormick, A. M. Marino, V. Boyer, and P. D. Lett, Strong relative intensity squeezing by four-wave mixing in rubidium vapor, Phys. Rev. A 78, 043816 (2008).
  31. Q. Glorieux, R. Dubessy, S. Guibal, L. Guidoni, J.-P. Likforman, T. Coudreau, and E. Arimondo, Double-Λ microscopic model for entangled light generation by four-wave mixing, Phys. Rev. A 82, 033819 (2010).
  32. W. C. Magno, R. B. Prandini, P. Nussenzveig, and S. S. Vianna, Four-wave mixing with Rydberg levels in rubidium vapor: Observation of interference fringes, Phys. Rev. A 63, 063406 (2001).
  33. J. Ferraz, D. Felinto, L. H. Acioli, and S. S. Vianna, Quantum interference in atomic vapor observed by four-wave mixing with incoherent light, Opt. Lett. 30, 1876 (2005).
  34. J. Ferraz, L. H. Acioli, and S. S. Vianna, One- and two-photon quantum interferences in parametric four-wave mixing, Opt. Lett. 35, 375 (2010).
  35. R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Observation of squeezed states generated by four-wave mixing in an optical cavity, Phys. Rev. Lett. 55, 2409 (1985).
  36. C. F. McCormick, V. Boyer, E. Arimondo, and P. D. Lett, Strong relative intensity squeezing by four-wave mixing in rubidium vapor, Opt. Lett. 32, 178 (2007).
  37. C. Liu, J. Jing, Z. Zhou, R. C. Pooser, F. Hudelist, L. Zhou, and W. Zhang, Realization of low frequency and controllable bandwidth squeezing based on a four-wave-mixing amplifier in rubidium vapor, Opt. Lett. 36, 2979 (2011).
  38. R. C. Pooser, A. M. Marino, and P. D. Lett, Quantum correlated light beams from non-degenerate four-wave mixing in an atomic vapor: The D1 and D2 lines of Rb85 and Rb87, Opt. Express 17, 16722 (2009).
  39. V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, Entangled images from four-wave mixing, Science 321, 544 (2008).
  40. M. Jasperse, L. D. Turner, and R. E. Scholten, Relative intensity squeezing by four-wave mixing with loss, Opt. Express 19, 3765 (2011).
  41. M. T. Turnbull, P. G. Petrov, C. S. Embrey, A. M. Marino, and V. Boyer, Role of the phase-matching condition in nondegenerate four-wave mixing in hot vapors for the generation of squeezed states of light, Phys. Rev. A 88, 033845 (2013).
  42. Q. Glorieux, L. Guidoni, S. Guibal, J. P. Likforman, and T. Coudreau, Quantum correlations by four-wave mixing in an atomic vapor in a nonamplifying regime: Quantum beam splitter for photons, Phys. Rev. A 84, 053826 (2011).
  43. N. V. Corzo, Q. Glorieux, A. M. Marino, J. B. Clark, R. T. Glasser, and P. D. Lett, Rotation of the noise ellipse for squeezed vacuum light generated via four-wave mixing, Phys. Rev. A 88, 043836 (2013).
  44. R. T. Willis, F. E. Becerra, L. A. Orozco, and S. L. Rolston, Four-wave mixing in the diamond configuration in an atomic vapor, Phys. Rev. A 79, 033814 (2009).
  45. H. G. Barros, J. Ferraz, W. Lozano B., L. H. Acioli, and S. S. Vianna, Coherent control of quantum interference in two-photon absorption, Phys. Rev. A 74, 055402 (2006).
  46. H. Friedmann and A. D. Wilson-Gordon, Asymmetry in pulsed four-wave mixing, Phys. Rev. A 57, 4854 (1998).
  47. A. S. Alvarez, A. A. C. de Almeida, and S. S. Vianna, Two symmetric four-wave mixing signals generated in a medium with anomalous refractive index, J. Phys. B: At. Mol. Opt. Phys. 54, 045403 (2021).
  48. U. Hinze, B. N. Chichkov, E. Tiemann, and B. Wellegehausen, Resonant continuous four-wave mixing and parametric amplification, J. Opt. Soc. Am. B 17, 2001 (2000).
  49. R. L. Abrams and R. C. Lind, Degenerate four-wave mixing in absorbing media, Opt. Lett. 2, 94 (1978).
  50. H. Kang, G. Hernandez, and Y. Zhu, Resonant four-wave mixing with slow light, Phys. Rev. A 70, 061804(R) (2004).
  51. F. E. Becerra, R. T. Willis, S. L. Rolston, and L. A. Orozco, Nondegenerate four-wave mixing in rubidium vapor, Phys. Rev. A 78, 013834 (2008).
  52. T. P. de Silans, C. S. L. Goncalves, D. Felinto, and J. W. R. Tabosa, Enhanced four-wave mixing via crossover resonance in cesium vapor, J. Opt. Soc. Am. B 28, 2220 (2011).
  53. B. Kleinmann, P. Grambow, D. Kolbe, and H. J. Eichler, Degenerate four-wave mixing in sodium vapor in the Rabi regime, J. Opt. Soc. Am. B 2, 704 (1985).
  54. M. O. Araújo, H. L. D. S. Cavalcante, M. Oriá, M. Chevrollier, T. Passerat de Silans, R. Castro, and D. Moretti, Measurement of the Kerr nonlinear refractive index of Cs vapor, Phys. Rev. A 88, 063818 (2013).
  55. D. A. Steck, Rubidium 85 D line data, 2019, http://steck.us/alkalidata.
  56. D. A. Steck, Rubidium 87 D line data, 2021, http://steck.us/alkalidata.
  57. A. Gallagher and E. L. Lewis, Determination of the vapor pressure of rubidium by optical absorption, J. Opt. Soc. Am. 63, 864 (1973).
  58. J. Baek, S. Park, M.-H. Lee, H.-R. Noh, and G. Moon, Polarization-selective four-wave mixing in a degenerate multi-level system, Sci. Rep. 14, 5705 (2024).
  59. M. Pinard, R. W. Boyd, and G. Grynberg, Third-order nonlinear optical response resulting from optical pumping: Effects of atomic motion, Phys. Rev. A 49, 1326 (1994).
  60. C. J. Foot, Atomic Physics, 1st ed. (Oxford University Press, Oxford, 2005).
  61. W. Happer, Optical pumping, Rev. Mod. Phys. 44, 169 (1972).
  62. D. Budker, D. F. J. Kimball, and D. DeMille, Nonlinear magneto-optics and reduced group velocity of light in atomic vapor with slow ground state relaxation, Rev. Mod. Phys. 74, 1153 (2002).
  63. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, San Diego, 2008).
  64. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, 1997).
  65. C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions: Basic Processes and Applications (Wiley, New York, 1998).
  66. B. W. Shore, The Theory of Coherent Atomic Excitation (Wiley, New York, 1990).
  67. D. P. Craig and T. Thirunamachandran, Molecular Quantum Electrodynamics (Dover, Mineola, New York, 1998).
  68. G. S. Agarwal, Quantum Optics (Cambridge University Press, Cambridge, 2012).
  69. R. Loudon, The Quantum Theory of Light, 3rd ed. (Oxford University Press, New York, 2000).
  70. P. W. Milonni and J. H. Eberly, Laser Physics (Wiley, New York, 2010).
  71. H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations (Springer, New York, 1999).
  72. K. Blum, Density Matrix Theory and Applications, 3rd ed. (Springer, New York, 2012).

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Phys. Rev. A 114, 013522– Published 29 July, 2026
  • Received 13 March 2026
  • Accepted 7 July 2026
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