Abstract

We report on observations of four-wave mixing in rubidium vapor via two-photon resonant excitation with nd and (n+2)s states, for n=1420. The intensity of the generated beam is studied in a collinear configuration, and the results are consistent with the predictions of a theoretical model based on the semiclassical Maxwell–Bloch equations, considering interference effects associated with multiple excitation pathways and with internally generated fields. We have also observed spectral broadening of the two-photon resonant lines, which is well described by a linear dependence on atomic density, in accordance with collision-induced broadening.

© 2014 Optical Society of America

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  1. R. W. Boyd, M. S. Malcuit, D. J. Gauthier, and K. Rzażewski, “Competition between amplified spontaneous emission and the four-wave-mixing process,” Phys. Rev. A 35, 1648–1658 (1987).
    [CrossRef]
  2. R. K. Wunderlich, D. J. Gauthier, R. C. Hart, M. A. Moore, and M. G. Payne, “Nonlinear optical processes near the sodium 4D two-photon resonance,” Phys. Rev. A 41, 6345–6360 (1990).
    [CrossRef]
  3. H. Nagai and T. Nakanaga, “Competition between two-photon-resonant three-photon ionization and four-wave mixing in Xe,” Phys. Rev. A 84, 063408 (2011).
    [CrossRef]
  4. D. J. Jackson and J. J. Wynne, “Interference effects between different optical harmonics,” Phys. Rev. Lett. 49, 543–546 (1982).
    [CrossRef]
  5. L. Deng, M. G. Payne, and W. R. Garrett, “Effects of multi-photon interferences from internally generated fields in strongly resonant systems,” Phys. Rep. 429, 123–241 (2006), and references therein.
    [CrossRef]
  6. A. Armyras, D. Pentaris, T. Efthimiopoulos, N. Merlemis, and A. Lyras, “Saturation and population transfer of a two-photon excited four-level potassium atom,” J. Phys. B 44, 165401 (2011).
    [CrossRef]
  7. T. Chaneliere, D. N. Matsukevich, S. D. Jenkins, T. A. B. Kennedy, M. S. Chapman, and A. Kuzmich, “Quantum telecommunication based on atomic cascade transitions,” Phys. Rev. Lett. 96, 093604 (2006).
    [CrossRef]
  8. A. S. Zibrov, M. D. Lukin, L. Hollberg, and M. O. Scully, “Efficient frequency up-conversion in resonant coherent media,” Phys. Rev. A 65, 051801 (2002).
    [CrossRef]
  9. T. Meijer, J. D. White, B. Smeets, M. Jeppesen, and R. E. Scholten, “Blue five-level frequency-upconversion system in rubidium,” Opt. Lett. 31, 1002–1004 (2006).
    [CrossRef]
  10. 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).
    [CrossRef]
  11. A. K. Mohapatra, T. R. Jackson, and C. S. Adams, “Coherent optical detection of highly excited Rydberg states using electromagnetically induced transparency,” Phys. Rev. Lett 98, 113003 (2007).
    [CrossRef]
  12. E. Brekke, J. O. Day, and T. G. Walker, “Four-wave mixing in ultracold atoms using intermediate Rydberg states,” Phys. Rev. A 78, 063830 (2008).
    [CrossRef]
  13. A. Kölle, G. Epple, H. Kübler, W. R. Lö, and T. Pfau, “Four-wave mixing involving Rydberg states in thermal vapor,” Phys. Rev. A 85, 063821 (2012).
    [CrossRef]
  14. Y. O. Dudin and A. Kuzmich, “Strongly interacting Rydberg excitations of a cold atomic gas,” Science 336, 887–889 (2012).
    [CrossRef]
  15. H. B. Zheng, X. Yao, Z. Y. Zhang, J. L. Che, Y. Q. Zhang, Y. P. Zhang, and M. Xiao, “Blockaded six- and eight-wave mixing processes tailored by electromagnetically induced transparency scissors,” Laser Phys. 24, 045404 (2014).
    [CrossRef]
  16. H. C. Tran, P. C. John, J. Gao, and J. G. Eden, “Interaction of atomic wave packets with four-wave mixing: detection of rubidium and potassium wave packets by coherent ultraviolet emission,” Opt. Lett. 23, 70–72 (1998).
    [CrossRef]
  17. R. R. T. Luna, G. H. Cavalcanti, L. H. Coutinho, and A. G. Trigueiros, “A compilation of wavelengths and energy levels for the spectrum of neutral rubidium (Rb I),” J. Quant. Spectrosc. Radiat. Transfer 75, 559–587 (2002).
    [CrossRef]
  18. S. S. Vianna, P. Nussenzveig, W. C. Magno, and J. W. Tabosa, “Polarization dependence and interference in four-wave mixing with Rydberg levels in rubidium vapor,” Phys. Rev. A 58, 3000–3003 (1998).
    [CrossRef]
  19. 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).
    [CrossRef]
  20. M. G. Payne, L. Deng, and W. R. Garret, “Theory of the effect of odd-photon destructive interference on optical shifts in resonantly enhanced multiphoton excitation and ionization,” Phys. Rev. A 58, 1361–1371 (1998).
    [CrossRef]
  21. E. L. Lewis, “Collisional relaxation of atomic excited states, line broadening and interatomic interactions,” Phys. Rep. 58, 1–71 (1980).
    [CrossRef]
  22. L. Weller, R. J. Bettles, P. Siddons, C. S. Adams, and I. G. Hughes, “Absolute absorption on rubidium D1 line: including resonant dipole-dipole interactions,” J. Phys. B 44, 195006 (2011).
    [CrossRef]
  23. V. A. Sautenkov, “Line shapes of atomic transitions in excited dense gas,” Laser Phys. Lett. 8, 771 (2011).
    [CrossRef]
  24. B. P. Stoicheff and E. Weinberger, “Frequency shifts, line broadenings, and phase-interference effects in Rb+Rb collisions, measured by Doppler-free two-photon spectroscopy,” Phys. Rev. Lett. 44, 733–736 (1980).
    [CrossRef]
  25. D. C. Thompson, E. Weinberger, G. X. Xu, and B. P. Stoicheff, “Frequency shifts and line broadenings in collisions between Rydberg and ground-state alkali-metal atoms,” Phys. Rev. A 35, 690–700 (1987).
    [CrossRef]
  26. F. Shen, J. Gao, A. A. Senin, C. J. Zhu, J. R. Allen, Z. H. Lu, Y. Xiao, and J. G. Eden, “Many-body dipole-dipole interactions between excited Rb atoms probed by wave packets and parametric four-wave mixing,” Phys. Rev. Lett. 99, 143201 (2007).
    [CrossRef]
  27. N. R. de Melo and S. S. Vianna, “Rydberg-Rydberg interaction in the second-harmonic generation from rubidium atoms,” Phys. Rev. A 77, 023418 (2008).
    [CrossRef]
  28. T. F. Gallagher, Rydberg Atoms (Cambridge University, 1994).
  29. M. Chret, L. Barbier, W. Lindinger, and R. Deloche, “Penning and associative ionisation of highly excited rubidium atoms,” J. Phys. B 15, 3463–3477 (1982).
    [CrossRef]

2014

H. B. Zheng, X. Yao, Z. Y. Zhang, J. L. Che, Y. Q. Zhang, Y. P. Zhang, and M. Xiao, “Blockaded six- and eight-wave mixing processes tailored by electromagnetically induced transparency scissors,” Laser Phys. 24, 045404 (2014).
[CrossRef]

2012

A. Kölle, G. Epple, H. Kübler, W. R. Lö, and T. Pfau, “Four-wave mixing involving Rydberg states in thermal vapor,” Phys. Rev. A 85, 063821 (2012).
[CrossRef]

Y. O. Dudin and A. Kuzmich, “Strongly interacting Rydberg excitations of a cold atomic gas,” Science 336, 887–889 (2012).
[CrossRef]

2011

H. Nagai and T. Nakanaga, “Competition between two-photon-resonant three-photon ionization and four-wave mixing in Xe,” Phys. Rev. A 84, 063408 (2011).
[CrossRef]

A. Armyras, D. Pentaris, T. Efthimiopoulos, N. Merlemis, and A. Lyras, “Saturation and population transfer of a two-photon excited four-level potassium atom,” J. Phys. B 44, 165401 (2011).
[CrossRef]

L. Weller, R. J. Bettles, P. Siddons, C. S. Adams, and I. G. Hughes, “Absolute absorption on rubidium D1 line: including resonant dipole-dipole interactions,” J. Phys. B 44, 195006 (2011).
[CrossRef]

V. A. Sautenkov, “Line shapes of atomic transitions in excited dense gas,” Laser Phys. Lett. 8, 771 (2011).
[CrossRef]

2009

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).
[CrossRef]

2008

E. Brekke, J. O. Day, and T. G. Walker, “Four-wave mixing in ultracold atoms using intermediate Rydberg states,” Phys. Rev. A 78, 063830 (2008).
[CrossRef]

N. R. de Melo and S. S. Vianna, “Rydberg-Rydberg interaction in the second-harmonic generation from rubidium atoms,” Phys. Rev. A 77, 023418 (2008).
[CrossRef]

2007

F. Shen, J. Gao, A. A. Senin, C. J. Zhu, J. R. Allen, Z. H. Lu, Y. Xiao, and J. G. Eden, “Many-body dipole-dipole interactions between excited Rb atoms probed by wave packets and parametric four-wave mixing,” Phys. Rev. Lett. 99, 143201 (2007).
[CrossRef]

A. K. Mohapatra, T. R. Jackson, and C. S. Adams, “Coherent optical detection of highly excited Rydberg states using electromagnetically induced transparency,” Phys. Rev. Lett 98, 113003 (2007).
[CrossRef]

2006

T. Meijer, J. D. White, B. Smeets, M. Jeppesen, and R. E. Scholten, “Blue five-level frequency-upconversion system in rubidium,” Opt. Lett. 31, 1002–1004 (2006).
[CrossRef]

L. Deng, M. G. Payne, and W. R. Garrett, “Effects of multi-photon interferences from internally generated fields in strongly resonant systems,” Phys. Rep. 429, 123–241 (2006), and references therein.
[CrossRef]

T. Chaneliere, D. N. Matsukevich, S. D. Jenkins, T. A. B. Kennedy, M. S. Chapman, and A. Kuzmich, “Quantum telecommunication based on atomic cascade transitions,” Phys. Rev. Lett. 96, 093604 (2006).
[CrossRef]

2002

A. S. Zibrov, M. D. Lukin, L. Hollberg, and M. O. Scully, “Efficient frequency up-conversion in resonant coherent media,” Phys. Rev. A 65, 051801 (2002).
[CrossRef]

R. R. T. Luna, G. H. Cavalcanti, L. H. Coutinho, and A. G. Trigueiros, “A compilation of wavelengths and energy levels for the spectrum of neutral rubidium (Rb I),” J. Quant. Spectrosc. Radiat. Transfer 75, 559–587 (2002).
[CrossRef]

2001

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).
[CrossRef]

1998

M. G. Payne, L. Deng, and W. R. Garret, “Theory of the effect of odd-photon destructive interference on optical shifts in resonantly enhanced multiphoton excitation and ionization,” Phys. Rev. A 58, 1361–1371 (1998).
[CrossRef]

S. S. Vianna, P. Nussenzveig, W. C. Magno, and J. W. Tabosa, “Polarization dependence and interference in four-wave mixing with Rydberg levels in rubidium vapor,” Phys. Rev. A 58, 3000–3003 (1998).
[CrossRef]

H. C. Tran, P. C. John, J. Gao, and J. G. Eden, “Interaction of atomic wave packets with four-wave mixing: detection of rubidium and potassium wave packets by coherent ultraviolet emission,” Opt. Lett. 23, 70–72 (1998).
[CrossRef]

1990

R. K. Wunderlich, D. J. Gauthier, R. C. Hart, M. A. Moore, and M. G. Payne, “Nonlinear optical processes near the sodium 4D two-photon resonance,” Phys. Rev. A 41, 6345–6360 (1990).
[CrossRef]

1987

R. W. Boyd, M. S. Malcuit, D. J. Gauthier, and K. Rzażewski, “Competition between amplified spontaneous emission and the four-wave-mixing process,” Phys. Rev. A 35, 1648–1658 (1987).
[CrossRef]

D. C. Thompson, E. Weinberger, G. X. Xu, and B. P. Stoicheff, “Frequency shifts and line broadenings in collisions between Rydberg and ground-state alkali-metal atoms,” Phys. Rev. A 35, 690–700 (1987).
[CrossRef]

1982

M. Chret, L. Barbier, W. Lindinger, and R. Deloche, “Penning and associative ionisation of highly excited rubidium atoms,” J. Phys. B 15, 3463–3477 (1982).
[CrossRef]

D. J. Jackson and J. J. Wynne, “Interference effects between different optical harmonics,” Phys. Rev. Lett. 49, 543–546 (1982).
[CrossRef]

1980

E. L. Lewis, “Collisional relaxation of atomic excited states, line broadening and interatomic interactions,” Phys. Rep. 58, 1–71 (1980).
[CrossRef]

B. P. Stoicheff and E. Weinberger, “Frequency shifts, line broadenings, and phase-interference effects in Rb+Rb collisions, measured by Doppler-free two-photon spectroscopy,” Phys. Rev. Lett. 44, 733–736 (1980).
[CrossRef]

Adams, C. S.

L. Weller, R. J. Bettles, P. Siddons, C. S. Adams, and I. G. Hughes, “Absolute absorption on rubidium D1 line: including resonant dipole-dipole interactions,” J. Phys. B 44, 195006 (2011).
[CrossRef]

A. K. Mohapatra, T. R. Jackson, and C. S. Adams, “Coherent optical detection of highly excited Rydberg states using electromagnetically induced transparency,” Phys. Rev. Lett 98, 113003 (2007).
[CrossRef]

Allen, J. R.

F. Shen, J. Gao, A. A. Senin, C. J. Zhu, J. R. Allen, Z. H. Lu, Y. Xiao, and J. G. Eden, “Many-body dipole-dipole interactions between excited Rb atoms probed by wave packets and parametric four-wave mixing,” Phys. Rev. Lett. 99, 143201 (2007).
[CrossRef]

Armyras, A.

A. Armyras, D. Pentaris, T. Efthimiopoulos, N. Merlemis, and A. Lyras, “Saturation and population transfer of a two-photon excited four-level potassium atom,” J. Phys. B 44, 165401 (2011).
[CrossRef]

Barbier, L.

M. Chret, L. Barbier, W. Lindinger, and R. Deloche, “Penning and associative ionisation of highly excited rubidium atoms,” J. Phys. B 15, 3463–3477 (1982).
[CrossRef]

Becerra, F. E.

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).
[CrossRef]

Bettles, R. J.

L. Weller, R. J. Bettles, P. Siddons, C. S. Adams, and I. G. Hughes, “Absolute absorption on rubidium D1 line: including resonant dipole-dipole interactions,” J. Phys. B 44, 195006 (2011).
[CrossRef]

Boyd, R. W.

R. W. Boyd, M. S. Malcuit, D. J. Gauthier, and K. Rzażewski, “Competition between amplified spontaneous emission and the four-wave-mixing process,” Phys. Rev. A 35, 1648–1658 (1987).
[CrossRef]

Brekke, E.

E. Brekke, J. O. Day, and T. G. Walker, “Four-wave mixing in ultracold atoms using intermediate Rydberg states,” Phys. Rev. A 78, 063830 (2008).
[CrossRef]

Cavalcanti, G. H.

R. R. T. Luna, G. H. Cavalcanti, L. H. Coutinho, and A. G. Trigueiros, “A compilation of wavelengths and energy levels for the spectrum of neutral rubidium (Rb I),” J. Quant. Spectrosc. Radiat. Transfer 75, 559–587 (2002).
[CrossRef]

Chaneliere, T.

T. Chaneliere, D. N. Matsukevich, S. D. Jenkins, T. A. B. Kennedy, M. S. Chapman, and A. Kuzmich, “Quantum telecommunication based on atomic cascade transitions,” Phys. Rev. Lett. 96, 093604 (2006).
[CrossRef]

Chapman, M. S.

T. Chaneliere, D. N. Matsukevich, S. D. Jenkins, T. A. B. Kennedy, M. S. Chapman, and A. Kuzmich, “Quantum telecommunication based on atomic cascade transitions,” Phys. Rev. Lett. 96, 093604 (2006).
[CrossRef]

Che, J. L.

H. B. Zheng, X. Yao, Z. Y. Zhang, J. L. Che, Y. Q. Zhang, Y. P. Zhang, and M. Xiao, “Blockaded six- and eight-wave mixing processes tailored by electromagnetically induced transparency scissors,” Laser Phys. 24, 045404 (2014).
[CrossRef]

Chret, M.

M. Chret, L. Barbier, W. Lindinger, and R. Deloche, “Penning and associative ionisation of highly excited rubidium atoms,” J. Phys. B 15, 3463–3477 (1982).
[CrossRef]

Coutinho, L. H.

R. R. T. Luna, G. H. Cavalcanti, L. H. Coutinho, and A. G. Trigueiros, “A compilation of wavelengths and energy levels for the spectrum of neutral rubidium (Rb I),” J. Quant. Spectrosc. Radiat. Transfer 75, 559–587 (2002).
[CrossRef]

Day, J. O.

E. Brekke, J. O. Day, and T. G. Walker, “Four-wave mixing in ultracold atoms using intermediate Rydberg states,” Phys. Rev. A 78, 063830 (2008).
[CrossRef]

de Melo, N. R.

N. R. de Melo and S. S. Vianna, “Rydberg-Rydberg interaction in the second-harmonic generation from rubidium atoms,” Phys. Rev. A 77, 023418 (2008).
[CrossRef]

Deloche, R.

M. Chret, L. Barbier, W. Lindinger, and R. Deloche, “Penning and associative ionisation of highly excited rubidium atoms,” J. Phys. B 15, 3463–3477 (1982).
[CrossRef]

Deng, L.

L. Deng, M. G. Payne, and W. R. Garrett, “Effects of multi-photon interferences from internally generated fields in strongly resonant systems,” Phys. Rep. 429, 123–241 (2006), and references therein.
[CrossRef]

M. G. Payne, L. Deng, and W. R. Garret, “Theory of the effect of odd-photon destructive interference on optical shifts in resonantly enhanced multiphoton excitation and ionization,” Phys. Rev. A 58, 1361–1371 (1998).
[CrossRef]

Dudin, Y. O.

Y. O. Dudin and A. Kuzmich, “Strongly interacting Rydberg excitations of a cold atomic gas,” Science 336, 887–889 (2012).
[CrossRef]

Eden, J. G.

F. Shen, J. Gao, A. A. Senin, C. J. Zhu, J. R. Allen, Z. H. Lu, Y. Xiao, and J. G. Eden, “Many-body dipole-dipole interactions between excited Rb atoms probed by wave packets and parametric four-wave mixing,” Phys. Rev. Lett. 99, 143201 (2007).
[CrossRef]

H. C. Tran, P. C. John, J. Gao, and J. G. Eden, “Interaction of atomic wave packets with four-wave mixing: detection of rubidium and potassium wave packets by coherent ultraviolet emission,” Opt. Lett. 23, 70–72 (1998).
[CrossRef]

Efthimiopoulos, T.

A. Armyras, D. Pentaris, T. Efthimiopoulos, N. Merlemis, and A. Lyras, “Saturation and population transfer of a two-photon excited four-level potassium atom,” J. Phys. B 44, 165401 (2011).
[CrossRef]

Epple, G.

A. Kölle, G. Epple, H. Kübler, W. R. Lö, and T. Pfau, “Four-wave mixing involving Rydberg states in thermal vapor,” Phys. Rev. A 85, 063821 (2012).
[CrossRef]

Gallagher, T. F.

T. F. Gallagher, Rydberg Atoms (Cambridge University, 1994).

Gao, J.

F. Shen, J. Gao, A. A. Senin, C. J. Zhu, J. R. Allen, Z. H. Lu, Y. Xiao, and J. G. Eden, “Many-body dipole-dipole interactions between excited Rb atoms probed by wave packets and parametric four-wave mixing,” Phys. Rev. Lett. 99, 143201 (2007).
[CrossRef]

H. C. Tran, P. C. John, J. Gao, and J. G. Eden, “Interaction of atomic wave packets with four-wave mixing: detection of rubidium and potassium wave packets by coherent ultraviolet emission,” Opt. Lett. 23, 70–72 (1998).
[CrossRef]

Garret, W. R.

M. G. Payne, L. Deng, and W. R. Garret, “Theory of the effect of odd-photon destructive interference on optical shifts in resonantly enhanced multiphoton excitation and ionization,” Phys. Rev. A 58, 1361–1371 (1998).
[CrossRef]

Garrett, W. R.

L. Deng, M. G. Payne, and W. R. Garrett, “Effects of multi-photon interferences from internally generated fields in strongly resonant systems,” Phys. Rep. 429, 123–241 (2006), and references therein.
[CrossRef]

Gauthier, D. J.

R. K. Wunderlich, D. J. Gauthier, R. C. Hart, M. A. Moore, and M. G. Payne, “Nonlinear optical processes near the sodium 4D two-photon resonance,” Phys. Rev. A 41, 6345–6360 (1990).
[CrossRef]

R. W. Boyd, M. S. Malcuit, D. J. Gauthier, and K. Rzażewski, “Competition between amplified spontaneous emission and the four-wave-mixing process,” Phys. Rev. A 35, 1648–1658 (1987).
[CrossRef]

Hart, R. C.

R. K. Wunderlich, D. J. Gauthier, R. C. Hart, M. A. Moore, and M. G. Payne, “Nonlinear optical processes near the sodium 4D two-photon resonance,” Phys. Rev. A 41, 6345–6360 (1990).
[CrossRef]

Hollberg, L.

A. S. Zibrov, M. D. Lukin, L. Hollberg, and M. O. Scully, “Efficient frequency up-conversion in resonant coherent media,” Phys. Rev. A 65, 051801 (2002).
[CrossRef]

Hughes, I. G.

L. Weller, R. J. Bettles, P. Siddons, C. S. Adams, and I. G. Hughes, “Absolute absorption on rubidium D1 line: including resonant dipole-dipole interactions,” J. Phys. B 44, 195006 (2011).
[CrossRef]

Jackson, D. J.

D. J. Jackson and J. J. Wynne, “Interference effects between different optical harmonics,” Phys. Rev. Lett. 49, 543–546 (1982).
[CrossRef]

Jackson, T. R.

A. K. Mohapatra, T. R. Jackson, and C. S. Adams, “Coherent optical detection of highly excited Rydberg states using electromagnetically induced transparency,” Phys. Rev. Lett 98, 113003 (2007).
[CrossRef]

Jenkins, S. D.

T. Chaneliere, D. N. Matsukevich, S. D. Jenkins, T. A. B. Kennedy, M. S. Chapman, and A. Kuzmich, “Quantum telecommunication based on atomic cascade transitions,” Phys. Rev. Lett. 96, 093604 (2006).
[CrossRef]

Jeppesen, M.

John, P. C.

Kennedy, T. A. B.

T. Chaneliere, D. N. Matsukevich, S. D. Jenkins, T. A. B. Kennedy, M. S. Chapman, and A. Kuzmich, “Quantum telecommunication based on atomic cascade transitions,” Phys. Rev. Lett. 96, 093604 (2006).
[CrossRef]

Kölle, A.

A. Kölle, G. Epple, H. Kübler, W. R. Lö, and T. Pfau, “Four-wave mixing involving Rydberg states in thermal vapor,” Phys. Rev. A 85, 063821 (2012).
[CrossRef]

Kübler, H.

A. Kölle, G. Epple, H. Kübler, W. R. Lö, and T. Pfau, “Four-wave mixing involving Rydberg states in thermal vapor,” Phys. Rev. A 85, 063821 (2012).
[CrossRef]

Kuzmich, A.

Y. O. Dudin and A. Kuzmich, “Strongly interacting Rydberg excitations of a cold atomic gas,” Science 336, 887–889 (2012).
[CrossRef]

T. Chaneliere, D. N. Matsukevich, S. D. Jenkins, T. A. B. Kennedy, M. S. Chapman, and A. Kuzmich, “Quantum telecommunication based on atomic cascade transitions,” Phys. Rev. Lett. 96, 093604 (2006).
[CrossRef]

Lewis, E. L.

E. L. Lewis, “Collisional relaxation of atomic excited states, line broadening and interatomic interactions,” Phys. Rep. 58, 1–71 (1980).
[CrossRef]

Lindinger, W.

M. Chret, L. Barbier, W. Lindinger, and R. Deloche, “Penning and associative ionisation of highly excited rubidium atoms,” J. Phys. B 15, 3463–3477 (1982).
[CrossRef]

Lö, W. R.

A. Kölle, G. Epple, H. Kübler, W. R. Lö, and T. Pfau, “Four-wave mixing involving Rydberg states in thermal vapor,” Phys. Rev. A 85, 063821 (2012).
[CrossRef]

Lu, Z. H.

F. Shen, J. Gao, A. A. Senin, C. J. Zhu, J. R. Allen, Z. H. Lu, Y. Xiao, and J. G. Eden, “Many-body dipole-dipole interactions between excited Rb atoms probed by wave packets and parametric four-wave mixing,” Phys. Rev. Lett. 99, 143201 (2007).
[CrossRef]

Lukin, M. D.

A. S. Zibrov, M. D. Lukin, L. Hollberg, and M. O. Scully, “Efficient frequency up-conversion in resonant coherent media,” Phys. Rev. A 65, 051801 (2002).
[CrossRef]

Luna, R. R. T.

R. R. T. Luna, G. H. Cavalcanti, L. H. Coutinho, and A. G. Trigueiros, “A compilation of wavelengths and energy levels for the spectrum of neutral rubidium (Rb I),” J. Quant. Spectrosc. Radiat. Transfer 75, 559–587 (2002).
[CrossRef]

Lyras, A.

A. Armyras, D. Pentaris, T. Efthimiopoulos, N. Merlemis, and A. Lyras, “Saturation and population transfer of a two-photon excited four-level potassium atom,” J. Phys. B 44, 165401 (2011).
[CrossRef]

Magno, W. C.

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).
[CrossRef]

S. S. Vianna, P. Nussenzveig, W. C. Magno, and J. W. Tabosa, “Polarization dependence and interference in four-wave mixing with Rydberg levels in rubidium vapor,” Phys. Rev. A 58, 3000–3003 (1998).
[CrossRef]

Malcuit, M. S.

R. W. Boyd, M. S. Malcuit, D. J. Gauthier, and K. Rzażewski, “Competition between amplified spontaneous emission and the four-wave-mixing process,” Phys. Rev. A 35, 1648–1658 (1987).
[CrossRef]

Matsukevich, D. N.

T. Chaneliere, D. N. Matsukevich, S. D. Jenkins, T. A. B. Kennedy, M. S. Chapman, and A. Kuzmich, “Quantum telecommunication based on atomic cascade transitions,” Phys. Rev. Lett. 96, 093604 (2006).
[CrossRef]

Meijer, T.

Merlemis, N.

A. Armyras, D. Pentaris, T. Efthimiopoulos, N. Merlemis, and A. Lyras, “Saturation and population transfer of a two-photon excited four-level potassium atom,” J. Phys. B 44, 165401 (2011).
[CrossRef]

Mohapatra, A. K.

A. K. Mohapatra, T. R. Jackson, and C. S. Adams, “Coherent optical detection of highly excited Rydberg states using electromagnetically induced transparency,” Phys. Rev. Lett 98, 113003 (2007).
[CrossRef]

Moore, M. A.

R. K. Wunderlich, D. J. Gauthier, R. C. Hart, M. A. Moore, and M. G. Payne, “Nonlinear optical processes near the sodium 4D two-photon resonance,” Phys. Rev. A 41, 6345–6360 (1990).
[CrossRef]

Nagai, H.

H. Nagai and T. Nakanaga, “Competition between two-photon-resonant three-photon ionization and four-wave mixing in Xe,” Phys. Rev. A 84, 063408 (2011).
[CrossRef]

Nakanaga, T.

H. Nagai and T. Nakanaga, “Competition between two-photon-resonant three-photon ionization and four-wave mixing in Xe,” Phys. Rev. A 84, 063408 (2011).
[CrossRef]

Nussenzveig, P.

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).
[CrossRef]

S. S. Vianna, P. Nussenzveig, W. C. Magno, and J. W. Tabosa, “Polarization dependence and interference in four-wave mixing with Rydberg levels in rubidium vapor,” Phys. Rev. A 58, 3000–3003 (1998).
[CrossRef]

Orozco, L. A.

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).
[CrossRef]

Payne, M. G.

L. Deng, M. G. Payne, and W. R. Garrett, “Effects of multi-photon interferences from internally generated fields in strongly resonant systems,” Phys. Rep. 429, 123–241 (2006), and references therein.
[CrossRef]

M. G. Payne, L. Deng, and W. R. Garret, “Theory of the effect of odd-photon destructive interference on optical shifts in resonantly enhanced multiphoton excitation and ionization,” Phys. Rev. A 58, 1361–1371 (1998).
[CrossRef]

R. K. Wunderlich, D. J. Gauthier, R. C. Hart, M. A. Moore, and M. G. Payne, “Nonlinear optical processes near the sodium 4D two-photon resonance,” Phys. Rev. A 41, 6345–6360 (1990).
[CrossRef]

Pentaris, D.

A. Armyras, D. Pentaris, T. Efthimiopoulos, N. Merlemis, and A. Lyras, “Saturation and population transfer of a two-photon excited four-level potassium atom,” J. Phys. B 44, 165401 (2011).
[CrossRef]

Pfau, T.

A. Kölle, G. Epple, H. Kübler, W. R. Lö, and T. Pfau, “Four-wave mixing involving Rydberg states in thermal vapor,” Phys. Rev. A 85, 063821 (2012).
[CrossRef]

Prandini, R. B.

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).
[CrossRef]

Rolston, S. L.

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).
[CrossRef]

Rzazewski, K.

R. W. Boyd, M. S. Malcuit, D. J. Gauthier, and K. Rzażewski, “Competition between amplified spontaneous emission and the four-wave-mixing process,” Phys. Rev. A 35, 1648–1658 (1987).
[CrossRef]

Sautenkov, V. A.

V. A. Sautenkov, “Line shapes of atomic transitions in excited dense gas,” Laser Phys. Lett. 8, 771 (2011).
[CrossRef]

Scholten, R. E.

Scully, M. O.

A. S. Zibrov, M. D. Lukin, L. Hollberg, and M. O. Scully, “Efficient frequency up-conversion in resonant coherent media,” Phys. Rev. A 65, 051801 (2002).
[CrossRef]

Senin, A. A.

F. Shen, J. Gao, A. A. Senin, C. J. Zhu, J. R. Allen, Z. H. Lu, Y. Xiao, and J. G. Eden, “Many-body dipole-dipole interactions between excited Rb atoms probed by wave packets and parametric four-wave mixing,” Phys. Rev. Lett. 99, 143201 (2007).
[CrossRef]

Shen, F.

F. Shen, J. Gao, A. A. Senin, C. J. Zhu, J. R. Allen, Z. H. Lu, Y. Xiao, and J. G. Eden, “Many-body dipole-dipole interactions between excited Rb atoms probed by wave packets and parametric four-wave mixing,” Phys. Rev. Lett. 99, 143201 (2007).
[CrossRef]

Siddons, P.

L. Weller, R. J. Bettles, P. Siddons, C. S. Adams, and I. G. Hughes, “Absolute absorption on rubidium D1 line: including resonant dipole-dipole interactions,” J. Phys. B 44, 195006 (2011).
[CrossRef]

Smeets, B.

Stoicheff, B. P.

D. C. Thompson, E. Weinberger, G. X. Xu, and B. P. Stoicheff, “Frequency shifts and line broadenings in collisions between Rydberg and ground-state alkali-metal atoms,” Phys. Rev. A 35, 690–700 (1987).
[CrossRef]

B. P. Stoicheff and E. Weinberger, “Frequency shifts, line broadenings, and phase-interference effects in Rb+Rb collisions, measured by Doppler-free two-photon spectroscopy,” Phys. Rev. Lett. 44, 733–736 (1980).
[CrossRef]

Tabosa, J. W.

S. S. Vianna, P. Nussenzveig, W. C. Magno, and J. W. Tabosa, “Polarization dependence and interference in four-wave mixing with Rydberg levels in rubidium vapor,” Phys. Rev. A 58, 3000–3003 (1998).
[CrossRef]

Thompson, D. C.

D. C. Thompson, E. Weinberger, G. X. Xu, and B. P. Stoicheff, “Frequency shifts and line broadenings in collisions between Rydberg and ground-state alkali-metal atoms,” Phys. Rev. A 35, 690–700 (1987).
[CrossRef]

Tran, H. C.

Trigueiros, A. G.

R. R. T. Luna, G. H. Cavalcanti, L. H. Coutinho, and A. G. Trigueiros, “A compilation of wavelengths and energy levels for the spectrum of neutral rubidium (Rb I),” J. Quant. Spectrosc. Radiat. Transfer 75, 559–587 (2002).
[CrossRef]

Vianna, S. S.

N. R. de Melo and S. S. Vianna, “Rydberg-Rydberg interaction in the second-harmonic generation from rubidium atoms,” Phys. Rev. A 77, 023418 (2008).
[CrossRef]

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).
[CrossRef]

S. S. Vianna, P. Nussenzveig, W. C. Magno, and J. W. Tabosa, “Polarization dependence and interference in four-wave mixing with Rydberg levels in rubidium vapor,” Phys. Rev. A 58, 3000–3003 (1998).
[CrossRef]

Walker, T. G.

E. Brekke, J. O. Day, and T. G. Walker, “Four-wave mixing in ultracold atoms using intermediate Rydberg states,” Phys. Rev. A 78, 063830 (2008).
[CrossRef]

Weinberger, E.

D. C. Thompson, E. Weinberger, G. X. Xu, and B. P. Stoicheff, “Frequency shifts and line broadenings in collisions between Rydberg and ground-state alkali-metal atoms,” Phys. Rev. A 35, 690–700 (1987).
[CrossRef]

B. P. Stoicheff and E. Weinberger, “Frequency shifts, line broadenings, and phase-interference effects in Rb+Rb collisions, measured by Doppler-free two-photon spectroscopy,” Phys. Rev. Lett. 44, 733–736 (1980).
[CrossRef]

Weller, L.

L. Weller, R. J. Bettles, P. Siddons, C. S. Adams, and I. G. Hughes, “Absolute absorption on rubidium D1 line: including resonant dipole-dipole interactions,” J. Phys. B 44, 195006 (2011).
[CrossRef]

White, J. D.

Willis, R. T.

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).
[CrossRef]

Wunderlich, R. K.

R. K. Wunderlich, D. J. Gauthier, R. C. Hart, M. A. Moore, and M. G. Payne, “Nonlinear optical processes near the sodium 4D two-photon resonance,” Phys. Rev. A 41, 6345–6360 (1990).
[CrossRef]

Wynne, J. J.

D. J. Jackson and J. J. Wynne, “Interference effects between different optical harmonics,” Phys. Rev. Lett. 49, 543–546 (1982).
[CrossRef]

Xiao, M.

H. B. Zheng, X. Yao, Z. Y. Zhang, J. L. Che, Y. Q. Zhang, Y. P. Zhang, and M. Xiao, “Blockaded six- and eight-wave mixing processes tailored by electromagnetically induced transparency scissors,” Laser Phys. 24, 045404 (2014).
[CrossRef]

Xiao, Y.

F. Shen, J. Gao, A. A. Senin, C. J. Zhu, J. R. Allen, Z. H. Lu, Y. Xiao, and J. G. Eden, “Many-body dipole-dipole interactions between excited Rb atoms probed by wave packets and parametric four-wave mixing,” Phys. Rev. Lett. 99, 143201 (2007).
[CrossRef]

Xu, G. X.

D. C. Thompson, E. Weinberger, G. X. Xu, and B. P. Stoicheff, “Frequency shifts and line broadenings in collisions between Rydberg and ground-state alkali-metal atoms,” Phys. Rev. A 35, 690–700 (1987).
[CrossRef]

Yao, X.

H. B. Zheng, X. Yao, Z. Y. Zhang, J. L. Che, Y. Q. Zhang, Y. P. Zhang, and M. Xiao, “Blockaded six- and eight-wave mixing processes tailored by electromagnetically induced transparency scissors,” Laser Phys. 24, 045404 (2014).
[CrossRef]

Zhang, Y. P.

H. B. Zheng, X. Yao, Z. Y. Zhang, J. L. Che, Y. Q. Zhang, Y. P. Zhang, and M. Xiao, “Blockaded six- and eight-wave mixing processes tailored by electromagnetically induced transparency scissors,” Laser Phys. 24, 045404 (2014).
[CrossRef]

Zhang, Y. Q.

H. B. Zheng, X. Yao, Z. Y. Zhang, J. L. Che, Y. Q. Zhang, Y. P. Zhang, and M. Xiao, “Blockaded six- and eight-wave mixing processes tailored by electromagnetically induced transparency scissors,” Laser Phys. 24, 045404 (2014).
[CrossRef]

Zhang, Z. Y.

H. B. Zheng, X. Yao, Z. Y. Zhang, J. L. Che, Y. Q. Zhang, Y. P. Zhang, and M. Xiao, “Blockaded six- and eight-wave mixing processes tailored by electromagnetically induced transparency scissors,” Laser Phys. 24, 045404 (2014).
[CrossRef]

Zheng, H. B.

H. B. Zheng, X. Yao, Z. Y. Zhang, J. L. Che, Y. Q. Zhang, Y. P. Zhang, and M. Xiao, “Blockaded six- and eight-wave mixing processes tailored by electromagnetically induced transparency scissors,” Laser Phys. 24, 045404 (2014).
[CrossRef]

Zhu, C. J.

F. Shen, J. Gao, A. A. Senin, C. J. Zhu, J. R. Allen, Z. H. Lu, Y. Xiao, and J. G. Eden, “Many-body dipole-dipole interactions between excited Rb atoms probed by wave packets and parametric four-wave mixing,” Phys. Rev. Lett. 99, 143201 (2007).
[CrossRef]

Zibrov, A. S.

A. S. Zibrov, M. D. Lukin, L. Hollberg, and M. O. Scully, “Efficient frequency up-conversion in resonant coherent media,” Phys. Rev. A 65, 051801 (2002).
[CrossRef]

J. Phys. B

A. Armyras, D. Pentaris, T. Efthimiopoulos, N. Merlemis, and A. Lyras, “Saturation and population transfer of a two-photon excited four-level potassium atom,” J. Phys. B 44, 165401 (2011).
[CrossRef]

L. Weller, R. J. Bettles, P. Siddons, C. S. Adams, and I. G. Hughes, “Absolute absorption on rubidium D1 line: including resonant dipole-dipole interactions,” J. Phys. B 44, 195006 (2011).
[CrossRef]

M. Chret, L. Barbier, W. Lindinger, and R. Deloche, “Penning and associative ionisation of highly excited rubidium atoms,” J. Phys. B 15, 3463–3477 (1982).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer

R. R. T. Luna, G. H. Cavalcanti, L. H. Coutinho, and A. G. Trigueiros, “A compilation of wavelengths and energy levels for the spectrum of neutral rubidium (Rb I),” J. Quant. Spectrosc. Radiat. Transfer 75, 559–587 (2002).
[CrossRef]

Laser Phys.

H. B. Zheng, X. Yao, Z. Y. Zhang, J. L. Che, Y. Q. Zhang, Y. P. Zhang, and M. Xiao, “Blockaded six- and eight-wave mixing processes tailored by electromagnetically induced transparency scissors,” Laser Phys. 24, 045404 (2014).
[CrossRef]

Laser Phys. Lett.

V. A. Sautenkov, “Line shapes of atomic transitions in excited dense gas,” Laser Phys. Lett. 8, 771 (2011).
[CrossRef]

Opt. Lett.

Phys. Rep.

E. L. Lewis, “Collisional relaxation of atomic excited states, line broadening and interatomic interactions,” Phys. Rep. 58, 1–71 (1980).
[CrossRef]

L. Deng, M. G. Payne, and W. R. Garrett, “Effects of multi-photon interferences from internally generated fields in strongly resonant systems,” Phys. Rep. 429, 123–241 (2006), and references therein.
[CrossRef]

Phys. Rev. A

A. S. Zibrov, M. D. Lukin, L. Hollberg, and M. O. Scully, “Efficient frequency up-conversion in resonant coherent media,” Phys. Rev. A 65, 051801 (2002).
[CrossRef]

R. W. Boyd, M. S. Malcuit, D. J. Gauthier, and K. Rzażewski, “Competition between amplified spontaneous emission and the four-wave-mixing process,” Phys. Rev. A 35, 1648–1658 (1987).
[CrossRef]

R. K. Wunderlich, D. J. Gauthier, R. C. Hart, M. A. Moore, and M. G. Payne, “Nonlinear optical processes near the sodium 4D two-photon resonance,” Phys. Rev. A 41, 6345–6360 (1990).
[CrossRef]

H. Nagai and T. Nakanaga, “Competition between two-photon-resonant three-photon ionization and four-wave mixing in Xe,” Phys. Rev. A 84, 063408 (2011).
[CrossRef]

D. C. Thompson, E. Weinberger, G. X. Xu, and B. P. Stoicheff, “Frequency shifts and line broadenings in collisions between Rydberg and ground-state alkali-metal atoms,” Phys. Rev. A 35, 690–700 (1987).
[CrossRef]

S. S. Vianna, P. Nussenzveig, W. C. Magno, and J. W. Tabosa, “Polarization dependence and interference in four-wave mixing with Rydberg levels in rubidium vapor,” Phys. Rev. A 58, 3000–3003 (1998).
[CrossRef]

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).
[CrossRef]

M. G. Payne, L. Deng, and W. R. Garret, “Theory of the effect of odd-photon destructive interference on optical shifts in resonantly enhanced multiphoton excitation and ionization,” Phys. Rev. A 58, 1361–1371 (1998).
[CrossRef]

E. Brekke, J. O. Day, and T. G. Walker, “Four-wave mixing in ultracold atoms using intermediate Rydberg states,” Phys. Rev. A 78, 063830 (2008).
[CrossRef]

A. Kölle, G. Epple, H. Kübler, W. R. Lö, and T. Pfau, “Four-wave mixing involving Rydberg states in thermal vapor,” Phys. Rev. A 85, 063821 (2012).
[CrossRef]

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).
[CrossRef]

N. R. de Melo and S. S. Vianna, “Rydberg-Rydberg interaction in the second-harmonic generation from rubidium atoms,” Phys. Rev. A 77, 023418 (2008).
[CrossRef]

Phys. Rev. Lett

A. K. Mohapatra, T. R. Jackson, and C. S. Adams, “Coherent optical detection of highly excited Rydberg states using electromagnetically induced transparency,” Phys. Rev. Lett 98, 113003 (2007).
[CrossRef]

Phys. Rev. Lett.

F. Shen, J. Gao, A. A. Senin, C. J. Zhu, J. R. Allen, Z. H. Lu, Y. Xiao, and J. G. Eden, “Many-body dipole-dipole interactions between excited Rb atoms probed by wave packets and parametric four-wave mixing,” Phys. Rev. Lett. 99, 143201 (2007).
[CrossRef]

B. P. Stoicheff and E. Weinberger, “Frequency shifts, line broadenings, and phase-interference effects in Rb+Rb collisions, measured by Doppler-free two-photon spectroscopy,” Phys. Rev. Lett. 44, 733–736 (1980).
[CrossRef]

D. J. Jackson and J. J. Wynne, “Interference effects between different optical harmonics,” Phys. Rev. Lett. 49, 543–546 (1982).
[CrossRef]

T. Chaneliere, D. N. Matsukevich, S. D. Jenkins, T. A. B. Kennedy, M. S. Chapman, and A. Kuzmich, “Quantum telecommunication based on atomic cascade transitions,” Phys. Rev. Lett. 96, 093604 (2006).
[CrossRef]

Science

Y. O. Dudin and A. Kuzmich, “Strongly interacting Rydberg excitations of a cold atomic gas,” Science 336, 887–889 (2012).
[CrossRef]

Other

T. F. Gallagher, Rydberg Atoms (Cambridge University, 1994).

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Figures (4)

Fig. 1.
Fig. 1.

(a) Experimental setup, where the symbols PMT and F stand for the photomultiplier tube and filter, respectively. (b) Schematic representation of the relevant energy levels of Rb, where ω1 represents the dye laser frequency.

Fig. 2.
Fig. 2.

FWM intensity as a function of dye laser detuning, Δω, relative to the two-photon transitions 5snd, for n=14, 15, 16, 17, and 20. The open circles corresponding to the experimental results and the solid lines are obtained from the numerical integration of Eq. (6). Note the varying multiplicative factors on the curves. The dye and IR beam intensities are Idye=1GW/cm2 and IIR=3GW/cm2, respectively. The atomic density is N5×1015cm3.

Fig. 3.
Fig. 3.

FWM spectra involving the 16d and 18s two-photon resonances for three atomic densities. The open circles are the experimental data and the solid lines are obtained from the numerical integration of Eq. (6), including a correction due to self-broadening. The dashed straight line shows the frequency position of the 6P3/2 peak. The inset is a magnification of the upper curve near the three-photon resonance region, where the dotted line represents the theoretical result with no odd-photon destructive interference.

Fig. 4.
Fig. 4.

(a) Intensity ratio between the 16d and 18s peaks and (b) the 16d peak intensity, as a function of the atomic density. The full circles with the error bars were obtained from the experimental spectra and the solid, dashed, and dotted lines are given by the theoretical model (see text).

Tables (1)

Tables Icon

Table 1. Values for Δab=(2ω1ω2ωba) With 2ω1 at the Two-Photon Resonant Transition (5snl), and b=6P1/2 (Δ6P1/2) or b=6P3/2 (Δ6P3/2) [17]

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

ρ^t=i[H^,ρ^]+(relaxation terms),
H^int=Ωad(2)ei2k1z|ad|Ωas(2)ei2k1z|as|Ωbdeik2z|bd|Ωbseik2z|bs|Ωabeik3z|ab|+H.c.
E(z,t)=i=1312[Eieiωit+ikiz+c.c].
σba=[Ωda(2)ΩbdΔda+iγad+Ωsa(2)ΩbsΔsa+iγas]eiΔkzΔba+iγabΩbaΔba+iγab,
Ωbaz=iκabσba,
Ωba=[Ωda(2)ΩbdΔda+iγad+Ωsa(2)ΩbsΔsa+iγas][1ei(κabΔba+iγab)z].
Γnl=γnl+Γself=γnl+βN,

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