Abstract

We present theoretically and experimentally a novel type of parametric fluorescence and oscillation in which four optical waves in off-axis, diagonal directions simultaneously build up under resonant counterpropagating pump-laser excitation in atomic sodium vapor. Two of the four are anti-Stokes shifted and the other two are Stokes shifted, and they are strongly coupled owing to electromagnetically induced diffraction. Theoretically, the Liouville–Maxwell equations are solved to derive four-wave propagation equations. Experimentally, we have observed four-wave parametric oscillation when a weak off-axis seed beam is introduced into otherwise axially symmetric parametric fluorescence patterns. Temporal waveforms are also analyzed, and strong correlations between any pairs of the four emissions are identified.

© 2008 Optical Society of America

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  1. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge U. Press, 1997).
  2. J. A. Giormaine and R. C. Miller, "Tunable coherent parametric oscillation in LiNbO3 at optical frequencies," Phys. Rev. Lett. 14, 973-976 (1965).
    [CrossRef]
  3. M. D. Lukin, P. R. Hemmer, M. Löffler, and M. O. Scully, "Resonant enhancement of parametric processes via radiative interference and induced coherence," Phys. Rev. Lett. 81, 2675-2678 (1998).
    [CrossRef]
  4. A. S. Zibrov, M. D. Lukin, and M. O. Scully, "Nondegenerate parametric self-oscillation via multiwave mixing in coherent atomic media," Phys. Rev. Lett. 83, 4049-4052 (1999).
    [CrossRef]
  5. M. D. Lukin, A. B. Matsko, M. Fleischhauer, and M. O. Scully, "Quantum noise and correlations in resonantly enhanced wave mixing based on atomic coherence," Phys. Rev. Lett. 82, 1847-1850 (1999).
    [CrossRef]
  6. 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, 2409-2412 (1985).
    [CrossRef] [PubMed]
  7. 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, 178-180 (2007).
    [CrossRef]
  8. A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, and C. Fabre, "Observation of quantum noise reduction on twin laser beams," Phys. Rev. Lett. 59, 2555-2557 (1987).
    [CrossRef] [PubMed]
  9. O. Aytür and P. Kumar, "Pulsed twin beams of light," Phys. Rev. Lett. 65, 1551-1554 (1990).
    [CrossRef] [PubMed]
  10. H. Zou, A. Zhai, J. Guo, R. Yang, and J. Gao, "Preparation and measurement of tunable high-power sub-Poissonian light using twin beams," Opt. Lett. 31, 1735-1737 (2006).
    [CrossRef] [PubMed]
  11. M. Fleischhauer and M. D. Lukin, "Quantum memory for photons: dark-state polaritons," Phys. Rev. A 65, 022314 (2002).
    [CrossRef]
  12. C. H. van der Wal, M. D. Eisaman, A. André, R. L. Walsworth, D. F. Phillips, A. S. Zibrov, and M. D. Lukin, "Atomic memory for correlated photon states," Science 301, 196-200 (2003).
    [CrossRef] [PubMed]
  13. A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L.-M. Duan, and H. J. Kimble, "Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles," Nature 423, 731-734 (2003).
    [CrossRef] [PubMed]
  14. K. Harada, M. Ogata, and M. Mitsunaga, "Four-wave parametric oscillation in sodium vapor by electromagnetically induced diffraction," Opt. Lett. 32, 1111-1113 (2007).
    [CrossRef] [PubMed]
  15. K. Harada, S. Tanaka, T. Kanbashi, M. Mitsunaga, and K. Motomura, "Electromagnetically induced diffraction in sodium vapor," Opt. Lett. 30, 2004-2006 (2005).
    [CrossRef] [PubMed]
  16. K.-J. Boller, A. Imamoglu, and S. E. Harris, "Observation of electromagnetically induced transparency," Phys. Rev. Lett. 66, 2593-2596 (1991).
    [CrossRef] [PubMed]
  17. J. E. Field, K. H. Hahn, and S. E. Harris, "Observation of electromagnetically induced transparency in collisionally broadened lead vapor," Phys. Rev. Lett. 67, 3062-3065 (1991).
    [CrossRef] [PubMed]
  18. A. F. Huss, N. Peer, R. Lammegger, E. A. Korsunsky, and L. Windholz, "Efficient Raman sideband generation in a coherent atomic medium," Phys. Rev. A 63, 013802 (2000).
    [CrossRef]
  19. V. Wong, R. S. Bennink, A. M. Marino, R. W. Boyd, C. R. Stroud, Jr., and F. A. Narducci, "Influence of coherent Raman scattering on coherent population trapping in atomic sodium vapor," Phys. Rev. A 70, 053811 (2004).
    [CrossRef]
  20. K. Harada, K. Motomura, T. Koshimizu, H. Ueno, and M. Mitsunaga, "Coherent Raman beats from dark states," J. Opt. Soc. Am. B 22, 1105-1111 (2005).
    [CrossRef]
  21. K. Harada, T. Kanbashi, M. Mitsunaga, and K. Motomura, "Competition between electromagnetically induced transparency and stimulated Raman scattering," Phys. Rev. A 73, 013807 (2006).
    [CrossRef]
  22. G. S. Agarwal, T. N. Dey, and D. J. Gauthier, "Competition between electromagnetically induced transparency and Raman processes," Phys. Rev. A 74, 043805 (2006).
    [CrossRef]
  23. P. R. Hemmer, D. P. Katz, J. Donoghue, M. Cronin-Golomb, M. S. Shahriar, and P. Kumar, "Efficient low-intensity optical phase conjugation based on coherent population trapping in sodium," Opt. Lett. 20, 982-984 (1995).
    [CrossRef] [PubMed]
  24. D. A. Braje, V. Balic, S. Goda, G. Y. Yin, and S. E. Harris, "Frequency mixing using electomagnetically induced transparency in cold atoms," Phys. Rev. Lett. 93, 183601 (2004).
    [CrossRef] [PubMed]
  25. V. Balic, D. A. Braje, P. Kolchin, G. Y. Yin, and S. E. Harris, "Generation of paired photons with controllable waveforms," Phys. Rev. Lett. 94, 183601 (2005).
    [CrossRef] [PubMed]
  26. D. N. Matsukevich, T. Chanelière, M. Bhattacharya, S.-Y. Lan, S. D. Jenkins, T. A. B. Kennedy, and A. Kuzmich, "Entanglement of a photon and a collective atomic excitation," Phys. Rev. Lett. 95, 040405 (2005).
    [CrossRef] [PubMed]
  27. T. Chanelière, D. N. Matsukevich, S. D. Jenkins, S.-Y. Lan, T. A. B. Kennedy, and A. Kuzmich, "Storage and retrieval of single photons transmitted between remote quantum memories," Nature 438, 833-836 (2005).
    [CrossRef] [PubMed]
  28. A. Petrossian, M. Pinard, A. Maître, J.-Y. Courtois, and G. Grynberg, "Transverse-pattern formation for counterpropagating laser beams in rubidium vapour," Europhys. Lett. 18, 689-695 (1992).
    [CrossRef]
  29. A. Maître, A. Petrossian, A. Blouin, M. Panard, and G. Grynberg, "Spatio-temporal instability for counterpropagating beams in rubidium vapor," Opt. Commun. 116, 153-158 (1995).
    [CrossRef]
  30. A. M. C. Dawes, L. Illing, S. M. Clark, and D. J. Gauthier, "All-optical switching in rubidium vapor," Science 308, 672-674 (2005).
    [CrossRef] [PubMed]
  31. K. Motomura, M. Tsukamoto, A. Wakiyama, K. Harada, and M. Mitsunaga, "Observation of correlated anti-Stokes emissions by multiwave mixing in sodium vapor," Phys. Rev. A 71, 043817 (2005).
    [CrossRef]
  32. Similar results were obtained if another definition of correlation function G(2)(τ)=⟨δf1(t)δf2(t+τ)⟩/⟨δf1(t)2⟩⟨δf2(t+τ)2⟩ was employed, where δf1,2(t)≡f1,2(t)−⟨f1,2⟩. See V. A. Sautenkov, Y. V. Rostovtsev, and M. O. Scully, "Switching between photon-photon correlations and Raman anticorrelations in a coherently prepared Rb vapor," Phys. Rev. A 72, 065801 (2005).
    [CrossRef]

2007 (2)

2006 (3)

H. Zou, A. Zhai, J. Guo, R. Yang, and J. Gao, "Preparation and measurement of tunable high-power sub-Poissonian light using twin beams," Opt. Lett. 31, 1735-1737 (2006).
[CrossRef] [PubMed]

K. Harada, T. Kanbashi, M. Mitsunaga, and K. Motomura, "Competition between electromagnetically induced transparency and stimulated Raman scattering," Phys. Rev. A 73, 013807 (2006).
[CrossRef]

G. S. Agarwal, T. N. Dey, and D. J. Gauthier, "Competition between electromagnetically induced transparency and Raman processes," Phys. Rev. A 74, 043805 (2006).
[CrossRef]

2005 (8)

K. Harada, K. Motomura, T. Koshimizu, H. Ueno, and M. Mitsunaga, "Coherent Raman beats from dark states," J. Opt. Soc. Am. B 22, 1105-1111 (2005).
[CrossRef]

V. Balic, D. A. Braje, P. Kolchin, G. Y. Yin, and S. E. Harris, "Generation of paired photons with controllable waveforms," Phys. Rev. Lett. 94, 183601 (2005).
[CrossRef] [PubMed]

D. N. Matsukevich, T. Chanelière, M. Bhattacharya, S.-Y. Lan, S. D. Jenkins, T. A. B. Kennedy, and A. Kuzmich, "Entanglement of a photon and a collective atomic excitation," Phys. Rev. Lett. 95, 040405 (2005).
[CrossRef] [PubMed]

T. Chanelière, D. N. Matsukevich, S. D. Jenkins, S.-Y. Lan, T. A. B. Kennedy, and A. Kuzmich, "Storage and retrieval of single photons transmitted between remote quantum memories," Nature 438, 833-836 (2005).
[CrossRef] [PubMed]

A. M. C. Dawes, L. Illing, S. M. Clark, and D. J. Gauthier, "All-optical switching in rubidium vapor," Science 308, 672-674 (2005).
[CrossRef] [PubMed]

K. Motomura, M. Tsukamoto, A. Wakiyama, K. Harada, and M. Mitsunaga, "Observation of correlated anti-Stokes emissions by multiwave mixing in sodium vapor," Phys. Rev. A 71, 043817 (2005).
[CrossRef]

Similar results were obtained if another definition of correlation function G(2)(τ)=⟨δf1(t)δf2(t+τ)⟩/⟨δf1(t)2⟩⟨δf2(t+τ)2⟩ was employed, where δf1,2(t)≡f1,2(t)−⟨f1,2⟩. See V. A. Sautenkov, Y. V. Rostovtsev, and M. O. Scully, "Switching between photon-photon correlations and Raman anticorrelations in a coherently prepared Rb vapor," Phys. Rev. A 72, 065801 (2005).
[CrossRef]

K. Harada, S. Tanaka, T. Kanbashi, M. Mitsunaga, and K. Motomura, "Electromagnetically induced diffraction in sodium vapor," Opt. Lett. 30, 2004-2006 (2005).
[CrossRef] [PubMed]

2004 (2)

D. A. Braje, V. Balic, S. Goda, G. Y. Yin, and S. E. Harris, "Frequency mixing using electomagnetically induced transparency in cold atoms," Phys. Rev. Lett. 93, 183601 (2004).
[CrossRef] [PubMed]

V. Wong, R. S. Bennink, A. M. Marino, R. W. Boyd, C. R. Stroud, Jr., and F. A. Narducci, "Influence of coherent Raman scattering on coherent population trapping in atomic sodium vapor," Phys. Rev. A 70, 053811 (2004).
[CrossRef]

2003 (2)

C. H. van der Wal, M. D. Eisaman, A. André, R. L. Walsworth, D. F. Phillips, A. S. Zibrov, and M. D. Lukin, "Atomic memory for correlated photon states," Science 301, 196-200 (2003).
[CrossRef] [PubMed]

A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L.-M. Duan, and H. J. Kimble, "Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles," Nature 423, 731-734 (2003).
[CrossRef] [PubMed]

2002 (1)

M. Fleischhauer and M. D. Lukin, "Quantum memory for photons: dark-state polaritons," Phys. Rev. A 65, 022314 (2002).
[CrossRef]

2000 (1)

A. F. Huss, N. Peer, R. Lammegger, E. A. Korsunsky, and L. Windholz, "Efficient Raman sideband generation in a coherent atomic medium," Phys. Rev. A 63, 013802 (2000).
[CrossRef]

1999 (2)

A. S. Zibrov, M. D. Lukin, and M. O. Scully, "Nondegenerate parametric self-oscillation via multiwave mixing in coherent atomic media," Phys. Rev. Lett. 83, 4049-4052 (1999).
[CrossRef]

M. D. Lukin, A. B. Matsko, M. Fleischhauer, and M. O. Scully, "Quantum noise and correlations in resonantly enhanced wave mixing based on atomic coherence," Phys. Rev. Lett. 82, 1847-1850 (1999).
[CrossRef]

1998 (1)

M. D. Lukin, P. R. Hemmer, M. Löffler, and M. O. Scully, "Resonant enhancement of parametric processes via radiative interference and induced coherence," Phys. Rev. Lett. 81, 2675-2678 (1998).
[CrossRef]

1995 (2)

P. R. Hemmer, D. P. Katz, J. Donoghue, M. Cronin-Golomb, M. S. Shahriar, and P. Kumar, "Efficient low-intensity optical phase conjugation based on coherent population trapping in sodium," Opt. Lett. 20, 982-984 (1995).
[CrossRef] [PubMed]

A. Maître, A. Petrossian, A. Blouin, M. Panard, and G. Grynberg, "Spatio-temporal instability for counterpropagating beams in rubidium vapor," Opt. Commun. 116, 153-158 (1995).
[CrossRef]

1992 (1)

A. Petrossian, M. Pinard, A. Maître, J.-Y. Courtois, and G. Grynberg, "Transverse-pattern formation for counterpropagating laser beams in rubidium vapour," Europhys. Lett. 18, 689-695 (1992).
[CrossRef]

1991 (2)

K.-J. Boller, A. Imamoglu, and S. E. Harris, "Observation of electromagnetically induced transparency," Phys. Rev. Lett. 66, 2593-2596 (1991).
[CrossRef] [PubMed]

J. E. Field, K. H. Hahn, and S. E. Harris, "Observation of electromagnetically induced transparency in collisionally broadened lead vapor," Phys. Rev. Lett. 67, 3062-3065 (1991).
[CrossRef] [PubMed]

1990 (1)

O. Aytür and P. Kumar, "Pulsed twin beams of light," Phys. Rev. Lett. 65, 1551-1554 (1990).
[CrossRef] [PubMed]

1987 (1)

A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, and C. Fabre, "Observation of quantum noise reduction on twin laser beams," Phys. Rev. Lett. 59, 2555-2557 (1987).
[CrossRef] [PubMed]

1985 (1)

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, 2409-2412 (1985).
[CrossRef] [PubMed]

1965 (1)

J. A. Giormaine and R. C. Miller, "Tunable coherent parametric oscillation in LiNbO3 at optical frequencies," Phys. Rev. Lett. 14, 973-976 (1965).
[CrossRef]

Europhys. Lett. (1)

A. Petrossian, M. Pinard, A. Maître, J.-Y. Courtois, and G. Grynberg, "Transverse-pattern formation for counterpropagating laser beams in rubidium vapour," Europhys. Lett. 18, 689-695 (1992).
[CrossRef]

J. Opt. Soc. Am. B (1)

Nature (2)

A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L.-M. Duan, and H. J. Kimble, "Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles," Nature 423, 731-734 (2003).
[CrossRef] [PubMed]

T. Chanelière, D. N. Matsukevich, S. D. Jenkins, S.-Y. Lan, T. A. B. Kennedy, and A. Kuzmich, "Storage and retrieval of single photons transmitted between remote quantum memories," Nature 438, 833-836 (2005).
[CrossRef] [PubMed]

Opt. Commun. (1)

A. Maître, A. Petrossian, A. Blouin, M. Panard, and G. Grynberg, "Spatio-temporal instability for counterpropagating beams in rubidium vapor," Opt. Commun. 116, 153-158 (1995).
[CrossRef]

Opt. Lett. (5)

Phys. Rev. A (7)

M. Fleischhauer and M. D. Lukin, "Quantum memory for photons: dark-state polaritons," Phys. Rev. A 65, 022314 (2002).
[CrossRef]

K. Harada, T. Kanbashi, M. Mitsunaga, and K. Motomura, "Competition between electromagnetically induced transparency and stimulated Raman scattering," Phys. Rev. A 73, 013807 (2006).
[CrossRef]

G. S. Agarwal, T. N. Dey, and D. J. Gauthier, "Competition between electromagnetically induced transparency and Raman processes," Phys. Rev. A 74, 043805 (2006).
[CrossRef]

A. F. Huss, N. Peer, R. Lammegger, E. A. Korsunsky, and L. Windholz, "Efficient Raman sideband generation in a coherent atomic medium," Phys. Rev. A 63, 013802 (2000).
[CrossRef]

V. Wong, R. S. Bennink, A. M. Marino, R. W. Boyd, C. R. Stroud, Jr., and F. A. Narducci, "Influence of coherent Raman scattering on coherent population trapping in atomic sodium vapor," Phys. Rev. A 70, 053811 (2004).
[CrossRef]

K. Motomura, M. Tsukamoto, A. Wakiyama, K. Harada, and M. Mitsunaga, "Observation of correlated anti-Stokes emissions by multiwave mixing in sodium vapor," Phys. Rev. A 71, 043817 (2005).
[CrossRef]

Similar results were obtained if another definition of correlation function G(2)(τ)=⟨δf1(t)δf2(t+τ)⟩/⟨δf1(t)2⟩⟨δf2(t+τ)2⟩ was employed, where δf1,2(t)≡f1,2(t)−⟨f1,2⟩. See V. A. Sautenkov, Y. V. Rostovtsev, and M. O. Scully, "Switching between photon-photon correlations and Raman anticorrelations in a coherently prepared Rb vapor," Phys. Rev. A 72, 065801 (2005).
[CrossRef]

Phys. Rev. Lett. (12)

D. A. Braje, V. Balic, S. Goda, G. Y. Yin, and S. E. Harris, "Frequency mixing using electomagnetically induced transparency in cold atoms," Phys. Rev. Lett. 93, 183601 (2004).
[CrossRef] [PubMed]

V. Balic, D. A. Braje, P. Kolchin, G. Y. Yin, and S. E. Harris, "Generation of paired photons with controllable waveforms," Phys. Rev. Lett. 94, 183601 (2005).
[CrossRef] [PubMed]

D. N. Matsukevich, T. Chanelière, M. Bhattacharya, S.-Y. Lan, S. D. Jenkins, T. A. B. Kennedy, and A. Kuzmich, "Entanglement of a photon and a collective atomic excitation," Phys. Rev. Lett. 95, 040405 (2005).
[CrossRef] [PubMed]

K.-J. Boller, A. Imamoglu, and S. E. Harris, "Observation of electromagnetically induced transparency," Phys. Rev. Lett. 66, 2593-2596 (1991).
[CrossRef] [PubMed]

J. E. Field, K. H. Hahn, and S. E. Harris, "Observation of electromagnetically induced transparency in collisionally broadened lead vapor," Phys. Rev. Lett. 67, 3062-3065 (1991).
[CrossRef] [PubMed]

A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, and C. Fabre, "Observation of quantum noise reduction on twin laser beams," Phys. Rev. Lett. 59, 2555-2557 (1987).
[CrossRef] [PubMed]

O. Aytür and P. Kumar, "Pulsed twin beams of light," Phys. Rev. Lett. 65, 1551-1554 (1990).
[CrossRef] [PubMed]

J. A. Giormaine and R. C. Miller, "Tunable coherent parametric oscillation in LiNbO3 at optical frequencies," Phys. Rev. Lett. 14, 973-976 (1965).
[CrossRef]

M. D. Lukin, P. R. Hemmer, M. Löffler, and M. O. Scully, "Resonant enhancement of parametric processes via radiative interference and induced coherence," Phys. Rev. Lett. 81, 2675-2678 (1998).
[CrossRef]

A. S. Zibrov, M. D. Lukin, and M. O. Scully, "Nondegenerate parametric self-oscillation via multiwave mixing in coherent atomic media," Phys. Rev. Lett. 83, 4049-4052 (1999).
[CrossRef]

M. D. Lukin, A. B. Matsko, M. Fleischhauer, and M. O. Scully, "Quantum noise and correlations in resonantly enhanced wave mixing based on atomic coherence," Phys. Rev. Lett. 82, 1847-1850 (1999).
[CrossRef]

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, 2409-2412 (1985).
[CrossRef] [PubMed]

Science (2)

C. H. van der Wal, M. D. Eisaman, A. André, R. L. Walsworth, D. F. Phillips, A. S. Zibrov, and M. D. Lukin, "Atomic memory for correlated photon states," Science 301, 196-200 (2003).
[CrossRef] [PubMed]

A. M. C. Dawes, L. Illing, S. M. Clark, and D. J. Gauthier, "All-optical switching in rubidium vapor," Science 308, 672-674 (2005).
[CrossRef] [PubMed]

Other (1)

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge U. Press, 1997).

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Figures (7)

Fig. 1
Fig. 1

Energy-level diagram of a three-level Λ atom interacting with pump ( ω L ) , Stokes ( ω S ) and anti-Stokes ( ω A S ) fields.

Fig. 2
Fig. 2

(a) Configuration of the two input beams (forward and backward) and the four output beams (F1, F2, B1, and B2) in this experiment. Four EID elementary processes show (b) EIT, (c) forward diffraction (FD), (d) backward diffraction (BD), and (e) phase conjugation (PC).

Fig. 3
Fig. 3

Analytical solutions [Eq. (6)] for the four signal waves as a function of distance z along the sample. Arrows indicate the propagation directions. Parameters are a 0 = 0.1 cm 1 , c E I T = q , c F D = 1.2 q , c B D = 0.5 q , and c P C = 2.0 q . (a) Weak pump case when q = 0.03 cm 1 . (b) Medium pump case when q = 0.052 cm 1 . (c) Strong pump case right before the threshold q = 0.056 cm 1 . Notice that the vertical scales are totally different for the three figure parts.

Fig. 4
Fig. 4

(a) Schematic of the experiment. PBSs, polarizing beam splitters; SMFs, single-mode fibers; PDs, photo-detectors. (b) and (c) Typical pictures of forward and backward beam profiles when (a) the seed beam is absent and (b) the seed beam is present.

Fig. 5
Fig. 5

Output powers of the four signal waves versus seed-to-coupling frequency difference ω 0 2 π , when probe is seeded. Vertical line indicates ω 21 2 π = 1771.6 MHz . Filled circles, Δ ν L = 600 MHz ; open circles; Δ ν L = 420 MHz .

Fig. 6
Fig. 6

(a) Output powers of the four signal waves versus input probe power, when the probe is seeded; Δ ν L = 600 MHz . (b) Output powers of the four signal waves versus input pump power (forward or backward), when probe is seeded; Δ ν L = 560 MHz .

Fig. 7
Fig. 7

(a) Typical temporal waveforms of the four signals when the seed beam is absent. Cross correlations of (b) F1 versus F2 and (c) F2 versus B1.

Tables (1)

Tables Icon

Table 1 Typical Input and Output Powers for the Two Methods

Equations (31)

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

{ ρ ̇ 13 = ( i ω 31 γ ) ρ 13 i H 13 ( n 3 n 1 ) + i H 23 ρ 12 ρ ̇ 23 = ( i ω 32 γ ) ρ 23 i H 23 ( n 3 n 2 ) + i H 13 ρ 21 , ρ ̇ 12 = ( i ω 21 γ s ) ρ 12 i H 13 ρ 32 + i H 32 ρ 13
H 3 m = p 3 m j [ E j exp [ i ( k j r ω j t ) ] + c.c. ] ,
z ( E F 1 E F 2 * E B 1 E B 2 * ) = ( α F 1 2 c F 1 F D E F 2 c F 1 B D E B * E F c F 1 P C E B E F ( c F 2 F D ) * E F * 2 α F 2 * 2 ( c F 2 P C ) * E B * E F * ( c F 2 B D ) * E B E F * c B 1 B D E B E F * c B 1 P C E B E F α B 1 2 c B 1 F D E B 2 ( c B 2 P C ) * E B * E F * ( c B 2 B D ) * E B * E F ( c B 2 F D ) * E B * 2 α B 2 * 2 ) ( E F 1 E F 2 * E B 1 E B 2 * ) ,
α j = α j 2 c j E I T E F 2 ( j = F 1 , F 2 ) ,
α j = α j 2 c j E I T E B 2 ( j = B 1 , B 2 ) ,
c F 1 E I T = β δ s δ F 1 , 1 ( n 1 δ F 1 , 1 + n 2 δ F , 2 * ) , c B 1 E I T = β δ s δ B 1 , 1 ( n 1 δ B 1 , 1 + n 2 δ B , 2 * ) ,
( c F 2 E I T ) * = β δ s δ F 2 , 2 * ( n 1 δ F , 1 + n 2 δ F 2 , 2 * ) ,
( c B 2 E I T ) * = β δ s δ B 2 , 2 * ( n 1 δ B , 1 + n 2 δ B 2 , 2 * ) ,
c F 1 F D = β δ s δ F 1 , 1 ( n 1 δ F , 1 + n 2 δ F 2 , 2 * ) , c B 1 F D = β δ s δ B 1 , 1 ( n 1 δ B , 1 + n 2 δ B 2 , 2 * ) ,
( c F 2 F D ) * = β δ s δ F 2 , 2 * ( n 1 δ F 1 , 1 + n 2 δ F , 2 * ) ,
( c B 2 F D ) * = β δ s δ B 2 , 2 * ( n 1 δ B 1 , 1 + n 2 δ B , 2 * ) ,
c F 1 B D = β δ s δ F 1 , 1 ( n 1 δ B 1 , 1 + n 2 δ B , 2 * ) , c B 1 B D = β δ s δ B 1 , 1 ( n 1 δ F 1 , 1 + n 2 δ F , 2 * ) ,
( c F 2 B D ) * = β δ s δ F 2 , 2 * ( n 1 δ B , 1 + n 2 δ B 2 , 2 * ) ,
( c B 2 B D ) * = β δ s δ B 2 , 2 * ( n 1 δ F , 1 + n 2 δ F 2 , 2 * ) ,
c F 1 P C = β δ s δ F 1 , 1 ( n 1 δ B , 1 + n 2 δ B 2 , 2 * ) , c B 1 P C = β δ s δ B 1 , 1 ( n 1 δ F , 1 + n 2 δ F 2 , 2 * ) ,
( c F 2 P C ) * = β δ s δ F 2 , 2 * ( n 1 δ B 1 , 1 + n 2 δ B , 2 * ) ,
( c B 2 P C ) * = β δ s δ B 2 , 2 * ( n 1 δ F 1 , 1 + n 2 δ F , 2 * ) ,
α j = α 1 n 1 δ j 1 + α 2 n 2 δ j 2 , α m = k N p 3 m 2 ϵ 0 γ , β = k N p 31 2 p 32 2 2 ϵ 0 3 γ 2 γ s ,
δ s = 1 + I F , 2 δ F 1 , 1 + I F , 1 δ F 2 , 2 * + I B , 2 δ B 1 , 1 + I B , 1 δ B 2 , 2 * i ω 0 ω 21 γ s ,
δ j , m = 1 i ω j ω 3 m γ ,
I j , m = Ω j m 2 4 γ γ s , Ω j m = 2 p 3 m E j , ω 0 = ω A S ω L ,
z ( E F 1 E F 2 * E B 1 E B 2 * ) = ( a c F D c B D c P C c F D a c P C c B D c B D c P C a c F D c P C c B D c F D a ) ( E F 1 E F 2 * E B 1 E B 2 * ) .
{ E F 1 ( z ) = E 0 [ f 1 ( z ) + g 1 ( z ) ] 2 E F 2 * ( z ) = E 0 [ f 1 ( z ) g 1 ( z ) ] 2 E B 1 ( z ) = E 0 [ f 2 ( z ) + g 2 ( z ) ] 2 E B 2 * ( z ) = E 0 [ f 2 ( z ) g 2 ( z ) ] 2 ,
f 1 ( z ) = [ ( a c F D ) sin [ s ( L z ) ] + s cos [ s ( L z ) ] ] F ,
g 1 ( z ) = [ ( a + c F D ) sinh [ β ( L z ) ] + β cosh [ β ( L z ) ] ] G ,
f 2 ( z ) = ( c B D + c P C ) sin [ s ( L z ) ] F ,
g 2 ( z ) = ( c B D c P C ) sinh [ β ( L z ) ] G ,
s = ( c B D + c P C ) 2 ( a c F D ) 2 ,
β = ( a + c F D ) 2 ( c B D c P C ) 2 ,
F = ( a c F D ) sin ( s L ) + s cos ( s L ) ,
G = ( a + c F D ) sinh ( β L ) + β cosh ( β L ) .

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