Abstract

We introduce here double electromagnetically induced grating (DEIG) using a tripod atomic structure, wherein two probe and signal fields with different frequencies simultaneously experience an atomic grating. Properties of presented DEIG can be substantially modified by the detuning of the applied fields. It has also been found that applying an incoherent pump field has a remarkable influence on the high-order diffraction efficiencies. Amplification of travelling weak lights via the incoherent pump field results in large diffraction efficiencies in the first-order and second-order directions. Such a novel scheme might open up the possibility for designing a two-qubit switch that would be advantageous to quantum information processing and quantum networking.

© 2020 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Beam splitter and router via an incoherent pump-assisted electromagnetically induced blazed grating

Yu-Yuan Chen, Zhuan-Zhuan Liu, and Ren-Gang Wan
Appl. Opt. 56(20) 5736-5744 (2017)

Phase-controlled electromagnetically induced symmetric and asymmetric grating in an asymmetric three-coupled quantum well

Ali Akbar Naeimi, Elham Darabi, Ali Mortezapour, and Ghasem Naeimi
Appl. Opt. 58(35) 9662-9669 (2019)

Electromagnetically induced grating in the microwave-driven four-level atomic systems

Rasoul Sadighi-Bonabi, Tayebeh Naseri, and Morteza Navadeh-Toupchi
Appl. Opt. 54(3) 368-377 (2015)

References

  • View by:
  • |
  • |
  • |

  1. K. J. Boller, A. Imamoglu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett. 66, 2593–2596 (1991).
    [Crossref]
  2. S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997).
    [Crossref]
  3. S. E. Harris, “Lasers without inversion: interference of lifetime-broadened resonances,” Phys. Rev. Lett. 62, 1033–1036 (1989).
    [Crossref]
  4. M. O. Scully, S.-Y. Zhu, and A. Gavrielides, “Degenerate quantum-beat laser: lasing without inversion and inversion without lasing,” Phys. Rev. Lett. 62, 2813–2816 (1989).
    [Crossref]
  5. M. O. Scully, “Enhancement of the index of refraction via quantum coherence,” Phys. Rev. Lett. 67, 1855–1858 (1991).
    [Crossref]
  6. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
    [Crossref]
  7. S. E. Schwarz and T. Y. Tan, “Wave interactions in saturable absorbers,” Appl. Phys. Lett. 10, 4 (1967).
    [Crossref]
  8. G. S. Agarwal and T. N. Dey, “Non-electromagnetically induced transparency mechanisms for slow light,” Laser Photon. Rev. 3, 287–300 (2009).
    [Crossref]
  9. M. Fleischhauer and M. D. Lukin, “Dark-state polaritons in electromagnetically induced transparency,” Phys. Rev. Lett. 84, 5094–5097 (2000).
    [Crossref]
  10. M. A. Maynard, F. Bretenaker, and F. Goldfarb, “Light storage in a room-temperature atomic vapor based on coherent population oscillations,” Phys. Rev. A 90, 061801 (2014).
    [Crossref]
  11. K. T. Kapale and G. S. Agarwal, “Subnanoscale resolution for microscopy via coherent population trapping,” Opt. Lett. 35, 2792–2794 (2010).
    [Crossref]
  12. M. O. Scully and M. Fleischhauer, “High-sensitivity magnetometer based on index-enhanced media,” Phys. Rev. Lett. 69, 1360–1363 (1992).
    [Crossref]
  13. H. Y. Ling, Y. Q. Li, and M. Xiao, “Electromagnetically induced grating: Homogeneously broadened medium,” Phys. Rev. A 57, 1338–1344 (1998).
    [Crossref]
  14. S. Q. Kuang, R. G. Wan, J. Kou, Y. Jiang, and J. Y. Gao, “Tunable double photonic bandgaps in a homogeneous atomic medium,” J. Opt. Soc. Am. B 27, 1518–1522 (2010).
    [Crossref]
  15. M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature 426, 638–641 (2003).
    [Crossref]
  16. D. Moretti, D. Felinto, J. W. R. Tabosa, and A. Lezama, “Dynamics of a stored Zeeman coherence grating in an external magnetic field,” J. Phys. B 43, 115502 (2010).
    [Crossref]
  17. L. Zhao, W. Duan, and S. F. Yelin, “All-optical beam control with high speed using image-induced blazed gratings in coherent media,” Phys. Rev. A 82, 013809 (2010).
    [Crossref]
  18. F. Wen, W. Wang, I. Ahmed, H. Wang, Y. Zhang, Y. Zhang, A. R. Mahesar, and M. Xiao, “Two-dimensional Talbot self-imaging via electromagnetically induced lattice,” Sci. Rep. 7, 41790 (2017).
    [Crossref]
  19. L. E. E. de Araujo, “Electromagnetically induced phase grating,” Opt. Lett. 35, 977–979 (2010).
    [Crossref]
  20. S. A. Carvalho and L. E. E. de Araujo, “Electromagnetically induced grating with maximal atomic coherence,” Phys. Rev. A 84, 043850 (2011).
    [Crossref]
  21. F. X. Zhou, Y. H. Qi, H. Sun, D. J. Chen, J. Yang, Y. P. Niu, and S. Q. Gong, “Electromagnetically induced grating in asymmetric quantum wells via Fano interference,” Opt. Express 21, 12249–12259 (2013).
    [Crossref]
  22. Y. M. Liu, F. Gao, C. H. Fan, and J. H. Wu, “Asymmetric light diffraction of an atomic grating with PT symmetry,” Opt. Lett. 42, 4283–4286 (2017).
    [Crossref]
  23. A. Vafafard and M. Mahmoudi, “Switching from electromagnetically induced absorption grating to electromagnetically induced phase grating in a closed-loop atomic system,” Appl. Opt. 54, 10613–10617 (2015).
    [Crossref]
  24. L. Wang, F. X. Zhou, P. D. Hu, Y. P. Niu, and S. Q. Gong, “Two-dimensional electromagnetically induced cross-grating in a four-level tripod-type atomic system,” J. Phys. B 47, 225501 (2014).
    [Crossref]
  25. A. Vafafard and M. Sahrai, “Electromagnetically induced grating based on Zeeman coherence oscillations in cases beyond the multi-photon resonance condition,” J. Opt. Soc. Am. B 35, 2118–2124 (2018).
    [Crossref]
  26. J. Wu and B. Ai, “Two-dimensional electromagnetically induced cross-grating in a four-level N-type atomic system,” J. Phys. B 48, 115504 (2015).
    [Crossref]
  27. J. C. Wu and T. T. Hu, “Two-dimensional electromagnetically induced gain-phase grating with an incoherent pump field,” Laser Phys. Lett. 15, 065202 (2018).
    [Crossref]
  28. Y. Y. Chen, Z. Z. Liu, and R. G. Wan, “Electromagnetically induced two-dimensional grating assisted by incoherent pump,” Phys. Lett. A 381, 1362–1368 (2017).
    [Crossref]
  29. E. Paspalakis and P. L. Knight, “Electromagnetically induced transparency and controlled group velocity in a multilevel system,” Phys. Rev. A 66, 015802 (2002).
    [Crossref]
  30. S. Rebić, D. Vitali, C. Ottaviani, P. Tombesi, M. Artoni, F. Cataliotti, and R. Corbalán, “Polarization phase gate with a tripod atomic system,” Phys. Rev. A 70, 032317 (2004).
    [Crossref]
  31. A. Joshi and M. Xiao, “Phase gate with a four-level inverted-Y system,” Phys. Rev. A 72, 062319 (2005).
    [Crossref]
  32. Z.-B. Wang, K.-P. Marzlin, and B. C. Sanders, “Large cross-phase modulation between slow copropagating weak pulses in 87Rb,” Phys. Rev. Lett. 97, 063901 (2006).
    [Crossref]
  33. C. Ottaviani, S. Rebić, D. Vitali, and P. Tombesi, “Quantum phase-gate operation based on nonlinear optics: full quantum analysis,” Phys. Rev. A 73, 010301 (2006).
    [Crossref]
  34. A. MacRae, G. Campbell, and A. I. Lvovsky, “Matched slow pulses using double electromagnetically induced transparency,” Opt. Lett. 33, 2659–2661 (2008).
    [Crossref]
  35. H. M. M. Alotaibi and B. C. Sanders, “Double-double electromagnetically induced transparency with amplification,” Phys. Rev. A 89, 021802 (2014).
    [Crossref]
  36. H. M. M. Alotaibi and B. C. Sanders, “Slowing the probe field in the second window of double-double electromagnetically induced transparency,” Phys. Rev. A 91, 043817 (2015).
    [Crossref]
  37. A. W. Brown and M. Xiao, “All-optical switching and routing based on an electromagnetically induced absorption grating,” Opt. Lett. 30, 699–701 (2005).
    [Crossref]
  38. K. Kapale, M. Scully, S. Y. Zhu, and M. Zubairy, “Quenching of spontaneous emission through interference of incoherent pump processes,” Phys. Rev. A 67, 023804 (2003).
    [Crossref]
  39. Y. R. Shen and N. Bloembergen, “Theory of stimulated Brillouin and Raman scattering,” Phys. Rev. A 137, A1787–A1805 (1965).
    [Crossref]

2018 (2)

A. Vafafard and M. Sahrai, “Electromagnetically induced grating based on Zeeman coherence oscillations in cases beyond the multi-photon resonance condition,” J. Opt. Soc. Am. B 35, 2118–2124 (2018).
[Crossref]

J. C. Wu and T. T. Hu, “Two-dimensional electromagnetically induced gain-phase grating with an incoherent pump field,” Laser Phys. Lett. 15, 065202 (2018).
[Crossref]

2017 (3)

Y. Y. Chen, Z. Z. Liu, and R. G. Wan, “Electromagnetically induced two-dimensional grating assisted by incoherent pump,” Phys. Lett. A 381, 1362–1368 (2017).
[Crossref]

Y. M. Liu, F. Gao, C. H. Fan, and J. H. Wu, “Asymmetric light diffraction of an atomic grating with PT symmetry,” Opt. Lett. 42, 4283–4286 (2017).
[Crossref]

F. Wen, W. Wang, I. Ahmed, H. Wang, Y. Zhang, Y. Zhang, A. R. Mahesar, and M. Xiao, “Two-dimensional Talbot self-imaging via electromagnetically induced lattice,” Sci. Rep. 7, 41790 (2017).
[Crossref]

2015 (3)

A. Vafafard and M. Mahmoudi, “Switching from electromagnetically induced absorption grating to electromagnetically induced phase grating in a closed-loop atomic system,” Appl. Opt. 54, 10613–10617 (2015).
[Crossref]

J. Wu and B. Ai, “Two-dimensional electromagnetically induced cross-grating in a four-level N-type atomic system,” J. Phys. B 48, 115504 (2015).
[Crossref]

H. M. M. Alotaibi and B. C. Sanders, “Slowing the probe field in the second window of double-double electromagnetically induced transparency,” Phys. Rev. A 91, 043817 (2015).
[Crossref]

2014 (3)

L. Wang, F. X. Zhou, P. D. Hu, Y. P. Niu, and S. Q. Gong, “Two-dimensional electromagnetically induced cross-grating in a four-level tripod-type atomic system,” J. Phys. B 47, 225501 (2014).
[Crossref]

M. A. Maynard, F. Bretenaker, and F. Goldfarb, “Light storage in a room-temperature atomic vapor based on coherent population oscillations,” Phys. Rev. A 90, 061801 (2014).
[Crossref]

H. M. M. Alotaibi and B. C. Sanders, “Double-double electromagnetically induced transparency with amplification,” Phys. Rev. A 89, 021802 (2014).
[Crossref]

2013 (1)

2011 (1)

S. A. Carvalho and L. E. E. de Araujo, “Electromagnetically induced grating with maximal atomic coherence,” Phys. Rev. A 84, 043850 (2011).
[Crossref]

2010 (5)

K. T. Kapale and G. S. Agarwal, “Subnanoscale resolution for microscopy via coherent population trapping,” Opt. Lett. 35, 2792–2794 (2010).
[Crossref]

S. Q. Kuang, R. G. Wan, J. Kou, Y. Jiang, and J. Y. Gao, “Tunable double photonic bandgaps in a homogeneous atomic medium,” J. Opt. Soc. Am. B 27, 1518–1522 (2010).
[Crossref]

L. E. E. de Araujo, “Electromagnetically induced phase grating,” Opt. Lett. 35, 977–979 (2010).
[Crossref]

D. Moretti, D. Felinto, J. W. R. Tabosa, and A. Lezama, “Dynamics of a stored Zeeman coherence grating in an external magnetic field,” J. Phys. B 43, 115502 (2010).
[Crossref]

L. Zhao, W. Duan, and S. F. Yelin, “All-optical beam control with high speed using image-induced blazed gratings in coherent media,” Phys. Rev. A 82, 013809 (2010).
[Crossref]

2009 (1)

G. S. Agarwal and T. N. Dey, “Non-electromagnetically induced transparency mechanisms for slow light,” Laser Photon. Rev. 3, 287–300 (2009).
[Crossref]

2008 (1)

2006 (2)

Z.-B. Wang, K.-P. Marzlin, and B. C. Sanders, “Large cross-phase modulation between slow copropagating weak pulses in 87Rb,” Phys. Rev. Lett. 97, 063901 (2006).
[Crossref]

C. Ottaviani, S. Rebić, D. Vitali, and P. Tombesi, “Quantum phase-gate operation based on nonlinear optics: full quantum analysis,” Phys. Rev. A 73, 010301 (2006).
[Crossref]

2005 (2)

2004 (1)

S. Rebić, D. Vitali, C. Ottaviani, P. Tombesi, M. Artoni, F. Cataliotti, and R. Corbalán, “Polarization phase gate with a tripod atomic system,” Phys. Rev. A 70, 032317 (2004).
[Crossref]

2003 (2)

K. Kapale, M. Scully, S. Y. Zhu, and M. Zubairy, “Quenching of spontaneous emission through interference of incoherent pump processes,” Phys. Rev. A 67, 023804 (2003).
[Crossref]

M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature 426, 638–641 (2003).
[Crossref]

2002 (1)

E. Paspalakis and P. L. Knight, “Electromagnetically induced transparency and controlled group velocity in a multilevel system,” Phys. Rev. A 66, 015802 (2002).
[Crossref]

2000 (1)

M. Fleischhauer and M. D. Lukin, “Dark-state polaritons in electromagnetically induced transparency,” Phys. Rev. Lett. 84, 5094–5097 (2000).
[Crossref]

1999 (1)

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[Crossref]

1998 (1)

H. Y. Ling, Y. Q. Li, and M. Xiao, “Electromagnetically induced grating: Homogeneously broadened medium,” Phys. Rev. A 57, 1338–1344 (1998).
[Crossref]

1997 (1)

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997).
[Crossref]

1992 (1)

M. O. Scully and M. Fleischhauer, “High-sensitivity magnetometer based on index-enhanced media,” Phys. Rev. Lett. 69, 1360–1363 (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]

M. O. Scully, “Enhancement of the index of refraction via quantum coherence,” Phys. Rev. Lett. 67, 1855–1858 (1991).
[Crossref]

1989 (2)

S. E. Harris, “Lasers without inversion: interference of lifetime-broadened resonances,” Phys. Rev. Lett. 62, 1033–1036 (1989).
[Crossref]

M. O. Scully, S.-Y. Zhu, and A. Gavrielides, “Degenerate quantum-beat laser: lasing without inversion and inversion without lasing,” Phys. Rev. Lett. 62, 2813–2816 (1989).
[Crossref]

1967 (1)

S. E. Schwarz and T. Y. Tan, “Wave interactions in saturable absorbers,” Appl. Phys. Lett. 10, 4 (1967).
[Crossref]

1965 (1)

Y. R. Shen and N. Bloembergen, “Theory of stimulated Brillouin and Raman scattering,” Phys. Rev. A 137, A1787–A1805 (1965).
[Crossref]

Agarwal, G. S.

K. T. Kapale and G. S. Agarwal, “Subnanoscale resolution for microscopy via coherent population trapping,” Opt. Lett. 35, 2792–2794 (2010).
[Crossref]

G. S. Agarwal and T. N. Dey, “Non-electromagnetically induced transparency mechanisms for slow light,” Laser Photon. Rev. 3, 287–300 (2009).
[Crossref]

Ahmed, I.

F. Wen, W. Wang, I. Ahmed, H. Wang, Y. Zhang, Y. Zhang, A. R. Mahesar, and M. Xiao, “Two-dimensional Talbot self-imaging via electromagnetically induced lattice,” Sci. Rep. 7, 41790 (2017).
[Crossref]

Ai, B.

J. Wu and B. Ai, “Two-dimensional electromagnetically induced cross-grating in a four-level N-type atomic system,” J. Phys. B 48, 115504 (2015).
[Crossref]

Alotaibi, H. M. M.

H. M. M. Alotaibi and B. C. Sanders, “Slowing the probe field in the second window of double-double electromagnetically induced transparency,” Phys. Rev. A 91, 043817 (2015).
[Crossref]

H. M. M. Alotaibi and B. C. Sanders, “Double-double electromagnetically induced transparency with amplification,” Phys. Rev. A 89, 021802 (2014).
[Crossref]

Artoni, M.

S. Rebić, D. Vitali, C. Ottaviani, P. Tombesi, M. Artoni, F. Cataliotti, and R. Corbalán, “Polarization phase gate with a tripod atomic system,” Phys. Rev. A 70, 032317 (2004).
[Crossref]

Bajcsy, M.

M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature 426, 638–641 (2003).
[Crossref]

Behroozi, C. H.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[Crossref]

Bloembergen, N.

Y. R. Shen and N. Bloembergen, “Theory of stimulated Brillouin and Raman scattering,” Phys. Rev. A 137, A1787–A1805 (1965).
[Crossref]

Boller, K. J.

K. J. Boller, A. Imamoglu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett. 66, 2593–2596 (1991).
[Crossref]

Bretenaker, F.

M. A. Maynard, F. Bretenaker, and F. Goldfarb, “Light storage in a room-temperature atomic vapor based on coherent population oscillations,” Phys. Rev. A 90, 061801 (2014).
[Crossref]

Brown, A. W.

Campbell, G.

Carvalho, S. A.

S. A. Carvalho and L. E. E. de Araujo, “Electromagnetically induced grating with maximal atomic coherence,” Phys. Rev. A 84, 043850 (2011).
[Crossref]

Cataliotti, F.

S. Rebić, D. Vitali, C. Ottaviani, P. Tombesi, M. Artoni, F. Cataliotti, and R. Corbalán, “Polarization phase gate with a tripod atomic system,” Phys. Rev. A 70, 032317 (2004).
[Crossref]

Chen, D. J.

Chen, Y. Y.

Y. Y. Chen, Z. Z. Liu, and R. G. Wan, “Electromagnetically induced two-dimensional grating assisted by incoherent pump,” Phys. Lett. A 381, 1362–1368 (2017).
[Crossref]

Corbalán, R.

S. Rebić, D. Vitali, C. Ottaviani, P. Tombesi, M. Artoni, F. Cataliotti, and R. Corbalán, “Polarization phase gate with a tripod atomic system,” Phys. Rev. A 70, 032317 (2004).
[Crossref]

de Araujo, L. E. E.

S. A. Carvalho and L. E. E. de Araujo, “Electromagnetically induced grating with maximal atomic coherence,” Phys. Rev. A 84, 043850 (2011).
[Crossref]

L. E. E. de Araujo, “Electromagnetically induced phase grating,” Opt. Lett. 35, 977–979 (2010).
[Crossref]

Dey, T. N.

G. S. Agarwal and T. N. Dey, “Non-electromagnetically induced transparency mechanisms for slow light,” Laser Photon. Rev. 3, 287–300 (2009).
[Crossref]

Duan, W.

L. Zhao, W. Duan, and S. F. Yelin, “All-optical beam control with high speed using image-induced blazed gratings in coherent media,” Phys. Rev. A 82, 013809 (2010).
[Crossref]

Dutton, Z.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[Crossref]

Fan, C. H.

Felinto, D.

D. Moretti, D. Felinto, J. W. R. Tabosa, and A. Lezama, “Dynamics of a stored Zeeman coherence grating in an external magnetic field,” J. Phys. B 43, 115502 (2010).
[Crossref]

Fleischhauer, M.

M. Fleischhauer and M. D. Lukin, “Dark-state polaritons in electromagnetically induced transparency,” Phys. Rev. Lett. 84, 5094–5097 (2000).
[Crossref]

M. O. Scully and M. Fleischhauer, “High-sensitivity magnetometer based on index-enhanced media,” Phys. Rev. Lett. 69, 1360–1363 (1992).
[Crossref]

Gao, F.

Gao, J. Y.

Gavrielides, A.

M. O. Scully, S.-Y. Zhu, and A. Gavrielides, “Degenerate quantum-beat laser: lasing without inversion and inversion without lasing,” Phys. Rev. Lett. 62, 2813–2816 (1989).
[Crossref]

Goldfarb, F.

M. A. Maynard, F. Bretenaker, and F. Goldfarb, “Light storage in a room-temperature atomic vapor based on coherent population oscillations,” Phys. Rev. A 90, 061801 (2014).
[Crossref]

Gong, S. Q.

L. Wang, F. X. Zhou, P. D. Hu, Y. P. Niu, and S. Q. Gong, “Two-dimensional electromagnetically induced cross-grating in a four-level tripod-type atomic system,” J. Phys. B 47, 225501 (2014).
[Crossref]

F. X. Zhou, Y. H. Qi, H. Sun, D. J. Chen, J. Yang, Y. P. Niu, and S. Q. Gong, “Electromagnetically induced grating in asymmetric quantum wells via Fano interference,” Opt. Express 21, 12249–12259 (2013).
[Crossref]

Harris, S. E.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[Crossref]

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997).
[Crossref]

K. J. Boller, A. Imamoglu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett. 66, 2593–2596 (1991).
[Crossref]

S. E. Harris, “Lasers without inversion: interference of lifetime-broadened resonances,” Phys. Rev. Lett. 62, 1033–1036 (1989).
[Crossref]

Hau, L. V.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[Crossref]

Hu, P. D.

L. Wang, F. X. Zhou, P. D. Hu, Y. P. Niu, and S. Q. Gong, “Two-dimensional electromagnetically induced cross-grating in a four-level tripod-type atomic system,” J. Phys. B 47, 225501 (2014).
[Crossref]

Hu, T. T.

J. C. Wu and T. T. Hu, “Two-dimensional electromagnetically induced gain-phase grating with an incoherent pump field,” Laser Phys. Lett. 15, 065202 (2018).
[Crossref]

Imamoglu, A.

K. J. Boller, A. Imamoglu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett. 66, 2593–2596 (1991).
[Crossref]

Jiang, Y.

Joshi, A.

A. Joshi and M. Xiao, “Phase gate with a four-level inverted-Y system,” Phys. Rev. A 72, 062319 (2005).
[Crossref]

Kapale, K.

K. Kapale, M. Scully, S. Y. Zhu, and M. Zubairy, “Quenching of spontaneous emission through interference of incoherent pump processes,” Phys. Rev. A 67, 023804 (2003).
[Crossref]

Kapale, K. T.

Knight, P. L.

E. Paspalakis and P. L. Knight, “Electromagnetically induced transparency and controlled group velocity in a multilevel system,” Phys. Rev. A 66, 015802 (2002).
[Crossref]

Kou, J.

Kuang, S. Q.

Lezama, A.

D. Moretti, D. Felinto, J. W. R. Tabosa, and A. Lezama, “Dynamics of a stored Zeeman coherence grating in an external magnetic field,” J. Phys. B 43, 115502 (2010).
[Crossref]

Li, Y. Q.

H. Y. Ling, Y. Q. Li, and M. Xiao, “Electromagnetically induced grating: Homogeneously broadened medium,” Phys. Rev. A 57, 1338–1344 (1998).
[Crossref]

Ling, H. Y.

H. Y. Ling, Y. Q. Li, and M. Xiao, “Electromagnetically induced grating: Homogeneously broadened medium,” Phys. Rev. A 57, 1338–1344 (1998).
[Crossref]

Liu, Y. M.

Liu, Z. Z.

Y. Y. Chen, Z. Z. Liu, and R. G. Wan, “Electromagnetically induced two-dimensional grating assisted by incoherent pump,” Phys. Lett. A 381, 1362–1368 (2017).
[Crossref]

Lukin, M. D.

M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature 426, 638–641 (2003).
[Crossref]

M. Fleischhauer and M. D. Lukin, “Dark-state polaritons in electromagnetically induced transparency,” Phys. Rev. Lett. 84, 5094–5097 (2000).
[Crossref]

Lvovsky, A. I.

MacRae, A.

Mahesar, A. R.

F. Wen, W. Wang, I. Ahmed, H. Wang, Y. Zhang, Y. Zhang, A. R. Mahesar, and M. Xiao, “Two-dimensional Talbot self-imaging via electromagnetically induced lattice,” Sci. Rep. 7, 41790 (2017).
[Crossref]

Mahmoudi, M.

Marzlin, K.-P.

Z.-B. Wang, K.-P. Marzlin, and B. C. Sanders, “Large cross-phase modulation between slow copropagating weak pulses in 87Rb,” Phys. Rev. Lett. 97, 063901 (2006).
[Crossref]

Maynard, M. A.

M. A. Maynard, F. Bretenaker, and F. Goldfarb, “Light storage in a room-temperature atomic vapor based on coherent population oscillations,” Phys. Rev. A 90, 061801 (2014).
[Crossref]

Moretti, D.

D. Moretti, D. Felinto, J. W. R. Tabosa, and A. Lezama, “Dynamics of a stored Zeeman coherence grating in an external magnetic field,” J. Phys. B 43, 115502 (2010).
[Crossref]

Niu, Y. P.

L. Wang, F. X. Zhou, P. D. Hu, Y. P. Niu, and S. Q. Gong, “Two-dimensional electromagnetically induced cross-grating in a four-level tripod-type atomic system,” J. Phys. B 47, 225501 (2014).
[Crossref]

F. X. Zhou, Y. H. Qi, H. Sun, D. J. Chen, J. Yang, Y. P. Niu, and S. Q. Gong, “Electromagnetically induced grating in asymmetric quantum wells via Fano interference,” Opt. Express 21, 12249–12259 (2013).
[Crossref]

Ottaviani, C.

C. Ottaviani, S. Rebić, D. Vitali, and P. Tombesi, “Quantum phase-gate operation based on nonlinear optics: full quantum analysis,” Phys. Rev. A 73, 010301 (2006).
[Crossref]

S. Rebić, D. Vitali, C. Ottaviani, P. Tombesi, M. Artoni, F. Cataliotti, and R. Corbalán, “Polarization phase gate with a tripod atomic system,” Phys. Rev. A 70, 032317 (2004).
[Crossref]

Paspalakis, E.

E. Paspalakis and P. L. Knight, “Electromagnetically induced transparency and controlled group velocity in a multilevel system,” Phys. Rev. A 66, 015802 (2002).
[Crossref]

Qi, Y. H.

Rebic, S.

C. Ottaviani, S. Rebić, D. Vitali, and P. Tombesi, “Quantum phase-gate operation based on nonlinear optics: full quantum analysis,” Phys. Rev. A 73, 010301 (2006).
[Crossref]

S. Rebić, D. Vitali, C. Ottaviani, P. Tombesi, M. Artoni, F. Cataliotti, and R. Corbalán, “Polarization phase gate with a tripod atomic system,” Phys. Rev. A 70, 032317 (2004).
[Crossref]

Sahrai, M.

Sanders, B. C.

H. M. M. Alotaibi and B. C. Sanders, “Slowing the probe field in the second window of double-double electromagnetically induced transparency,” Phys. Rev. A 91, 043817 (2015).
[Crossref]

H. M. M. Alotaibi and B. C. Sanders, “Double-double electromagnetically induced transparency with amplification,” Phys. Rev. A 89, 021802 (2014).
[Crossref]

Z.-B. Wang, K.-P. Marzlin, and B. C. Sanders, “Large cross-phase modulation between slow copropagating weak pulses in 87Rb,” Phys. Rev. Lett. 97, 063901 (2006).
[Crossref]

Schwarz, S. E.

S. E. Schwarz and T. Y. Tan, “Wave interactions in saturable absorbers,” Appl. Phys. Lett. 10, 4 (1967).
[Crossref]

Scully, M.

K. Kapale, M. Scully, S. Y. Zhu, and M. Zubairy, “Quenching of spontaneous emission through interference of incoherent pump processes,” Phys. Rev. A 67, 023804 (2003).
[Crossref]

Scully, M. O.

M. O. Scully and M. Fleischhauer, “High-sensitivity magnetometer based on index-enhanced media,” Phys. Rev. Lett. 69, 1360–1363 (1992).
[Crossref]

M. O. Scully, “Enhancement of the index of refraction via quantum coherence,” Phys. Rev. Lett. 67, 1855–1858 (1991).
[Crossref]

M. O. Scully, S.-Y. Zhu, and A. Gavrielides, “Degenerate quantum-beat laser: lasing without inversion and inversion without lasing,” Phys. Rev. Lett. 62, 2813–2816 (1989).
[Crossref]

Shen, Y. R.

Y. R. Shen and N. Bloembergen, “Theory of stimulated Brillouin and Raman scattering,” Phys. Rev. A 137, A1787–A1805 (1965).
[Crossref]

Sun, H.

Tabosa, J. W. R.

D. Moretti, D. Felinto, J. W. R. Tabosa, and A. Lezama, “Dynamics of a stored Zeeman coherence grating in an external magnetic field,” J. Phys. B 43, 115502 (2010).
[Crossref]

Tan, T. Y.

S. E. Schwarz and T. Y. Tan, “Wave interactions in saturable absorbers,” Appl. Phys. Lett. 10, 4 (1967).
[Crossref]

Tombesi, P.

C. Ottaviani, S. Rebić, D. Vitali, and P. Tombesi, “Quantum phase-gate operation based on nonlinear optics: full quantum analysis,” Phys. Rev. A 73, 010301 (2006).
[Crossref]

S. Rebić, D. Vitali, C. Ottaviani, P. Tombesi, M. Artoni, F. Cataliotti, and R. Corbalán, “Polarization phase gate with a tripod atomic system,” Phys. Rev. A 70, 032317 (2004).
[Crossref]

Vafafard, A.

Vitali, D.

C. Ottaviani, S. Rebić, D. Vitali, and P. Tombesi, “Quantum phase-gate operation based on nonlinear optics: full quantum analysis,” Phys. Rev. A 73, 010301 (2006).
[Crossref]

S. Rebić, D. Vitali, C. Ottaviani, P. Tombesi, M. Artoni, F. Cataliotti, and R. Corbalán, “Polarization phase gate with a tripod atomic system,” Phys. Rev. A 70, 032317 (2004).
[Crossref]

Wan, R. G.

Y. Y. Chen, Z. Z. Liu, and R. G. Wan, “Electromagnetically induced two-dimensional grating assisted by incoherent pump,” Phys. Lett. A 381, 1362–1368 (2017).
[Crossref]

S. Q. Kuang, R. G. Wan, J. Kou, Y. Jiang, and J. Y. Gao, “Tunable double photonic bandgaps in a homogeneous atomic medium,” J. Opt. Soc. Am. B 27, 1518–1522 (2010).
[Crossref]

Wang, H.

F. Wen, W. Wang, I. Ahmed, H. Wang, Y. Zhang, Y. Zhang, A. R. Mahesar, and M. Xiao, “Two-dimensional Talbot self-imaging via electromagnetically induced lattice,” Sci. Rep. 7, 41790 (2017).
[Crossref]

Wang, L.

L. Wang, F. X. Zhou, P. D. Hu, Y. P. Niu, and S. Q. Gong, “Two-dimensional electromagnetically induced cross-grating in a four-level tripod-type atomic system,” J. Phys. B 47, 225501 (2014).
[Crossref]

Wang, W.

F. Wen, W. Wang, I. Ahmed, H. Wang, Y. Zhang, Y. Zhang, A. R. Mahesar, and M. Xiao, “Two-dimensional Talbot self-imaging via electromagnetically induced lattice,” Sci. Rep. 7, 41790 (2017).
[Crossref]

Wang, Z.-B.

Z.-B. Wang, K.-P. Marzlin, and B. C. Sanders, “Large cross-phase modulation between slow copropagating weak pulses in 87Rb,” Phys. Rev. Lett. 97, 063901 (2006).
[Crossref]

Wen, F.

F. Wen, W. Wang, I. Ahmed, H. Wang, Y. Zhang, Y. Zhang, A. R. Mahesar, and M. Xiao, “Two-dimensional Talbot self-imaging via electromagnetically induced lattice,” Sci. Rep. 7, 41790 (2017).
[Crossref]

Wu, J.

J. Wu and B. Ai, “Two-dimensional electromagnetically induced cross-grating in a four-level N-type atomic system,” J. Phys. B 48, 115504 (2015).
[Crossref]

Wu, J. C.

J. C. Wu and T. T. Hu, “Two-dimensional electromagnetically induced gain-phase grating with an incoherent pump field,” Laser Phys. Lett. 15, 065202 (2018).
[Crossref]

Wu, J. H.

Xiao, M.

F. Wen, W. Wang, I. Ahmed, H. Wang, Y. Zhang, Y. Zhang, A. R. Mahesar, and M. Xiao, “Two-dimensional Talbot self-imaging via electromagnetically induced lattice,” Sci. Rep. 7, 41790 (2017).
[Crossref]

A. W. Brown and M. Xiao, “All-optical switching and routing based on an electromagnetically induced absorption grating,” Opt. Lett. 30, 699–701 (2005).
[Crossref]

A. Joshi and M. Xiao, “Phase gate with a four-level inverted-Y system,” Phys. Rev. A 72, 062319 (2005).
[Crossref]

H. Y. Ling, Y. Q. Li, and M. Xiao, “Electromagnetically induced grating: Homogeneously broadened medium,” Phys. Rev. A 57, 1338–1344 (1998).
[Crossref]

Yang, J.

Yelin, S. F.

L. Zhao, W. Duan, and S. F. Yelin, “All-optical beam control with high speed using image-induced blazed gratings in coherent media,” Phys. Rev. A 82, 013809 (2010).
[Crossref]

Zhang, Y.

F. Wen, W. Wang, I. Ahmed, H. Wang, Y. Zhang, Y. Zhang, A. R. Mahesar, and M. Xiao, “Two-dimensional Talbot self-imaging via electromagnetically induced lattice,” Sci. Rep. 7, 41790 (2017).
[Crossref]

F. Wen, W. Wang, I. Ahmed, H. Wang, Y. Zhang, Y. Zhang, A. R. Mahesar, and M. Xiao, “Two-dimensional Talbot self-imaging via electromagnetically induced lattice,” Sci. Rep. 7, 41790 (2017).
[Crossref]

Zhao, L.

L. Zhao, W. Duan, and S. F. Yelin, “All-optical beam control with high speed using image-induced blazed gratings in coherent media,” Phys. Rev. A 82, 013809 (2010).
[Crossref]

Zhou, F. X.

L. Wang, F. X. Zhou, P. D. Hu, Y. P. Niu, and S. Q. Gong, “Two-dimensional electromagnetically induced cross-grating in a four-level tripod-type atomic system,” J. Phys. B 47, 225501 (2014).
[Crossref]

F. X. Zhou, Y. H. Qi, H. Sun, D. J. Chen, J. Yang, Y. P. Niu, and S. Q. Gong, “Electromagnetically induced grating in asymmetric quantum wells via Fano interference,” Opt. Express 21, 12249–12259 (2013).
[Crossref]

Zhu, S. Y.

K. Kapale, M. Scully, S. Y. Zhu, and M. Zubairy, “Quenching of spontaneous emission through interference of incoherent pump processes,” Phys. Rev. A 67, 023804 (2003).
[Crossref]

Zhu, S.-Y.

M. O. Scully, S.-Y. Zhu, and A. Gavrielides, “Degenerate quantum-beat laser: lasing without inversion and inversion without lasing,” Phys. Rev. Lett. 62, 2813–2816 (1989).
[Crossref]

Zibrov, A. S.

M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature 426, 638–641 (2003).
[Crossref]

Zubairy, M.

K. Kapale, M. Scully, S. Y. Zhu, and M. Zubairy, “Quenching of spontaneous emission through interference of incoherent pump processes,” Phys. Rev. A 67, 023804 (2003).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

S. E. Schwarz and T. Y. Tan, “Wave interactions in saturable absorbers,” Appl. Phys. Lett. 10, 4 (1967).
[Crossref]

J. Opt. Soc. Am. B (2)

J. Phys. B (3)

L. Wang, F. X. Zhou, P. D. Hu, Y. P. Niu, and S. Q. Gong, “Two-dimensional electromagnetically induced cross-grating in a four-level tripod-type atomic system,” J. Phys. B 47, 225501 (2014).
[Crossref]

J. Wu and B. Ai, “Two-dimensional electromagnetically induced cross-grating in a four-level N-type atomic system,” J. Phys. B 48, 115504 (2015).
[Crossref]

D. Moretti, D. Felinto, J. W. R. Tabosa, and A. Lezama, “Dynamics of a stored Zeeman coherence grating in an external magnetic field,” J. Phys. B 43, 115502 (2010).
[Crossref]

Laser Photon. Rev. (1)

G. S. Agarwal and T. N. Dey, “Non-electromagnetically induced transparency mechanisms for slow light,” Laser Photon. Rev. 3, 287–300 (2009).
[Crossref]

Laser Phys. Lett. (1)

J. C. Wu and T. T. Hu, “Two-dimensional electromagnetically induced gain-phase grating with an incoherent pump field,” Laser Phys. Lett. 15, 065202 (2018).
[Crossref]

Nature (2)

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[Crossref]

M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature 426, 638–641 (2003).
[Crossref]

Opt. Express (1)

Opt. Lett. (5)

Phys. Lett. A (1)

Y. Y. Chen, Z. Z. Liu, and R. G. Wan, “Electromagnetically induced two-dimensional grating assisted by incoherent pump,” Phys. Lett. A 381, 1362–1368 (2017).
[Crossref]

Phys. Rev. A (12)

E. Paspalakis and P. L. Knight, “Electromagnetically induced transparency and controlled group velocity in a multilevel system,” Phys. Rev. A 66, 015802 (2002).
[Crossref]

S. Rebić, D. Vitali, C. Ottaviani, P. Tombesi, M. Artoni, F. Cataliotti, and R. Corbalán, “Polarization phase gate with a tripod atomic system,” Phys. Rev. A 70, 032317 (2004).
[Crossref]

A. Joshi and M. Xiao, “Phase gate with a four-level inverted-Y system,” Phys. Rev. A 72, 062319 (2005).
[Crossref]

M. A. Maynard, F. Bretenaker, and F. Goldfarb, “Light storage in a room-temperature atomic vapor based on coherent population oscillations,” Phys. Rev. A 90, 061801 (2014).
[Crossref]

S. A. Carvalho and L. E. E. de Araujo, “Electromagnetically induced grating with maximal atomic coherence,” Phys. Rev. A 84, 043850 (2011).
[Crossref]

H. Y. Ling, Y. Q. Li, and M. Xiao, “Electromagnetically induced grating: Homogeneously broadened medium,” Phys. Rev. A 57, 1338–1344 (1998).
[Crossref]

L. Zhao, W. Duan, and S. F. Yelin, “All-optical beam control with high speed using image-induced blazed gratings in coherent media,” Phys. Rev. A 82, 013809 (2010).
[Crossref]

C. Ottaviani, S. Rebić, D. Vitali, and P. Tombesi, “Quantum phase-gate operation based on nonlinear optics: full quantum analysis,” Phys. Rev. A 73, 010301 (2006).
[Crossref]

H. M. M. Alotaibi and B. C. Sanders, “Double-double electromagnetically induced transparency with amplification,” Phys. Rev. A 89, 021802 (2014).
[Crossref]

H. M. M. Alotaibi and B. C. Sanders, “Slowing the probe field in the second window of double-double electromagnetically induced transparency,” Phys. Rev. A 91, 043817 (2015).
[Crossref]

K. Kapale, M. Scully, S. Y. Zhu, and M. Zubairy, “Quenching of spontaneous emission through interference of incoherent pump processes,” Phys. Rev. A 67, 023804 (2003).
[Crossref]

Y. R. Shen and N. Bloembergen, “Theory of stimulated Brillouin and Raman scattering,” Phys. Rev. A 137, A1787–A1805 (1965).
[Crossref]

Phys. Rev. Lett. (7)

M. O. Scully and M. Fleischhauer, “High-sensitivity magnetometer based on index-enhanced media,” Phys. Rev. Lett. 69, 1360–1363 (1992).
[Crossref]

M. Fleischhauer and M. D. Lukin, “Dark-state polaritons in electromagnetically induced transparency,” Phys. Rev. Lett. 84, 5094–5097 (2000).
[Crossref]

K. J. Boller, A. Imamoglu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett. 66, 2593–2596 (1991).
[Crossref]

S. E. Harris, “Lasers without inversion: interference of lifetime-broadened resonances,” Phys. Rev. Lett. 62, 1033–1036 (1989).
[Crossref]

M. O. Scully, S.-Y. Zhu, and A. Gavrielides, “Degenerate quantum-beat laser: lasing without inversion and inversion without lasing,” Phys. Rev. Lett. 62, 2813–2816 (1989).
[Crossref]

M. O. Scully, “Enhancement of the index of refraction via quantum coherence,” Phys. Rev. Lett. 67, 1855–1858 (1991).
[Crossref]

Z.-B. Wang, K.-P. Marzlin, and B. C. Sanders, “Large cross-phase modulation between slow copropagating weak pulses in 87Rb,” Phys. Rev. Lett. 97, 063901 (2006).
[Crossref]

Phys. Today (1)

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997).
[Crossref]

Sci. Rep. (1)

F. Wen, W. Wang, I. Ahmed, H. Wang, Y. Zhang, Y. Zhang, A. R. Mahesar, and M. Xiao, “Two-dimensional Talbot self-imaging via electromagnetically induced lattice,” Sci. Rep. 7, 41790 (2017).
[Crossref]

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1.
Fig. 1. (a) Sketch of experimental arrangement. (b) A schematic diagram of the tripod atomic system contains three lower states, $ | 1\rangle = 5 {S_{1/2}}(F = 1, {m_F} = 0) $, $ | 2\rangle = 5{S_{1/2}}(F = 2, {m_F} = 0) $, and $ | 3\rangle = 5{S_{1/2}}(F = 2, {m_F} = 2) $, and one excited state, $ | 4\rangle = 5p_{1/2}(F = 2, {m_F} = 1) $, in $ D1 $ line of $ ^{87}{\rm Rb} $.
Fig. 2.
Fig. 2. (a), (d) Absorption spectrum; (b), (e) corresponding amplitude of the transmission function; and (c), (f) the normalized diffraction intensity of the probe (first row) and signal fields (second row) for (a)–(c) $ {\delta _s} = 0 $, $ {\Omega _p} = 0.01 {\gamma _4} $, and $ {\Omega _s} = 0.1{\gamma _4} $, and (d, e, f) $ {\delta _p} = 0 $, $ {\Omega _s} = 0.01 {\gamma _4} $, and $ {\Omega _p} = 0.1{\gamma _4} $. Other parameters are $ {\gamma _{14}} = {\gamma _{24}} = {\gamma _{34}} = {\gamma _4}/3 $, $ \Omega = {\gamma _4} $, $ {\gamma _2} = 40\;{\rm kHz} $, $ {\gamma _3} = 10\;{\rm kHz} $, $ {\delta _c} = 0 $, $ L = 8 $, $ M = 5 $, and $ \Lambda /{\lambda _p} = 4 $; (b), (c) $ {\delta _p} = 0 $; and (e), (f) $ {\delta _s} = 0 $.
Fig. 3.
Fig. 3. (a) Amplitude and (b) phase parts of the transmission function and (c) normalized diffraction intensity of the probe field for $ {\delta _s} = 9\;{\rm MHz} $. Other parameters are the same as for Figs. 2(b) and 2(c).
Fig. 4.
Fig. 4. Amplitude (solid line) and phase (dashed line) parts of the transmission function of the probe field for $ R = 0.1\;{\rm MHz} $ and $ \delta s = 9\;{\rm MHz} $ in (a) $ \delta p = 0 $ and (b) $ {\delta _p} = 9\;{\rm MHz} $. (c) The corresponding normalized diffraction intensity as a function of $ \sin (\theta ) $. Other parameters are the same as for Figs. 2(b) and 2(c).
Fig. 5.
Fig. 5. (a) First-order and (b) second-order diffraction intensities as a function of the incoherent pump rate and $ {\delta _p} $ for $ {\delta _s} = 9\;{\rm MHz} $. Other parameters are the same as for Fig. 4(c).
Fig. 6.
Fig. 6. First-order diffraction intensity of the probe field for different values of $ {\delta _s} $. $ R = 0.1\;{\rm MHz} $, and other parameters are the same as for Fig. 4(c).
Fig. 7.
Fig. 7. Normalized diffraction intensity of the signal field as a function of $ (\sin \theta ) $ and $ {\delta _s} $ for $ {\delta _p} = 9\;{\rm MHz} $, (a) $ R = 0 $ and (b) $ R = 0.1\;{\rm MHz} $. Panel (b) is produced in a range from $ {\delta _s} = 8\;{\rm MHz} $ to $ {\delta _s} = 10\;{\rm MHz} $.

Equations (13)

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

H = 2 ( Ω p e i δ p t | 1 4 | + Ω c e i δ c t | 2 4 | + Ω s e i δ s t | 3 4 | + c . c . ) ,
ρ ˙ 11 = i 2 ( Ω p ρ 41 Ω p ρ 14 ) R ρ 11 + γ 41 ρ 44 , ρ ˙ 22 = i 2 Ω sin ( π x Λ ) ( ρ 42 ρ 24 ) + γ 42 ρ 44 , ρ ˙ 44 = i 2 ( Ω p ρ 41 Ω p ρ 14 ) + i 2 Ω sin ( π x Λ ) ( ρ 42 ρ 24 ) + i 2 ( Ω s ρ 43 Ω s ρ 34 ) + R ρ 11 γ 4 ρ 44 , ρ ˙ 12 = i ( δ p δ c ) ρ 12 i 2 ( Ω p ρ 42 Ω sin ( π x Λ ) ρ 14 ) 1 2 ( γ 2 + R ) ρ 12 , ρ ˙ 13 = i ( δ p δ s ) ρ 13 i 2 ( Ω p ρ 43 Ω s ρ 14 ) 1 2 ( γ 3 + R ) ρ 13 , ρ ˙ 14 = i δ p ρ 14 + i 2 ( Ω sin ( π x Λ ) ρ 12 + Ω s ρ 13 ) + i 2 Ω p ( ρ 11 ρ 44 ) 1 2 ( γ 4 + 2 R ) ρ 14 , ρ ˙ 23 = i ( δ c δ s ) ρ 23 i 2 ( Ω sin ( π x Λ ) ρ 43 Ω s ρ 24 ) 1 2 ( γ 2 + γ 3 ) ρ 23 , ρ ˙ 24 = i δ c ρ 24 + i 2 ( Ω p ρ 21 Ω s ρ 23 ) i 2 Ω sin ( π x Λ ) ( ρ 44 ρ 22 ) 1 2 ( γ 2 + γ 4 ) ρ 24 , ρ ˙ 34 = i δ s ρ 34 + i 2 ( Ω sin ( π x Λ ) ρ 32 + Ω p ρ 31 ) i 2 Ω s ( ρ 44 ρ 33 ) 1 2 ( ( γ 3 + γ 4 ) + R ) ρ 34 , ρ 11 + ρ 22 + ρ 33 + ρ 44 = 1 ,
ρ 14 = Ω p { i ( γ 2 γ 3 + 4 δ p δ ps ) 2 ( δ p γ 3 + δ ps γ 2 ) } A ,
ρ 34 = Ω s { i ( Γ 32 γ 3 + 4 δ s δ ps ) + 2 ( δ s γ 3 δ ps Γ 32 ) } B ,
A = 4 γ 4 δ p δ ps γ 3 | Ω c | 2 γ 2 γ 3 γ 4 γ 2 | Ω s | 2 + 2 i ( 4 δ p 2 δ ps + δ ps | Ω c | 2 + δ p | Ω s | 2 ) ,
B = 4 γ 4 δ s δ ps + γ 3 | Ω c | 2 Γ 32 γ 3 γ 4 + Γ 32 | Ω p | 2 + 2 i ( 4 δ s 2 δ ps + δ ps | Ω c | 2 δ s | Ω p | 2 ) .
ρ 14 = ( i γ 2 + 2 δ p ) ( 2 δ s | Ω s | 2 Ω p ) { R X + δ s Y } A C ,
X = ( 2 γ 4 δ s + 4 i δ s 2 i | Ω c | 2 ) Y = ( 2 γ 3 γ 4 4 i γ 4 δ ps ) C = δ s 2 ( 4 R γ 4 2 + 16 R δ s 2 8 R | Ω c | 2 + 32 R | Ω s | 2 + 4 γ 4 Ω s | 2 ) + R ( | Ω c | 4 + | Ω c | 2 | Ω s | 2 + 32 | Ω s | 2 ) .
χ p ( ω p ) = 2 α 0 γ 41 k p ρ 14 ( ω p ) Ω p ,
E p z = ( α + i β ) E p ,
T p ( x ) = e α ( x ) L + i β ( x ) L .
I p ( θ ) = | E p ( θ ) | 2 sin 2 ( M π Λ sin θ / λ p ) M 2 sin 2 ( π Λ sin θ / λ p ) .
E p ( θ ) = 0 1 T ( x ) e ( 2 i π x Λ sin θ / λ p ) d z .

Metrics