Abstract

An analytical method based on a two-mode approximation is here developed to study the optical response of a periodically modulated medium of ultracold atoms driven into a regime of standing-wave electromagnetically induced transparency. A systematic comparison with the usual approach based on the coupled Maxwell–Liouville equations shows that our method is very accurate in the frequency region of interest. Our method, in particular, explains in a straightforward manner the formation of a well-developed photonic bandgap in the optical Bloch wave vector dispersion. For ultracold Rb87 atoms nearly perfect reflectivity may be attained and a light pulse whose frequency components are contained within the gap is seen to be reflected with little loss and deformation.

© 2008 Optical Society of America

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  1. S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50, 36-42 (1997).
    [CrossRef]
  2. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633-673 (2005).
    [CrossRef]
  3. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meterspersecond in an ultracold atomic gas,” Nature (London) 397, 594-598 (1999).
    [CrossRef]
  4. A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2001).
    [CrossRef]
  5. M. Fleischhauer and M. D. Lukin, “Dark-state polaritons in electromagnetically induced transparency,” Phys. Rev. Lett. 84, 5094-5097 (2000).
    [CrossRef] [PubMed]
  6. C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature (London) 409, 490-493 (2001).
    [CrossRef]
  7. J. J. Longdell, E. Fraval, M. J. Sellars, and N. B. Manson, “Stopped light with storage times greater than one second using electromagnetically induced transparency in a solid,” Phys. Rev. Lett. 95, 063601 (2005).
    [CrossRef] [PubMed]
  8. S. E. Harris and L. V. Hau, “Nonlinear optics at low light levels,” Phys. Rev. Lett. 82, 4611-4614 (1999).
    [CrossRef]
  9. D. A. Braje, V. Balic, G. Y. Yin, and S. E. Harris, “Low-light-level nonlinear optics with slow light,” Phys. Rev. A 68, 041801(R) (2003).
    [CrossRef]
  10. A. B. Matsko, A. Novikova, G. R. Welch, and M. S. Zubairy, “Enhancement of Kerr nonlinearity by multiphoton coherence,” Opt. Lett. 28, 96-98 (2003).
    [CrossRef] [PubMed]
  11. C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization qubit phase gate in driven atomic media,” Phys. Rev. Lett. 90, 197902 (2003).
    [CrossRef] [PubMed]
  12. S. Rebic, D. Vitali, C. Ottaviani, P. Tombesi, M. Artoni, F. Cataliotti, and R. Corbalan, “Polarization phase gate with a tripod atomic system,” Phys. Rev. A 70, 032317 (2004).
    [CrossRef]
  13. S. E. Harris and Y. Yamamoto, “Photon switching by quantum interference,” Phys. Rev. Lett. 81, 3611-3614 (1998).
    [CrossRef]
  14. A. Andre and M. D. Lukin, “Manipulating light pulses via dynamically controlled photonic band gap,” Phys. Rev. Lett. 89, 143602 (2002).
    [CrossRef] [PubMed]
  15. M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature (London) 426, 638-641 (2003).
    [CrossRef]
  16. A. Andre, M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Nonlinear optics with stationary pulses of light,” Phys. Rev. Lett. 94, 063902 (2005).
    [CrossRef] [PubMed]
  17. B. S. Ham, “Spatiotemporal quantum manipulation of traveling light: quantum transport,” Appl. Phys. Lett. 88, 121117 (2006).
    [CrossRef]
  18. F. E. Zimmer, A. Andre, M. D. Lukin, and M. Fleischhauer, “Coherent control of stationary light pulses,” Opt. Commun. 264, 441-453 (2006).
    [CrossRef]
  19. S. A. Moiseev and B. S. Ham, “Generation of entangled lights with temporally reversed photon wave packets,” Phys. Rev. A 71, 053802 (2005).
    [CrossRef]
  20. K. R. Hansen and K. Molmer, “Trapping of light pulses in ensembles of stationary lambda atoms,” Phys. Rev. A 75, 053802 (2007).
    [CrossRef]
  21. K. R. Hansen and K. Molmer, “Stationary light pulses in ultracold atomic gases,” Phys. Rev. A 75, 065804 (2007).
    [CrossRef]
  22. K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2001).
  23. X.-M. Su and B. S. Ham, “Dynamic control of the photonic band gap using quantum coherence,” Phys. Rev. A 71, 013821 (2005).
    [CrossRef]
  24. D. Petrosyan, “Tunable photonic band gaps with coherently driven atoms in optical lattices,” Phys. Rev. A 76, 053823 (2007).
    [CrossRef]
  25. M. Artoni and G. C. La Rocca, “Optically tunable photonic stop bands in homogeneous absorbing media,” Phys. Rev. Lett. 96, 073905 (2006).
    [CrossRef] [PubMed]
  26. Q.-Y. He, Y. Xue, M. Artoni, G. C. La Rocca, J.-H. Xu, and J.-Y. Gao, “Coherently induced stop-bands in resonantly absorbing and inhomogeneously broadened doped crystals,” Phys. Rev. B 73, 195124 (2006).
    [CrossRef]
  27. Q.-Y. He, J.-H. Wu, T.-J. Wang, and J.-Y. Gao, “Dynamic control of the photonic stop bands formed by a standing wave in inhomogeneous broadening solids,” Phys. Rev. A 73, 053813 (2006).
    [CrossRef]
  28. J. P. Prineas, J. Y. Zhou, J. Kuhl, H. M. Gibbs, G. Khitrova, S. W. Koch, and A. Knorr, “Ultrafast ac Stark effect switching of the active photonic band gap from Bragg-periodic semiconductor quantum wells,” Appl. Phys. Lett. 81, 4332-4334 (2002).
    [CrossRef]
  29. S. M. Sadeghi, W. Li, and H. M. van Driel, “Coherently induced one-dimensional photonic band gap,” Phys. Rev. B 69, 073304 (2004).
    [CrossRef]
  30. F. Silva, J. Mompart, V. Ahufinger, and R. Corbalan, “Electromagnetically induced transparency in Doppler-broadened three-level systems with resonant standing-wave drive,” Europhys. Lett. 51, 286-292 (2000).
    [CrossRef]
  31. For the atomic samples considered here λc coincides, however, with the vacuum pump wave vector as the dielectric constant experienced by the pump is essentially unity.
  32. K. Riley, M. Hobson, and S. Bence, Mathematical Methods for Physics and Engineering, 3rd ed. (Cambridge U. Press, 2006).
  33. N. Ashcroft and D. Mermin, Solid State Physics (Saunders College, 1976).
  34. M. Artoni, G. La Rocca, and F. Bassani, “Resonantly absorbing one-dimensional photonic crystals,” Phys. Rev. E 72, 046604 (2005).
    [CrossRef]
  35. By replacing kp-->(ωp/c) and κ-->kc in Eq. in the simplified case where χ0 and χ1 are real and frequency independent, one obtains ωp2-->(kcc)2/(1+χ0∓χ1) yielding the upper and lower edge of the frequency stop band at the Brillouin zone boundary π/a. The width of such a photonic bandgap is directly proportional to χ1.
  36. M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999).
  37. The optical coherence in Eq. contributes with the two components ρ31{−1,0} while the probe polarization Pp has an expression analogous to the one for Ep in Eq. where the “forward” polarization component Pp+=ϵ0(χ0Ep++χ1Ep−) and a similar one for the “backward” component Pp−, which is obtained upon interchanging {+ ↔ −} in Pp+.

2007 (3)

D. Petrosyan, “Tunable photonic band gaps with coherently driven atoms in optical lattices,” Phys. Rev. A 76, 053823 (2007).
[CrossRef]

K. R. Hansen and K. Molmer, “Trapping of light pulses in ensembles of stationary lambda atoms,” Phys. Rev. A 75, 053802 (2007).
[CrossRef]

K. R. Hansen and K. Molmer, “Stationary light pulses in ultracold atomic gases,” Phys. Rev. A 75, 065804 (2007).
[CrossRef]

2006 (5)

M. Artoni and G. C. La Rocca, “Optically tunable photonic stop bands in homogeneous absorbing media,” Phys. Rev. Lett. 96, 073905 (2006).
[CrossRef] [PubMed]

Q.-Y. He, Y. Xue, M. Artoni, G. C. La Rocca, J.-H. Xu, and J.-Y. Gao, “Coherently induced stop-bands in resonantly absorbing and inhomogeneously broadened doped crystals,” Phys. Rev. B 73, 195124 (2006).
[CrossRef]

Q.-Y. He, J.-H. Wu, T.-J. Wang, and J.-Y. Gao, “Dynamic control of the photonic stop bands formed by a standing wave in inhomogeneous broadening solids,” Phys. Rev. A 73, 053813 (2006).
[CrossRef]

B. S. Ham, “Spatiotemporal quantum manipulation of traveling light: quantum transport,” Appl. Phys. Lett. 88, 121117 (2006).
[CrossRef]

F. E. Zimmer, A. Andre, M. D. Lukin, and M. Fleischhauer, “Coherent control of stationary light pulses,” Opt. Commun. 264, 441-453 (2006).
[CrossRef]

2005 (6)

S. A. Moiseev and B. S. Ham, “Generation of entangled lights with temporally reversed photon wave packets,” Phys. Rev. A 71, 053802 (2005).
[CrossRef]

A. Andre, M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Nonlinear optics with stationary pulses of light,” Phys. Rev. Lett. 94, 063902 (2005).
[CrossRef] [PubMed]

X.-M. Su and B. S. Ham, “Dynamic control of the photonic band gap using quantum coherence,” Phys. Rev. A 71, 013821 (2005).
[CrossRef]

M. Artoni, G. La Rocca, and F. Bassani, “Resonantly absorbing one-dimensional photonic crystals,” Phys. Rev. E 72, 046604 (2005).
[CrossRef]

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633-673 (2005).
[CrossRef]

J. J. Longdell, E. Fraval, M. J. Sellars, and N. B. Manson, “Stopped light with storage times greater than one second using electromagnetically induced transparency in a solid,” Phys. Rev. Lett. 95, 063601 (2005).
[CrossRef] [PubMed]

2004 (2)

S. M. Sadeghi, W. Li, and H. M. van Driel, “Coherently induced one-dimensional photonic band gap,” Phys. Rev. B 69, 073304 (2004).
[CrossRef]

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

2003 (4)

C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization qubit phase gate in driven atomic media,” Phys. Rev. Lett. 90, 197902 (2003).
[CrossRef] [PubMed]

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

D. A. Braje, V. Balic, G. Y. Yin, and S. E. Harris, “Low-light-level nonlinear optics with slow light,” Phys. Rev. A 68, 041801(R) (2003).
[CrossRef]

A. B. Matsko, A. Novikova, G. R. Welch, and M. S. Zubairy, “Enhancement of Kerr nonlinearity by multiphoton coherence,” Opt. Lett. 28, 96-98 (2003).
[CrossRef] [PubMed]

2002 (2)

A. Andre and M. D. Lukin, “Manipulating light pulses via dynamically controlled photonic band gap,” Phys. Rev. Lett. 89, 143602 (2002).
[CrossRef] [PubMed]

J. P. Prineas, J. Y. Zhou, J. Kuhl, H. M. Gibbs, G. Khitrova, S. W. Koch, and A. Knorr, “Ultrafast ac Stark effect switching of the active photonic band gap from Bragg-periodic semiconductor quantum wells,” Appl. Phys. Lett. 81, 4332-4334 (2002).
[CrossRef]

2001 (2)

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature (London) 409, 490-493 (2001).
[CrossRef]

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2001).
[CrossRef]

2000 (2)

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

F. Silva, J. Mompart, V. Ahufinger, and R. Corbalan, “Electromagnetically induced transparency in Doppler-broadened three-level systems with resonant standing-wave drive,” Europhys. Lett. 51, 286-292 (2000).
[CrossRef]

1999 (2)

S. E. Harris and L. V. Hau, “Nonlinear optics at low light levels,” Phys. Rev. Lett. 82, 4611-4614 (1999).
[CrossRef]

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meterspersecond in an ultracold atomic gas,” Nature (London) 397, 594-598 (1999).
[CrossRef]

1998 (1)

S. E. Harris and Y. Yamamoto, “Photon switching by quantum interference,” Phys. Rev. Lett. 81, 3611-3614 (1998).
[CrossRef]

1997 (1)

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

Ahufinger, V.

F. Silva, J. Mompart, V. Ahufinger, and R. Corbalan, “Electromagnetically induced transparency in Doppler-broadened three-level systems with resonant standing-wave drive,” Europhys. Lett. 51, 286-292 (2000).
[CrossRef]

Andre, A.

F. E. Zimmer, A. Andre, M. D. Lukin, and M. Fleischhauer, “Coherent control of stationary light pulses,” Opt. Commun. 264, 441-453 (2006).
[CrossRef]

A. Andre, M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Nonlinear optics with stationary pulses of light,” Phys. Rev. Lett. 94, 063902 (2005).
[CrossRef] [PubMed]

A. Andre and M. D. Lukin, “Manipulating light pulses via dynamically controlled photonic band gap,” Phys. Rev. Lett. 89, 143602 (2002).
[CrossRef] [PubMed]

Artoni, M.

M. Artoni and G. C. La Rocca, “Optically tunable photonic stop bands in homogeneous absorbing media,” Phys. Rev. Lett. 96, 073905 (2006).
[CrossRef] [PubMed]

Q.-Y. He, Y. Xue, M. Artoni, G. C. La Rocca, J.-H. Xu, and J.-Y. Gao, “Coherently induced stop-bands in resonantly absorbing and inhomogeneously broadened doped crystals,” Phys. Rev. B 73, 195124 (2006).
[CrossRef]

M. Artoni, G. La Rocca, and F. Bassani, “Resonantly absorbing one-dimensional photonic crystals,” Phys. Rev. E 72, 046604 (2005).
[CrossRef]

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

C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization qubit phase gate in driven atomic media,” Phys. Rev. Lett. 90, 197902 (2003).
[CrossRef] [PubMed]

Ashcroft, N.

N. Ashcroft and D. Mermin, Solid State Physics (Saunders College, 1976).

Bajcsy, M.

A. Andre, M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Nonlinear optics with stationary pulses of light,” Phys. Rev. Lett. 94, 063902 (2005).
[CrossRef] [PubMed]

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

Balic, V.

D. A. Braje, V. Balic, G. Y. Yin, and S. E. Harris, “Low-light-level nonlinear optics with slow light,” Phys. Rev. A 68, 041801(R) (2003).
[CrossRef]

Bassani, F.

M. Artoni, G. La Rocca, and F. Bassani, “Resonantly absorbing one-dimensional photonic crystals,” Phys. Rev. E 72, 046604 (2005).
[CrossRef]

Behroozi, C. H.

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature (London) 409, 490-493 (2001).
[CrossRef]

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meterspersecond in an ultracold atomic gas,” Nature (London) 397, 594-598 (1999).
[CrossRef]

Bence, S.

K. Riley, M. Hobson, and S. Bence, Mathematical Methods for Physics and Engineering, 3rd ed. (Cambridge U. Press, 2006).

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999).

Braje, D. A.

D. A. Braje, V. Balic, G. Y. Yin, and S. E. Harris, “Low-light-level nonlinear optics with slow light,” Phys. Rev. A 68, 041801(R) (2003).
[CrossRef]

Cataliotti, F.

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

C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization qubit phase gate in driven atomic media,” Phys. Rev. Lett. 90, 197902 (2003).
[CrossRef] [PubMed]

Corbalan, R.

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

F. Silva, J. Mompart, V. Ahufinger, and R. Corbalan, “Electromagnetically induced transparency in Doppler-broadened three-level systems with resonant standing-wave drive,” Europhys. Lett. 51, 286-292 (2000).
[CrossRef]

Dutton, Z.

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature (London) 409, 490-493 (2001).
[CrossRef]

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meterspersecond in an ultracold atomic gas,” Nature (London) 397, 594-598 (1999).
[CrossRef]

Fleischhauer, M.

F. E. Zimmer, A. Andre, M. D. Lukin, and M. Fleischhauer, “Coherent control of stationary light pulses,” Opt. Commun. 264, 441-453 (2006).
[CrossRef]

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633-673 (2005).
[CrossRef]

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

Fraval, E.

J. J. Longdell, E. Fraval, M. J. Sellars, and N. B. Manson, “Stopped light with storage times greater than one second using electromagnetically induced transparency in a solid,” Phys. Rev. Lett. 95, 063601 (2005).
[CrossRef] [PubMed]

Gao, J.-Y.

Q.-Y. He, J.-H. Wu, T.-J. Wang, and J.-Y. Gao, “Dynamic control of the photonic stop bands formed by a standing wave in inhomogeneous broadening solids,” Phys. Rev. A 73, 053813 (2006).
[CrossRef]

Q.-Y. He, Y. Xue, M. Artoni, G. C. La Rocca, J.-H. Xu, and J.-Y. Gao, “Coherently induced stop-bands in resonantly absorbing and inhomogeneously broadened doped crystals,” Phys. Rev. B 73, 195124 (2006).
[CrossRef]

Gibbs, H. M.

J. P. Prineas, J. Y. Zhou, J. Kuhl, H. M. Gibbs, G. Khitrova, S. W. Koch, and A. Knorr, “Ultrafast ac Stark effect switching of the active photonic band gap from Bragg-periodic semiconductor quantum wells,” Appl. Phys. Lett. 81, 4332-4334 (2002).
[CrossRef]

Ham, B. S.

B. S. Ham, “Spatiotemporal quantum manipulation of traveling light: quantum transport,” Appl. Phys. Lett. 88, 121117 (2006).
[CrossRef]

S. A. Moiseev and B. S. Ham, “Generation of entangled lights with temporally reversed photon wave packets,” Phys. Rev. A 71, 053802 (2005).
[CrossRef]

X.-M. Su and B. S. Ham, “Dynamic control of the photonic band gap using quantum coherence,” Phys. Rev. A 71, 013821 (2005).
[CrossRef]

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2001).
[CrossRef]

Hansen, K. R.

K. R. Hansen and K. Molmer, “Stationary light pulses in ultracold atomic gases,” Phys. Rev. A 75, 065804 (2007).
[CrossRef]

K. R. Hansen and K. Molmer, “Trapping of light pulses in ensembles of stationary lambda atoms,” Phys. Rev. A 75, 053802 (2007).
[CrossRef]

Harris, S. E.

D. A. Braje, V. Balic, G. Y. Yin, and S. E. Harris, “Low-light-level nonlinear optics with slow light,” Phys. Rev. A 68, 041801(R) (2003).
[CrossRef]

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meterspersecond in an ultracold atomic gas,” Nature (London) 397, 594-598 (1999).
[CrossRef]

S. E. Harris and L. V. Hau, “Nonlinear optics at low light levels,” Phys. Rev. Lett. 82, 4611-4614 (1999).
[CrossRef]

S. E. Harris and Y. Yamamoto, “Photon switching by quantum interference,” Phys. Rev. Lett. 81, 3611-3614 (1998).
[CrossRef]

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

Hau, L. V.

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature (London) 409, 490-493 (2001).
[CrossRef]

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meterspersecond in an ultracold atomic gas,” Nature (London) 397, 594-598 (1999).
[CrossRef]

S. E. Harris and L. V. Hau, “Nonlinear optics at low light levels,” Phys. Rev. Lett. 82, 4611-4614 (1999).
[CrossRef]

He, Q.-Y.

Q.-Y. He, Y. Xue, M. Artoni, G. C. La Rocca, J.-H. Xu, and J.-Y. Gao, “Coherently induced stop-bands in resonantly absorbing and inhomogeneously broadened doped crystals,” Phys. Rev. B 73, 195124 (2006).
[CrossRef]

Q.-Y. He, J.-H. Wu, T.-J. Wang, and J.-Y. Gao, “Dynamic control of the photonic stop bands formed by a standing wave in inhomogeneous broadening solids,” Phys. Rev. A 73, 053813 (2006).
[CrossRef]

Hemmer, P. R.

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2001).
[CrossRef]

Hobson, M.

K. Riley, M. Hobson, and S. Bence, Mathematical Methods for Physics and Engineering, 3rd ed. (Cambridge U. Press, 2006).

Imamoglu, A.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633-673 (2005).
[CrossRef]

Khitrova, G.

J. P. Prineas, J. Y. Zhou, J. Kuhl, H. M. Gibbs, G. Khitrova, S. W. Koch, and A. Knorr, “Ultrafast ac Stark effect switching of the active photonic band gap from Bragg-periodic semiconductor quantum wells,” Appl. Phys. Lett. 81, 4332-4334 (2002).
[CrossRef]

Knorr, A.

J. P. Prineas, J. Y. Zhou, J. Kuhl, H. M. Gibbs, G. Khitrova, S. W. Koch, and A. Knorr, “Ultrafast ac Stark effect switching of the active photonic band gap from Bragg-periodic semiconductor quantum wells,” Appl. Phys. Lett. 81, 4332-4334 (2002).
[CrossRef]

Koch, S. W.

J. P. Prineas, J. Y. Zhou, J. Kuhl, H. M. Gibbs, G. Khitrova, S. W. Koch, and A. Knorr, “Ultrafast ac Stark effect switching of the active photonic band gap from Bragg-periodic semiconductor quantum wells,” Appl. Phys. Lett. 81, 4332-4334 (2002).
[CrossRef]

Kuhl, J.

J. P. Prineas, J. Y. Zhou, J. Kuhl, H. M. Gibbs, G. Khitrova, S. W. Koch, and A. Knorr, “Ultrafast ac Stark effect switching of the active photonic band gap from Bragg-periodic semiconductor quantum wells,” Appl. Phys. Lett. 81, 4332-4334 (2002).
[CrossRef]

La Rocca, G.

M. Artoni, G. La Rocca, and F. Bassani, “Resonantly absorbing one-dimensional photonic crystals,” Phys. Rev. E 72, 046604 (2005).
[CrossRef]

La Rocca, G. C.

Q.-Y. He, Y. Xue, M. Artoni, G. C. La Rocca, J.-H. Xu, and J.-Y. Gao, “Coherently induced stop-bands in resonantly absorbing and inhomogeneously broadened doped crystals,” Phys. Rev. B 73, 195124 (2006).
[CrossRef]

M. Artoni and G. C. La Rocca, “Optically tunable photonic stop bands in homogeneous absorbing media,” Phys. Rev. Lett. 96, 073905 (2006).
[CrossRef] [PubMed]

Li, W.

S. M. Sadeghi, W. Li, and H. M. van Driel, “Coherently induced one-dimensional photonic band gap,” Phys. Rev. B 69, 073304 (2004).
[CrossRef]

Liu, C.

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature (London) 409, 490-493 (2001).
[CrossRef]

Longdell, J. J.

J. J. Longdell, E. Fraval, M. J. Sellars, and N. B. Manson, “Stopped light with storage times greater than one second using electromagnetically induced transparency in a solid,” Phys. Rev. Lett. 95, 063601 (2005).
[CrossRef] [PubMed]

Lukin, M. D.

F. E. Zimmer, A. Andre, M. D. Lukin, and M. Fleischhauer, “Coherent control of stationary light pulses,” Opt. Commun. 264, 441-453 (2006).
[CrossRef]

A. Andre, M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Nonlinear optics with stationary pulses of light,” Phys. Rev. Lett. 94, 063902 (2005).
[CrossRef] [PubMed]

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

A. Andre and M. D. Lukin, “Manipulating light pulses via dynamically controlled photonic band gap,” Phys. Rev. Lett. 89, 143602 (2002).
[CrossRef] [PubMed]

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

Manson, N. B.

J. J. Longdell, E. Fraval, M. J. Sellars, and N. B. Manson, “Stopped light with storage times greater than one second using electromagnetically induced transparency in a solid,” Phys. Rev. Lett. 95, 063601 (2005).
[CrossRef] [PubMed]

Marangos, J. P.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633-673 (2005).
[CrossRef]

Matsko, A. B.

Mermin, D.

N. Ashcroft and D. Mermin, Solid State Physics (Saunders College, 1976).

Moiseev, S. A.

S. A. Moiseev and B. S. Ham, “Generation of entangled lights with temporally reversed photon wave packets,” Phys. Rev. A 71, 053802 (2005).
[CrossRef]

Molmer, K.

K. R. Hansen and K. Molmer, “Stationary light pulses in ultracold atomic gases,” Phys. Rev. A 75, 065804 (2007).
[CrossRef]

K. R. Hansen and K. Molmer, “Trapping of light pulses in ensembles of stationary lambda atoms,” Phys. Rev. A 75, 053802 (2007).
[CrossRef]

Mompart, J.

F. Silva, J. Mompart, V. Ahufinger, and R. Corbalan, “Electromagnetically induced transparency in Doppler-broadened three-level systems with resonant standing-wave drive,” Europhys. Lett. 51, 286-292 (2000).
[CrossRef]

Musser, J. A.

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2001).
[CrossRef]

Novikova, A.

Ottaviani, C.

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

C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization qubit phase gate in driven atomic media,” Phys. Rev. Lett. 90, 197902 (2003).
[CrossRef] [PubMed]

Petrosyan, D.

D. Petrosyan, “Tunable photonic band gaps with coherently driven atoms in optical lattices,” Phys. Rev. A 76, 053823 (2007).
[CrossRef]

Prineas, J. P.

J. P. Prineas, J. Y. Zhou, J. Kuhl, H. M. Gibbs, G. Khitrova, S. W. Koch, and A. Knorr, “Ultrafast ac Stark effect switching of the active photonic band gap from Bragg-periodic semiconductor quantum wells,” Appl. Phys. Lett. 81, 4332-4334 (2002).
[CrossRef]

Rebic, S.

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

Riley, K.

K. Riley, M. Hobson, and S. Bence, Mathematical Methods for Physics and Engineering, 3rd ed. (Cambridge U. Press, 2006).

Sadeghi, S. M.

S. M. Sadeghi, W. Li, and H. M. van Driel, “Coherently induced one-dimensional photonic band gap,” Phys. Rev. B 69, 073304 (2004).
[CrossRef]

Sakoda, K.

K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2001).

Sellars, M. J.

J. J. Longdell, E. Fraval, M. J. Sellars, and N. B. Manson, “Stopped light with storage times greater than one second using electromagnetically induced transparency in a solid,” Phys. Rev. Lett. 95, 063601 (2005).
[CrossRef] [PubMed]

Shahriar, M. S.

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2001).
[CrossRef]

Silva, F.

F. Silva, J. Mompart, V. Ahufinger, and R. Corbalan, “Electromagnetically induced transparency in Doppler-broadened three-level systems with resonant standing-wave drive,” Europhys. Lett. 51, 286-292 (2000).
[CrossRef]

Su, X.-M.

X.-M. Su and B. S. Ham, “Dynamic control of the photonic band gap using quantum coherence,” Phys. Rev. A 71, 013821 (2005).
[CrossRef]

Sudarshanam, V. S.

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2001).
[CrossRef]

Tombesi, P.

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

C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization qubit phase gate in driven atomic media,” Phys. Rev. Lett. 90, 197902 (2003).
[CrossRef] [PubMed]

Turukhin, A. V.

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2001).
[CrossRef]

van Driel, H. M.

S. M. Sadeghi, W. Li, and H. M. van Driel, “Coherently induced one-dimensional photonic band gap,” Phys. Rev. B 69, 073304 (2004).
[CrossRef]

Vitali, D.

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

C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization qubit phase gate in driven atomic media,” Phys. Rev. Lett. 90, 197902 (2003).
[CrossRef] [PubMed]

Wang, T.-J.

Q.-Y. He, J.-H. Wu, T.-J. Wang, and J.-Y. Gao, “Dynamic control of the photonic stop bands formed by a standing wave in inhomogeneous broadening solids,” Phys. Rev. A 73, 053813 (2006).
[CrossRef]

Welch, G. R.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999).

Wu, J.-H.

Q.-Y. He, J.-H. Wu, T.-J. Wang, and J.-Y. Gao, “Dynamic control of the photonic stop bands formed by a standing wave in inhomogeneous broadening solids,” Phys. Rev. A 73, 053813 (2006).
[CrossRef]

Xu, J.-H.

Q.-Y. He, Y. Xue, M. Artoni, G. C. La Rocca, J.-H. Xu, and J.-Y. Gao, “Coherently induced stop-bands in resonantly absorbing and inhomogeneously broadened doped crystals,” Phys. Rev. B 73, 195124 (2006).
[CrossRef]

Xue, Y.

Q.-Y. He, Y. Xue, M. Artoni, G. C. La Rocca, J.-H. Xu, and J.-Y. Gao, “Coherently induced stop-bands in resonantly absorbing and inhomogeneously broadened doped crystals,” Phys. Rev. B 73, 195124 (2006).
[CrossRef]

Yamamoto, Y.

S. E. Harris and Y. Yamamoto, “Photon switching by quantum interference,” Phys. Rev. Lett. 81, 3611-3614 (1998).
[CrossRef]

Yin, G. Y.

D. A. Braje, V. Balic, G. Y. Yin, and S. E. Harris, “Low-light-level nonlinear optics with slow light,” Phys. Rev. A 68, 041801(R) (2003).
[CrossRef]

Zhou, J. Y.

J. P. Prineas, J. Y. Zhou, J. Kuhl, H. M. Gibbs, G. Khitrova, S. W. Koch, and A. Knorr, “Ultrafast ac Stark effect switching of the active photonic band gap from Bragg-periodic semiconductor quantum wells,” Appl. Phys. Lett. 81, 4332-4334 (2002).
[CrossRef]

Zibrov, A. S.

A. Andre, M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Nonlinear optics with stationary pulses of light,” Phys. Rev. Lett. 94, 063902 (2005).
[CrossRef] [PubMed]

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

Zimmer, F. E.

F. E. Zimmer, A. Andre, M. D. Lukin, and M. Fleischhauer, “Coherent control of stationary light pulses,” Opt. Commun. 264, 441-453 (2006).
[CrossRef]

Zubairy, M. S.

Appl. Phys. Lett. (2)

B. S. Ham, “Spatiotemporal quantum manipulation of traveling light: quantum transport,” Appl. Phys. Lett. 88, 121117 (2006).
[CrossRef]

J. P. Prineas, J. Y. Zhou, J. Kuhl, H. M. Gibbs, G. Khitrova, S. W. Koch, and A. Knorr, “Ultrafast ac Stark effect switching of the active photonic band gap from Bragg-periodic semiconductor quantum wells,” Appl. Phys. Lett. 81, 4332-4334 (2002).
[CrossRef]

Europhys. Lett. (1)

F. Silva, J. Mompart, V. Ahufinger, and R. Corbalan, “Electromagnetically induced transparency in Doppler-broadened three-level systems with resonant standing-wave drive,” Europhys. Lett. 51, 286-292 (2000).
[CrossRef]

Nature (London) (3)

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

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meterspersecond in an ultracold atomic gas,” Nature (London) 397, 594-598 (1999).
[CrossRef]

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature (London) 409, 490-493 (2001).
[CrossRef]

Opt. Commun. (1)

F. E. Zimmer, A. Andre, M. D. Lukin, and M. Fleischhauer, “Coherent control of stationary light pulses,” Opt. Commun. 264, 441-453 (2006).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (8)

Q.-Y. He, J.-H. Wu, T.-J. Wang, and J.-Y. Gao, “Dynamic control of the photonic stop bands formed by a standing wave in inhomogeneous broadening solids,” Phys. Rev. A 73, 053813 (2006).
[CrossRef]

D. A. Braje, V. Balic, G. Y. Yin, and S. E. Harris, “Low-light-level nonlinear optics with slow light,” Phys. Rev. A 68, 041801(R) (2003).
[CrossRef]

S. A. Moiseev and B. S. Ham, “Generation of entangled lights with temporally reversed photon wave packets,” Phys. Rev. A 71, 053802 (2005).
[CrossRef]

K. R. Hansen and K. Molmer, “Trapping of light pulses in ensembles of stationary lambda atoms,” Phys. Rev. A 75, 053802 (2007).
[CrossRef]

K. R. Hansen and K. Molmer, “Stationary light pulses in ultracold atomic gases,” Phys. Rev. A 75, 065804 (2007).
[CrossRef]

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

X.-M. Su and B. S. Ham, “Dynamic control of the photonic band gap using quantum coherence,” Phys. Rev. A 71, 013821 (2005).
[CrossRef]

D. Petrosyan, “Tunable photonic band gaps with coherently driven atoms in optical lattices,” Phys. Rev. A 76, 053823 (2007).
[CrossRef]

Phys. Rev. B (2)

Q.-Y. He, Y. Xue, M. Artoni, G. C. La Rocca, J.-H. Xu, and J.-Y. Gao, “Coherently induced stop-bands in resonantly absorbing and inhomogeneously broadened doped crystals,” Phys. Rev. B 73, 195124 (2006).
[CrossRef]

S. M. Sadeghi, W. Li, and H. M. van Driel, “Coherently induced one-dimensional photonic band gap,” Phys. Rev. B 69, 073304 (2004).
[CrossRef]

Phys. Rev. E (1)

M. Artoni, G. La Rocca, and F. Bassani, “Resonantly absorbing one-dimensional photonic crystals,” Phys. Rev. E 72, 046604 (2005).
[CrossRef]

Phys. Rev. Lett. (9)

J. J. Longdell, E. Fraval, M. J. Sellars, and N. B. Manson, “Stopped light with storage times greater than one second using electromagnetically induced transparency in a solid,” Phys. Rev. Lett. 95, 063601 (2005).
[CrossRef] [PubMed]

S. E. Harris and L. V. Hau, “Nonlinear optics at low light levels,” Phys. Rev. Lett. 82, 4611-4614 (1999).
[CrossRef]

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2001).
[CrossRef]

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

C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization qubit phase gate in driven atomic media,” Phys. Rev. Lett. 90, 197902 (2003).
[CrossRef] [PubMed]

M. Artoni and G. C. La Rocca, “Optically tunable photonic stop bands in homogeneous absorbing media,” Phys. Rev. Lett. 96, 073905 (2006).
[CrossRef] [PubMed]

S. E. Harris and Y. Yamamoto, “Photon switching by quantum interference,” Phys. Rev. Lett. 81, 3611-3614 (1998).
[CrossRef]

A. Andre and M. D. Lukin, “Manipulating light pulses via dynamically controlled photonic band gap,” Phys. Rev. Lett. 89, 143602 (2002).
[CrossRef] [PubMed]

A. Andre, M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Nonlinear optics with stationary pulses of light,” Phys. Rev. Lett. 94, 063902 (2005).
[CrossRef] [PubMed]

Phys. Today (1)

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

Rev. Mod. Phys. (1)

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633-673 (2005).
[CrossRef]

Other (7)

By replacing kp-->(ωp/c) and κ-->kc in Eq. in the simplified case where χ0 and χ1 are real and frequency independent, one obtains ωp2-->(kcc)2/(1+χ0∓χ1) yielding the upper and lower edge of the frequency stop band at the Brillouin zone boundary π/a. The width of such a photonic bandgap is directly proportional to χ1.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999).

The optical coherence in Eq. contributes with the two components ρ31{−1,0} while the probe polarization Pp has an expression analogous to the one for Ep in Eq. where the “forward” polarization component Pp+=ϵ0(χ0Ep++χ1Ep−) and a similar one for the “backward” component Pp−, which is obtained upon interchanging {+ ↔ −} in Pp+.

K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2001).

For the atomic samples considered here λc coincides, however, with the vacuum pump wave vector as the dielectric constant experienced by the pump is essentially unity.

K. Riley, M. Hobson, and S. Bence, Mathematical Methods for Physics and Engineering, 3rd ed. (Cambridge U. Press, 2006).

N. Ashcroft and D. Mermin, Solid State Physics (Saunders College, 1976).

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Figures (14)

Fig. 1
Fig. 1

Schematic of a probe ω p impinging on a three-level atom driven by an external coupling beam ω c , which is here retroreflected upon impinging on a mirror to form a SW pattern within the atomic sample. By slightly reducing the mirror reflectivity the FD and BD components of the SW intensity no longer vanishes at the nodes that are replaced by quasi-nodes. Furthermore, a misalignment θ between the FD and BD beam components, measured with respect to the direction z where the two coupling beams are perfectly aligned, modifies the lattice periodicity as λ c 2 λ c 2 cos ( θ 2 ) .

Fig. 2
Fig. 2

Photonic bandgap structures in a sample of ultracold Rb 87 atoms as obtained from Eq. (12). The black-solid and red-dashed curves correspond to κ + and κ . The atomic parameters are Γ 31 = Γ 32 = 6.0 MHz , Γ 21 = 2.0 kHz , and N = 10 13 cm 3 , while the FD and BD coupling beam components have Rabi frequencies Ω c + = 30.0 MHz and Ω c = 25.0 MHz and are slightly misaligned ( θ = 45.0 mrad ) . The probe transition wavelength is λ 31 = 780.792 nm while for the coupling beam we have λ 32 = 780.778 nm .

Fig. 3
Fig. 3

Probe reflectivity ( R ) and transmissivity ( T ) as obtained from Eq. (19) within the two-mode approximation for an ultracold Rb 87 sample of width L = 2.0 mm (black-solid curve), L = 1.0 mm (red-dashed curve) and L = 0.5 mm (blue-dotted curve). Other parameters are the same as in Fig. 2.

Fig. 4
Fig. 4

Probe reflectivity ( R ) and transmissivity ( T ) as obtained from Eq. (19) in an ultracold Rb 87 sample of width L = 2.0 mm (black-solid curve), L = 1.0 mm (red-dashed curve) and L = 0.5 mm (blue-dotted curve). Ω c + = 60.0 MHz , Ω c = 50.0 MHz , and θ = 55.0 mrad , while other parameters are the same as in Fig. 2.

Fig. 5
Fig. 5

Difference between the photonic energy bands obtained within the two-mode approximation (Fig. 2) and the exact ones obtained from the transfer-matrix approach [25, 36]. All parameters are the same as in Fig. 2.

Fig. 6
Fig. 6

Difference between the probe reflectivity ( R ) and transmissivity ( T ) obtained within the two-mode approximation (Fig. 3) and the exact ones obtained from the transfer-matrix approach for a L = 2.0 mm long sample [25, 36]. Other parameters are the same as in Fig. 2.

Fig. 7
Fig. 7

Zero- and first-order components of the probe susceptibility of a sample of ultracold Rb 87 atoms for the same parameters as in Fig. 2. The black-solid and red-dashed curves are, respectively, derived from Eqs. (23) truncated at n = 15 and 30, while the blue-dotted curves are from Eq. (8).

Fig. 8
Fig. 8

Central region blow-up of the probe susceptibility profiles of Fig. 7.

Fig. 9
Fig. 9

Reflected probe pulses seen at z = 0.0 and transmitted probe pulses seen at z = L for a long square pulse in a L = 2.0 mm long sample of ultracold Rb 87 atoms with Δ p = 0.0 MHz (black-solid curve), 0.4 MHz (red-dashed curve), 0.8 MHz (blue-dotted curve), and 1.2 MHz (green-dash-dotted curve), respectively. Other parameters are the same as in Fig. 2.

Fig. 10
Fig. 10

Steady-state reflectivity and transmissivity spectra in a L = 2.0 mm long sample of ultracold Rb 87 atoms. The black-solid curves are derived from the Maxwell–Liouville equations, i.e., from Eqs. (23) truncated at n = 30 and Eqs. (27), while the red-dashed curves are from Eqs. (19) within the two-mode approximation. Other parameters are the same as in Fig. 2.

Fig. 11
Fig. 11

Transmitted components at z = L (upper) and reflected components at z = 0 (lower) of a weak probe pulse incident upon a L = 2.0 mm long sample of ultracold Rb 87 atoms with T 0 = 25.0 μ s and δ T = 4.0 μ s . Other parameters are the same as in Fig. 2.

Fig. 12
Fig. 12

Scaled intensity distributions of the FD (upper) and BD (lower) probe pulses inside a sample of ultracold Rb 87 atoms with Δ p 0 = 0.4 MHz . Other parameters are the same as in Fig. 11.

Fig. 13
Fig. 13

Scaled intensity distributions of the FD (upper) and BD (lower) probe pulses inside a sample of ultracold Rb 87 atoms with Δ p 0 = 0.0 MHz . Other parameters are the same as in Fig. 11.

Fig. 14
Fig. 14

Reflected probe pulses seen at z = 0 as obtained by using the dressed susceptibility (red-dashed curve) and the Maxwell–Liouville (blue-dotted curve) approach. The incident probe pulse (black-solid curve), which is here scaled to unity, has a carrier frequency Δ p 0 = ( a ) 0.4 and (b) 0.0 MHz while other parameters are the same as in Fig. 11. The insets show the relative difference between the reflection spectra obtained with the two approaches.

Equations (50)

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H = ω 31 3 3 + ω 21 2 2 + H int ,
H int = Ω p e i Δ p t 3 1 Ω c e i Δ c t 3 2 + H.c.
ρ 11 t = Γ 31 ρ 33 + Γ 21 ρ 22 + i Ω p * ρ 31 i Ω p ρ 13 ,
ρ 22 t = Γ 32 ρ 33 Γ 21 ρ 22 + i Ω c * ρ 32 i Ω c ρ 23 ,
ρ 12 t = [ γ 12 + i ( Δ p Δ c ) ] ρ 12 + i Ω p * ρ 32 i Ω c ρ 13 ,
ρ 13 t = [ γ 13 + i Δ p ] ρ 13 i Ω c * ρ 12 + i Ω p * ( ρ 33 ρ 11 ) ,
ρ 23 t = [ γ 23 + i Δ c ] ρ 23 i Ω p * ρ 21 + i Ω c * ( ρ 33 ρ 22 ) ,
χ p = N d 13 2 2 ε 0 Δ p + i γ 12 ( γ 12 i Δ p ) ( γ 13 i Δ p ) + Ω c 2 .
χ p ( z ) = N d 13 2 2 ε 0 A 1 + B cos ( 2 k c z ) ,
A = Δ p + i γ 12 Ω c + 2 + Ω c 2 Δ p 2 i ( γ 12 + γ 13 ) Δ p + γ 12 γ 13 ,
B = 2 Ω c + Ω c Ω c + 2 + Ω c 2 Δ p 2 i ( γ 12 + γ 13 ) Δ p + γ 12 γ 13 .
χ p ( z ) = χ 0 + 2 n = 1 χ n cos ( 2 n k c z ) ,
χ n = k c π N A d 13 2 2 ϵ o 0 π k c cos ( 2 n k c z ) d z 1 + B cos ( 2 k c z ) = N A d 13 2 2 ϵ 0 1 1 B 2 ( 1 B 2 1 B ) n .
E ( z ) ϵ ( z ) e i κ z = [ n = ϵ n e i 2 n k c z ] e i κ z ,
2 E ( z ) z 2 + k p 2 [ 1 + χ p ( z ) ] E ( z ) = 0 .
( k p 2 ( 1 + χ 0 ) κ 2 k p 2 χ 1 k p 2 χ 1 k p 2 ( 1 + χ 0 ) ( κ 2 k c ) 2 ) ( ϵ 0 ϵ 1 ) = 0 .
q ± ± 1 2 k c [ k p 2 ( 1 + χ 0 ) k c 2 ] 2 k p 4 χ 1 2 ± q ,
E + ( z ) = ϵ 0 + e i κ + z + ϵ 1 + e i κ z ,
E ( z ) = ϵ 0 e i κ z + ϵ 1 e i κ + z ,
B + ( z ) = ( ϵ 0 + κ + e i κ + z ϵ 1 + κ e i κ z ) k p ,
B ( z ) = ( ϵ 0 κ e i κ z ϵ 1 κ + e i κ + z ) k p ,
E p ( z ) = α E + ( z ) + β E ( z ) ,
B p ( z ) = α B + ( z ) + β B ( z ) ,
E p ( 0 ) = ( 1 + r ) E in ( 0 ) ,
B p ( 0 ) = ( 1 r ) B in ( 0 ) ,
E p ( L ) = t E in ( 0 ) ,
B p ( L ) = t B in ( 0 ) ,
r = α ( ϵ 0 + + ϵ 1 + ) + β ( ϵ 0 + ϵ 1 ) 1 ,
r = 1 α κ + ϵ 0 + κ ϵ 1 + k p β κ ϵ 0 κ + ϵ 1 k p ,
t = α ( ϵ 0 + + ϵ 1 + ) e i q L + β ( ϵ 0 + ϵ 1 ) e i q L ,
t = α κ + ϵ 0 + κ ϵ 1 + k p e i q L + β κ ϵ 0 κ + ϵ 1 k p e i q L ,
X ± ϵ 0 ± ϵ 1 ± = χ 1 ( 1 + χ 0 ) ( 2 κ ± k c + k c 2 ) k p 2 ,
r = 2 A ( 1 + X + ) e i q L 2 B ( 1 + X ) e i q L A B + e i q L A + B e i q L 1 ,
t = 2 A ( 1 + X + ) 2 B ( 1 + X ) A B + e i q L A + B e i q L ,
ρ 12 t = [ γ 12 + i ( Δ p Δ c ) ] ρ 12 i Ω c ρ 13 ,
ρ 13 t = [ γ 13 + i Δ p ] ρ 13 i Ω c * ρ 12 i Ω p * ,
E p ( z , t ) = ( E p + e i k c z + E p e i k c z ) e i ω p t ,
E c ( z , t ) = ( E c + e i k c z + E c e i k c z ) e i ω c t .
ρ 12 = n = ρ 12 ( n ) e i 2 n k c z ,
ρ 13 = n = ρ 13 ( n ) e i ( 2 n + 1 ) k c z ,
ρ 12 ( n ) t = γ 12 ρ 12 ( n ) i Ω c ρ 13 ( n ) i Ω c + ρ 13 ( n 1 ) ,
ρ 13 ( n ) t = γ 13 ρ 13 ( n ) i Ω c * ρ 12 ( n ) i Ω c + * ρ 12 ( n + 1 ) i Ω p * δ n , 0 i Ω p + * δ n , 1 ,
Ω p + χ 0 + Ω p χ 1 = N d 13 2 2 ε 0 ρ 31 ( 1 ) ,
Ω p χ 0 + Ω p + χ 1 = N d 13 2 2 ε 0 ρ 31 ( 0 ) ,
2 E p z 2 1 c 2 2 E p t 2 = μ 0 2 P t 2 ,
( E p + z + k p c k c E p + t i ( k p 2 k c 2 ) 2 k c E p + ) e + i k c z ( E p z k p c k c E p t + i ( k p 2 k c 2 ) 2 k c E p ) e i k c z = i k p 2 k c N d 13 2 ε 0 ρ 31 ,
E p + z + 1 c E p + t = + i Δ k E p + + i N d 13 k p 2 ε 0 ρ 31 ( 0 ) ,
E p z 1 c E p t = i Δ k E p i N d 13 k p 2 ε 0 ρ 31 ( 1 ) ,
E r t ( t ) = E r f ( Δ p ) e i ( t T 0 ) Δ p d ( Δ p ) ,
E t t ( t ) = E t f ( Δ p ) e i ( t T 0 ) Δ p d ( Δ p ) ,

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