Abstract

We investigate, both theoretically and experimentally, the cross-Kerr nonlinearity generated in a four-level tripod system in the Rb87 D1 line. The system exhibits simultaneous electromagnetically induced transparency (EIT) windows for two weak (probe and trigger) fields and the enhanced cross-Kerr nonlinearity by EIT. The cross-Kerr nonlinear phase shifts for probe (trigger) field are measured at several different intensities of trigger (probe) field. The results show that large cross-phase modulation (XPM) coefficients can be achieved at low light intensities.

© 2009 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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  35. B. S. Ham and P. R. Hemmer, “Coherence switching in a four-level system: quantum switching,” Phys. Rev. Lett. 84, 4080-4083 (2000).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  37. Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, “Measurement of conditional phase shifts for quantum logic,” Phys. Rev. Lett. 75, 4710-4713 (1995).
    [CrossRef] [PubMed]
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    [CrossRef]
  39. S. D. Barrett, P. Kok, K. Nemoto, R. G. Beausoleil, W. J. Munro, and T. P. Spiller, “Symmetry analyzer for nondestructive Bell-state detection using weak nonlinearities,” Phys. Rev. A 71, 060302(R) (2005).
    [CrossRef]
  40. W. J. Munro, K. Nemoto, and T. P. Spiller, “Weak nonlinearities: a new route to optical quantum computation,” New J. Phys. 7, 137 (2005).
    [CrossRef]
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    [CrossRef] [PubMed]
  42. S. Li, X. Yang, X. Cao, C. Xie, and H. Wang, “Two electromagnetically induced transparency windows and an enhanced electromagnetically induced transparency signal in a four-level tripod atomic system,” J. Phys. B 40, 3211-3219 (2007).
    [CrossRef]
  43. H. Wang, D. Goorskey, and M. Xiao, “Atomic coherence induced Kerr nonlinearity enhancement in Rb vapor,” J. Mod. Opt. 49, 335-347 (2002).
    [CrossRef]
  44. M. Xiao, Y. Li, S. Jin, and J. Gea-Banacloche, “Measurement of dispersive properties of electromagnetically induced transparency in Rubidium atoms,” Phys. Rev. Lett. 74, 666-669 (1995).
    [CrossRef] [PubMed]
  45. Y. Han, J. Xiao, Y. Liu, C. Zhang, H. Wang, M. Xiao, and K. Peng, “Interacting dark states with enhanced nonlinearity in an ideal four-level tripod atomic system,” Phys. Rev. A 77, 023824 (2008).
    [CrossRef]
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    [CrossRef]

2008 (2)

S. Li, X. Yang, X. Cao, C. Zhang, C. Xie, and H. Wang, “Enhanced cross-phase modulation based on a double electromagnetically induced transparency in a four-level tripod atomic system,” Phys. Rev. Lett. 101, 073602 (2008).
[CrossRef] [PubMed]

Y. Han, J. Xiao, Y. Liu, C. Zhang, H. Wang, M. Xiao, and K. Peng, “Interacting dark states with enhanced nonlinearity in an ideal four-level tripod atomic system,” Phys. Rev. A 77, 023824 (2008).
[CrossRef]

2007 (3)

S. Li, X. Yang, X. Cao, C. Xie, and H. Wang, “Two electromagnetically induced transparency windows and an enhanced electromagnetically induced transparency signal in a four-level tripod atomic system,” J. Phys. B 40, 3211-3219 (2007).
[CrossRef]

L. Deng and M. G. Payne, “Gain-assisted large and rapidly responding Kerr effect using a room-temperature active Raman gain medium,” Phys. Rev. Lett. 98, 253902 (2007).
[CrossRef] [PubMed]

J. Zhang, G. Hernandez, and Y. Zhu, “All-optical switching at ultralow light levels,” Opt. Lett. 32, 1317-1319 (2007).
[CrossRef] [PubMed]

2006 (1)

Z.-B. Wang, K.-P. Marzlin, and B. C. Sanders, “Large cross-phase modulation between slow copropagating weak pulses in Rb87,” Phys. Rev. Lett. 97, 063901 (2006).
[CrossRef] [PubMed]

2005 (6)

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

D. Petrosyan, “Towards deterministic optical quantum computation with coherently driven atomic ensembles,” J. Opt. B: Quantum Semiclassical Opt. 7, S141-S151 (2005).
[CrossRef]

S. D. Barrett, P. Kok, K. Nemoto, R. G. Beausoleil, W. J. Munro, and T. P. Spiller, “Symmetry analyzer for nondestructive Bell-state detection using weak nonlinearities,” Phys. Rev. A 71, 060302(R) (2005).
[CrossRef]

W. J. Munro, K. Nemoto, and T. P. Spiller, “Weak nonlinearities: a new route to optical quantum computation,” New J. Phys. 7, 137 (2005).
[CrossRef]

Y. F. Chen, Z. H. Tsai, Y. C. Liu, and I. A. Yu, “Low-light-level photon switching by quantum interference,” Opt. Lett. 30, 3207-3209 (2005).
[CrossRef] [PubMed]

Y. Niu, S. Gong, R. Li, Z. Xu, and X. Liang, “Giant Kerr nonlinearity induced by interacting dark resonances,” Opt. Lett. 30, 3371-3373 (2005).
[CrossRef]

2004 (4)

K. Nemoto and W. J. Munro, “Nearly deterministic linear optical controlled-NOT gate,” Phys. Rev. Lett. 93, 250502 (2004).
[CrossRef]

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

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]

D. Petrosyan and Y. P. Malakyan, “Magneto-optical rotation and cross-phase modulation via coherently driven four-level atoms in a tripod configuration,” Phys. Rev. A 70, 023822 (2004).
[CrossRef]

2003 (5)

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]

H. Kang and Y. Zhu, “Observation of large Kerr nonlinearity at low light intensities,” Phys. Rev. Lett. 91, 093601 (2003).
[CrossRef] [PubMed]

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

S. F. Yelin, V. A. Sautenkov, M. M. Kash, G. R. Welch, and M. D. Lukin, “Nonlinear optics via double dark resonances,” Phys. Rev. A 68, 063801 (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 (2003).
[CrossRef]

2002 (3)

C. Y. Ye and A. S. Zibrov, “Width of the electromagnetically induced transparency resonance in atomic vapor,” Phys. Rev. A 65, 023806 (2002).
[CrossRef]

H. Wang, D. Goorskey, and M. Xiao, “Atomic coherence induced Kerr nonlinearity enhancement in Rb vapor,” J. Mod. Opt. 49, 335-347 (2002).
[CrossRef]

D. Petrosyan and G. Kurizki, “Symmetric photon-photon coupling by atoms with Zeeman-split sublevels,” Phys. Rev. A 65, 033833 (2002).
[CrossRef]

2001 (2)

H. Wang, D. Goorskey, and M. Xiao, “Enhanced Kerr nonlinearity via atomic coherence in a three-level atomic system,” Phys. Rev. Lett. 87, 073601 (2001).
[CrossRef] [PubMed]

D. F. Phillips, A. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin, “Storage of light in atomic vapor,” Phys. Rev. Lett. 86, 783-786 (2001).
[CrossRef] [PubMed]

2000 (3)

M. D. Lukin and A. Imamoglu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Phys. Rev. Lett. 84, 1419-1422 (2000).
[CrossRef] [PubMed]

B. S. Ham and P. R. Hemmer, “Coherence switching in a four-level system: quantum switching,” Phys. Rev. Lett. 84, 4080-4083 (2000).
[CrossRef] [PubMed]

A. J. Marriam, S. J. Sharpe, M. Shverdin, D. Manuszak, G. Y. Yin, and S. E. Harris, “Efficient nonlinear frequency conversion in an all-resonant double-Λ system,” Phys. Rev. Lett. 84, 5308-5311 (2000).
[CrossRef]

1999 (3)

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229-5232 (1999).
[CrossRef]

D. Budker, D. F. Kimball, S. M. Rochester, and V. V. Yashchuk, “Nonlinear magneto-optics and reduced group velocity of light in atomic vapor with slow ground state relaxation,” Phys. Rev. Lett. 83, 1767-1770 (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 397, 594-598 (1999).
[CrossRef]

1998 (1)

1997 (2)

A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467-1470 (1997).
[CrossRef]

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

1996 (3)

1995 (6)

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]

M. Xiao, Y. Li, S. Jin, and J. Gea-Banacloche, “Measurement of dispersive properties of electromagnetically induced transparency in Rubidium atoms,” Phys. Rev. Lett. 74, 666-669 (1995).
[CrossRef] [PubMed]

Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, “Measurement of conditional phase shifts for quantum logic,” Phys. Rev. Lett. 75, 4710-4713 (1995).
[CrossRef] [PubMed]

J. Gea-Banacloche, Y. Li, S. Jin, and M. Xiao, “Electromagnetically induced transparency in ladder-type inhomogeneously broadened media: theory and experiment,” Phys. Rev. A 51, 576-584 (1995).
[CrossRef] [PubMed]

Y. Li and M. Xiao, “Electromagnetically induced transparency in a three-level Λ-type system in rubidium atoms,” Phys. Rev. A 51, R2703-R2706 (1995).
[CrossRef] [PubMed]

A. Kasapi, M. Jain, G. Y. Yin, and S. E. Harris, “Electromagnetically induced transparency: propagation dynamics,” Phys. Rev. Lett. 74, 2447-2450 (1995).
[CrossRef] [PubMed]

1991 (2)

K. L. 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]

Artoni, M.

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]

Balic, V.

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

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 (2003).
[CrossRef]

Barrett, S. D.

S. D. Barrett, P. Kok, K. Nemoto, R. G. Beausoleil, W. J. Munro, and T. P. Spiller, “Symmetry analyzer for nondestructive Bell-state detection using weak nonlinearities,” Phys. Rev. A 71, 060302(R) (2005).
[CrossRef]

Beausoleil, R. G.

S. D. Barrett, P. Kok, K. Nemoto, R. G. Beausoleil, W. J. Munro, and T. P. Spiller, “Symmetry analyzer for nondestructive Bell-state detection using weak nonlinearities,” Phys. Rev. A 71, 060302(R) (2005).
[CrossRef]

Behroozi, C. H.

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

Boller, K. L.

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

Braje, D. A.

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

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 (2003).
[CrossRef]

Budker, D.

D. Budker, D. F. Kimball, S. M. Rochester, and V. V. Yashchuk, “Nonlinear magneto-optics and reduced group velocity of light in atomic vapor with slow ground state relaxation,” Phys. Rev. Lett. 83, 1767-1770 (1999).
[CrossRef]

Burkett, W. H.

Cao, X.

S. Li, X. Yang, X. Cao, C. Zhang, C. Xie, and H. Wang, “Enhanced cross-phase modulation based on a double electromagnetically induced transparency in a four-level tripod atomic system,” Phys. Rev. Lett. 101, 073602 (2008).
[CrossRef] [PubMed]

S. Li, X. Yang, X. Cao, C. Xie, and H. Wang, “Two electromagnetically induced transparency windows and an enhanced electromagnetically induced transparency signal in a four-level tripod atomic system,” J. Phys. B 40, 3211-3219 (2007).
[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]

Chen, Y. F.

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]

Cronin-Golomb, M.

Deng, L.

L. Deng and M. G. Payne, “Gain-assisted large and rapidly responding Kerr effect using a room-temperature active Raman gain medium,” Phys. Rev. Lett. 98, 253902 (2007).
[CrossRef] [PubMed]

Deutsch, M.

A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467-1470 (1997).
[CrossRef]

Donoghue, J.

Dutton, Z.

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

Field, J. E.

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]

Fleischhauer, A.

D. F. Phillips, A. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin, “Storage of light in atomic vapor,” Phys. Rev. Lett. 86, 783-786 (2001).
[CrossRef] [PubMed]

Fry, E. S.

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229-5232 (1999).
[CrossRef]

Gea-Banacloche, J.

M. Xiao, Y. Li, S. Jin, and J. Gea-Banacloche, “Measurement of dispersive properties of electromagnetically induced transparency in Rubidium atoms,” Phys. Rev. Lett. 74, 666-669 (1995).
[CrossRef] [PubMed]

J. Gea-Banacloche, Y. Li, S. Jin, and M. Xiao, “Electromagnetically induced transparency in ladder-type inhomogeneously broadened media: theory and experiment,” Phys. Rev. A 51, 576-584 (1995).
[CrossRef] [PubMed]

Goda, S.

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

Gong, S.

Goorskey, D.

H. Wang, D. Goorskey, and M. Xiao, “Atomic coherence induced Kerr nonlinearity enhancement in Rb vapor,” J. Mod. Opt. 49, 335-347 (2002).
[CrossRef]

H. Wang, D. Goorskey, and M. Xiao, “Enhanced Kerr nonlinearity via atomic coherence in a three-level atomic system,” Phys. Rev. Lett. 87, 073601 (2001).
[CrossRef] [PubMed]

Hahn, K. H.

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]

Ham, B. S.

B. S. Ham and P. R. Hemmer, “Coherence switching in a four-level system: quantum switching,” Phys. Rev. Lett. 84, 4080-4083 (2000).
[CrossRef] [PubMed]

Han, Y.

Y. Han, J. Xiao, Y. Liu, C. Zhang, H. Wang, M. Xiao, and K. Peng, “Interacting dark states with enhanced nonlinearity in an ideal four-level tripod atomic system,” Phys. Rev. A 77, 023824 (2008).
[CrossRef]

Harris, S. E.

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

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 (2003).
[CrossRef]

A. J. Marriam, S. J. Sharpe, M. Shverdin, D. Manuszak, G. Y. Yin, and S. E. Harris, “Efficient nonlinear frequency conversion in an all-resonant double-Λ system,” Phys. Rev. Lett. 84, 5308-5311 (2000).
[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 397, 594-598 (1999).
[CrossRef]

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

M. Jain, H. Xia, G. Y. Yin, A. J. Marriam, and S. E. Harris, “Efficient nonlinear frequency conversion with maximal atomic coherence,” Phys. Rev. Lett. 77, 4326-4329 (1996).
[CrossRef] [PubMed]

A. Kasapi, M. Jain, G. Y. Yin, and S. E. Harris, “Electromagnetically induced transparency: propagation dynamics,” Phys. Rev. Lett. 74, 2447-2450 (1995).
[CrossRef] [PubMed]

K. L. 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]

Hau, L. V.

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

Hemmer, P. R.

Hernandez, G.

Hollberg, L.

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229-5232 (1999).
[CrossRef]

Hood, C. J.

Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, “Measurement of conditional phase shifts for quantum logic,” Phys. Rev. Lett. 75, 4710-4713 (1995).
[CrossRef] [PubMed]

Imamoglu, A.

M. D. Lukin and A. Imamoglu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Phys. Rev. Lett. 84, 1419-1422 (2000).
[CrossRef] [PubMed]

A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467-1470 (1997).
[CrossRef]

H. Schmidt and A. Imamoglu, “Giant Kerr nonlinearities obtained by electromagnetically induced transparency,” Opt. Lett. 21, 1936-1938 (1996).
[CrossRef] [PubMed]

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

Jain, M.

M. Jain, H. Xia, G. Y. Yin, A. J. Marriam, and S. E. Harris, “Efficient nonlinear frequency conversion with maximal atomic coherence,” Phys. Rev. Lett. 77, 4326-4329 (1996).
[CrossRef] [PubMed]

A. Kasapi, M. Jain, G. Y. Yin, and S. E. Harris, “Electromagnetically induced transparency: propagation dynamics,” Phys. Rev. Lett. 74, 2447-2450 (1995).
[CrossRef] [PubMed]

Jin, S.

J. Gea-Banacloche, Y. Li, S. Jin, and M. Xiao, “Electromagnetically induced transparency in ladder-type inhomogeneously broadened media: theory and experiment,” Phys. Rev. A 51, 576-584 (1995).
[CrossRef] [PubMed]

M. Xiao, Y. Li, S. Jin, and J. Gea-Banacloche, “Measurement of dispersive properties of electromagnetically induced transparency in Rubidium atoms,” Phys. Rev. Lett. 74, 666-669 (1995).
[CrossRef] [PubMed]

Joshi, A.

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

Kang, H.

H. Kang and Y. Zhu, “Observation of large Kerr nonlinearity at low light intensities,” Phys. Rev. Lett. 91, 093601 (2003).
[CrossRef] [PubMed]

Kasapi, A.

A. Kasapi, M. Jain, G. Y. Yin, and S. E. Harris, “Electromagnetically induced transparency: propagation dynamics,” Phys. Rev. Lett. 74, 2447-2450 (1995).
[CrossRef] [PubMed]

Kash, M. M.

S. F. Yelin, V. A. Sautenkov, M. M. Kash, G. R. Welch, and M. D. Lukin, “Nonlinear optics via double dark resonances,” Phys. Rev. A 68, 063801 (2003).
[CrossRef]

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229-5232 (1999).
[CrossRef]

Katz, D. P.

Kimball, D. F.

D. Budker, D. F. Kimball, S. M. Rochester, and V. V. Yashchuk, “Nonlinear magneto-optics and reduced group velocity of light in atomic vapor with slow ground state relaxation,” Phys. Rev. Lett. 83, 1767-1770 (1999).
[CrossRef]

Kimble, H. J.

Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, “Measurement of conditional phase shifts for quantum logic,” Phys. Rev. Lett. 75, 4710-4713 (1995).
[CrossRef] [PubMed]

Kok, P.

S. D. Barrett, P. Kok, K. Nemoto, R. G. Beausoleil, W. J. Munro, and T. P. Spiller, “Symmetry analyzer for nondestructive Bell-state detection using weak nonlinearities,” Phys. Rev. A 71, 060302(R) (2005).
[CrossRef]

Kumar, P.

Kurizki, G.

D. Petrosyan and G. Kurizki, “Symmetric photon-photon coupling by atoms with Zeeman-split sublevels,” Phys. Rev. A 65, 033833 (2002).
[CrossRef]

Lange, W.

Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, “Measurement of conditional phase shifts for quantum logic,” Phys. Rev. Lett. 75, 4710-4713 (1995).
[CrossRef] [PubMed]

Li, R.

Li, S.

S. Li, X. Yang, X. Cao, C. Zhang, C. Xie, and H. Wang, “Enhanced cross-phase modulation based on a double electromagnetically induced transparency in a four-level tripod atomic system,” Phys. Rev. Lett. 101, 073602 (2008).
[CrossRef] [PubMed]

S. Li, X. Yang, X. Cao, C. Xie, and H. Wang, “Two electromagnetically induced transparency windows and an enhanced electromagnetically induced transparency signal in a four-level tripod atomic system,” J. Phys. B 40, 3211-3219 (2007).
[CrossRef]

Li, Y.

Y. Li and M. Xiao, “Enhancement of nondegenerate four-wave mixing based on electromagnetically induced transparency in rubidium atoms,” Opt. Lett. 21, 1064-1066 (1996).
[CrossRef] [PubMed]

M. Xiao, Y. Li, S. Jin, and J. Gea-Banacloche, “Measurement of dispersive properties of electromagnetically induced transparency in Rubidium atoms,” Phys. Rev. Lett. 74, 666-669 (1995).
[CrossRef] [PubMed]

Y. Li and M. Xiao, “Electromagnetically induced transparency in a three-level Λ-type system in rubidium atoms,” Phys. Rev. A 51, R2703-R2706 (1995).
[CrossRef] [PubMed]

J. Gea-Banacloche, Y. Li, S. Jin, and M. Xiao, “Electromagnetically induced transparency in ladder-type inhomogeneously broadened media: theory and experiment,” Phys. Rev. A 51, 576-584 (1995).
[CrossRef] [PubMed]

Liang, X.

Liu, Y.

Y. Han, J. Xiao, Y. Liu, C. Zhang, H. Wang, M. Xiao, and K. Peng, “Interacting dark states with enhanced nonlinearity in an ideal four-level tripod atomic system,” Phys. Rev. A 77, 023824 (2008).
[CrossRef]

Liu, Y. C.

Lu, B.

Lukin, M. D.

S. F. Yelin, V. A. Sautenkov, M. M. Kash, G. R. Welch, and M. D. Lukin, “Nonlinear optics via double dark resonances,” Phys. Rev. A 68, 063801 (2003).
[CrossRef]

D. F. Phillips, A. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin, “Storage of light in atomic vapor,” Phys. Rev. Lett. 86, 783-786 (2001).
[CrossRef] [PubMed]

M. D. Lukin and A. Imamoglu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Phys. Rev. Lett. 84, 1419-1422 (2000).
[CrossRef] [PubMed]

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229-5232 (1999).
[CrossRef]

Mabuchi, H.

Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, “Measurement of conditional phase shifts for quantum logic,” Phys. Rev. Lett. 75, 4710-4713 (1995).
[CrossRef] [PubMed]

Mair, A.

D. F. Phillips, A. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin, “Storage of light in atomic vapor,” Phys. Rev. Lett. 86, 783-786 (2001).
[CrossRef] [PubMed]

Malakyan, Y. P.

D. Petrosyan and Y. P. Malakyan, “Magneto-optical rotation and cross-phase modulation via coherently driven four-level atoms in a tripod configuration,” Phys. Rev. A 70, 023822 (2004).
[CrossRef]

Manuszak, D.

A. J. Marriam, S. J. Sharpe, M. Shverdin, D. Manuszak, G. Y. Yin, and S. E. Harris, “Efficient nonlinear frequency conversion in an all-resonant double-Λ system,” Phys. Rev. Lett. 84, 5308-5311 (2000).
[CrossRef]

Marriam, A. J.

A. J. Marriam, S. J. Sharpe, M. Shverdin, D. Manuszak, G. Y. Yin, and S. E. Harris, “Efficient nonlinear frequency conversion in an all-resonant double-Λ system,” Phys. Rev. Lett. 84, 5308-5311 (2000).
[CrossRef]

M. Jain, H. Xia, G. Y. Yin, A. J. Marriam, and S. E. Harris, “Efficient nonlinear frequency conversion with maximal atomic coherence,” Phys. Rev. Lett. 77, 4326-4329 (1996).
[CrossRef] [PubMed]

Marzlin, K.-P.

Z.-B. Wang, K.-P. Marzlin, and B. C. Sanders, “Large cross-phase modulation between slow copropagating weak pulses in Rb87,” Phys. Rev. Lett. 97, 063901 (2006).
[CrossRef] [PubMed]

Matsko, A. B.

Munro, W. J.

W. J. Munro, K. Nemoto, and T. P. Spiller, “Weak nonlinearities: a new route to optical quantum computation,” New J. Phys. 7, 137 (2005).
[CrossRef]

S. D. Barrett, P. Kok, K. Nemoto, R. G. Beausoleil, W. J. Munro, and T. P. Spiller, “Symmetry analyzer for nondestructive Bell-state detection using weak nonlinearities,” Phys. Rev. A 71, 060302(R) (2005).
[CrossRef]

K. Nemoto and W. J. Munro, “Nearly deterministic linear optical controlled-NOT gate,” Phys. Rev. Lett. 93, 250502 (2004).
[CrossRef]

Nemoto, K.

W. J. Munro, K. Nemoto, and T. P. Spiller, “Weak nonlinearities: a new route to optical quantum computation,” New J. Phys. 7, 137 (2005).
[CrossRef]

S. D. Barrett, P. Kok, K. Nemoto, R. G. Beausoleil, W. J. Munro, and T. P. Spiller, “Symmetry analyzer for nondestructive Bell-state detection using weak nonlinearities,” Phys. Rev. A 71, 060302(R) (2005).
[CrossRef]

K. Nemoto and W. J. Munro, “Nearly deterministic linear optical controlled-NOT gate,” Phys. Rev. Lett. 93, 250502 (2004).
[CrossRef]

Niu, Y.

Novikova, I.

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]

Payne, M. G.

L. Deng and M. G. Payne, “Gain-assisted large and rapidly responding Kerr effect using a room-temperature active Raman gain medium,” Phys. Rev. Lett. 98, 253902 (2007).
[CrossRef] [PubMed]

Peng, K.

Y. Han, J. Xiao, Y. Liu, C. Zhang, H. Wang, M. Xiao, and K. Peng, “Interacting dark states with enhanced nonlinearity in an ideal four-level tripod atomic system,” Phys. Rev. A 77, 023824 (2008).
[CrossRef]

Petrosyan, D.

D. Petrosyan, “Towards deterministic optical quantum computation with coherently driven atomic ensembles,” J. Opt. B: Quantum Semiclassical Opt. 7, S141-S151 (2005).
[CrossRef]

D. Petrosyan and Y. P. Malakyan, “Magneto-optical rotation and cross-phase modulation via coherently driven four-level atoms in a tripod configuration,” Phys. Rev. A 70, 023822 (2004).
[CrossRef]

D. Petrosyan and G. Kurizki, “Symmetric photon-photon coupling by atoms with Zeeman-split sublevels,” Phys. Rev. A 65, 033833 (2002).
[CrossRef]

Phillips, D. F.

D. F. Phillips, A. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin, “Storage of light in atomic vapor,” Phys. Rev. Lett. 86, 783-786 (2001).
[CrossRef] [PubMed]

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]

Rochester, S. M.

D. Budker, D. F. Kimball, S. M. Rochester, and V. V. Yashchuk, “Nonlinear magneto-optics and reduced group velocity of light in atomic vapor with slow ground state relaxation,” Phys. Rev. Lett. 83, 1767-1770 (1999).
[CrossRef]

Rostovtsev, Y.

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229-5232 (1999).
[CrossRef]

Sanders, B. C.

Z.-B. Wang, K.-P. Marzlin, and B. C. Sanders, “Large cross-phase modulation between slow copropagating weak pulses in Rb87,” Phys. Rev. Lett. 97, 063901 (2006).
[CrossRef] [PubMed]

Sautenkov, V. A.

S. F. Yelin, V. A. Sautenkov, M. M. Kash, G. R. Welch, and M. D. Lukin, “Nonlinear optics via double dark resonances,” Phys. Rev. A 68, 063801 (2003).
[CrossRef]

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229-5232 (1999).
[CrossRef]

Schmidt, H.

A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467-1470 (1997).
[CrossRef]

H. Schmidt and A. Imamoglu, “Giant Kerr nonlinearities obtained by electromagnetically induced transparency,” Opt. Lett. 21, 1936-1938 (1996).
[CrossRef] [PubMed]

Scully, M. O.

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229-5232 (1999).
[CrossRef]

Shahriar, M. S.

Sharpe, S. J.

A. J. Marriam, S. J. Sharpe, M. Shverdin, D. Manuszak, G. Y. Yin, and S. E. Harris, “Efficient nonlinear frequency conversion in an all-resonant double-Λ system,” Phys. Rev. Lett. 84, 5308-5311 (2000).
[CrossRef]

Shverdin, M.

A. J. Marriam, S. J. Sharpe, M. Shverdin, D. Manuszak, G. Y. Yin, and S. E. Harris, “Efficient nonlinear frequency conversion in an all-resonant double-Λ system,” Phys. Rev. Lett. 84, 5308-5311 (2000).
[CrossRef]

Spiller, T. P.

W. J. Munro, K. Nemoto, and T. P. Spiller, “Weak nonlinearities: a new route to optical quantum computation,” New J. Phys. 7, 137 (2005).
[CrossRef]

S. D. Barrett, P. Kok, K. Nemoto, R. G. Beausoleil, W. J. Munro, and T. P. Spiller, “Symmetry analyzer for nondestructive Bell-state detection using weak nonlinearities,” Phys. Rev. A 71, 060302(R) (2005).
[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]

Tsai, Z. H.

Turchette, Q. A.

Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, “Measurement of conditional phase shifts for quantum logic,” Phys. Rev. Lett. 75, 4710-4713 (1995).
[CrossRef] [PubMed]

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]

Walsworth, R. L.

D. F. Phillips, A. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin, “Storage of light in atomic vapor,” Phys. Rev. Lett. 86, 783-786 (2001).
[CrossRef] [PubMed]

Wang, H.

S. Li, X. Yang, X. Cao, C. Zhang, C. Xie, and H. Wang, “Enhanced cross-phase modulation based on a double electromagnetically induced transparency in a four-level tripod atomic system,” Phys. Rev. Lett. 101, 073602 (2008).
[CrossRef] [PubMed]

Y. Han, J. Xiao, Y. Liu, C. Zhang, H. Wang, M. Xiao, and K. Peng, “Interacting dark states with enhanced nonlinearity in an ideal four-level tripod atomic system,” Phys. Rev. A 77, 023824 (2008).
[CrossRef]

S. Li, X. Yang, X. Cao, C. Xie, and H. Wang, “Two electromagnetically induced transparency windows and an enhanced electromagnetically induced transparency signal in a four-level tripod atomic system,” J. Phys. B 40, 3211-3219 (2007).
[CrossRef]

H. Wang, D. Goorskey, and M. Xiao, “Atomic coherence induced Kerr nonlinearity enhancement in Rb vapor,” J. Mod. Opt. 49, 335-347 (2002).
[CrossRef]

H. Wang, D. Goorskey, and M. Xiao, “Enhanced Kerr nonlinearity via atomic coherence in a three-level atomic system,” Phys. Rev. Lett. 87, 073601 (2001).
[CrossRef] [PubMed]

Wang, Z.-B.

Z.-B. Wang, K.-P. Marzlin, and B. C. Sanders, “Large cross-phase modulation between slow copropagating weak pulses in Rb87,” Phys. Rev. Lett. 97, 063901 (2006).
[CrossRef] [PubMed]

Welch, G. R.

S. F. Yelin, V. A. Sautenkov, M. M. Kash, G. R. Welch, and M. D. Lukin, “Nonlinear optics via double dark resonances,” Phys. Rev. A 68, 063801 (2003).
[CrossRef]

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

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229-5232 (1999).
[CrossRef]

Woods, G.

A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467-1470 (1997).
[CrossRef]

Xia, H.

M. Jain, H. Xia, G. Y. Yin, A. J. Marriam, and S. E. Harris, “Efficient nonlinear frequency conversion with maximal atomic coherence,” Phys. Rev. Lett. 77, 4326-4329 (1996).
[CrossRef] [PubMed]

Xiao, J.

Y. Han, J. Xiao, Y. Liu, C. Zhang, H. Wang, M. Xiao, and K. Peng, “Interacting dark states with enhanced nonlinearity in an ideal four-level tripod atomic system,” Phys. Rev. A 77, 023824 (2008).
[CrossRef]

Xiao, M.

Y. Han, J. Xiao, Y. Liu, C. Zhang, H. Wang, M. Xiao, and K. Peng, “Interacting dark states with enhanced nonlinearity in an ideal four-level tripod atomic system,” Phys. Rev. A 77, 023824 (2008).
[CrossRef]

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

H. Wang, D. Goorskey, and M. Xiao, “Atomic coherence induced Kerr nonlinearity enhancement in Rb vapor,” J. Mod. Opt. 49, 335-347 (2002).
[CrossRef]

H. Wang, D. Goorskey, and M. Xiao, “Enhanced Kerr nonlinearity via atomic coherence in a three-level atomic system,” Phys. Rev. Lett. 87, 073601 (2001).
[CrossRef] [PubMed]

B. Lu, W. H. Burkett, and M. Xiao, “Nondegenerate four-wave mixing in a double-L system under the influence of coherent population trapping,” Opt. Lett. 23, 804-806 (1998).
[CrossRef]

Y. Li and M. Xiao, “Enhancement of nondegenerate four-wave mixing based on electromagnetically induced transparency in rubidium atoms,” Opt. Lett. 21, 1064-1066 (1996).
[CrossRef] [PubMed]

M. Xiao, Y. Li, S. Jin, and J. Gea-Banacloche, “Measurement of dispersive properties of electromagnetically induced transparency in Rubidium atoms,” Phys. Rev. Lett. 74, 666-669 (1995).
[CrossRef] [PubMed]

Y. Li and M. Xiao, “Electromagnetically induced transparency in a three-level Λ-type system in rubidium atoms,” Phys. Rev. A 51, R2703-R2706 (1995).
[CrossRef] [PubMed]

J. Gea-Banacloche, Y. Li, S. Jin, and M. Xiao, “Electromagnetically induced transparency in ladder-type inhomogeneously broadened media: theory and experiment,” Phys. Rev. A 51, 576-584 (1995).
[CrossRef] [PubMed]

Xie, C.

S. Li, X. Yang, X. Cao, C. Zhang, C. Xie, and H. Wang, “Enhanced cross-phase modulation based on a double electromagnetically induced transparency in a four-level tripod atomic system,” Phys. Rev. Lett. 101, 073602 (2008).
[CrossRef] [PubMed]

S. Li, X. Yang, X. Cao, C. Xie, and H. Wang, “Two electromagnetically induced transparency windows and an enhanced electromagnetically induced transparency signal in a four-level tripod atomic system,” J. Phys. B 40, 3211-3219 (2007).
[CrossRef]

Xu, Z.

Yang, X.

S. Li, X. Yang, X. Cao, C. Zhang, C. Xie, and H. Wang, “Enhanced cross-phase modulation based on a double electromagnetically induced transparency in a four-level tripod atomic system,” Phys. Rev. Lett. 101, 073602 (2008).
[CrossRef] [PubMed]

S. Li, X. Yang, X. Cao, C. Xie, and H. Wang, “Two electromagnetically induced transparency windows and an enhanced electromagnetically induced transparency signal in a four-level tripod atomic system,” J. Phys. B 40, 3211-3219 (2007).
[CrossRef]

Yashchuk, V. V.

D. Budker, D. F. Kimball, S. M. Rochester, and V. V. Yashchuk, “Nonlinear magneto-optics and reduced group velocity of light in atomic vapor with slow ground state relaxation,” Phys. Rev. Lett. 83, 1767-1770 (1999).
[CrossRef]

Ye, C. Y.

C. Y. Ye and A. S. Zibrov, “Width of the electromagnetically induced transparency resonance in atomic vapor,” Phys. Rev. A 65, 023806 (2002).
[CrossRef]

Yelin, S. F.

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A. J. Marriam, S. J. Sharpe, M. Shverdin, D. Manuszak, G. Y. Yin, and S. E. Harris, “Efficient nonlinear frequency conversion in an all-resonant double-Λ system,” Phys. Rev. Lett. 84, 5308-5311 (2000).
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Zhang, C.

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[CrossRef]

S. F. Yelin, V. A. Sautenkov, M. M. Kash, G. R. Welch, and M. D. Lukin, “Nonlinear optics via double dark resonances,” Phys. Rev. A 68, 063801 (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 (2003).
[CrossRef]

Y. Han, J. Xiao, Y. Liu, C. Zhang, H. Wang, M. Xiao, and K. Peng, “Interacting dark states with enhanced nonlinearity in an ideal four-level tripod atomic system,” Phys. Rev. A 77, 023824 (2008).
[CrossRef]

C. Y. Ye and A. S. Zibrov, “Width of the electromagnetically induced transparency resonance in atomic vapor,” Phys. Rev. A 65, 023806 (2002).
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A. Kasapi, M. Jain, G. Y. Yin, and S. E. Harris, “Electromagnetically induced transparency: propagation dynamics,” Phys. Rev. Lett. 74, 2447-2450 (1995).
[CrossRef] [PubMed]

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229-5232 (1999).
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M. D. Lukin and A. Imamoglu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Phys. Rev. Lett. 84, 1419-1422 (2000).
[CrossRef] [PubMed]

H. Kang and Y. Zhu, “Observation of large Kerr nonlinearity at low light intensities,” Phys. Rev. Lett. 91, 093601 (2003).
[CrossRef] [PubMed]

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

M. Jain, H. Xia, G. Y. Yin, A. J. Marriam, and S. E. Harris, “Efficient nonlinear frequency conversion with maximal atomic coherence,” Phys. Rev. Lett. 77, 4326-4329 (1996).
[CrossRef] [PubMed]

A. J. Marriam, S. J. Sharpe, M. Shverdin, D. Manuszak, G. Y. Yin, and S. E. Harris, “Efficient nonlinear frequency conversion in an all-resonant double-Λ system,” Phys. Rev. Lett. 84, 5308-5311 (2000).
[CrossRef]

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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Phys. Today (1)

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

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Figures (12)

Fig. 1
Fig. 1

Relevant atomic energy level diagram of the D1 line in Rb 87 atom.

Fig. 2
Fig. 2

Theoretically calculated EIT windows for the (a) probe beam and (b) trigger beam created by the coupling beam with a Rabi frequency Ω C = 70 MHz at Δ C = 0 . (c),(d) Corresponding dispersions. The other parameters are Ω P = 8 MHz , Ω T = 8 MHz , γ e = 3.5 MHz , γ 1 = 0.5 MHz , γ 2 = 1.5 MHz , γ 3 = 1.0 MHz , and N = 0.6 × 10 11 cm 3 .

Fig. 3
Fig. 3

Calculated Im ( χ P ) and Re ( χ P ) curves as the function of Δ P . (a),(b) Im ( χ P ) and Re ( χ P ) curves without the trigger beam, respectively. (c),(d) Im ( χ P ) and Re ( χ P ) curves with the trigger beam ( Ω T = 8 MHz ) , respectively. The parameters are Ω C = 70 MHz , Ω P = 8 MHz , Δ C = Δ T = 0 , γ e = 3.5 MHz , γ 1 = 0.5 MHz , γ 2 = 1.5 MHz , γ 3 = 1.0 MHz , and N = 0.6 × 10 11 cm 3 .

Fig. 4
Fig. 4

Calculated Im ( χ T ) and Re ( χ T ) curves as the function of Δ P . (a),(b) Im ( χ T ) and Re ( χ T ) curves without the probe beam, respectively. (c), (d) Im ( χ T ) and Re ( χ T ) curves with the probe beam ( Ω P = 8 MHz ) , respectively. The parameters are Ω C = 70 MHz , Ω T = 8 MHz , Δ C = Δ T = 0 , γ e = 3.5 MHz , γ 1 = 0.5 MHz , γ 2 = 1.5 MHz , γ 3 = 1.0 MHz , and N = 0.6 × 10 11 cm 3 .

Fig. 5
Fig. 5

Experimental setup. Q1 and Q2 are λ 4 wave plates.

Fig. 6
Fig. 6

The polarization directions of the (a) probe and (b) trigger beams.

Fig. 7
Fig. 7

Measured simultaneous EIT windows for the (a) probe and (b) trigger beams created by the coupling beam with a power of 14 mW and detuning Δ C = 0 .

Fig. 8
Fig. 8

Measured probe absorption and dispersion signals as the function of Δ P for Δ C = Δ T = 0 , P C = 14 mW . Curves I in (a1)–(a4) [ I in (b1)–(b4)] are the absorption (dispersion) curves without the trigger beam ( P T = 0 ) . Curves II in (a1)–(a4) [ II in (b1)–(b4)] are the absorption (dispersion) curves measured at the trigger powers P T = 300 μ W , 200 μ W , 100 μ W , and 50 μ W , respectively.

Fig. 9
Fig. 9

Measured trigger absorption and dispersion signals as the function of Δ P for Δ C = Δ T = 0 , P C = 14 mW . Curves I in (a1)–(a4) [ I in (b1)–(b4)] are the absorption (dispersion) curves without the probe beam ( P P = 0 ) . Curves II in (a1)–(a4) [ II in (b1)–(b4)] are the absorption (dispersion) curves measured at the probe powers P P = 300 μ W , 200 μ W , 100 μ W , and 50 μ W , respectively.

Fig. 10
Fig. 10

Measured probe absorptions (a1) and dispersions (b1), as well as trigger absorptions (a2) and dispersions (b2) versus Δ P for Δ C = Δ T = 0 and P C = 14 mW . The curves I of (a1) and I of (b1) are without the trigger beam, and the curves II of (a1) and II of (b1) are with the trigger beam. The curves i of (a2) and i of (b2) are without the probe beam, and the curves ii of (a2) and ii of (b2) are with the probe beam.

Fig. 11
Fig. 11

Measured probe XPM coefficient (A) at Δ P 0.7 MHz and (B) at Δ P 0.6 MHz versus the power of the applied trigger beam ( P T ) , respectively. Curves A and B are corresponding theoretical results with experimental parameters γ e = 3.5 MHz , γ 1 = 0.5 MHz , γ 2 = 1.5 MHz , γ 3 = 1.0 MHz , Ω C = 70 MHz , Ω P = 4 MHz , and N = 3.72 × 10 11 cm 3 .

Fig. 12
Fig. 12

Measured trigger XPM coefficient (A) at Δ P 0.5 MHz and (B) at Δ P 0.5 MHz versus the power of the applied probe beam ( P P ) , respectively. Curves A and B are corresponding theoretical results with experimental parameters γ e = 3.5 MHz , γ 1 = 0.5 MHz , γ 2 = 1.5 MHz , γ 3 = 1.0 MHz , Ω C = 70 MHz , Ω T = 4 MHz , and N = 3.72 × 10 11 cm 3 .

Equations (76)

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H int = H int , 1 + H int , 2 ,
H int , 1 = Δ P | e 1 e 1 | + ( Δ P Δ T ) | b 1 b 1 | + ( Δ P Δ C ) | b 3 b 3 | 2 [ Ω T 1 | e 1 b 1 | + Ω C 1 | e 1 b 3 | + Ω P 1 | e 1 a 2 | + c.c. ]
H int , 2 = Δ P | e 2 e 2 | + ( Δ P Δ T ) | b 2 b 2 | + ( Δ P Δ C ) | b 4 b 4 | 2 [ Ω T 2 | e 2 b 2 | + Ω C 2 | e 2 b 4 | + Ω P 2 | e 2 a 3 | + c.c. ]
ρ ̃ ̇ a 2 a 2 = Γ 3 ρ ̃ e 1 e 1 + Λ ( ρ ̃ b 1 b 1 + ρ ̃ b 3 b 3 ) 2 Λ ρ ̃ a 2 a 2 i ( Ω P 1 2 ρ ̃ a 2 e 1 c.c. ) ,
ρ ̃ ̇ b 1 b 1 = Γ 3 ρ ̃ e 1 e 1 + Λ ( ρ ̃ b 3 b 3 + ρ ̃ a 2 a 2 ) 2 Λ ρ ̃ b 1 b 1 i ( Ω T 1 2 ρ ̃ b 1 e 1 c.c. ) ,
ρ ̃ ̇ b 3 b 3 = Γ 3 ρ ̃ e 1 e 1 + Λ ( ρ ̃ b 1 b 1 + ρ ̃ a 2 a 2 ) 2 Λ ρ ̃ b 3 b 3 i ( Ω C 1 2 ρ ̃ b 3 e 1 c.c. ) ,
ρ ̃ ̇ e 1 e 1 = Γ ρ ̃ e 1 e 1 i ( Ω P 1 * 2 ρ ̃ e 1 a 2 + Ω T 1 * 2 ρ ̃ e 1 b 1 + Ω C 1 * 2 ρ ̃ e 1 b 3 c.c. ) ,
ρ ̃ ̇ a 2 e 1 = ( i Δ P γ e ) ρ ̃ a 2 e 1 + i Ω p 1 * 2 ( ρ ̃ e 1 e 1 ρ ̃ a 2 a 2 ) i Ω T 1 * 2 ρ ̃ a 2 b 1 i Ω c 1 * 2 ρ ̃ a 2 b 3 ,
ρ ̃ ̇ b 1 e 1 = ( i Δ T γ e ) ρ ̃ b 1 e 1 + i Ω T 1 * 2 ( ρ ̃ e 1 e 1 ρ ̃ b 1 b 1 ) i Ω P 1 * 2 ρ ̃ b 1 a 2 i Ω c 1 * 2 ρ ̃ b 1 b 3 ,
ρ ̃ ̇ b 3 e 1 = ( i Δ C γ e ) ρ ̃ b 3 e 1 + i Ω C 1 * 2 ( ρ ̃ e 1 e 1 ρ ̃ b 3 b 3 ) i Ω P 1 * 2 ρ ̃ b 3 a 2 i Ω T 1 * 2 ρ ̃ b 3 b 1 ,
ρ ̃ ̇ a 2 b 1 = [ i ( Δ P Δ T ) γ 1 ] ρ ̃ a 2 b 1 i Ω T 1 2 ρ ̃ a 2 e 1 + i Ω p 1 * 2 ρ ̃ e 1 b 1 ,
ρ ̃ ̇ a 2 b 3 = [ i ( Δ P Δ C ) γ 2 ] ρ ̃ a 2 b 3 i Ω C 1 2 ρ ̃ a 2 e 1 + i Ω P 1 * 2 ρ ̃ e 1 b 3 ,
ρ ̃ ̇ b 1 b 3 = [ i ( Δ T Δ C ) γ 3 ] ρ ̃ b 1 b 3 i Ω C 1 2 ρ ̃ b 1 e 1 + i Ω T 1 2 ρ ̃ e 1 b 3 ,
ρ ̃ ̇ a 3 a 3 = Γ 3 ρ ̃ e 2 e 2 + Λ ( ρ ̃ b 2 b 2 + ρ ̃ b 4 b 4 ) 2 Λ ρ ̃ a 3 a 3 i ( Ω P 2 2 ρ ̃ a 3 e 2 c.c. ) ,
ρ ̃ ̇ b 2 b 2 = Γ 3 ρ ̃ e 2 e 2 + Λ ( ρ ̃ b 4 b 4 + ρ ̃ a 3 a 3 ) 2 Λ ρ ̃ b 2 b 2 i ( Ω T 2 2 ρ ̃ b 2 e 2 c.c. ) ,
ρ ̃ ̇ b 4 b 4 = Γ 3 ρ ̃ e 2 e 2 + Λ ( ρ ̃ b 2 b 2 + ρ ̃ a 3 a 3 ) 2 Λ ρ ̃ b 4 b 4 i ( Ω C 2 2 ρ ̃ b 4 e 2 c.c. ) ,
ρ ̃ ̇ e 2 e 2 = Γ ρ ̃ e 2 e 2 i ( Ω P 2 * 2 ρ ̃ e 2 a 3 + Ω T 2 * 2 ρ ̃ e 2 b 2 + Ω C 2 * 2 ρ ̃ e 2 b 4 c.c. ) ,
ρ ̃ ̇ a 3 e 2 = ( i Δ P γ e ) ρ ̃ a 3 e 2 + i Ω p 2 * 2 ( ρ ̃ e 2 e 2 ρ ̃ a 3 a 3 ) i Ω T 2 * 2 ρ ̃ a 3 b 2 i Ω c 2 * 2 ρ ̃ a 3 b 4 ,
ρ ̃ ̇ b 2 e 2 = ( i Δ T γ e ) ρ ̃ b 2 e 2 + i Ω T 2 * 2 ( ρ ̃ e 2 e 2 ρ ̃ b 2 b 2 ) i Ω P 2 * 2 ρ ̃ b 2 a 3 i Ω c 2 * 2 ρ ̃ b 2 b 4 ,
ρ ̃ ̇ b 4 e 2 = ( i Δ C γ e ) ρ ̃ b 4 e 2 + i Ω C 2 * 2 ( ρ ̃ e 2 e 2 ρ ̃ b 4 b 4 ) i Ω P 2 * 2 ρ ̃ b 4 a 3 i Ω T 2 * 2 ρ ̃ b 4 b 2 ,
ρ ̃ ̇ a 3 b 2 = [ i ( Δ P Δ T ) γ 1 ] ρ ̃ a 3 b 2 i Ω T 2 2 ρ ̃ a 3 e 2 + i Ω p 2 * 2 ρ ̃ e 2 b 2 ,
ρ ̃ ̇ a 3 b 4 = [ i ( Δ P Δ C ) γ 2 ] ρ ̃ a 3 b 4 i Ω C 2 2 ρ ̃ a 3 e 2 + i Ω P 2 * 2 ρ ̃ e 2 b 4 ,
ρ ̃ ̇ b 2 b 4 = [ i ( Δ T Δ C ) γ 3 ] ρ ̃ b 2 b 4 i Ω C 2 2 ρ ̃ b 2 e 2 + i Ω T 2 2 ρ ̃ e 2 b 4 ,
χ P 1 = 2 N | μ a 2 , e 1 | 2 ρ ̃ a 2 , e 1 ɛ 0 Ω P 1 ,
χ P 2 = 2 N | μ a 3 , e 2 | 2 ρ ̃ a 3 , e 2 ɛ 0 Ω P 2 ,
χ T 1 = 2 N | μ b 1 , e 1 | 2 ρ ̃ b 1 , e 1 ɛ 0 Ω T 1 ,
χ T 2 = 2 N | μ b 2 , e 2 | 2 ρ ̃ b 2 , e 2 ɛ 0 Ω T 2 ,
χ P i = N | μ a i + 1 , e i | 2 ɛ 0 { ρ a i + 1 , a i + 1 ρ e i , e i Δ P i γ e + | Ω T i | 2 4 ( Δ P Δ T + i γ 1 ) + | Ω C i | 2 4 ( Δ P Δ C + i γ 2 ) | Ω T i | 2 ( ρ b i , b i ρ e i , e i ) 4 ( Δ P + Δ T i γ 1 ) ( Δ T + i γ e + | Ω c i | 2 4 ( Δ T Δ C i γ 3 ) ) ( Δ P i γ e + | Ω C i | 2 4 ( Δ P Δ C + i γ 2 ) ) } ,
χ T i = N | μ b i , e i | 2 ɛ 0 { ρ b i , b i ρ e i , e i Δ T i γ e | Ω P i | 2 4 ( Δ P Δ T i γ 1 ) + | Ω C i | 2 4 ( Δ T Δ C + i γ 3 ) | Ω P i | 2 ( ρ a i + 1 , a i + 1 ρ e i , e i ) 4 ( Δ P Δ T i γ 1 ) ( Δ P + i γ e + | Ω c i | 2 4 ( Δ P Δ C i γ 2 ) ) ( Δ T i γ e + | Ω C i | 2 4 ( Δ T Δ C + i γ 3 ) ) } ,
χ P i = | μ a i + 1 , e i | 2 ɛ 0 [ ρ a i + 1 , a i + 1 ρ e i , e i A i | Ω T i | 2 ( ρ b i , b i ρ e i , e i ) 4 B i ] N ( v ) d v ,
χ T i = | μ b i , e i | 2 ɛ 0 [ ρ b i , b i ρ e i , e i C i | Ω P i | 2 ( ρ a i + 1 , a i + 1 ρ e i , e i ) 4 D i ] N ( v ) d v ,
A i = Δ P i γ e ω p v c + | Ω T i | 2 4 ( Δ P Δ T + i γ 1 ) + | Ω C i | 2 4 ( Δ P Δ C + i γ 2 ) ,
B i = ( Δ P + Δ T i γ 1 ) ( Δ T + i γ e ω T v c + | Ω c i | 2 4 ( Δ T Δ C i γ 3 ) ) ( Δ P i γ e ω p v c + | Ω C i | 2 4 ( Δ P Δ C + i γ 2 ) ) ,
C i = Δ T i γ e ω T v c | Ω P i | 2 4 ( Δ P Δ T i γ 1 ) + | Ω C i | 2 4 ( Δ T Δ C + i γ 3 ) ,
D i = ( Δ P Δ T i γ 1 ) ( Δ P + i γ e ω P v c + | Ω c i | 2 4 ( Δ P Δ C i γ 2 ) ) ( Δ T i γ e ω T v c + | Ω C i | 2 4 ( Δ T Δ C + i γ 3 ) ) .
v g P d ω P d k P = c n + ω P ( d n d ω P ) = c n + ω P 2 d [ Re ( χ P ) ] d ω P ,
v g T d ω T d k T = c n + ω T ( d n d ω T ) = c n + ω T 2 d [ Re ( χ T ) ] d ω T .
n P ( 2 ) = [ Re ( χ P ) ( Ω T 0 ) Re ( χ P ) ( Ω T = 0 ) ] 2 I T ,
n T ( 2 ) = [ Re ( χ T ) ( Ω P 0 ) Re ( χ T ) ( Ω P = 0 ) ] 2 I P .
E P = ε P e i ( ω P t + Ψ p ) e α P l 2 ,
E R P = ε R P e i ( ω P t + Ψ P R ) e α R P l 2 ,
E T = ε T e i ( ω T t + Ψ T ) e α T l 2 ,
E R T = ε R T e i ( ω T t + Ψ T R ) e α R T l 2 ,
I D 1 = 1 2 c ε 0 ( 1 T 2 ) ε R P 2 e α R P l ,
I D 2 = 1 2 c ε 0 ( 1 T 2 ) ε T 2 e α T l ,
I D 3 = 1 2 c ε 0 ( 1 T 1 ) ε P 2 e α P l ,
I D 4 = 1 2 c ε 0 ( 1 T 1 ) ε R T 2 e α R T l ,
E D 5 = T 2 2 ε T e i ( ω T t + Ψ T + Ψ T ) e α T l 2 + T 1 2 ε R T e i ( ω T t + Ψ T R ) ,
E D 6 = T 2 2 ε T e i ( ω T t + Ψ T + Ψ T ) e α T l 2 + T 1 2 ε R T e i ( ω T t + Ψ T R + π ) ,
E D 7 = T 1 2 ε P e i ( ω P t + Ψ P + Ψ P ) e α P l 2 + T 2 2 ε R P e i ( ω P t + Ψ P R ) ,
E D 8 = T 1 2 ε P e i ( ω P t + Ψ P + Ψ P ) e α P l 2 + T 2 2 ε R P e i ( ω P t + Ψ P R + π ) ,
Δ I H 1 = I D 7 I D 8 = c ε 0 T 1 T 2 ε R P ε P e α P ( ω P ) l 2 cos ( Ψ H 1 + k P n P ( ω P ) l ) ,
Δ I H 2 = I D 5 I D 6 = c ε 0 T 2 T 1 ε R T ε T e α T ( ω T ) l 2 cos ( Ψ H 2 + k T n T ( ω T ) l ) ,
Ψ P = arcsin ( ( 1 T 1 ) ( 1 T 2 ) Δ I H 1 2 T 1 T 2 I D 3 I D 1 ) ,
Ψ T = arcsin ( ( 1 T 2 ) ( 1 T 1 ) Δ I H 2 2 T 2 T 1 I D 4 I D 2 ) .
ρ ̃ ̇ a 2 a 2 = Γ 3 ρ ̃ e 1 e 1 + Γ 3 ρ ̃ e 3 e 3 + Λ ( ρ ̃ b 1 b 1 + ρ ̃ b 3 b 3 ) 2 Λ ρ ̃ a 2 a 2 i ( Ω P 1 2 ρ ̃ a 2 e 1 c.c. ) ,
ρ ̃ ̇ b 1 b 1 = Γ 3 ρ ̃ e 1 e 1 + Λ ( ρ ̃ b 3 b 3 + ρ ̃ a 2 a 2 ) 2 Λ ρ ̃ b 1 b 1 i ( Ω T 1 2 ρ ̃ b 1 e 1 c.c. ) ,
ρ ̃ ̇ b 3 b 3 = Γ 3 ρ ̃ e 1 e 1 + Γ 3 ρ ̃ e 3 e 3 + Λ ( ρ ̃ b 1 b 1 + ρ ̃ a 2 a 2 + ρ ̃ b 5 b 5 ) 3 Λ ρ ̃ b 3 b 3 i ( Ω C 1 2 ρ ̃ b 3 e 1 c.c. ) ,
ρ ̃ ̇ b 5 b 5 = Γ 3 ρ ̃ e 3 e 3 + Λ ρ ̃ b 3 b 3 Λ ρ ̃ b 5 b 5 i ( Ω C 3 2 ρ ̃ b 5 e 3 c.c. ) ,
ρ ̃ ̇ e 1 e 1 = Γ ρ ̃ e 1 e 1 i ( Ω P 1 * 2 ρ ̃ e 1 a 2 + Ω T 1 * 2 ρ ̃ e 1 b 1 + Ω C 1 * 2 ρ ̃ e 1 b 3 c.c. ) ,
ρ ̃ ̇ e 3 e 3 = Γ ρ ̃ e 3 e 3 i ( Ω C 3 * 2 ρ ̃ e 3 b 5 + Ω T 3 * 2 ρ ̃ e 3 b 3 c.c. ) ,
ρ ̃ ̇ a 2 e 1 = ( i Δ P γ e ) ρ ̃ a 2 e 1 + i Ω p 1 * 2 ( ρ ̃ e 1 e 1 ρ ̃ a 2 a 2 ) i Ω T 1 * 2 ρ ̃ a 2 b 1 i Ω c 1 * 2 ρ ̃ a 2 b 3 ,
ρ ̃ ̇ b 1 e 1 = ( i Δ T γ e ) ρ ̃ b 1 e 1 + i Ω T 1 * 2 ( ρ ̃ e 1 e 1 ρ ̃ b 1 b 1 ) i Ω P 1 * 2 ρ ̃ b 1 a 2 i Ω c 1 * 2 ρ ̃ b 1 b 3 ,
ρ ̃ ̇ b 3 e 1 = ( i Δ C γ e ) ρ ̃ b 3 e 1 + i Ω C 1 * 2 ( ρ ̃ e 1 e 1 ρ ̃ b 3 b 3 ) i Ω P 1 * 2 ρ ̃ b 3 a 2 i Ω T 1 * 2 ρ ̃ b 3 b 1 ,
ρ ̃ ̇ a 2 b 1 = [ i ( Δ P Δ T ) γ 1 ] ρ ̃ a 2 b 1 i Ω T 1 2 ρ ̃ a 2 e 1 + i Ω p 1 * 2 ρ ̃ e 1 b 1 ,
ρ ̃ ̇ a 2 b 3 = [ i ( Δ P Δ C ) γ 2 ] ρ ̃ a 2 b 3 i Ω C 1 2 ρ ̃ a 2 e 1 i Ω T 3 2 ρ ̃ a 2 e 3 + i Ω P 1 * 2 ρ ̃ e 1 b 3 ,
ρ ̃ ̇ b 1 b 3 = [ i ( Δ T Δ C ) γ 3 ] ρ ̃ b 1 b 3 i Ω C 1 2 ρ ̃ b 1 e 1 i Ω T 3 2 ρ ̃ b 1 e 3 + i Ω T 1 * 2 ρ ̃ e 1 b 3 ,
ρ ̃ ̇ b 1 b 5 = [ i ( 2 Δ T 2 Δ C ) γ 3 ] ρ ̃ b 1 b 5 i Ω C 3 2 ρ ̃ b 1 e 3 + i Ω T 1 * 2 ρ ̃ e 1 b 5 ,
ρ ̃ ̇ a 2 b 5 = [ i ( Δ P + Δ T 2 Δ C ) γ 2 ] ρ ̃ a 2 b 5 i Ω C 3 2 ρ ̃ a 2 e 3 + i Ω P 1 * 2 ρ ̃ e 1 b 5 ,
ρ ̃ ̇ b 1 e 3 = [ i ( 2 Δ T Δ C ) γ e ] ρ ̃ b 1 e 3 i Ω T 3 * 2 ρ ̃ b 1 b 3 i Ω C 3 * 2 ρ ̃ b 1 b 5 + i Ω T 1 * 2 ρ ̃ e 1 e 3 ,
ρ ̃ ̇ a 2 e 3 = [ i ( Δ P Δ C + Δ T ) γ e ] ρ ̃ a 2 e 3 i Ω T 3 * 2 ρ ̃ a 2 b 3 i Ω C 3 * 2 ρ ̃ a 2 b 5 + i Ω P 1 * 2 ρ ̃ e 1 e 3 ,
ρ ̃ ̇ b 5 e 1 = [ i ( 2 Δ C Δ T ) γ e ] ρ ̃ b 5 e 1 i Ω P 1 * 2 ρ ̃ b 5 a 2 i Ω T 1 * 2 ρ ̃ b 5 b 1 i Ω C 1 * 2 ρ ̃ b 5 b 3 + i Ω C 3 * 2 ρ ̃ e 3 e 1 ,
ρ ̃ ̇ e 3 e 1 = [ i ( Δ C Δ T ) 2 γ e ] ρ ̃ e 3 e 1 i Ω P 1 * 2 ρ ̃ e 3 a 2 i Ω T 1 * 2 ρ ̃ e 3 b 1 i Ω C 1 * 2 ρ ̃ e 3 b 3 + i Ω T 3 * 2 ρ ̃ e 3 e 1 + i Ω C 3 * 2 ρ ̃ b 5 e 1 ,
ρ ̃ ̇ b 3 b 5 = [ i ( Δ T Δ C ) γ 3 ] ρ ̃ b 3 b 5 i Ω C 3 * 2 ρ ̃ b 3 e 3 + i Ω C 1 * 2 ρ ̃ e 1 b 5 + i Ω T 3 * 2 ρ ̃ e 3 b 5 ,
ρ ̃ ̇ b 3 e 3 = ( i Δ T γ e ) ρ ̃ b 3 e 3 i Ω T 3 * 2 ( ρ ̃ b 3 b 3 ρ ̃ e 3 e 3 ) i Ω C 3 * 2 ρ ̃ b 3 b 5 + i Ω C 1 * 2 ρ ̃ e 1 e 3     ,
ρ ̃ ̇ b 5 e 3 = ( i Δ C γ e ) ρ ̃ b 5 e 3 i Ω C 3 * 2 ( ρ ̃ b 5 b 5 ρ ̃ e 3 e 3 ) i Ω T 3 * 2 ρ ̃ b 5 b 3 .

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