Abstract

We experimentally demonstrate dressed multi-wave mixing (MWM) and the reflection of the probe beam due to electromagnetically induced absorption (EIA) grating can coexist in a five-level atomic ensemble. The reflection is derived from the photonic band gap (PBG) of EIA grating, which is much broader than the PBG of EIT grating. Therefore, EIA-type PBG can reflect more energy from probe than EIT-type PBG does, which can effectively affect the MWM signal. The EIA-type as well as EIT-type PBG can be controlled by multiple parameters including the frequency detunings, propagation angles and powers of the involved light fields. Also, the EIA-type PBG by considering both the linear and third-order nonlinear refractive indices is also investigated. The theoretical analysis agrees well with the experimental results. This investigation has potential applications in all-optical communication and information processing.

© 2013 Optical Society of America

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2013 (2)

Y. Zhang, C. Yuan, Y. Zhang, H. Zheng, H. Chen, C. Li, Z. Wang, and M. Xiao, “Surface solitons of four-wave mixing in an electromagnetically induced lattice,” Laser Phys. Lett.10, 055406 (2013).
[CrossRef]

D.-W. Wang, H.-T. Zhou, M.-J. Guo, J.-X. Zhang, J. Evers, and S.-Y. Zhu, “Optical diode made from a moving photonic crystal,” Phys. Rev. Lett.110, 093901 (2013).
[CrossRef] [PubMed]

2012 (1)

Y. Zhang, X. Yao, C. Yuan, P. Li, J. Yuan, W. Feng, S. Jia, and Y. Zhang, “Controllable multiwave mixing Talbot effect,” IEEE Photon. J.4, 2057–2065 (2012).
[CrossRef]

2011 (3)

Y. Zhang, Z. Wang, Z. Nie, C. Li, H. Chen, K. Lu, and M. Xiao, “Four-wave mixing dipole soliton in laser-induced atomic gratings,” Phys. Rev. Lett.106, 093904 (2011).
[CrossRef] [PubMed]

R.-G. Wan, J. Kou, L. Jiang, Y. Jiang, and J.-Y. Gao, “Electromagnetically induced grating via enhanced nonlinear modulation by spontaneously generated coherence,” Phys. Rev. A83, 033824 (2011).
[CrossRef]

J. Wen, S. Du, H. Chen, and M. Xiao, “Electromagnetically induced Talbot effect,” Appl. Phys. Lett.98, 081108 (2011).
[CrossRef]

2010 (2)

2009 (1)

Y. Zhang, C. Zuo, H. Zheng, C. Li, Z. Nie, J. Song, H. Chang, and M. Xiao, “Controlled spatial beam splitter using four-wave-mixing images,” Phys. Rev. A80, 055804 (2009).
[CrossRef]

2008 (1)

2007 (3)

Y. Zhang, A. W. Brown, and M. Xiao, “Opening four-wave mixing and six-wave mixing channels via dual electromagnetically induced transparency windows,” Phys. Rev. Lett.99, 123603 (2007).
[CrossRef] [PubMed]

D. Petrosyan, “Tunable photonic band gaps with coherently driven atoms in optical lattices,” Phys. Rev. A76, 053823 (2007).
[CrossRef]

J. Fuchs, G. J. Duffy, W. J. Rowlands, A. Lezama, P. Hannaford, and A. M. Akulshin, “Electromagnetically induced transparency and absorption due to optical and ground-state coherences in 6Li,” J. Phys. B: At. Mol. Opt. Phys.40, 1117–1129 (2007).
[CrossRef]

2006 (2)

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

Z. Zuo, J. Sun, X. Liu, Q. Jiang, G. Fu, L.-A. Wu, and P. Fu, “Generalized n-photon resonant 2n-wave mixing in an (n+ 1)-level system with phase-conjugate geometry,” Phys. Rev. Lett.97, 193904 (2006).
[CrossRef]

2005 (2)

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

A. M. Akulshin, A. Lezama, A. I. Sidorov, R. J. McLean, and P. Hannaford, “’storage of light’ in an atomic medium using electromagnetically induced absorption,” J. Phys. B: At. Mol. Opt. Phys.38, L365–L374 (2005).
[CrossRef]

2004 (2)

H. Kang, G. Hernandez, and Y. Zhu, “Resonant four-wave mixing with slow light,” Phys. Rev. A70, 061804 (2004).
[CrossRef]

J. Sun, Z. Zuo, X. Mi, Z. Yu, Q. Jiang, Y. Wang, L.-A. Wu, and P. Fu, “Two-photon resonant four-wave mixing in a dressed atomic system,” Phys. Rev. A70, 053820 (2004).
[CrossRef]

2002 (2)

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

G. C. Cardoso and J. W. R. Tabosa, “Electromagnetically induced gratings in a degenerate open two-level system,” Phys. Rev. A65, 033803 (2002).
[CrossRef]

1999 (4)

M. Mitsunaga and N. Imoto, “Observation of an electromagnetically induced grating in cold sodium atoms,” Phys. Rev. A59, 4773–4776 (1999).
[CrossRef]

G. C. Cardoso, V. R. de Carvalho, S. S. Vianna, and J. W. R. Tabosa, “Population-grating transfer in cold cesium atoms,” Phys. Rev. A59, 1408–1412 (1999).
[CrossRef]

A. Lezama, S. Barreiro, and A. M. Akulshin, “Electromagnetically induced absorption,” Phys. Rev. A59, 4732–4735 (1999).
[CrossRef]

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

1998 (1)

A. M. Akulshin, S. Barreiro, and A. Lezama, “Electromagnetically induced absorption and transparency due to resonant two-field excitation of quasidegenerate levels in rb vapor,” Phys. Rev. A57, 2996–3002 (1998).
[CrossRef]

1996 (1)

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

1995 (2)

J. Gea-Banacloche, Y.-q. Li, S.-z. Jin, and M. Xiao, “Electromagnetically induced transparency in ladder-type inhomogeneously broadened media: Theory and experiment,” Phys. Rev. A51, 576–584 (1995).
[CrossRef] [PubMed]

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

1991 (1)

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

Akulshin, A. M.

J. Fuchs, G. J. Duffy, W. J. Rowlands, A. Lezama, P. Hannaford, and A. M. Akulshin, “Electromagnetically induced transparency and absorption due to optical and ground-state coherences in 6Li,” J. Phys. B: At. Mol. Opt. Phys.40, 1117–1129 (2007).
[CrossRef]

A. M. Akulshin, A. Lezama, A. I. Sidorov, R. J. McLean, and P. Hannaford, “’storage of light’ in an atomic medium using electromagnetically induced absorption,” J. Phys. B: At. Mol. Opt. Phys.38, L365–L374 (2005).
[CrossRef]

A. Lezama, S. Barreiro, and A. M. Akulshin, “Electromagnetically induced absorption,” Phys. Rev. A59, 4732–4735 (1999).
[CrossRef]

A. M. Akulshin, S. Barreiro, and A. Lezama, “Electromagnetically induced absorption and transparency due to resonant two-field excitation of quasidegenerate levels in rb vapor,” Phys. Rev. A57, 2996–3002 (1998).
[CrossRef]

André, A.

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

Artoni, M.

J.-H. Wu, M. Artoni, and G. C. L. Rocca, “Controlling the photonic band structure of optically driven cold atoms,” J. Opt. Soc. Am. B25, 1840–1849 (2008).
[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]

Ba, N.

Barreiro, S.

A. Lezama, S. Barreiro, and A. M. Akulshin, “Electromagnetically induced absorption,” Phys. Rev. A59, 4732–4735 (1999).
[CrossRef]

A. M. Akulshin, S. Barreiro, and A. Lezama, “Electromagnetically induced absorption and transparency due to resonant two-field excitation of quasidegenerate levels in rb vapor,” Phys. Rev. A57, 2996–3002 (1998).
[CrossRef]

Boller, K.-J.

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

Brown, A. W.

Y. Zhang, A. W. Brown, and M. Xiao, “Opening four-wave mixing and six-wave mixing channels via dual electromagnetically induced transparency windows,” Phys. Rev. Lett.99, 123603 (2007).
[CrossRef] [PubMed]

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

Cardoso, G. C.

G. C. Cardoso and J. W. R. Tabosa, “Electromagnetically induced gratings in a degenerate open two-level system,” Phys. Rev. A65, 033803 (2002).
[CrossRef]

G. C. Cardoso, V. R. de Carvalho, S. S. Vianna, and J. W. R. Tabosa, “Population-grating transfer in cold cesium atoms,” Phys. Rev. A59, 1408–1412 (1999).
[CrossRef]

Chang, H.

Y. Zhang, C. Zuo, H. Zheng, C. Li, Z. Nie, J. Song, H. Chang, and M. Xiao, “Controlled spatial beam splitter using four-wave-mixing images,” Phys. Rev. A80, 055804 (2009).
[CrossRef]

Chen, H.

Y. Zhang, C. Yuan, Y. Zhang, H. Zheng, H. Chen, C. Li, Z. Wang, and M. Xiao, “Surface solitons of four-wave mixing in an electromagnetically induced lattice,” Laser Phys. Lett.10, 055406 (2013).
[CrossRef]

Y. Zhang, Z. Wang, Z. Nie, C. Li, H. Chen, K. Lu, and M. Xiao, “Four-wave mixing dipole soliton in laser-induced atomic gratings,” Phys. Rev. Lett.106, 093904 (2011).
[CrossRef] [PubMed]

J. Wen, S. Du, H. Chen, and M. Xiao, “Electromagnetically induced Talbot effect,” Appl. Phys. Lett.98, 081108 (2011).
[CrossRef]

Cronin-Golomb, M.

Cui, C.-L.

de Carvalho, V. R.

G. C. Cardoso, V. R. de Carvalho, S. S. Vianna, and J. W. R. Tabosa, “Population-grating transfer in cold cesium atoms,” Phys. Rev. A59, 1408–1412 (1999).
[CrossRef]

Donoghue, J.

Du, S.

J. Wen, S. Du, H. Chen, and M. Xiao, “Electromagnetically induced Talbot effect,” Appl. Phys. Lett.98, 081108 (2011).
[CrossRef]

Duffy, G. J.

J. Fuchs, G. J. Duffy, W. J. Rowlands, A. Lezama, P. Hannaford, and A. M. Akulshin, “Electromagnetically induced transparency and absorption due to optical and ground-state coherences in 6Li,” J. Phys. B: At. Mol. Opt. Phys.40, 1117–1129 (2007).
[CrossRef]

Evers, J.

D.-W. Wang, H.-T. Zhou, M.-J. Guo, J.-X. Zhang, J. Evers, and S.-Y. Zhu, “Optical diode made from a moving photonic crystal,” Phys. Rev. Lett.110, 093901 (2013).
[CrossRef] [PubMed]

Feng, W.

Y. Zhang, X. Yao, C. Yuan, P. Li, J. Yuan, W. Feng, S. Jia, and Y. Zhang, “Controllable multiwave mixing Talbot effect,” IEEE Photon. J.4, 2057–2065 (2012).
[CrossRef]

Fleischhauer, M.

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

Fu, G.

Z. Zuo, J. Sun, X. Liu, Q. Jiang, G. Fu, L.-A. Wu, and P. Fu, “Generalized n-photon resonant 2n-wave mixing in an (n+ 1)-level system with phase-conjugate geometry,” Phys. Rev. Lett.97, 193904 (2006).
[CrossRef]

Fu, P.

Z. Zuo, J. Sun, X. Liu, Q. Jiang, G. Fu, L.-A. Wu, and P. Fu, “Generalized n-photon resonant 2n-wave mixing in an (n+ 1)-level system with phase-conjugate geometry,” Phys. Rev. Lett.97, 193904 (2006).
[CrossRef]

J. Sun, Z. Zuo, X. Mi, Z. Yu, Q. Jiang, Y. Wang, L.-A. Wu, and P. Fu, “Two-photon resonant four-wave mixing in a dressed atomic system,” Phys. Rev. A70, 053820 (2004).
[CrossRef]

Fuchs, J.

J. Fuchs, G. J. Duffy, W. J. Rowlands, A. Lezama, P. Hannaford, and A. M. Akulshin, “Electromagnetically induced transparency and absorption due to optical and ground-state coherences in 6Li,” J. Phys. B: At. Mol. Opt. Phys.40, 1117–1129 (2007).
[CrossRef]

Gao, J.-W.

Gao, J.-Y.

R.-G. Wan, J. Kou, L. Jiang, Y. Jiang, and J.-Y. Gao, “Electromagnetically induced grating via enhanced nonlinear modulation by spontaneously generated coherence,” Phys. Rev. A83, 033824 (2011).
[CrossRef]

Gea-Banacloche, J.

J. Gea-Banacloche, Y.-q. Li, S.-z. Jin, and M. Xiao, “Electromagnetically induced transparency in ladder-type inhomogeneously broadened media: Theory and experiment,” Phys. Rev. A51, 576–584 (1995).
[CrossRef] [PubMed]

Guo, M.-J.

D.-W. Wang, H.-T. Zhou, M.-J. Guo, J.-X. Zhang, J. Evers, and S.-Y. Zhu, “Optical diode made from a moving photonic crystal,” Phys. Rev. Lett.110, 093901 (2013).
[CrossRef] [PubMed]

Hannaford, P.

J. Fuchs, G. J. Duffy, W. J. Rowlands, A. Lezama, P. Hannaford, and A. M. Akulshin, “Electromagnetically induced transparency and absorption due to optical and ground-state coherences in 6Li,” J. Phys. B: At. Mol. Opt. Phys.40, 1117–1129 (2007).
[CrossRef]

A. M. Akulshin, A. Lezama, A. I. Sidorov, R. J. McLean, and P. Hannaford, “’storage of light’ in an atomic medium using electromagnetically induced absorption,” J. Phys. B: At. Mol. Opt. Phys.38, L365–L374 (2005).
[CrossRef]

Harris, S. E.

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

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

Hemmer, P. R.

Hernandez, G.

H. Kang, G. Hernandez, and Y. Zhu, “Resonant four-wave mixing with slow light,” Phys. Rev. A70, 061804 (2004).
[CrossRef]

Imamolu, A.

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

Imoto, N.

M. Mitsunaga and N. Imoto, “Observation of an electromagnetically induced grating in cold sodium atoms,” Phys. Rev. A59, 4773–4776 (1999).
[CrossRef]

Jain, M.

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

Jia, S.

Y. Zhang, X. Yao, C. Yuan, P. Li, J. Yuan, W. Feng, S. Jia, and Y. Zhang, “Controllable multiwave mixing Talbot effect,” IEEE Photon. J.4, 2057–2065 (2012).
[CrossRef]

Jiang, L.

R.-G. Wan, J. Kou, L. Jiang, Y. Jiang, and J.-Y. Gao, “Electromagnetically induced grating via enhanced nonlinear modulation by spontaneously generated coherence,” Phys. Rev. A83, 033824 (2011).
[CrossRef]

Jiang, Q.

Z. Zuo, J. Sun, X. Liu, Q. Jiang, G. Fu, L.-A. Wu, and P. Fu, “Generalized n-photon resonant 2n-wave mixing in an (n+ 1)-level system with phase-conjugate geometry,” Phys. Rev. Lett.97, 193904 (2006).
[CrossRef]

J. Sun, Z. Zuo, X. Mi, Z. Yu, Q. Jiang, Y. Wang, L.-A. Wu, and P. Fu, “Two-photon resonant four-wave mixing in a dressed atomic system,” Phys. Rev. A70, 053820 (2004).
[CrossRef]

Jiang, Y.

R.-G. Wan, J. Kou, L. Jiang, Y. Jiang, and J.-Y. Gao, “Electromagnetically induced grating via enhanced nonlinear modulation by spontaneously generated coherence,” Phys. Rev. A83, 033824 (2011).
[CrossRef]

Jin, S.-z.

J. Gea-Banacloche, Y.-q. Li, S.-z. Jin, and M. Xiao, “Electromagnetically induced transparency in ladder-type inhomogeneously broadened media: Theory and experiment,” Phys. Rev. A51, 576–584 (1995).
[CrossRef] [PubMed]

Kang, H.

H. Kang, G. Hernandez, and Y. Zhu, “Resonant four-wave mixing with slow light,” Phys. Rev. A70, 061804 (2004).
[CrossRef]

Katz, D. P.

Kou, J.

R.-G. Wan, J. Kou, L. Jiang, Y. Jiang, and J.-Y. Gao, “Electromagnetically induced grating via enhanced nonlinear modulation by spontaneously generated coherence,” Phys. Rev. A83, 033824 (2011).
[CrossRef]

Kumar, P.

La Rocca, G. C.

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

Lezama, A.

J. Fuchs, G. J. Duffy, W. J. Rowlands, A. Lezama, P. Hannaford, and A. M. Akulshin, “Electromagnetically induced transparency and absorption due to optical and ground-state coherences in 6Li,” J. Phys. B: At. Mol. Opt. Phys.40, 1117–1129 (2007).
[CrossRef]

A. M. Akulshin, A. Lezama, A. I. Sidorov, R. J. McLean, and P. Hannaford, “’storage of light’ in an atomic medium using electromagnetically induced absorption,” J. Phys. B: At. Mol. Opt. Phys.38, L365–L374 (2005).
[CrossRef]

A. Lezama, S. Barreiro, and A. M. Akulshin, “Electromagnetically induced absorption,” Phys. Rev. A59, 4732–4735 (1999).
[CrossRef]

A. M. Akulshin, S. Barreiro, and A. Lezama, “Electromagnetically induced absorption and transparency due to resonant two-field excitation of quasidegenerate levels in rb vapor,” Phys. Rev. A57, 2996–3002 (1998).
[CrossRef]

Li, C.

Y. Zhang, C. Yuan, Y. Zhang, H. Zheng, H. Chen, C. Li, Z. Wang, and M. Xiao, “Surface solitons of four-wave mixing in an electromagnetically induced lattice,” Laser Phys. Lett.10, 055406 (2013).
[CrossRef]

Y. Zhang, Z. Wang, Z. Nie, C. Li, H. Chen, K. Lu, and M. Xiao, “Four-wave mixing dipole soliton in laser-induced atomic gratings,” Phys. Rev. Lett.106, 093904 (2011).
[CrossRef] [PubMed]

Y. Zhang, C. Zuo, H. Zheng, C. Li, Z. Nie, J. Song, H. Chang, and M. Xiao, “Controlled spatial beam splitter using four-wave-mixing images,” Phys. Rev. A80, 055804 (2009).
[CrossRef]

Li, P.

Y. Zhang, X. Yao, C. Yuan, P. Li, J. Yuan, W. Feng, S. Jia, and Y. Zhang, “Controllable multiwave mixing Talbot effect,” IEEE Photon. J.4, 2057–2065 (2012).
[CrossRef]

Li, Y.-q.

J. Gea-Banacloche, Y.-q. Li, S.-z. Jin, and M. Xiao, “Electromagnetically induced transparency in ladder-type inhomogeneously broadened media: Theory and experiment,” Phys. Rev. A51, 576–584 (1995).
[CrossRef] [PubMed]

Liu, X.

Z. Zuo, J. Sun, X. Liu, Q. Jiang, G. Fu, L.-A. Wu, and P. Fu, “Generalized n-photon resonant 2n-wave mixing in an (n+ 1)-level system with phase-conjugate geometry,” Phys. Rev. Lett.97, 193904 (2006).
[CrossRef]

Lu, K.

Y. Zhang, Z. Wang, Z. Nie, C. Li, H. Chen, K. Lu, and M. Xiao, “Four-wave mixing dipole soliton in laser-induced atomic gratings,” Phys. Rev. Lett.106, 093904 (2011).
[CrossRef] [PubMed]

Lukin, M. D.

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

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

Matsko, A. B.

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

McLean, R. J.

A. M. Akulshin, A. Lezama, A. I. Sidorov, R. J. McLean, and P. Hannaford, “’storage of light’ in an atomic medium using electromagnetically induced absorption,” J. Phys. B: At. Mol. Opt. Phys.38, L365–L374 (2005).
[CrossRef]

Merriam, A. J.

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

Mi, X.

J. Sun, Z. Zuo, X. Mi, Z. Yu, Q. Jiang, Y. Wang, L.-A. Wu, and P. Fu, “Two-photon resonant four-wave mixing in a dressed atomic system,” Phys. Rev. A70, 053820 (2004).
[CrossRef]

Mitsunaga, M.

M. Mitsunaga and N. Imoto, “Observation of an electromagnetically induced grating in cold sodium atoms,” Phys. Rev. A59, 4773–4776 (1999).
[CrossRef]

Nie, Z.

Y. Zhang, Z. Wang, Z. Nie, C. Li, H. Chen, K. Lu, and M. Xiao, “Four-wave mixing dipole soliton in laser-induced atomic gratings,” Phys. Rev. Lett.106, 093904 (2011).
[CrossRef] [PubMed]

Y. Zhang, C. Zuo, H. Zheng, C. Li, Z. Nie, J. Song, H. Chang, and M. Xiao, “Controlled spatial beam splitter using four-wave-mixing images,” Phys. Rev. A80, 055804 (2009).
[CrossRef]

Petrosyan, D.

D. Petrosyan, “Tunable photonic band gaps with coherently driven atoms in optical lattices,” Phys. Rev. A76, 053823 (2007).
[CrossRef]

Rocca, G. C. L.

Rowlands, W. J.

J. Fuchs, G. J. Duffy, W. J. Rowlands, A. Lezama, P. Hannaford, and A. M. Akulshin, “Electromagnetically induced transparency and absorption due to optical and ground-state coherences in 6Li,” J. Phys. B: At. Mol. Opt. Phys.40, 1117–1129 (2007).
[CrossRef]

Scully, M. O.

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

Shahriar, M. S.

Sidorov, A. I.

A. M. Akulshin, A. Lezama, A. I. Sidorov, R. J. McLean, and P. Hannaford, “’storage of light’ in an atomic medium using electromagnetically induced absorption,” J. Phys. B: At. Mol. Opt. Phys.38, L365–L374 (2005).
[CrossRef]

Song, J.

Y. Zhang, C. Zuo, H. Zheng, C. Li, Z. Nie, J. Song, H. Chang, and M. Xiao, “Controlled spatial beam splitter using four-wave-mixing images,” Phys. Rev. A80, 055804 (2009).
[CrossRef]

Sun, J.

Z. Zuo, J. Sun, X. Liu, Q. Jiang, G. Fu, L.-A. Wu, and P. Fu, “Generalized n-photon resonant 2n-wave mixing in an (n+ 1)-level system with phase-conjugate geometry,” Phys. Rev. Lett.97, 193904 (2006).
[CrossRef]

J. Sun, Z. Zuo, X. Mi, Z. Yu, Q. Jiang, Y. Wang, L.-A. Wu, and P. Fu, “Two-photon resonant four-wave mixing in a dressed atomic system,” Phys. Rev. A70, 053820 (2004).
[CrossRef]

Tabosa, J. W. R.

G. C. Cardoso and J. W. R. Tabosa, “Electromagnetically induced gratings in a degenerate open two-level system,” Phys. Rev. A65, 033803 (2002).
[CrossRef]

G. C. Cardoso, V. R. de Carvalho, S. S. Vianna, and J. W. R. Tabosa, “Population-grating transfer in cold cesium atoms,” Phys. Rev. A59, 1408–1412 (1999).
[CrossRef]

Vianna, S. S.

G. C. Cardoso, V. R. de Carvalho, S. S. Vianna, and J. W. R. Tabosa, “Population-grating transfer in cold cesium atoms,” Phys. Rev. A59, 1408–1412 (1999).
[CrossRef]

Wan, R.-G.

R.-G. Wan, J. Kou, L. Jiang, Y. Jiang, and J.-Y. Gao, “Electromagnetically induced grating via enhanced nonlinear modulation by spontaneously generated coherence,” Phys. Rev. A83, 033824 (2011).
[CrossRef]

Wang, D.-W.

D.-W. Wang, H.-T. Zhou, M.-J. Guo, J.-X. Zhang, J. Evers, and S.-Y. Zhu, “Optical diode made from a moving photonic crystal,” Phys. Rev. Lett.110, 093901 (2013).
[CrossRef] [PubMed]

Wang, Y.

J. Sun, Z. Zuo, X. Mi, Z. Yu, Q. Jiang, Y. Wang, L.-A. Wu, and P. Fu, “Two-photon resonant four-wave mixing in a dressed atomic system,” Phys. Rev. A70, 053820 (2004).
[CrossRef]

Wang, Z.

Y. Zhang, C. Yuan, Y. Zhang, H. Zheng, H. Chen, C. Li, Z. Wang, and M. Xiao, “Surface solitons of four-wave mixing in an electromagnetically induced lattice,” Laser Phys. Lett.10, 055406 (2013).
[CrossRef]

Y. Zhang, Z. Wang, Z. Nie, C. Li, H. Chen, K. Lu, and M. Xiao, “Four-wave mixing dipole soliton in laser-induced atomic gratings,” Phys. Rev. Lett.106, 093904 (2011).
[CrossRef] [PubMed]

Wen, J.

J. Wen, S. Du, H. Chen, and M. Xiao, “Electromagnetically induced Talbot effect,” Appl. Phys. Lett.98, 081108 (2011).
[CrossRef]

Wu, J.-H.

Wu, L.-A.

Z. Zuo, J. Sun, X. Liu, Q. Jiang, G. Fu, L.-A. Wu, and P. Fu, “Generalized n-photon resonant 2n-wave mixing in an (n+ 1)-level system with phase-conjugate geometry,” Phys. Rev. Lett.97, 193904 (2006).
[CrossRef]

J. Sun, Z. Zuo, X. Mi, Z. Yu, Q. Jiang, Y. Wang, L.-A. Wu, and P. Fu, “Two-photon resonant four-wave mixing in a dressed atomic system,” Phys. Rev. A70, 053820 (2004).
[CrossRef]

Xia, H.

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

Xiao, M.

Y. Zhang, C. Yuan, Y. Zhang, H. Zheng, H. Chen, C. Li, Z. Wang, and M. Xiao, “Surface solitons of four-wave mixing in an electromagnetically induced lattice,” Laser Phys. Lett.10, 055406 (2013).
[CrossRef]

Y. Zhang, Z. Wang, Z. Nie, C. Li, H. Chen, K. Lu, and M. Xiao, “Four-wave mixing dipole soliton in laser-induced atomic gratings,” Phys. Rev. Lett.106, 093904 (2011).
[CrossRef] [PubMed]

J. Wen, S. Du, H. Chen, and M. Xiao, “Electromagnetically induced Talbot effect,” Appl. Phys. Lett.98, 081108 (2011).
[CrossRef]

Y. Zhang, C. Zuo, H. Zheng, C. Li, Z. Nie, J. Song, H. Chang, and M. Xiao, “Controlled spatial beam splitter using four-wave-mixing images,” Phys. Rev. A80, 055804 (2009).
[CrossRef]

Y. Zhang, A. W. Brown, and M. Xiao, “Opening four-wave mixing and six-wave mixing channels via dual electromagnetically induced transparency windows,” Phys. Rev. Lett.99, 123603 (2007).
[CrossRef] [PubMed]

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

J. Gea-Banacloche, Y.-q. Li, S.-z. Jin, and M. Xiao, “Electromagnetically induced transparency in ladder-type inhomogeneously broadened media: Theory and experiment,” Phys. Rev. A51, 576–584 (1995).
[CrossRef] [PubMed]

Yao, X.

Y. Zhang, X. Yao, C. Yuan, P. Li, J. Yuan, W. Feng, S. Jia, and Y. Zhang, “Controllable multiwave mixing Talbot effect,” IEEE Photon. J.4, 2057–2065 (2012).
[CrossRef]

Yin, G. Y.

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

Yu, Z.

J. Sun, Z. Zuo, X. Mi, Z. Yu, Q. Jiang, Y. Wang, L.-A. Wu, and P. Fu, “Two-photon resonant four-wave mixing in a dressed atomic system,” Phys. Rev. A70, 053820 (2004).
[CrossRef]

Yuan, C.

Y. Zhang, C. Yuan, Y. Zhang, H. Zheng, H. Chen, C. Li, Z. Wang, and M. Xiao, “Surface solitons of four-wave mixing in an electromagnetically induced lattice,” Laser Phys. Lett.10, 055406 (2013).
[CrossRef]

Y. Zhang, X. Yao, C. Yuan, P. Li, J. Yuan, W. Feng, S. Jia, and Y. Zhang, “Controllable multiwave mixing Talbot effect,” IEEE Photon. J.4, 2057–2065 (2012).
[CrossRef]

Yuan, J.

Y. Zhang, X. Yao, C. Yuan, P. Li, J. Yuan, W. Feng, S. Jia, and Y. Zhang, “Controllable multiwave mixing Talbot effect,” IEEE Photon. J.4, 2057–2065 (2012).
[CrossRef]

Zhang, J.-X.

D.-W. Wang, H.-T. Zhou, M.-J. Guo, J.-X. Zhang, J. Evers, and S.-Y. Zhu, “Optical diode made from a moving photonic crystal,” Phys. Rev. Lett.110, 093901 (2013).
[CrossRef] [PubMed]

Zhang, Y.

Y. Zhang, C. Yuan, Y. Zhang, H. Zheng, H. Chen, C. Li, Z. Wang, and M. Xiao, “Surface solitons of four-wave mixing in an electromagnetically induced lattice,” Laser Phys. Lett.10, 055406 (2013).
[CrossRef]

Y. Zhang, C. Yuan, Y. Zhang, H. Zheng, H. Chen, C. Li, Z. Wang, and M. Xiao, “Surface solitons of four-wave mixing in an electromagnetically induced lattice,” Laser Phys. Lett.10, 055406 (2013).
[CrossRef]

Y. Zhang, X. Yao, C. Yuan, P. Li, J. Yuan, W. Feng, S. Jia, and Y. Zhang, “Controllable multiwave mixing Talbot effect,” IEEE Photon. J.4, 2057–2065 (2012).
[CrossRef]

Y. Zhang, X. Yao, C. Yuan, P. Li, J. Yuan, W. Feng, S. Jia, and Y. Zhang, “Controllable multiwave mixing Talbot effect,” IEEE Photon. J.4, 2057–2065 (2012).
[CrossRef]

Y. Zhang, Z. Wang, Z. Nie, C. Li, H. Chen, K. Lu, and M. Xiao, “Four-wave mixing dipole soliton in laser-induced atomic gratings,” Phys. Rev. Lett.106, 093904 (2011).
[CrossRef] [PubMed]

C.-L. Cui, J.-H. Wu, J.-W. Gao, Y. Zhang, and N. Ba, “Double photonic bandgaps dynamically induced in a tripod system of cold atoms,” Opt. Express18, 4538–4546 (2010).
[CrossRef] [PubMed]

J.-W. Gao, Y. Zhang, N. Ba, C.-L. Cui, and J.-H. Wu, “Dynamically induced double photonic bandgaps in the presence of spontaneously generated coherence,” Opt. Lett.35, 709–711 (2010).
[CrossRef] [PubMed]

Y. Zhang, C. Zuo, H. Zheng, C. Li, Z. Nie, J. Song, H. Chang, and M. Xiao, “Controlled spatial beam splitter using four-wave-mixing images,” Phys. Rev. A80, 055804 (2009).
[CrossRef]

Y. Zhang, A. W. Brown, and M. Xiao, “Opening four-wave mixing and six-wave mixing channels via dual electromagnetically induced transparency windows,” Phys. Rev. Lett.99, 123603 (2007).
[CrossRef] [PubMed]

Zheng, H.

Y. Zhang, C. Yuan, Y. Zhang, H. Zheng, H. Chen, C. Li, Z. Wang, and M. Xiao, “Surface solitons of four-wave mixing in an electromagnetically induced lattice,” Laser Phys. Lett.10, 055406 (2013).
[CrossRef]

Y. Zhang, C. Zuo, H. Zheng, C. Li, Z. Nie, J. Song, H. Chang, and M. Xiao, “Controlled spatial beam splitter using four-wave-mixing images,” Phys. Rev. A80, 055804 (2009).
[CrossRef]

Zhou, H.-T.

D.-W. Wang, H.-T. Zhou, M.-J. Guo, J.-X. Zhang, J. Evers, and S.-Y. Zhu, “Optical diode made from a moving photonic crystal,” Phys. Rev. Lett.110, 093901 (2013).
[CrossRef] [PubMed]

Zhu, S.-Y.

D.-W. Wang, H.-T. Zhou, M.-J. Guo, J.-X. Zhang, J. Evers, and S.-Y. Zhu, “Optical diode made from a moving photonic crystal,” Phys. Rev. Lett.110, 093901 (2013).
[CrossRef] [PubMed]

Zhu, Y.

H. Kang, G. Hernandez, and Y. Zhu, “Resonant four-wave mixing with slow light,” Phys. Rev. A70, 061804 (2004).
[CrossRef]

Zuo, C.

Y. Zhang, C. Zuo, H. Zheng, C. Li, Z. Nie, J. Song, H. Chang, and M. Xiao, “Controlled spatial beam splitter using four-wave-mixing images,” Phys. Rev. A80, 055804 (2009).
[CrossRef]

Zuo, Z.

Z. Zuo, J. Sun, X. Liu, Q. Jiang, G. Fu, L.-A. Wu, and P. Fu, “Generalized n-photon resonant 2n-wave mixing in an (n+ 1)-level system with phase-conjugate geometry,” Phys. Rev. Lett.97, 193904 (2006).
[CrossRef]

J. Sun, Z. Zuo, X. Mi, Z. Yu, Q. Jiang, Y. Wang, L.-A. Wu, and P. Fu, “Two-photon resonant four-wave mixing in a dressed atomic system,” Phys. Rev. A70, 053820 (2004).
[CrossRef]

Appl. Phys. Lett. (1)

J. Wen, S. Du, H. Chen, and M. Xiao, “Electromagnetically induced Talbot effect,” Appl. Phys. Lett.98, 081108 (2011).
[CrossRef]

IEEE Photon. J. (1)

Y. Zhang, X. Yao, C. Yuan, P. Li, J. Yuan, W. Feng, S. Jia, and Y. Zhang, “Controllable multiwave mixing Talbot effect,” IEEE Photon. J.4, 2057–2065 (2012).
[CrossRef]

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

J. Phys. B: At. Mol. Opt. Phys. (2)

A. M. Akulshin, A. Lezama, A. I. Sidorov, R. J. McLean, and P. Hannaford, “’storage of light’ in an atomic medium using electromagnetically induced absorption,” J. Phys. B: At. Mol. Opt. Phys.38, L365–L374 (2005).
[CrossRef]

J. Fuchs, G. J. Duffy, W. J. Rowlands, A. Lezama, P. Hannaford, and A. M. Akulshin, “Electromagnetically induced transparency and absorption due to optical and ground-state coherences in 6Li,” J. Phys. B: At. Mol. Opt. Phys.40, 1117–1129 (2007).
[CrossRef]

Laser Phys. Lett. (1)

Y. Zhang, C. Yuan, Y. Zhang, H. Zheng, H. Chen, C. Li, Z. Wang, and M. Xiao, “Surface solitons of four-wave mixing in an electromagnetically induced lattice,” Laser Phys. Lett.10, 055406 (2013).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. A (11)

Y. Zhang, C. Zuo, H. Zheng, C. Li, Z. Nie, J. Song, H. Chang, and M. Xiao, “Controlled spatial beam splitter using four-wave-mixing images,” Phys. Rev. A80, 055804 (2009).
[CrossRef]

J. Sun, Z. Zuo, X. Mi, Z. Yu, Q. Jiang, Y. Wang, L.-A. Wu, and P. Fu, “Two-photon resonant four-wave mixing in a dressed atomic system,” Phys. Rev. A70, 053820 (2004).
[CrossRef]

R.-G. Wan, J. Kou, L. Jiang, Y. Jiang, and J.-Y. Gao, “Electromagnetically induced grating via enhanced nonlinear modulation by spontaneously generated coherence,” Phys. Rev. A83, 033824 (2011).
[CrossRef]

J. Gea-Banacloche, Y.-q. Li, S.-z. Jin, and M. Xiao, “Electromagnetically induced transparency in ladder-type inhomogeneously broadened media: Theory and experiment,” Phys. Rev. A51, 576–584 (1995).
[CrossRef] [PubMed]

A. M. Akulshin, S. Barreiro, and A. Lezama, “Electromagnetically induced absorption and transparency due to resonant two-field excitation of quasidegenerate levels in rb vapor,” Phys. Rev. A57, 2996–3002 (1998).
[CrossRef]

A. Lezama, S. Barreiro, and A. M. Akulshin, “Electromagnetically induced absorption,” Phys. Rev. A59, 4732–4735 (1999).
[CrossRef]

D. Petrosyan, “Tunable photonic band gaps with coherently driven atoms in optical lattices,” Phys. Rev. A76, 053823 (2007).
[CrossRef]

H. Kang, G. Hernandez, and Y. Zhu, “Resonant four-wave mixing with slow light,” Phys. Rev. A70, 061804 (2004).
[CrossRef]

G. C. Cardoso, V. R. de Carvalho, S. S. Vianna, and J. W. R. Tabosa, “Population-grating transfer in cold cesium atoms,” Phys. Rev. A59, 1408–1412 (1999).
[CrossRef]

G. C. Cardoso and J. W. R. Tabosa, “Electromagnetically induced gratings in a degenerate open two-level system,” Phys. Rev. A65, 033803 (2002).
[CrossRef]

M. Mitsunaga and N. Imoto, “Observation of an electromagnetically induced grating in cold sodium atoms,” Phys. Rev. A59, 4773–4776 (1999).
[CrossRef]

Phys. Rev. Lett. (9)

Y. Zhang, Z. Wang, Z. Nie, C. Li, H. Chen, K. Lu, and M. Xiao, “Four-wave mixing dipole soliton in laser-induced atomic gratings,” Phys. Rev. Lett.106, 093904 (2011).
[CrossRef] [PubMed]

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

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

Y. Zhang, A. W. Brown, and M. Xiao, “Opening four-wave mixing and six-wave mixing channels via dual electromagnetically induced transparency windows,” Phys. Rev. Lett.99, 123603 (2007).
[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]

D.-W. Wang, H.-T. Zhou, M.-J. Guo, J.-X. Zhang, J. Evers, and S.-Y. Zhu, “Optical diode made from a moving photonic crystal,” Phys. Rev. Lett.110, 093901 (2013).
[CrossRef] [PubMed]

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

Z. Zuo, J. Sun, X. Liu, Q. Jiang, G. Fu, L.-A. Wu, and P. Fu, “Generalized n-photon resonant 2n-wave mixing in an (n+ 1)-level system with phase-conjugate geometry,” Phys. Rev. Lett.97, 193904 (2006).
[CrossRef]

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

Other (1)

D. A. Steck, “Alkali D line data,” available online at http://steck.us/alkalidata .

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Figures (5)

Fig. 2
Fig. 2

(a1)–(a3) Measured PT, FWM, and FS signal versus Δ2 for different Δ1, respectively. (b1)–(b3) Similar to (a1)–(a3) but versus Δ2 for different Δ4. (c1) and (c2) Measured PT and FWM signal with a larger power of E1 and dressing effects from E2 and E4. (d1)–(d3) Theoretical PBGs versus Δ12 and Δ24 correspond to (a)–(c), respectively.

Fig. 3
Fig. 3

(a1)–(a3) PT, FWM and FS signals versus Δ2 with different Δ1 and Δ4 = 0. (b1)–(b3) Similar to (a1)–(a3) but with different Δ4 and Δ1 = 0. The other parameters are P1 =3.6 mW, P2 =30.6 mW, P′2 =5.4 mW and P3 =14.0 mW. (c1)–(c2) PT and FWM signals versus Δ2 with Δ1 fixed. (d1)–(d2) Similar to (c1)–(c2) but with Δ1 =100 MHz when E1, E2 and E′2 are turned on. Other parameters are P1 =4 mW and P′2 =8 mW with P2 =2.4, 8, 12, 16 and 20 mW from bottom to top. (e1)–(e3) Theoretical PBGs correspond to (a)–(d), respectively.

Fig. 4
Fig. 4

PT (a), SWM (b) and FS signals (c) versus Δ2 at Δ1 = −40 MHz with ΔΦ = −3π/4, −5π/8, −π/2, −π/4, 0 and π/4 from the bottom to top, respectively. (d) Theoretical PBG versus Δ2 and ΔΦ, and the dashed lines correspond to the curves in (a). Other parameters are P1 =7 mW, P2 =17 mW, and P′2 =8 mW.

Fig. 5
Fig. 5

(a1) PT (first row), FWM (second row) signals versus Δ2 and PT images (third row) with Δ1 =10, 15, 20, 25, and 30 MHz from left to right, respectively. (a2) Similar to (a1) but with Δ1 far away from TPR. The x–labels just display the values of Δ1, at which the experimental results are obtained (have noting to do with the x–scale). (b1) and (b2) Theoretical PBGs correspond to (a1) and (a2), respectively.

Equations (8)

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

χ = i N μ 10 2 ε 0 ( 1 d 1 + | G 2 | 2 / d 2 1 d 1 + | G 3 | 2 / d 3 ) i N μ 10 2 ε 0 ( G 2 2 + G 2 2 d 2 ( d 1 + | G 2 | 2 / d 2 ) 2 G 3 2 + G 3 2 d 3 ( d 1 + | G 3 | 2 / d 3 ) 2 ) ,
q = ± 1 2 k [ k 2 ( 1 + c 0 ) k 2 ] 2 k 1 4 c 1 2 ,
c 0 = i N μ 10 2 ε 0 [ ( A 1 B 2 A 1 B 2 ) ( C ( 1 B 2 ) 3 C ( 1 B 2 ) 3 ) ] ,
c 1 = i N μ 10 2 ε 0 [ ( A 1 B 2 1 B 1 B 2 A 1 B 2 1 B 1 B 2 ) + ( B C ( 1 B 2 ) 3 B C ( 1 B 2 ) 3 ) ] ,
A = d 2 G 2 2 + G 2 2 + d 1 d 2 , B = 2 G 2 G 2 G 2 2 + G 2 2 + d 1 d 2 , C = ( G 2 2 + G 2 2 ) d 2 ( G 2 2 + G 2 2 + d 1 d 2 ) 2 , A = d 3 G 3 2 + G 3 2 + d 1 d 3 , B = 2 G 3 G 3 G 3 2 + G 3 2 + d 1 d 3 , C = ( G 3 2 + G 3 2 ) d 3 ( G 3 2 + G 3 2 + d 1 d 3 ) 2 .
Δ gap = ( Δ 1 + Ω 10 ) [ 2 + 2 Re ( c 0 ) + Re ( c 1 ) ] 2 1 + Re ( c 0 ) ( 1 + 1 2 [ Im ( c 0 ) 1 + Re ( c 0 ) ] 2 ) .
Δ gap = ω 0 n 0 3 [ 2 n 0 2 + Re ( c 1 ) ] 2 n 0 4 + Im 2 ( c 0 ) ,
1 I in ( I R 1 + I R 2 + I R 3 + I M + R + T ) = 1 ,

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