Abstract

Self-induced transparency pulses propagating in a random medium embedded in a two-level system can transfer energy to localized Anderson states. This allows the onset of two-level laserlike action.

© 2011 Optical Society of America

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  1. V. V. Kozlov, Phys. Rev. A 56, 1607 (1997).
    [CrossRef]
  2. S. L. McCall and E. L. Hahn, Phys. Rev. Lett. 18, 908 (1967).
    [CrossRef]
  3. G. Kurizki, A. E. Kozhekin, and M. B. A. T. Opatrny, Prog. Opt. 42, 93 (2001).
    [CrossRef]
  4. B. Mantsyzov, I. Mel’nikov, and J. Aitchison, IEEE J. Sel. Top. Quantum Electron. 10, 893 (2004).
    [CrossRef]
  5. N. N. Rosanov, V. V. Kozlov, and S. Wabnitz, Phys. Rev. A 81, 043815 (2010).
    [CrossRef]
  6. A. I. Maimistov, A. M. Basharov, S. O. Elyutin, and Y. M. Sklyarov, Phys. Rep. 191, 1 (1990).
    [CrossRef]
  7. H. Cao, X. Jiang, Y. Ling, J. X. Xu, and C. M. Soukoulis, Phys. Rev. B 67, 161101R (2003).
    [CrossRef]
  8. D. S. Wiersma, Nat. Phys. 4, 359 (2008).
    [CrossRef]
  9. R. G. S. El-Dardiry, A. P. Mosk, O. L. Muskens, and A. Lagendijk, Phys. Rev. A 81, 043830 (2010).
    [CrossRef]
  10. J. Andreasen, A. Asatryan, L. Botten, M. Byrne, H. Cao, L. Ge, L. Labonté, P. Sebbah, A. D. Stone, H. Türeci, and C. Vanneste, ““Modes of random lasers,” arXiv e-prints, arxiv.org/abs/1001.4671.
  11. T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
    [CrossRef] [PubMed]
  12. I. V. Shadrivov, K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, and Y. S. Kivshar, Phys. Rev. Lett. 104, 123902 (2010).
    [CrossRef] [PubMed]
  13. C. Conti and A. Fratalocchi, Nat. Phys. 4, 794 (2008).
    [CrossRef]
  14. C. Conti, E. D’Asaro, S. Stivala, A. Busacca, and G. Assanto, Opt. Lett. 35, 3760 (2010).
    [CrossRef] [PubMed]
  15. C. Conti and L. Leuzzi, Phys. Rev. B 83, 134204 (2011).
    [CrossRef]
  16. S. John, Phys. Rev. Lett. 58, 2486 (1987).
    [CrossRef] [PubMed]
  17. A. Szameit, Y. V. Kartashov, P. Zeil, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, V. A. Vysloukh, and L. Torner, Opt. Lett. 35, 1172 (2010).
    [CrossRef] [PubMed]
  18. A. Kozhekin and G. Kurizki, Phys. Rev. Lett. 74, 5020(1995).
    [CrossRef] [PubMed]
  19. V. Folli and C. Conti, Phys. Rev. Lett. 104, 193901 (2010).
    [CrossRef] [PubMed]
  20. V. V. Kozlov and A. B. Matsko, J. Opt. Soc. Am. B 17, 1031 (2000).
    [CrossRef]
  21. B. Alfassi, T. Schwartz, and M. Segev, in CLEO/QELS 2009 Proceedings (Optical Society of America, 2009), p. 2285.
  22. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. A 77, 051802 (2008).
    [CrossRef]
  23. R. W. Ziolkowski, J. M. Arnold, and D. M. Gogny, Phys. Rev. A 52, 3082 (1995).
    [CrossRef] [PubMed]

2011

C. Conti and L. Leuzzi, Phys. Rev. B 83, 134204 (2011).
[CrossRef]

2010

A. Szameit, Y. V. Kartashov, P. Zeil, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, V. A. Vysloukh, and L. Torner, Opt. Lett. 35, 1172 (2010).
[CrossRef] [PubMed]

R. G. S. El-Dardiry, A. P. Mosk, O. L. Muskens, and A. Lagendijk, Phys. Rev. A 81, 043830 (2010).
[CrossRef]

I. V. Shadrivov, K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, and Y. S. Kivshar, Phys. Rev. Lett. 104, 123902 (2010).
[CrossRef] [PubMed]

N. N. Rosanov, V. V. Kozlov, and S. Wabnitz, Phys. Rev. A 81, 043815 (2010).
[CrossRef]

V. Folli and C. Conti, Phys. Rev. Lett. 104, 193901 (2010).
[CrossRef] [PubMed]

C. Conti, E. D’Asaro, S. Stivala, A. Busacca, and G. Assanto, Opt. Lett. 35, 3760 (2010).
[CrossRef] [PubMed]

2008

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. A 77, 051802 (2008).
[CrossRef]

D. S. Wiersma, Nat. Phys. 4, 359 (2008).
[CrossRef]

C. Conti and A. Fratalocchi, Nat. Phys. 4, 794 (2008).
[CrossRef]

2007

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef] [PubMed]

2004

B. Mantsyzov, I. Mel’nikov, and J. Aitchison, IEEE J. Sel. Top. Quantum Electron. 10, 893 (2004).
[CrossRef]

2003

H. Cao, X. Jiang, Y. Ling, J. X. Xu, and C. M. Soukoulis, Phys. Rev. B 67, 161101R (2003).
[CrossRef]

2001

G. Kurizki, A. E. Kozhekin, and M. B. A. T. Opatrny, Prog. Opt. 42, 93 (2001).
[CrossRef]

2000

1997

V. V. Kozlov, Phys. Rev. A 56, 1607 (1997).
[CrossRef]

1995

A. Kozhekin and G. Kurizki, Phys. Rev. Lett. 74, 5020(1995).
[CrossRef] [PubMed]

R. W. Ziolkowski, J. M. Arnold, and D. M. Gogny, Phys. Rev. A 52, 3082 (1995).
[CrossRef] [PubMed]

1990

A. I. Maimistov, A. M. Basharov, S. O. Elyutin, and Y. M. Sklyarov, Phys. Rep. 191, 1 (1990).
[CrossRef]

1987

S. John, Phys. Rev. Lett. 58, 2486 (1987).
[CrossRef] [PubMed]

1967

S. L. McCall and E. L. Hahn, Phys. Rev. Lett. 18, 908 (1967).
[CrossRef]

Aitchison, J.

B. Mantsyzov, I. Mel’nikov, and J. Aitchison, IEEE J. Sel. Top. Quantum Electron. 10, 893 (2004).
[CrossRef]

Alfassi, B.

B. Alfassi, T. Schwartz, and M. Segev, in CLEO/QELS 2009 Proceedings (Optical Society of America, 2009), p. 2285.

Andreasen, J.

J. Andreasen, A. Asatryan, L. Botten, M. Byrne, H. Cao, L. Ge, L. Labonté, P. Sebbah, A. D. Stone, H. Türeci, and C. Vanneste, ““Modes of random lasers,” arXiv e-prints, arxiv.org/abs/1001.4671.

Arnold, J. M.

R. W. Ziolkowski, J. M. Arnold, and D. M. Gogny, Phys. Rev. A 52, 3082 (1995).
[CrossRef] [PubMed]

Asatryan, A.

J. Andreasen, A. Asatryan, L. Botten, M. Byrne, H. Cao, L. Ge, L. Labonté, P. Sebbah, A. D. Stone, H. Türeci, and C. Vanneste, ““Modes of random lasers,” arXiv e-prints, arxiv.org/abs/1001.4671.

Assanto, G.

Bartal, G.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef] [PubMed]

Basharov, A. M.

A. I. Maimistov, A. M. Basharov, S. O. Elyutin, and Y. M. Sklyarov, Phys. Rep. 191, 1 (1990).
[CrossRef]

Bliokh, K. Y.

I. V. Shadrivov, K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, and Y. S. Kivshar, Phys. Rev. Lett. 104, 123902 (2010).
[CrossRef] [PubMed]

Bliokh, Y. P.

I. V. Shadrivov, K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, and Y. S. Kivshar, Phys. Rev. Lett. 104, 123902 (2010).
[CrossRef] [PubMed]

Botten, L.

J. Andreasen, A. Asatryan, L. Botten, M. Byrne, H. Cao, L. Ge, L. Labonté, P. Sebbah, A. D. Stone, H. Türeci, and C. Vanneste, ““Modes of random lasers,” arXiv e-prints, arxiv.org/abs/1001.4671.

Busacca, A.

Byrne, M.

J. Andreasen, A. Asatryan, L. Botten, M. Byrne, H. Cao, L. Ge, L. Labonté, P. Sebbah, A. D. Stone, H. Türeci, and C. Vanneste, ““Modes of random lasers,” arXiv e-prints, arxiv.org/abs/1001.4671.

Cao, H.

H. Cao, X. Jiang, Y. Ling, J. X. Xu, and C. M. Soukoulis, Phys. Rev. B 67, 161101R (2003).
[CrossRef]

J. Andreasen, A. Asatryan, L. Botten, M. Byrne, H. Cao, L. Ge, L. Labonté, P. Sebbah, A. D. Stone, H. Türeci, and C. Vanneste, ““Modes of random lasers,” arXiv e-prints, arxiv.org/abs/1001.4671.

Conti, C.

C. Conti and L. Leuzzi, Phys. Rev. B 83, 134204 (2011).
[CrossRef]

C. Conti, E. D’Asaro, S. Stivala, A. Busacca, and G. Assanto, Opt. Lett. 35, 3760 (2010).
[CrossRef] [PubMed]

V. Folli and C. Conti, Phys. Rev. Lett. 104, 193901 (2010).
[CrossRef] [PubMed]

C. Conti and A. Fratalocchi, Nat. Phys. 4, 794 (2008).
[CrossRef]

D’Asaro, E.

Dreisow, F.

El-Dardiry, R. G. S.

R. G. S. El-Dardiry, A. P. Mosk, O. L. Muskens, and A. Lagendijk, Phys. Rev. A 81, 043830 (2010).
[CrossRef]

Elyutin, S. O.

A. I. Maimistov, A. M. Basharov, S. O. Elyutin, and Y. M. Sklyarov, Phys. Rep. 191, 1 (1990).
[CrossRef]

Fishman, S.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef] [PubMed]

Folli, V.

V. Folli and C. Conti, Phys. Rev. Lett. 104, 193901 (2010).
[CrossRef] [PubMed]

Fratalocchi, A.

C. Conti and A. Fratalocchi, Nat. Phys. 4, 794 (2008).
[CrossRef]

Freilikher, V.

I. V. Shadrivov, K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, and Y. S. Kivshar, Phys. Rev. Lett. 104, 123902 (2010).
[CrossRef] [PubMed]

Ge, L.

J. Andreasen, A. Asatryan, L. Botten, M. Byrne, H. Cao, L. Ge, L. Labonté, P. Sebbah, A. D. Stone, H. Türeci, and C. Vanneste, ““Modes of random lasers,” arXiv e-prints, arxiv.org/abs/1001.4671.

Gogny, D. M.

R. W. Ziolkowski, J. M. Arnold, and D. M. Gogny, Phys. Rev. A 52, 3082 (1995).
[CrossRef] [PubMed]

Hahn, E. L.

S. L. McCall and E. L. Hahn, Phys. Rev. Lett. 18, 908 (1967).
[CrossRef]

Heinrich, M.

Jiang, X.

H. Cao, X. Jiang, Y. Ling, J. X. Xu, and C. M. Soukoulis, Phys. Rev. B 67, 161101R (2003).
[CrossRef]

John, S.

S. John, Phys. Rev. Lett. 58, 2486 (1987).
[CrossRef] [PubMed]

Kartashov, Y. V.

Keil, R.

Kivshar, Y. S.

I. V. Shadrivov, K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, and Y. S. Kivshar, Phys. Rev. Lett. 104, 123902 (2010).
[CrossRef] [PubMed]

Kozhekin, A.

A. Kozhekin and G. Kurizki, Phys. Rev. Lett. 74, 5020(1995).
[CrossRef] [PubMed]

Kozhekin, A. E.

G. Kurizki, A. E. Kozhekin, and M. B. A. T. Opatrny, Prog. Opt. 42, 93 (2001).
[CrossRef]

Kozlov, V. V.

N. N. Rosanov, V. V. Kozlov, and S. Wabnitz, Phys. Rev. A 81, 043815 (2010).
[CrossRef]

V. V. Kozlov and A. B. Matsko, J. Opt. Soc. Am. B 17, 1031 (2000).
[CrossRef]

V. V. Kozlov, Phys. Rev. A 56, 1607 (1997).
[CrossRef]

Kurizki, G.

G. Kurizki, A. E. Kozhekin, and M. B. A. T. Opatrny, Prog. Opt. 42, 93 (2001).
[CrossRef]

A. Kozhekin and G. Kurizki, Phys. Rev. Lett. 74, 5020(1995).
[CrossRef] [PubMed]

Labonté, L.

J. Andreasen, A. Asatryan, L. Botten, M. Byrne, H. Cao, L. Ge, L. Labonté, P. Sebbah, A. D. Stone, H. Türeci, and C. Vanneste, ““Modes of random lasers,” arXiv e-prints, arxiv.org/abs/1001.4671.

Lagendijk, A.

R. G. S. El-Dardiry, A. P. Mosk, O. L. Muskens, and A. Lagendijk, Phys. Rev. A 81, 043830 (2010).
[CrossRef]

Leuzzi, L.

C. Conti and L. Leuzzi, Phys. Rev. B 83, 134204 (2011).
[CrossRef]

Ling, Y.

H. Cao, X. Jiang, Y. Ling, J. X. Xu, and C. M. Soukoulis, Phys. Rev. B 67, 161101R (2003).
[CrossRef]

Maimistov, A. I.

A. I. Maimistov, A. M. Basharov, S. O. Elyutin, and Y. M. Sklyarov, Phys. Rep. 191, 1 (1990).
[CrossRef]

Mantsyzov, B.

B. Mantsyzov, I. Mel’nikov, and J. Aitchison, IEEE J. Sel. Top. Quantum Electron. 10, 893 (2004).
[CrossRef]

Matsko, A. B.

McCall, S. L.

S. L. McCall and E. L. Hahn, Phys. Rev. Lett. 18, 908 (1967).
[CrossRef]

Mel’nikov, I.

B. Mantsyzov, I. Mel’nikov, and J. Aitchison, IEEE J. Sel. Top. Quantum Electron. 10, 893 (2004).
[CrossRef]

Mosk, A. P.

R. G. S. El-Dardiry, A. P. Mosk, O. L. Muskens, and A. Lagendijk, Phys. Rev. A 81, 043830 (2010).
[CrossRef]

Muskens, O. L.

R. G. S. El-Dardiry, A. P. Mosk, O. L. Muskens, and A. Lagendijk, Phys. Rev. A 81, 043830 (2010).
[CrossRef]

Nolte, S.

Opatrny, M. B. A. T.

G. Kurizki, A. E. Kozhekin, and M. B. A. T. Opatrny, Prog. Opt. 42, 93 (2001).
[CrossRef]

Rosanov, N. N.

N. N. Rosanov, V. V. Kozlov, and S. Wabnitz, Phys. Rev. A 81, 043815 (2010).
[CrossRef]

Schwartz, T.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef] [PubMed]

B. Alfassi, T. Schwartz, and M. Segev, in CLEO/QELS 2009 Proceedings (Optical Society of America, 2009), p. 2285.

Sebbah, P.

J. Andreasen, A. Asatryan, L. Botten, M. Byrne, H. Cao, L. Ge, L. Labonté, P. Sebbah, A. D. Stone, H. Türeci, and C. Vanneste, ““Modes of random lasers,” arXiv e-prints, arxiv.org/abs/1001.4671.

Segev, M.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef] [PubMed]

B. Alfassi, T. Schwartz, and M. Segev, in CLEO/QELS 2009 Proceedings (Optical Society of America, 2009), p. 2285.

Shadrivov, I. V.

I. V. Shadrivov, K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, and Y. S. Kivshar, Phys. Rev. Lett. 104, 123902 (2010).
[CrossRef] [PubMed]

Sklyarov, Y. M.

A. I. Maimistov, A. M. Basharov, S. O. Elyutin, and Y. M. Sklyarov, Phys. Rep. 191, 1 (1990).
[CrossRef]

Soukoulis, C. M.

H. Cao, X. Jiang, Y. Ling, J. X. Xu, and C. M. Soukoulis, Phys. Rev. B 67, 161101R (2003).
[CrossRef]

Stivala, S.

Stone, A. D.

J. Andreasen, A. Asatryan, L. Botten, M. Byrne, H. Cao, L. Ge, L. Labonté, P. Sebbah, A. D. Stone, H. Türeci, and C. Vanneste, ““Modes of random lasers,” arXiv e-prints, arxiv.org/abs/1001.4671.

Szameit, A.

Torner, L.

Tünnermann, A.

Türeci, H.

J. Andreasen, A. Asatryan, L. Botten, M. Byrne, H. Cao, L. Ge, L. Labonté, P. Sebbah, A. D. Stone, H. Türeci, and C. Vanneste, ““Modes of random lasers,” arXiv e-prints, arxiv.org/abs/1001.4671.

Vanneste, C.

J. Andreasen, A. Asatryan, L. Botten, M. Byrne, H. Cao, L. Ge, L. Labonté, P. Sebbah, A. D. Stone, H. Türeci, and C. Vanneste, ““Modes of random lasers,” arXiv e-prints, arxiv.org/abs/1001.4671.

Vysloukh, V. A.

Wabnitz, S.

N. N. Rosanov, V. V. Kozlov, and S. Wabnitz, Phys. Rev. A 81, 043815 (2010).
[CrossRef]

Wiersma, D. S.

D. S. Wiersma, Nat. Phys. 4, 359 (2008).
[CrossRef]

Xu, J. X.

H. Cao, X. Jiang, Y. Ling, J. X. Xu, and C. M. Soukoulis, Phys. Rev. B 67, 161101R (2003).
[CrossRef]

Zeil, P.

Ziolkowski, R. W.

R. W. Ziolkowski, J. M. Arnold, and D. M. Gogny, Phys. Rev. A 52, 3082 (1995).
[CrossRef] [PubMed]

IEEE J. Sel. Top. Quantum Electron.

B. Mantsyzov, I. Mel’nikov, and J. Aitchison, IEEE J. Sel. Top. Quantum Electron. 10, 893 (2004).
[CrossRef]

J. Opt. Soc. Am. B

Nat. Phys.

D. S. Wiersma, Nat. Phys. 4, 359 (2008).
[CrossRef]

C. Conti and A. Fratalocchi, Nat. Phys. 4, 794 (2008).
[CrossRef]

Nature

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef] [PubMed]

Opt. Lett.

Phys. Rep.

A. I. Maimistov, A. M. Basharov, S. O. Elyutin, and Y. M. Sklyarov, Phys. Rep. 191, 1 (1990).
[CrossRef]

Phys. Rev. A

R. G. S. El-Dardiry, A. P. Mosk, O. L. Muskens, and A. Lagendijk, Phys. Rev. A 81, 043830 (2010).
[CrossRef]

N. N. Rosanov, V. V. Kozlov, and S. Wabnitz, Phys. Rev. A 81, 043815 (2010).
[CrossRef]

V. V. Kozlov, Phys. Rev. A 56, 1607 (1997).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. A 77, 051802 (2008).
[CrossRef]

R. W. Ziolkowski, J. M. Arnold, and D. M. Gogny, Phys. Rev. A 52, 3082 (1995).
[CrossRef] [PubMed]

Phys. Rev. B

H. Cao, X. Jiang, Y. Ling, J. X. Xu, and C. M. Soukoulis, Phys. Rev. B 67, 161101R (2003).
[CrossRef]

C. Conti and L. Leuzzi, Phys. Rev. B 83, 134204 (2011).
[CrossRef]

Phys. Rev. Lett.

S. John, Phys. Rev. Lett. 58, 2486 (1987).
[CrossRef] [PubMed]

A. Kozhekin and G. Kurizki, Phys. Rev. Lett. 74, 5020(1995).
[CrossRef] [PubMed]

V. Folli and C. Conti, Phys. Rev. Lett. 104, 193901 (2010).
[CrossRef] [PubMed]

I. V. Shadrivov, K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, and Y. S. Kivshar, Phys. Rev. Lett. 104, 123902 (2010).
[CrossRef] [PubMed]

S. L. McCall and E. L. Hahn, Phys. Rev. Lett. 18, 908 (1967).
[CrossRef]

Prog. Opt.

G. Kurizki, A. E. Kozhekin, and M. B. A. T. Opatrny, Prog. Opt. 42, 93 (2001).
[CrossRef]

Other

J. Andreasen, A. Asatryan, L. Botten, M. Byrne, H. Cao, L. Ge, L. Labonté, P. Sebbah, A. D. Stone, H. Türeci, and C. Vanneste, ““Modes of random lasers,” arXiv e-prints, arxiv.org/abs/1001.4671.

B. Alfassi, T. Schwartz, and M. Segev, in CLEO/QELS 2009 Proceedings (Optical Society of America, 2009), p. 2285.

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Figures (4)

Fig. 1
Fig. 1

(a) Structure length of 200 μm , homogeneous two-level medium from 7.5 μm to 50 μm ; disordered two-level medium ( ϵ r = 1 ) from 50 μm to 150 μm ; initial population ρ 30 = 1 and randomly alternated by nonresonant layers ( ρ 30 = 0 , ϵ r = 11 ). A snapshot of the SIT pulse is shown before impinging on the structure; (b) input pulse spectrum for τ = 100 fs (dashed line); transmitted spectrum without the resonant medium showing the photonic bandgap (continuous line, γ = 0 ) in the ordered case, and disorder-induced resonances in the bandgap for γ = 0.5

Fig. 2
Fig. 2

SIT population inversion ρ 3 in (a) low-index ordered case ( ϵ r = 1.5 , γ = 0 ) and in (b) high-index disordered case ( ϵ r = 11 , γ = 1 ). The insets show the corresponding electric field. ρ 3 appears periodically modulated as ρ 3 = 0 in the absence of the two-level system (dielectric layers with ϵ r > 1 ).

Fig. 3
Fig. 3

(a) Structure length of 200 μm , homogeneous two-level medium from 7.5 μm to 50 μm ; disordered two-level medium ( ϵ r = 1 ) from 50 μm to 150 μm ; initial population ρ 30 = 1 and randomly alternated by nonresonant layers ( ρ 30 = 0 , ϵ r = 11 ). A snapshot of the SIT pulse is shown before impinging on the structure; (b) input pulse spectrum for τ = 100 fs (dashed line); transmitted spectrum without the resonant medium showing the photonic bandgap (continuous line, γ = 0 ) in the ordered case, and disorder-induced resonances in the bandgap for γ = 0.5

Fig. 4
Fig. 4

(a) Temporal profiles of the reflected electric field in the highly scattering case; (b) corresponding spectrum (continuous line) compared with the input SIT pulse (dashed line).

Equations (3)

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[ z + ( 1 c 1 v g ) ξ ] A = k B + S 1 ,
ξ B = 1 2 A N + S 2 , ξ N = ( B * A + B A * ) + S 3 ,
δ X ( z ) 2 = V B 2 z 4 β 4 ,

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