Abstract

A new scheme is suggested to manipulate the probe transitions (and hence the optical properties of atomic vapors) via double-control destructive and constructive quantum interferences. The influence of phase coherence between the two control transitions on the probe transition is also studied. The most remarkable feature of the present scheme is that the optical properties (absorption, transparency and dispersion) of an atomic system can be manipulated using this double-control multi-pathway interferences (multiple routes to excitation). It is also shown that a four-level system will exhibit a two-level resonant absorption because the two control levels (driven by the two control fields) form a dark state (and hence a destructive quantum interference occurs between the two control transitions). However, the present four-level system will exhibit electromagnetically induced transparency to the probe field when the three lower levels (including the probe level and the two control levels) form a three-level dark state. The present scenario has potential applications in new devices (e.g. logic gates and sensitive optical switches) and new techniques (e.g. quantum coherent information storage).

© 2007 Optical Society of America

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References

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  1. M. Fleischhauer andM. O. Scully, "Quantum sensitivity limits of an optical magnetometer based on atomic phase coherence," Phys. Rev. A 49, 1973-1986 (1994).
    [CrossRef] [PubMed]
  2. S. E. Harris, "Electromagnetically induced transparency," Phys. Today 50(7), 36-42 (1997) and references therein.
    [CrossRef]
  3. S. Y. Zhu and M. O. Scully, "Spectral line elimination and spontaneous emission cancellation via quantum interference," Phys. Rev. Lett. 76, 388-391 (1996).
    [CrossRef] [PubMed]
  4. J. Q. Shen, "Quantum-vacuum geometric phases in the noncoplanarly curved fiber system," Eur. Phys. J. D 30,259-264 (2004).
    [CrossRef]
  5. J. Q. Shen, "Negative refractive index in gyrotropically magnetoelectric media," Phys. Rev. B 73, 045113(1-5) (2006).
    [CrossRef]
  6. J. P. Marangos, "Electromagnetically induced transparency," J. Mod. Opt. 45,471-503 (1998).
    [CrossRef]
  7. J. L. Cohen and P. R. Berman, "Amplification without inversion: Understanding probability amplitudes, quantum interference, and Feynman rules in a strongly driven system," Phys. Rev. A 55,3900-3917 (1997).
    [CrossRef]
  8. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature 397, 594-598 (1999) and references therein.
    [CrossRef]
  9. R. R. Moseley, S. Shepherd, D. J. Fulton, B. D. Sinclair, and M. H. Dunn, "Spatial consequences of electromagnetically induced transparency: Observation of electromagnetically induced focusing," Phys. Rev. Lett. 74,670-673 (1995).
    [CrossRef] [PubMed]
  10. L. J. Wang, A. Kuzmich, and A. Dogariu, "Gain-assisted superluminal pulse propagation," Nature 406, 277-279 (2000) and references therein.
    [CrossRef] [PubMed]
  11. P. Arve, P. J¨anes, and Lars Thyl´en, "Propagation of two-dimensional pulses in electromagnetically induced transparency media," Phys. Rev. A 69, 063809(1-8) (2004).
    [CrossRef]
  12. J. Q. Shen and S. He, "Dimension-sensitive optical responses of electromagnetically induced transparency vapor in a waveguide," Phys. Rev. A 74, 063831(1-6) (2006).
  13. Y. Q. Li andM. Xiao, "Transient properties of an electromagnetically induced transparency in three-level atoms," Opt. Lett. 20, 1489-1491 (1995).
    [CrossRef] [PubMed]
  14. H. Schmidt and A. Imamo¡glu, "Giant Kerr nonlinearities obtained by electromagnetically induced transparency," Opt. Lett. 21,1936-1938 (1996).
    [CrossRef] [PubMed]
  15. R.Y. Chiao, "Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations," Phys. Rev. A 48, R34-R37 (1993).
    [CrossRef] [PubMed]
  16. E. L. Bolda, J. C. Garrison, and R. Y. Chiao, "Optical pulse propagation at negative group velocities due to a nearby gain line," Phys. Rev. A 49, 2938-2947 (1994).
    [CrossRef] [PubMed]
  17. S. Sangu, K. Kobayashi, A. Shojiguchi, and M. Ohtsu, "Logic and functional operations using a near-field optically coupled quantum-dot system," Phys. Rev. B 69, 115334(1-13) (2004).
    [CrossRef]
  18. T. Kawazoe, K. Kobayashi, S. Sangu, and M. Ohtsu, "Demonstration of a nanophotonic switching operation by optical near-field energy transfer," Appl. Phys. Lett. 82, 2957-2959 (2003).
    [CrossRef]
  19. T. Kawazoe, K. Kobayashi, andM. Ohtsu,"A nanophotonic NOT-gate using near-field optically coupled quantum dots," 2005 Conference on Lasers & Electro-Optics (CLEO), Baltimore, MD, USA, 728-730 (2005).
    [CrossRef]
  20. J. Q. Yao, H. B. Wu, and H. Wang, "The transient optical properties in four-level atomic medium induced by quantum interference effect,"Acta Sin. Quantum Opt. 9,121-125 (2003).

2004 (1)

J. Q. Shen, "Quantum-vacuum geometric phases in the noncoplanarly curved fiber system," Eur. Phys. J. D 30,259-264 (2004).
[CrossRef]

2003 (2)

T. Kawazoe, K. Kobayashi, S. Sangu, and M. Ohtsu, "Demonstration of a nanophotonic switching operation by optical near-field energy transfer," Appl. Phys. Lett. 82, 2957-2959 (2003).
[CrossRef]

J. Q. Yao, H. B. Wu, and H. Wang, "The transient optical properties in four-level atomic medium induced by quantum interference effect,"Acta Sin. Quantum Opt. 9,121-125 (2003).

1998 (1)

J. P. Marangos, "Electromagnetically induced transparency," J. Mod. Opt. 45,471-503 (1998).
[CrossRef]

1997 (1)

J. L. Cohen and P. R. Berman, "Amplification without inversion: Understanding probability amplitudes, quantum interference, and Feynman rules in a strongly driven system," Phys. Rev. A 55,3900-3917 (1997).
[CrossRef]

1996 (2)

S. Y. Zhu and M. O. Scully, "Spectral line elimination and spontaneous emission cancellation via quantum interference," Phys. Rev. Lett. 76, 388-391 (1996).
[CrossRef] [PubMed]

H. Schmidt and A. Imamo¡glu, "Giant Kerr nonlinearities obtained by electromagnetically induced transparency," Opt. Lett. 21,1936-1938 (1996).
[CrossRef] [PubMed]

1995 (2)

Y. Q. Li andM. Xiao, "Transient properties of an electromagnetically induced transparency in three-level atoms," Opt. Lett. 20, 1489-1491 (1995).
[CrossRef] [PubMed]

R. R. Moseley, S. Shepherd, D. J. Fulton, B. D. Sinclair, and M. H. Dunn, "Spatial consequences of electromagnetically induced transparency: Observation of electromagnetically induced focusing," Phys. Rev. Lett. 74,670-673 (1995).
[CrossRef] [PubMed]

1994 (2)

M. Fleischhauer andM. O. Scully, "Quantum sensitivity limits of an optical magnetometer based on atomic phase coherence," Phys. Rev. A 49, 1973-1986 (1994).
[CrossRef] [PubMed]

E. L. Bolda, J. C. Garrison, and R. Y. Chiao, "Optical pulse propagation at negative group velocities due to a nearby gain line," Phys. Rev. A 49, 2938-2947 (1994).
[CrossRef] [PubMed]

1993 (1)

R.Y. Chiao, "Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations," Phys. Rev. A 48, R34-R37 (1993).
[CrossRef] [PubMed]

Arve, P.

P. Arve, P. J¨anes, and Lars Thyl´en, "Propagation of two-dimensional pulses in electromagnetically induced transparency media," Phys. Rev. A 69, 063809(1-8) (2004).
[CrossRef]

Behroozi, C. H.

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

Berman, P. R.

J. L. Cohen and P. R. Berman, "Amplification without inversion: Understanding probability amplitudes, quantum interference, and Feynman rules in a strongly driven system," Phys. Rev. A 55,3900-3917 (1997).
[CrossRef]

Bolda, E. L.

E. L. Bolda, J. C. Garrison, and R. Y. Chiao, "Optical pulse propagation at negative group velocities due to a nearby gain line," Phys. Rev. A 49, 2938-2947 (1994).
[CrossRef] [PubMed]

Chiao, R. Y.

E. L. Bolda, J. C. Garrison, and R. Y. Chiao, "Optical pulse propagation at negative group velocities due to a nearby gain line," Phys. Rev. A 49, 2938-2947 (1994).
[CrossRef] [PubMed]

Chiao, R.Y.

R.Y. Chiao, "Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations," Phys. Rev. A 48, R34-R37 (1993).
[CrossRef] [PubMed]

Cohen, J. L.

J. L. Cohen and P. R. Berman, "Amplification without inversion: Understanding probability amplitudes, quantum interference, and Feynman rules in a strongly driven system," Phys. Rev. A 55,3900-3917 (1997).
[CrossRef]

Dogariu, A.

L. J. Wang, A. Kuzmich, and A. Dogariu, "Gain-assisted superluminal pulse propagation," Nature 406, 277-279 (2000) and references therein.
[CrossRef] [PubMed]

Dunn, M. H.

R. R. Moseley, S. Shepherd, D. J. Fulton, B. D. Sinclair, and M. H. Dunn, "Spatial consequences of electromagnetically induced transparency: Observation of electromagnetically induced focusing," Phys. Rev. Lett. 74,670-673 (1995).
[CrossRef] [PubMed]

Dutton, Z.

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

Fleischhauer, M.

M. Fleischhauer andM. O. Scully, "Quantum sensitivity limits of an optical magnetometer based on atomic phase coherence," Phys. Rev. A 49, 1973-1986 (1994).
[CrossRef] [PubMed]

Fulton, D. J.

R. R. Moseley, S. Shepherd, D. J. Fulton, B. D. Sinclair, and M. H. Dunn, "Spatial consequences of electromagnetically induced transparency: Observation of electromagnetically induced focusing," Phys. Rev. Lett. 74,670-673 (1995).
[CrossRef] [PubMed]

Garrison, J. C.

E. L. Bolda, J. C. Garrison, and R. Y. Chiao, "Optical pulse propagation at negative group velocities due to a nearby gain line," Phys. Rev. A 49, 2938-2947 (1994).
[CrossRef] [PubMed]

Harris, S. E.

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

S. E. Harris, "Electromagnetically induced transparency," Phys. Today 50(7), 36-42 (1997) and references therein.
[CrossRef]

Hau, L. V.

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

He, S.

J. Q. Shen and S. He, "Dimension-sensitive optical responses of electromagnetically induced transparency vapor in a waveguide," Phys. Rev. A 74, 063831(1-6) (2006).

Kawazoe, T.

T. Kawazoe, K. Kobayashi, S. Sangu, and M. Ohtsu, "Demonstration of a nanophotonic switching operation by optical near-field energy transfer," Appl. Phys. Lett. 82, 2957-2959 (2003).
[CrossRef]

Kobayashi, K.

T. Kawazoe, K. Kobayashi, S. Sangu, and M. Ohtsu, "Demonstration of a nanophotonic switching operation by optical near-field energy transfer," Appl. Phys. Lett. 82, 2957-2959 (2003).
[CrossRef]

S. Sangu, K. Kobayashi, A. Shojiguchi, and M. Ohtsu, "Logic and functional operations using a near-field optically coupled quantum-dot system," Phys. Rev. B 69, 115334(1-13) (2004).
[CrossRef]

Kuzmich, A.

L. J. Wang, A. Kuzmich, and A. Dogariu, "Gain-assisted superluminal pulse propagation," Nature 406, 277-279 (2000) and references therein.
[CrossRef] [PubMed]

Li, Y. Q.

Marangos, J. P.

J. P. Marangos, "Electromagnetically induced transparency," J. Mod. Opt. 45,471-503 (1998).
[CrossRef]

Moseley, R. R.

R. R. Moseley, S. Shepherd, D. J. Fulton, B. D. Sinclair, and M. H. Dunn, "Spatial consequences of electromagnetically induced transparency: Observation of electromagnetically induced focusing," Phys. Rev. Lett. 74,670-673 (1995).
[CrossRef] [PubMed]

Ohtsu, M.

T. Kawazoe, K. Kobayashi, S. Sangu, and M. Ohtsu, "Demonstration of a nanophotonic switching operation by optical near-field energy transfer," Appl. Phys. Lett. 82, 2957-2959 (2003).
[CrossRef]

S. Sangu, K. Kobayashi, A. Shojiguchi, and M. Ohtsu, "Logic and functional operations using a near-field optically coupled quantum-dot system," Phys. Rev. B 69, 115334(1-13) (2004).
[CrossRef]

Sangu, S.

T. Kawazoe, K. Kobayashi, S. Sangu, and M. Ohtsu, "Demonstration of a nanophotonic switching operation by optical near-field energy transfer," Appl. Phys. Lett. 82, 2957-2959 (2003).
[CrossRef]

S. Sangu, K. Kobayashi, A. Shojiguchi, and M. Ohtsu, "Logic and functional operations using a near-field optically coupled quantum-dot system," Phys. Rev. B 69, 115334(1-13) (2004).
[CrossRef]

Schmidt, H.

Scully, M. O.

S. Y. Zhu and M. O. Scully, "Spectral line elimination and spontaneous emission cancellation via quantum interference," Phys. Rev. Lett. 76, 388-391 (1996).
[CrossRef] [PubMed]

M. Fleischhauer andM. O. Scully, "Quantum sensitivity limits of an optical magnetometer based on atomic phase coherence," Phys. Rev. A 49, 1973-1986 (1994).
[CrossRef] [PubMed]

Shen, J. Q.

J. Q. Shen, "Quantum-vacuum geometric phases in the noncoplanarly curved fiber system," Eur. Phys. J. D 30,259-264 (2004).
[CrossRef]

J. Q. Shen and S. He, "Dimension-sensitive optical responses of electromagnetically induced transparency vapor in a waveguide," Phys. Rev. A 74, 063831(1-6) (2006).

J. Q. Shen, "Negative refractive index in gyrotropically magnetoelectric media," Phys. Rev. B 73, 045113(1-5) (2006).
[CrossRef]

Shepherd, S.

R. R. Moseley, S. Shepherd, D. J. Fulton, B. D. Sinclair, and M. H. Dunn, "Spatial consequences of electromagnetically induced transparency: Observation of electromagnetically induced focusing," Phys. Rev. Lett. 74,670-673 (1995).
[CrossRef] [PubMed]

Shojiguchi, A.

S. Sangu, K. Kobayashi, A. Shojiguchi, and M. Ohtsu, "Logic and functional operations using a near-field optically coupled quantum-dot system," Phys. Rev. B 69, 115334(1-13) (2004).
[CrossRef]

Sinclair, B. D.

R. R. Moseley, S. Shepherd, D. J. Fulton, B. D. Sinclair, and M. H. Dunn, "Spatial consequences of electromagnetically induced transparency: Observation of electromagnetically induced focusing," Phys. Rev. Lett. 74,670-673 (1995).
[CrossRef] [PubMed]

Wang, H.

J. Q. Yao, H. B. Wu, and H. Wang, "The transient optical properties in four-level atomic medium induced by quantum interference effect,"Acta Sin. Quantum Opt. 9,121-125 (2003).

Wang, L. J.

L. J. Wang, A. Kuzmich, and A. Dogariu, "Gain-assisted superluminal pulse propagation," Nature 406, 277-279 (2000) and references therein.
[CrossRef] [PubMed]

Wu, H. B.

J. Q. Yao, H. B. Wu, and H. Wang, "The transient optical properties in four-level atomic medium induced by quantum interference effect,"Acta Sin. Quantum Opt. 9,121-125 (2003).

Xiao, M.

Yao, J. Q.

J. Q. Yao, H. B. Wu, and H. Wang, "The transient optical properties in four-level atomic medium induced by quantum interference effect,"Acta Sin. Quantum Opt. 9,121-125 (2003).

Zhu, S. Y.

S. Y. Zhu and M. O. Scully, "Spectral line elimination and spontaneous emission cancellation via quantum interference," Phys. Rev. Lett. 76, 388-391 (1996).
[CrossRef] [PubMed]

Acta Sin. Quantum Opt. (1)

J. Q. Yao, H. B. Wu, and H. Wang, "The transient optical properties in four-level atomic medium induced by quantum interference effect,"Acta Sin. Quantum Opt. 9,121-125 (2003).

Appl. Phys. Lett. (1)

T. Kawazoe, K. Kobayashi, S. Sangu, and M. Ohtsu, "Demonstration of a nanophotonic switching operation by optical near-field energy transfer," Appl. Phys. Lett. 82, 2957-2959 (2003).
[CrossRef]

Eur. Phys. J. D (1)

J. Q. Shen, "Quantum-vacuum geometric phases in the noncoplanarly curved fiber system," Eur. Phys. J. D 30,259-264 (2004).
[CrossRef]

J. Mod. Opt. (1)

J. P. Marangos, "Electromagnetically induced transparency," J. Mod. Opt. 45,471-503 (1998).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (4)

J. L. Cohen and P. R. Berman, "Amplification without inversion: Understanding probability amplitudes, quantum interference, and Feynman rules in a strongly driven system," Phys. Rev. A 55,3900-3917 (1997).
[CrossRef]

R.Y. Chiao, "Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations," Phys. Rev. A 48, R34-R37 (1993).
[CrossRef] [PubMed]

E. L. Bolda, J. C. Garrison, and R. Y. Chiao, "Optical pulse propagation at negative group velocities due to a nearby gain line," Phys. Rev. A 49, 2938-2947 (1994).
[CrossRef] [PubMed]

M. Fleischhauer andM. O. Scully, "Quantum sensitivity limits of an optical magnetometer based on atomic phase coherence," Phys. Rev. A 49, 1973-1986 (1994).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

R. R. Moseley, S. Shepherd, D. J. Fulton, B. D. Sinclair, and M. H. Dunn, "Spatial consequences of electromagnetically induced transparency: Observation of electromagnetically induced focusing," Phys. Rev. Lett. 74,670-673 (1995).
[CrossRef] [PubMed]

S. Y. Zhu and M. O. Scully, "Spectral line elimination and spontaneous emission cancellation via quantum interference," Phys. Rev. Lett. 76, 388-391 (1996).
[CrossRef] [PubMed]

Other (8)

L. J. Wang, A. Kuzmich, and A. Dogariu, "Gain-assisted superluminal pulse propagation," Nature 406, 277-279 (2000) and references therein.
[CrossRef] [PubMed]

P. Arve, P. J¨anes, and Lars Thyl´en, "Propagation of two-dimensional pulses in electromagnetically induced transparency media," Phys. Rev. A 69, 063809(1-8) (2004).
[CrossRef]

J. Q. Shen and S. He, "Dimension-sensitive optical responses of electromagnetically induced transparency vapor in a waveguide," Phys. Rev. A 74, 063831(1-6) (2006).

S. E. Harris, "Electromagnetically induced transparency," Phys. Today 50(7), 36-42 (1997) and references therein.
[CrossRef]

J. Q. Shen, "Negative refractive index in gyrotropically magnetoelectric media," Phys. Rev. B 73, 045113(1-5) (2006).
[CrossRef]

S. Sangu, K. Kobayashi, A. Shojiguchi, and M. Ohtsu, "Logic and functional operations using a near-field optically coupled quantum-dot system," Phys. Rev. B 69, 115334(1-13) (2004).
[CrossRef]

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

T. Kawazoe, K. Kobayashi, andM. Ohtsu,"A nanophotonic NOT-gate using near-field optically coupled quantum dots," 2005 Conference on Lasers & Electro-Optics (CLEO), Baltimore, MD, USA, 728-730 (2005).
[CrossRef]

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Figures (3)

Fig. 1.
Fig. 1.

The schematic diagram of a double-control four-level system. The two control laser beams, Ωc and Ωc′, drive the |2〉-|3〉 and |2′〉-|3〉) transitions, respectively. The probe transition |1〉-|3〉 can be controllably manipulated via the destructive and constructive quantum interferences between the |2〉-|3〉 and |2′〉-|3〉 transitions. If levels |1〉, |2〉 and |2′〉 form a three-level dark state, then the atomic vapor is transparent to the probe field, whereas it is opaque to the probe field when levels |2〉 and |2′〉) form a two-level dark state.

Fig. 2.
Fig. 2.

The dispersive behavior of the real and imaginary parts of the electric susceptibility as the probe frequency detuning varies. Both the real and imaginary parts of χ(∆p) tend to zero at probe frequency detunings ∆p = 3.0×107 s-1 and ∆p = 8.0×107 s-1.

Fig. 3.
Fig. 3.

The dispersive behavior of the absorption coefficients of both the three- and the four-level atomic vapors as the probe frequency detuning varies. There are two resonant frequencies (∆p = 3.0 × 107 s-1 and ∆p = 8.0 × 107 s-1), where the four-level system exhibits zero absorption.

Equations (30)

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

a 1 ˙ = i 2 Ω p * a 3 ,
a 2 ˙ = [ γ 2 2 + i ( Δ p Δ c ) ] a 2 + i 2 Ω c * a 3 ,
a 2′ ˙ = [ γ′ 2 2 + i ( Δ p Δ c′ ) ] a 2′ + i 2 Ω c′ * a 3 ,
a 3 ˙ = ( Γ 3 2 + i Δ p ) a 3 + i 2 ( Ω p a 1 + Ω c a 2 + Ω c′ a 2′ ) ,
t ( a 1 ( t ) a 2 ( t ) a 2′ ( t ) a 3 ( t ) ) = 𝓐 ( a 1 ( t ) a 2 ( t ) a 2′ ( t ) a 2 ( t ) ) ,
𝓐 = ( 0 0 0 i 2 Ω p * 0 0 0 i 2 Ω c * 0 0 0 i 2 Ω c′ * i 2 Ω p i 2 Ω c i 2 Ω c′ 0 ) .
( a 1 ( t ) a 2 ( t ) a 2′ ( t ) a 3 ( t ) ) = ( a 1 ( 0 ) a 2 ( 0 ) a 2′ ( 0 ) a 2 ( 0 ) ) e λt .
( 𝓐 λ 𝓢 ) ( a 1 ( 0 ) a 2 ( 0 ) a 2′ ( 0 ) a 3 ( 0 ) ) ( λ 0 0 i 2 Ω p * 0 λ 0 i 2 Ω c * 0 0 λ i 2 Ω c′ * i 2 Ω p i 2 Ω c i 2 Ω c′ λ ) ( a 1 ( 0 ) a 2 ( 0 ) a 2′ ( 0 ) a 3 ( 0 ) ) = 0 ,
λ 2 [ λ 2 + Ω c * Ω c + Ω c′ * Ω c′ + Ω p * Ω p 4 ] = 0 .
λ 1 = 0 , λ 2 = 0 , λ ± = ± i Ω c * Ω c + Ω c′ * Ω c′ + Ω p * Ω p 2 .
Ω p a 1 + Ω c a 2 + Ω c′ a 2′ = 0 .
a 2 ˙ = [ γ 2 2 + i ( Δ p Δ c ) ] a 2 + i 2 Ω c * a 3 ,
a 2′ ˙ = [ γ′ 2 2 + i ( Δ p Δ c′ ) ] a 2′ + i 2 Ω c′ * a 3 ,
a 3 ˙ = ( Γ 3 2 + i Δ p ) a 3 + i 2 ( Ω c a 2 + Ω c′ a 2′ ) + i 2 Ω p .
a 2 = 1 4 𝓓 Ω p Ω c * [ γ′ 2 2 + i ( Δ p Δ c′ ) ] ,
a 2′ = 1 4 𝓓 Ω p Ω c′ * [ γ 2 2 + i ( Δ p Δ c ) ] ,
a 3 = 1 2 𝓓 Ω p [ γ 2 2 + i ( Δ p Δ c ) ] [ γ′ 2 2 + i ( Δ p Δ c′ ) ] ,
𝓢 = ( Γ 3 2 + i Δ p ) [ γ 2 2 + i ( Δ p Δ c ) ] [ γ′ 2 2 + i ( Δ p Δ c′ ) ]
+ 1 4 Ω c′ * Ω c′ [ γ 2 2 + i ( Δ p Δ c ) ] + 1 4 Ω c * Ω c [ γ′ 2 2 + i ( Δ p Δ c′ ) ] .
β ( Δ p ) = 13 2 ε 0 h ¯ 1 𝓓 [ γ 2 2 + i ( Δ p Δ c ) ] [ γ′ 2 2 + i ( Δ p Δ c′ ) ] .
a 2 a 2′ = Ω c * Ω c′ * γ′ 2 2 + i ( Δ p Δ c′ ) γ 2 2 + i ( Δ p Δ c ) .
β ( Δ p ) i 13 2 ε 0 h ¯ 1 Γ 3 2 + i Δ p ,
Ω c′ * Ω c′ Ω c * Ω c γ′ 2 2 + i ( Δ p Δ c′ ) γ 2 2 + i ( Δ p Δ c ) .
a 2 a 2′ Ω c′ Ω c or Ω c a 2 + Ω c′ a 2′ 0 .
Ω c a 2 𝓒 e Ω c′ a 2′ = 0 ,
𝓒 e = Ω c * Ω c Ω c′ * Ω c′ γ′ 2 2 + i ( Δ p Δ c′ ) γ 2 2 + i ( Δ p Δ c ) .
β ( Δ p ) = i 13 2 ε 0 h ¯ γ 2 2 + i ( Δ p Δ c ) ( Γ 3 2 + i Δ p ) [ γ 2 2 + i ( Δ p Δ c ) ] + 1 4 Ω c * Ω c ( 1 + 1 𝓒 e ) .
β ( Δ p ) = i 13 2 ε 0 h ¯ γ′ 2 2 + i ( Δ p Δ c′ ) ( Γ 3 2 + i Δ p ) [ γ′ 2 2 + i ( Δ p Δ c′ ) ] + 1 4 Ω c′ * Ω c′ ( 1 + 𝓒 e ) .
𝓒 e = Ω c * Ω c ( Δ p Δ c′ ) Ω c′ * Ω c′ ( Δ p Δ c ) .
β ( Δ p ) = i 13 2 ε 0 h ¯ γ 2 2 + i ( Δ p Δ c ) ( Γ 3 2 + i Δ p ) [ γ 2 2 + i ( Δ p Δ c ) ] + 1 4 Ω c * Ω c ,

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