Abstract

Collective coupling of multiple atoms with a cavity mode produces two normal modes that are separated in energy by Vacuum Rabi splitting. We show that the normal mode excitation of the cavity-atom system can be suppressed by coupling a control laser to the atomic system from free space. The control laser splits the normal mode of the cavity-atoms system and opens two excitation channels. The destructive quantum interference between the two channels renders the cavity-atoms system opaque to the light coupled to the cavity-atom system. We demonstrate suppression of the normal mode excitation by the destructive quantum interference in an experiment using cold Rb atoms confined in an optical cavity.

© 2008 Optical Society of America

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References

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  1. P. R. Berman, Cavity Quantum Electrodynamics (Academic, San Diego, 1994).
  2. E. T. Jaynes and F. W. Cummings, "Comparison of quantum and semiclassical radiation theories with application to the beam maser," Proc. IEEE 51, 89-109(1963).
    [CrossRef]
  3. J. J. Sanchez-Mondragon, N. B. Narozhny, and J. H. Eberly, "Theory of Spontaneous-Emission Line Shape in an Ideal Cavity," Phys. Rev. Lett. 51, 550-553(1983).
    [CrossRef]
  4. A. Boca, R. Miller, K. M. Birnbaum, A. D. Boozer, J. McKeever, and H. J. Kimble, "Observation of the Vacuum Rabi Spectrum for One Trapped Atom," Phys. Rev. Lett. 93, 233603(1-4) (2004).
    [CrossRef] [PubMed]
  5. G. S. Agarwal, "Vacuum-Field Rabi Splittings in Microwave Absorption by Rydberg Atoms in a Cavity," Phys. Rev. Lett. 53, 1732-1735(1984).
    [CrossRef]
  6. M. G. Raizen, R. J. Thompson, R. J. Brecha, H. J. Kimble, and H. J. Carmichael, "Normal-mode splitting and linewidth averaging for two-state atoms in an optical cavity," Phys. Rev. Lett. 63, 240 - 243 (1989).
    [CrossRef] [PubMed]
  7. Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, "Vacuum Rabi splitting as a feature of linear-dispersion theory: Analysis and experimental observations," Phys. Rev. Lett. 64, 2499-2502 (1990).
    [CrossRef] [PubMed]
  8. J. Gripp, S. L. Mielke, and L. A. Orozco, "Evolution of the vacuum Rabi peaks in a detuned atom-cavity system," Phys. Rev. A 56, 3262-3273 (1997).
    [CrossRef]
  9. J. Klinner, M. Lindholdt, B. Nagorny, and A. Hemmerich, "Normal Mode Splitting and Mechanical Effects of an Optical Lattice in a Ring Cavity," Phys. Rev. Lett. 96, 023002(1-4) (2006)
    [CrossRef] [PubMed]
  10. G. Rempe, R. J. Thompson, R. J. Brecha, W. D. Lee, and H. J. Kimble,"Optical bistability and photon statistics in cavity quantum electrodynamics," Phys. Rev. Lett. 67, 1727 - 1730 (1991).
    [CrossRef] [PubMed]
  11. P. Grangier, J. F. Roch, J. Roger. L. A. Lugiato, E. M. Pessina, G. Scandroglio, P. Galatola, "2-photon double-beam optical bistability in the dispersive regime," Phys. Rev. A 46, 2735-2743 (1992).
    [CrossRef] [PubMed]
  12. S. E. Harris, "Electromagnetically Induced Transparency," Phys. Today 50, 36-42 (1997).
    [CrossRef]
  13. E. Arimondo, "Coherent population trapping in laser spectroscopy," in Progress in Optics, E. Wolf, ed., (Elsevier, Amsterdam, 1996) Vol. 31, pp. 257-354.
  14. M. D. Lukin, M. Fleischhauer, M. O. Scully, and V. L. Velichansky, "Intracavity electromagnetically induced transparency," Opt. Lett. 23, 295-297 (1998).
    [CrossRef]
  15. H. Wang, D. J. Goorskey, W. H. Burkett, and M. Xiao, "Cavity-linewidth narrowing by means of electromagnetically induced transparency," Opt. Lett. 25, 1732-1735 (2000).
    [CrossRef]
  16. G. Hernandez, J. Zhang, and Y. Zhu, "Vacuum Rabi splitting and intracavity dark state in a cavity-atoms system," Phys. Rev. A 76, 053814 (1-4) (2007).
    [CrossRef]
  17. A. Joshi and M. Xiao, "Optical multistability in three-level atoms inside an optical ring cavity", Phys. Rev. Lett. 91, 143904-(1-4) (2003).
    [CrossRef] [PubMed]
  18. J. Zhang, G. Hernandez, and Y. Zhu, "Slow light with cavity electromagnetically induced transparency," Opt. Lett. 33, 46-48 (2008).
    [CrossRef]
  19. V. S. C. Rao, S. D. Gupta, and G. S. Agarwal, "Atomic absorbers for controlling pulse propagation in resonators," Opt. Lett. 29, 307-309 (2004).
    [CrossRef]
  20. V. S. C. Manga Rao and S. D. Gupta, "Sub and superluminal propagation through stratified media," IEEE Proc.-Circuits Syst. IEE Proceedings-Circuits Devices and Systems 152, 527-531 (2005).
  21. H. Kang, G. Hernadez, and Y. Zhu, "Superluminal and slow light propagation in cold atoms" Phys. Rev. A (Rapid Commun) 70, 011801(1-4) (R) (2004)
  22. H. Kang, L. Wen, and Y. Zhu, "Normal or anomalous dispersion and gain in a resonant coherent medium," Phys. Rev. A 68, 063806(1-5) (2003).
    [CrossRef]

2008

2005

V. S. C. Manga Rao and S. D. Gupta, "Sub and superluminal propagation through stratified media," IEEE Proc.-Circuits Syst. IEE Proceedings-Circuits Devices and Systems 152, 527-531 (2005).

2004

2000

1998

1997

S. E. Harris, "Electromagnetically Induced Transparency," Phys. Today 50, 36-42 (1997).
[CrossRef]

J. Gripp, S. L. Mielke, and L. A. Orozco, "Evolution of the vacuum Rabi peaks in a detuned atom-cavity system," Phys. Rev. A 56, 3262-3273 (1997).
[CrossRef]

1992

P. Grangier, J. F. Roch, J. Roger. L. A. Lugiato, E. M. Pessina, G. Scandroglio, P. Galatola, "2-photon double-beam optical bistability in the dispersive regime," Phys. Rev. A 46, 2735-2743 (1992).
[CrossRef] [PubMed]

1991

G. Rempe, R. J. Thompson, R. J. Brecha, W. D. Lee, and H. J. Kimble,"Optical bistability and photon statistics in cavity quantum electrodynamics," Phys. Rev. Lett. 67, 1727 - 1730 (1991).
[CrossRef] [PubMed]

1990

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, "Vacuum Rabi splitting as a feature of linear-dispersion theory: Analysis and experimental observations," Phys. Rev. Lett. 64, 2499-2502 (1990).
[CrossRef] [PubMed]

1989

M. G. Raizen, R. J. Thompson, R. J. Brecha, H. J. Kimble, and H. J. Carmichael, "Normal-mode splitting and linewidth averaging for two-state atoms in an optical cavity," Phys. Rev. Lett. 63, 240 - 243 (1989).
[CrossRef] [PubMed]

1984

G. S. Agarwal, "Vacuum-Field Rabi Splittings in Microwave Absorption by Rydberg Atoms in a Cavity," Phys. Rev. Lett. 53, 1732-1735(1984).
[CrossRef]

1983

J. J. Sanchez-Mondragon, N. B. Narozhny, and J. H. Eberly, "Theory of Spontaneous-Emission Line Shape in an Ideal Cavity," Phys. Rev. Lett. 51, 550-553(1983).
[CrossRef]

1963

E. T. Jaynes and F. W. Cummings, "Comparison of quantum and semiclassical radiation theories with application to the beam maser," Proc. IEEE 51, 89-109(1963).
[CrossRef]

Agarwal, G. S.

V. S. C. Rao, S. D. Gupta, and G. S. Agarwal, "Atomic absorbers for controlling pulse propagation in resonators," Opt. Lett. 29, 307-309 (2004).
[CrossRef]

G. S. Agarwal, "Vacuum-Field Rabi Splittings in Microwave Absorption by Rydberg Atoms in a Cavity," Phys. Rev. Lett. 53, 1732-1735(1984).
[CrossRef]

Brecha, R. J.

G. Rempe, R. J. Thompson, R. J. Brecha, W. D. Lee, and H. J. Kimble,"Optical bistability and photon statistics in cavity quantum electrodynamics," Phys. Rev. Lett. 67, 1727 - 1730 (1991).
[CrossRef] [PubMed]

M. G. Raizen, R. J. Thompson, R. J. Brecha, H. J. Kimble, and H. J. Carmichael, "Normal-mode splitting and linewidth averaging for two-state atoms in an optical cavity," Phys. Rev. Lett. 63, 240 - 243 (1989).
[CrossRef] [PubMed]

Burkett, W. H.

Carmichael, H. J.

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, "Vacuum Rabi splitting as a feature of linear-dispersion theory: Analysis and experimental observations," Phys. Rev. Lett. 64, 2499-2502 (1990).
[CrossRef] [PubMed]

M. G. Raizen, R. J. Thompson, R. J. Brecha, H. J. Kimble, and H. J. Carmichael, "Normal-mode splitting and linewidth averaging for two-state atoms in an optical cavity," Phys. Rev. Lett. 63, 240 - 243 (1989).
[CrossRef] [PubMed]

Cummings, F. W.

E. T. Jaynes and F. W. Cummings, "Comparison of quantum and semiclassical radiation theories with application to the beam maser," Proc. IEEE 51, 89-109(1963).
[CrossRef]

Eberly, J. H.

J. J. Sanchez-Mondragon, N. B. Narozhny, and J. H. Eberly, "Theory of Spontaneous-Emission Line Shape in an Ideal Cavity," Phys. Rev. Lett. 51, 550-553(1983).
[CrossRef]

Fleischhauer, M.

Gauthier, D. J.

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, "Vacuum Rabi splitting as a feature of linear-dispersion theory: Analysis and experimental observations," Phys. Rev. Lett. 64, 2499-2502 (1990).
[CrossRef] [PubMed]

Goorskey, D. J.

Grangier, P.

P. Grangier, J. F. Roch, J. Roger. L. A. Lugiato, E. M. Pessina, G. Scandroglio, P. Galatola, "2-photon double-beam optical bistability in the dispersive regime," Phys. Rev. A 46, 2735-2743 (1992).
[CrossRef] [PubMed]

Gripp, J.

J. Gripp, S. L. Mielke, and L. A. Orozco, "Evolution of the vacuum Rabi peaks in a detuned atom-cavity system," Phys. Rev. A 56, 3262-3273 (1997).
[CrossRef]

Gupta, S. D.

V. S. C. Manga Rao and S. D. Gupta, "Sub and superluminal propagation through stratified media," IEEE Proc.-Circuits Syst. IEE Proceedings-Circuits Devices and Systems 152, 527-531 (2005).

V. S. C. Rao, S. D. Gupta, and G. S. Agarwal, "Atomic absorbers for controlling pulse propagation in resonators," Opt. Lett. 29, 307-309 (2004).
[CrossRef]

Harris, S. E.

S. E. Harris, "Electromagnetically Induced Transparency," Phys. Today 50, 36-42 (1997).
[CrossRef]

Hernandez, G.

Jaynes, E. T.

E. T. Jaynes and F. W. Cummings, "Comparison of quantum and semiclassical radiation theories with application to the beam maser," Proc. IEEE 51, 89-109(1963).
[CrossRef]

Kimble, H. J.

G. Rempe, R. J. Thompson, R. J. Brecha, W. D. Lee, and H. J. Kimble,"Optical bistability and photon statistics in cavity quantum electrodynamics," Phys. Rev. Lett. 67, 1727 - 1730 (1991).
[CrossRef] [PubMed]

M. G. Raizen, R. J. Thompson, R. J. Brecha, H. J. Kimble, and H. J. Carmichael, "Normal-mode splitting and linewidth averaging for two-state atoms in an optical cavity," Phys. Rev. Lett. 63, 240 - 243 (1989).
[CrossRef] [PubMed]

Lee, W. D.

G. Rempe, R. J. Thompson, R. J. Brecha, W. D. Lee, and H. J. Kimble,"Optical bistability and photon statistics in cavity quantum electrodynamics," Phys. Rev. Lett. 67, 1727 - 1730 (1991).
[CrossRef] [PubMed]

Lukin, M. D.

Manga Rao, V. S. C.

V. S. C. Manga Rao and S. D. Gupta, "Sub and superluminal propagation through stratified media," IEEE Proc.-Circuits Syst. IEE Proceedings-Circuits Devices and Systems 152, 527-531 (2005).

Mielke, S. L.

J. Gripp, S. L. Mielke, and L. A. Orozco, "Evolution of the vacuum Rabi peaks in a detuned atom-cavity system," Phys. Rev. A 56, 3262-3273 (1997).
[CrossRef]

Morin, S. E.

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, "Vacuum Rabi splitting as a feature of linear-dispersion theory: Analysis and experimental observations," Phys. Rev. Lett. 64, 2499-2502 (1990).
[CrossRef] [PubMed]

Mossberg, T. W.

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, "Vacuum Rabi splitting as a feature of linear-dispersion theory: Analysis and experimental observations," Phys. Rev. Lett. 64, 2499-2502 (1990).
[CrossRef] [PubMed]

Narozhny, N. B.

J. J. Sanchez-Mondragon, N. B. Narozhny, and J. H. Eberly, "Theory of Spontaneous-Emission Line Shape in an Ideal Cavity," Phys. Rev. Lett. 51, 550-553(1983).
[CrossRef]

Orozco, L. A.

J. Gripp, S. L. Mielke, and L. A. Orozco, "Evolution of the vacuum Rabi peaks in a detuned atom-cavity system," Phys. Rev. A 56, 3262-3273 (1997).
[CrossRef]

Raizen, M. G.

M. G. Raizen, R. J. Thompson, R. J. Brecha, H. J. Kimble, and H. J. Carmichael, "Normal-mode splitting and linewidth averaging for two-state atoms in an optical cavity," Phys. Rev. Lett. 63, 240 - 243 (1989).
[CrossRef] [PubMed]

Rao, V. S. C.

Rempe, G.

G. Rempe, R. J. Thompson, R. J. Brecha, W. D. Lee, and H. J. Kimble,"Optical bistability and photon statistics in cavity quantum electrodynamics," Phys. Rev. Lett. 67, 1727 - 1730 (1991).
[CrossRef] [PubMed]

Roch, J. F.

P. Grangier, J. F. Roch, J. Roger. L. A. Lugiato, E. M. Pessina, G. Scandroglio, P. Galatola, "2-photon double-beam optical bistability in the dispersive regime," Phys. Rev. A 46, 2735-2743 (1992).
[CrossRef] [PubMed]

Roger, J.

P. Grangier, J. F. Roch, J. Roger. L. A. Lugiato, E. M. Pessina, G. Scandroglio, P. Galatola, "2-photon double-beam optical bistability in the dispersive regime," Phys. Rev. A 46, 2735-2743 (1992).
[CrossRef] [PubMed]

Sanchez-Mondragon, J. J.

J. J. Sanchez-Mondragon, N. B. Narozhny, and J. H. Eberly, "Theory of Spontaneous-Emission Line Shape in an Ideal Cavity," Phys. Rev. Lett. 51, 550-553(1983).
[CrossRef]

Scully, M. O.

Thompson, R. J.

G. Rempe, R. J. Thompson, R. J. Brecha, W. D. Lee, and H. J. Kimble,"Optical bistability and photon statistics in cavity quantum electrodynamics," Phys. Rev. Lett. 67, 1727 - 1730 (1991).
[CrossRef] [PubMed]

M. G. Raizen, R. J. Thompson, R. J. Brecha, H. J. Kimble, and H. J. Carmichael, "Normal-mode splitting and linewidth averaging for two-state atoms in an optical cavity," Phys. Rev. Lett. 63, 240 - 243 (1989).
[CrossRef] [PubMed]

Velichansky, V. L.

Wang, H.

Wu, Q.

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, "Vacuum Rabi splitting as a feature of linear-dispersion theory: Analysis and experimental observations," Phys. Rev. Lett. 64, 2499-2502 (1990).
[CrossRef] [PubMed]

Xiao, M.

Zhang, J.

Zhu, Y.

J. Zhang, G. Hernandez, and Y. Zhu, "Slow light with cavity electromagnetically induced transparency," Opt. Lett. 33, 46-48 (2008).
[CrossRef]

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, "Vacuum Rabi splitting as a feature of linear-dispersion theory: Analysis and experimental observations," Phys. Rev. Lett. 64, 2499-2502 (1990).
[CrossRef] [PubMed]

IEE Proceedings-Circuits Devices and Systems

V. S. C. Manga Rao and S. D. Gupta, "Sub and superluminal propagation through stratified media," IEEE Proc.-Circuits Syst. IEE Proceedings-Circuits Devices and Systems 152, 527-531 (2005).

Opt. Lett.

Phys. Rev. A

P. Grangier, J. F. Roch, J. Roger. L. A. Lugiato, E. M. Pessina, G. Scandroglio, P. Galatola, "2-photon double-beam optical bistability in the dispersive regime," Phys. Rev. A 46, 2735-2743 (1992).
[CrossRef] [PubMed]

J. Gripp, S. L. Mielke, and L. A. Orozco, "Evolution of the vacuum Rabi peaks in a detuned atom-cavity system," Phys. Rev. A 56, 3262-3273 (1997).
[CrossRef]

Phys. Rev. Lett.

G. S. Agarwal, "Vacuum-Field Rabi Splittings in Microwave Absorption by Rydberg Atoms in a Cavity," Phys. Rev. Lett. 53, 1732-1735(1984).
[CrossRef]

M. G. Raizen, R. J. Thompson, R. J. Brecha, H. J. Kimble, and H. J. Carmichael, "Normal-mode splitting and linewidth averaging for two-state atoms in an optical cavity," Phys. Rev. Lett. 63, 240 - 243 (1989).
[CrossRef] [PubMed]

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, "Vacuum Rabi splitting as a feature of linear-dispersion theory: Analysis and experimental observations," Phys. Rev. Lett. 64, 2499-2502 (1990).
[CrossRef] [PubMed]

J. J. Sanchez-Mondragon, N. B. Narozhny, and J. H. Eberly, "Theory of Spontaneous-Emission Line Shape in an Ideal Cavity," Phys. Rev. Lett. 51, 550-553(1983).
[CrossRef]

G. Rempe, R. J. Thompson, R. J. Brecha, W. D. Lee, and H. J. Kimble,"Optical bistability and photon statistics in cavity quantum electrodynamics," Phys. Rev. Lett. 67, 1727 - 1730 (1991).
[CrossRef] [PubMed]

Phys. Today

S. E. Harris, "Electromagnetically Induced Transparency," Phys. Today 50, 36-42 (1997).
[CrossRef]

Proc. IEEE

E. T. Jaynes and F. W. Cummings, "Comparison of quantum and semiclassical radiation theories with application to the beam maser," Proc. IEEE 51, 89-109(1963).
[CrossRef]

Other

P. R. Berman, Cavity Quantum Electrodynamics (Academic, San Diego, 1994).

A. Boca, R. Miller, K. M. Birnbaum, A. D. Boozer, J. McKeever, and H. J. Kimble, "Observation of the Vacuum Rabi Spectrum for One Trapped Atom," Phys. Rev. Lett. 93, 233603(1-4) (2004).
[CrossRef] [PubMed]

J. Klinner, M. Lindholdt, B. Nagorny, and A. Hemmerich, "Normal Mode Splitting and Mechanical Effects of an Optical Lattice in a Ring Cavity," Phys. Rev. Lett. 96, 023002(1-4) (2006)
[CrossRef] [PubMed]

E. Arimondo, "Coherent population trapping in laser spectroscopy," in Progress in Optics, E. Wolf, ed., (Elsevier, Amsterdam, 1996) Vol. 31, pp. 257-354.

G. Hernandez, J. Zhang, and Y. Zhu, "Vacuum Rabi splitting and intracavity dark state in a cavity-atoms system," Phys. Rev. A 76, 053814 (1-4) (2007).
[CrossRef]

A. Joshi and M. Xiao, "Optical multistability in three-level atoms inside an optical ring cavity", Phys. Rev. Lett. 91, 143904-(1-4) (2003).
[CrossRef] [PubMed]

H. Kang, G. Hernadez, and Y. Zhu, "Superluminal and slow light propagation in cold atoms" Phys. Rev. A (Rapid Commun) 70, 011801(1-4) (R) (2004)

H. Kang, L. Wen, and Y. Zhu, "Normal or anomalous dispersion and gain in a resonant coherent medium," Phys. Rev. A 68, 063806(1-5) (2003).
[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

(a). Three-level atoms coupled by a cavity field and a control field. (b) The coupled cavity-atom system forms the ground state |a,0>, and the first excited states |φ+> and |φ-> referred to as two normal modes. (c) The control laser coupled to the normal mode |φ+> splits it into two dressed states |Ψ+> and |Ψ->. Destructive interference between two excitation paths, |b,0>-|Ψ+> and |b,0>-|Ψ->, suppresses the normal mode excitation and renders the cavity-atom system opaque to the incident probe light.

Fig. 2.
Fig. 2.

Cavity transmitted intensity of the probe laser versus the normalized probe frequency detuning Δp/Γ. Γ is the spontaneous decay rate of the state |e>. Parameters used in the calculations are: γ=0, optical depth nσeaℓ=3, L=5 cm, and R=0.98. The resulting vacuum Rabi frequency 2√N g=16.1Γ. (a) Without control laser (Ω=0). (b) With the control laser (the control laser detuning Δ=√N g=8.05Γ). Black line, Ω=0.1Γ; red line, Ω=0.5Γ; and blue line, Ω=Γ. The inset figure highlights the right peak between Δp/Γ=6-10.

Fig. 3.
Fig. 3.

Cavity transmitted intensity of the probe laser versus the normalized probe frequency detuning Δp/Γ. Parameters used in the calculations are: Ω=0.2Γ, Δ=8.05Γ, nσeaℓ=3, L=5 cm, and R=0.98. Black line, γ=0; red line, γ=0.05Γ; and blue line, γ=0.2Γ.

Fig. 4.
Fig. 4.

(a). 85Rb atoms interacting with a control field and a cavity field, which forms a threelevel Λ-type system. The spontaneous decay rate of the excited state |e> is Γ=2π×5.4×106 s-1. (b) Schematic drawing of the cavity apparatus. The control laser is circularly polarized and the probe laser is linearly polarized along the x direction.

Fig. 5.
Fig. 5.

Cavity transmission versus the probe detuning Δp. Red dotted lines are experimental data and blue lines are calculations. The control laser detuning Δ≈44 MHz. (a) Ω=0; (b) Ω≈2 MHz; (c) Ω≈5 MHz; (d) Ω≈9 MHz.

Fig. 6.
Fig. 6.

Free-space probe transmission versus the probe detuning Δ. Red dotted lines are experimental data and blue lines are calculations. The control laser detuning Δ≈44 MHz. (a) Ω=0; (b) Ω≈2 MHz; (c) Ω≈5 MHz; (d) Ω≈9 MHz.

Equations (2)

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P = < a , 0 μ ea Ψ + > v p ( v ea + N g + Ω ) + < a , 0 μ ea Ψ > v p ( v ea + N g Ω ) 2 = 1 4 < a , 0 μ ea e , 0 > Ω + < a , 0 μ ea e , 0 > Ω 2 = 0
I T ( υ p ) = I i ( 1 R ) 2 · exp ( 2 k χ ( υ p ) ) 1 R · exp ( 2 i k ( L + χ ( υ p ) + i χ ( υ p ) ) ) 2

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