Abstract

We first investigate the nonlinear optical response of a five-level tripod atomic system in electromagnetically induced transparency and then propose a 3-qubit phase gate protocol based on the fifth-order nonlinearity. The cross-phase modulation among the three weak fields induced by the fifth-order nonlinearity in electromagnetically induced transparency can produce the phase shift of order π, which can be used to realize the 3-qubit polarization phase gate. This protocol should have potential applications in quantum information processing.

© 2008 Optical Society of America

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References

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  1. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge U. Press, 2000).
  2. H. Schmidt and A. Imamoglu, “Giant Kerr nonlinearities obtained by electromagnetically induced transparency,” Opt. Lett. 21, 1936-1938 (1996).
    [Crossref] [PubMed]
  3. S. E. Harris, J. E. Field, and A. Imamoglu, “Nonlinear optical process using electromagnetically induced transparency,” Phys. Rev. Lett. 64, 1107-1110 (1990).
    [Crossref] [PubMed]
  4. M. S. Zubairy, A. B. Matsko, and M. O. Scully, “Resonant enhancement of high-order optical nonlinearities based on atomic coherence,” Phys. Rev. A 65, 043804 (2002).
    [Crossref]
  5. A. B. Matsko, I. Novikova, G. R. Welch, and M. S. Zubairy, “Enhancement of Kerr nonlinearity by multiphoton coherence,” Opt. Lett. 28, 96-98 (2002).
    [Crossref]
  6. A. B. Matsko, I. Novikova, M. S. Zubairy, and G. R. Welch, “Nonlinear magneto-optical rotation of elliptically polarized light,” Phys. Rev. A 67, 043805 (2003).
    [Crossref]
  7. A. D. Greentree, D. Richards, J. A. Vaccaro, A. V. Durrant, S. R. de Echaniz, D. M. Segal, and J. P. Marangos, “Intensity-dependent dispersion under conditions of electromagnetically induced transparency in coherently prepared multistate atoms,” Phys. Rev. A 67, 023818 (2003).
    [Crossref]
  8. D. Petrosyan and G. Kurizki, “Symmetric photon-photon coupling by atoms with Zeeman-split sublevels,” Phys. Rev. A 65, 033833 (2002).
    [Crossref]
  9. H. Wang, D. Goorskey, and M. Xiao, “Enhanced Kerr nonlinearity via atomic coherence in a three-level atomic system,” Phys. Rev. Lett. 87, 073601 (2001).
    [Crossref] [PubMed]
  10. J.-F. Roch, K. Vigneron, P. Grelu, A. Sinatra, J.-P. Poizat, and P. Grangier, “Quantum nondemolition measurements using cold trapped atoms,” Phys. Rev. Lett. 78, 634-637 (1997).
    [Crossref]
  11. H. Kang and Y. Zhu, “Observation of large Kerr nonlinearity at low light intensities,” Phys. Rev. Lett. 91, 093601 (2003).
    [Crossref] [PubMed]
  12. S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50, 36-42 (1997).
    [Crossref]
  13. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633-673 (2005).
    [Crossref]
  14. Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, “Measurement of conditional phase shifts for quantum logic,” Phys. Rev. Lett. 75, 4710-4713 (1995).
    [Crossref] [PubMed]
  15. K. J. Resch, J. S. Lundeen, and A. M. Steinberg, “Conditional-phase switch at the single-photon level,” Phys. Rev. Lett. 89, 037904 (2002).
    [Crossref] [PubMed]
  16. C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization qubit phase gate in driven atomic media,” Phys. Rev. Lett. 90, 197902 (2003).
    [Crossref] [PubMed]
  17. C. Ottaviani, S. Rebic, D. Vitali, and P. Tombesi, “Quantum phase-gate operation based on nonlinear optics: full quantum analysis,” Phys. Rev. A 73, 010301(R) (2006).
    [Crossref]
  18. S. Rebic, D. Vitali, C. Ottaviani, P. Tombesi, M. Artoni, F. Cataliotti, and R. Corbalan, “Polarization phase gate with a tripod atomic system,” Phys. Rev. A 70, 032317 (2004).
    [Crossref]
  19. A. Joshi and M. Xiao, “Phase gate with a four-level inverted-Y system,” Phys. Rev. A 72, 062319 (2005).
    [Crossref]
  20. M. D. Lukin and A. Imamoglu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Phys. Rev. Lett. 84, 1419-1422 (2000).
    [Crossref] [PubMed]
  21. H. Goto and K. Ichimura, “Multiqubit controlled unitary gate by adiabatic passage with an optical cavity,” Phys. Rev. A 70, 012305 (2004).
    [Crossref]
  22. C. Hang, Y. Li, L. Ma, and G. X. Huang, “Three-way entanglement and three-qubit phase gate based on a coherent six-level atomic system,” Phys. Rev. A 74, 012319 (2006).
    [Crossref]
  23. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge U. Press, 1997).
  24. R. W. Boyd, Nonlinear Optics (Academic, 1992).
  25. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145-195 (2002).
    [Crossref]
  26. D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575-579 (1997).
    [Crossref]
  27. K. J. Blow, R. Loudon, S. J. D. Phoenix, and T. J. Shepherd, “Continuum fields in quantum optics,” Phys. Rev. A 42, 4102-4114 (1990).
    [Crossref] [PubMed]
  28. C. Tai, W. Happer, and R. Gupta, “Hyperfine structure and lifetime measurements of the second-excited D states of rubidium and cesium by cascade fluorescence spectroscopy,” Phys. Rev. A 12, 736-747 (1975).
    [Crossref]
  29. E. Arimondo, M. Inguscio, and P. Violino, “Experimental determinations of the hyperfine structure in the alkali atoms,” Rev. Mod. Phys. 49, 31-75 (1977).
    [Crossref]
  30. M. Yan, E. G. Rickey, and Y. Zhu, “Suppression of two-photon absorption by quantum interference,” Phys. Rev. A 64, 043807 (2001).
    [Crossref]

2006 (2)

C. Ottaviani, S. Rebic, D. Vitali, and P. Tombesi, “Quantum phase-gate operation based on nonlinear optics: full quantum analysis,” Phys. Rev. A 73, 010301(R) (2006).
[Crossref]

C. Hang, Y. Li, L. Ma, and G. X. Huang, “Three-way entanglement and three-qubit phase gate based on a coherent six-level atomic system,” Phys. Rev. A 74, 012319 (2006).
[Crossref]

2005 (2)

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

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633-673 (2005).
[Crossref]

2004 (2)

S. Rebic, D. Vitali, C. Ottaviani, P. Tombesi, M. Artoni, F. Cataliotti, and R. Corbalan, “Polarization phase gate with a tripod atomic system,” Phys. Rev. A 70, 032317 (2004).
[Crossref]

H. Goto and K. Ichimura, “Multiqubit controlled unitary gate by adiabatic passage with an optical cavity,” Phys. Rev. A 70, 012305 (2004).
[Crossref]

2003 (4)

C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization qubit phase gate in driven atomic media,” Phys. Rev. Lett. 90, 197902 (2003).
[Crossref] [PubMed]

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

A. B. Matsko, I. Novikova, M. S. Zubairy, and G. R. Welch, “Nonlinear magneto-optical rotation of elliptically polarized light,” Phys. Rev. A 67, 043805 (2003).
[Crossref]

A. D. Greentree, D. Richards, J. A. Vaccaro, A. V. Durrant, S. R. de Echaniz, D. M. Segal, and J. P. Marangos, “Intensity-dependent dispersion under conditions of electromagnetically induced transparency in coherently prepared multistate atoms,” Phys. Rev. A 67, 023818 (2003).
[Crossref]

2002 (5)

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

M. S. Zubairy, A. B. Matsko, and M. O. Scully, “Resonant enhancement of high-order optical nonlinearities based on atomic coherence,” Phys. Rev. A 65, 043804 (2002).
[Crossref]

A. B. Matsko, I. Novikova, G. R. Welch, and M. S. Zubairy, “Enhancement of Kerr nonlinearity by multiphoton coherence,” Opt. Lett. 28, 96-98 (2002).
[Crossref]

K. J. Resch, J. S. Lundeen, and A. M. Steinberg, “Conditional-phase switch at the single-photon level,” Phys. Rev. Lett. 89, 037904 (2002).
[Crossref] [PubMed]

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145-195 (2002).
[Crossref]

2001 (2)

M. Yan, E. G. Rickey, and Y. Zhu, “Suppression of two-photon absorption by quantum interference,” Phys. Rev. A 64, 043807 (2001).
[Crossref]

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

2000 (1)

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

1997 (3)

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575-579 (1997).
[Crossref]

J.-F. Roch, K. Vigneron, P. Grelu, A. Sinatra, J.-P. Poizat, and P. Grangier, “Quantum nondemolition measurements using cold trapped atoms,” Phys. Rev. Lett. 78, 634-637 (1997).
[Crossref]

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

1996 (1)

1995 (1)

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

1990 (2)

S. E. Harris, J. E. Field, and A. Imamoglu, “Nonlinear optical process using electromagnetically induced transparency,” Phys. Rev. Lett. 64, 1107-1110 (1990).
[Crossref] [PubMed]

K. J. Blow, R. Loudon, S. J. D. Phoenix, and T. J. Shepherd, “Continuum fields in quantum optics,” Phys. Rev. A 42, 4102-4114 (1990).
[Crossref] [PubMed]

1977 (1)

E. Arimondo, M. Inguscio, and P. Violino, “Experimental determinations of the hyperfine structure in the alkali atoms,” Rev. Mod. Phys. 49, 31-75 (1977).
[Crossref]

1975 (1)

C. Tai, W. Happer, and R. Gupta, “Hyperfine structure and lifetime measurements of the second-excited D states of rubidium and cesium by cascade fluorescence spectroscopy,” Phys. Rev. A 12, 736-747 (1975).
[Crossref]

Nature (1)

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575-579 (1997).
[Crossref]

Opt. Lett. (2)

Phys. Rev. A (12)

M. S. Zubairy, A. B. Matsko, and M. O. Scully, “Resonant enhancement of high-order optical nonlinearities based on atomic coherence,” Phys. Rev. A 65, 043804 (2002).
[Crossref]

K. J. Blow, R. Loudon, S. J. D. Phoenix, and T. J. Shepherd, “Continuum fields in quantum optics,” Phys. Rev. A 42, 4102-4114 (1990).
[Crossref] [PubMed]

C. Tai, W. Happer, and R. Gupta, “Hyperfine structure and lifetime measurements of the second-excited D states of rubidium and cesium by cascade fluorescence spectroscopy,” Phys. Rev. A 12, 736-747 (1975).
[Crossref]

M. Yan, E. G. Rickey, and Y. Zhu, “Suppression of two-photon absorption by quantum interference,” Phys. Rev. A 64, 043807 (2001).
[Crossref]

A. B. Matsko, I. Novikova, M. S. Zubairy, and G. R. Welch, “Nonlinear magneto-optical rotation of elliptically polarized light,” Phys. Rev. A 67, 043805 (2003).
[Crossref]

A. D. Greentree, D. Richards, J. A. Vaccaro, A. V. Durrant, S. R. de Echaniz, D. M. Segal, and J. P. Marangos, “Intensity-dependent dispersion under conditions of electromagnetically induced transparency in coherently prepared multistate atoms,” Phys. Rev. A 67, 023818 (2003).
[Crossref]

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

C. Ottaviani, S. Rebic, D. Vitali, and P. Tombesi, “Quantum phase-gate operation based on nonlinear optics: full quantum analysis,” Phys. Rev. A 73, 010301(R) (2006).
[Crossref]

S. Rebic, D. Vitali, C. Ottaviani, P. Tombesi, M. Artoni, F. Cataliotti, and R. Corbalan, “Polarization phase gate with a tripod atomic system,” Phys. Rev. A 70, 032317 (2004).
[Crossref]

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

H. Goto and K. Ichimura, “Multiqubit controlled unitary gate by adiabatic passage with an optical cavity,” Phys. Rev. A 70, 012305 (2004).
[Crossref]

C. Hang, Y. Li, L. Ma, and G. X. Huang, “Three-way entanglement and three-qubit phase gate based on a coherent six-level atomic system,” Phys. Rev. A 74, 012319 (2006).
[Crossref]

Phys. Rev. Lett. (8)

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

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

J.-F. Roch, K. Vigneron, P. Grelu, A. Sinatra, J.-P. Poizat, and P. Grangier, “Quantum nondemolition measurements using cold trapped atoms,” Phys. Rev. Lett. 78, 634-637 (1997).
[Crossref]

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

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

K. J. Resch, J. S. Lundeen, and A. M. Steinberg, “Conditional-phase switch at the single-photon level,” Phys. Rev. Lett. 89, 037904 (2002).
[Crossref] [PubMed]

C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization qubit phase gate in driven atomic media,” Phys. Rev. Lett. 90, 197902 (2003).
[Crossref] [PubMed]

S. E. Harris, J. E. Field, and A. Imamoglu, “Nonlinear optical process using electromagnetically induced transparency,” Phys. Rev. Lett. 64, 1107-1110 (1990).
[Crossref] [PubMed]

Phys. Today (1)

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

Rev. Mod. Phys. (3)

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633-673 (2005).
[Crossref]

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145-195 (2002).
[Crossref]

E. Arimondo, M. Inguscio, and P. Violino, “Experimental determinations of the hyperfine structure in the alkali atoms,” Rev. Mod. Phys. 49, 31-75 (1977).
[Crossref]

Other (3)

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge U. Press, 2000).

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge U. Press, 1997).

R. W. Boyd, Nonlinear Optics (Academic, 1992).

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Figures (5)

Fig. 1
Fig. 1

Schematic of a five-level tripod atomic system. Probe, signal, and trigger fields have Rabi frequencies Ω 1 , Ω 2 , and Ω 3 , and polarizations σ + , σ , and σ + . The Rabi frequency of the control field is Ω 4 .

Fig. 2
Fig. 2

(a) Linear susceptibility χ P ( 1 ) , (b) χ S ( 1 ) , and (c) χ T ( 1 ) versus respective detuning. The common parameters are Ω 1 = Ω 2 = Ω 3 = 0.5 , Ω 4 = 5 . The detunings are (a) Δ 2 = Δ 3 = 1.5 , Δ 4 = 1.45 , (b) Δ 1 = Δ 3 = 1.5 , Δ 4 = 1.45 , and (c) Δ 1 = Δ 2 = 1.5 , Δ 4 = 1.45 . The dispersion and absorption are represented by solid and dashed curves. Exact numerical results (solid triangle) including all orders of contributions are shown together with the analytical results.

Fig. 3
Fig. 3

Fifth-order susceptibility of (a) probe, (b) signal, and (c) trigger fields. Parameters are chosen as those in Fig. 2.

Fig. 4
Fig. 4

Absorption properties of the three weak pulses propagating in the EIT media. (a) Transmitted field amplitude as a function of the detuning through a length of L 0.7 mm gases. (b) Field amplitude versus the propagating length in the EIT media. Other parameters are the same as those in Fig. 2.

Fig. 5
Fig. 5

Total fifth-order nonlinear phase shifts of probe, signal, and trigger fields under different Rabi frequencies of the control field.

Equations (60)

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H ̂ I = Δ 1 σ d d + ( Δ 1 Δ 2 ) σ b b + ( Δ 1 Δ 3 ) σ c c + ( Δ 1 + Δ 4 ) σ e e + Ω 1 σ d a + Ω 2 σ d b + Ω 3 σ d c + Ω 4 σ e d + H.c. ,
ρ ̇ a a = 2 γ d a ρ d d + 2 γ b a ρ b b + 2 γ c a ρ c c + i ( Ω 1 ρ a d Ω 1 * ρ d a ) ,
ρ ̇ b b = 2 γ d b ρ d d 2 γ b a ρ b b + 2 γ c b ρ c c + i ( Ω 2 ρ b d Ω 2 * ρ d b ) ,
ρ ̇ c c = 2 γ d c ρ d d 2 γ c b ρ c c 2 γ c a ρ c c + i ( Ω 3 ρ c d Ω 3 * ρ d c ) ,
ρ ̇ d d = 2 ( γ d a + γ d b + γ d c ) ρ d d + 2 γ e d ρ e e i ( Ω 1 ρ a d Ω 1 * ρ d a ) i ( Ω 2 ρ b d Ω 2 * ρ d b ) i ( Ω 3 ρ c d Ω 3 * ρ d c ) + i ( Ω 4 ρ d e Ω 4 * ρ e d ) ,
ρ ̇ e e = 2 γ e d ρ e e i ( Ω 4 ρ d e Ω 4 * ρ e d ) ,
ρ ̇ a d = ( γ d a + γ d b + γ d c i Δ 1 ) ρ a d + i Ω 1 * ( ρ a a ρ d d ) + i Ω 2 * ρ a b + i Ω 3 * ρ a c + i Ω 4 ρ a e ,
ρ ̇ a b = [ γ b a i ( Δ 1 Δ 2 ) ] ρ a b + i Ω 2 ρ a d i Ω 1 * ρ d b ,
ρ ̇ a c = [ γ c a + γ c b i ( Δ 1 Δ 3 ) ] ρ a c + i Ω 3 ρ a d i Ω 1 * ρ d c ,
ρ ̇ a e = [ γ e d i ( Δ 1 + Δ 4 ) ] ρ a e + i Ω 4 * ρ a d i Ω 1 * ρ d e ,
ρ ̇ b c = [ γ c a + γ c b + γ b a i ( Δ 2 Δ 3 ) ] ρ b c + i Ω 3 ρ b d i Ω 2 * ρ d c ,
ρ ̇ b d = ( γ d a + γ d b + γ d c + γ b a i Δ 2 ) ρ b d + i Ω 2 * ( ρ b b ρ d d ) + i Ω 4 ρ b e + i Ω 1 * ρ b a + i Ω 3 * ρ b c ,
ρ ̇ b e = [ γ e d + γ b a i ( Δ 2 + Δ 4 ) ] ρ b e + i Ω 4 * ρ b d i Ω 2 * ρ d e ,
ρ ̇ c d = ( γ d a + γ d b + γ d c + γ c a + γ c b i Δ 3 ) ρ c d + i Ω 3 * ( ρ c c ρ d d ) + i Ω 1 * ρ c a + i Ω 2 * ρ c b + i Ω 4 ρ c e ,
ρ ̇ c e = [ γ e d + γ c a + γ c b i ( Δ 3 + Δ 4 ) ] ρ c e + i Ω 4 * ρ c d i Ω 3 * ρ d e ,
ρ ̇ d e = ( γ d a + γ d b + γ d c + γ e d i Δ 4 ) ρ d e + i Ω 4 * ( ρ d d ρ e e ) i Ω 1 ρ a e i Ω 2 ρ b e i Ω 3 ρ c e ,
ρ a d i Ω 1 * ( ρ a a ( 0 ) ρ d d ( 0 ) ) B i Ω 1 * B [ Ω 2 2 ( ρ b b ( 0 ) ρ d d ( 0 ) ) C D + Ω 3 2 ( ρ c c ( 0 ) ρ d d ( 0 ) ) E F ] + i Ω 1 * Ω 2 2 Ω 3 2 B [ ( ρ c c ( 0 ) ρ d d ( 0 ) ) C D F * G + ( ρ b b ( 0 ) ρ d d ( 0 ) ) D * E F G * ] ,
B = γ i Δ 1 + Ω 4 2 γ e d i ( Δ 1 + Δ 4 ) ,
C = γ b a i ( Δ 1 Δ 2 ) ,
D = γ + i Δ 2 + Ω 4 2 γ e d + γ b a + i ( Δ 2 + Δ 4 ) ,
E = γ c a + γ c b i ( Δ 1 Δ 3 ) ,
F = γ + i Δ 3 + Ω 4 2 γ e d + γ c a + γ c b + i ( Δ 3 + Δ 4 ) ,
G = γ c a + γ c b + γ b a + i ( Δ 2 Δ 3 ) ,
χ P = N μ 1 2 ϵ 0 ρ d a Ω 1 ,
χ S = N μ 2 2 ϵ 0 ρ d b Ω 2 ,
χ T = N μ 3 2 ϵ 0 ρ d c Ω 3 ,
χ P χ P ( 1 ) + χ P 1 ( 3 ) E 2 2 + χ P 2 ( 3 ) E 3 2 + χ P ( 5 ) E 2 2 E 3 2 ,
χ S χ S ( 1 ) + χ S 1 ( 3 ) E 1 2 + χ S 2 ( 3 ) E 3 2 + χ S ( 5 ) E 1 2 E 3 2 ,
χ T χ T ( 1 ) + χ T 1 ( 3 ) E 1 2 + χ T 2 ( 3 ) E 2 2 + χ T ( 5 ) E 1 2 E 2 2 .
χ P ( 1 ) N μ 1 2 3 ϵ 0 i B * ,
χ S ( 1 ) N μ 2 2 3 ϵ 0 i D ,
χ T ( 1 ) N μ 3 2 3 ϵ 0 i F .
χ P 1 ( 3 ) N μ 1 2 μ 2 2 3 3 ϵ 0 i B * C * D * ,
χ P 2 ( 3 ) N μ 1 2 μ 3 2 3 3 ϵ 0 i B * E * F * .
χ P ( 5 ) N μ 1 2 μ 2 2 μ 3 2 3 5 ϵ 0 i B * × [ 1 C * D * F G * + 1 D E * F * G ] .
( υ g ) P 4 c ϵ 0 ω 1 N μ 1 2 ( Ω 2 2 + Ω 3 2 + Ω 4 2 ) ,
( υ g ) S 4 c ϵ 0 ω 2 N μ 2 2 ( Ω 1 2 + Ω 3 2 + Ω 4 2 ) ,
( υ g ) T 4 c ϵ 0 ω 3 N μ 3 2 ( Ω 1 2 + Ω 2 2 + Ω 4 2 ) .
ψ i = α i + σ + i + α i σ i , i = { P , S , T } ,
σ ± i = d ω ξ i ( ω ) a ̂ ± ( ω ) 0 ,
a ̂ ± ( ω ) a ̂ ± ( ω ) exp { i ω c 0 l d z n ± ( ω , z ) } ,
σ ± P σ ± S σ ± T e i ( ϕ ± P + ϕ ± S + ϕ ± T ) σ ± P σ ± S σ ± T .
σ P σ S σ + T e i ( ϕ 0 P + ϕ K S + ϕ K T ) σ P σ S σ + T ,
σ + P σ + S σ + T e i ( ϕ K P + ϕ 0 S + ϕ K T ) σ + P σ + S σ + T ,
σ + P σ S σ T e i ( ϕ K P + ϕ K S + ϕ 0 T ) σ + P σ S σ T ,
σ P σ + S σ + T e i ( ϕ 0 P + ϕ 0 S + ϕ l i n T ) σ P σ + S σ + T ,
σ + P σ + S σ T e i ( ϕ l i n P + ϕ 0 S + ϕ 0 T ) σ + P σ + S σ T ,
σ P σ S σ T e i ( ϕ 0 P + ϕ l i n S + ϕ 0 T ) σ P σ S σ T ,
σ P σ + S σ T e i ( ϕ 0 P + ϕ 0 S + ϕ 0 T ) σ P σ + S σ T ,
σ + P σ S σ + T e i ( ϕ + P + ϕ S + ϕ + T ) σ + P σ S σ + T ,
ϕ F P k P l π 3 2 4 Ω 2 2 Ω 3 2 4 μ 2 2 μ 3 2 erf [ ζ P ] ζ P Re [ χ P ( 5 ) ] ,
ϕ F S k S l π 3 2 4 Ω 1 2 Ω 3 2 4 μ 1 2 μ 3 2 erf [ ζ S ] ζ S Re [ χ S ( 5 ) ] ,
ϕ F T k T l π 3 2 4 Ω 1 2 Ω 2 2 4 μ 1 2 μ 2 2 erf [ ζ T ] ζ T Re [ χ T ( 5 ) ] ,
E j z = i 2 π ω j c P j ( j = 1 , 2 , 3 ) ,
χ S 1 ( 3 ) N μ 1 2 μ 2 2 3 3 ϵ 0 i γ + i Δ 2 + Ω 4 2 γ e d + i ( Δ 2 + Δ 4 ) × 1 [ γ b a i ( Δ 1 Δ 2 ) ] [ γ i Δ 1 + Ω 4 2 γ e d i ( Δ 1 + Δ 4 ) ] ,
χ S 2 ( 3 ) N μ 2 2 μ 3 2 3 3 ϵ 0 i γ + i Δ 2 + Ω 4 2 γ e d + i ( Δ 2 + Δ 4 ) 1 [ γ c a + γ c b + γ b a + i ( Δ 2 Δ 3 ) ] [ γ i Δ 3 + Ω 4 2 γ e d i ( Δ 3 + Δ 4 ) ] ,
χ T 1 ( 3 ) N μ 1 2 μ 3 2 3 3 ϵ 0 i γ + i Δ 3 + Ω 4 2 γ e d + i ( Δ 3 + Δ 4 ) 1 [ γ c a + γ c b i ( Δ 1 Δ 3 ) ] [ γ i Δ 1 + Ω 4 2 γ e d i ( Δ 1 + Δ 4 ) ] ,
χ T 2 ( 3 ) N μ 2 2 μ 3 2 3 3 ϵ 0 i γ + i Δ 3 + Ω 4 2 γ e d + i ( Δ 3 + Δ 4 ) 1 [ γ c a + γ c b + γ b a i ( Δ 2 Δ 3 ) ] [ γ i Δ 2 + Ω 4 2 γ e d i ( Δ 2 + Δ 4 ) ] ,
χ S ( 5 ) N μ 1 2 μ 2 2 μ 3 2 3 5 ϵ 0 i γ + i Δ 2 + Ω 4 2 γ e d + i ( Δ 2 + Δ 4 ) { 1 [ γ c a + γ c b i ( Δ 1 Δ 3 ) ] [ γ + i Δ 3 + Ω 4 2 γ e d + i ( Δ 3 + Δ 4 ) ] 1 [ γ i Δ 1 + Ω 4 2 γ e d i ( Δ 1 + Δ 4 ) ] [ γ b a i ( Δ 1 Δ 2 ) ] + 1 [ γ c a + γ c b + γ b a + i ( Δ 2 Δ 3 ) ] [ γ i Δ 3 + Ω 4 2 γ e d i ( Δ 3 + Δ 4 ) ] 1 [ γ c a + γ c b + i ( Δ 1 Δ 3 ) ] [ γ + i Δ 1 + Ω 4 2 γ e d + i ( Δ 1 + Δ 4 ) ] } ,
χ T ( 5 ) N μ 1 2 μ 2 2 μ 3 2 3 5 ϵ 0 i γ + i Δ 3 + Ω 4 2 γ e d + i ( Δ 3 + Δ 4 ) × { 1 [ γ b a i ( Δ 1 Δ 2 ) ] [ γ + i Δ 2 + Ω 4 2 γ e d + i ( Δ 2 + Δ 4 ) ] ] × 1 [ γ c a + γ c b i ( Δ 1 Δ 3 ) ] [ γ i Δ 1 + Ω 4 2 γ e d i ( Δ 1 + Δ 4 ) ] + 1 [ γ b a + i ( Δ 1 Δ 2 ) ] [ γ + i Δ 1 + Ω 4 2 γ e d + i ( Δ 1 + Δ 4 ) ] × 1 [ γ c a + γ c b + γ b a i ( Δ 2 Δ 3 ) ] [ γ i Δ 2 + Ω 4 2 γ e d i ( Δ 2 + Δ 4 ) ] } .

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