Abstract

A spatially varying field mode is included in calculating the squeezing effect for a system of two-level atoms in the good-cavity limit. Two examples of a Gaussian mode field in a ring cavity and a plane-wave field in a standing-wave interferometer are used to demonstrate the quite general method. In qualitative terms, the squeezing predicted for plane waves is preserved. However, for a given value of atomic cooperativity parameter C, there is a degradation in squeezing because of the spatially varying field structure.

© 1987 Optical Society of America

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  1. M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, and H. J. Kimble, Phys. Rev. Lett. 59, 198 (1987).
    [CrossRef] [PubMed]
  2. R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
    [CrossRef] [PubMed]
  3. M. W. Maeda, P. Kumar, and J. H. Shapiro, Opt. Lett. 12, 161 (1987).
    [CrossRef] [PubMed]
  4. M. D. Reid and D. F. Walls, Phys. Rev. A 31, 1622 (1985); Phys. Rev. A 34, 4929 (1986).
    [CrossRef] [PubMed]
  5. M. D. Reid and D. F. Walls, Phys. Rev. A 32, 396 (1985).
    [CrossRef] [PubMed]
  6. H. J. Carmichael, Phys. Rev. A 33, 3262 (1985).
    [CrossRef]
  7. D. A. Holm and M. Sargent, Phys. Rev. A 35, 2150 (1987).
    [CrossRef] [PubMed]
  8. L. A. Lugiato and G. Strini, Opt. Commun. 41, 67 (1982).
    [CrossRef]
  9. M. Xiao, H. J. Kimble, and H. J. Carmichael, Phys. Rev. A 35, 3832 (1987).
    [CrossRef] [PubMed]
  10. W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1983).
  11. P. D. Drummond and D. F. Walls, Phys. Rev. A 23, 2563 (1981).
    [CrossRef]
  12. L. A. Lugiato, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1984), vol. 21.
    [CrossRef]
  13. P. D. Drummond, IEEE J. Quantum Electron. QE-17, 301 (1981).
    [CrossRef]
  14. H. Haken, Encyclopedia of Physics (Springer-Verlag, Berlin, 1970), Vol. XXV/2C.
  15. P. D. Drummond and C. W. Gardiner, J. Phys. A 13, 2353 (1980); P. D. Drummond, C. W. Gardiner, and D. F. Walls, Phys. Rev. A 24, 914 (1981).
    [CrossRef]
  16. L. Arnold, Stochastic Differential Equations (Wiley, New York, 1974).
  17. H. Risken, The Fokker–Planck Equation (Springer-Verlag, Berlin, 1984).
    [CrossRef]
  18. S. Chaturvedi, C. W. Gardiner, I. Matheson, and D. F. Walls, J. Stat. Phys. 17, 469 (1977).
    [CrossRef]
  19. C. W. Gardiner, Handbook of Stochastic Processes (Springer-Verlag, New York, 1983).

1987 (4)

M. W. Maeda, P. Kumar, and J. H. Shapiro, Opt. Lett. 12, 161 (1987).
[CrossRef] [PubMed]

M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, and H. J. Kimble, Phys. Rev. Lett. 59, 198 (1987).
[CrossRef] [PubMed]

D. A. Holm and M. Sargent, Phys. Rev. A 35, 2150 (1987).
[CrossRef] [PubMed]

M. Xiao, H. J. Kimble, and H. J. Carmichael, Phys. Rev. A 35, 3832 (1987).
[CrossRef] [PubMed]

1985 (4)

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
[CrossRef] [PubMed]

M. D. Reid and D. F. Walls, Phys. Rev. A 31, 1622 (1985); Phys. Rev. A 34, 4929 (1986).
[CrossRef] [PubMed]

M. D. Reid and D. F. Walls, Phys. Rev. A 32, 396 (1985).
[CrossRef] [PubMed]

H. J. Carmichael, Phys. Rev. A 33, 3262 (1985).
[CrossRef]

1982 (1)

L. A. Lugiato and G. Strini, Opt. Commun. 41, 67 (1982).
[CrossRef]

1981 (2)

P. D. Drummond and D. F. Walls, Phys. Rev. A 23, 2563 (1981).
[CrossRef]

P. D. Drummond, IEEE J. Quantum Electron. QE-17, 301 (1981).
[CrossRef]

1980 (1)

P. D. Drummond and C. W. Gardiner, J. Phys. A 13, 2353 (1980); P. D. Drummond, C. W. Gardiner, and D. F. Walls, Phys. Rev. A 24, 914 (1981).
[CrossRef]

1977 (1)

S. Chaturvedi, C. W. Gardiner, I. Matheson, and D. F. Walls, J. Stat. Phys. 17, 469 (1977).
[CrossRef]

Arnold, L.

L. Arnold, Stochastic Differential Equations (Wiley, New York, 1974).

Boyd, T. L.

M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, and H. J. Kimble, Phys. Rev. Lett. 59, 198 (1987).
[CrossRef] [PubMed]

Carmichael, H. J.

M. Xiao, H. J. Kimble, and H. J. Carmichael, Phys. Rev. A 35, 3832 (1987).
[CrossRef] [PubMed]

H. J. Carmichael, Phys. Rev. A 33, 3262 (1985).
[CrossRef]

Chaturvedi, S.

S. Chaturvedi, C. W. Gardiner, I. Matheson, and D. F. Walls, J. Stat. Phys. 17, 469 (1977).
[CrossRef]

Drummond, P. D.

P. D. Drummond and D. F. Walls, Phys. Rev. A 23, 2563 (1981).
[CrossRef]

P. D. Drummond, IEEE J. Quantum Electron. QE-17, 301 (1981).
[CrossRef]

P. D. Drummond and C. W. Gardiner, J. Phys. A 13, 2353 (1980); P. D. Drummond, C. W. Gardiner, and D. F. Walls, Phys. Rev. A 24, 914 (1981).
[CrossRef]

Gardiner, C. W.

P. D. Drummond and C. W. Gardiner, J. Phys. A 13, 2353 (1980); P. D. Drummond, C. W. Gardiner, and D. F. Walls, Phys. Rev. A 24, 914 (1981).
[CrossRef]

S. Chaturvedi, C. W. Gardiner, I. Matheson, and D. F. Walls, J. Stat. Phys. 17, 469 (1977).
[CrossRef]

C. W. Gardiner, Handbook of Stochastic Processes (Springer-Verlag, New York, 1983).

Haken, H.

H. Haken, Encyclopedia of Physics (Springer-Verlag, Berlin, 1970), Vol. XXV/2C.

Hollberg, L. W.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
[CrossRef] [PubMed]

Holm, D. A.

D. A. Holm and M. Sargent, Phys. Rev. A 35, 2150 (1987).
[CrossRef] [PubMed]

Kimble, H. J.

M. Xiao, H. J. Kimble, and H. J. Carmichael, Phys. Rev. A 35, 3832 (1987).
[CrossRef] [PubMed]

M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, and H. J. Kimble, Phys. Rev. Lett. 59, 198 (1987).
[CrossRef] [PubMed]

Kumar, P.

Louisell, W. H.

W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1983).

Lugiato, L. A.

L. A. Lugiato and G. Strini, Opt. Commun. 41, 67 (1982).
[CrossRef]

L. A. Lugiato, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1984), vol. 21.
[CrossRef]

Maeda, M. W.

Matheson, I.

S. Chaturvedi, C. W. Gardiner, I. Matheson, and D. F. Walls, J. Stat. Phys. 17, 469 (1977).
[CrossRef]

Mertz, J. C.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
[CrossRef] [PubMed]

Orozco, L. A.

M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, and H. J. Kimble, Phys. Rev. Lett. 59, 198 (1987).
[CrossRef] [PubMed]

Raizen, M. G.

M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, and H. J. Kimble, Phys. Rev. Lett. 59, 198 (1987).
[CrossRef] [PubMed]

Reid, M. D.

M. D. Reid and D. F. Walls, Phys. Rev. A 31, 1622 (1985); Phys. Rev. A 34, 4929 (1986).
[CrossRef] [PubMed]

M. D. Reid and D. F. Walls, Phys. Rev. A 32, 396 (1985).
[CrossRef] [PubMed]

Risken, H.

H. Risken, The Fokker–Planck Equation (Springer-Verlag, Berlin, 1984).
[CrossRef]

Sargent, M.

D. A. Holm and M. Sargent, Phys. Rev. A 35, 2150 (1987).
[CrossRef] [PubMed]

Shapiro, J. H.

Slusher, R. E.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
[CrossRef] [PubMed]

Strini, G.

L. A. Lugiato and G. Strini, Opt. Commun. 41, 67 (1982).
[CrossRef]

Valley, J. F.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
[CrossRef] [PubMed]

Walls, D. F.

M. D. Reid and D. F. Walls, Phys. Rev. A 32, 396 (1985).
[CrossRef] [PubMed]

M. D. Reid and D. F. Walls, Phys. Rev. A 31, 1622 (1985); Phys. Rev. A 34, 4929 (1986).
[CrossRef] [PubMed]

P. D. Drummond and D. F. Walls, Phys. Rev. A 23, 2563 (1981).
[CrossRef]

S. Chaturvedi, C. W. Gardiner, I. Matheson, and D. F. Walls, J. Stat. Phys. 17, 469 (1977).
[CrossRef]

Xiao, M.

M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, and H. J. Kimble, Phys. Rev. Lett. 59, 198 (1987).
[CrossRef] [PubMed]

M. Xiao, H. J. Kimble, and H. J. Carmichael, Phys. Rev. A 35, 3832 (1987).
[CrossRef] [PubMed]

Yurke, B.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
[CrossRef] [PubMed]

IEEE J. Quantum Electron. (1)

P. D. Drummond, IEEE J. Quantum Electron. QE-17, 301 (1981).
[CrossRef]

J. Phys. A (1)

P. D. Drummond and C. W. Gardiner, J. Phys. A 13, 2353 (1980); P. D. Drummond, C. W. Gardiner, and D. F. Walls, Phys. Rev. A 24, 914 (1981).
[CrossRef]

J. Stat. Phys. (1)

S. Chaturvedi, C. W. Gardiner, I. Matheson, and D. F. Walls, J. Stat. Phys. 17, 469 (1977).
[CrossRef]

Opt. Commun. (1)

L. A. Lugiato and G. Strini, Opt. Commun. 41, 67 (1982).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (6)

M. D. Reid and D. F. Walls, Phys. Rev. A 31, 1622 (1985); Phys. Rev. A 34, 4929 (1986).
[CrossRef] [PubMed]

M. D. Reid and D. F. Walls, Phys. Rev. A 32, 396 (1985).
[CrossRef] [PubMed]

H. J. Carmichael, Phys. Rev. A 33, 3262 (1985).
[CrossRef]

D. A. Holm and M. Sargent, Phys. Rev. A 35, 2150 (1987).
[CrossRef] [PubMed]

M. Xiao, H. J. Kimble, and H. J. Carmichael, Phys. Rev. A 35, 3832 (1987).
[CrossRef] [PubMed]

P. D. Drummond and D. F. Walls, Phys. Rev. A 23, 2563 (1981).
[CrossRef]

Phys. Rev. Lett. (2)

M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, and H. J. Kimble, Phys. Rev. Lett. 59, 198 (1987).
[CrossRef] [PubMed]

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
[CrossRef] [PubMed]

Other (6)

W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1983).

L. A. Lugiato, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1984), vol. 21.
[CrossRef]

H. Haken, Encyclopedia of Physics (Springer-Verlag, Berlin, 1970), Vol. XXV/2C.

C. W. Gardiner, Handbook of Stochastic Processes (Springer-Verlag, New York, 1983).

L. Arnold, Stochastic Differential Equations (Wiley, New York, 1974).

H. Risken, The Fokker–Planck Equation (Springer-Verlag, Berlin, 1984).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Optimal squeezing Sout(0) versus atomic cooperativity parameter C. Curve i, plane wave in a ring cavity; curve ii, Gaussian mode in a ring cavity. δ, ϕ, and X are optimized at each point to give best squeezing. Representative values are as follows: C = 10: i, δ = 7.20, ϕ = 1.92, X = 8.64; ii δ = 7.18, ϕ = 2.10, X = 6.92. C = 102: i, δ = 28.2, ϕ = 6.00, X = 73.9; ii, δ = 26.5, ϕ = 6.60, X = 54.2. C = 103: i, δ = 111, ϕ = 16.7, X = 556; ii, δ = 101, ϕ = 18.6, X = 388. C = 104: i, δ = 438, ϕ = 44.2, X = 3.87 × 103; ii, δ = 392, ϕ = 49.7, X = 2.64 × 103. C = 105: i, δ = 1.73 × 103, ϕ = 114, X = 2.58 × 104; ii, δ = 1.52 × 103, ϕ = 130, X = 1.70 × 104. C = 106: i, δ = 6.86 × 103, ϕ = 290, X = 1.70 × 105; ii, δ = 5.72 × 103, ϕ = 348, X = 9.97 × 104. C = 107: i, δ = 2.73 × 104, ϕ = 731, X = 1.11 × 106; ii, δ = 2.54 × 104, ϕ = 785, X = 9.05 × 105. C = 108: i, δ = 1.07 × 105, ϕ = 1.87 × 103, X = 6.76 × 106; ii, δ = 1.03 × 105, ϕ = 1.94 × 103, X = 5.90 × 106.

Fig. 2
Fig. 2

Squeezing Sout(0) versus intensity X for the plane-wave case. Curve i: C = 103, δ = 111 (optimal for given C), ϕ = 15.0; curve ii: C = 103, δ = 111, ϕ = 16.7 (optimal); curve iii: C = 103, δ = 111, ϕ = 18.0.

Fig. 3
Fig. 3

Squeezing Sout(0) versus intensity X for the Gaussian case. Curve i: C = 103, δ = 101 (optimal for given C), ϕ = 17.0; curve ii: C = 103, δ = 101, ϕ = 18.6 (optimal); curve iii: C = 103, δ = 101, ϕ = 20.0.

Fig. 4
Fig. 4

Optimal squeezing Sout(0) versus atomic cooperativity parameter C. Curve i, plane wave in a ring cavity; curve ii, standing-wave longitudinal structure with plane-wave transverse profile. δ, ϕ, and X are optimized at each point to give best squeezing. Representative values are as follows: C = 10: i, δ = 7.20, ϕ = 1.92, X = 8.64; ii δ = 7.19, ϕ = 1.99, X = 7.98. C = 102: i, δ = 28.2, ϕ = 6.00, X = 73.9; ii, δ = 27.5, ϕ = 6.22, X = 65.8. C = 103: i, δ = 111, ϕ = 16.7, X = 556; ii, δ = 107, ϕ = 17.4, X = 486. C = 104: i, δ = 438, ϕ = 44.2, X = 3.87 × 103; ii, δ = 420, ϕ = 46.2, X = 3.36 × 103. C = 105: i, δ = 1.73 × 103, ϕ = 114, X = 2.58 × 104; ii, δ = 1.66 × 103, ϕ = 119, X = 2.30 × 104. C = 106: i, δ = 6.86 × 103, ϕ = 290, X = 1.70 × 105; ii, δ = 6.64 × 103, ϕ = 300, X = 1.51 × 105. C = 107: i, δ = 2.73 × 104, ϕ = 731, X = 1.11 × 106; ii, δ = 2.70 × 104, ϕ = 738, X = 1.07 × 105. C = 108: i, δ = 1.07 × 105, ϕ = 1.87 × 103, X = 6.76 × 106; ii, δ = 1.00 × 105, ϕ = 2.00 × 103, X = 6.11 × 106.

Fig. 5
Fig. 5

Squeezing Sout(0) versus intensity X for the standing-wave case. Curve i, C = 103, δ = 107 (optimal for given C), ϕ = 16.0; curve ii, C = 103, δ = 107, ϕ = 17.4 (optimal); curve iii, C = 103, δ = 107, ϕ = 19.0.

Equations (24)

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ρ ^ t = 1 i { ω c [ a ^ a ^ , ρ ^ ] + 1 2 ω a μ = 1 N [ σ ^ μ z , ρ ^ ] } + μ = 1 N { g μ exp ( - i k · r μ ) [ a ^ σ ^ μ - , ρ ^ ] - g μ exp ( i k · r μ ) × [ a ^ σ ^ μ + , ρ ^ ] } + μ = 1 N { ( ½ γ ) ( [ σ ^ μ - ρ ^ , σ ^ μ + ] + [ σ ^ μ - , ρ ^ σ ^ μ + ] ) } + ( ¼ γ P ) ( [ σ ^ μ z ρ ^ , σ ^ μ z ] + [ σ ^ μ z , ρ ^ σ ^ μ z ] ) + κ { [ a ^ ρ ^ , a ^ ] + [ a ^ , ρ ^ a ^ ] } + κ { exp ( - i ω I t ) [ a ^ , ρ ^ ] - * exp ( i ω I t ) [ a ^ , ρ ^ ] } .
g μ = ( μ 2 ω c 2 0 ) 1 / 2 U ( r μ ) g 0 U ( r μ ) .
Y = X | 1 + i ϕ + 2 ( 1 - i δ ) 1 + δ 2 · C s j = 1 M U ( r j ) 2 Δ V j 1 + V eff s X U ( r j ) 2 1 + δ 2 | 2 .
V eff = s 2 / V ¯ U ( r ) 4 d 3 r ,
lim M Δ V j 0 j = 1 M Δ V j = V ¯ d 3 r .
( δ x ¯ ˙ δ x ¯ ˙ ) = - M ss ( δ x ¯ δ x ¯ ) + 1 n 0 B ( ξ 1 ( τ ) ξ 2 ( τ ) ) ,
M ss ( a b b * a * ) ,             D ss ( - P s s Q s s Q s s - P ss * ) .
a = 1 + i ϕ + 2 ( 1 - i δ ) s V ¯ U ( r ) 2 d 3 r 1 + δ 2 + V eff X s U ( r ) 2 - 2 ( 1 - i δ ) X C V eff s 2 V ¯ U ( r ) 4 d 3 r [ 1 + δ 2 + V eff X s U ( r ) 2 ] 2 , b = - 2 ( 1 - i δ ) x 2 C V eff s 2 V ¯ U ( r ) 4 d 3 r [ 1 + δ 2 + V eff X s U ( r ) 2 ] 2 .
P ss = 2 C x 2 V eff s 2 V ¯ d 3 r U ( r ) 4 [ 1 + δ 2 + X V eff s U ( r ) 2 ] 3 [ ( 1 - i δ ) 3 f + i δ ( 1 - i δ ) ( 1 - f ) X V eff s U ( r ) 2 + 1 2 · X 2 V eff 2 s 2 U ( r ) 4 ] , Q ss = 2 C X V eff s 2 V ¯ d 3 r U ( r ) 4 [ 1 + δ 2 + X V eff s U ( r ) 2 ] 3 { ( 1 + δ 2 ) × ( 1 - f ) + [ 2 + ( 1 - f ) δ 2 ] X V eff s U ( r ) 2 + 1 2 · X 2 V eff 2 s 2 U ( r ) 4 } ,
A ^ θ ( t ) = [ a ^ ( t ) e - i θ + a ^ ( t ) e i θ ] ,
S out ( Ω , θ ) = 2 κ 1 - + d τ T : A ^ θ ( t ) A ^ θ ( t + τ ) : e - i Ω τ = 2 κ 1 [ S 12 ( Ω ) + S 21 ( Ω ) + S 11 ( Ω ) e - 2 i θ + S 22 ( Ω ) e 2 i θ ] ,
S ( Ω ) = 1 n 0 ( M ss + i Ω I ) - 1 D ss ( M ss T - i Ω I ) - 1 .
e 2 i θ = - S 22 * ( Ω 0 ) S 22 ( Ω 0 ) ,
S out ( Ω ) = 2 κ 1 { S 12 ( Ω ) + S 21 ( Ω ) - 2 Re [ S 22 * ( Ω 0 ) S 22 ( Ω ) S 22 ( Ω 0 ) ] } .
U ( r ) = ( 2 π L W 2 ) 1 / 2 exp ( - r 2 / W 2 ) ,
d 3 r = ( 2 π r d r ) L s
V eff = ( π W 2 L ) s .
a = ( 1 + i ϕ ) + 2 C ( 1 - i δ ) 1 + δ 2 + 2 X , b = ( 1 - i δ ) C x 2 X [ 2 1 + δ 2 + 2 X - 1 X ln ( 1 + 2 X 1 + δ 2 ) ]
P ss = C x 2 ( 1 ( 1 + δ 2 + 2 X ) 2 { 2 ( 1 - i δ ) 3 f 1 + δ 2 + 4 X + 3 [ δ 2 ( 1 + 2 f ) - 2 i δ ( 1 + f ) + 3 ] + 1 + δ 2 X [ ( 1 + 2 f ) δ 2 - 2 i δ ( 1 - f ) + 3 ] } - ( 1 + 2 f ) δ 2 - 2 i δ ( 1 - f ) + 3 2 X 2 ln ( 1 + 2 X 1 + δ 2 ) ) , Q ss = C { 4 X 2 - ( 1 + 2 f ) ( 1 - 3 δ 2 ) X - ( 1 + δ 2 ) [ 1 - δ 2 ( 1 + 2 f ) ] ( 1 + δ 2 + 2 X ) 2 + 1 - δ 2 ( 1 + 2 f ) 2 X ln ( 1 + 2 X 1 + δ 2 ) } .
U ( r ) = ( 2 A L ) 1 / 2 cos ( k z )
d 3 r = A d z ,
V eff = ( 2 3 A L ) s .
a = ( 1 + i ϕ ) + 2 C ( 1 - i δ ) ( 1 + δ 2 ) [ 1 + 4 X 3 ( 1 + δ 2 ) ] 3 / 2 , b = ( 1 - i δ ) 3 C x * 2 { 2 X + 1 + δ 2 ( 1 + δ 2 ) [ 1 + 4 X 3 ( 1 + δ 2 ) ] 3 / 2 - 1 } ,
P ss = C x 2 { ( 1 + 2 f ) - δ 2 [ 4 f + δ 2 ( 1 - 2 f ) ] - i δ [ 4 f ( 1 - δ 2 ) + 2 ( 1 + δ 2 ) ] ( 1 + δ 2 ) 3 [ 1 + 4 X 3 ( 1 + δ 2 ) ] 5 / 2 + 1 X 2 [ 1 + f δ 2 - i δ ( 1 - f ) ] X ( 1 + δ 2 ) [ 1 + 4 X 3 ( 1 + δ 2 ) ] 3 / 2 - 3 X 2 [ 1 - 1 1 + 4 X 3 ( 1 + δ 2 ) ] [ 3 2 + δ 2 2 ( 1 + 2 f ) - i δ ( 1 - f ) ] } , Q ss = C { 1 - X ( 1 + δ 2 ) 2 [ 1 + 4 X 3 ( 1 + δ 2 ) ] 5 / 2 [ ( 1 + 2 f ) + δ 2 ( 1 - 2 f ) ] - 2 ( 1 - f δ 2 ) ( 1 + δ 2 ) [ 1 + 4 X 3 ( 1 + δ 2 ) ] 3 / 2 + 3 X [ 1 - 1 1 + 4 X 3 ( 1 + δ 2 ) ] [ 1 2 - δ 2 2 ( 1 - 2 f ) ] } .

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