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, H. J. Kimble, Phys. Rev. Lett. 59, 198 (1987).
    [CrossRef] [PubMed]
  2. R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
    [CrossRef] [PubMed]
  3. M. W. Maeda, P. Kumar, J. H. Shapiro, Opt. Lett. 12, 161 (1987).
    [CrossRef] [PubMed]
  4. M. D. Reid, D. F. Walls, Phys. Rev. A 31, 1622 (1985); Phys. Rev. A 34, 4929 (1986).
    [CrossRef] [PubMed]
  5. M. D. Reid, 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, M. Sargent, Phys. Rev. A 35, 2150 (1987).
    [CrossRef] [PubMed]
  8. L. A. Lugiato, G. Strini, Opt. Commun. 41, 67 (1982).
    [CrossRef]
  9. M. Xiao, H. J. Kimble, 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, 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, C. W. Gardiner, J. Phys. A 13, 2353 (1980); P. D. Drummond, C. W. Gardiner, 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, 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, J. H. Shapiro, Opt. Lett. 12, 161 (1987).
[CrossRef] [PubMed]

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

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

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

1985 (4)

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

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

M. D. Reid, 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, G. Strini, Opt. Commun. 41, 67 (1982).
[CrossRef]

1981 (2)

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

1977 (1)

S. Chaturvedi, C. W. Gardiner, I. Matheson, 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, H. J. Kimble, Phys. Rev. Lett. 59, 198 (1987).
[CrossRef] [PubMed]

Carmichael, H. J.

M. Xiao, H. J. Kimble, 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, D. F. Walls, J. Stat. Phys. 17, 469 (1977).
[CrossRef]

Drummond, P. D.

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

Gardiner, C. W.

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

S. Chaturvedi, C. W. Gardiner, I. Matheson, 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, J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
[CrossRef] [PubMed]

Holm, D. A.

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

Kimble, H. J.

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

M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, 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, 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, D. F. Walls, J. Stat. Phys. 17, 469 (1977).
[CrossRef]

Mertz, J. C.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, 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, 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, H. J. Kimble, Phys. Rev. Lett. 59, 198 (1987).
[CrossRef] [PubMed]

Reid, M. D.

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

M. D. Reid, 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, 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, J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
[CrossRef] [PubMed]

Strini, G.

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

Valley, J. F.

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

Walls, D. F.

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

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

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

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

Xiao, M.

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

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

Yurke, B.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, 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, C. W. Gardiner, J. Phys. A 13, 2353 (1980); P. D. Drummond, C. W. Gardiner, D. F. Walls, Phys. Rev. A 24, 914 (1981).
[CrossRef]

J. Stat. Phys. (1)

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

Opt. Commun. (1)

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

Opt. Lett. (1)

Phys. Rev. A (6)

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

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

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

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

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

P. D. Drummond, 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, H. J. Kimble, Phys. Rev. Lett. 59, 198 (1987).
[CrossRef] [PubMed]

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, 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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