Abstract

We calculate the squeezing spectra for a detuned degenerate four-wave mixing system. In the detuning case sizable squeezing still exists for both the pump and the signal fields. Especially in the limits for which the ratio of the damping rates becomes very small and large, the squeezing for the signal and the pump fields, respectively, is nearly perfect.

© 1993 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. D. Drummond, K. J. McNeil, and D. F. Walls, Opt. Acta 27, 321 (1980); Opt. Acta 28, 211 (1981).
    [CrossRef]
  2. M. J. Collett and C. W. Gardiner, Phys. Rev. A 30, 1386 (1984).
    [CrossRef]
  3. B. Yurke, Phys. Rev. A 29, 408 (1984).
    [CrossRef]
  4. M. J. Collett and D. F. Walls, Phys. Rev. A 32, 2887 (1985).
    [CrossRef] [PubMed]
  5. L. Wu, H. J. Kimble, J. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
    [CrossRef] [PubMed]
  6. H. J. Carmichael, J. Opt. Soc. Am. B 4, 1588 (1987).
    [CrossRef]
  7. C. Fabre, E. Giacobino, A. Heidmann, L. Lugiato, S. Reynaud, M. Vadacchino, and K. Wang, Quantum Opt. 2, 159 (1990).
    [CrossRef]
  8. P. Galatola, L. A. Lugiato, M. Vadacchino, and N. B. Abraham, Opt. Commun. 69, 419 (1989).
    [CrossRef]
  9. P. Mandel and X.-G. Wu, J. Opt. Soc. Am. B 3, 940 (1986).
    [CrossRef]
  10. P. García-Fernandez, L. A. Lugiato, F. J. Bermejo, and P. Galatola, Quantum Opt. 2, 49 (1990).
    [CrossRef]
  11. H. P. Yuen and J. H. Shapiro, Opt. Lett. 4, 334 (1979).
    [CrossRef] [PubMed]
  12. P. Kumar and J. H. Shapiro, Phys. Rev. A 30, 1568 (1984).
    [CrossRef]
  13. R. E. Slusher, L. M. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
    [CrossRef] [PubMed]
  14. M. Reid and D. F. Walls, Phys. Rev. A 31, 1622 (1986); Phys. Rev. A 33, 4465 (1986); Phys. Rev. A 34, 4929 (1986).
    [CrossRef]
  15. R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. De Voe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
    [CrossRef] [PubMed]
  16. M. W. Maeda, P. Kumar, and J. H. Shapiro, Opt. Lett. 12, 161 (1987).
    [CrossRef] [PubMed]
  17. C. M. Savage and D. F. Walls, J. Opt. Soc. Am. B 4, 1514 (1987).
    [CrossRef]
  18. L. A. Lugiato and K. Wang, Ann. Phys. 7, 539 (1990).
    [CrossRef]
  19. P. Mandel, N. P. Pettiaux, K. Wang, P. Galatola, and L. A. Lugiato, Phys. Rev. A 43, 424 (1991).
    [CrossRef] [PubMed]
  20. K. Wang, P. Galatola, L. A. Lugiato, M. Vadacchino, N. B. Abraham, and L. M. Narducci, Coherence and Quantum Optics VI, J. H. Eberly, L. Mandel, and E. Wolf, eds. (Plenum, New York, 1989) p. 577.
  21. L. A. Lugiato, P. Galatola, and L. M. Narducci, Opt. Commun. 76, 276 (1990).
    [CrossRef]
  22. R. J. Glauber, Phys. Rev. 130, 2529 (1963); Phys. Rev. 131, 2766 (1963).
    [CrossRef]

1991 (1)

P. Mandel, N. P. Pettiaux, K. Wang, P. Galatola, and L. A. Lugiato, Phys. Rev. A 43, 424 (1991).
[CrossRef] [PubMed]

1990 (4)

L. A. Lugiato, P. Galatola, and L. M. Narducci, Opt. Commun. 76, 276 (1990).
[CrossRef]

L. A. Lugiato and K. Wang, Ann. Phys. 7, 539 (1990).
[CrossRef]

C. Fabre, E. Giacobino, A. Heidmann, L. Lugiato, S. Reynaud, M. Vadacchino, and K. Wang, Quantum Opt. 2, 159 (1990).
[CrossRef]

P. García-Fernandez, L. A. Lugiato, F. J. Bermejo, and P. Galatola, Quantum Opt. 2, 49 (1990).
[CrossRef]

1989 (1)

P. Galatola, L. A. Lugiato, M. Vadacchino, and N. B. Abraham, Opt. Commun. 69, 419 (1989).
[CrossRef]

1987 (3)

1986 (4)

M. Reid and D. F. Walls, Phys. Rev. A 31, 1622 (1986); Phys. Rev. A 33, 4465 (1986); Phys. Rev. A 34, 4929 (1986).
[CrossRef]

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. De Voe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
[CrossRef] [PubMed]

L. Wu, H. J. Kimble, J. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

P. Mandel and X.-G. Wu, J. Opt. Soc. Am. B 3, 940 (1986).
[CrossRef]

1985 (2)

M. J. Collett and D. F. Walls, Phys. Rev. A 32, 2887 (1985).
[CrossRef] [PubMed]

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

1984 (3)

P. Kumar and J. H. Shapiro, Phys. Rev. A 30, 1568 (1984).
[CrossRef]

M. J. Collett and C. W. Gardiner, Phys. Rev. A 30, 1386 (1984).
[CrossRef]

B. Yurke, Phys. Rev. A 29, 408 (1984).
[CrossRef]

1980 (1)

P. D. Drummond, K. J. McNeil, and D. F. Walls, Opt. Acta 27, 321 (1980); Opt. Acta 28, 211 (1981).
[CrossRef]

1979 (1)

1963 (1)

R. J. Glauber, Phys. Rev. 130, 2529 (1963); Phys. Rev. 131, 2766 (1963).
[CrossRef]

Abraham, N. B.

P. Galatola, L. A. Lugiato, M. Vadacchino, and N. B. Abraham, Opt. Commun. 69, 419 (1989).
[CrossRef]

K. Wang, P. Galatola, L. A. Lugiato, M. Vadacchino, N. B. Abraham, and L. M. Narducci, Coherence and Quantum Optics VI, J. H. Eberly, L. Mandel, and E. Wolf, eds. (Plenum, New York, 1989) p. 577.

Bermejo, F. J.

P. García-Fernandez, L. A. Lugiato, F. J. Bermejo, and P. Galatola, Quantum Opt. 2, 49 (1990).
[CrossRef]

Carmichael, H. J.

Collett, M. J.

M. J. Collett and D. F. Walls, Phys. Rev. A 32, 2887 (1985).
[CrossRef] [PubMed]

M. J. Collett and C. W. Gardiner, Phys. Rev. A 30, 1386 (1984).
[CrossRef]

De Voe, R. G.

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. De Voe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
[CrossRef] [PubMed]

Drummond, P. D.

P. D. Drummond, K. J. McNeil, and D. F. Walls, Opt. Acta 27, 321 (1980); Opt. Acta 28, 211 (1981).
[CrossRef]

Fabre, C.

C. Fabre, E. Giacobino, A. Heidmann, L. Lugiato, S. Reynaud, M. Vadacchino, and K. Wang, Quantum Opt. 2, 159 (1990).
[CrossRef]

Galatola, P.

P. Mandel, N. P. Pettiaux, K. Wang, P. Galatola, and L. A. Lugiato, Phys. Rev. A 43, 424 (1991).
[CrossRef] [PubMed]

P. García-Fernandez, L. A. Lugiato, F. J. Bermejo, and P. Galatola, Quantum Opt. 2, 49 (1990).
[CrossRef]

L. A. Lugiato, P. Galatola, and L. M. Narducci, Opt. Commun. 76, 276 (1990).
[CrossRef]

P. Galatola, L. A. Lugiato, M. Vadacchino, and N. B. Abraham, Opt. Commun. 69, 419 (1989).
[CrossRef]

K. Wang, P. Galatola, L. A. Lugiato, M. Vadacchino, N. B. Abraham, and L. M. Narducci, Coherence and Quantum Optics VI, J. H. Eberly, L. Mandel, and E. Wolf, eds. (Plenum, New York, 1989) p. 577.

García-Fernandez, P.

P. García-Fernandez, L. A. Lugiato, F. J. Bermejo, and P. Galatola, Quantum Opt. 2, 49 (1990).
[CrossRef]

Gardiner, C. W.

M. J. Collett and C. W. Gardiner, Phys. Rev. A 30, 1386 (1984).
[CrossRef]

Giacobino, E.

C. Fabre, E. Giacobino, A. Heidmann, L. Lugiato, S. Reynaud, M. Vadacchino, and K. Wang, Quantum Opt. 2, 159 (1990).
[CrossRef]

Glauber, R. J.

R. J. Glauber, Phys. Rev. 130, 2529 (1963); Phys. Rev. 131, 2766 (1963).
[CrossRef]

Hall, J.

L. Wu, H. J. Kimble, J. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

Heidmann, A.

C. Fabre, E. Giacobino, A. Heidmann, L. Lugiato, S. Reynaud, M. Vadacchino, and K. Wang, Quantum Opt. 2, 159 (1990).
[CrossRef]

Hollberg, L. M.

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

Kimble, H. J.

L. Wu, H. J. Kimble, J. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

Kumar, P.

Levenson, M. D.

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. De Voe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
[CrossRef] [PubMed]

Lugiato, L.

C. Fabre, E. Giacobino, A. Heidmann, L. Lugiato, S. Reynaud, M. Vadacchino, and K. Wang, Quantum Opt. 2, 159 (1990).
[CrossRef]

Lugiato, L. A.

P. Mandel, N. P. Pettiaux, K. Wang, P. Galatola, and L. A. Lugiato, Phys. Rev. A 43, 424 (1991).
[CrossRef] [PubMed]

L. A. Lugiato, P. Galatola, and L. M. Narducci, Opt. Commun. 76, 276 (1990).
[CrossRef]

P. García-Fernandez, L. A. Lugiato, F. J. Bermejo, and P. Galatola, Quantum Opt. 2, 49 (1990).
[CrossRef]

L. A. Lugiato and K. Wang, Ann. Phys. 7, 539 (1990).
[CrossRef]

P. Galatola, L. A. Lugiato, M. Vadacchino, and N. B. Abraham, Opt. Commun. 69, 419 (1989).
[CrossRef]

K. Wang, P. Galatola, L. A. Lugiato, M. Vadacchino, N. B. Abraham, and L. M. Narducci, Coherence and Quantum Optics VI, J. H. Eberly, L. Mandel, and E. Wolf, eds. (Plenum, New York, 1989) p. 577.

Maeda, M. W.

Mandel, P.

P. Mandel, N. P. Pettiaux, K. Wang, P. Galatola, and L. A. Lugiato, Phys. Rev. A 43, 424 (1991).
[CrossRef] [PubMed]

P. Mandel and X.-G. Wu, J. Opt. Soc. Am. B 3, 940 (1986).
[CrossRef]

McNeil, K. J.

P. D. Drummond, K. J. McNeil, and D. F. Walls, Opt. Acta 27, 321 (1980); Opt. Acta 28, 211 (1981).
[CrossRef]

Mertz, J. C.

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

Narducci, L. M.

L. A. Lugiato, P. Galatola, and L. M. Narducci, Opt. Commun. 76, 276 (1990).
[CrossRef]

K. Wang, P. Galatola, L. A. Lugiato, M. Vadacchino, N. B. Abraham, and L. M. Narducci, Coherence and Quantum Optics VI, J. H. Eberly, L. Mandel, and E. Wolf, eds. (Plenum, New York, 1989) p. 577.

Perlmutter, S. H.

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. De Voe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
[CrossRef] [PubMed]

Pettiaux, N. P.

P. Mandel, N. P. Pettiaux, K. Wang, P. Galatola, and L. A. Lugiato, Phys. Rev. A 43, 424 (1991).
[CrossRef] [PubMed]

Reid, M.

M. Reid and D. F. Walls, Phys. Rev. A 31, 1622 (1986); Phys. Rev. A 33, 4465 (1986); Phys. Rev. A 34, 4929 (1986).
[CrossRef]

Reynaud, S.

C. Fabre, E. Giacobino, A. Heidmann, L. Lugiato, S. Reynaud, M. Vadacchino, and K. Wang, Quantum Opt. 2, 159 (1990).
[CrossRef]

Savage, C. M.

Shapiro, J. H.

Shelby, R. M.

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. De Voe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
[CrossRef] [PubMed]

Slusher, R. E.

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

Vadacchino, M.

C. Fabre, E. Giacobino, A. Heidmann, L. Lugiato, S. Reynaud, M. Vadacchino, and K. Wang, Quantum Opt. 2, 159 (1990).
[CrossRef]

P. Galatola, L. A. Lugiato, M. Vadacchino, and N. B. Abraham, Opt. Commun. 69, 419 (1989).
[CrossRef]

K. Wang, P. Galatola, L. A. Lugiato, M. Vadacchino, N. B. Abraham, and L. M. Narducci, Coherence and Quantum Optics VI, J. H. Eberly, L. Mandel, and E. Wolf, eds. (Plenum, New York, 1989) p. 577.

Valley, J. F.

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

Walls, D. F.

C. M. Savage and D. F. Walls, J. Opt. Soc. Am. B 4, 1514 (1987).
[CrossRef]

M. Reid and D. F. Walls, Phys. Rev. A 31, 1622 (1986); Phys. Rev. A 33, 4465 (1986); Phys. Rev. A 34, 4929 (1986).
[CrossRef]

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. De Voe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
[CrossRef] [PubMed]

M. J. Collett and D. F. Walls, Phys. Rev. A 32, 2887 (1985).
[CrossRef] [PubMed]

P. D. Drummond, K. J. McNeil, and D. F. Walls, Opt. Acta 27, 321 (1980); Opt. Acta 28, 211 (1981).
[CrossRef]

Wang, K.

P. Mandel, N. P. Pettiaux, K. Wang, P. Galatola, and L. A. Lugiato, Phys. Rev. A 43, 424 (1991).
[CrossRef] [PubMed]

C. Fabre, E. Giacobino, A. Heidmann, L. Lugiato, S. Reynaud, M. Vadacchino, and K. Wang, Quantum Opt. 2, 159 (1990).
[CrossRef]

L. A. Lugiato and K. Wang, Ann. Phys. 7, 539 (1990).
[CrossRef]

K. Wang, P. Galatola, L. A. Lugiato, M. Vadacchino, N. B. Abraham, and L. M. Narducci, Coherence and Quantum Optics VI, J. H. Eberly, L. Mandel, and E. Wolf, eds. (Plenum, New York, 1989) p. 577.

Wu, H.

L. Wu, H. J. Kimble, J. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

Wu, L.

L. Wu, H. J. Kimble, J. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

Wu, X.-G.

Yuen, H. P.

Yurke, B.

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

B. Yurke, Phys. Rev. A 29, 408 (1984).
[CrossRef]

Ann. Phys. (1)

L. A. Lugiato and K. Wang, Ann. Phys. 7, 539 (1990).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Acta (1)

P. D. Drummond, K. J. McNeil, and D. F. Walls, Opt. Acta 27, 321 (1980); Opt. Acta 28, 211 (1981).
[CrossRef]

Opt. Commun. (2)

P. Galatola, L. A. Lugiato, M. Vadacchino, and N. B. Abraham, Opt. Commun. 69, 419 (1989).
[CrossRef]

L. A. Lugiato, P. Galatola, and L. M. Narducci, Opt. Commun. 76, 276 (1990).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. (1)

R. J. Glauber, Phys. Rev. 130, 2529 (1963); Phys. Rev. 131, 2766 (1963).
[CrossRef]

Phys. Rev. A (6)

P. Kumar and J. H. Shapiro, Phys. Rev. A 30, 1568 (1984).
[CrossRef]

M. Reid and D. F. Walls, Phys. Rev. A 31, 1622 (1986); Phys. Rev. A 33, 4465 (1986); Phys. Rev. A 34, 4929 (1986).
[CrossRef]

P. Mandel, N. P. Pettiaux, K. Wang, P. Galatola, and L. A. Lugiato, Phys. Rev. A 43, 424 (1991).
[CrossRef] [PubMed]

M. J. Collett and C. W. Gardiner, Phys. Rev. A 30, 1386 (1984).
[CrossRef]

B. Yurke, Phys. Rev. A 29, 408 (1984).
[CrossRef]

M. J. Collett and D. F. Walls, Phys. Rev. A 32, 2887 (1985).
[CrossRef] [PubMed]

Phys. Rev. Lett. (3)

L. Wu, H. J. Kimble, J. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. De Voe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
[CrossRef] [PubMed]

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

Quantum Opt. (2)

P. García-Fernandez, L. A. Lugiato, F. J. Bermejo, and P. Galatola, Quantum Opt. 2, 49 (1990).
[CrossRef]

C. Fabre, E. Giacobino, A. Heidmann, L. Lugiato, S. Reynaud, M. Vadacchino, and K. Wang, Quantum Opt. 2, 159 (1990).
[CrossRef]

Other (1)

K. Wang, P. Galatola, L. A. Lugiato, M. Vadacchino, N. B. Abraham, and L. M. Narducci, Coherence and Quantum Optics VI, J. H. Eberly, L. Mandel, and E. Wolf, eds. (Plenum, New York, 1989) p. 577.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Intensity |A2|2 of the signal mode as a function of the input intensity |Ç|2 for Δ1 = 1.0 and Δ2 = 3.0. ABCD is a Instability loop, and point E is defined as | A 2 | 2 = 1 + Δ 1 2.

Fig. 2
Fig. 2

Optimum squeezing at ω ¯ = 10 5 as a function of the detunings Δ1 and Δ2 at the transition point D: a, For the pump mode; b, for the signal mode. The damping ratio is μ = 0.1.

Fig. 3
Fig. 3

Same as Fig. 2a, but μ = 1.0.

Fig. 4
Fig. 4

Same as Fig. 2, but μ = 10.0.

Fig. 5
Fig. 5

Variation of squeezing spectra for the signal mode as μ varies from 1.0 to 0.001. The parameters are Δ1 = 1.0, Δ2 = 3.0, and | A 2 | 2 = 2 ( Δ 1 Δ 2 1 ) / 1 + Δ 2 2 1.265 (point C in Fig. 1), and ω is normalized by γ2. Curves 1, 2, 3, and 4 correspond to μ equal to 1, 0.1, 0.01, and 0.001, respectively. Curve 5 represents perfect squeezing according to Eq. (21).

Fig. 6
Fig. 6

Variation of squeezing spectra for the pump mode as μ varies from 1 to 1000. The parameters are Δ1 = 1.0, Δ2 = 3.0, and | A 2 | 2 = 1 + Δ 1 2 1.414 (point E in Fig. 1), and ω is normalized by γ1. Curves 1, 2, 3, and 4 correspond to μ equal to 1, 10, 100, and 1000, respectively. Curve 5 is perfect squeezing.

Equations (42)

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

H = i = 1 2 ћ Δ i γ i a i a i + i ћ E ( a 1 a 1 ) + i ћ κ 2 ( a 1 2 a 2 2 a 1 2 a 2 2 ) ,
Λ i = γ i ( [ a i , ρ a i ] + [ a i ρ , a i ] ) ( i = 1 , 2 ) ,
t P ( α 1 , α 1 * , α 2 , α 2 * , t ) = { α 1 [ E γ 1 ( 1 + i Δ 1 ) α 1 κ α 1 * α 1 2 ] α 2 [ γ 2 ( 1 + i Δ 2 ) α 2 + κ α 2 * α 1 2 ] + κ 2 ( α 2 2 α 1 2 α 1 2 α 2 2 ) } P ( α 1 , α 1 * , α 2 , α 2 * , t ) + c . c .,
A 1 = κ / γ 2 α 1 , A 2 = κ / γ 1 α 2 , E ¯ = ( E / γ 1 ) κ / γ 2 , τ = γ 2 t , μ = γ 1 / γ 2 .
P t = { μ A 1 [ E ¯ ( 1 + i Δ 1 ) A 1 A 1 * A 2 2 ] A 2 [ ( 1 + i Δ 2 ) A 2 + A 2 * A 1 2 ] + κ 2 γ 2 ( 1 μ 2 A 2 2 A 1 2 μ 2 A 1 2 A 2 2 ) } P + c . c .
1 μ d A 1 d τ = E ¯ ( 1 + i Δ 1 ) A 1 A 1 * A 1 2 , d A 2 d τ = ( 1 + i Δ 2 ) A 2 + A 1 2 A 2 * .
A 1 = E ¯ ( 1 + i Δ 1 ) , A 2 = 0.
| E ¯ | 2 ( 1 + Δ 1 2 ) 1 + Δ 2 2 .
| E ¯ | th 2 = ( 1 + Δ 1 2 ) 1 + Δ 2 2 .
| A 1 | 2 = 1 + Δ 2 2 ,
| A 2 | 2 = 1 Δ 1 Δ 2 1 + Δ 2 2 ± [ ( 1 Δ 1 Δ 2 ) 2 1 + Δ 2 2 ( 1 + Δ 1 2 | E ¯ | 2 1 + Δ 2 2 ) ] 1 / 2 .
A 1 = E ¯ 1 + i Δ 1 + | A 2 | 2 [ ( 1 i Δ 2 ) / 1 + Δ 2 2 ] ,
A 2 = ± [ E ¯ ( 1 + i Δ 1 ) A 1 A 1 * ] 1 / 2 .
Δ 1 Δ 2 > 1.
τ P ( q , τ ) = i , j = 1 4 ( q i M i j q j + 1 2 2 q i q j D i j ) P ( q , τ ) ,
q = ( δ A 1 δ A 1 * δ A 2 δ A 2 * ) .
M = [ μ ( 1 + i Δ 1 ) μ A 2 2 2 μ A 1 * A 2 0 μ A 2 * 2 μ ( 1 i Δ 1 ) 0 2 μ A 1 A 2 * 2 A 1 A 2 * 0 1 + i Δ 2 A 1 2 0 2 A 1 * A 2 A 1 * 2 1 i Δ 2 ] ,
D = κ γ 2 diag ( μ A 2 2 , μ A 2 * 2 , A 1 2 μ , A 1 * 2 μ ) .
S ( ω ¯ ) = ( M + i ω ¯ I ) 1 D ( M T i ω ¯ I ) 1 ,
S 1 ( ω ) = ( 2 γ 1 ) ( γ 2 κ ) ( 1 γ 2 ) { S 12 ( ω ¯ ) + S 21 ( ω ¯ ) + Re [ S 11 ( ω ¯ ) exp ( 2 i θ ) ] } + 1 = 2 γ 1 κ { S 12 ( ω ¯ ) + S 21 ( ω ¯ ) + 2 Re [ S 11 ( ω ¯ ) exp ( 2 i θ ) ] } + 1 ,
S 2 ( ω ) = ( 2 γ 2 ) ( γ 1 κ ) ( 1 γ 2 ) { S 34 ( ω ¯ ) + S 43 ( ω ¯ ) + 2 Re [ S 33 ( ω ¯ ) exp ( 2 i θ ) ] } + 1 = 2 γ 1 κ { S 34 ( ω ¯ ) + S 43 ( ω ¯ ) + 2 Re [ S 33 ( ω ¯ ) exp ( 2 i θ ) ] } + 1 ,
S ( ω , Δ 1 , Δ 2 , θ ) = S ( ω , Δ 1 , Δ 2 , θ ) .
M = [ M 1 0 0 M 2 ] , D = [ D 1 0 0 D 2 ] ,
M 2 = [ 1 + i Δ 2 A 1 2 A 1 * 2 1 i Δ 2 ] , D 2 = [ A 1 2 0 0 A 1 * 2 ] .
S 2 ( ω ¯ ) = [ ( 1 + Δ 2 2 ω ¯ 2 | A 1 | 4 ) 2 + 4 ω ¯ 2 ] 1 4 { 2 | A 1 | 4 + Re [ A 1 2 ( 1 Δ 2 2 + ω ¯ 2 + | A 1 | 4 2 i Δ 2 ) exp ( 2 i θ ) ] } + 1.
S 2 ( ω ¯ ) = [ ω ¯ 2 ( ω ¯ 2 + 4 ) ] 1 4 { 2 ( 1 + Δ 2 2 ) + Re [ ( 2 + ω ¯ 2 2 i Δ 2 ) A 1 2 exp ( 2 i θ ) ] } + 1.
exp ( 2 i θ ) = S 33 * ( 0 ) | S 33 ( 0 ) | = A 1 * 2 ( 1 + i Δ 2 ) | A 1 | 4 ,
S 2 ( ω ¯ ) = 4 ω ¯ 2 + 4 + 1.
M 1 = [ 1 + i Δ 1 A 2 2 A 2 * 2 1 i Δ 1 ] , D 1 = κ γ 2 [ A 2 2 0 0 A 2 * 2 ] .
S 1 ( ω ) = [ ( 1 + Δ 1 2 ω 2 | A 2 | 4 ) 2 + 4 ω 2 ] 1 4 { 2 | A 2 | 4 + Re [ A 2 2 ( 1 Δ 1 2 + ω 2 + | A 2 | 4 2 i Δ 1 ) × exp ( 2 i θ ) ] } + 1 ,
exp ( 2 i θ ) = S 11 * ( 0 ) | S 11 ( 0 ) | = A 2 * 2 ( 1 Δ 1 2 + | A 2 | 4 + 2 i Δ 1 ) | A 2 | 2 ( 1 Δ 1 2 + | A 2 | 4 ) 2 + 4 Δ 1 2 .
S 1 ( ω ) = 4 | A 2 | 2 ( 1 + Δ 1 2 | A 2 | 4 ω 2 ) 2 + 4 ω 2 { 2 | A 2 | 2 ( 1 Δ 1 2 + | A 2 | 4 ) ( 1 Δ 1 2 + | A 2 | 4 + ω 2 ) + 4 Δ 1 2 [ ( 1 Δ 1 2 + | A 2 | 4 ) 2 + 4 Δ 1 2 ] 1 / 2 } + 1.
M λ I = λ 4 + c 1 λ 3 + c 2 λ 2 + c 3 λ + c 4 = 0.
c 1 = 2 ( 1 + μ ) ,
c 2 = μ [ 4 + μ ( 1 + Δ 1 2 ) + 8 1 + Δ 2 2 | A 2 | 2 μ | A 2 | 4 ] ,
c 3 = 2 μ [ μ ( 1 + Δ 1 2 ) + 4 ( 1 μ ) 1 + Δ 2 2 | A 2 | 2 μ | A 2 | 4 ] ,
c 4 = 8 μ 2 [ ( 1 Δ 1 Δ 2 ) 1 + Δ 2 2 | A 2 | 2 + ( 1 + Δ 2 2 ) | A 2 | 4 ] .
M ( Δ 1 , Δ 2 ) = M * ( Δ 1 , Δ 2 ) , D ( Δ 1 , Δ 2 ) = D * ( Δ 1 , Δ 2 ) .
S ( ω , Δ 1 , Δ 2 ) = [ M ( Δ 1 , Δ 2 ) + i ω I ) ] 1 D ( Δ 1 , Δ 2 ) × [ M T ( Δ 1 , Δ 2 ) i ω I ] 1 = [ M * ( Δ 1 , Δ 2 ) + i ω I ) ] 1 D * ( Δ 1 , Δ 2 ) [ M * T ( Δ 1 , Δ 2 ) + i ω I ) ] 1 .
S + ( ω , Δ 1 , Δ 2 ) = S ( ω , Δ 1 , Δ 2 ) ,
S i j ( ω , Δ 1 , Δ 2 ) = S i j * ( ω , Δ 1 , Δ 2 ) .
S ( ω , Δ 1 , Δ 2 , θ ) = S 12 ( ω , Δ 1 , Δ 2 ) + S 21 ( ω , Δ 1 , Δ 2 ) + 2 Re [ S 11 ( ω , Δ 1 , Δ 2 ) exp ( 2 i θ ) ] = S 21 * ( ω , Δ 1 , Δ 2 ) S 12 * ( ω , Δ 1 , Δ 2 ) + 2 Re [ S 11 * ( ω , Δ 1 , Δ 2 ) exp ( 2 i θ ) ] = S 12 ( ω , Δ 1 , Δ 2 ) + S 21 ( ω , Δ 1 , Δ 2 ) + 2 Re [ S 11 ( ω , Δ 1 , Δ 2 ) exp ( 2 i θ ) ] = S ( ω , Δ 1 , Δ 2 , θ ) .

Metrics