Abstract

Squeezed states of the electromagnetic field are generated by degenerate parametric downconversion in a sub-threshold optical parametric oscillator. Reductions in photocurrent noise greater than 60% (−4 dB) below the limit set by the vacuum fluctuations of the field are observed in a balanced homodyne detector. A quantitative comparison with theory suggests that the observed noise reductions result from a field that in the absence of avoidable linear attenuation would be squeezed more than tenfold. A degree of squeezing of approximately fivefold is inferred for the actual field emitted through one mirror of the optical parametric oscillator. An explicit demonstration of the Heisenberg uncertainty principle for the electromagnetic field is made from the measurements, which show that the field state produced by the downconversion process is a state of minimum uncertainty.

© 1987 Optical Society of America

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1987 (5)

B. Yurke, P. Grangier, R. E. Slusher, and M. J. Potasek, Phys. Rev. A 35, 3586 (1987).
[CrossRef] [PubMed]

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

S. Machida, Y. Yamamoto, and Y. Itaya, Phys. Rev. Lett. 58, 1000 (1987).
[CrossRef] [PubMed]

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

M. Xiao, L.-A. Wu, and H. J. Kimble, Phys. Rev. Lett. 41, 278 (1987).
[CrossRef]

1986 (2)

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986); B. L. Schumaker, S. H. Perlmutter, R. M. Shelby, and M. D. Levenson, Phys. Rev. Lett. 58, 357 (1987).
[CrossRef] [PubMed]

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

1985 (9)

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

J. H. Shapiro, IEEE J. Quantum Electron. 21, 237 (1985).
[CrossRef]

L.-A. Wu and H. J. Kimble, J. Opt. Soc. Am. B 2, 697 (1985).
[CrossRef]

K.-C. Peng, L.-A. Wu, and H. J. Kimble, Appl. Opt. 24, 938 (1985).
[CrossRef] [PubMed]

M. D. Levenson, R. M. Shelby, and S. H. Perlmutter, Opt. Lett. 10, 514 (1985).
[CrossRef] [PubMed]

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

C. M. Caves and B. L. Schumaker, Phys. Rev. A 31, 3068 (1985); Phys. Rev. A 31, 3093 (1985); in Quantum Optics IV, J. H. Harvey and D. F. Walls, eds. (Springer-Verlag, New York, 1986), p. 20.
[CrossRef] [PubMed]

M. M. Fejer, G. A. Magel, and R. L. Byer, Appl. Opt. 24, 2362 (1985); Y. S. Luh, R. S. Feigelson, M. M. Fejer, and R. L. Byer, J. Cryst. Growth 78, 135 (1986).
[CrossRef] [PubMed]

S. Friberg, C. K. Hong, and L. Mandel, Phys. Rev. Lett. 54, 2011 (1985).
[CrossRef] [PubMed]

1984 (4)

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

M. J. Collett and C. W. Gardiner, Phys. Rev. A 30, 1386 (1984); Phys. Rev. A 31, 3761 (1985).
[CrossRef]

C. W. Gardiner and C. M. Savage, Opt. Commun. 50, 173 (1984).
[CrossRef]

H. J. Kimble and L. Mandel, Phys. Rev. A 30, 844 (1984).
[CrossRef]

1983 (3)

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, Appl. Phys. B 31, 97 (1983).
[CrossRef]

H. P. Yuen and V. W. S. Chan, Opt. Lett. 8, 177 (1983); B. L. Schumaker, Opt. Lett. 9, 189 (1984).
[CrossRef] [PubMed]

D. F. Walls, Nature 306, 141 (1983).
[CrossRef]

1981 (1)

C. M. Caves, Phys. Rev. D 23, 1693 (1981).
[CrossRef]

1980 (1)

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

1976 (1)

H. P. Yuen, Phys. Rev. A 13, 2226 (1976).
[CrossRef]

1975 (1)

1973 (1)

B. R. Mollow, Phys. Rev. A 8, 2684 (1973).
[CrossRef]

1971 (1)

E. Y. C. Lu, Lett. Nuovo Cimento 2, 1241 (1971).
[CrossRef]

1970 (2)

D. Stoler, Phys. Rev. D 1, 3217 (1970); Phys. Rev. D 4, 1925 (1971).
[CrossRef]

D. C. Burnham and D. L. Weinberg, Phys. Rev. Lett. 25, 84 (1970).
[CrossRef]

1968 (1)

G. D. Boyd and D. A. Kleinman, J. Appl. Phys. 39, 3597 (1968).
[CrossRef]

1967 (1)

B. R. Mollow and R. J. Glauber, Phys. Rev. 160, 1076 (1967).
[CrossRef]

1965 (1)

H. Takahasi, Adv. Commun. Syst. 1, 227 (1965).

1961 (1)

W. H. Louisell, A. Yariv, and A. E. Siegman, Phys. Rev. 124, 1646 (1961).
[CrossRef]

Boyd, G. D.

G. D. Boyd and D. A. Kleinman, J. Appl. Phys. 39, 3597 (1968).
[CrossRef]

Boyd, T. L.

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

Burnham, D. C.

D. C. Burnham and D. L. Weinberg, Phys. Rev. Lett. 25, 84 (1970).
[CrossRef]

Byer, R. L.

M. M. Fejer, G. A. Magel, and R. L. Byer, Appl. Opt. 24, 2362 (1985); Y. S. Luh, R. S. Feigelson, M. M. Fejer, and R. L. Byer, J. Cryst. Growth 78, 135 (1986).
[CrossRef] [PubMed]

R. L. Byer, in Quantum Electronics: A Treatise, H. Rabin and C. L. Tang, eds. (Academic, New York, 1975), Vol. 1, Part B, p. 587.

Caves, C. M.

C. M. Caves and B. L. Schumaker, Phys. Rev. A 31, 3068 (1985); Phys. Rev. A 31, 3093 (1985); in Quantum Optics IV, J. H. Harvey and D. F. Walls, eds. (Springer-Verlag, New York, 1986), p. 20.
[CrossRef] [PubMed]

C. M. Caves, Phys. Rev. D 23, 1693 (1981).
[CrossRef]

Chan, V. W. S.

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); Phys. Rev. A 31, 3761 (1985).
[CrossRef]

DeVoe, R. G.

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986); B. L. Schumaker, S. H. Perlmutter, R. M. Shelby, and M. D. Levenson, Phys. Rev. Lett. 58, 357 (1987).
[CrossRef] [PubMed]

Drever, R. W. P.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, Appl. Phys. B 31, 97 (1983).
[CrossRef]

Drummond, P. D.

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

Fejer, M. M.

Ford, G. M.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, Appl. Phys. B 31, 97 (1983).
[CrossRef]

Friberg, S.

S. Friberg, C. K. Hong, and L. Mandel, Phys. Rev. Lett. 54, 2011 (1985).
[CrossRef] [PubMed]

Gardiner, C. W.

M. J. Collett and C. W. Gardiner, Phys. Rev. A 30, 1386 (1984); Phys. Rev. A 31, 3761 (1985).
[CrossRef]

C. W. Gardiner and C. M. Savage, Opt. Commun. 50, 173 (1984).
[CrossRef]

Glauber, R. J.

B. R. Mollow and R. J. Glauber, Phys. Rev. 160, 1076 (1967).
[CrossRef]

R. J. Glauber, in Quantum Optics and Electronics, C. deWitt, A. Blandim, and C. Cohen-Tannoudji, eds. (Gordon & Breach, New York, 1965).

Graham, R.

R. Graham, Springer Tracts in Modern Physics (Springer-Verlag, Berlin, 1973), Vol. 66.

Grangier, P.

B. Yurke, P. Grangier, R. E. Slusher, and M. J. Potasek, Phys. Rev. A 35, 3586 (1987).
[CrossRef] [PubMed]

Hall, J. L.

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

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, Appl. Phys. B 31, 97 (1983).
[CrossRef]

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]

Hong, C. K.

S. Friberg, C. K. Hong, and L. Mandel, Phys. Rev. Lett. 54, 2011 (1985).
[CrossRef] [PubMed]

Hough, J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, Appl. Phys. B 31, 97 (1983).
[CrossRef]

Itaya, Y.

S. Machida, Y. Yamamoto, and Y. Itaya, Phys. Rev. Lett. 58, 1000 (1987).
[CrossRef] [PubMed]

Kimble, H. J.

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

M. Xiao, L.-A. Wu, and H. J. Kimble, Phys. Rev. Lett. 41, 278 (1987).
[CrossRef]

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

K.-C. Peng, L.-A. Wu, and H. J. Kimble, Appl. Opt. 24, 938 (1985).
[CrossRef] [PubMed]

L.-A. Wu and H. J. Kimble, J. Opt. Soc. Am. B 2, 697 (1985).
[CrossRef]

H. J. Kimble and L. Mandel, Phys. Rev. A 30, 844 (1984).
[CrossRef]

Kleinman, D. A.

G. D. Boyd and D. A. Kleinman, J. Appl. Phys. 39, 3597 (1968).
[CrossRef]

Kowalski, F. V.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, Appl. Phys. B 31, 97 (1983).
[CrossRef]

Kumar, P.

Levenson, M. D.

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986); B. L. Schumaker, S. H. Perlmutter, R. M. Shelby, and M. D. Levenson, Phys. Rev. Lett. 58, 357 (1987).
[CrossRef] [PubMed]

M. D. Levenson, R. M. Shelby, and S. H. Perlmutter, Opt. Lett. 10, 514 (1985).
[CrossRef] [PubMed]

Louisell, W. H.

W. H. Louisell, A. Yariv, and A. E. Siegman, Phys. Rev. 124, 1646 (1961).
[CrossRef]

Lu, E. Y. C.

E. Y. C. Lu, Lett. Nuovo Cimento 2, 1241 (1971).
[CrossRef]

Machida, S.

S. Machida, Y. Yamamoto, and Y. Itaya, Phys. Rev. Lett. 58, 1000 (1987).
[CrossRef] [PubMed]

Maeda, M. W.

Magel, G. A.

Mandel, L.

S. Friberg, C. K. Hong, and L. Mandel, Phys. Rev. Lett. 54, 2011 (1985).
[CrossRef] [PubMed]

H. J. Kimble and L. Mandel, Phys. Rev. A 30, 844 (1984).
[CrossRef]

L. Mandel and E. Wolf, J. Opt. Soc. Am. 65, 413 (1975).
[CrossRef]

L. Mandel, in Optics in Four Dimension—1980, M. A. Machado and L. M. Narducci, eds., AIP Conf. Proc. No. 65 (American Institute of Physics, New York, 1981).

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

Midwinter, J. E.

F. Zernike and J. E. Midwinter, Applied Nonlinear Optics (Wiley, New York, 1973).

Mollow, B. R.

B. R. Mollow, Phys. Rev. A 8, 2684 (1973).
[CrossRef]

B. R. Mollow and R. J. Glauber, Phys. Rev. 160, 1076 (1967).
[CrossRef]

Munley, A. J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, Appl. Phys. B 31, 97 (1983).
[CrossRef]

Orozco, L. A.

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

Peng, K.-C.

Perlmutter, S. H.

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986); B. L. Schumaker, S. H. Perlmutter, R. M. Shelby, and M. D. Levenson, Phys. Rev. Lett. 58, 357 (1987).
[CrossRef] [PubMed]

M. D. Levenson, R. M. Shelby, and S. H. Perlmutter, Opt. Lett. 10, 514 (1985).
[CrossRef] [PubMed]

Potasek, M. J.

B. Yurke, P. Grangier, R. E. Slusher, and M. J. Potasek, Phys. Rev. A 35, 3586 (1987).
[CrossRef] [PubMed]

Raizen, M. G.

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

Savage, C. M.

C. W. Gardiner and C. M. Savage, Opt. Commun. 50, 173 (1984).
[CrossRef]

Schumaker, B. L.

C. M. Caves and B. L. Schumaker, Phys. Rev. A 31, 3068 (1985); Phys. Rev. A 31, 3093 (1985); in Quantum Optics IV, J. H. Harvey and D. F. Walls, eds. (Springer-Verlag, New York, 1986), p. 20.
[CrossRef] [PubMed]

Shapiro, J. H.

Shelby, R. M.

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986); B. L. Schumaker, S. H. Perlmutter, R. M. Shelby, and M. D. Levenson, Phys. Rev. Lett. 58, 357 (1987).
[CrossRef] [PubMed]

M. D. Levenson, R. M. Shelby, and S. H. Perlmutter, Opt. Lett. 10, 514 (1985).
[CrossRef] [PubMed]

Siegman, A. E.

W. H. Louisell, A. Yariv, and A. E. Siegman, Phys. Rev. 124, 1646 (1961).
[CrossRef]

Slusher, R. E.

B. Yurke, P. Grangier, R. E. Slusher, and M. J. Potasek, Phys. Rev. A 35, 3586 (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]

Smith, R. G.

R. G. Smith, in Lasers: A Series of Advances, A. K. Levine and A. DeMaria, eds. (Dekker and Marcel, New York, 1976), Vol. 4.

Stoler, D.

D. Stoler, Phys. Rev. D 1, 3217 (1970); Phys. Rev. D 4, 1925 (1971).
[CrossRef]

Takahasi, H.

H. Takahasi, Adv. Commun. Syst. 1, 227 (1965).

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.

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986); B. L. Schumaker, S. H. Perlmutter, R. M. Shelby, and M. D. Levenson, Phys. Rev. Lett. 58, 357 (1987).
[CrossRef] [PubMed]

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

D. F. Walls, Nature 306, 141 (1983).
[CrossRef]

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

Ward, H.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, Appl. Phys. B 31, 97 (1983).
[CrossRef]

Weinberg, D. L.

D. C. Burnham and D. L. Weinberg, Phys. Rev. Lett. 25, 84 (1970).
[CrossRef]

Wolf, E.

Wu, H.

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

Wu, L.-A.

M. Xiao, L.-A. Wu, and H. J. Kimble, Phys. Rev. Lett. 41, 278 (1987).
[CrossRef]

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

L.-A. Wu and H. J. Kimble, J. Opt. Soc. Am. B 2, 697 (1985).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Sketch of basic arrangement of the optical parametric oscillator. The pump beam P2 at frequency 2ω excites the cavity through the mirror M. The downconverted field of approximate frequency ω decays predominantly through the mirror M′. The system can be used as an amplifier by injecting a coherent field ain at the subharmonic frequency through M′.

Fig. 2
Fig. 2

Diagram of principal elements of the apparatus.

Fig. 3
Fig. 3

Schematic of balanced homodyne detector. The signal field ÊS from the optical parametric oscillator and a strong local-oscillator field ÊLO are combined at the surface of a beam splitter. The composite fields from the beam splitter are directed to photodiodes D1 and D2. The photocurrents are amplified with gains (g1, g2) and summed in a hybrid junction with either a 0° (∑+) or a 180° (∑) phase shift.

Fig. 4
Fig. 4

Dependence of rms noise voltage V(θ) on local-oscillator phase θ for the signal from the balanced homodyne detector shown in Fig. 3. With the output of the OPO blocked, the vacuum field entering the detector produces the noise voltage V0 labeled by (i) with no sensitivity on θ. With the OPO input present, trace (ii) exhibits phase-sensitive deviations both below and above the vacuum level, with the dips below trace (i) representing a 61% reduction in noise power relative to the vacuum level. Trace (iii) is the amplifier noise level. Note that for traces (i)–(iii) the ordinate is linear in noise voltage (amplitude). Trace (iv) is actually two curves almost superimposed that give the levels of dc photocurrent (with zero at the bottom of the figure) during the acquisition of traces (i) and (ii). For traces (i)–(iii), ν/2π = 1.6 MHz. The sharp feature is generated by the flyback of the piezoelectric ceramic used to scan the local-oscillator phase. The time for the entire sweep is 0.2 sec, the analysis bandwidth is 100 kHz, and time constants for the post-detection video filters are 1.5 × 10−4 and 5.0 × 10−4 sec, respectively.

Fig. 5
Fig. 5

Dependence of noise voltage V(θ) on local-oscillator phase θ for the signal beam produced by the subthreshold OPO. Operating conditions are as in Fig. 4, with traces (i) and (iv) deleted; the dashed line is the vacuum level obtained by multiple averaging.

Fig. 6
Fig. 6

Dependence of noise voltage V(θ) on local-oscillator phase θ for the signal beam produced by the subthreshold OPO. The linear scale in rms noise voltage is expanded relative to Figs. 4 and 5 to display the deviation below the vacuum level (dashed line) more clearly. Reductions in noise power of 63% relative to the vacuum level are shown. ν/2π = 1.6 MHz. Analysis bandwidth and video filter time constants are as in Fig. 4.

Fig. 7
Fig. 7

Comparisons of the values S(r) derived from the measurements with those given by theory (solid curve) for fixed Ω = 0.21 with the measured values ρT0βη2 = 0.52 and P0 = 30 mW. Perfect squeezing occurs for S(Ω) = −1 and is indicated by the dashed line. The uncertainties shown for each point derive from uncertainties in the determination of R and of P2 at a given operating point and do not include the overall uncertainties in ρ, T0, β, η2, or P0.

Fig. 8
Fig. 8

Variances (1 + S+) and (1 + S) determined from measurements such as those shown in Figs. 46, as discussed in the text. The solid curve is the hyperbola (1 + S+)(1 + S) = 1, which defines a class of minimum-uncertainty states. Squeezed states are those states for which (1 + S±) < 1 and that lie in the region bounded by the hyperbola and the dashed lines.

Equations (37)

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α ˙ 1 = - i Δ 1 α 1 - Γ 1 α 1 + κ α 1 * α 2 + E 1 , α ˙ 2 = - i Δ 2 α 2 - Γ 2 α 2 - 1 / 2 κ α 1 2 + E 2 .
α 1 0 = [ 2 κ ( E 2 - E 2 c ) ] 1 / 2 ,             α 2 0 = Γ 1 / κ .
G + = 1 + d 1 - d , G - = 1 - d 1 + d ,
X ^ θ ( t ) e - i θ α ^ ( t ) + e i θ α ^ ( t ) ,
A ( τ , θ ) T : X ^ θ ( t ) , X ^ θ ( t + τ ) : ,
S ( ν , θ ) = 2 Γ 1 A ( ν , θ ) ,
A ( ν , θ ) = - d τ e - i ν τ A ( τ , θ ) .
S ± ( ν ) = ± 4 Γ 1 2 ( Γ 2 2 + ν 2 ) [ Γ 2 ( Γ 1 2 ) + 1 2 - ν 2 ] 2 + ν 2 ( Γ 1 2 + Γ 2 ) 2 ,
S ± ( Ω ) = ± 4 E 2 / E 2 c Ω 2 + ( 1 E 2 / E 2 c ) 2 ,
κ = 2 d e n 3 ( h ω 1 3 0 3 ) 1 / 2 Γ Ψ 2 * ( r ) [ Ψ 1 ( r ) ] 2 d 3 x ,
κ 2 = π c 4 L 3 λ 2 n 3 E NL .
E NL = 16 π 2 l d e 2 0 3 c n 2 λ 1 3 g ( σ , ζ ) ,
g ( σ , ζ ) = π 2 ζ Im { H ( σ , ζ ) exp [ i ( ϕ 2 - 2 ϕ 1 ) ] } 2 ,
H ( σ , ζ ) 1 2 π - ζ ζ d x e i σ x 1 + i x .
g ( σ , ζ ) ζ ( sin σ ζ σ ζ ) 2 sin 2 ( ϕ 2 - 2 ϕ 1 ) .
ϕ 2 - 2 ϕ 1 = ± ( π / 2 ) + ξ ,
P 2 c = π 2 4 F ω 1 2 B ω 2 E NL .
Φ ( ν ) = Δ i ( t ) Δ i ( t + τ ) e - i ν τ d τ ,
Δ i ( t ) Δ i ( t + τ ) = R 1 Q 1 2 δ ( τ ) + R 2 Q 2 2 δ ( τ ) + R 1 2 Q 1 2 λ 11 ( τ ) + R 2 2 Q 2 2 λ 22 ( τ ) - R 1 R 2 Q 1 λ 12 ( τ ) - R 1 R 2 Q 1 Q 2 λ 21 ( τ ) ,
λ i j ( τ ) = T : E ^ i ( t ) E ^ j ( t + τ ) E ^ j ( t + τ ) E ^ i ( t ) : E ^ i ( t ) E ^ i ( t ) E ^ j ( t + τ ) E ^ j ( t + τ ) ,         i , j = 1 , 2.
Φ ( Ω , θ ) = Q 1 i 1 + Q 2 i 2 [ 1 + ρ T 0 β η 2 S ( Ω , θ ) ] ,
Φ ( Ω , θ ) S = 0 = ( Q 1 i 1 + Q 2 i 2 ) ,
R ( Ω , θ ) Φ ( Ω , θ ) ( Q 1 i 1 + Q 2 i 2 ) = 1 + ρ T 0 β η 2 S ( Ω , θ ) ,
β R T ( α 1 g 1 + α 2 g 2 ) 2 ( R α 1 g 1 2 + T α 2 g 2 2 ) .
η = | d 2 r U * ( r ) V ( r ) | ,
η 2 = D 2 ω α P s Δ f ,
S LO ( ν ) d τ T : Y ^ ( t ) , Y ^ ( t + τ ) : > e - i ν τ ,
( α 1 g 1 R - α 2 g 2 T ) 2 α 1 g 1 2 R + α 2 g 2 2 T S LO ( ν ) δ - S LO ( ν ) ,
δ - α 1 ( 2 - δ α 2 α 1 - δ g 2 g 1 ) 2 .
χ + ( ν ) = ( Q 1 i 1 + Q 2 i 2 ) [ 1 + δ + S LO ( ν ) ]
χ - ( ν ) = ( Q 1 i 1 + Q 2 i 2 ) [ 1 + δ - S LO ( ν ) ] ,
δ ± = ( α 1 g 1 R ± α 2 g 2 T ) 2 ( α 1 g 1 2 R + α 2 g 2 2 T ) = α ( R ± T ) 2 ,
S LO = χ + / χ - - 1 δ + ( 1 - χ + δ - χ - δ + ) ( χ + / χ - - 1 ) / α ,
R ( Ω , θ ) = 1 + ρ T 0 β η 2 S ( Ω , θ ) ,
R ( θ ) = V 2 ( θ ) - V A 2 V 0 2 - V A 2 ,
[ 1 + S + ( Ω ) ] [ 1 + S - ( Ω ) ] = 1.
[ 1 + S + ( Ω ) ] [ 1 + S - ( Ω ) ] 1.

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