Abstract

We study the possibility of employing phase-sensitive parametric amplifiers to compensate for losses incurred by solitons propagating in optical fibers. As phase-sensitive amplification is free from the excess spontaneous noise associated with laser amplifiers, the only quantum noise source in such a system arises from the fiber losses. In addition, under phase-sensitive amplification, fluctuations in the soliton frequency are squeezed to one half of the vacuum level, thus producing a negligible contribution to the fluctuations in the soliton arrival time. This greatly reduces the Gordon–Haus jitter of the solitons. The performance of the phase-sensitive amplifier is compared with that of alternative noise-reduction schemes employing phase-insensitive amplifiers and spectral filters as well as with the limits on long-distance data transmission imposed by quantum mechanics.

© 1994 Optical Society of America

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  1. R. J. Mears, L. Reekie, I. M. Jauncey, and D. N. Payne, Electron. Lett. 23, 1026 (1987); E. Desurvire, J. R. Simpson, and P. C. Becker, Opt. Lett. 12, 888 (1987).
    [CrossRef] [PubMed]
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    [CrossRef]
  3. A. Mecozzi, J. D. Moores, H. A. Haus, and Y. Lai, Opt. Lett. 16, 1841 (1991); Y. Kodama and A. Hasegawa, Opt. Lett. 17, 31 (1992).
    [CrossRef] [PubMed]
  4. M. Nakazawa, E. Yamada, H. Kubota, and K. Suzuki, Electron. Lett. 27, 1270 (1991).
    [CrossRef]
  5. A. Mecozzi, J. D. Moores, H. A. Haus, and Y. Lai, J. Opt. Soc. Am. B 9, 1350 (1992).
    [CrossRef]
  6. L. F. Mollenauer, E. Lichtman, G. T. Harvey, M. J. Neubelt, and B. M. Nyman, Electron. Lett. 28, 792 (1992).
    [CrossRef]
  7. L. F. Mollenauer, E. Lichtman, M. J. Neubelt, and G. T. Harvey, Electron. Lett. 29, 910 (1993).
    [CrossRef]
  8. H. P. Yuen, Opt. Lett. 17, 73 (1992).
    [CrossRef] [PubMed]
  9. Yi Mu and C. M. Savage, J. Opt. Soc. Am. B 9, 65 (1992).
    [CrossRef]
  10. J. N. Kutz, W. L. Kath, R. Li, and P. Kumar, Opt. Lett. 18, 18 (1993).
    [CrossRef]
  11. C. M. Caves, Phys. Rev. D 26, 1817 (1982).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  14. B. Huttner and S. M. Barnett, Phys. Rev. A 46, 4306 (1992).
    [CrossRef] [PubMed]
  15. A. Hasegawa and Y. Kodama, Opt. Lett. 15, 1443 (1990); L. F. Mollenauer, S. G. Evangelides, and H. A. Haus, J. Lightwave Technol. 9, 194 (1991); K. J. Blow and N. J. Doran, Photon. Technol. Lett. 3, 369 (1991).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  19. P. D. Drummond and S. J. Carter, J. Opt. Soc. Am. B 4, 1565 (1987); P. D. Drummond, S. J. Carter, and R. M. Shelby, Opt. Lett. 14, 373 (1989).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  21. G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, Mass., 1989), p. 116.
  22. L. F. Mollenauer, J. P. Gordon, and S. G. Evangelides, Opt. Lett. 17, 1575 (1992).
    [CrossRef] [PubMed]
  23. I. H. Deutsch, R. Y. Chiao, and J. C. Garrison, Phys. Rev. Lett. 69, 3627 (1992).
    [CrossRef] [PubMed]
  24. E. M. Wright, Phys. Rev. A 43, 3836 (1991).
    [CrossRef] [PubMed]
  25. I. H. Deutsch, Am. J. Phys. 59, 834 (1991).
    [CrossRef]
  26. N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981).

1993 (3)

L. F. Mollenauer, E. Lichtman, M. J. Neubelt, and G. T. Harvey, Electron. Lett. 29, 910 (1993).
[CrossRef]

J. N. Kutz, W. L. Kath, R. Li, and P. Kumar, Opt. Lett. 18, 18 (1993).
[CrossRef]

J. A. Levenson, I. Abram, T. Rivera, P. Fayolle, J. C. Garreau, and P. Grangier, Phys. Rev. Lett. 70, 267 (1993).
[CrossRef] [PubMed]

1992 (7)

B. Huttner and S. M. Barnett, Phys. Rev. A 46, 4306 (1992).
[CrossRef] [PubMed]

H. P. Yuen, Opt. Lett. 17, 73 (1992).
[CrossRef] [PubMed]

Yi Mu and C. M. Savage, J. Opt. Soc. Am. B 9, 65 (1992).
[CrossRef]

A. Mecozzi, J. D. Moores, H. A. Haus, and Y. Lai, J. Opt. Soc. Am. B 9, 1350 (1992).
[CrossRef]

L. F. Mollenauer, E. Lichtman, G. T. Harvey, M. J. Neubelt, and B. M. Nyman, Electron. Lett. 28, 792 (1992).
[CrossRef]

L. F. Mollenauer, J. P. Gordon, and S. G. Evangelides, Opt. Lett. 17, 1575 (1992).
[CrossRef] [PubMed]

I. H. Deutsch, R. Y. Chiao, and J. C. Garrison, Phys. Rev. Lett. 69, 3627 (1992).
[CrossRef] [PubMed]

1991 (4)

E. M. Wright, Phys. Rev. A 43, 3836 (1991).
[CrossRef] [PubMed]

I. H. Deutsch, Am. J. Phys. 59, 834 (1991).
[CrossRef]

A. Mecozzi, J. D. Moores, H. A. Haus, and Y. Lai, Opt. Lett. 16, 1841 (1991); Y. Kodama and A. Hasegawa, Opt. Lett. 17, 31 (1992).
[CrossRef] [PubMed]

M. Nakazawa, E. Yamada, H. Kubota, and K. Suzuki, Electron. Lett. 27, 1270 (1991).
[CrossRef]

1990 (2)

1989 (1)

Y. Lai and H. A. Haus, Phys. Rev. A 40, 844, 1138 (1989).
[CrossRef] [PubMed]

1987 (3)

1986 (1)

J. P. Gordon and H. A. Haus, Opt. Lett. 10, 665 (1986).
[CrossRef]

1984 (1)

M. Hillery and L. Mlodinow, Phys. Rev. A 30, 1860 (1984).
[CrossRef]

1982 (1)

C. M. Caves, Phys. Rev. D 26, 1817 (1982).
[CrossRef]

1974 (1)

J. Satsuma and N. Yajima, Suppl. Prog. Theor. Phys. 55, 284 (1974); V. I. Karpman and V. V. Solovev, Physica 3D, 487 (1981); D. J. Kaup, Phys. Rev. A 42, 5689 (1990).
[CrossRef] [PubMed]

Abram, I.

J. A. Levenson, I. Abram, T. Rivera, P. Fayolle, J. C. Garreau, and P. Grangier, Phys. Rev. Lett. 70, 267 (1993).
[CrossRef] [PubMed]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, Mass., 1989), p. 116.

Barnett, S. M.

B. Huttner and S. M. Barnett, Phys. Rev. A 46, 4306 (1992).
[CrossRef] [PubMed]

Carter, S. J.

Caves, C. M.

C. M. Caves, Phys. Rev. D 26, 1817 (1982).
[CrossRef]

Chiao, R. Y.

I. H. Deutsch, R. Y. Chiao, and J. C. Garrison, Phys. Rev. Lett. 69, 3627 (1992).
[CrossRef] [PubMed]

Deutsch, I. H.

I. H. Deutsch, R. Y. Chiao, and J. C. Garrison, Phys. Rev. Lett. 69, 3627 (1992).
[CrossRef] [PubMed]

I. H. Deutsch, Am. J. Phys. 59, 834 (1991).
[CrossRef]

Drummond, P. D.

Evangelides, S. G.

Fayolle, P.

J. A. Levenson, I. Abram, T. Rivera, P. Fayolle, J. C. Garreau, and P. Grangier, Phys. Rev. Lett. 70, 267 (1993).
[CrossRef] [PubMed]

Garreau, J. C.

J. A. Levenson, I. Abram, T. Rivera, P. Fayolle, J. C. Garreau, and P. Grangier, Phys. Rev. Lett. 70, 267 (1993).
[CrossRef] [PubMed]

Garrison, J. C.

I. H. Deutsch, R. Y. Chiao, and J. C. Garrison, Phys. Rev. Lett. 69, 3627 (1992).
[CrossRef] [PubMed]

Gordon, J. P.

Grangier, P.

J. A. Levenson, I. Abram, T. Rivera, P. Fayolle, J. C. Garreau, and P. Grangier, Phys. Rev. Lett. 70, 267 (1993).
[CrossRef] [PubMed]

Harvey, G. T.

L. F. Mollenauer, E. Lichtman, M. J. Neubelt, and G. T. Harvey, Electron. Lett. 29, 910 (1993).
[CrossRef]

L. F. Mollenauer, E. Lichtman, G. T. Harvey, M. J. Neubelt, and B. M. Nyman, Electron. Lett. 28, 792 (1992).
[CrossRef]

Hasegawa, A.

Haus, H. A.

Hillery, M.

M. Hillery and L. Mlodinow, Phys. Rev. A 30, 1860 (1984).
[CrossRef]

Huttner, B.

B. Huttner and S. M. Barnett, Phys. Rev. A 46, 4306 (1992).
[CrossRef] [PubMed]

Jauncey, I. M.

R. J. Mears, L. Reekie, I. M. Jauncey, and D. N. Payne, Electron. Lett. 23, 1026 (1987); E. Desurvire, J. R. Simpson, and P. C. Becker, Opt. Lett. 12, 888 (1987).
[CrossRef] [PubMed]

Kath, W. L.

J. N. Kutz, W. L. Kath, R. Li, and P. Kumar, Opt. Lett. 18, 18 (1993).
[CrossRef]

Kodama, Y.

Kubota, H.

M. Nakazawa, E. Yamada, H. Kubota, and K. Suzuki, Electron. Lett. 27, 1270 (1991).
[CrossRef]

Kumar, P.

J. N. Kutz, W. L. Kath, R. Li, and P. Kumar, Opt. Lett. 18, 18 (1993).
[CrossRef]

Kutz, J. N.

J. N. Kutz, W. L. Kath, R. Li, and P. Kumar, Opt. Lett. 18, 18 (1993).
[CrossRef]

Lai, Y.

Levenson, J. A.

J. A. Levenson, I. Abram, T. Rivera, P. Fayolle, J. C. Garreau, and P. Grangier, Phys. Rev. Lett. 70, 267 (1993).
[CrossRef] [PubMed]

Li, R.

J. N. Kutz, W. L. Kath, R. Li, and P. Kumar, Opt. Lett. 18, 18 (1993).
[CrossRef]

Lichtman, E.

L. F. Mollenauer, E. Lichtman, M. J. Neubelt, and G. T. Harvey, Electron. Lett. 29, 910 (1993).
[CrossRef]

L. F. Mollenauer, E. Lichtman, G. T. Harvey, M. J. Neubelt, and B. M. Nyman, Electron. Lett. 28, 792 (1992).
[CrossRef]

Mears, R. J.

R. J. Mears, L. Reekie, I. M. Jauncey, and D. N. Payne, Electron. Lett. 23, 1026 (1987); E. Desurvire, J. R. Simpson, and P. C. Becker, Opt. Lett. 12, 888 (1987).
[CrossRef] [PubMed]

Mecozzi, A.

Mlodinow, L.

M. Hillery and L. Mlodinow, Phys. Rev. A 30, 1860 (1984).
[CrossRef]

Mollenauer, L. F.

L. F. Mollenauer, E. Lichtman, M. J. Neubelt, and G. T. Harvey, Electron. Lett. 29, 910 (1993).
[CrossRef]

L. F. Mollenauer, E. Lichtman, G. T. Harvey, M. J. Neubelt, and B. M. Nyman, Electron. Lett. 28, 792 (1992).
[CrossRef]

L. F. Mollenauer, J. P. Gordon, and S. G. Evangelides, Opt. Lett. 17, 1575 (1992).
[CrossRef] [PubMed]

Moores, J. D.

Mu, Yi

Nakazawa, M.

M. Nakazawa, E. Yamada, H. Kubota, and K. Suzuki, Electron. Lett. 27, 1270 (1991).
[CrossRef]

Neubelt, M. J.

L. F. Mollenauer, E. Lichtman, M. J. Neubelt, and G. T. Harvey, Electron. Lett. 29, 910 (1993).
[CrossRef]

L. F. Mollenauer, E. Lichtman, G. T. Harvey, M. J. Neubelt, and B. M. Nyman, Electron. Lett. 28, 792 (1992).
[CrossRef]

Nyman, B. M.

L. F. Mollenauer, E. Lichtman, G. T. Harvey, M. J. Neubelt, and B. M. Nyman, Electron. Lett. 28, 792 (1992).
[CrossRef]

Payne, D. N.

R. J. Mears, L. Reekie, I. M. Jauncey, and D. N. Payne, Electron. Lett. 23, 1026 (1987); E. Desurvire, J. R. Simpson, and P. C. Becker, Opt. Lett. 12, 888 (1987).
[CrossRef] [PubMed]

Reekie, L.

R. J. Mears, L. Reekie, I. M. Jauncey, and D. N. Payne, Electron. Lett. 23, 1026 (1987); E. Desurvire, J. R. Simpson, and P. C. Becker, Opt. Lett. 12, 888 (1987).
[CrossRef] [PubMed]

Rivera, T.

J. A. Levenson, I. Abram, T. Rivera, P. Fayolle, J. C. Garreau, and P. Grangier, Phys. Rev. Lett. 70, 267 (1993).
[CrossRef] [PubMed]

Satsuma, J.

J. Satsuma and N. Yajima, Suppl. Prog. Theor. Phys. 55, 284 (1974); V. I. Karpman and V. V. Solovev, Physica 3D, 487 (1981); D. J. Kaup, Phys. Rev. A 42, 5689 (1990).
[CrossRef] [PubMed]

Savage, C. M.

Suzuki, K.

M. Nakazawa, E. Yamada, H. Kubota, and K. Suzuki, Electron. Lett. 27, 1270 (1991).
[CrossRef]

van Kampen, N. G.

N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981).

Wright, E. M.

E. M. Wright, Phys. Rev. A 43, 3836 (1991).
[CrossRef] [PubMed]

Yajima, N.

J. Satsuma and N. Yajima, Suppl. Prog. Theor. Phys. 55, 284 (1974); V. I. Karpman and V. V. Solovev, Physica 3D, 487 (1981); D. J. Kaup, Phys. Rev. A 42, 5689 (1990).
[CrossRef] [PubMed]

Yamada, E.

M. Nakazawa, E. Yamada, H. Kubota, and K. Suzuki, Electron. Lett. 27, 1270 (1991).
[CrossRef]

Yuen, H. P.

Am. J. Phys. (1)

I. H. Deutsch, Am. J. Phys. 59, 834 (1991).
[CrossRef]

Electron. Lett. (4)

L. F. Mollenauer, E. Lichtman, G. T. Harvey, M. J. Neubelt, and B. M. Nyman, Electron. Lett. 28, 792 (1992).
[CrossRef]

L. F. Mollenauer, E. Lichtman, M. J. Neubelt, and G. T. Harvey, Electron. Lett. 29, 910 (1993).
[CrossRef]

M. Nakazawa, E. Yamada, H. Kubota, and K. Suzuki, Electron. Lett. 27, 1270 (1991).
[CrossRef]

R. J. Mears, L. Reekie, I. M. Jauncey, and D. N. Payne, Electron. Lett. 23, 1026 (1987); E. Desurvire, J. R. Simpson, and P. C. Becker, Opt. Lett. 12, 888 (1987).
[CrossRef] [PubMed]

J. Opt. Soc. Am. B (5)

Opt. Lett. (6)

Phys. Rev. A (4)

M. Hillery and L. Mlodinow, Phys. Rev. A 30, 1860 (1984).
[CrossRef]

B. Huttner and S. M. Barnett, Phys. Rev. A 46, 4306 (1992).
[CrossRef] [PubMed]

Y. Lai and H. A. Haus, Phys. Rev. A 40, 844, 1138 (1989).
[CrossRef] [PubMed]

E. M. Wright, Phys. Rev. A 43, 3836 (1991).
[CrossRef] [PubMed]

Phys. Rev. D (1)

C. M. Caves, Phys. Rev. D 26, 1817 (1982).
[CrossRef]

Phys. Rev. Lett. (2)

J. A. Levenson, I. Abram, T. Rivera, P. Fayolle, J. C. Garreau, and P. Grangier, Phys. Rev. Lett. 70, 267 (1993).
[CrossRef] [PubMed]

I. H. Deutsch, R. Y. Chiao, and J. C. Garrison, Phys. Rev. Lett. 69, 3627 (1992).
[CrossRef] [PubMed]

Suppl. Prog. Theor. Phys. (1)

J. Satsuma and N. Yajima, Suppl. Prog. Theor. Phys. 55, 284 (1974); V. I. Karpman and V. V. Solovev, Physica 3D, 487 (1981); D. J. Kaup, Phys. Rev. A 42, 5689 (1990).
[CrossRef] [PubMed]

Other (2)

G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, Mass., 1989), p. 116.

N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981).

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Figures (1)

Fig. 1
Fig. 1

Asymptotic output fluctuations in the soliton arrival time ΔT, normalized to the soliton full width at half-maximum Ts, versus the distance propagated, normalized to the soliton period zs. The four graphs correspond to the propagation of solitons in (solid line) a PSA-compensated lossy fiber, given by relation (3.27); (short-dashed line) a PIA-compensated lossy fiber, corresponding to the standard G-H limit, given by Eq. (3.28); (long-dashed line) a PIA system with sliding filters, given by Eq. (4.2); and (dotted line) a lossless fiber, corresponding to free soliton propagation, given by Eq. (4.13). The fiber parameters chosen are D = 0.45 (ps/nm)/km, n2 = 3.2 × 10−20 m2/W, Aeff = 50 μm2, λ = 1.55 μm, and 2γ = 0.048 km−1. In the case of the filters the strength parameter [Eq. (4.4)] is η = 0.52. The soliton parameters are Ts = 20 ps, corresponding to dispersion length z0 = 225 km, soliton period zs = 353 km, and mean number of photons N ¯ = 3.07 × 105. For 10 Gbits/s the timing jitter causes a BER grater than 10−9 when the normalized ratio is greater than 4.1 × 10−1 [see inequality (4.6)]. The PSA, as well as the spectral filtering scheme with PIA, leads to jitter with a linear asymptotic growth rate; the standard G-H limit has an asymptotic cubic dependence, whereas the jitter growth is quadratic for free-soliton propagation. The linearization approximation breaks down at distances of the order of 104zs, clearly outside the range of this graph.

Equations (102)

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i ψ z = - 1 2 2 ψ τ 2 - 2 N ¯ ψ 2 ψ .
z 0 = τ 0 2 / k 0 ,
E 0 = ( 8 π ω 0 A eff v τ 0 ) 1 / 2 ,
N ¯ = 2 k 0 A eff c n 2 ω 0 2 τ 0
ψ 0 ( τ , z ) = ( N ¯ / 2 ) 1 / 2 sech ( τ ) exp ( i z / 2 ) .
ψ ( τ , z ) = [ ψ 1 ( τ , z ) + i ψ 2 ( τ , z ) ] exp ( i z / 2 ) .
ψ 1 z = - 1 2 2 ψ 2 τ 2 + [ 1 2 - 2 N ¯ ( ψ 1 2 + ψ 2 2 ) ] ψ 2 ,
ψ 2 z = 1 2 2 ψ 1 τ 2 + [ - 1 2 - 2 N ¯ ( ψ 1 2 + ψ 2 2 ) ] ψ 1 .
2 ψ 1 τ 2 - ψ 1 + 4 N ¯ ψ 1 3 = 0 ,
[ ψ ( τ , z ) , ψ ( τ , z ) ] = δ ( τ - τ )
ψ 1 z = ( g - γ ) ψ 1 - 1 2 2 ψ 2 τ 2 + [ 1 2 - 2 N ¯ ( ψ 1 2 + ψ 2 2 ) ] ψ 2 + Γ 1 ,
ψ 2 z = - ( g - γ ) ψ 2 - 1 2 2 ψ 1 τ 2 + [ - 1 2 - 2 N ¯ ( ψ 1 2 + ψ 2 2 ) ] ψ 1 + Γ 2 ,
Γ 1 ( τ , z ) Γ 1 ( τ , z ) = Γ 2 ( τ , z ) Γ 2 ( τ , z ) = ( γ / 2 ) δ ( τ - τ ) δ ( z - z ) .
ψ ( τ , z ) = [ N ¯ f ( τ ) + v ( τ , z ) ] exp ( i z / 2 ) ,
f ( τ ) = sech ( τ ) / 2
v 1 ( τ , 0 ) v 1 ( τ , 0 ) = v 2 ( τ , 0 ) v 2 ( τ , 0 ) = ¼ δ ( τ - τ ) .
v 1 z = - 1 2 2 v 2 τ 2 + ( 1 2 - 2 f 2 ) v 2 + Γ 1 ,
v 2 z = - 2 γ v 2 + 1 2 2 v 1 τ 2 + ( - 1 2 + 6 f 2 ) v 1 + Γ 2 .
T ( z ) = [ d τ τ ψ ( τ , z ) ψ ( τ , z ) ] N - 1 ,
Ω ( z ) = - i 2 d τ [ ψ ( τ , z ) ψ τ - ψ τ ψ ( τ , z ) ] ,
N ( z ) = d τ ψ ( τ , z ) ψ ( τ , z )
δ T ( z ) = 2 N ¯ d τ τ f ( τ ) v 1 ( τ , z ) ,
δ Ω ( z ) = - 2 N ¯ d τ f τ v 2 ( τ , z ) ,
δ N ( z ) = 2 N ¯ d τ f ( τ ) v 1 ( τ , z ) .
δ T 0 2 [ δ T ( 0 ) ] 2 = Δ τ 2 / N ¯ ,
Δ τ 2 = d τ τ 2 f ( τ ) 2 = π 2 / 12
δ T 0 2 [ δ Ω ( 0 ) ] 2 = N ¯ Δ ω 2 ,
Δ ω 2 = - d τ f ( τ ) 2 τ 2 f ( τ ) = 1 3
δ N 0 2 = [ δ N ( 0 ) ] 2 = N ¯
d d z δ T = 1 N ¯ δ Ω + Γ T ,
d d z δ Ω = - 2 γ δ Ω + Γ Ω ,
d d z δ N = Γ N .
Γ T ( z ) = 2 N ¯ d τ [ τ f ( τ ) ] Γ 1 ( τ , z ) ,
Γ Ω ( z ) = - 2 N ¯ d τ [ τ f ( τ ) ] Γ 2 ( τ , z ) ,
Γ N = d τ f ( τ ) Γ 1 ( τ , z ) ,
Γ T ( z ) Γ T ( z ) = 2 γ δ T 0 2 δ ( z - z ) = π 2 γ 6 N ¯ δ ( z - z ) ,
Γ Ω ( z ) Γ Ω ( z ) = 2 γ δ Ω 0 2 δ ( z - z ) = 2 γ N ¯ 3 δ ( z - z ) ,
Γ N ( z ) Γ N ( z ) = 2 γ δ N 0 2 δ ( z - z ) = 2 γ N ¯ δ ( z - z ) .
[ δ T ( L ) ] 2 = [ δ T ( 0 ) ] 2 + 0 L d z 0 L d z Γ T ( z ) Γ T ( z ) + 1 N ¯ 2 0 L d z 0 L d z δ Ω ( z ) δ Ω ( z ) .
δ Ω ( z ) δ Ω ( z ) = exp [ - 2 γ ( z + z ) ] δ Ω 0 2 + [ exp ( - 2 γ z - z ) - exp [ - 2 γ ( z + z ) ] δ Ω 0 2 2 ,
[ δ Ω ( L ) ] 2 = exp ( - 4 γ L ) δ Ω 0 2 + [ 1 - exp ( - 4 γ L ) ] δ Ω 0 2 2 .
[ δ Ω ( L ) ] 2 δ Ω 0 2 2 = N ¯ 6 ,
[ δ T ( L ) ] 2 = δ T 0 2 + 2 γ L δ T 0 2 + δ Ω 0 2 N ¯ 2 [ 1 - exp ( 2 γ L ) 2 γ ] 2 + δ Ω 0 2 4 γ 2 N ¯ 2 × [ 2 γ L - 3 2 + 2 exp ( - 2 γ L ) - 1 2 exp ( - 4 γ L ) ] .
[ δ T ( L ) ] 2 = ( 1 + 2 γ L ) δ T 0 2 + ( - 1 2 + 2 γ L ) δ Ω 0 2 4 γ 2 N ¯ 2 .
[ δ T ( L ) ] 2 ( δ T 0 2 + δ Ω 0 2 4 γ 2 N ¯ 2 ) 2 γ L = [ 1 + ( z l z s ) 2 ] L z l π 2 τ 0 2 12 N ¯ ,
[ δ T ( L ) ] 2 GH = 2 9 L 3 z l z 0 2 τ 0 2 N ¯ .
[ δ N ( L ) ] 2 = N ¯ + ( 2 γ L ) N ¯ .
L N ¯ / ( 2 γ )
Δ T PSA Δ T GH = ( 3 2 ) 1 / 2 z s L = 1.22 z s L
[ δ T ( L ) ] 2 F = [ 1 + ( z f z s ) 2 ] L z l π 2 τ 0 2 6 N ¯ = [ 1 + ( 3 π η ) 2 ] L z l π 2 τ 0 2 6 N ¯ ,
z f = 3 Ω f 2 τ 0 2 l / 4
η = 3 z 0 2 z f
Δ T F Δ T GH = { 3 [ 1 + ( 3 π η ) 2 ] } 1 / 2 z s L .
[ δ T ( L ) ] 2 < ( T w 6.1 ) 2 .
L GH max = [ 0.375 ( T w T s ) 2 N ¯ z l z 0 2 ] 1 / 3 .
L PSA max = 0.101 ( T w T s ) 2 N ¯ z l ,
Δ N ( L PSA , δ T max ) N ¯ 2 γ L PSA , δ T max N ¯ = 0.79.
[ δ N ( L ) ] 2 < ( N ¯ / 6.1 ) 2 ,
L PSA , δ N max = N ¯ ( 6.1 ) 2 z l = 1.72 × 10 5 km .
L F max = 0.0506 1 + ( 3 / π η ) 2 ( T w T s ) 2 N ¯ z l .
[ δ T ( L ) ] 2 vac = δ T 0 2 + ( L z 0 ) 2 δ Ω 0 2 N ¯ 2 = [ 1 + ( L z s ) 2 ] π 2 τ 0 2 12 N ¯ .
Δ T vac Δ T PSA = L z l z s .
L vac max = 0.318 ( T w T s ) N ¯ z s .
v 1 ( τ , 0 ) v 1 ( τ , 0 ) = G 4 δ ( τ - τ ) , v 2 ( τ , 0 ) v 2 ( τ , 0 ) = G - 1 4 δ ( τ - τ ) .
δ T 0 2 = G δ T 0 2 vac ,             δ Ω 0 2 = G - 1 δ Ω 0 2 vac .
[ δ T ( L ) ] 2 sq = G δ T 0 2 vac + G - 1 ( L z 0 ) 2 δ Ω 0 2 vac N ¯ 2 = [ G + G - 1 ( L z s ) 2 ] π 2 τ 0 2 12 N ¯ .
ψ ( τ , z ) = Tr [ ρ ψ ( τ , z ) ] = F ( τ , z ) = N ¯ f ˜ ( τ , z ) = N ¯ exp ( i z / 2 ) f ( τ ) ,
Δ ψ 2 d τ [ 1 2 ψ ψ + ψ ψ - ψ ψ ] N ¯ ,
ψ ( τ , z ) = F ( τ , z ) + v ˜ ( τ , z ) ,
v ˜ ( τ , z ) = D [ F ] ψ ( τ , z ) D [ F ] .
D [ F ] = exp { d τ [ F ( τ , z ) ψ ( τ , z ) + F * ( τ , z ) ψ ( τ , z ) ] }
v ˜ ( τ , z ) = Tr ( ρ v ˜ ) = Tr ( ρ ˜ ψ ) = 0 ,
Δ ψ 2 = d τ 1 2 v ˜ v ˜ + v ˜ v ˜ N ¯ ,
ρ ˜ = D [ F ] ρ D [ F ] .
i v ˜ z = - i γ v ˜ + i g exp ( i θ pump ) v ˜ - 1 2 τ 2 v ˜ - 2 f ˜ 2 v ˜ - 4 f ˜ 2 v ˜ + i Γ - 4 N ¯ f ˜ v ˜ v ˜ - 2 N ¯ f ˜ * v ˜ 2 - 2 N ¯ v ˜ v ˜ v ˜ .
d a d t = - γ a + g a + Γ ,
Γ ( t ) Γ ( t ) = 2 γ δ ( t - t ) ,
Γ ( t ) Γ ( t ) = Γ ( t ) Γ ( t ) = Γ ( t ) Γ ( t ) = 0.
Γ 1 ( t ) Γ 1 ( t ) = Γ 2 ( t ) Γ 2 ( t ) = ( γ / 2 ) δ ( t - t ) ,
a ˙ 1 = ( g - γ ) a 1 + Γ 1 ,
a ˙ 2 = - ( g + γ ) a 2 + Γ 2 .
a 1 ( t ) = exp [ ( g - γ ) t ] a 1 ( 0 ) + 0 t d t exp [ ( g - γ ) ( t - t ) ] Γ 1 ( t ) ,
a 2 ( t ) = exp [ - ( g + γ ) t ] a 2 ( 0 ) + 0 t d t exp [ - ( g + γ ) ( t - t ) ] Γ 2 ( t ) .
[ Δ a 1 ( t ) ] 2 = η G [ Δ a 1 ( 0 ) ] 2 + γ 4 ( g - γ ) ( 1 - η G ) ,
[ Δ a 2 ( t ) ] 2 = η G - 1 [ Δ a 2 ( 0 ) ] 2 + γ 4 ( g + γ ) ( 1 - η G - 1 ) ,
[ Δ a 1 ( t ) ] 2 = [ Δ a 1 ( 0 ) ] 2 + g t / 2 ,
[ Δ a 2 ( t ) ] 2 = G - 1 [ Δ a 2 ( 0 ) ] 2 + ( 1 - G - 2 ) .
[ Δ a 1 ( t ) ] 2 g t / 2 ,
[ Δ a 2 ( t ) ] 2 1 / 8.
[ Δ a 1 ( t ) ] 2 = [ Δ a 2 ( t ) ] 2 g t .
d d z δ T ( z ) = 2 N ¯ d τ τ f ( τ ) z v 1 ( τ , z ) ,
d d z δ Ω ( z ) = - 2 N ¯ d τ f τ z v 2 ( τ , z ) ,
d d z δ N ( z ) = 2 N ¯ d τ f ( τ ) z v 1 ( τ , z ) .
d d z δ T ( z ) = 2 N ¯ d τ [ - 1 2 τ 2 ( τ f ) + τ f ( 1 2 - 2 f 2 ) ] v 2 + Γ T ( z ) ,
Γ T ( z ) = 2 N ¯ d τ τ f ( τ ) Γ 1 ( τ , z ) ,
Γ T ( z ) Γ T ( z ) = 2 γ N ¯ d τ τ 2 f 2 ( τ ) δ ( z - z ) = 2 γ [ δ T ( 0 ) ] 2 δ ( z - z ) = π 2 γ 3 N ¯ δ ( z - z ) .
d d z δ T ( z ) = - 2 N ¯ d τ f τ v 2 ( τ , z ) - 1 2 d τ τ ( τ 2 f - f + 4 f 3 ) v 2 + Γ T ( z ) .
d d z δ T = 1 N ¯ δ Ω + Γ T .
d d z δ Ω = - 2 γ δ Ω + Γ Ω ,
Γ Ω ( z ) Γ Ω ( z ) = 2 γ [ δ Ω ( 0 ) ] 2 δ ( z - z ) = 2 γ N ¯ 3 δ ( z - z ) ,
d d z δ N = Γ N ,
Γ N ( z ) Γ N ( z ) = 2 γ [ δ N ( 0 ) ] 2 δ ( z - z ) = 2 γ N ¯ δ ( z - z ) .

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