Abstract

The spectrum of a two-level atom in the presence of a multimode laser pulse is calculated. The field is allowed to fluctuate in amplitude or in phase; the emitted spectrum has richer emission lines than in the case of the nonfluctuating field and shows peaks shifted toward the blue with respect to the traditional harmonic peaks. The position of the lines is predicted by the formula ω2n+1=(2n+1)(1+Δ)ωL with Δ being a parameter that can be found numerically. In this way the fluctuations seem to result in an effective increase of the laser frequency.

© 1998 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  4. C.-G. Wahlström, J. Larsson, A. Persson, T. Starczewski, S. Svanberg, P. Salières, Ph. Balcou, and A. L’Huillier, Phys. Rev. A 48, 4709 (1993).
    [CrossRef]
  5. P. Balcou and A. L’Huillier, Phys. Rev. Lett. 70, 774 (1993).
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    [CrossRef] [PubMed]
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1996 (8)

C. Lyngå, A. L’Huillier, and C.-G. Wahlström, J. Phys. B 29, 3293 (1996).
[CrossRef]

P. Salières, T. Ditmire, M. D. Perry, A. L’Huillier, and M. Lewenstein, J. Phys. B 29, 4771 (1996).
[CrossRef]

N. E. Karapanagioti, D. Xenakis, D. Charalambidis, and C. Fotakis, J. Phys. B 29, 3599 (1996).
[CrossRef]

S. De Luca and E. Fiordilino, J. Phys. B 29, 3277 (1996).
[CrossRef]

J. Zhou, J. Peatross, M. M. Murnane, H. C. Kapteyn, and I. P. Christov, Phys. Rev. Lett. 77, 1743 (1996).
[CrossRef]

I. P. Christov, J. Zhou, J. Peatross, A. Rundquist, M. M. Murnane, and H. C. Kapteyn, Phys. Rev. Lett. 76, 752 (1996).
[CrossRef]

E. Fiordilino and V. Miceli, J. Mod. Opt. 43, 735 (1996).
[CrossRef]

E. Fiordilino and V. Miceli, Laser Phys. 6, 1204 (1996).

1995 (3)

P. Antoine, B. Piraux, and A. Maquet, Phys. Rev. A 51, 1750 (1995).
[CrossRef]

K. Miyazaki and H. Takada, Phys. Rev. A 52, 3007 (1995).
[CrossRef] [PubMed]

C. Kan, C. E. Capjack, R. Rankin, and N. H. Burnett, Phys. Rev. A 52, 4336 (1995).
[CrossRef]

1994 (2)

M. Lewenstein, P. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, Phys. Rev. A 48, 2117 (1994).
[CrossRef]

E. Fiordilino and V. Miceli, J. Mod. Opt. 41, 1415 (1994).
[CrossRef]

1993 (6)

P. B. Corkum, Phys. Rev. Lett. 71, 1994 (1993).
[CrossRef] [PubMed]

V. C. Reed, K. Burnett, and P. L. Knight, Phys. Rev. A 47, 34 (1993).
[CrossRef]

L. Plaja and L. Roso, J. Mod. Opt. 40, 793 (1993).
[CrossRef]

R. Haight and D. R. Peale, Phys. Rev. Lett. 70, 3979 (1993).
[CrossRef] [PubMed]

C.-G. Wahlström, J. Larsson, A. Persson, T. Starczewski, S. Svanberg, P. Salières, Ph. Balcou, and A. L’Huillier, Phys. Rev. A 48, 4709 (1993).
[CrossRef]

P. Balcou and A. L’Huillier, Phys. Rev. Lett. 70, 774 (1993).
[CrossRef]

1992 (4)

K. Miyazaki and H. Sakai, J. Phys. B 25, L83 (1992).
[CrossRef]

K. Burnett, V. C. Reed, J. Cooper, and P. L. Knight, Phys. Rev. A 45, 3347 (1992).
[CrossRef] [PubMed]

T. F. Jiang and S. I. Chu, Phys. Rev. A 46, 7322 (1992).
[CrossRef] [PubMed]

L. Plaja and L. Roso-Franco, J. Opt. Soc. Am. B 9, 2210 (1992).
[CrossRef]

1991 (1)

A. L’Huillier, K. J. Schafer, and K. C. Kulander, J. Phys. B 24, 3315 (1991).
[CrossRef]

1990 (1)

B. Sundaram and P. W. Milonni, Phys. Rev. A 41, 6571 (1990).
[CrossRef] [PubMed]

1988 (2)

M. Ferray, A. L’Huillier, X. F. Li, L. A. Lomprè, and C. Manus, J. Phys. B 21, L31 (1988).
[CrossRef]

E. Fiordilino and F. Trombetta, J. Mod. Opt. 35, 1635 (1988).
[CrossRef]

1987 (1)

E. Fiordilino, Nuovo Cimento D 9, 599 (1987).
[CrossRef]

Antoine, P.

P. Antoine, B. Piraux, and A. Maquet, Phys. Rev. A 51, 1750 (1995).
[CrossRef]

Balcou, P.

M. Lewenstein, P. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, Phys. Rev. A 48, 2117 (1994).
[CrossRef]

P. Balcou and A. L’Huillier, Phys. Rev. Lett. 70, 774 (1993).
[CrossRef]

Balcou, Ph.

C.-G. Wahlström, J. Larsson, A. Persson, T. Starczewski, S. Svanberg, P. Salières, Ph. Balcou, and A. L’Huillier, Phys. Rev. A 48, 4709 (1993).
[CrossRef]

Burnett, K.

V. C. Reed, K. Burnett, and P. L. Knight, Phys. Rev. A 47, 34 (1993).
[CrossRef]

K. Burnett, V. C. Reed, J. Cooper, and P. L. Knight, Phys. Rev. A 45, 3347 (1992).
[CrossRef] [PubMed]

Burnett, N. H.

C. Kan, C. E. Capjack, R. Rankin, and N. H. Burnett, Phys. Rev. A 52, 4336 (1995).
[CrossRef]

Capjack, C. E.

C. Kan, C. E. Capjack, R. Rankin, and N. H. Burnett, Phys. Rev. A 52, 4336 (1995).
[CrossRef]

Charalambidis, D.

N. E. Karapanagioti, D. Xenakis, D. Charalambidis, and C. Fotakis, J. Phys. B 29, 3599 (1996).
[CrossRef]

Christov, I. P.

J. Zhou, J. Peatross, M. M. Murnane, H. C. Kapteyn, and I. P. Christov, Phys. Rev. Lett. 77, 1743 (1996).
[CrossRef]

I. P. Christov, J. Zhou, J. Peatross, A. Rundquist, M. M. Murnane, and H. C. Kapteyn, Phys. Rev. Lett. 76, 752 (1996).
[CrossRef]

Chu, S. I.

T. F. Jiang and S. I. Chu, Phys. Rev. A 46, 7322 (1992).
[CrossRef] [PubMed]

Cooper, J.

K. Burnett, V. C. Reed, J. Cooper, and P. L. Knight, Phys. Rev. A 45, 3347 (1992).
[CrossRef] [PubMed]

Corkum, P. B.

M. Lewenstein, P. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, Phys. Rev. A 48, 2117 (1994).
[CrossRef]

P. B. Corkum, Phys. Rev. Lett. 71, 1994 (1993).
[CrossRef] [PubMed]

De Luca, S.

S. De Luca and E. Fiordilino, J. Phys. B 29, 3277 (1996).
[CrossRef]

Ditmire, T.

P. Salières, T. Ditmire, M. D. Perry, A. L’Huillier, and M. Lewenstein, J. Phys. B 29, 4771 (1996).
[CrossRef]

Ferray, M.

M. Ferray, A. L’Huillier, X. F. Li, L. A. Lomprè, and C. Manus, J. Phys. B 21, L31 (1988).
[CrossRef]

Fiordilino, E.

S. De Luca and E. Fiordilino, J. Phys. B 29, 3277 (1996).
[CrossRef]

E. Fiordilino and V. Miceli, J. Mod. Opt. 43, 735 (1996).
[CrossRef]

E. Fiordilino and V. Miceli, Laser Phys. 6, 1204 (1996).

E. Fiordilino and V. Miceli, J. Mod. Opt. 41, 1415 (1994).
[CrossRef]

E. Fiordilino and F. Trombetta, J. Mod. Opt. 35, 1635 (1988).
[CrossRef]

E. Fiordilino, Nuovo Cimento D 9, 599 (1987).
[CrossRef]

Fotakis, C.

N. E. Karapanagioti, D. Xenakis, D. Charalambidis, and C. Fotakis, J. Phys. B 29, 3599 (1996).
[CrossRef]

Haight, R.

R. Haight and D. R. Peale, Phys. Rev. Lett. 70, 3979 (1993).
[CrossRef] [PubMed]

Ivanov, M. Yu.

M. Lewenstein, P. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, Phys. Rev. A 48, 2117 (1994).
[CrossRef]

Jiang, T. F.

T. F. Jiang and S. I. Chu, Phys. Rev. A 46, 7322 (1992).
[CrossRef] [PubMed]

Kan, C.

C. Kan, C. E. Capjack, R. Rankin, and N. H. Burnett, Phys. Rev. A 52, 4336 (1995).
[CrossRef]

Kapteyn, H. C.

I. P. Christov, J. Zhou, J. Peatross, A. Rundquist, M. M. Murnane, and H. C. Kapteyn, Phys. Rev. Lett. 76, 752 (1996).
[CrossRef]

J. Zhou, J. Peatross, M. M. Murnane, H. C. Kapteyn, and I. P. Christov, Phys. Rev. Lett. 77, 1743 (1996).
[CrossRef]

Karapanagioti, N. E.

N. E. Karapanagioti, D. Xenakis, D. Charalambidis, and C. Fotakis, J. Phys. B 29, 3599 (1996).
[CrossRef]

Knight, P. L.

V. C. Reed, K. Burnett, and P. L. Knight, Phys. Rev. A 47, 34 (1993).
[CrossRef]

K. Burnett, V. C. Reed, J. Cooper, and P. L. Knight, Phys. Rev. A 45, 3347 (1992).
[CrossRef] [PubMed]

Kulander, K. C.

A. L’Huillier, K. J. Schafer, and K. C. Kulander, J. Phys. B 24, 3315 (1991).
[CrossRef]

L’Huillier, A.

P. Salières, T. Ditmire, M. D. Perry, A. L’Huillier, and M. Lewenstein, J. Phys. B 29, 4771 (1996).
[CrossRef]

C. Lyngå, A. L’Huillier, and C.-G. Wahlström, J. Phys. B 29, 3293 (1996).
[CrossRef]

M. Lewenstein, P. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, Phys. Rev. A 48, 2117 (1994).
[CrossRef]

P. Balcou and A. L’Huillier, Phys. Rev. Lett. 70, 774 (1993).
[CrossRef]

C.-G. Wahlström, J. Larsson, A. Persson, T. Starczewski, S. Svanberg, P. Salières, Ph. Balcou, and A. L’Huillier, Phys. Rev. A 48, 4709 (1993).
[CrossRef]

A. L’Huillier, K. J. Schafer, and K. C. Kulander, J. Phys. B 24, 3315 (1991).
[CrossRef]

M. Ferray, A. L’Huillier, X. F. Li, L. A. Lomprè, and C. Manus, J. Phys. B 21, L31 (1988).
[CrossRef]

Larsson, J.

C.-G. Wahlström, J. Larsson, A. Persson, T. Starczewski, S. Svanberg, P. Salières, Ph. Balcou, and A. L’Huillier, Phys. Rev. A 48, 4709 (1993).
[CrossRef]

Lewenstein, M.

P. Salières, T. Ditmire, M. D. Perry, A. L’Huillier, and M. Lewenstein, J. Phys. B 29, 4771 (1996).
[CrossRef]

M. Lewenstein, P. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, Phys. Rev. A 48, 2117 (1994).
[CrossRef]

Li, X. F.

M. Ferray, A. L’Huillier, X. F. Li, L. A. Lomprè, and C. Manus, J. Phys. B 21, L31 (1988).
[CrossRef]

Lomprè, L. A.

M. Ferray, A. L’Huillier, X. F. Li, L. A. Lomprè, and C. Manus, J. Phys. B 21, L31 (1988).
[CrossRef]

Lyngå, C.

C. Lyngå, A. L’Huillier, and C.-G. Wahlström, J. Phys. B 29, 3293 (1996).
[CrossRef]

Manus, C.

M. Ferray, A. L’Huillier, X. F. Li, L. A. Lomprè, and C. Manus, J. Phys. B 21, L31 (1988).
[CrossRef]

Maquet, A.

P. Antoine, B. Piraux, and A. Maquet, Phys. Rev. A 51, 1750 (1995).
[CrossRef]

Miceli, V.

E. Fiordilino and V. Miceli, Laser Phys. 6, 1204 (1996).

E. Fiordilino and V. Miceli, J. Mod. Opt. 43, 735 (1996).
[CrossRef]

E. Fiordilino and V. Miceli, J. Mod. Opt. 41, 1415 (1994).
[CrossRef]

Milonni, P. W.

B. Sundaram and P. W. Milonni, Phys. Rev. A 41, 6571 (1990).
[CrossRef] [PubMed]

Miyazaki, K.

K. Miyazaki and H. Takada, Phys. Rev. A 52, 3007 (1995).
[CrossRef] [PubMed]

K. Miyazaki and H. Sakai, J. Phys. B 25, L83 (1992).
[CrossRef]

Murnane, M. M.

I. P. Christov, J. Zhou, J. Peatross, A. Rundquist, M. M. Murnane, and H. C. Kapteyn, Phys. Rev. Lett. 76, 752 (1996).
[CrossRef]

J. Zhou, J. Peatross, M. M. Murnane, H. C. Kapteyn, and I. P. Christov, Phys. Rev. Lett. 77, 1743 (1996).
[CrossRef]

Peale, D. R.

R. Haight and D. R. Peale, Phys. Rev. Lett. 70, 3979 (1993).
[CrossRef] [PubMed]

Peatross, J.

J. Zhou, J. Peatross, M. M. Murnane, H. C. Kapteyn, and I. P. Christov, Phys. Rev. Lett. 77, 1743 (1996).
[CrossRef]

I. P. Christov, J. Zhou, J. Peatross, A. Rundquist, M. M. Murnane, and H. C. Kapteyn, Phys. Rev. Lett. 76, 752 (1996).
[CrossRef]

Perry, M. D.

P. Salières, T. Ditmire, M. D. Perry, A. L’Huillier, and M. Lewenstein, J. Phys. B 29, 4771 (1996).
[CrossRef]

Persson, A.

C.-G. Wahlström, J. Larsson, A. Persson, T. Starczewski, S. Svanberg, P. Salières, Ph. Balcou, and A. L’Huillier, Phys. Rev. A 48, 4709 (1993).
[CrossRef]

Piraux, B.

P. Antoine, B. Piraux, and A. Maquet, Phys. Rev. A 51, 1750 (1995).
[CrossRef]

Plaja, L.

Rankin, R.

C. Kan, C. E. Capjack, R. Rankin, and N. H. Burnett, Phys. Rev. A 52, 4336 (1995).
[CrossRef]

Reed, V. C.

V. C. Reed, K. Burnett, and P. L. Knight, Phys. Rev. A 47, 34 (1993).
[CrossRef]

K. Burnett, V. C. Reed, J. Cooper, and P. L. Knight, Phys. Rev. A 45, 3347 (1992).
[CrossRef] [PubMed]

Roso, L.

L. Plaja and L. Roso, J. Mod. Opt. 40, 793 (1993).
[CrossRef]

Roso-Franco, L.

Rundquist, A.

I. P. Christov, J. Zhou, J. Peatross, A. Rundquist, M. M. Murnane, and H. C. Kapteyn, Phys. Rev. Lett. 76, 752 (1996).
[CrossRef]

Sakai, H.

K. Miyazaki and H. Sakai, J. Phys. B 25, L83 (1992).
[CrossRef]

Salières, P.

P. Salières, T. Ditmire, M. D. Perry, A. L’Huillier, and M. Lewenstein, J. Phys. B 29, 4771 (1996).
[CrossRef]

C.-G. Wahlström, J. Larsson, A. Persson, T. Starczewski, S. Svanberg, P. Salières, Ph. Balcou, and A. L’Huillier, Phys. Rev. A 48, 4709 (1993).
[CrossRef]

Schafer, K. J.

A. L’Huillier, K. J. Schafer, and K. C. Kulander, J. Phys. B 24, 3315 (1991).
[CrossRef]

Starczewski, T.

C.-G. Wahlström, J. Larsson, A. Persson, T. Starczewski, S. Svanberg, P. Salières, Ph. Balcou, and A. L’Huillier, Phys. Rev. A 48, 4709 (1993).
[CrossRef]

Sundaram, B.

B. Sundaram and P. W. Milonni, Phys. Rev. A 41, 6571 (1990).
[CrossRef] [PubMed]

Svanberg, S.

C.-G. Wahlström, J. Larsson, A. Persson, T. Starczewski, S. Svanberg, P. Salières, Ph. Balcou, and A. L’Huillier, Phys. Rev. A 48, 4709 (1993).
[CrossRef]

Takada, H.

K. Miyazaki and H. Takada, Phys. Rev. A 52, 3007 (1995).
[CrossRef] [PubMed]

Trombetta, F.

E. Fiordilino and F. Trombetta, J. Mod. Opt. 35, 1635 (1988).
[CrossRef]

Wahlström, C.-G.

C. Lyngå, A. L’Huillier, and C.-G. Wahlström, J. Phys. B 29, 3293 (1996).
[CrossRef]

C.-G. Wahlström, J. Larsson, A. Persson, T. Starczewski, S. Svanberg, P. Salières, Ph. Balcou, and A. L’Huillier, Phys. Rev. A 48, 4709 (1993).
[CrossRef]

Xenakis, D.

N. E. Karapanagioti, D. Xenakis, D. Charalambidis, and C. Fotakis, J. Phys. B 29, 3599 (1996).
[CrossRef]

Zhou, J.

J. Zhou, J. Peatross, M. M. Murnane, H. C. Kapteyn, and I. P. Christov, Phys. Rev. Lett. 77, 1743 (1996).
[CrossRef]

I. P. Christov, J. Zhou, J. Peatross, A. Rundquist, M. M. Murnane, and H. C. Kapteyn, Phys. Rev. Lett. 76, 752 (1996).
[CrossRef]

J. Mod. Opt. (4)

L. Plaja and L. Roso, J. Mod. Opt. 40, 793 (1993).
[CrossRef]

E. Fiordilino and V. Miceli, J. Mod. Opt. 41, 1415 (1994).
[CrossRef]

E. Fiordilino and F. Trombetta, J. Mod. Opt. 35, 1635 (1988).
[CrossRef]

E. Fiordilino and V. Miceli, J. Mod. Opt. 43, 735 (1996).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Phys. B (7)

S. De Luca and E. Fiordilino, J. Phys. B 29, 3277 (1996).
[CrossRef]

N. E. Karapanagioti, D. Xenakis, D. Charalambidis, and C. Fotakis, J. Phys. B 29, 3599 (1996).
[CrossRef]

M. Ferray, A. L’Huillier, X. F. Li, L. A. Lomprè, and C. Manus, J. Phys. B 21, L31 (1988).
[CrossRef]

A. L’Huillier, K. J. Schafer, and K. C. Kulander, J. Phys. B 24, 3315 (1991).
[CrossRef]

K. Miyazaki and H. Sakai, J. Phys. B 25, L83 (1992).
[CrossRef]

C. Lyngå, A. L’Huillier, and C.-G. Wahlström, J. Phys. B 29, 3293 (1996).
[CrossRef]

P. Salières, T. Ditmire, M. D. Perry, A. L’Huillier, and M. Lewenstein, J. Phys. B 29, 4771 (1996).
[CrossRef]

Laser Phys. (1)

E. Fiordilino and V. Miceli, Laser Phys. 6, 1204 (1996).

Nuovo Cimento D (1)

E. Fiordilino, Nuovo Cimento D 9, 599 (1987).
[CrossRef]

Phys. Rev. A (9)

K. Miyazaki and H. Takada, Phys. Rev. A 52, 3007 (1995).
[CrossRef] [PubMed]

M. Lewenstein, P. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, Phys. Rev. A 48, 2117 (1994).
[CrossRef]

B. Sundaram and P. W. Milonni, Phys. Rev. A 41, 6571 (1990).
[CrossRef] [PubMed]

V. C. Reed, K. Burnett, and P. L. Knight, Phys. Rev. A 47, 34 (1993).
[CrossRef]

C. Kan, C. E. Capjack, R. Rankin, and N. H. Burnett, Phys. Rev. A 52, 4336 (1995).
[CrossRef]

C.-G. Wahlström, J. Larsson, A. Persson, T. Starczewski, S. Svanberg, P. Salières, Ph. Balcou, and A. L’Huillier, Phys. Rev. A 48, 4709 (1993).
[CrossRef]

K. Burnett, V. C. Reed, J. Cooper, and P. L. Knight, Phys. Rev. A 45, 3347 (1992).
[CrossRef] [PubMed]

T. F. Jiang and S. I. Chu, Phys. Rev. A 46, 7322 (1992).
[CrossRef] [PubMed]

P. Antoine, B. Piraux, and A. Maquet, Phys. Rev. A 51, 1750 (1995).
[CrossRef]

Phys. Rev. Lett. (5)

J. Zhou, J. Peatross, M. M. Murnane, H. C. Kapteyn, and I. P. Christov, Phys. Rev. Lett. 77, 1743 (1996).
[CrossRef]

I. P. Christov, J. Zhou, J. Peatross, A. Rundquist, M. M. Murnane, and H. C. Kapteyn, Phys. Rev. Lett. 76, 752 (1996).
[CrossRef]

P. B. Corkum, Phys. Rev. Lett. 71, 1994 (1993).
[CrossRef] [PubMed]

P. Balcou and A. L’Huillier, Phys. Rev. Lett. 70, 774 (1993).
[CrossRef]

R. Haight and D. R. Peale, Phys. Rev. Lett. 70, 3979 (1993).
[CrossRef] [PubMed]

Other (2)

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

M. H. Mittleman, Theory of Laser–Atom Interactions (Plenum, New York, 1982).

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Figures (3)

Fig. 1
Fig. 1

Fourier acceleration spectrum for a two-level atom in the presence of a laser field without fluctuations. The parameters for the calculations are ω10=0.12 a.u., ω10/ωL=10, W0/ωL =8.3×10-2, and τ=100 optical cycles.

Fig. 2
Fig. 2

Fourier acceleration spectrum for a two-level atom in the presence of a laser electric field fluctuating with Gaussian statistics; the width of the Gaussian is σ=10-3. The parameters for the calculations are K=10, ω10/ωL=10, W0/ωL=8.3 ×10-2, and τ=100 optical cycles.

Fig. 3
Fig. 3

Fourier acceleration spectrum for a two-level atom in the presence of a laser field with fluctuating phase. The phase fluctuates with Gaussian statistics; the width of the Gaussian is σ =10-3. The parameters for the calculations are K=10, ω10/ωL=10, W0/ωL=8.3×10-2, and τ=100 optical cycles.

Equations (17)

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ddω S(ω)=4e23c3 |aF(ω)|2,
n(ω)=1ω ddω S(ω).
ωM=UI+βUp,
ω2n+1=(2n+1)(1+Δ)ωL,
E(t)=zˆ 1K F(t)k=1Kηk(t)cos[ωLt+ϕk(t)],
F(t)=tτ2 exp1-tτ2,t0,
R1(t)=R0,
Rk+1(t)=Rk(t)[1+gk(t)],1k<K,
η(t)=η¯(t)cos[ωt+ϕ¯(t)],
η¯2(t)=k=1Kηk2(t)+k>j=1K2ηk(t)ηj(t)cos(ϕk-ϕj),
tan ϕ¯=k=1Kηk(t)sin ϕkk=1Kηk(t)cos ϕk.
|t=c0 exp(-iω0t)|0+c1 exp(-iω1t)|1.
ic˙0=E(t)r10 exp(-iω10t)c1(t),
ic˙1=E(t)r10 exp(iω10t)c0(t),
r+{Δ(z)+E(z)cos[2(z+ϕ)]}r
=-2 ω10ωL V01(z)ωL r01+4 V01(z)ωL 0zr(z1) V01(z1)ωL dz1,
E(z)=2 W022ωL2 F2(z),Δ(z)=2ω1022ωL2+E(z).

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