Abstract

A quantum theory of high-order harmonic generation by a strong laser field in the presence of more bound states is formulated. The obtained numerical and analytical results for a two-state hydrogenlike atom model show that the harmonic spectrum consists of two parts: a usual single-state harmonic spectrum of odd harmonics having the energies (2k+1)ω and a resonant part with the peaks around the excitation energy Δω. The energy of the harmonics in the resonant part of the spectrum is equal to Δω±ω,Δω±3ω, …. For energies higher than the excitation energy, the resonant part forms a plateau, followed by a cutoff. The emission rate of the harmonics in this resonant plateau is many orders of magnitude higher than that of the harmonics generated in the presence of the ground state alone. The influence of the depletion of the initial states, as well as of the pulse shape and intensity, is analyzed.

© 2006 Optical Society of America

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  1. W. Becker, F. Grasbon, R. Kopold, D. B. Milosevic, G. G. Paulus, and H. Walther, "Above-threshold ionization: from classical features to quantum effects," Adv. At. Mol. Opt. Phys. 48, 35-98 (2002).
    [CrossRef]
  2. D. B. Milosevic and F. Ehlotzky, "Scattering and reaction processes in powerful laser fields," Adv. At. Mol. Opt. Phys. 49, 373-532 (2003).
  3. A. Becker and F. H. M. Faisal, "Intense field many-body S-matrix theory," J. Phys. B 38, R1-R56 (2005).
    [CrossRef]
  4. P. Salières, A. L'Huillier, Ph. Antoine, and M. Lewenstein, "Study of the spatial and temporal coherence of high-order harmonics," Adv. At. Mol. Opt. Phys. 41, 83-142 (1999).
    [CrossRef]
  5. J. L. Krause, K. J. Schafer, and K. C. Kulander, "High-order harmonic generation from atoms and ions in the high intensity regime," Phys. Rev. Lett. 68, 3535-3538 (1992).
    [CrossRef] [PubMed]
  6. P. B. Corkum, "Plasma perspective on strong field multiphoton ionization," Phys. Rev. Lett. 71, 1994-1997 (1993).
    [CrossRef] [PubMed]
  7. K. C. Kulander, K. J. Schafer, and J. L. Krause, "Dynamics of short-pulse excitation, ionization and atomic conversion," in Super-Intense Laser-Atom Physics, B.Piraux, A.L'Huillier, and K.Rzazewski, eds., Vol. 316 of NATO Advanced Science Institutes Series Series B (Plenum, 1993), pp. 95-110.
  8. M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L'Huillier, and P. B. Corkum, "Theory of high-harmonic generation by low-frequency laser fields," Phys. Rev. A 49, 2117-2132 (1994).
    [CrossRef] [PubMed]
  9. D. B. Milosevic and B. Piraux, "High-order harmonic generation in a bichromatic elliptically polarized laser field," Phys. Rev. A 54, 1522-1531 (1996).
    [CrossRef] [PubMed]
  10. P. M. Paul, T. O. Clatterbuck, C. Lynga, P. Colosimo, L. F. DiMauro, P. Agostini, and K. C. Kulander, "Enhanced high harmonic generation from an optically prepared excited medium," Phys. Rev. Lett. 94, 113906 (2005).
    [CrossRef] [PubMed]
  11. J. B. Watson, A. Sanpera, X. Chen, and K. Burnett, "Harmonic generation from a coherent superposition of states," Phys. Rev. A 53, R1962-R1965 (1996).
    [CrossRef] [PubMed]
  12. A. Sanpera, J. B. Watson, M. Lewenstein, and K. Burnett, "Harmonic-generation control," Phys. Rev. A 54, 4320-4326 (1996).
    [CrossRef] [PubMed]
  13. Z. Zeng, R. Li, Y. Cheng, W. Yu, and Z. Xu, "Resonance-enhanced high-order harmonic generation and frequency mixing in two-color laser field," Phys. Scr. 66, 321-325 (2002).
    [CrossRef]
  14. H. Niikura, D. M. Villeneuve, and P. B. Corkum, "Mapping attosecond electron wave packet motion," Phys. Rev. Lett. 94, 083003 (2005).
    [CrossRef] [PubMed]
  15. T. Brabec and F. Krausz, "Intense few-cycle laser fields: frontiers of nonlinear optics," Rev. Mod. Phys. 72, 545-591 (2000).
    [CrossRef]
  16. P. Agostini and L. F. DiMauro, "The physics of attosecond light pulses," Rep. Prog. Phys. 67, 813-855 (2004); see corrigendum, Ref. .
    [CrossRef]
  17. P. Agostini and L. F. DiMauro, "Corrigendum: the physics of attosecond light pulses," Rep. Prog. Phys. 67, 1563 (2004).
    [CrossRef]
  18. R. J. Jones, K. D. Moll, M. J. Thorpe, and J. Ye, "Phase-coherent frequency combs in the vacuum ultraviolet via high-harmonic generation inside a femtosecond enhancement cavity," Phys. Rev. Lett. 94, 193201 (2005).
    [CrossRef] [PubMed]
  19. C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and T. W. Hänsch, "A frequency comb in the extreme ultraviolet," Nature 436, 234-237 (2005).
    [CrossRef] [PubMed]
  20. C. Kan, C. E. Capjack, R. Rankin, and N. H. Burnett, "Spectral and temporal structure in high harmonic emission from ionizing atomic gases," Phys. Rev. A 52, R4336-R4339 (1995).
    [CrossRef] [PubMed]
  21. L. V. Keldysh, "Ionization in the field of a strong electromagnetic wave," Sov. Phys. JETP 20, 1307-1319 (1965) L. V. Keldysh, [Zh. Eksp. Teor. Fiz. 47, 1945-1957 (1964)].
  22. F. A. Ilkov, J. E. Decker, and S. L. Chin, "Ionization of atoms in the tunnelling regime with experimental evidence using Hg atoms," J. Phys. B 25, 4005-4020 (1992).
    [CrossRef]
  23. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1964), 7.1.17, Fig. 7.2.

2005 (5)

A. Becker and F. H. M. Faisal, "Intense field many-body S-matrix theory," J. Phys. B 38, R1-R56 (2005).
[CrossRef]

P. M. Paul, T. O. Clatterbuck, C. Lynga, P. Colosimo, L. F. DiMauro, P. Agostini, and K. C. Kulander, "Enhanced high harmonic generation from an optically prepared excited medium," Phys. Rev. Lett. 94, 113906 (2005).
[CrossRef] [PubMed]

H. Niikura, D. M. Villeneuve, and P. B. Corkum, "Mapping attosecond electron wave packet motion," Phys. Rev. Lett. 94, 083003 (2005).
[CrossRef] [PubMed]

R. J. Jones, K. D. Moll, M. J. Thorpe, and J. Ye, "Phase-coherent frequency combs in the vacuum ultraviolet via high-harmonic generation inside a femtosecond enhancement cavity," Phys. Rev. Lett. 94, 193201 (2005).
[CrossRef] [PubMed]

C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and T. W. Hänsch, "A frequency comb in the extreme ultraviolet," Nature 436, 234-237 (2005).
[CrossRef] [PubMed]

2004 (2)

P. Agostini and L. F. DiMauro, "The physics of attosecond light pulses," Rep. Prog. Phys. 67, 813-855 (2004); see corrigendum, Ref. .
[CrossRef]

P. Agostini and L. F. DiMauro, "Corrigendum: the physics of attosecond light pulses," Rep. Prog. Phys. 67, 1563 (2004).
[CrossRef]

2003 (1)

D. B. Milosevic and F. Ehlotzky, "Scattering and reaction processes in powerful laser fields," Adv. At. Mol. Opt. Phys. 49, 373-532 (2003).

2002 (2)

W. Becker, F. Grasbon, R. Kopold, D. B. Milosevic, G. G. Paulus, and H. Walther, "Above-threshold ionization: from classical features to quantum effects," Adv. At. Mol. Opt. Phys. 48, 35-98 (2002).
[CrossRef]

Z. Zeng, R. Li, Y. Cheng, W. Yu, and Z. Xu, "Resonance-enhanced high-order harmonic generation and frequency mixing in two-color laser field," Phys. Scr. 66, 321-325 (2002).
[CrossRef]

2000 (1)

T. Brabec and F. Krausz, "Intense few-cycle laser fields: frontiers of nonlinear optics," Rev. Mod. Phys. 72, 545-591 (2000).
[CrossRef]

1999 (1)

P. Salières, A. L'Huillier, Ph. Antoine, and M. Lewenstein, "Study of the spatial and temporal coherence of high-order harmonics," Adv. At. Mol. Opt. Phys. 41, 83-142 (1999).
[CrossRef]

1996 (3)

J. B. Watson, A. Sanpera, X. Chen, and K. Burnett, "Harmonic generation from a coherent superposition of states," Phys. Rev. A 53, R1962-R1965 (1996).
[CrossRef] [PubMed]

A. Sanpera, J. B. Watson, M. Lewenstein, and K. Burnett, "Harmonic-generation control," Phys. Rev. A 54, 4320-4326 (1996).
[CrossRef] [PubMed]

D. B. Milosevic and B. Piraux, "High-order harmonic generation in a bichromatic elliptically polarized laser field," Phys. Rev. A 54, 1522-1531 (1996).
[CrossRef] [PubMed]

1995 (1)

C. Kan, C. E. Capjack, R. Rankin, and N. H. Burnett, "Spectral and temporal structure in high harmonic emission from ionizing atomic gases," Phys. Rev. A 52, R4336-R4339 (1995).
[CrossRef] [PubMed]

1994 (1)

M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L'Huillier, and P. B. Corkum, "Theory of high-harmonic generation by low-frequency laser fields," Phys. Rev. A 49, 2117-2132 (1994).
[CrossRef] [PubMed]

1993 (1)

P. B. Corkum, "Plasma perspective on strong field multiphoton ionization," Phys. Rev. Lett. 71, 1994-1997 (1993).
[CrossRef] [PubMed]

1992 (2)

J. L. Krause, K. J. Schafer, and K. C. Kulander, "High-order harmonic generation from atoms and ions in the high intensity regime," Phys. Rev. Lett. 68, 3535-3538 (1992).
[CrossRef] [PubMed]

F. A. Ilkov, J. E. Decker, and S. L. Chin, "Ionization of atoms in the tunnelling regime with experimental evidence using Hg atoms," J. Phys. B 25, 4005-4020 (1992).
[CrossRef]

1986 (1)

L. V. Keldysh, "Ionization in the field of a strong electromagnetic wave," Sov. Phys. JETP 20, 1307-1319 (1965) L. V. Keldysh, [Zh. Eksp. Teor. Fiz. 47, 1945-1957 (1964)].

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1964), 7.1.17, Fig. 7.2.

Agostini, P.

P. M. Paul, T. O. Clatterbuck, C. Lynga, P. Colosimo, L. F. DiMauro, P. Agostini, and K. C. Kulander, "Enhanced high harmonic generation from an optically prepared excited medium," Phys. Rev. Lett. 94, 113906 (2005).
[CrossRef] [PubMed]

P. Agostini and L. F. DiMauro, "The physics of attosecond light pulses," Rep. Prog. Phys. 67, 813-855 (2004); see corrigendum, Ref. .
[CrossRef]

P. Agostini and L. F. DiMauro, "Corrigendum: the physics of attosecond light pulses," Rep. Prog. Phys. 67, 1563 (2004).
[CrossRef]

Ammosov, M. V.

L. V. Keldysh, "Ionization in the field of a strong electromagnetic wave," Sov. Phys. JETP 20, 1307-1319 (1965) L. V. Keldysh, [Zh. Eksp. Teor. Fiz. 47, 1945-1957 (1964)].

Antoine, Ph.

P. Salières, A. L'Huillier, Ph. Antoine, and M. Lewenstein, "Study of the spatial and temporal coherence of high-order harmonics," Adv. At. Mol. Opt. Phys. 41, 83-142 (1999).
[CrossRef]

Balcou, Ph.

M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L'Huillier, and P. B. Corkum, "Theory of high-harmonic generation by low-frequency laser fields," Phys. Rev. A 49, 2117-2132 (1994).
[CrossRef] [PubMed]

Becker, A.

A. Becker and F. H. M. Faisal, "Intense field many-body S-matrix theory," J. Phys. B 38, R1-R56 (2005).
[CrossRef]

Becker, W.

W. Becker, F. Grasbon, R. Kopold, D. B. Milosevic, G. G. Paulus, and H. Walther, "Above-threshold ionization: from classical features to quantum effects," Adv. At. Mol. Opt. Phys. 48, 35-98 (2002).
[CrossRef]

Brabec, T.

T. Brabec and F. Krausz, "Intense few-cycle laser fields: frontiers of nonlinear optics," Rev. Mod. Phys. 72, 545-591 (2000).
[CrossRef]

Burnett, K.

J. B. Watson, A. Sanpera, X. Chen, and K. Burnett, "Harmonic generation from a coherent superposition of states," Phys. Rev. A 53, R1962-R1965 (1996).
[CrossRef] [PubMed]

A. Sanpera, J. B. Watson, M. Lewenstein, and K. Burnett, "Harmonic-generation control," Phys. Rev. A 54, 4320-4326 (1996).
[CrossRef] [PubMed]

Burnett, N. H.

C. Kan, C. E. Capjack, R. Rankin, and N. H. Burnett, "Spectral and temporal structure in high harmonic emission from ionizing atomic gases," Phys. Rev. A 52, R4336-R4339 (1995).
[CrossRef] [PubMed]

Capjack, C. E.

C. Kan, C. E. Capjack, R. Rankin, and N. H. Burnett, "Spectral and temporal structure in high harmonic emission from ionizing atomic gases," Phys. Rev. A 52, R4336-R4339 (1995).
[CrossRef] [PubMed]

Chen, X.

J. B. Watson, A. Sanpera, X. Chen, and K. Burnett, "Harmonic generation from a coherent superposition of states," Phys. Rev. A 53, R1962-R1965 (1996).
[CrossRef] [PubMed]

Cheng, Y.

Z. Zeng, R. Li, Y. Cheng, W. Yu, and Z. Xu, "Resonance-enhanced high-order harmonic generation and frequency mixing in two-color laser field," Phys. Scr. 66, 321-325 (2002).
[CrossRef]

Chin, S. L.

F. A. Ilkov, J. E. Decker, and S. L. Chin, "Ionization of atoms in the tunnelling regime with experimental evidence using Hg atoms," J. Phys. B 25, 4005-4020 (1992).
[CrossRef]

Clatterbuck, T. O.

P. M. Paul, T. O. Clatterbuck, C. Lynga, P. Colosimo, L. F. DiMauro, P. Agostini, and K. C. Kulander, "Enhanced high harmonic generation from an optically prepared excited medium," Phys. Rev. Lett. 94, 113906 (2005).
[CrossRef] [PubMed]

Colosimo, P.

P. M. Paul, T. O. Clatterbuck, C. Lynga, P. Colosimo, L. F. DiMauro, P. Agostini, and K. C. Kulander, "Enhanced high harmonic generation from an optically prepared excited medium," Phys. Rev. Lett. 94, 113906 (2005).
[CrossRef] [PubMed]

Corkum, P. B.

H. Niikura, D. M. Villeneuve, and P. B. Corkum, "Mapping attosecond electron wave packet motion," Phys. Rev. Lett. 94, 083003 (2005).
[CrossRef] [PubMed]

M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L'Huillier, and P. B. Corkum, "Theory of high-harmonic generation by low-frequency laser fields," Phys. Rev. A 49, 2117-2132 (1994).
[CrossRef] [PubMed]

P. B. Corkum, "Plasma perspective on strong field multiphoton ionization," Phys. Rev. Lett. 71, 1994-1997 (1993).
[CrossRef] [PubMed]

Decker, J. E.

F. A. Ilkov, J. E. Decker, and S. L. Chin, "Ionization of atoms in the tunnelling regime with experimental evidence using Hg atoms," J. Phys. B 25, 4005-4020 (1992).
[CrossRef]

Delone, N. B.

L. V. Keldysh, "Ionization in the field of a strong electromagnetic wave," Sov. Phys. JETP 20, 1307-1319 (1965) L. V. Keldysh, [Zh. Eksp. Teor. Fiz. 47, 1945-1957 (1964)].

DiMauro, L. F.

P. M. Paul, T. O. Clatterbuck, C. Lynga, P. Colosimo, L. F. DiMauro, P. Agostini, and K. C. Kulander, "Enhanced high harmonic generation from an optically prepared excited medium," Phys. Rev. Lett. 94, 113906 (2005).
[CrossRef] [PubMed]

P. Agostini and L. F. DiMauro, "The physics of attosecond light pulses," Rep. Prog. Phys. 67, 813-855 (2004); see corrigendum, Ref. .
[CrossRef]

P. Agostini and L. F. DiMauro, "Corrigendum: the physics of attosecond light pulses," Rep. Prog. Phys. 67, 1563 (2004).
[CrossRef]

Ehlotzky, F.

D. B. Milosevic and F. Ehlotzky, "Scattering and reaction processes in powerful laser fields," Adv. At. Mol. Opt. Phys. 49, 373-532 (2003).

Faisal, F. H.

A. Becker and F. H. M. Faisal, "Intense field many-body S-matrix theory," J. Phys. B 38, R1-R56 (2005).
[CrossRef]

Gohle, C.

C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and T. W. Hänsch, "A frequency comb in the extreme ultraviolet," Nature 436, 234-237 (2005).
[CrossRef] [PubMed]

Grasbon, F.

W. Becker, F. Grasbon, R. Kopold, D. B. Milosevic, G. G. Paulus, and H. Walther, "Above-threshold ionization: from classical features to quantum effects," Adv. At. Mol. Opt. Phys. 48, 35-98 (2002).
[CrossRef]

Hänsch, T. W.

C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and T. W. Hänsch, "A frequency comb in the extreme ultraviolet," Nature 436, 234-237 (2005).
[CrossRef] [PubMed]

Herrmann, M.

C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and T. W. Hänsch, "A frequency comb in the extreme ultraviolet," Nature 436, 234-237 (2005).
[CrossRef] [PubMed]

Holzwarth, R.

C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and T. W. Hänsch, "A frequency comb in the extreme ultraviolet," Nature 436, 234-237 (2005).
[CrossRef] [PubMed]

Ilkov, F. A.

F. A. Ilkov, J. E. Decker, and S. L. Chin, "Ionization of atoms in the tunnelling regime with experimental evidence using Hg atoms," J. Phys. B 25, 4005-4020 (1992).
[CrossRef]

Ivanov, M. Yu.

M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L'Huillier, and P. B. Corkum, "Theory of high-harmonic generation by low-frequency laser fields," Phys. Rev. A 49, 2117-2132 (1994).
[CrossRef] [PubMed]

Jones, R. J.

R. J. Jones, K. D. Moll, M. J. Thorpe, and J. Ye, "Phase-coherent frequency combs in the vacuum ultraviolet via high-harmonic generation inside a femtosecond enhancement cavity," Phys. Rev. Lett. 94, 193201 (2005).
[CrossRef] [PubMed]

Kan, C.

C. Kan, C. E. Capjack, R. Rankin, and N. H. Burnett, "Spectral and temporal structure in high harmonic emission from ionizing atomic gases," Phys. Rev. A 52, R4336-R4339 (1995).
[CrossRef] [PubMed]

Kopold, R.

W. Becker, F. Grasbon, R. Kopold, D. B. Milosevic, G. G. Paulus, and H. Walther, "Above-threshold ionization: from classical features to quantum effects," Adv. At. Mol. Opt. Phys. 48, 35-98 (2002).
[CrossRef]

Krainov, V. P.

L. V. Keldysh, "Ionization in the field of a strong electromagnetic wave," Sov. Phys. JETP 20, 1307-1319 (1965) L. V. Keldysh, [Zh. Eksp. Teor. Fiz. 47, 1945-1957 (1964)].

Krause, J. L.

J. L. Krause, K. J. Schafer, and K. C. Kulander, "High-order harmonic generation from atoms and ions in the high intensity regime," Phys. Rev. Lett. 68, 3535-3538 (1992).
[CrossRef] [PubMed]

K. C. Kulander, K. J. Schafer, and J. L. Krause, "Dynamics of short-pulse excitation, ionization and atomic conversion," in Super-Intense Laser-Atom Physics, B.Piraux, A.L'Huillier, and K.Rzazewski, eds., Vol. 316 of NATO Advanced Science Institutes Series Series B (Plenum, 1993), pp. 95-110.

Krausz, F.

C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and T. W. Hänsch, "A frequency comb in the extreme ultraviolet," Nature 436, 234-237 (2005).
[CrossRef] [PubMed]

T. Brabec and F. Krausz, "Intense few-cycle laser fields: frontiers of nonlinear optics," Rev. Mod. Phys. 72, 545-591 (2000).
[CrossRef]

Kulander, K. C.

P. M. Paul, T. O. Clatterbuck, C. Lynga, P. Colosimo, L. F. DiMauro, P. Agostini, and K. C. Kulander, "Enhanced high harmonic generation from an optically prepared excited medium," Phys. Rev. Lett. 94, 113906 (2005).
[CrossRef] [PubMed]

J. L. Krause, K. J. Schafer, and K. C. Kulander, "High-order harmonic generation from atoms and ions in the high intensity regime," Phys. Rev. Lett. 68, 3535-3538 (1992).
[CrossRef] [PubMed]

K. C. Kulander, K. J. Schafer, and J. L. Krause, "Dynamics of short-pulse excitation, ionization and atomic conversion," in Super-Intense Laser-Atom Physics, B.Piraux, A.L'Huillier, and K.Rzazewski, eds., Vol. 316 of NATO Advanced Science Institutes Series Series B (Plenum, 1993), pp. 95-110.

Lewenstein, M.

P. Salières, A. L'Huillier, Ph. Antoine, and M. Lewenstein, "Study of the spatial and temporal coherence of high-order harmonics," Adv. At. Mol. Opt. Phys. 41, 83-142 (1999).
[CrossRef]

A. Sanpera, J. B. Watson, M. Lewenstein, and K. Burnett, "Harmonic-generation control," Phys. Rev. A 54, 4320-4326 (1996).
[CrossRef] [PubMed]

M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L'Huillier, and P. B. Corkum, "Theory of high-harmonic generation by low-frequency laser fields," Phys. Rev. A 49, 2117-2132 (1994).
[CrossRef] [PubMed]

L'Huillier, A.

P. Salières, A. L'Huillier, Ph. Antoine, and M. Lewenstein, "Study of the spatial and temporal coherence of high-order harmonics," Adv. At. Mol. Opt. Phys. 41, 83-142 (1999).
[CrossRef]

M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L'Huillier, and P. B. Corkum, "Theory of high-harmonic generation by low-frequency laser fields," Phys. Rev. A 49, 2117-2132 (1994).
[CrossRef] [PubMed]

Li, R.

Z. Zeng, R. Li, Y. Cheng, W. Yu, and Z. Xu, "Resonance-enhanced high-order harmonic generation and frequency mixing in two-color laser field," Phys. Scr. 66, 321-325 (2002).
[CrossRef]

Lynga, C.

P. M. Paul, T. O. Clatterbuck, C. Lynga, P. Colosimo, L. F. DiMauro, P. Agostini, and K. C. Kulander, "Enhanced high harmonic generation from an optically prepared excited medium," Phys. Rev. Lett. 94, 113906 (2005).
[CrossRef] [PubMed]

Milosevic, D. B.

D. B. Milosevic and F. Ehlotzky, "Scattering and reaction processes in powerful laser fields," Adv. At. Mol. Opt. Phys. 49, 373-532 (2003).

W. Becker, F. Grasbon, R. Kopold, D. B. Milosevic, G. G. Paulus, and H. Walther, "Above-threshold ionization: from classical features to quantum effects," Adv. At. Mol. Opt. Phys. 48, 35-98 (2002).
[CrossRef]

D. B. Milosevic and B. Piraux, "High-order harmonic generation in a bichromatic elliptically polarized laser field," Phys. Rev. A 54, 1522-1531 (1996).
[CrossRef] [PubMed]

Moll, K. D.

R. J. Jones, K. D. Moll, M. J. Thorpe, and J. Ye, "Phase-coherent frequency combs in the vacuum ultraviolet via high-harmonic generation inside a femtosecond enhancement cavity," Phys. Rev. Lett. 94, 193201 (2005).
[CrossRef] [PubMed]

Niikura, H.

H. Niikura, D. M. Villeneuve, and P. B. Corkum, "Mapping attosecond electron wave packet motion," Phys. Rev. Lett. 94, 083003 (2005).
[CrossRef] [PubMed]

Paul, P. M.

P. M. Paul, T. O. Clatterbuck, C. Lynga, P. Colosimo, L. F. DiMauro, P. Agostini, and K. C. Kulander, "Enhanced high harmonic generation from an optically prepared excited medium," Phys. Rev. Lett. 94, 113906 (2005).
[CrossRef] [PubMed]

Paulus, G. G.

W. Becker, F. Grasbon, R. Kopold, D. B. Milosevic, G. G. Paulus, and H. Walther, "Above-threshold ionization: from classical features to quantum effects," Adv. At. Mol. Opt. Phys. 48, 35-98 (2002).
[CrossRef]

Piraux, B.

D. B. Milosevic and B. Piraux, "High-order harmonic generation in a bichromatic elliptically polarized laser field," Phys. Rev. A 54, 1522-1531 (1996).
[CrossRef] [PubMed]

Rankin, R.

C. Kan, C. E. Capjack, R. Rankin, and N. H. Burnett, "Spectral and temporal structure in high harmonic emission from ionizing atomic gases," Phys. Rev. A 52, R4336-R4339 (1995).
[CrossRef] [PubMed]

Rauschenberger, J.

C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and T. W. Hänsch, "A frequency comb in the extreme ultraviolet," Nature 436, 234-237 (2005).
[CrossRef] [PubMed]

Salières, P.

P. Salières, A. L'Huillier, Ph. Antoine, and M. Lewenstein, "Study of the spatial and temporal coherence of high-order harmonics," Adv. At. Mol. Opt. Phys. 41, 83-142 (1999).
[CrossRef]

Sanpera, A.

J. B. Watson, A. Sanpera, X. Chen, and K. Burnett, "Harmonic generation from a coherent superposition of states," Phys. Rev. A 53, R1962-R1965 (1996).
[CrossRef] [PubMed]

A. Sanpera, J. B. Watson, M. Lewenstein, and K. Burnett, "Harmonic-generation control," Phys. Rev. A 54, 4320-4326 (1996).
[CrossRef] [PubMed]

Schafer, K. J.

J. L. Krause, K. J. Schafer, and K. C. Kulander, "High-order harmonic generation from atoms and ions in the high intensity regime," Phys. Rev. Lett. 68, 3535-3538 (1992).
[CrossRef] [PubMed]

K. C. Kulander, K. J. Schafer, and J. L. Krause, "Dynamics of short-pulse excitation, ionization and atomic conversion," in Super-Intense Laser-Atom Physics, B.Piraux, A.L'Huillier, and K.Rzazewski, eds., Vol. 316 of NATO Advanced Science Institutes Series Series B (Plenum, 1993), pp. 95-110.

Schuessler, H. A.

C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and T. W. Hänsch, "A frequency comb in the extreme ultraviolet," Nature 436, 234-237 (2005).
[CrossRef] [PubMed]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1964), 7.1.17, Fig. 7.2.

Thorpe, M. J.

R. J. Jones, K. D. Moll, M. J. Thorpe, and J. Ye, "Phase-coherent frequency combs in the vacuum ultraviolet via high-harmonic generation inside a femtosecond enhancement cavity," Phys. Rev. Lett. 94, 193201 (2005).
[CrossRef] [PubMed]

Udem, T.

C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and T. W. Hänsch, "A frequency comb in the extreme ultraviolet," Nature 436, 234-237 (2005).
[CrossRef] [PubMed]

Villeneuve, D. M.

H. Niikura, D. M. Villeneuve, and P. B. Corkum, "Mapping attosecond electron wave packet motion," Phys. Rev. Lett. 94, 083003 (2005).
[CrossRef] [PubMed]

Walther, H.

W. Becker, F. Grasbon, R. Kopold, D. B. Milosevic, G. G. Paulus, and H. Walther, "Above-threshold ionization: from classical features to quantum effects," Adv. At. Mol. Opt. Phys. 48, 35-98 (2002).
[CrossRef]

Watson, J. B.

J. B. Watson, A. Sanpera, X. Chen, and K. Burnett, "Harmonic generation from a coherent superposition of states," Phys. Rev. A 53, R1962-R1965 (1996).
[CrossRef] [PubMed]

A. Sanpera, J. B. Watson, M. Lewenstein, and K. Burnett, "Harmonic-generation control," Phys. Rev. A 54, 4320-4326 (1996).
[CrossRef] [PubMed]

Xu, Z.

Z. Zeng, R. Li, Y. Cheng, W. Yu, and Z. Xu, "Resonance-enhanced high-order harmonic generation and frequency mixing in two-color laser field," Phys. Scr. 66, 321-325 (2002).
[CrossRef]

Ye, J.

R. J. Jones, K. D. Moll, M. J. Thorpe, and J. Ye, "Phase-coherent frequency combs in the vacuum ultraviolet via high-harmonic generation inside a femtosecond enhancement cavity," Phys. Rev. Lett. 94, 193201 (2005).
[CrossRef] [PubMed]

Yu, W.

Z. Zeng, R. Li, Y. Cheng, W. Yu, and Z. Xu, "Resonance-enhanced high-order harmonic generation and frequency mixing in two-color laser field," Phys. Scr. 66, 321-325 (2002).
[CrossRef]

Zeng, Z.

Z. Zeng, R. Li, Y. Cheng, W. Yu, and Z. Xu, "Resonance-enhanced high-order harmonic generation and frequency mixing in two-color laser field," Phys. Scr. 66, 321-325 (2002).
[CrossRef]

Adv. At. Mol. Opt. Phys. (3)

P. Salières, A. L'Huillier, Ph. Antoine, and M. Lewenstein, "Study of the spatial and temporal coherence of high-order harmonics," Adv. At. Mol. Opt. Phys. 41, 83-142 (1999).
[CrossRef]

W. Becker, F. Grasbon, R. Kopold, D. B. Milosevic, G. G. Paulus, and H. Walther, "Above-threshold ionization: from classical features to quantum effects," Adv. At. Mol. Opt. Phys. 48, 35-98 (2002).
[CrossRef]

D. B. Milosevic and F. Ehlotzky, "Scattering and reaction processes in powerful laser fields," Adv. At. Mol. Opt. Phys. 49, 373-532 (2003).

J. Phys. B (2)

A. Becker and F. H. M. Faisal, "Intense field many-body S-matrix theory," J. Phys. B 38, R1-R56 (2005).
[CrossRef]

F. A. Ilkov, J. E. Decker, and S. L. Chin, "Ionization of atoms in the tunnelling regime with experimental evidence using Hg atoms," J. Phys. B 25, 4005-4020 (1992).
[CrossRef]

Nature (1)

C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and T. W. Hänsch, "A frequency comb in the extreme ultraviolet," Nature 436, 234-237 (2005).
[CrossRef] [PubMed]

Phys. Rev. A (5)

C. Kan, C. E. Capjack, R. Rankin, and N. H. Burnett, "Spectral and temporal structure in high harmonic emission from ionizing atomic gases," Phys. Rev. A 52, R4336-R4339 (1995).
[CrossRef] [PubMed]

J. B. Watson, A. Sanpera, X. Chen, and K. Burnett, "Harmonic generation from a coherent superposition of states," Phys. Rev. A 53, R1962-R1965 (1996).
[CrossRef] [PubMed]

A. Sanpera, J. B. Watson, M. Lewenstein, and K. Burnett, "Harmonic-generation control," Phys. Rev. A 54, 4320-4326 (1996).
[CrossRef] [PubMed]

M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L'Huillier, and P. B. Corkum, "Theory of high-harmonic generation by low-frequency laser fields," Phys. Rev. A 49, 2117-2132 (1994).
[CrossRef] [PubMed]

D. B. Milosevic and B. Piraux, "High-order harmonic generation in a bichromatic elliptically polarized laser field," Phys. Rev. A 54, 1522-1531 (1996).
[CrossRef] [PubMed]

Phys. Rev. Lett. (5)

P. M. Paul, T. O. Clatterbuck, C. Lynga, P. Colosimo, L. F. DiMauro, P. Agostini, and K. C. Kulander, "Enhanced high harmonic generation from an optically prepared excited medium," Phys. Rev. Lett. 94, 113906 (2005).
[CrossRef] [PubMed]

J. L. Krause, K. J. Schafer, and K. C. Kulander, "High-order harmonic generation from atoms and ions in the high intensity regime," Phys. Rev. Lett. 68, 3535-3538 (1992).
[CrossRef] [PubMed]

P. B. Corkum, "Plasma perspective on strong field multiphoton ionization," Phys. Rev. Lett. 71, 1994-1997 (1993).
[CrossRef] [PubMed]

H. Niikura, D. M. Villeneuve, and P. B. Corkum, "Mapping attosecond electron wave packet motion," Phys. Rev. Lett. 94, 083003 (2005).
[CrossRef] [PubMed]

R. J. Jones, K. D. Moll, M. J. Thorpe, and J. Ye, "Phase-coherent frequency combs in the vacuum ultraviolet via high-harmonic generation inside a femtosecond enhancement cavity," Phys. Rev. Lett. 94, 193201 (2005).
[CrossRef] [PubMed]

Phys. Scr. (1)

Z. Zeng, R. Li, Y. Cheng, W. Yu, and Z. Xu, "Resonance-enhanced high-order harmonic generation and frequency mixing in two-color laser field," Phys. Scr. 66, 321-325 (2002).
[CrossRef]

Rep. Prog. Phys. (2)

P. Agostini and L. F. DiMauro, "The physics of attosecond light pulses," Rep. Prog. Phys. 67, 813-855 (2004); see corrigendum, Ref. .
[CrossRef]

P. Agostini and L. F. DiMauro, "Corrigendum: the physics of attosecond light pulses," Rep. Prog. Phys. 67, 1563 (2004).
[CrossRef]

Rev. Mod. Phys. (1)

T. Brabec and F. Krausz, "Intense few-cycle laser fields: frontiers of nonlinear optics," Rev. Mod. Phys. 72, 545-591 (2000).
[CrossRef]

Sov. Phys. JETP (1)

L. V. Keldysh, "Ionization in the field of a strong electromagnetic wave," Sov. Phys. JETP 20, 1307-1319 (1965) L. V. Keldysh, [Zh. Eksp. Teor. Fiz. 47, 1945-1957 (1964)].

Other (2)

K. C. Kulander, K. J. Schafer, and J. L. Krause, "Dynamics of short-pulse excitation, ionization and atomic conversion," in Super-Intense Laser-Atom Physics, B.Piraux, A.L'Huillier, and K.Rzazewski, eds., Vol. 316 of NATO Advanced Science Institutes Series Series B (Plenum, 1993), pp. 95-110.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1964), 7.1.17, Fig. 7.2.

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Figures (6)

Fig. 1
Fig. 1

Harmonic intensity as a function of the harmonic order for ω = 0.047 a.u. , I = 10 14 W cm 2 , I p = 2 a.u. , and a flat pulse. The solid curves correspond to an initial coherent superposition of ground and excited states with equally weighted populations, and the double-dotted–dashed curve with the filled circles corresponds to the initial presence of an excited 2 s -state alone.

Fig. 2
Fig. 2

Survival probability of the 2 s state as a function of time for two different laser field intensities: I = 10 14 W cm 2 (solid curve) and I = 2 × 10 14 W cm 2 (short-dashed curve). The pulse envelope has a 6 T long sin 2 turn-on, a flat part ( 20 T ) , and a sin 2 turn-off ( 6 T ) . The laser field frequency is the same as in Fig. 1.

Fig. 3
Fig. 3

Comparison of the numerical results of Fig. 1 (for equally weighted populations) with the more realistic results (solid curves), which include the depletion of ground and excited states and the pulse shape (same as in Fig. 2). The double-dotted–dashed curve with the filled circles corresponds to the single- 2 s -state part of the harmonic spectrum (for the flat pulse envelope of Fig. 1), and the long-dashed curve with open triangles corresponds to the resonant part.

Fig. 4
Fig. 4

Same as in Fig. 3 but for I = 2 × 10 14 W cm 2 .

Fig. 5
Fig. 5

Harmonic intensity as a function of the laser field intensity for different values of the harmonic order Ω ω = 3 (squares), 32.91 (diamonds), 39 (triangles), and 74.91 (circles). The isolated symbols and the symbols connected by a dashed curve (for Ω = 3 ω ) correspond to the results obtained taking into account the depletion effect and the pulse shape as in Fig. 2. Other parameters are as in Fig. 1.

Fig. 6
Fig. 6

Same as in Fig. 3 but for I = 8.85 × 10 13 W cm 2 and ω = 0.06 a.u. .

Equations (58)

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a ( Ω ) = Ω 2 D ( Ω ) , D ( Ω ) = 0 T p d t T p d ( t ) exp ( i Ω t ) ,
d ( t ) = Ψ ( t ) r Ψ ( t ) .
[ i t p 2 2 E ( t ) r V ( r ) ] Ψ ( t ) = 0 ,
Ψ ( t 0 ) = j a j u j ( t 0 ) , u j ( t ) = j exp ( i E j t ) , j a j 2 = 1 .
Ψ ( t ) = j a j u j ( t ) + t 0 t d t G ( t , t ) E ( t ) r j a j u j ( t ) .
G ( t , t ) = [ i t p 2 2 E ( t ) r V ( r ) ] 1 δ ( t t ) i d 3 q Ψ q ( t ) Ψ q ( t ) ,
d ( t ) = j , j a j * a j [ u j ( t ) + i t 0 t d t u j ( t ) E ( t ) r d 3 q Ψ q ( t ) Ψ q ( t ) ] r [ u j ( t ) i t 0 t d t d 3 q Ψ q ( t ) Ψ q ( t ) E ( t ) r u j ( t ) ] .
d ( t ) j , j a j * a j j r j exp [ i ( E j E j ) t ] + d ( 1 ) ( t ) ,
d ( 1 ) ( t ) = i j , j a j * a j t 0 t d t d 3 q u j ( t ) r Ψ q ( t ) Ψ q ( t ) E ( t ) r u j ( t ) + c.c. ,
χ q ( t ) = exp [ i A ( t ) r ] q exp { i [ q α ( t ) + U ( t ) + E q t ] } ,
α ( t ) = t 0 t d t A ( t ) , U ( t ) = 1 2 t 0 t d t A 2 ( t ) , E q = q 2 2 .
d ( 1 ) ( t ) 2 Re { i j , j a j * a j exp [ i ( E j E j ) t ] t 0 t d t d 3 q j r q + A ( t ) q + A ( t ) E ( t ) r j exp [ i S j ( q ; t , t ) ] } ,
S j ( q ; t , t ) = t t d τ { [ q + A ( τ ) ] 2 2 E j }
d ( t ) = j , j a j * a j j r j exp [ i ( E j E j ) t ] 2 Re { i j , j a j * a j exp [ i ( E j E j ) t ] × t 0 t d t d 3 qd j * [ q + A ( t ) ] E ( t ) d j [ q + A ( t ) ] exp [ i S j ( q ; t , t ) ] } .
d ( t ) = i t 0 t d t d 3 qd * [ q + A ( t ) ] E ( t ) d [ q + A ( t ) ] exp [ i S ( q ; t , t ) ] + c.c. ,
S ( q ; t , t ) = t t d τ { [ q + A ( τ ) ] 2 2 + I p } .
a j ( t ) 2 = 1 t 0 t d t w ( t ) ,
d ( t ) = i j , j a j * ( t ) exp ( i E j t ) 0 t t 0 d τ ( 2 π ϵ + i τ ) 3 2 d j * [ q s + A ( t ) ] × exp { i t τ t d t [ q s + A ( t ) ] 2 2 } E ( t τ ) d j [ q s + A ( t τ ) ] a j ( t τ ) exp [ i E j ( t τ ) ] + c.c. ,
q s ( t , τ ) = 1 τ t τ t d t A ( t ) .
d ( t ) = 4 π 3 2 j , j a j ( t ) 0 t t 0 d τ τ 3 2 ( sin S j j + cos S j j ) d j * [ q s + A ( t ) ] E ( t τ ) d j [ q s + A ( t τ ) ] a j ( t τ ) ,
S j j S j j ( t , τ ) = E j t E j ( t τ ) 1 2 t τ t d t [ q s + A ( t ) ] 2 .
D ( Ω ) = a 1 2 ( t 0 ) D 11 ( Ω ) + a 2 2 ( t 0 ) D 22 ( Ω ) + a 1 ( t 0 ) a 2 ( t 0 ) [ D 21 ( Ω ) + D 12 ( Ω ) ] ,
D j j ( Ω ) = 0 T p d t T p exp ( i Ω t ) [ d j j ( t ) + d j j * ( t ) ] ,
d j j ( t ) = i 0 t t 0 d τ ( 2 π ϵ + i τ ) 3 2 d j * [ q s + A ( t ) ] E ( t τ ) d j [ q s + A ( t τ ) ] exp [ i S j j ( t , τ ) ] ,
Ω = Δ ω ± ( 2 k + 1 ) ω , k = 0 , 1 , 2 , .
Φ 1 s ( p ) = 2 3 2 π ( 2 I p ) 5 4 ( p 2 + 2 I p ) 2 , Φ 2 s ( p ) = ( 2 I p ) 5 4 π p 2 I p 2 ( p 2 + I p 2 ) 3 ,
p r 1 s = i 2 7 2 ( 2 I p ) 5 4 π p ( p 2 + 2 I p ) 3 ,
p r 2 s = 4 i ( 2 I p ) 5 4 π ( p 2 I p ) p ( p 2 + I p 2 ) 4 .
E ( t ; φ ) = { E 0 cos ( ω t + φ ) if t [ 0 , n p T ] 0 otherwise } ,
w n = ( 3 2 π ) 1 2 2 E n C n * l * 2 [ 2 F 0 E ( 1 + γ 2 ) 1 2 ] 2 n * 3 2 A 0 ( ω , γ ) exp [ 2 F 0 3 E g ( γ ) ] ,
A 0 ( ω , γ ) = 4 ( 3 π ) 1 2 γ 2 1 + γ 2 k ν exp [ ( k ν ) α ] F { [ ( k ν ) β ] 1 2 } ,
ν = E n ω ( 1 + 1 2 γ 2 ) = E j + U p ω , F 0 = ( 2 E n ) 3 2 ,
β ( γ ) = 2 γ ( 1 + γ 2 ) 1 2 , α ( γ ) = 2 [ sinh 1 γ γ ( 1 + γ 2 ) 1 2 ] ,
g ( γ ) = 3 2 γ [ ( 1 + 1 2 γ 2 ) sinh 1 γ ( 1 + γ 2 ) 1 2 2 γ ] ,
F ( x ) = exp ( x 2 ) 0 x d y exp ( y 2 ) = π 1 2 Im w ( x ) 2 .
C n * l * 2 = 2 2 n * n * Γ ( n * + l * + 1 ) Γ ( n * l * ) ,
d j j ( t + n T 2 ) = ( 1 ) n d j j ( t ) exp [ i ( E j E j ) n T 2 ] .
d j j ( t ; φ ) = d j j ( t + φ ω ) exp [ i ( E j E j ) φ ω ] .
d j j ( t + n T 2 ; φ ) = ( 1 ) n d j j ( t + φ ω ) exp [ i ( E j E j ) ( φ n π ) ω ] ,
D ( Ω ; φ ) = 0 T p d t T p exp ( i Ω t ) 2 Re { a 1 2 ( t 0 ) d 11 ( t + φ ω ) + a 2 2 ( t 0 ) d 22 ( t + φ ω ) + a 1 ( t 0 ) a 2 ( t 0 ) [ d 21 ( t + φ ω ) exp ( i Δ ω φ ω ) + d 12 ( t + φ ω ) exp ( i Δ ω φ ω ) ] } ,
Re d j j ( t + φ ω + n T 2 ) = ( 1 ) n Re d j j ( t + φ ω ) ( j = 1 , 2 ) ,
Re [ d 21 ( t + φ ω + n T 2 ) exp ( i Δ ω φ ω ) + d 12 ( t + φ ω + n T 2 ) exp ( i Δ ω φ ω ) ] = ( 1 ) n { cos Δ ω ( n π φ ) ω Re [ d 21 ( t + φ ω ) + d 12 ( t + φ ω ) ] sin Δ ω ( n π φ ) ω Im [ d 21 ( t + φ ω ) d 12 ( t + φ ω ) ] } .
D ( Ω ) = 1 T p n = 0 2 n p 1 exp ( i n Ω T 2 ) 0 T 2 d t exp ( i Ω t ) 2 Re { ( 1 ) n [ a 1 2 ( t 0 ) d 11 ( t ) + a 2 2 ( t 0 ) d 22 ( t ) ] + a 1 ( t 0 ) a 2 ( t 0 ) [ d 21 ( t ) + d 12 ( t ) ] } .
D ( 2 k ω ) = a 1 ( t 0 ) a 2 ( t 0 ) 2 T 0 T 2 d t exp ( i Ω t ) 2 Re [ d 21 ( t ) + d 12 ( t ) ] ,
D [ ( 2 k + 1 ) ω ] = 2 T 0 T 2 d t exp ( i Ω t ) 2 Re [ a 1 2 ( t 0 ) d 11 ( t ) + a 2 2 ( t 0 ) d 22 ( t ) ] .
D ( Ω ) = j , j a j a j D j j ( Ω ) ,
D j j ( Ω ) = 0 T p d t T p exp ( i Ω t ) 2 Re d j j ( t ) ,
R j j ( Ω ) = 2 T 0 T 2 d t exp ( i Ω t ) 2 Re d j j ( t ) ,
I j j ( Ω ) = 2 T 0 T 2 d t exp ( i Ω t ) 2 Im d j j ( t ) .
D j j ( Ω ) = 1 2 n p n = 0 2 n p 1 exp [ i n π ( 1 + Ω ω ) ] 2 T 0 T 2 d t exp ( i Ω t ) 2 Re d j j ( t ) exp [ i ( E j E j ) n π ω ] = 1 2 n p n = 0 2 n p 1 exp [ i n π ( 1 + Ω ω ) ] { cos [ ( E j E j ) n π ω ] R j j ( Ω ) sin [ ( E j E j ) n π ω ] I j j ( Ω ) } .
lim n p 1 2 n p n = 0 2 n p 1 exp ( i n π { 1 + [ Ω ± ( E j E j ) ] ω } ) = δ [ Ω ± ( E j E j ) ] Ω + 1 , 2 k ( k = 0 , ± 1 , ± 2 , ) ,
D j j ( Ω ) = 1 2 [ δ Ω , ( 2 k 1 ) ω E j + E j + δ Ω , ( 2 k 1 ) ω + E j E j ] R j j ( Ω ) + i 2 [ δ Ω , ( 2 k 1 ) ω E j + E j + δ Ω , ( 2 k 1 ) ω + E j E j ] I j j ( Ω ) ,
D j j ( Ω ) = δ Ω , ( 2 k 1 ) ω R j j ( Ω ) ( j = 1 , 2 ) ,
D 21 ( Ω ) + D 12 ( Ω ) = 1 2 [ δ Ω , ( 2 k 1 ) ω Δ ω + δ Ω , ( 2 k 1 ) ω + Δ ω ] [ R 21 ( Ω ) + R 12 ( Ω ) ] + i 2 [ δ Ω , ( 2 k 1 ) ω Δ ω δ Ω , ( 2 k 1 ) ω + Δ ω ] [ I 21 ( Ω ) I 12 ( Ω ) ] .
R j j ( Ω ; φ ) = 2 T 0 T 2 d t exp ( i Ω t ) 2 Re d j j ( t + φ ω ) exp [ i ( E j E j ) φ ω ] = 2 T 0 T 2 d t { 2 Re d j j ( t + φ ω ) cos [ ( E j E j ) φ ω ] 2 Im d j j ( t + φ ω ) sin [ ( E j E j ) φ ω ] } ,
I j j ( Ω ; φ ) = 2 T 0 T 2 d t { 2 Re d j j ( t + φ ω ) sin [ ( E j E j ) φ ω ] + 2 Im d j j ( t + φ ω ) cos [ ( E j E j ) φ ω ] } .
Re d j j ( t + φ ω ) = Re d j j ( t + φ ω T 2 ) cos [ ( E j E j ) T 2 ] Im d j j ( t + φ ω T 2 ) sin [ ( E j E j ) T 2 ] ,
D ( Ω ; φ + n π ) = ( 1 ) n D ( Ω ; φ ) ( 0 φ < π ) .

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