Abstract

The physics of laser-mater interactions beyond the perturbative limit configures the field of extreme non-linear optics. Although most experiments have been done in the near infrared (λ≤1µm), the situation is changing nowadays with the development of sources at longer wavelengths (<5µm), opening new perspectives in the synthesis of shorter XUV attosecond pulses and higher frequencies. The theory of intense-field interactions is based either on the exact numerical integration of the time-dependent Schrödinger equation or in the development of models, mostly based on the strong-field approximation. Recent studies in the short-wave infrared show a divergence between the predictions of these models and the exact results. In this paper we will show that this discrepancy reveals the incompleteness of our present understanding of high-order harmonic generation. We discuss the physical grounds, provide a theoretical framework beyond the standard approximations and develop a compact approach that accounts for the correct scaling of the harmonic yield.

© 2009 Optical Society of America

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2008

P. Colosimo, G. Doumy, C. I. Blaga, J. Wheeler, C. Hauri, F. Catoire, J. Tate, R. Chirla, A. M. March, G. G. Paulus, H. G. Muller, P. Agostini, and L. F. Di Mauro, "Scaling strong-field interactions towards the classical limit," Nature Phys. 4, 386-389 (2008).
[CrossRef]

2007

2006

O. Smirnova, M. Spanner, and M. Ivanov, "Coulomb and polarization effects in sub-cycle dynamics of strong field ionization," J. Phys. B: At. Mol. Opt. Phys. 39, S307-321 (2006).
[CrossRef]

G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villores, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, "Isolated Single-Cycle Attosecond Pulses," Science 314, 443-446 (2006).
[CrossRef] [PubMed]

1997

W. Becker, A. Lohr, M. Kleber, and M. Lewenstein, "A unified theory of high-harmonic generation: application to polarization properties of the harmonics," Phys. Rev. A 56, 645-656 (1997).
[CrossRef]

V. P. Krainov, "Ionization rates and energy and angular distributions at the barrier-suppression ionization of complex atoms and atomic ions," J. Opt. Soc. Am. B 14, 425431 (1997).
[CrossRef]

1995

M. Lewenstein, P. Sali`eres, and A. L’Huiller, "Phase of the atomic polarization in high-order harmonic generation," Phys. Rev. A 52, 4747-4754 (1995).
[CrossRef] [PubMed]

1994

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

1993

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

1990

1980

H. R. Reiss, "Effect of an intense electromagnetic field on a weakly bound system," Phys. Rev. A 22,1786-1813 (1980).
[CrossRef]

1973

F. H. M. Faisal, "Multiple Absorption of Laser Photons by Atoms," J. Phys. B 6, L89-L92 (1973).
[CrossRef]

1964

L. V. Keldysh, "Ionization in the field of a strong electromagnetic wave," Zh. Eksp. Teor. Fiz. 47, 1945-1956 (1964) [ Sov. Phys. JETP 20 1307-1314 (1965)].

Agostini, P.

P. Colosimo, G. Doumy, C. I. Blaga, J. Wheeler, C. Hauri, F. Catoire, J. Tate, R. Chirla, A. M. March, G. G. Paulus, H. G. Muller, P. Agostini, and L. F. Di Mauro, "Scaling strong-field interactions towards the classical limit," Nature Phys. 4, 386-389 (2008).
[CrossRef]

Altucci, C.

G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villores, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, "Isolated Single-Cycle Attosecond Pulses," Science 314, 443-446 (2006).
[CrossRef] [PubMed]

Avaldi, L.

G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villores, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, "Isolated Single-Cycle Attosecond Pulses," Science 314, 443-446 (2006).
[CrossRef] [PubMed]

Balcou, Ph.

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

Becker, W.

W. Becker, A. Lohr, M. Kleber, and M. Lewenstein, "A unified theory of high-harmonic generation: application to polarization properties of the harmonics," Phys. Rev. A 56, 645-656 (1997).
[CrossRef]

Benedetti, E.

G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villores, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, "Isolated Single-Cycle Attosecond Pulses," Science 314, 443-446 (2006).
[CrossRef] [PubMed]

Blaga, C. I.

P. Colosimo, G. Doumy, C. I. Blaga, J. Wheeler, C. Hauri, F. Catoire, J. Tate, R. Chirla, A. M. March, G. G. Paulus, H. G. Muller, P. Agostini, and L. F. Di Mauro, "Scaling strong-field interactions towards the classical limit," Nature Phys. 4, 386-389 (2008).
[CrossRef]

Calegari, F.

G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villores, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, "Isolated Single-Cycle Attosecond Pulses," Science 314, 443-446 (2006).
[CrossRef] [PubMed]

Catoire, F.

P. Colosimo, G. Doumy, C. I. Blaga, J. Wheeler, C. Hauri, F. Catoire, J. Tate, R. Chirla, A. M. March, G. G. Paulus, H. G. Muller, P. Agostini, and L. F. Di Mauro, "Scaling strong-field interactions towards the classical limit," Nature Phys. 4, 386-389 (2008).
[CrossRef]

Chirla, R.

P. Colosimo, G. Doumy, C. I. Blaga, J. Wheeler, C. Hauri, F. Catoire, J. Tate, R. Chirla, A. M. March, G. G. Paulus, H. G. Muller, P. Agostini, and L. F. Di Mauro, "Scaling strong-field interactions towards the classical limit," Nature Phys. 4, 386-389 (2008).
[CrossRef]

Colosimo, P.

P. Colosimo, G. Doumy, C. I. Blaga, J. Wheeler, C. Hauri, F. Catoire, J. Tate, R. Chirla, A. M. March, G. G. Paulus, H. G. Muller, P. Agostini, and L. F. Di Mauro, "Scaling strong-field interactions towards the classical limit," Nature Phys. 4, 386-389 (2008).
[CrossRef]

Corkum, P. B.

M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huiller, 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]

De Silvestri, S.

G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villores, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, "Isolated Single-Cycle Attosecond Pulses," Science 314, 443-446 (2006).
[CrossRef] [PubMed]

Di Mauro, L. F.

P. Colosimo, G. Doumy, C. I. Blaga, J. Wheeler, C. Hauri, F. Catoire, J. Tate, R. Chirla, A. M. March, G. G. Paulus, H. G. Muller, P. Agostini, and L. F. Di Mauro, "Scaling strong-field interactions towards the classical limit," Nature Phys. 4, 386-389 (2008).
[CrossRef]

Doumy, G.

P. Colosimo, G. Doumy, C. I. Blaga, J. Wheeler, C. Hauri, F. Catoire, J. Tate, R. Chirla, A. M. March, G. G. Paulus, H. G. Muller, P. Agostini, and L. F. Di Mauro, "Scaling strong-field interactions towards the classical limit," Nature Phys. 4, 386-389 (2008).
[CrossRef]

Faisal, F. H. M.

F. H. M. Faisal, "Multiple Absorption of Laser Photons by Atoms," J. Phys. B 6, L89-L92 (1973).
[CrossRef]

Flammini, R.

G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villores, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, "Isolated Single-Cycle Attosecond Pulses," Science 314, 443-446 (2006).
[CrossRef] [PubMed]

Hahn, Y.

Hauri, C.

P. Colosimo, G. Doumy, C. I. Blaga, J. Wheeler, C. Hauri, F. Catoire, J. Tate, R. Chirla, A. M. March, G. G. Paulus, H. G. Muller, P. Agostini, and L. F. Di Mauro, "Scaling strong-field interactions towards the classical limit," Nature Phys. 4, 386-389 (2008).
[CrossRef]

Ivanov, M.

O. Smirnova, M. Spanner, and M. Ivanov, "Coulomb and polarization effects in sub-cycle dynamics of strong field ionization," J. Phys. B: At. Mol. Opt. Phys. 39, S307-321 (2006).
[CrossRef]

Ivanov, M. Yu.

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

Janjusevic, M.

Keldysh, L. V.

L. V. Keldysh, "Ionization in the field of a strong electromagnetic wave," Zh. Eksp. Teor. Fiz. 47, 1945-1956 (1964) [ Sov. Phys. JETP 20 1307-1314 (1965)].

Kleber, M.

W. Becker, A. Lohr, M. Kleber, and M. Lewenstein, "A unified theory of high-harmonic generation: application to polarization properties of the harmonics," Phys. Rev. A 56, 645-656 (1997).
[CrossRef]

Krainov, V. P.

V. P. Krainov, "Ionization rates and energy and angular distributions at the barrier-suppression ionization of complex atoms and atomic ions," J. Opt. Soc. Am. B 14, 425431 (1997).
[CrossRef]

L’Huiller, A.

M. Lewenstein, P. Sali`eres, and A. L’Huiller, "Phase of the atomic polarization in high-order harmonic generation," Phys. Rev. A 52, 4747-4754 (1995).
[CrossRef] [PubMed]

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

Lewenstein, M.

W. Becker, A. Lohr, M. Kleber, and M. Lewenstein, "A unified theory of high-harmonic generation: application to polarization properties of the harmonics," Phys. Rev. A 56, 645-656 (1997).
[CrossRef]

M. Lewenstein, P. Sali`eres, and A. L’Huiller, "Phase of the atomic polarization in high-order harmonic generation," Phys. Rev. A 52, 4747-4754 (1995).
[CrossRef] [PubMed]

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

Lohr, A.

W. Becker, A. Lohr, M. Kleber, and M. Lewenstein, "A unified theory of high-harmonic generation: application to polarization properties of the harmonics," Phys. Rev. A 56, 645-656 (1997).
[CrossRef]

March, A. M.

P. Colosimo, G. Doumy, C. I. Blaga, J. Wheeler, C. Hauri, F. Catoire, J. Tate, R. Chirla, A. M. March, G. G. Paulus, H. G. Muller, P. Agostini, and L. F. Di Mauro, "Scaling strong-field interactions towards the classical limit," Nature Phys. 4, 386-389 (2008).
[CrossRef]

Muller, H. G.

P. Colosimo, G. Doumy, C. I. Blaga, J. Wheeler, C. Hauri, F. Catoire, J. Tate, R. Chirla, A. M. March, G. G. Paulus, H. G. Muller, P. Agostini, and L. F. Di Mauro, "Scaling strong-field interactions towards the classical limit," Nature Phys. 4, 386-389 (2008).
[CrossRef]

Nisoli, M.

G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villores, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, "Isolated Single-Cycle Attosecond Pulses," Science 314, 443-446 (2006).
[CrossRef] [PubMed]

Paulus, G. G.

P. Colosimo, G. Doumy, C. I. Blaga, J. Wheeler, C. Hauri, F. Catoire, J. Tate, R. Chirla, A. M. March, G. G. Paulus, H. G. Muller, P. Agostini, and L. F. Di Mauro, "Scaling strong-field interactions towards the classical limit," Nature Phys. 4, 386-389 (2008).
[CrossRef]

Plaja, L.

Poletto, L.

G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villores, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, "Isolated Single-Cycle Attosecond Pulses," Science 314, 443-446 (2006).
[CrossRef] [PubMed]

Reiss, H. R.

H. R. Reiss, "Effect of an intense electromagnetic field on a weakly bound system," Phys. Rev. A 22,1786-1813 (1980).
[CrossRef]

Sali`eres, P.

M. Lewenstein, P. Sali`eres, and A. L’Huiller, "Phase of the atomic polarization in high-order harmonic generation," Phys. Rev. A 52, 4747-4754 (1995).
[CrossRef] [PubMed]

Sansone, G.

G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villores, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, "Isolated Single-Cycle Attosecond Pulses," Science 314, 443-446 (2006).
[CrossRef] [PubMed]

Smirnova, O.

O. Smirnova, M. Spanner, and M. Ivanov, "Coulomb and polarization effects in sub-cycle dynamics of strong field ionization," J. Phys. B: At. Mol. Opt. Phys. 39, S307-321 (2006).
[CrossRef]

Spanner, M.

O. Smirnova, M. Spanner, and M. Ivanov, "Coulomb and polarization effects in sub-cycle dynamics of strong field ionization," J. Phys. B: At. Mol. Opt. Phys. 39, S307-321 (2006).
[CrossRef]

Stagira, S.

G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villores, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, "Isolated Single-Cycle Attosecond Pulses," Science 314, 443-446 (2006).
[CrossRef] [PubMed]

Tate, J.

P. Colosimo, G. Doumy, C. I. Blaga, J. Wheeler, C. Hauri, F. Catoire, J. Tate, R. Chirla, A. M. March, G. G. Paulus, H. G. Muller, P. Agostini, and L. F. Di Mauro, "Scaling strong-field interactions towards the classical limit," Nature Phys. 4, 386-389 (2008).
[CrossRef]

Velotta, R.

G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villores, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, "Isolated Single-Cycle Attosecond Pulses," Science 314, 443-446 (2006).
[CrossRef] [PubMed]

Villores, P.

G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villores, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, "Isolated Single-Cycle Attosecond Pulses," Science 314, 443-446 (2006).
[CrossRef] [PubMed]

Vozzi, C.

G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villores, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, "Isolated Single-Cycle Attosecond Pulses," Science 314, 443-446 (2006).
[CrossRef] [PubMed]

Wheeler, J.

P. Colosimo, G. Doumy, C. I. Blaga, J. Wheeler, C. Hauri, F. Catoire, J. Tate, R. Chirla, A. M. March, G. G. Paulus, H. G. Muller, P. Agostini, and L. F. Di Mauro, "Scaling strong-field interactions towards the classical limit," Nature Phys. 4, 386-389 (2008).
[CrossRef]

J. Opt. Soc. Am. B

V. P. Krainov, "Ionization rates and energy and angular distributions at the barrier-suppression ionization of complex atoms and atomic ions," J. Opt. Soc. Am. B 14, 425431 (1997).
[CrossRef]

P. Krsti’c and M. H. Mittleman, "S-matrix theory of multiphoton ionization," J. Opt. Soc. Am. B 7, 587-591 (1990).
[CrossRef]

M. Janjusevic and Y. Hahn, "Testing of laser-dressed atomic states," J. Opt. Soc. Am. B 7, 592-597 (1990).
[CrossRef]

J. Phys. B

F. H. M. Faisal, "Multiple Absorption of Laser Photons by Atoms," J. Phys. B 6, L89-L92 (1973).
[CrossRef]

J. Phys. B: At. Mol. Opt. Phys.

O. Smirnova, M. Spanner, and M. Ivanov, "Coulomb and polarization effects in sub-cycle dynamics of strong field ionization," J. Phys. B: At. Mol. Opt. Phys. 39, S307-321 (2006).
[CrossRef]

Nature Phys.

P. Colosimo, G. Doumy, C. I. Blaga, J. Wheeler, C. Hauri, F. Catoire, J. Tate, R. Chirla, A. M. March, G. G. Paulus, H. G. Muller, P. Agostini, and L. F. Di Mauro, "Scaling strong-field interactions towards the classical limit," Nature Phys. 4, 386-389 (2008).
[CrossRef]

Opt. Express

Phys. Rev. A

H. R. Reiss, "Effect of an intense electromagnetic field on a weakly bound system," Phys. Rev. A 22,1786-1813 (1980).
[CrossRef]

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Figures (3)

Fig. 1.
Fig. 1.

Harmonic spectra resulting from the exact integration of the 3D TDSE (blue), S-Matrix SFA models in E·r and p·A gauges (red and orange) and our model beyond the SFA (green). Part (a) shows the location of the spectral windows used to compute the harmonic yield in [9, 10, 11] (enclosed between red dashed lines) and in this paper (enclosed between black dashed lines). Part (b) shows the results of the exact 3D TDSE and the three models in the former case (red dashed window). The computation corresponds to a 6-cycle pulse of intensity I=1.58×1014 and wavelength 1600 nm (see Appendix A for more details).

Fig. 2.
Fig. 2.

Detail of the higher frequency part of the harmonic spectrum for a 6 cycles driving field of intensity 1.58×1014 W/cm2 and wavelengths 800 nm (a) and 1600 nm (b). Part (b) corresponds to the same case shown in Fig. 1. As before, 3D TDSE results are plotted in blue, S-Matrix SFA in red (length gauge) and orange (velocity gauge), and the results of the present formulation are plotted in green. The limits of the window used to compute the yields of table 2 are shown in black dashed lines.

Fig. 3.
Fig. 3.

Harmonic yields computed using the window at the end of the plateau for a 6-cycle pulse of I=1.58×1014 and for wavelengths ranging from 800 to 1600 nm. (a) Comparison between results from the 3D TDSE (blue) and the present model (green), (b) comparison of the 3D TDSE (blue) with the S-Matrix SFA in length (red) and velocity (orange) gauges.

Tables (2)

Tables Icon

Table 1. Exponents n of the scalings λ n of the harmonic yields computed from the exact integration of the 3D TDSE at an intensity I=1.58×1014 using a fixed energy window (20–50 eV, enclosed between red lines in Fig. 1a), using a 10.2 eV window below the cutoff frequency ωc (enclosed between black lines in Fig. 1a) and of the harmonic power at the nearest harmonic to the cut-off frequency. Our result for the case of the fixed window and the field of 9 cycles agrees with the one reported in [10].

Tables Icon

Table 2. Scaling exponents of λ n extracted from the harmonic yields computed at the final part of the plateau for the three models considered in this paper and the 3D TDSE (also shown in the third column of table 1). The cases labelled as 6-cycles correspond to the results shown in figure 3.

Equations (37)

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iψ(t)=Ga+(t,t0)ψ(t0)+1h¯t0tdtG+(t,t)Vi(t)Ga+(t,t0)ψ(t0)
iQ̂ψ(t)=Ga+ (t,t0) ψ(t0)
+1h¯ t0tdtQ̂G+(t,t)Q̂Vi(t)Q̂Ga+(t,t0)ψ(t0)
iP̂ψ(t)= 1h¯ t0t dtP̂G+(t,t)P̂Vi(t)Q̂Ga+(t,t0)ψ(t0)
a(t)=ψ(t)âψ(t)ψ(t)Q̂âP̂ψ(t)+c.c.=ab(t)+ad(t)+c.c.
ab(t)=1h¯ t0tdt1 ψ(t0)Ga(t0,t)âP̂G+(t,t1)P̂Vi(t1)Q̂Ga+(t1,t0)ψ(t0)
ad(t)=1h¯2 t0tdt2ψ(t0)Ga(t0,t2)Q̂Vi(t2)Q̂G(t2,t)Q̂â×
t0t2dt1P̂G+(t,t1)P̂Vi(t1)Q̂Ga+(t1,t0)ψ(t0)
Q̂Vi(t2)Q̂=Q̂[H(t2)Ha]Q̂p̂22m+ΔsÎHa
Q̂G+(t,t2)Q̂exp [(ih¯)(ε0+Δs)(tt2)]
P̂dkkk
P̂Vi(t1)Q̂dkVi(k,t1)kkQ̂
P̂G+(t,t1)P̂=iCFrnexp[(ih¯)t1tdτP̂H(τ)P̂]
P̂H(τ)P̂dkε(k,τ)kk
a(t)=dk[ab(k,t)+ad(k,t)]+c.c.
ab(k,t)=ih¯CFt0tdt1eiε0(tt1)h¯ei1h¯t1tε(k,τ)dτϕ0âkVi(k,t1)krnϕ0
ad(k,t)=[1+k22mε0Δs]ab(k,t)
a(t)=dkh¯2k22mε0Δsab(k,th¯Δs)+c.c.
ddtab(k,t)=ih¯[ε0ε(k,t)]ab(k,t)ih¯CFϕ0âkVi(k,t)krnϕ0
E(t)=E0(t)sinω0t
Pq=4q23c3 aq2
δ ts(3π2ω0) ε0/3.17Up
Vi(t)=(1δts) tδtst[(qh¯mc)A(τ)kz+(q22mc2)A2(τ)] d τ
ΔsUp+2h¯ω0δtsUpmkzsinω0δts+Up2ω0δtssin2ω0δts
[h¯kz+2mUpcos(ω0t+ϕ)]22m=[ε0+Upcos2(ω0t+ϕ)] 3
h¯2kz22m+2h¯kzUpmcos(ω0t+ϕ)+23Upcos2(ω0t+ϕ)+ε03=0
kz=2h¯mUpsinω0δtsω0δts[1116(1+ε0Up+sin2ω0δts2ω0δts)(sinω0δtsω0δts)2]
ab(t)=1h¯ t0tdt1 dkψ(t0)Ga(t0,t)âkkP̂G+(t,t1)P̂P̂Vi(t1)Q̂Ga+(t1,t0)ψ(t0)
ad(t)=1h¯2 tδtstdt2ψ(t0)Ga(t0,t2)Q̂Vi(t2)Q̂Q̂G(t2,t)Q̂â×
dkkk t0t2dt1P̂G+(t,t1)P̂P̂Vi(t1)Q̂Ga+(t1,t0)ψ(t0)
ad(t)=1h¯2 CF tδtstdt2ϕ0eiε0(t2t0)h¯ei(ε0+Δs)(tt2)h¯[p22m+ΔsHa]â×
dkk t0t2dt1eit1tdτε(k,τ)Vi(k,t1)eiε0(t1t0)h¯krnψ(t0)
ad(t)=dk1h¯2CFtδtstdt2ei(ε0+Δs)(tt2)h¯[k22m+Δsε0]×
t0t2dt1eiε0(t2t1)h¯eit1tdτε(k,τ)ϕ0âkVi(k,t1)krnψ(t0)
ad(t)=ih¯ dk tδtstdt2eiΔs(tt2)h¯[k22m+Δsεs]eiε0(tt2)h¯eit22dτε(k,τ)ab(k,t2)
ad(t)ih¯ dk[k22m+Δsε0]ab(k,t)tδtstdt2eiΔs(tt2)h¯
ad(t)dk[1+k22mε0Δs]ab(k,t)

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