Abstract

We present a perturbative analysis of Floquet eigenstates in the context of two delayed laser processes (STIRAP) in three level systems. We show the efficiency of a systematic perturbative development which can be applied as long as no non-linear resonances occur.

© 1999 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. U. Gaubatz, P. Rudecki, S. Schiemann, and K. Bergmann, “Population transfer between molecular vibrational levels by stimulated Raman scattering with partially overlapping laserfields. A new concept and experimental results,” J. Chem. Phys. 92, 5363 (1990).
    [Crossref]
  2. J. Martin, B. W. Shore, and K. Bergmann, “Coherent population transfer in multilevel systems with magnetic sublevels. II. Algebraic analysis,” Phys. Rev. A 52, 583 (1995).
    [Crossref] [PubMed]
  3. S. Guérin and H. R. Jauslin, “Two-laser multiphoton adiabatic passage in the frame of the Floquet theory. Applications to (1+1) and (2+1) STIRAP,” Eur. Phys. J. D 2, 99 (1998).
  4. L. P. Yatsenko, S. Guérin, T. Halfmann, K. Böhmer, B. W. Shore, and K. Bergmann, “Stimulated hyper-Raman adiabatic passage. I. The basic problem and examples,” Phys. Rev. A 58, 4683 (1998).
    [Crossref]
  5. S. Guérin, L. P. Yatsenko, T. Halfmann, B. W. Shore, and K. Bergmann, “Stimulated hyper-Raman adiabatic passage. II. Static compensation of dynamic Stark shifts,”Phys. Rev. A 58, 4691 (1998).
    [Crossref]
  6. N. V. Vitanov and S. Stenholm, “Analytic properties and effective two-level problems in stimulated Raman adiabatic passage,” Phys. Rev. A 55, 648 (1997).
    [Crossref]
  7. S.-I. Chu, “Generalized Floquet theoretical approaches to intense-field multiphoton and nonlinear optical processes,” Adv. Chem. Phys. 73, 739 (1987).
    [Crossref]
  8. S. Guérin, F. Monti, J. M. Dupont, and H. R. Jauslin, “On the relation between cavity-dressed states, Floquet states,RWA and semiclassical models,” J. Phys. A 30, 7193 (1997).
    [Crossref]
  9. M. Combescure, “The quantum stability problem for time-periodic perturbations of the harmonic oscillator”, Ann. Inst. H. Poincaré 47, 63 (1987).
  10. P. Blekher, H. R. Jauslin, and J. L. Lebowitz, “Floquet spectrum for two-level systems in quasiperiodic time-dependent fields,” J. Stat. Phys. 68271 (1992).
    [Crossref]
  11. W. Scherer, “Superconvergent perturbative method in quantum mechanics,” Phys. Rev. Lett. 74, 1495 (1995).
    [Crossref] [PubMed]
  12. T. P. Grozdanov and M. J. Raković, “Quantum system driven by rapidly varying periodic perturbation,” Phys. Rev. A 38, 1739 (1988).
    [Crossref] [PubMed]
  13. R. G. Unanyan, S. Guérin, B. W. Shore, and K. Bergmann (unpublished).
  14. M. V. Berry, “Histories of adiabatic quantum transitions,” Proc. R. Soc. Lond. A 429, 61 (1990).
    [Crossref]
  15. A. Joye and C.-E. Pfister, “Superadiabatic evolution and adiabatic transition probability between two nondegenerate levels isolated in the spectrum,” J. Math. Phys. 34, 454 (1993).
    [Crossref]
  16. M. Elk, “Adiabatic transition histories of population transfer in the Λ system,” Phys. Rev. A 52, 4017 (1995).
    [Crossref] [PubMed]
  17. K. Drese and M. Holthaus, “Perturbative and nonperturbative processes in adiabatic population transfer,” Eur. Phys. J. D,  3, 73 (1998)
    [Crossref]
  18. B. W. Shore, The Theory of Coherent Atomic Excitation II. Multi-level Atoms and Incoherence (Wiley, New York, 1990), Chap. 18.7, pp. 1165–66.

1998 (4)

S. Guérin and H. R. Jauslin, “Two-laser multiphoton adiabatic passage in the frame of the Floquet theory. Applications to (1+1) and (2+1) STIRAP,” Eur. Phys. J. D 2, 99 (1998).

L. P. Yatsenko, S. Guérin, T. Halfmann, K. Böhmer, B. W. Shore, and K. Bergmann, “Stimulated hyper-Raman adiabatic passage. I. The basic problem and examples,” Phys. Rev. A 58, 4683 (1998).
[Crossref]

S. Guérin, L. P. Yatsenko, T. Halfmann, B. W. Shore, and K. Bergmann, “Stimulated hyper-Raman adiabatic passage. II. Static compensation of dynamic Stark shifts,”Phys. Rev. A 58, 4691 (1998).
[Crossref]

K. Drese and M. Holthaus, “Perturbative and nonperturbative processes in adiabatic population transfer,” Eur. Phys. J. D,  3, 73 (1998)
[Crossref]

1997 (2)

N. V. Vitanov and S. Stenholm, “Analytic properties and effective two-level problems in stimulated Raman adiabatic passage,” Phys. Rev. A 55, 648 (1997).
[Crossref]

S. Guérin, F. Monti, J. M. Dupont, and H. R. Jauslin, “On the relation between cavity-dressed states, Floquet states,RWA and semiclassical models,” J. Phys. A 30, 7193 (1997).
[Crossref]

1995 (3)

J. Martin, B. W. Shore, and K. Bergmann, “Coherent population transfer in multilevel systems with magnetic sublevels. II. Algebraic analysis,” Phys. Rev. A 52, 583 (1995).
[Crossref] [PubMed]

M. Elk, “Adiabatic transition histories of population transfer in the Λ system,” Phys. Rev. A 52, 4017 (1995).
[Crossref] [PubMed]

W. Scherer, “Superconvergent perturbative method in quantum mechanics,” Phys. Rev. Lett. 74, 1495 (1995).
[Crossref] [PubMed]

1993 (1)

A. Joye and C.-E. Pfister, “Superadiabatic evolution and adiabatic transition probability between two nondegenerate levels isolated in the spectrum,” J. Math. Phys. 34, 454 (1993).
[Crossref]

1992 (1)

P. Blekher, H. R. Jauslin, and J. L. Lebowitz, “Floquet spectrum for two-level systems in quasiperiodic time-dependent fields,” J. Stat. Phys. 68271 (1992).
[Crossref]

1990 (2)

M. V. Berry, “Histories of adiabatic quantum transitions,” Proc. R. Soc. Lond. A 429, 61 (1990).
[Crossref]

U. Gaubatz, P. Rudecki, S. Schiemann, and K. Bergmann, “Population transfer between molecular vibrational levels by stimulated Raman scattering with partially overlapping laserfields. A new concept and experimental results,” J. Chem. Phys. 92, 5363 (1990).
[Crossref]

1988 (1)

T. P. Grozdanov and M. J. Raković, “Quantum system driven by rapidly varying periodic perturbation,” Phys. Rev. A 38, 1739 (1988).
[Crossref] [PubMed]

1987 (2)

M. Combescure, “The quantum stability problem for time-periodic perturbations of the harmonic oscillator”, Ann. Inst. H. Poincaré 47, 63 (1987).

S.-I. Chu, “Generalized Floquet theoretical approaches to intense-field multiphoton and nonlinear optical processes,” Adv. Chem. Phys. 73, 739 (1987).
[Crossref]

Bergmann, K.

L. P. Yatsenko, S. Guérin, T. Halfmann, K. Böhmer, B. W. Shore, and K. Bergmann, “Stimulated hyper-Raman adiabatic passage. I. The basic problem and examples,” Phys. Rev. A 58, 4683 (1998).
[Crossref]

S. Guérin, L. P. Yatsenko, T. Halfmann, B. W. Shore, and K. Bergmann, “Stimulated hyper-Raman adiabatic passage. II. Static compensation of dynamic Stark shifts,”Phys. Rev. A 58, 4691 (1998).
[Crossref]

J. Martin, B. W. Shore, and K. Bergmann, “Coherent population transfer in multilevel systems with magnetic sublevels. II. Algebraic analysis,” Phys. Rev. A 52, 583 (1995).
[Crossref] [PubMed]

U. Gaubatz, P. Rudecki, S. Schiemann, and K. Bergmann, “Population transfer between molecular vibrational levels by stimulated Raman scattering with partially overlapping laserfields. A new concept and experimental results,” J. Chem. Phys. 92, 5363 (1990).
[Crossref]

R. G. Unanyan, S. Guérin, B. W. Shore, and K. Bergmann (unpublished).

Berry, M. V.

M. V. Berry, “Histories of adiabatic quantum transitions,” Proc. R. Soc. Lond. A 429, 61 (1990).
[Crossref]

Blekher, P.

P. Blekher, H. R. Jauslin, and J. L. Lebowitz, “Floquet spectrum for two-level systems in quasiperiodic time-dependent fields,” J. Stat. Phys. 68271 (1992).
[Crossref]

Böhmer, K.

L. P. Yatsenko, S. Guérin, T. Halfmann, K. Böhmer, B. W. Shore, and K. Bergmann, “Stimulated hyper-Raman adiabatic passage. I. The basic problem and examples,” Phys. Rev. A 58, 4683 (1998).
[Crossref]

Chu, S.-I.

S.-I. Chu, “Generalized Floquet theoretical approaches to intense-field multiphoton and nonlinear optical processes,” Adv. Chem. Phys. 73, 739 (1987).
[Crossref]

Combescure, M.

M. Combescure, “The quantum stability problem for time-periodic perturbations of the harmonic oscillator”, Ann. Inst. H. Poincaré 47, 63 (1987).

Drese, K.

K. Drese and M. Holthaus, “Perturbative and nonperturbative processes in adiabatic population transfer,” Eur. Phys. J. D,  3, 73 (1998)
[Crossref]

Dupont, J. M.

S. Guérin, F. Monti, J. M. Dupont, and H. R. Jauslin, “On the relation between cavity-dressed states, Floquet states,RWA and semiclassical models,” J. Phys. A 30, 7193 (1997).
[Crossref]

Elk, M.

M. Elk, “Adiabatic transition histories of population transfer in the Λ system,” Phys. Rev. A 52, 4017 (1995).
[Crossref] [PubMed]

Gaubatz, U.

U. Gaubatz, P. Rudecki, S. Schiemann, and K. Bergmann, “Population transfer between molecular vibrational levels by stimulated Raman scattering with partially overlapping laserfields. A new concept and experimental results,” J. Chem. Phys. 92, 5363 (1990).
[Crossref]

Grozdanov, T. P.

T. P. Grozdanov and M. J. Raković, “Quantum system driven by rapidly varying periodic perturbation,” Phys. Rev. A 38, 1739 (1988).
[Crossref] [PubMed]

Guérin, S.

S. Guérin and H. R. Jauslin, “Two-laser multiphoton adiabatic passage in the frame of the Floquet theory. Applications to (1+1) and (2+1) STIRAP,” Eur. Phys. J. D 2, 99 (1998).

L. P. Yatsenko, S. Guérin, T. Halfmann, K. Böhmer, B. W. Shore, and K. Bergmann, “Stimulated hyper-Raman adiabatic passage. I. The basic problem and examples,” Phys. Rev. A 58, 4683 (1998).
[Crossref]

S. Guérin, L. P. Yatsenko, T. Halfmann, B. W. Shore, and K. Bergmann, “Stimulated hyper-Raman adiabatic passage. II. Static compensation of dynamic Stark shifts,”Phys. Rev. A 58, 4691 (1998).
[Crossref]

S. Guérin, F. Monti, J. M. Dupont, and H. R. Jauslin, “On the relation between cavity-dressed states, Floquet states,RWA and semiclassical models,” J. Phys. A 30, 7193 (1997).
[Crossref]

R. G. Unanyan, S. Guérin, B. W. Shore, and K. Bergmann (unpublished).

Halfmann, T.

S. Guérin, L. P. Yatsenko, T. Halfmann, B. W. Shore, and K. Bergmann, “Stimulated hyper-Raman adiabatic passage. II. Static compensation of dynamic Stark shifts,”Phys. Rev. A 58, 4691 (1998).
[Crossref]

L. P. Yatsenko, S. Guérin, T. Halfmann, K. Böhmer, B. W. Shore, and K. Bergmann, “Stimulated hyper-Raman adiabatic passage. I. The basic problem and examples,” Phys. Rev. A 58, 4683 (1998).
[Crossref]

Holthaus, M.

K. Drese and M. Holthaus, “Perturbative and nonperturbative processes in adiabatic population transfer,” Eur. Phys. J. D,  3, 73 (1998)
[Crossref]

Jauslin, H. R.

S. Guérin and H. R. Jauslin, “Two-laser multiphoton adiabatic passage in the frame of the Floquet theory. Applications to (1+1) and (2+1) STIRAP,” Eur. Phys. J. D 2, 99 (1998).

S. Guérin, F. Monti, J. M. Dupont, and H. R. Jauslin, “On the relation between cavity-dressed states, Floquet states,RWA and semiclassical models,” J. Phys. A 30, 7193 (1997).
[Crossref]

P. Blekher, H. R. Jauslin, and J. L. Lebowitz, “Floquet spectrum for two-level systems in quasiperiodic time-dependent fields,” J. Stat. Phys. 68271 (1992).
[Crossref]

Joye, A.

A. Joye and C.-E. Pfister, “Superadiabatic evolution and adiabatic transition probability between two nondegenerate levels isolated in the spectrum,” J. Math. Phys. 34, 454 (1993).
[Crossref]

Lebowitz, J. L.

P. Blekher, H. R. Jauslin, and J. L. Lebowitz, “Floquet spectrum for two-level systems in quasiperiodic time-dependent fields,” J. Stat. Phys. 68271 (1992).
[Crossref]

Martin, J.

J. Martin, B. W. Shore, and K. Bergmann, “Coherent population transfer in multilevel systems with magnetic sublevels. II. Algebraic analysis,” Phys. Rev. A 52, 583 (1995).
[Crossref] [PubMed]

Monti, F.

S. Guérin, F. Monti, J. M. Dupont, and H. R. Jauslin, “On the relation between cavity-dressed states, Floquet states,RWA and semiclassical models,” J. Phys. A 30, 7193 (1997).
[Crossref]

Pfister, C.-E.

A. Joye and C.-E. Pfister, “Superadiabatic evolution and adiabatic transition probability between two nondegenerate levels isolated in the spectrum,” J. Math. Phys. 34, 454 (1993).
[Crossref]

Rakovic, M. J.

T. P. Grozdanov and M. J. Raković, “Quantum system driven by rapidly varying periodic perturbation,” Phys. Rev. A 38, 1739 (1988).
[Crossref] [PubMed]

Rudecki, P.

U. Gaubatz, P. Rudecki, S. Schiemann, and K. Bergmann, “Population transfer between molecular vibrational levels by stimulated Raman scattering with partially overlapping laserfields. A new concept and experimental results,” J. Chem. Phys. 92, 5363 (1990).
[Crossref]

Scherer, W.

W. Scherer, “Superconvergent perturbative method in quantum mechanics,” Phys. Rev. Lett. 74, 1495 (1995).
[Crossref] [PubMed]

Schiemann, S.

U. Gaubatz, P. Rudecki, S. Schiemann, and K. Bergmann, “Population transfer between molecular vibrational levels by stimulated Raman scattering with partially overlapping laserfields. A new concept and experimental results,” J. Chem. Phys. 92, 5363 (1990).
[Crossref]

Shore, B. W.

L. P. Yatsenko, S. Guérin, T. Halfmann, K. Böhmer, B. W. Shore, and K. Bergmann, “Stimulated hyper-Raman adiabatic passage. I. The basic problem and examples,” Phys. Rev. A 58, 4683 (1998).
[Crossref]

S. Guérin, L. P. Yatsenko, T. Halfmann, B. W. Shore, and K. Bergmann, “Stimulated hyper-Raman adiabatic passage. II. Static compensation of dynamic Stark shifts,”Phys. Rev. A 58, 4691 (1998).
[Crossref]

J. Martin, B. W. Shore, and K. Bergmann, “Coherent population transfer in multilevel systems with magnetic sublevels. II. Algebraic analysis,” Phys. Rev. A 52, 583 (1995).
[Crossref] [PubMed]

R. G. Unanyan, S. Guérin, B. W. Shore, and K. Bergmann (unpublished).

B. W. Shore, The Theory of Coherent Atomic Excitation II. Multi-level Atoms and Incoherence (Wiley, New York, 1990), Chap. 18.7, pp. 1165–66.

Stenholm, S.

N. V. Vitanov and S. Stenholm, “Analytic properties and effective two-level problems in stimulated Raman adiabatic passage,” Phys. Rev. A 55, 648 (1997).
[Crossref]

Unanyan, R. G.

R. G. Unanyan, S. Guérin, B. W. Shore, and K. Bergmann (unpublished).

Vitanov, N. V.

N. V. Vitanov and S. Stenholm, “Analytic properties and effective two-level problems in stimulated Raman adiabatic passage,” Phys. Rev. A 55, 648 (1997).
[Crossref]

Yatsenko, L. P.

L. P. Yatsenko, S. Guérin, T. Halfmann, K. Böhmer, B. W. Shore, and K. Bergmann, “Stimulated hyper-Raman adiabatic passage. I. The basic problem and examples,” Phys. Rev. A 58, 4683 (1998).
[Crossref]

S. Guérin, L. P. Yatsenko, T. Halfmann, B. W. Shore, and K. Bergmann, “Stimulated hyper-Raman adiabatic passage. II. Static compensation of dynamic Stark shifts,”Phys. Rev. A 58, 4691 (1998).
[Crossref]

Adv. Chem. Phys. (1)

S.-I. Chu, “Generalized Floquet theoretical approaches to intense-field multiphoton and nonlinear optical processes,” Adv. Chem. Phys. 73, 739 (1987).
[Crossref]

Ann. Inst. H. Poincaré (1)

M. Combescure, “The quantum stability problem for time-periodic perturbations of the harmonic oscillator”, Ann. Inst. H. Poincaré 47, 63 (1987).

Eur. Phys. J. D (2)

K. Drese and M. Holthaus, “Perturbative and nonperturbative processes in adiabatic population transfer,” Eur. Phys. J. D,  3, 73 (1998)
[Crossref]

S. Guérin and H. R. Jauslin, “Two-laser multiphoton adiabatic passage in the frame of the Floquet theory. Applications to (1+1) and (2+1) STIRAP,” Eur. Phys. J. D 2, 99 (1998).

J. Chem. Phys. (1)

U. Gaubatz, P. Rudecki, S. Schiemann, and K. Bergmann, “Population transfer between molecular vibrational levels by stimulated Raman scattering with partially overlapping laserfields. A new concept and experimental results,” J. Chem. Phys. 92, 5363 (1990).
[Crossref]

J. Math. Phys. (1)

A. Joye and C.-E. Pfister, “Superadiabatic evolution and adiabatic transition probability between two nondegenerate levels isolated in the spectrum,” J. Math. Phys. 34, 454 (1993).
[Crossref]

J. Phys. A (1)

S. Guérin, F. Monti, J. M. Dupont, and H. R. Jauslin, “On the relation between cavity-dressed states, Floquet states,RWA and semiclassical models,” J. Phys. A 30, 7193 (1997).
[Crossref]

J. Stat. Phys. (1)

P. Blekher, H. R. Jauslin, and J. L. Lebowitz, “Floquet spectrum for two-level systems in quasiperiodic time-dependent fields,” J. Stat. Phys. 68271 (1992).
[Crossref]

Phys. Rev. A (6)

T. P. Grozdanov and M. J. Raković, “Quantum system driven by rapidly varying periodic perturbation,” Phys. Rev. A 38, 1739 (1988).
[Crossref] [PubMed]

M. Elk, “Adiabatic transition histories of population transfer in the Λ system,” Phys. Rev. A 52, 4017 (1995).
[Crossref] [PubMed]

J. Martin, B. W. Shore, and K. Bergmann, “Coherent population transfer in multilevel systems with magnetic sublevels. II. Algebraic analysis,” Phys. Rev. A 52, 583 (1995).
[Crossref] [PubMed]

L. P. Yatsenko, S. Guérin, T. Halfmann, K. Böhmer, B. W. Shore, and K. Bergmann, “Stimulated hyper-Raman adiabatic passage. I. The basic problem and examples,” Phys. Rev. A 58, 4683 (1998).
[Crossref]

S. Guérin, L. P. Yatsenko, T. Halfmann, B. W. Shore, and K. Bergmann, “Stimulated hyper-Raman adiabatic passage. II. Static compensation of dynamic Stark shifts,”Phys. Rev. A 58, 4691 (1998).
[Crossref]

N. V. Vitanov and S. Stenholm, “Analytic properties and effective two-level problems in stimulated Raman adiabatic passage,” Phys. Rev. A 55, 648 (1997).
[Crossref]

Phys. Rev. Lett. (1)

W. Scherer, “Superconvergent perturbative method in quantum mechanics,” Phys. Rev. Lett. 74, 1495 (1995).
[Crossref] [PubMed]

Proc. R. Soc. Lond. A (1)

M. V. Berry, “Histories of adiabatic quantum transitions,” Proc. R. Soc. Lond. A 429, 61 (1990).
[Crossref]

Other (2)

R. G. Unanyan, S. Guérin, B. W. Shore, and K. Bergmann (unpublished).

B. W. Shore, The Theory of Coherent Atomic Excitation II. Multi-level Atoms and Incoherence (Wiley, New York, 1990), Chap. 18.7, pp. 1165–66.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (1)

Figure 1.
Figure 1.

For δ = 2 and squared trig function pulse (of length 1 and delay 0.33): a) Exact (full lines) and second order (dashed lines) eigenvalue curves; b) Differences between the exact eigevalues and: the fourth order ones (full lines), the second order ones (dashed lines), and the ones from adiabatic elimination (dotted lines).

Equations (26)

Equations on this page are rendered with MathJax. Learn more.

H α ̅ ( t ) ( θ ̅ + ω ̅ t ) = H 0 + μ [ α p ( t ) cos ( θ p + ω p t ) + α s ( t ) cos ( θ s + ω s t ) ] ,
K α ̅ ( t ) ( θ ̅ ) = H α ̅ ( t ) ( θ ̅ ) i ω ̅ · θ ̅ .
R 0 ( θ ̅ ) = diag [ e i θ p , 1 , e i θ s ] .
R 0 1 K R 0 = i ω ̅ · θ ¯ + 1 2 [ 0 α p 0 α p 0 α s 0 α s 0 ] + V 1 ( θ ̅ ) i ω ̅ · θ ̅ + H ( 0 ) + V 1 ( θ ̅ )
2 V 1 = [ 0 α p e 2 i θ p 0 α p e 2 i θ p 0 α s e 2 i θ s 0 α s e 2 i θ s 0 ] + [ 0 α s e i ( θ p + θ s ) 0 α s e i ( θ p + θ s ) 0 α p e i ( θ p + θ s ) 0 α p e i ( θ p + θ s ) 0 ]
+ [ 0 α s e i ( θ p θ s ) 0 α s e i ( θ p θ s ) 0 α p e i ( θ p θ s ) 0 α p e i ( θ p θ s ) 0 ] .
K ˜ T 0 1 R 0 1 K R 0 T 0 = K ( 0 ) + T 0 1 V 1 T 0
K ( 0 ) = i ω ̅ · θ ̅ + diag [ λ 1 ( 0 ) , λ 2 ( 0 ) , λ 3 ( 0 ) ]
λ 1 ( 0 ) = 1 2 α p 2 + α s 2 , λ 2 ( 0 ) = 0 , λ 3 ( 0 ) = 1 2 α p 2 + α s 2 .
K ˜ ( θ ̅ ) = K ( 0 ) + ε V ( 1 ) ( θ ̅ ) , K ( 0 ) = i ω ̅ · θ ̅ + D ( 0 ) ,
e εW K ˜ e εW = K ( 0 ) + D ( 1 ) [ 𝓞 ( ε ) ] + V ( 2 ) [ 𝓞 ( ε 2 ) , θ ̅ ] ,
[ K ( 0 ) , W ] + V ( 1 ) = D ( 1 ) , [ K ( 0 ) , D ( 1 ) ] = 0 .
D ( 1 ) = m m m V ( 1 ) m m , W = m , m m m m V ( 1 ) m m λ m ( 0 ) λ m ( 0 ) ,
V ( 2 ) = ε 2 1 2 [ V ( 1 ) , W ] + ε 3 1 3 [ [ V ( 1 ) , W ] , W ] + ε 4 1 8 [ [ [ V ( 1 ) , W ] , W ] , W ] + 𝓞 ( ε 5 ) .
α max ω p , ω s .
V ( 1 ) = k ̅ V k ̅ ( 1 ) e i k ̅ · θ ̅ , W = k ̅ W k ̅ e i k · θ ̅ ̅
W k ̅ = n , n n n n V k ̅ ( 1 ) n n λ n ( 0 ) λ n ( 0 ) k ̅ · ω ̅ ,
max t { α p 2 + α s 2 } ~ α max approaches ω p ω s .
V ( 2 ) = ε 2 2 { [ V k ̂ , W k ̂ ] + [ V k ̂ , W k ̂ ] + [ V k ̂ , W k ̂ ] e 2 i ( θ p θ s ) + [ V k ̂ , W k ̂ ] e 2 i ( θ p θ s ) } .
λ 1 ( 2 ) = λ 0 + 1 32 λ 0 2 ( α s 4 λ 0 + δ + α p 4 λ 0 δ ) , λ 3 ( 2 ) = λ 0 1 32 λ 0 2 ( α s 4 λ 0 δ + α p 4 λ 0 + δ ) ,
λ 2 ( 2 ) = δ 16 λ 0 2 ( λ 0 2 δ 2 ) ( α s 4 α p 4 ) .
2 λ 0 = α p 2 + α s 2
δ = ω p ω s .
Ψ n ( 1 ) = T 0 e εW n e i k ̅ · θ ̅ .
K a . e . = 1 2 [ α s 2 ( 2 δ ) α p 0 α p ( α s 2 α p 2 ) ( 2 δ ) α s 0 α s α p 2 ( 2 δ ) ] .
{ 1 e i θ p , 2 , 3 e i θ s , 1 e i θ s , 3 e i θ p , 2 e i ( θ p θ s ) , 2 e i ( θ p θ s ) } .

Metrics