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.

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References

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  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. Guerin 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. Guerin, T. Halfmann, K. Bohmer, 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. Poincare 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. 68 271 (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. Rakovic, "Quantum system driven by rapidly varying periodic perturbation," Phys. Rev. A 38, 1739 (1988).
    [CrossRef] [PubMed]
  13. R. G. Unanyan, S. Guerin, 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. Pfster, "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 _ 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.

Other (18)

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. 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]

S. Guerin 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. Guerin, T. Halfmann, K. Bohmer, 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]

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

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

P. Blekher, H. R. Jauslin and J. L. Lebowitz, "Floquet spectrum for two-level systems in quasiperiodic time-dependent fields," J. Stat. Phys. 68 271 (1992).
[CrossRef]

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

T. P. Grozdanov and M. J. Rakovic, "Quantum system driven by rapidly varying periodic perturbation," Phys. Rev. A 38, 1739 (1988).
[CrossRef] [PubMed]

R. G. Unanyan, S. Guerin, B. W. Shore and K. Bergmann (unpublished).

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

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

M. Elk, "Adiabatic transition histories of population transfer in the _ system," Phys. Rev. A 52, 4017 (1995).
[CrossRef] [PubMed]

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

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.

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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)

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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 ) } .

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