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Optics Express

  • Editor: J. H. Eberly
  • Vol. 4, Iss. 2 — Jan. 18, 1999
  • pp: 84–90
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Floquet perturbative analysis for STIRAP beyond the rotating wave approximation

S. Guérin, R. G. Unanyan, L. P. Yatsenko, and H. R. Jauslin  »View Author Affiliations


Optics Express, Vol. 4, Issue 2, pp. 84-90 (1999)
http://dx.doi.org/10.1364/OE.4.000084


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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.

© Optical Society of America

1. Introduction

The STIRAP process allows efficient population transfer in three level systems using two delayed laser pulses [1

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]

]. We consider the usual three level Λ-system {∣1⟩, ∣2⟩, ∣3⟩}, of respective energies E 1 < E 2 < E 3, with no coupling between ∣1⟩ and ∣3⟩. The population is initially in level ∣1⟩. Units are chosen such that ħ = 1.

The STIRAP process consists in applying the Stokes laser pulse (approximately tuned to the Bohr frequency E 3 - E 2) before the pump laser pulse (tuned to E 2 - E 1). (The opposite sequence does not lead to complete transfer). We consider here for simplicity lasers exactly tuned to the one-photon resonances. At the initial and final times (when the fields are off), the dressed states (or Floquet states) are in resonance, and hence degenerate. For any system, the key of the transfer for this process is (i) the initial and final liftings of degeneracy which give rise to a transfer state connecting level ∣1⟩ to ∣3⟩, (ii) the adiabatic following of the dynamics on the transfer state [2

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

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

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

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

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]

].

The usual rotating wave approximation (RWA) allows to treat the initial and final resonances as the lowest order of the stationary perturbative theory on Floquet states: it determines the lifting of the degeneracy. It gives the first order terms for the dressed eigenvalues and the zeroth order terms for the eigenvectors. (The dimensionless parameters of the expansion correspond to the ratios between Rabi and Bohr frequencies.)

If we consider ideal adiabatic evolution, the corrections of the dynamics during the process are given by the counter-rotating terms, which are often neglected when considering the STIRAP process. We can study the corrections perturbatively as long as they do not induce new resonances between dressed states. These new resonances, which would appear as avoided crossing at non-zero fields, are called nonlinear resonances. We are in particular interested in the case when one of the peak Rabi frequencies approaches the difference of the two frequencies.

We restrict ourselves to this case of absence of nonlinear resonances and study a systematic perturbative development to improve the quantitative description of the dynamics during the process. We also study the limitations of this perturbative development when we approach a non-linear resonance.

2. The full Hamiltonian

The free three-level system is decribed by a Hamiltonian H 0 on the Hilbert space 𝛨 = ℂ3 spanned by the vector set {∣1⟩, ∣2⟩, ∣3⟩}. It is driven by the two smooth pulsed-shaped monochromatic fields, with the dipole moment μ,

Hα̅(t)(θ̅+ω̅t)=H0+μ[αp(t)cos(θp+ωpt)+αs(t)cos(θs+ωst)],
(1)

where the time-dependent field envelopes, carrier frequencies and initial phases of the fields are respectively denoted α̱ = (αp ,αs ), ω̱ = (ωp ,ωs ) and θ̱ = (θp ,θs ). For each fixed value of the fields, we can solve the time-dependent Schrödinger equation by the multi-mode Floquet theory [7

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

, 3

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).

], which includes photon exchanges between matter and light [8

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]

]. This gives rise to the quasi-energy operator

Kα̅(t)(θ̅)=Hα̅(t)(θ̅)iω̅·θ̅.
(2)

It is defined in the enlarged space 𝛫 = 𝛨 ⊗ ℒ2(p /2π) ⊗ ℒ2(s /2π) where each ℒ2(i /2π) is a space of square integrable functions of an angle θi , corresponding to a monochromatic photon field.

The eigenelements can be indexed with two indices: one, denoted n, refers to levels of the (dressed) molecule, and another one, denoted = (kp , ks ), for the relative photon number in each mode. The eigenvalues, which are two-mode periodic (quasiperiodic), are denoted λn,k _ = λn,0 + ω and the eigenvectors ∣n,⟩.

Since the envelopes of the pulses vary slowly, we expand the solution of the time-dependent Schrödinger equation in the Floquet basis and apply adiabatic principles. If we consider as a first approximation “exact” adiabatic following of the transfer state, the time evolution can be written in terms of the eigenelements of K α_. In the following, we develop a systematic method to determine perturbatively the eigenelements of K α_. We consider here for simplicity that the field peak amplitudes are both α max and equal couplings μ 12 = μ 23 (μ 13 = 0).

3. The perturbative analysis

3.1 Preparing the Hamiltonian: The Rotating Wave Transformation

We start from the full Floquet Hamiltonian (2). It can be expressed as a 3 by 3 matrix (in the basis of H 0), whose elements are θ-dependent. To calculate the exact eigenelements of K, we have to diagonalize the full Hamiltonian. That can be done numerically in a truncated Fourier decomposition for each frequency (this comes down to a discretization of the variables θ̱). The idea is to extract from the full Hamiltonian the dominant θ̱-independent terms in a perturbative series.

Because of the initial and final degeneracies, perturbative series cannot converge without a preliminary treatment of K. This treatment is the usual Rotating Wave Transformation (RWT) represented by the diagonal matrix:

R0(θ̅)=diag[eiθp,1,eiθs].
(3)

It is denoted RWT as oposed to RWA because the counter-rotating terms are not discarded. We obtain (setting E 2 = 0)

R01KR0=iω̅·θ¯+12[0αp0αp0αs0αs0]+V1(θ̅)iω̅·θ̅+H(0)+V1(θ̅)
(4)

with

2V1=[0αpe2iθp0αpe2iθp0αse2iθs0αse2iθs0]+[0αsei(θp+θs)0αsei(θp+θs)0αpei(θp+θs)0αpei(θp+θs)0]
+[0αsei(θpθs)0αsei(θpθs)0αpei(θpθs)0αpei(θpθs)0].
(5)

The usual RWA consists in neglecting the θ̱-dependent operator V 1, i. e. the counter-rotating terms. We remark that the RWA is equivalent to the application (in one Floquet block) of quasi-degenerate stationary perturbation theory on the Floquet Hamiltonian to lowest order, i.e. just to take the good linear combinations in the degenerate subspace. The first term of Eq. (5) contains the counter-rotating terms of the pump laser on the 1–2 transition and of the Stokes laser on the 2–3 transition. The other terms correspond to the interactions of the pump laser on the 2–3 transition and of the Stokes laser on the 1–2 transition.

K˜T01R01KR0T0=K(0)+T01V1T0
(6)

where K (0) is the diagonalized usual STIRAP Hamiltonian

K(0)=iω̅·θ̅+diag[λ1(0),λ2(0),λ3(0)]
(7)

with the eigenvalues (including all the Brillouin zones) λn,k(0) _ = ω̱ + λn(0) , for n = {1, 2, 3},

λ1(0)=12αp2+αs2,λ2(0)=0,λ3(0)=12αp2+αs2.
(8)

The orthogonal matrix T 0 contains the normalized eigenvectors of H (0) as column vectors.

We have thus written the transformed operator as = K (0) + εV (l) with K(0)=iω̅·θ̅+D(0) , D (0) being diagonal and εV (l) = T01 V 1 T 0. We have introduced the formal parameter ε in order to treat eV (l) perturbatively.

3.2 The perturbative algorithm

We start with a quasienergy Hamiltonian K written (exactly) as

K˜(θ̅)=K(0)+εV(1)(θ̅),K(0)=iω̅·θ̅+D(0),
(9)

where ε is a small parameter. D (0) is diagonal and independent of θ̱.

We construct a unitary transformation exp(εW), with W† = -W antihermitian, such that

eεWK˜eεW=K(0)+D(1)[𝓞(ε)]+V(2)[𝓞(ε2),θ̅],
(10)

where D (l) is a diagonal part, of order e and independent of θ̱, and V (2) is a remaining correction of order ε 2 (or higher). The unitary transformation reduces the size of the perturbation from order e to order e2. This method is known under different names, like “contact transformation”, KAM transformation, or van Vleck method [9

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

, 10

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

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

]. Iterating this procedure is an alternative to expansions in power series (see e.g. [12

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

]) which yields improved convergence [11

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

]. In the present context, we will only do one step, which yields eigenvalues that contain all the corrections up to order ε 2 and eigenvectors up to order ε. Maybe more importantly, this method allows one to distinguish in a systematic way the dominant contributions of the perturbation.

Inserting the unitary transformation in (10), expanding the exponential and identifying the terms of order ε, we obtain the equations that determine the unknown W and D (1):

[K(0),W]+V(1)=D(1),[K(0),D(1)]=0.
(11)

Expressing these equations in terms of the matrix elements with respect to the basis {∣m⟩} of eigenvectors of K (0) (we use a unique integer index m for simplicity), the solution of (11) can be written as

D(1)=mmmV(1)mm,W=m,mmmmV(1)mmλm(0)λm(0),
(12)

where we have denoted the eigenvalues of K (0) as λm(0). The choice of W is not unique: one could add to it in (11) an arbitrary operator A that commutes with K (0). We choose A = 0.

In the present case, we have D (l) = 0, since we have already absorded the diagonal part into K (0).

The first three terms of the remaining correction of order ε can be written as:

V(2)=ε212[V(1),W]+ε313[[V(1),W],W]+ε418[[[V(1),W],W],W]+𝓞(ε5).
(13)

4. The first corrections to the usual STIRAP

αmaxωp,ωs.
(14)

4.1 Dominant corrections

K (0) is defined by Eq. (7) and we have εV (1) = T01 V 1 T 0.

By construction, we have the following two-mode Fourier developments

V(1)=k̅Vk̅(1)eik̅·θ̅,W=k̅Wk̅eik·θ̅̅
(15)

with the set = {(-2, 0); (2, 0); (0, -2); (0, 2); (-1, -1); (1,1); (-1,1); (1, - 1)} and, from the definition (12) of W,

Wk̅=n,nnnnVk̅(1)nnλn(0)λn(0)k̅·ω̅,
(16)

for n= 1, 2, 3 and the eigenvalues λn(0) defined in (8).

Taking into account the hypothesis (14), it appears clearly that the denominators appearing in W carry the dominant contribution for the set = {-̱; } = {(-1,1); (1, -1)}. More precisely, these denominators become small when

maxt{αp2+αs2}~αmaxapproachesωpωs.
(17)

Hence the second order gives the dominant contribution for the part of V 1 corresponding to the modes {-̱;̱, i.e. for the last term of (5) [13

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

].

4.2 Treatment of the corrections without nonlinear resonances

Keeping the dominant modes {-̱;̱} we obtain for the second order correction (the first commutator of (13)):

V(2)=ε22{[Vk̂,Wk̂]+[Vk̂,Wk̂]+[Vk̂,Wk̂]e2i(θpθs)+[Vk̂,Wk̂]e2i(θpθs)}.
(18)

The second order corrections of the eigenvalues are given by the diagonal part of V (2):

λ1(2)=λ0+132λ02(αs4λ0+δ+αp4λ0δ),λ3(2)=λ0132λ02(αs4λ0δ+αp4λ0+δ),
(19)
λ2(2)=δ16λ02(λ02δ2)(αs4αp4).
(20)

with

2λ0=αp2+αs2
(21)

and

δ=ωpωs.
(22)

The first order eigenvectors (θ̱-dependent) of R01 KR 0 are given by:

Ψn(1)=T0eεWneik̅·θ̅.
(23)

This scheme is correct if the left-hand side of (17) does not approach too closely to ∣ωp - ωs ∣, otherwise the corresponding denominators become very small (and even zero) and induces the divergence of the perturbative scheme: this produces nonlinear resonances, that have to be tretated specifically with a second local RWT.

4.3 Population transfer in the adiabatic regime

It has been shown that at the first order the middle eigenvalue λ2(0) is always connected to level 1 at the beginning and to level 3 at the end of the process [3

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).

]. The second order eigenvalue (20) also connects 1 to 3, in the regime of absence of nonlinear resonances. In the adiabatic regime, this eigenvalue characterizes the transfer state.

Fig. 1a displays, for δ = 2 and α max = 1 the second order eigenvalue curves (19) and (20), in comparison with the true quasienergies (obtained numerically): They are in quite good agreement. On Fig. 1b, the differences are plotted. We have also plotted the differences taking into account the diagonal part of the fourth order of V (2) (18). The accuracy is improved.

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).

We remark that a full description of the dynamics requires, besides the corrections due to counterrotating terms discussed here, corrections due to deviations from the adiabatic limit (nonadiabatic corrections). These deviations can be analyzed in terms of superadiabatic expansions [14

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

, 15

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

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

, 17

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

]. We notice that the corrections we obtain from the counterrotating terms in the present regime are larger than the nonadiabatic corrections obtained in Ref. [16

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

]. For the present example, the corrections due to counterrotating terms do not affect the connectivity, but they can modify population of level 2 during the process.

5. Comparison with adiabatic elimination of dressed states

Ka.e.=12[αs2(2δ)αp0αp(αs2αp2)(2δ)αs0αsαp2(2δ)].
(24)

More precisely, we obtain it following Ref. [18

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.

] for adiabatic elimination that is applied on the quasienergy operator projected in the relevant basis

{1eiθp,2,3eiθs,1eiθs,3eiθp,2ei(θpθs),2ei(θpθs)}.
(25)

The result is the STIRAP process with time-dependent Stark shifts (on-diagonal elements).

On Fig. 1b, we show the comparison of the eigenvalues obtained with different approximations.

6. Conclusion

In summary, we have discussed a systematic method to do perturbation analysis in the Floquet representation, based on an iterative scheme. We have calculated an explicit formula for corrections to second order of the eigenvalues. The comparison with the exact eigenvalues (computed numerically) shows a good agreement, provided that the peak intensities are sufficiently small to avoid nonlinear resonances. The results allow one to conclude that in this regime, the complete transfer of population is still possible. However, the transfer state contains a component on level 2 during the process. This may cause a partial loss of population, if level 2 is lossy.

Acknowledgments

We would like to thank Klaas Bergmann and Bruce Shore for many usefull discussions. RU and SG thank M. Fleischhauer and N. Vitanov for stimulating discussions. SG thanks the European Union HCM network “Laser controlled Dynamics of Molecular Processes and Applications” , 4050PL93-2602, and “La Fondation Carnot” for support. RU would like to thank the Alexander-von-Humboldt Foundation for financical support. LY is grateful to the Deutsche Forschungsgemeinschaft for support of his visit to Kaiserslautern.

References

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.

OCIS Codes
(020.4180) Atomic and molecular physics : Multiphoton processes
(270.6620) Quantum optics : Strong-field processes

ToC Category:
Focus Issue: Laser controlled dynamics

History
Original Manuscript: December 7, 1998
Published: January 18, 1999

Citation
S. Guerin, H. Jauslin, R. Unanyan, and L. Yatsenko, "Floquet perturbative analysis for STIRAP beyond the rotating wave approximation," Opt. Express 4, 84-90 (1999)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-4-2-84


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References

  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.

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