## Floquet perturbative analysis for STIRAP beyond the rotating wave approximation

Optics Express, Vol. 4, Issue 2, pp. 84-90 (1999)

http://dx.doi.org/10.1364/OE.4.000084

Acrobat PDF (239 KB)

### 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

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]

*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.

*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]

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]

*nonlinear resonances*. We are in particular interested in the case when one of the peak Rabi frequencies approaches the difference of the two frequencies.

## 2. The full Hamiltonian

*H*

_{0}on the Hilbert space 𝛨 = ℂ

^{3}spanned by the vector set {∣1⟩, ∣2⟩, ∣3⟩}. It is driven by the two smooth pulsed-shaped monochromatic fields, with the dipole moment

*μ*,

*α̱*= (

*α*

_{p},

*α*

_{s}),

*ω̱*= (

*ω*

_{p},

*ω*

_{s}) and

*θ̱*= (

*θ*

_{p},

*θ*

_{s}). 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]

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]

_{2}(

*dθ*

_{p}/2

*π*) ⊗ ℒ

_{2}(

*dθ*

_{s}/2

*π*) where each ℒ

_{2}(

*dθ*

_{i}/2

*π*) is a space of square integrable functions of an angle

*θ*

_{i}, corresponding to a monochromatic photon field.

*n*, refers to levels of the (dressed) molecule, and another one, denoted

*ḵ*= (

*k*

_{p},

*k*

_{s}), for the relative photon number in each mode. The eigenvalues, which are two-mode periodic (quasiperiodic), are denoted λ

_{n,k}

_{_}= λ

_{n,0}+

*ḵ*∙

*ω*and the eigenvectors ∣

*n*,

*ḵ*⟩.

*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

*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.

*K*. This treatment is the usual

*Rotating Wave Transformation*(RWT) represented by the diagonal matrix:

*E*

_{2}= 0)

*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.

*θ̱*-independent part of the Hamiltonian (4)

*K*

^{(0)}is the diagonalized usual STIRAP Hamiltonian

_{_}=

*ḵ*∙

*ω̱*+

*n*= {1, 2, 3},

*T*

_{0}contains the normalized eigenvectors of

*H*

^{(0)}as column vectors.

*K̃*=

*K*

^{(0)}+

*εV*

^{(l)}with

*D*

^{(0)}being diagonal and

*εV*

^{(l)}=

*V*

_{1}

*T*

_{0}. We have introduced the formal parameter

*ε*in order to treat

*eV*

^{(l)}perturbatively.

### 3.2 The perturbative algorithm

*K*written (exactly) as

*ε*is a small parameter.

*D*

^{(0)}is diagonal and independent of

*θ̱*.

*εW*), with

*W*† = -

*W*antihermitian, such that

*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, 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. **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. Raković, “Quantum system driven by rapidly varying periodic perturbation,” Phys. Rev. A **38**, 1739 (1988). [CrossRef] [PubMed]

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

*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.

*ε*, we obtain the equations that determine the unknown

*W*and

*D*

^{(1)}:

*m*⟩} of eigenvectors of

*K*

^{(0)}(we use a unique integer index

*m*for simplicity), the solution of (11) can be written as

*K*

^{(0)}as

*W*is not unique: one could add to it in (11) an arbitrary operator

*A*that commutes with

*K*

^{(0)}. We choose

*A*= 0.

*D*

^{(l)}= 0, since we have already absorded the diagonal part into

*K*

^{(0)}.

*ε*can be written as:

## 4. The first corrections to the usual STIRAP

### 4.1 Dominant corrections

*ḵ*= {(-2, 0); (2, 0); (0, -2); (0, 2); (-1, -1); (1,1); (-1,1); (1, - 1)} and, from the definition (12) of

*W*,

*n*= 1, 2, 3 and the eigenvalues

*W*carry the dominant contribution for the set

*ḵ*= {-

*k̂*̱;

*k̂*} = {(-1,1); (1, -1)}. More precisely, these denominators become small when

*V*

_{1}corresponding to the modes {-

*k̂*̱;

*k̂*̱, i.e. for the last term of (5) [13].

### 4.2 Treatment of the corrections without nonlinear resonances

*k̂*̱;

*k̂*̱} we obtain for the second order correction (the first commutator of (13)):

*V*

^{(2)}:

*θ̱*-dependent) of

*KR*

_{0}are given by:

*ω*

_{p}-

*ω*

_{s}∣, 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

*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.

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]

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

## 5. Comparison with adiabatic elimination of dressed states

*adiabatic elimination*under the hypothesis

## 6. Conclusion

## Acknowledgments

## 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. |

2. | J. Martin, B. W. Shore, and K. Bergmann, “Coherent population transfer in multilevel systems with magnetic sublevels. II. Algebraic analysis,” Phys. Rev. A |

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 |

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 |

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 |

6. | N. V. Vitanov and S. Stenholm, “Analytic properties and effective two-level problems in stimulated Raman adiabatic passage,” Phys. Rev. A |

7. | S.-I. Chu, “Generalized Floquet theoretical approaches to intense-field multiphoton and nonlinear optical processes,” Adv. Chem. Phys. |

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 |

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

10. | P. Blekher, H. R. Jauslin, and J. L. Lebowitz, “Floquet spectrum for two-level systems in quasiperiodic time-dependent fields,” J. Stat. Phys. |

11. | W. Scherer, “Superconvergent perturbative method in quantum mechanics,” Phys. Rev. Lett. |

12. | T. P. Grozdanov and M. J. Raković, “Quantum system driven by rapidly varying periodic perturbation,” Phys. Rev. A |

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 |

15. | A. Joye and C.-E. Pfister, “Superadiabatic evolution and adiabatic transition probability between two nondegenerate levels isolated in the spectrum,” J. Math. Phys. |

16. | M. Elk, “Adiabatic transition histories of population transfer in the Λ system,” Phys. Rev. A |

17. | K. Drese and M. Holthaus, “Perturbative and nonperturbative processes in adiabatic population transfer,” Eur. Phys. J. D , |

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

- 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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