## Complete population transfer to and from a continuum and the
radiative association of cold Na atoms to produce translationally cold
Na_{2} molecules in specific vib-rotational states

Optics Express, Vol. 4, Issue 2, pp. 91-106 (1999)

http://dx.doi.org/10.1364/OE.4.000091

Acrobat PDF (556 KB)

### Abstract

We demonstrate the feasibility of a laser induced complete population transfer to
and from a continuum of states. We study the two-photon dissociation of
*υ* = 28, *J* = 1,…, 10
sodium dimers. We demonstrate that using just a pair of “counter
intuitively” ordered pulses we can dissociate 100% of the molecules
in an ensemble. The scheme is shown to be stable with respect to the initial
choice of rotational level and to fluctuations in the laser frequency and
intensity. We also study the reverse phenomenon of complete population transfer
*from* the continuum. We perform calculations on the
radiative association of Na atoms to form the Na_{2} molecule in
specific vib-rotational states. It is shown that two pulses of 20 nsec duration
and as little as 6 MW/cm^{2} peak power can photoassociate more than 98%
of the atoms within a (pulse and velocity determined) relative effective
distance, to yield Na_{2} molecules in the chosen
*υ* = 28, *J* = 10 vib-rotational
state. This means that given a density of 10^{16} atoms/cm^{3}
and a temperature of 7*K*, a 10Hz pulsed laser source of the
above parameters can convert *half* of all the Na atoms in the
ensemble to *υ* = 28, *J* = 10
Na_{2} molecules within 15 seconds of operation.

© Optical Society of America

## 1. Introduction

1. J. M. Doyle, B. Friedrich, J. Kim, and D. Patterson, “Buffer gas loading of atoms and molecules into a magnetic trap”, Phys. Rev. A **52**, R2515 (1995). [CrossRef] [PubMed]

1. J. M. Doyle, B. Friedrich, J. Kim, and D. Patterson, “Buffer gas loading of atoms and molecules into a magnetic trap”, Phys. Rev. A **52**, R2515 (1995). [CrossRef] [PubMed]

2. J. T. Bahns, W. C. Stwalley, and P. L. Gould, “Laser cooling of molecules: A sequential scheme for rotation, translation, and vibration”, J. Chem. Phys. **104**, 9689 (1996). [CrossRef]

4. B. Friedrich and D. R. Herschbach, “Alignment and trapping of molecules in intense laser fields”, Phys. Rev. Lett. **74**, 4623 (1995). [CrossRef] [PubMed]

5. H. R. Thorsheim, J. Weiner, and P. S. Julienne, “Laser-induced photoassociation of ultracold sodium atoms”, Phys. Rev. Lett. **58**, 2420 (1987). [CrossRef] [PubMed]

10. A. Fioretti, D. Comparat, A. Crubellier, O. Dulieu, F. Masnou-Seeuws, and P. Pillet, “Formation of cold Cs2 molecules through photoassociation”, Phys. Rev. Lett. **80**, 4402 (1998). [CrossRef]

5. H. R. Thorsheim, J. Weiner, and P. S. Julienne, “Laser-induced photoassociation of ultracold sodium atoms”, Phys. Rev. Lett. **58**, 2420 (1987). [CrossRef] [PubMed]

11. A. Vardi, D. Abrashkevich, E. Frishman, and M. Shapiro, “Theory of radiative recombination with strong laser pulses and the formation of ultracold molecules via stimulated photo-recombination of cold atoms”, J. Chem. Phys. **107**, 6166 (1997). [CrossRef]

12. P. S. Julienne, K. Burnett, Y. B. Band, and W. C. Stwalley, “Stimulated Raman molecule production in Bose-Einstein condensates”,Phys. Rev. A **58**, R797 (1998). [CrossRef]

13. M. Shapiro, “Theory of one- and two-photon dissociation with strong laser pulses”, J. Chem. Phys. **101**, 3844 (1994). [CrossRef]

11. A. Vardi, D. Abrashkevich, E. Frishman, and M. Shapiro, “Theory of radiative recombination with strong laser pulses and the formation of ultracold molecules via stimulated photo-recombination of cold atoms”, J. Chem. Phys. **107**, 6166 (1997). [CrossRef]

16. U. Gaubatz, P. Rudecki, M. Becker, S. Schiemann, M. Külz, and K. Bergmann, “Population switching between vibrational levels in molecular beams”, Chem. Phys. Lett. **149**, 463 (1988). [CrossRef]

*υ*= 0,

*J*= 0 translationally cold Na

_{2}molecules.

*υ*= 28,

*J*= 0 to 10 Na

_{2}molecules is achievable using nanosecond laser pulses no stronger than a few MW/cm

^{2}, and that stimulated Raman photoassociation of sodium atoms in a beam (translational temperature of 5–10 K) can be utilized for the production of translationally cold molecules in specific vib-rotational levels. These findings prove that coherent population transfer is possible, even for a final (or an initial) continuum state.

**2**we review the theory of two-photon dissociation and association by strong laser pulses and in section

**3**we apply this formalism to the simulation of resonantly enhanced two-photon dissociation of sodium dimers and stimulated two-photon association of Na atoms.

## 2. Theory of Two Photon Dissociation and Association

### 2.1 The Slowly Varying Continuum Approximation

*E*,

**n**

^{±}) (it is convenient to use “incoming” scattering states ∣

*E*,

**n**

^{-}) for the dissociation problem and “outgoing” scattering states ∣

*E*,

**n**

^{+}) for the association process), subjected to the combined action of two laser pulses of central frequencies

*ω*

_{1}and

*ω*

_{2}. We assume that

*ω*

_{1}is in near resonance with the bound-bound transition, ∣ 1⟩ ↔ ∣ 2⟩, and that

*ω*

_{2}is in near resonance with the bound-free transition, ∣ 2⟩ ↔ ∣

*E*,

**n**

^{±}). Depending on the initial state of the system and on the pulse configuration, molecules in the bound manifold can dissociate to the continuum, or colliding atoms initialy in the continuum, may associate to form a bound molecular states. The situation is depicted in Fig. 1 for a Λ-type configuration. Other configurations such as the ladder system, may be equally treated.

*H*is the radiation-free Hamiltonian,

*ϵ*

_{1}(

*t*) and

*ϵ*

_{2}(

*t*) are “slowly varying” electric field amplitudes and

*iħ∂*Ψ/

*∂t*=

*H*

_{tot}Ψ, and use of the orthogonality of the ∣1⟩, ∣2⟩ and ∣

*E*,

**n**

^{±}) basis states, results in an (indenumerable) set of first-order differential equations for the expansion coefficients. In the rotating wave approximation, and neglecting the low amplitude inter-continuum transitions, this set of equations is of the form,

*N*is the number of (asymptotically) open channels,

*μ*

_{1}∣1⟩

*ϵ*

_{1}(

*t*) and

*ϵ*

_{2}(

*t*) are real. In the above we ignored spontaneous emission from ∣ 2⟩, assuming that the pulse intensities are such that the stimulated emission rates are much faster than the spontaneous emission rates.

13. M. Shapiro, “Theory of one- and two-photon dissociation with strong laser pulses”, J. Chem. Phys. **101**, 3844 (1994). [CrossRef]

*E*

_{L}=

*E*

_{2}-

*ħ*

*ω*

_{2},

14. E. Frishman and M. Shapiro, “Reversibility of bound-to-continuum transitions induced by a strong short laser pulse and the semiclassical uniform approximation”, Phys. Rev. A **54**, 3310 (1996). [CrossRef] [PubMed]

*E*and

*t*′ analytically. We obtain that,

*F*

_{2}(

*t*) is given as,

### 2.2 The Adiabatic Approximation

*adiabatic*basis set. In the above, the eigenvalue matrix,

*ε̂*, is given as,

*complex*“mixing angle”

*θ*[15

15. A. Vardi and M. Shapiro, “Two-photon dissociation/ionization beyond the adiabatic approximation”, J. Chem. Phys. **104**, 5490 (1996). [CrossRef]

*t*) on Eq. (16), and defining,

### 2.3 Adiabatic Two-Photon Dissociation

*E*and

**m**. As a result

*t*) of Eq. (15) is zero and g(

*t*) vanishes for all times. The adiabatic solution of Eq. (32) now becomes

_{0}= (1,0) and

### 2.4 Adiabatic Two-Photon Association

_{0}= 0 (the entire population is initialy in the continuum). Hence the adiabatic solutions of Eq. (32) are of the form,

*b*

_{2}(

*t*), the (channel specific) continuum coefficients

*b*

_{E,n}(

*t*) are obtained directly via Eq. (10).

## 3. Numerical Results

### 3.1 Photodissociation of Na_{2} Molecules

**2**enables an easy computation of PD and PA processes. In this section, we study the pulsed two-photon dissociation of Na

_{2}molecules in characteristic molecular-beam conditions. In order to perform the calculation, the transition-dipole matrix elements of Eq. (7), obtained by solving the radial Schrödinger equation with known [23] Na

_{2}potential curves, need be computed only once for all pulse configurations.

*X*

^{1}

*υ*= 28,

*J*) state with

*J*in the range of 0 to 10, to the (

*E*, 3

**+3**

*s***) continuum, with the bound**

*s**A*

^{1}

*υ*′ = 37,

*J*+1) state acting as an intermediate resonance. Given the

*ab-initio*[23] electronic dipole-moments and potential curves of Fig. 2, the bound eigenfunctions and eigenenergies are obtained using the renormalized Numerov method [24

24. B. R. Johnson, “New numerical methods applied to solving the one-dimensional eigenvalue problem”, J. Chem. Phys. **67**, 4086 (1977). [CrossRef]

25. R. E. Langer, “On the connection formulas and the solutions of the wave equation”, Phys. Rev. **51**, 669 (1937). [CrossRef]

*υ*′ are plotted in Fig. 3. Choosing

*υ*′ = 37 for the intermediate state clearly maximizes the bound-free transition probability without compromising the bound-bound transitions.

*J*in the range of 0 to 10 are plotted in Fig. 4. Both vibrational states, which lie well below their respective dissociation thresholds, are hardly affected by the centrifugal barrier. As a result, the variation in the bound-bound transition matrix elements with

*J*is less than 1%.

*J*by as much as 5%. These small variations are found, however, to have only a marginal effect on the overall population transfer probabilities, which are very insensitive to changes in the Rabi frequencies Ω

_{1}and Ω

_{2}.

*indistinguishable*from the numerically-exact RKM solutions.

*ϵ*

_{1}(

*t*) pulse is applied before the “pump”

*ϵ*

_{2}(

*t*) pulse, are shown in Fig. 5. As shown by Bergmann et al. [16–22

16. U. Gaubatz, P. Rudecki, M. Becker, S. Schiemann, M. Külz, and K. Bergmann, “Population switching between vibrational levels in molecular beams”, Chem. Phys. Lett. **149**, 463 (1988). [CrossRef]

15. A. Vardi and M. Shapiro, “Two-photon dissociation/ionization beyond the adiabatic approximation”, J. Chem. Phys. **104**, 5490 (1996). [CrossRef]

*passage*scenario, population transfer to the continuum is nevertheless adiabatic. [As pointed out above, the adiabatic solutions of Eq. (16) (Eq. (38) and Eq. (39)) are in perfect agreement with the RKM solutions].

### 3.2 Photoassociation of a Coherent Na+Na Wavepacket

*t*

_{0}is the peak time of the Na+Na wave packet (i.e. the time of maximum overlap with the ∣ 2⟩ state). In our simulations we have chosen the mean initial collision energies to be

*E*

_{0}= 1 - 10

*K*and wave packet widths

*δ*

_{E}= 10

^{-4}- 10

^{-3}cm

^{-1}. Radial waves with

*J*in the range of 0 to 10 were considered, keeping in mind that individual rotational transitions could be resolved due to the energetic-narrowness of the wavepacket and laser profiles.

*ω*

_{2}and

*ω*

_{1}(taken to be in resonance with the

*X*

^{1}

*υ*= 28,

*J*) to

*A*

^{1}

*υ*′ = 37,

*J*+1) transition), is the transfer of population from the continuum to a single vib-rotational state

*X*

^{1}

*υ*= 28,

*J*), with the bound

*A*

^{1}

*υ*′ = 37,

*J*+1) state acting as an intermediate resonance.

*μ*

_{2,E}bound-continuum matrix elements are practically constant over the pulse spectral bandwidth (typically in the range of 5–10 mK), thus justifying the use of the SVCA of Eq. (12).

*t*

_{1,2}pulse parameters used in the calculations. In all the results presented here the adiabatic solutions are found to be virtually identical to the RKM numerical-solutions.

_{2}molecules in the

*υ*= 28,

*J*= 10 level of the ground electronic state is shown in Fig. 8. Contrary to the PD process, in the PA process “counter-intuitive” pulse ordering means that the

*ϵ*

_{1}(

*t*) pulse, coupling the two bound states, is made to precede

*ϵ*

_{2}(

*t*) pulse, which couples the bound to the continuum states.

*A*

^{1}

*υ*′ = 37,

*J*′ = 11) level. As in the adiabatic PD process, in this way the spontaneous emission from the intermediate state is eliminated, thus preventing the formation of molecules in vib-rotational states other than the the

*X*

^{1}

*υ*= 28,

*J*= 10) level of choice. We observe that 98% of all

*J*= 10 atom-pairs that collide during the pulse form

*υ*= 28,

*J*= 10 Na

_{2}molecules.

*J*, we use the semiclassical relation between

*J*and the impact parameter

*m*is the reduced mass of the collision pair and

*υ*is their relative velocity. Due to the rotational selection rules for optical transitions, by tuning the laser central frequencies to a specific

*J*→

*J*± 1 →

*J*sequence, only those colliding pairs whose impact parameter lies between

*b*

_{J-1}and

*b*

_{J}are affected by the laser. Hence all atoms contained in a cylinder (see Fig. 9) whose height is

*υ*Δ

*t*

_{2}(where Δ

*t*

_{2}is the duration of the pump pulse), and whose area is

*π*(

*η*

_{J}(

*T*) - the fraction of recombining atoms per pulse at temperature

*T*is therefore given as,

*n*is the number-density of Na atoms in the beam and

*δ*

_{i,j}is the Kronecker delta function. Taking the atom density and average lateral velocity in a typical Na atomic beam to be

*n*= 10

^{16}cm

^{-3}and

*υ*= 1 × 10

^{4}cm/sec (corresponding to a translational temperature of ~ 7

*K*), and using a 20 nsec pump pulse, we find that the association yield to form

*J*= 10 molecules is

*η*

_{10}(7

*K*) = 4 × 10

^{-3}per pulse. This means that with a 10Hz pulsed laser source we can recombine half of all the ensemble of Na atoms in about 15 seconds.

*t*are identical to the unscaled coefficients at times

*t*/

*. Thus, pulses’ durations can be made longer and their intensities concomitantly scaled down, without changing the final population-transfer yields. This behavior is demonstrated in Fig. 10 where pulse widths and intensities are scaled as above, with*

**s***= 10. It is evident that the resulting time evolution of the system is scaled up by a factor of 10, with the same final populations. As mentioned above, longer pulses increase the Na-Na distances for which collisions are effective in bringing about radiative recombination. Since the intermediate state population is low at all times, one needn’t worry about spontaneous emission losses when pulse durations are taken beyond the radiative lifetime of that state.*

**s**## 4. Conclusions

30. J. Javanainen and M. Mackie “Probability of photoassociation from a quasicontinuum approach”, Phys. Rev. A **58**, R789 (1998). [CrossRef]

11. A. Vardi, D. Abrashkevich, E. Frishman, and M. Shapiro, “Theory of radiative recombination with strong laser pulses and the formation of ultracold molecules via stimulated photo-recombination of cold atoms”, J. Chem. Phys. **107**, 6166 (1997). [CrossRef]

**107**, 6166 (1997). [CrossRef]

*X*

^{1}

*υ*= 28,

*J*= 10) molecules. Due to more favorable transition dipole matrix elements, the required intensities for a given pulse duration are almost two orders of magnitude lower than those required to produce

*X*

^{1}

*υ*= 0,

*J*= 0) ultracold molecules, calculated in our previous work [11

**107**, 6166 (1997). [CrossRef]

^{-3}PA yield per 20 nsec pulse is three orders of magnitude higher than the efficiency we obtained for PA in a MOT.

_{2}molecules in the ground

*X*

^{1}

*υ*= 0,

*J*= 0) state. Starting from a translationally cold ensemble of Na atoms, the first stage is the two-photon association process outlined in this article. Once enough

*X*

^{1}

*υ*= 28,

*J*= 0) molecules are formed, a second 3-level STIRAP stage may be employed to transfer molecules from the

*X*

^{1}

*υ*= 28,

*J*= 0) level into the ground

*X*

^{1}

*J*= 0) state. This four-photon, two-step approach is admittedly more complex from an experimental point of view than the two-photon one-step scheme [11

**107**, 6166 (1997). [CrossRef]

## Acknowledgments

## References

1. | J. M. Doyle, B. Friedrich, J. Kim, and D. Patterson, “Buffer gas loading of atoms and molecules into a magnetic trap”, Phys. Rev. A |

2. | J. T. Bahns, W. C. Stwalley, and P. L. Gould, “Laser cooling of molecules: A sequential scheme for rotation, translation, and vibration”, J. Chem. Phys. |

3. | A. Bartana, R. Kosloff, and D. J. Tannor, “Laser cooling of molecular internal degrees of freedom by a series of shaped pulses”, J. Chem. Phys. |

4. | B. Friedrich and D. R. Herschbach, “Alignment and trapping of molecules in intense laser fields”, Phys. Rev. Lett. |

5. | H. R. Thorsheim, J. Weiner, and P. S. Julienne, “Laser-induced photoassociation of ultracold sodium atoms”, Phys. Rev. Lett. |

6. | Y. B. Band and P. S. Julienne, “Ultracold-molecule production by laser-cooled atom photoassociation”, Phys. Rev. A |

7. | K. M. Jones, S. Maleki, L. P. Ratliff, and P. D. Lett, “Two-color photoassociation spectroscopy of ultracold sodium”, J. Phys. B , |

8. | R. Coté and A. Dalgarno, “Mechanism for the production of vibrationally excited ultracold molecules of Li |

9. | R. Coté and A. Dalgarno, “Photoassociation intensities and radiative trap loss in lithium”, Phys. Rev. A |

10. | A. Fioretti, D. Comparat, A. Crubellier, O. Dulieu, F. Masnou-Seeuws, and P. Pillet, “Formation of cold Cs2 molecules through photoassociation”, Phys. Rev. Lett. |

11. | A. Vardi, D. Abrashkevich, E. Frishman, and M. Shapiro, “Theory of radiative recombination with strong laser pulses and the formation of ultracold molecules via stimulated photo-recombination of cold atoms”, J. Chem. Phys. |

12. | P. S. Julienne, K. Burnett, Y. B. Band, and W. C. Stwalley, “Stimulated Raman molecule production in Bose-Einstein condensates”,Phys. Rev. A |

13. | M. Shapiro, “Theory of one- and two-photon dissociation with strong laser pulses”, J. Chem. Phys. |

14. | E. Frishman and M. Shapiro, “Reversibility of bound-to-continuum transitions induced by a strong short laser pulse and the semiclassical uniform approximation”, Phys. Rev. A |

15. | A. Vardi and M. Shapiro, “Two-photon dissociation/ionization beyond the adiabatic approximation”, J. Chem. Phys. |

16. | U. Gaubatz, P. Rudecki, M. Becker, S. Schiemann, M. Külz, and K. Bergmann, “Population switching between vibrational levels in molecular beams”, Chem. Phys. Lett. |

17. | U. Gaubatz, P. Rudecki, S. Schiemann, and K. Bergmann, “Population transfer between molecular vibrational levels by stimulated Raman scattering with partially overlapping laser fields. A new concept and experimental results”, J. Chem. Phys. |

18. | J. R. Kuklinski, U. Gaubatz, F. T. Hioe, and K. Bergmann, “Adiabatic population transfer in a three-level system driven by delayed laser pulses”, Phys. Rev. A |

19. | B. W. Shore, K. Bergmann, J. Oreg, and S. Rosenwaks, “Multilevel adiabatic population transfer”, Phys. Rev. A |

20. | S. Schiemann, A. Kuhn, S. Steuerwald, and K. Bergmann, “Efficient coherent population transfer in NO molecules using pulsed lasers”, Phys. Rev. Lett. |

21. | T. Halfmann and K. Bergmann, “Coherent population transfer and dark resonances in SO |

22. | K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules”, Rev. Mod. Phys. |

23. | The Na-Na potential curves and the relevant electronic dipole moments are from I. Schmidt, Ph.D. Thesis, Kaiserslautern University, 1987. |

24. | B. R. Johnson, “New numerical methods applied to solving the one-dimensional eigenvalue problem”, J. Chem. Phys. |

25. | R. E. Langer, “On the connection formulas and the solutions of the wave equation”, Phys. Rev. |

26. | R. E. Langer, “On the asymptotic solutions of differential equations, with an application to the Bessel functions of large complex order”, Trans. Am. Math. Soc. |

27. | R. E. Langer, Trans. Am. Math. Soc. |

28. | R. E. Langer, Bull. Am. Math. Soc. |

29. | W. H. Miller, “Uniform semiclassical approximations for elastic scattering and eigenvalue problems”, J. Chem. Phys. |

30. | J. Javanainen and M. Mackie “Probability of photoassociation from a quasicontinuum approach”, Phys. Rev. A |

**OCIS Codes**

(020.1670) Atomic and molecular physics : Coherent optical effects

(020.4180) Atomic and molecular physics : Multiphoton processes

(140.3440) Lasers and laser optics : Laser-induced breakdown

(290.5910) Scattering : Scattering, stimulated Raman

**ToC Category:**

Focus Issue: Laser controlled dynamics

**History**

Original Manuscript: November 16, 1998

Published: January 18, 1999

**Citation**

A. Vardi, M. Shapiro, and K. Bergmann, "Complete population transfer to and from a continuum and the radiative association of cold Na atoms to produce translationally cold Na2 molecules in specific vib-rotational states," Opt. Express **4**, 91-106 (1999)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-4-2-91

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### References

- J. M. Doyle, B. Friedrich, J. Kim, and D. Patterson, "Buffer gas loading of atoms and molecules into a magnetic trap," Phys. Rev. A 52, R2515 (1995). [CrossRef] [PubMed]
- J. T. Bahns, W. C. Stwalley, and P. L. Gould, "Laser cooling of molecules: A sequential scheme for rotation, translation, and vibration," J. Chem. Phys. 104, 9689 (1996). [CrossRef]
- A. Bartana, R. Kosloff and D. J. Tannor, "Laser cooling of molecular internal degrees of freedom by a series of shaped pulses," J. Chem. Phys. 99, 196 (1993). [CrossRef]
- B. Friedrich and D. R. Herschbach, "Alignment and trapping of molecules in intense laser fields", Phys. Rev. Lett. 74, 4623 (1995). [CrossRef] [PubMed]
- H. R. Thorsheim, J. Weiner, and P. S. Julienne, "Laser-induced photoassociation of ultracold sodium atoms," Phys. Rev. Lett. 58, 2420 (1987). [CrossRef] [PubMed]
- Y. B. Band and P. S. Julienne, "Ultracold-molecule production by laser-cooled atom photoassociation," Phys. Rev. A 51, R4317 (1995). [CrossRef] [PubMed]
- K. M. Jones, S. Maleki, L. P. Ratliff, and P. D. Lett, "Two-color photoassociation spectroscopy of ultracold sodium," J. Phys. B, 30, 289 (1997). [CrossRef]
- R. Cote and A. Dalgarno, "Mechanism for the production of vibrationally excited ultracold molecules of Li2," Chem. Phys. Lett. 279, 50 (1997). [CrossRef]
- R. Cote and A. Dalgarno, "Photoassociation intensities and radiative trap loss in lithium," Phys. Rev. A 58, 498 (1998). [CrossRef]
- A. Fioretti, D. Comparat, A. Crubellier, O. Dulieu, F. Masnou-Seeuws, and P. Pillet, "Formation of cold Cs2 molecules through photoassociation," Phys. Rev. Lett. 80, 4402 (1998). [CrossRef]
- A. Vardi, D. Abrashkevich, E. Frishman, and M. Shapiro, "Theory of radiative recombination with strong laser pulses and the formation of ultracold molecules via stimulated photorecombination of cold atoms," J. Chem. Phys. 107, 6166 (1997). [CrossRef]
- P. S. Julienne, K. Burnett, Y. B. Band, and W. C. Stwalley, "Stimulated Raman molecule production in Bose-Einstein condensates," Phys. Rev. A 58, R797 (1998). [CrossRef]
- M. Shapiro, "Theory of one- and two-photon dissociation with strong laser pulses," J. Chem. Phys. 101, 3844 (1994). [CrossRef]
- E. Frishman and M. Shapiro, "Reversibility of bound-to-continuum transitions induced by a strong short laser pulse and the semiclassical uniform approximation," Phys. Rev. A 54, 3310 (1996). [CrossRef] [PubMed]
- A. Vardi and M. Shapiro, "Two-photon dissociation/ionization beyond the adiabatic approximation," J. Chem. Phys. 104, 5490 (1996). [CrossRef]
- U. Gaubatz, P. Rudecki, M. Becker, S. Schiemann, M. Kulz, and K. Bergmann, "Population switching between vibrational levels in molecular beams," Chem. Phys. Lett. 149, 463 (1988). [CrossRef]
- U. Gaubatz, P. Rudecki, S. Schiemann, and K. Bergmann, "Population transfer between molecular vibrational levels by stimulated Raman scattering with partially overlapping laser fields. A new concept and experimental results," J. Chem. Phys. 92, 5363 (1990). [CrossRef]
- J. R. Kuklinski, U. Gaubatz, F. T. Hioe, and K. Bergmann, "Adiabatic population transfer in a three-level system driven by delayed laser pulses," Phys. Rev. A 40, 6741 (1989). [CrossRef] [PubMed]
- B. W. Shore, K. Bergmann, J. Oreg and S. Rosenwaks, "Multilevel adiabatic population transfer", Phys. Rev. A 44, 7442 (1991). [CrossRef] [PubMed]
- S. Schiemann, A. Kuhn, S. Steuerwald, and K. Bergmann, "Efficient coherent population transfer in NO molecules using pulsed lasers," Phys. Rev. Lett. 71, 3637 (1993). [CrossRef] [PubMed]
- T. Halfmann and K. Bergmann, "Coherent population transfer and dark resonances in SO2," J. Chem. Phys. 104, 7068 (1996). [CrossRef]
- K. Bergmann, H. Theuer, and B. W. Shore, "Coherent population transfer among quantum states of atoms and molecules," Rev. Mod. Phys. 70, 1003 (1998). [CrossRef]
- The Na-Na potential curves and the relevant electronic dipole moments are from I. Schmidt, Ph.D. Thesis, Kaiserslautern University, 1987.
- B. R. Johnson, "New numerical methods applied to solving the one-dimensional eigenvalue problem," J. Chem. Phys. 67, 4086 (1977). [CrossRef]
- R. E. Langer, "On the connection formulas and the solutions of the wave equation," Phys. Rev. 51, 669 (1937). [CrossRef]
- R. E. Langer, "On the asymptotic solutions of differential equations, with an application to the Bessel functions of large complex order," Trans. Am. Math. Soc. 34, 447 (1932). [CrossRef]
- R. E. Langer, Trans. Am. Math. Soc. 37, 937 (1935).
- R. E. Langer, Bull. Am. Math. Soc. 40, 545 (1934). [CrossRef]
- W. H. Miller, "Uniform semiclassical approximations for elastic scattering and eigenvalue problems", J. Chem. Phys. 48, 464 (1968). [CrossRef]
- J. Javanainen and M. Mackie "Probability of photoassociation from a quasicontinuum approach," Phys. Rev. A 58, R789 (1998). [CrossRef]

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