## Quantum control of non-circular Trojan states in hydrogen

Optics Express, Vol. 3, Issue 3, pp. 124-129 (1998)

http://dx.doi.org/10.1364/OE.3.000124

Acrobat PDF (341 KB)

### Abstract

Control of non-circular and non-spreading wave packet states by a resonant radiation field is predicted and numerically confirmed for hydrogen.

© Optical Society of America

## 1. Introduction

1. I. Bialynicki-Birula, M. Kalinski, and J. H. Eberly, “Lagrange equilibrium points in celestial mechanics and nonspreading wave packets for strongly driven Rydberg electrons,” Phys. Rev. Lett. **73**, 1777 (1994). [CrossRef] [PubMed]

2. D. Farrelly, E. Lee, and T. Uzer, “Magnetic field stabilization of Rydberg, Gaussian wave packets in a circularly polarized microwave field,” Phys. Lett. A **204**, 359–372 (1995). [CrossRef]

3. J. Zakrzewski, D. Delande, and A. Buchleitner, “Nonspreading electronic wave packets and conductance fluctuations,” Phys. Rev. Lett. **75**, 4015 (1995). [CrossRef] [PubMed]

4. J. Zakrzewski, D. Delande, and A. Buchleitner, “Nondispersing wave packets as solitonic solutions of level dynamics,” Z. Phys B **103**, 115 (1997). [CrossRef]

5. H. P. Breuer and M. Holthaus, “A semiclassical theory of quasienergies and Floquet wave functions,” Ann. Phys. **211**, 249 (1991). [CrossRef]

6. J. Henkel and M. Holthaus, “Classical resonances in quantum mechanics,” Phys. Rev. A **45**, 1978 (1992). [CrossRef] [PubMed]

7. M. Holthaus, “On the classical-quantum correspondence for periodically time dependent systems,” Chaos, Solitons and Fractals **5**, 1143 (1995). [CrossRef]

8. D. Delande and A. Buchleitner, “Classical and quantum chaos in atomic systems,” Adv. At. Mol. Opt. Phys **35**, 85 (1994). [CrossRef]

9. M. Kalinski and J. H. Eberly, “Guiding electron orbits with chirped light,” Opt. Express **1**, 216 (1997); M. Kalinski, J. H. Eberly, and E. A. Shapiro, to appear. [CrossRef] [PubMed]

1. I. Bialynicki-Birula, M. Kalinski, and J. H. Eberly, “Lagrange equilibrium points in celestial mechanics and nonspreading wave packets for strongly driven Rydberg electrons,” Phys. Rev. Lett. **73**, 1777 (1994). [CrossRef] [PubMed]

10. M. Kalinski and J. H. Eberly, “Trojan wave packets: Mathieu theory and generation from circular states,” Phys. Rev. A **53**, 1715 (1996). [CrossRef] [PubMed]

11. B. V. Chirikov, “A universal instability of many-dimensional oscillator systems,” Phys. Rep. **52**, 263 (1979). [CrossRef]

14. e.g. See and J. E. Howard, “Stochastic ionization of hydrogen atoms in a circularly polarized microwave field,” Phys. Rev. A **46**, 364 (1992) and references therein. [CrossRef]

*n, l*. Trojan wave packets are a particular case for circular states,

*l*=

*n*- 1.

## 2. Classical Resonance Theory

*ω*= 1/

*n*

_{0}is the principal quantum number.

*l*=

*p*

_{φ}and the corresponding angle

*φ*-the angle of periapse of the ellipse of electron motion. The second pair is the action

*n*=

*p*

_{φ}+ 1/2

*π*∮

*p*

_{r}

*dr*and the corresponding angle

*θ*- the mean anomaly of the orbit, denoting an electron coordinate on the ellipse.

*θ*is proportional to the surface covered by electron radius-vector during electron motion along the orbit. In action-angle variables, the Hamiltonian is

*𝛨*

_{0}= -l/2

*n*

^{2}and the equations of motion have the form

*ω*and field strength

*ε*, the Hamiltonian takes the form

*x*and

*y*in action-angle variables [14

14. e.g. See and J. E. Howard, “Stochastic ionization of hydrogen atoms in a circularly polarized microwave field,” Phys. Rev. A **46**, 364 (1992) and references therein. [CrossRef]

*F*=

*ñ*(

*θ*+

*φ*-

*ωt*) +

*l̃φ*so that the new coordinates are

*η*

_{k}(

*n*,

*l*) and

*ζ*

_{k}(

*n*,

*l*) are expansion coefficients dependent only on

*n*and

*l*[14

14. e.g. See and J. E. Howard, “Stochastic ionization of hydrogen atoms in a circularly polarized microwave field,” Phys. Rev. A **46**, 364 (1992) and references therein. [CrossRef]

*n*

_{0}up to the second order, namely

*ω*= 1/

*ω*. After the averaging, only the term with

*k*= 1 will remain for consideration. The Hamiltonian then takes the form

## 3. Mathieu Wave Packets

*n*by the operator -

*i∂*/

*∂θ*+1/2,

*l*by -

*i∂*/

*∂φ*[16

16. e.g. See, S. D. Augustin, and H. Rabitz, “Action-angle variables in quantum mechanics,” J. Chem. Phys. **71**, 4956 (1979). [CrossRef]

17. M. Moshinsky and T. H. Seligman, “Canonical transformations to action and angle variables and their representation in quantum mechanics. II. The Coulomb problem,” Ann. Phys. , **120**, 402 (1979). [CrossRef]

18. R. A. Leacock and M. J. Pladgett, “Quantum action-angle-variable analysis of basic systems,” Am. J. Phys. , **55**, 261 (1986). [CrossRef]

19. B. Mirbach and H. J. Korsch, “Semiclassical quantization of KAM resonances in time-periodic systems,” J. Phys. A **27**, 6579 (1994). [CrossRef]

20. R. A. Marcus, “Theory of semiclassical transition probabilities (S-matrix) for inelastic and reactive collisions,” J. Chem. Phys. **54**, 3065 (1971). [CrossRef]

21. T. Uzer, D. W. Noid, and R. A. Marcus, “Uniform semiclassical theory of avoided crossings”, J. Chem. Phys. **79**, 4412 (1983). [CrossRef]

*ñ*→ -

*i∂*/

*∂θ̃*+1/2, and

*l̃*→ -

*i∂*/

*∂φ̃*-1/2. The stationary Schrödinger equation

*𝛨ψ*=

*Eψ*then takes the form

*π*-periodic in both

*θ̃*and

*φ̃*, and this defines the set of possible “dressed” energy values

*E*

^{dr}as well as the corresponding dressed eigenfunctions

*ψ*

^{dr}.

*n*

_{0}and

*l*in (

*θ̃*,

*φ̃*)-representation has the form exp[

*i L φ̃*] exp[

*i N θ̃*], where

*N*and

*L*are integers [16

16. e.g. See, S. D. Augustin, and H. Rabitz, “Action-angle variables in quantum mechanics,” J. Chem. Phys. **71**, 4956 (1979). [CrossRef]

17. M. Moshinsky and T. H. Seligman, “Canonical transformations to action and angle variables and their representation in quantum mechanics. II. The Coulomb problem,” Ann. Phys. , **120**, 402 (1979). [CrossRef]

18. R. A. Leacock and M. J. Pladgett, “Quantum action-angle-variable analysis of basic systems,” Am. J. Phys. , **55**, 261 (1986). [CrossRef]

*N*=

*n*

_{0}- 1/2,

*L*=

*l*-

*n*

_{0}+ 1/2. By seeking a dressed solution in the form

*i L φ̃*]exp[

*i N θ̃*]

*g*(

*θ̃*), we get the Mathieu equation for the function

*g*(

*θ̃*):

*N*+ 1/2)

^{2}-

*ω*)(

*N*+ 1/2), is the exact eigenenergy of unperturbed 2D hydrogen in the frame (3), with

*N*= 0,1, 2,… [23

23. X. L. Yang, S. H. Guo, and F. T. Chan, “Analytic solution of a two-dimensional hydrogen atom. I. Nonrelativistic theory”, Phys. Rev. A **43**, 1186–1205 (1991). [CrossRef] [PubMed]

10. M. Kalinski and J. H. Eberly, “Trojan wave packets: Mathieu theory and generation from circular states,” Phys. Rev. A **53**, 1715 (1996). [CrossRef] [PubMed]

*η*

_{1}(

*n*,

*l*) +

*ζ*

_{1}(

*n*,

*l*)] whereas for a circular state [

*η*

_{1}+

*ζ*

_{1}] =

*n*

^{2}. The total dependence of [

*η*

_{1}+

*ζ*

_{1}] on

*l*/

*n*is shown in Fig. 1. This figure indicates that the larger the ratio

*l*/

*n*of an initial state the larger effect a dressing resonant CP field has on this state. From Eq. (8) one can immediately derive the dressed eigenfunctions [22] as:

*M*(

*θ̃*) is a

*π*-periodic Mathieu function of the argument (

*θ̃*-

*π*)/2.

*ψ*

^{0}

_{n0l}for which exact resonance is fulfilled. In the rotating frame such a localized state has the form

*const**

*e -*

^{iLφ̃}

*e*

^{iNθ̃}

*e*

_{0}[(

*θ̃*-

*π*)/2] where

*e*

_{0}is the zero-order even

*π*-periodic Mathieu function . In the laboratory frame this state is expressed is

*θ*+

*φ*-

*ωt*.

*e*

_{0}can be approximated by a Gaussian [22]. In addition, recall that in classical mechanics the combination

*θ*+

*φ*can be approximated by

*ϕ*- sin

*θ*(2

*e*-

*e*

^{3}/4 + …) + … where

*ϕ*is the usual polar angle, and

*e*= (1 -

*l*

^{2}/

*n*

^{2})

^{1/2}is the eccentricity of the orbit [15]. With this approximation, and using the Gaussian approximation of the Mathieu function, the dressed wave function can be rewriten in the form

*ω*. The second exponential term is responsible for radial localization of the state near the points where sin

*θ*= 0, that is, for localization near turning points of clasical motion. The third term is responsible for an additional radius-dependent angular redistribution at radii at which sin

*θ*≠ 0. Thus a dressed state at exact resonance is an angular wave packet that is strongly modulated in radius and rotates around the nucleus with the field axis.

10. M. Kalinski and J. H. Eberly, “Trojan wave packets: Mathieu theory and generation from circular states,” Phys. Rev. A **53**, 1715 (1996). [CrossRef] [PubMed]

*εn*

^{2}changed to

*ε*[

*η*

_{1}+

*ζ*

_{1}]. The energies of the states adjacent to the resonant one are determined by the other stability lines of Mathieu equation (8), and the dressed wavefunctions are the corresponding

*π*-periodic Mathieu functions of

*θ*+

*φ*-

*ωt*.

## 4. Numerical Calculations

*l*=

*m*), which is approximately equivalent to considering two-dimensional hydrogen [23

23. X. L. Yang, S. H. Guo, and F. T. Chan, “Analytic solution of a two-dimensional hydrogen atom. I. Nonrelativistic theory”, Phys. Rev. A **43**, 1186–1205 (1991). [CrossRef] [PubMed]

*ω*= 1/

*n*

^{3}, for the state

*n*= 20,

*l*=

*m*= 14 and adjacent states versus predictions of the Mathieu theory. The coincidence of the calculated spectrum with the theoretical pedictions confirms that the 2D theory describes well-aligned states of 3D hydrogen.

*n*= 20,

*l*=

*m*= 14, and the dressed state obtained by numerical calculation for the field strength

*ε*= 0.02/

*ω*. On Fig. 2, one can see that the dressed state is the initial state localized both angularly and radially, as predicted by the theory.

*n*

_{0}= 20 and

*l*

_{0}= 14 as the initial state. A CP field with frequency

*ω*= 1/

*ε*=

*ε*

_{0}

*e*

^{0.2(t-30)}until

*t*= 30 when the value

*ε*

_{0}= 0.02/

*ε*

_{0}.

## 5. Conclusions

9. M. Kalinski and J. H. Eberly, “Guiding electron orbits with chirped light,” Opt. Express **1**, 216 (1997); M. Kalinski, J. H. Eberly, and E. A. Shapiro, to appear. [CrossRef] [PubMed]

## Acknowledgements

## References

1. | I. Bialynicki-Birula, M. Kalinski, and J. H. Eberly, “Lagrange equilibrium points in celestial mechanics and nonspreading wave packets for strongly driven Rydberg electrons,” Phys. Rev. Lett. |

2. | D. Farrelly, E. Lee, and T. Uzer, “Magnetic field stabilization of Rydberg, Gaussian wave packets in a circularly polarized microwave field,” Phys. Lett. A |

3. | J. Zakrzewski, D. Delande, and A. Buchleitner, “Nonspreading electronic wave packets and conductance fluctuations,” Phys. Rev. Lett. |

4. | J. Zakrzewski, D. Delande, and A. Buchleitner, “Nondispersing wave packets as solitonic solutions of level dynamics,” Z. Phys B |

5. | H. P. Breuer and M. Holthaus, “A semiclassical theory of quasienergies and Floquet wave functions,” Ann. Phys. |

6. | J. Henkel and M. Holthaus, “Classical resonances in quantum mechanics,” Phys. Rev. A |

7. | M. Holthaus, “On the classical-quantum correspondence for periodically time dependent systems,” Chaos, Solitons and Fractals |

8. | D. Delande and A. Buchleitner, “Classical and quantum chaos in atomic systems,” Adv. At. Mol. Opt. Phys |

9. | M. Kalinski and J. H. Eberly, “Guiding electron orbits with chirped light,” Opt. Express |

10. | M. Kalinski and J. H. Eberly, “Trojan wave packets: Mathieu theory and generation from circular states,” Phys. Rev. A |

11. | B. V. Chirikov, “A universal instability of many-dimensional oscillator systems,” Phys. Rep. |

12. | G. P. Berman and A. R. Kolovsky, “Quantum chaos in interactions of multilevel quantum systems with coherent radiation field,” Sov. Phys. Usp. |

13. | K. Sacha and J. Zakrzewski, “Resonance overlap criterion for H atom ionization by circularly polarized microwave fields,” Phys. Rev. A 55 , |

14. | e.g. See and J. E. Howard, “Stochastic ionization of hydrogen atoms in a circularly polarized microwave field,” Phys. Rev. A |

15. | D. Brouwer and G. Clemence, |

16. | e.g. See, S. D. Augustin, and H. Rabitz, “Action-angle variables in quantum mechanics,” J. Chem. Phys. |

17. | M. Moshinsky and T. H. Seligman, “Canonical transformations to action and angle variables and their representation in quantum mechanics. II. The Coulomb problem,” Ann. Phys. , |

18. | R. A. Leacock and M. J. Pladgett, “Quantum action-angle-variable analysis of basic systems,” Am. J. Phys. , |

19. | B. Mirbach and H. J. Korsch, “Semiclassical quantization of KAM resonances in time-periodic systems,” J. Phys. A |

20. | R. A. Marcus, “Theory of semiclassical transition probabilities (S-matrix) for inelastic and reactive collisions,” J. Chem. Phys. |

21. | T. Uzer, D. W. Noid, and R. A. Marcus, “Uniform semiclassical theory of avoided crossings”, J. Chem. Phys. |

22. | N. W. McLachlan, |

23. | X. L. Yang, S. H. Guo, and F. T. Chan, “Analytic solution of a two-dimensional hydrogen atom. I. Nonrelativistic theory”, Phys. Rev. A |

**OCIS Codes**

(020.0020) Atomic and molecular physics : Atomic and molecular physics

(020.5780) Atomic and molecular physics : Rydberg states

(270.6620) Quantum optics : Strong-field processes

**ToC Category:**

Research Papers

**History**

Original Manuscript: June 23, 1998

Revised Manuscript: June 22, 1998

Published: August 3, 1998

**Citation**

E. Shapiro, Maciej Kalinski, and J. Eberly, "Quantum control of non-circular Trojan states in hydrogen," Opt. Express **3**, 124-129 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-3-3-124

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### References

- I. Bialynicki-Birula, M. Kalinski and J. H. Eberly, "Lagrange equilibrium points in celestial mechanics and nonspreading wave packets for strongly driven Rydberg electrons," Phys. Rev. Lett. 73, 1777 (1994). [CrossRef] [PubMed]
- D. Farrelly, E. Lee and T. Uzer, "Magnetic field stabilization of Rydberg, Gaussian wave packets in a circularly polarized microwave field," Phys. Lett. A 204, 359-372 (1995). [CrossRef]
- J. Zakrzewski, D. Delande and A. Buchleitner, "Nonspreading electronic wave packets and con- ductance fluctuations," Phys. Rev. Lett. 75, 4015 (1995). [CrossRef] [PubMed]
- J. Zakrzewski, D. Delande and A. Buchleitner, "Nondispersing wave packets as solitonic solutions of level dynamics," Z. Phys B 103, 115 (1997). [CrossRef]
- H. P. Breuer and M. Holthaus, "A semiclassical theory of quasienergies and Floquet wave functions," Ann. Phys. 211, 249 (1991). [CrossRef]
- J. Henkel and M. Holthaus, "Classical resonances in quantum mechanics," Phys. Rev. A 45, 1978 (1992). [CrossRef] [PubMed]
- M. Holthaus, "On the classical-quantum correspondence for periodically time dependent systems," Chaos, Solitons and Fractals 5, 1143 (1995). [CrossRef]
- D. Delande and A. Buchleitner, "Classical and quantum chaos in atomic systems," Adv. At. Mol. Opt. Phys 35, 85 (1994). [CrossRef]
- M. Kalinski and J. H. Eberly, "Guiding electron orbits with chirped light," Opt. Express 1, 216 (1997); M. Kalinski, J. H. Eberly and E. A. Shapiro, to appear. [CrossRef] [PubMed]
- M. Kalinski and J. H. Eberly, "Trojan wave packets: Mathieu theory and generation from circular states," Phys. Rev. A 53, 1715 (1996). [CrossRef] [PubMed]
- B. V. Chirikov, "A universal instability of many-dimensional oscillator systems," Phys. Rep. 52, 263 (1979). [CrossRef]
- G. P. Berman and A. R. Kolovsky, "Quantum chaos in interactions of multilevel quantum systems with coherent radiation field," Sov. Phys. Usp. 162, 95 (1992).
- K. Sacha and J. Zakrzewski, "Resonance overlap criterion for H atom ionization by circularly polarized microwave fields," Phys. Rev. A 55, 568 (1997).
- See, e.g., J. E. Howard, "Stochastic ionization of hydrogen atoms in a circularly polarized microwave field," Phys. Rev. A 46, 364 (1992) and references therein. [CrossRef]
- D. Brouwer and G. Clemence, Methods of celestial mechanics (Academic Press, New York and London, 1961).
- See, e.g., S. D. Augustin and H. Rabitz, "Action-angle variables in quantum mechanics," J. Chem. Phys. 71, 4956 (1979). [CrossRef]
- M. Moshinsky and T. H. Seligman, "Canonical transformations to action and angle variables and their representation in quantum mechanics. II. The Coulomb problem," Ann. Phys. 120, 402 (1979). [CrossRef]
- R. A. Leacock and M. J. Pladgett, "Quantum action-angle-variable analysis of basic systems," Am. J. Phys. 55, 261 (1986). [CrossRef]
- B. Mirbach and H. J. Korsch, "Semiclassical quantization of KAM resonances in time-periodic systems," J. Phys. A 27, 6579 (1994). [CrossRef]
- R. A. Marcus, "Theory of semiclassical transition probabilities (S-matrix) for inelastic and reactive collisions," J. Chem. Phys. 54, 3065 (1971). [CrossRef]
- T. Uzer, D. W. Noid, and R. A. Marcus, "Uniform semiclassical theory of avoided crossings", J. Chem. Phys. 79, 4412 (1983). [CrossRef]
- N. W. McLachlan, Theory and Application of Mathieu Functions, Oxford University Press (1947).
- X. L. Yang, S. H. Guo and F. T. Chan, "Analytic solution of a two-dimensional hydrogen atom. I. Nonrelativistic theory", Phys. Rev. A 43, 1186-1205 (1991). [CrossRef] [PubMed]

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