## Quantum localization in circularly polarized electromagnetic field in ultra-strong field limit

Optics Express, Vol. 8, Issue 2, pp. 112-117 (2001)

http://dx.doi.org/10.1364/OE.8.000112

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### Abstract

We predict analytically and confirm numerically the existence of sharply localized quantum states in an ultra-strong circularly polarized electromagnetic field with the probability density that represents non-classical wave packets moving around strongly unstable classical circular orbits.

© Optical Society of America

9. I. Bialynicki-Birula and Z. Bialynicka-Birula, “Radiative decay of Trojan wave packets,” Phys. Rev. A **56**, 3623 (1997). [CrossRef]

4. J. H. Eberly, I. Bialynicki-Birula, and M. Kalinski, “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, “New states of Hydrogen in circularly polarized electromagnetic field,” Phys. Rev. Lett. **77**, 2420 (1996). [CrossRef] [PubMed]

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

*a*, and

*b*are annihilation operators for collective excitations in the phase space,

*ω*

_{-}/

*ω*,

*ω*

_{+}/

*ω*are functions of the parameter

*q*/1

*ω*

^{2}

*r*

^{3}and

*r*is the radius of the orbit. The circular orbit is stable (Fig. 1) for moderate field strength

*ε*<0.11

*ω*

^{4/3}. Weakly unstable orbits are also capable to support non-dispersing wave packets for the fields

*ε*<3/4

*ω*

^{5/3}. They have been called anti-Trojan as moving in anti-phase with Trojan states around the circular orbit. Note that the orbit regains formally the stability for ultra-strong field when

*q*→0 which corresponds to very large circular orbit comparing to the size of field-free Coulomb orbit with the motion of the same frequency. We will look for long-living states in this limit.

12. W. C. Henneberger, “Perturbation method for atoms in intense light beams,” Phys. Rev. Lett. **21**, 838 (1968). [CrossRef]

**E**(

*t*)=

*ε*(

*x*̂cos

*ωt*+

*ŷ*sin

*ωt*) and

*ε*is the strength of the CP field. It describes the electron motion in a uniformly rotating Coulomb potential, with the singularity moving around a circle with a radius

*a*. The Fourier decomposition of the K-H potential

13. M. Gavrila and J. Z. Kaminski, “Free-free transitions in intense high-frequency laser fields,” Phys. Rev. Lett. **52**, 613 (1984). [CrossRef]

*E*is the complete elliptic integral of the first type and

*a*=

*E*/

*ω*

^{2}. The latter expression allows to recognise the zero frequency component of K-H potential as the potential of a uniformly charged ring.

*r*=0. They correspond to localized wave packets moving around a circular orbit in the laboratory frame. There is a strong intuitive argument for such localization. The zero-order component of the K-H potential extended in one dimension may be treated as a potential of a deformed quantum pendulum with no closed boundary conditions and the effective angle

*ϕ*=

*πr/a*. It is able to support nonclassical states localized on the unstable equilibrium point with the energy close to the upside-down position or

*E*≈

*V*(0) [10

_{eff}10. M. Kalinski and J. H. Eberly, “New states of Hydrogen in circularly polarized electromagnetic field,” Phys. Rev. Lett. **77**, 2420 (1996). [CrossRef] [PubMed]

10. M. Kalinski and J. H. Eberly, “New states of Hydrogen in circularly polarized electromagnetic field,” Phys. Rev. Lett. **77**, 2420 (1996). [CrossRef] [PubMed]

*a*. States strongly localized are those with energies near the deformed separatrix

*E*=

*V*(0). Fig. 3 shows the localized eigenstate and some other eigenstates of the extended potential.

_{eff}*z*=0 plane of motion and therefore we consider a two dimensional model of localization. While compared to the one dimensional case the localization is strongly enhanced due to the cylindrical symmetry. We start from the stationary Schrödinger equation

*m*=0. We also expect the solution to be the superposition of Coulomb continuum states and we postulate the approximation for the Coulomb potential of the ring as [14

14. K. J. LaGattuta, “Laser effect in photoionization: Numerical solution for one-dimensional *δ* potential,” Phys. Rev. A **40**, 683 (1989) [CrossRef] [PubMed]

15. K. J. LaGattuta, “Laser-assisted scattering from a one-dimensional *δ*-potential: An exact solution,” Phys. Rev. A **49**, 1745 (1994) [CrossRef] [PubMed]

*J*

_{0}is the Bessel function of zero order. This is a discrete set of the contintinum states which are be being scattered by the ring potential with no phase shifts. The total scattering wave function for the delta ring can be expanded as

*N*in the cylindrical Neumann function of order

_{m}*m*. For (7) sin

*δ*=0 and this is the Ramsauer-Townsend effect [16] for this potential when the electron motion in the K-H frame can be considered as rescattering from the rotating target [17

_{m}17. N. Meyer, L. Benet, C. Lipp, D. Trautmann, C. Jung, and T. H. Seliman, “Chaotic scattering off a rotating target,” J. Phys. A **28**, 2529 (1995). [CrossRef]

*r*=0 and the solution represents a well localized wave packet moving round the circular orbit in the laboratory frame from the construction. We are interested in the lowest continuum state of this kind namely the state with the smallest wave vector

*k*. We expect it to be the best approximation for the states near the separatrix of true

*V*except for the cut-off normalization envelope

_{eff}*α*(

*t*)=-

*ε*/

*ω*

^{2}(

*x̂*cos

*ωt*+

*ŷ*sin

*ωt*). In order to check our predictions numerically we have solved the exact time-dependent Schrödinger equation in two spatial dimensions using the split-operator method. We took the wave function (10) for

*t*=0 as the initial condition and the radial part was modulated by the slowly varying envelope to preserve the norm of the state. Fig. 4 shows the probability density during 20-cycle long time evolution. We observe clear trapping and focusing of the electron with small decay due to ionization. This is caused by higher-order terms in the expansion (3) which is negligible on the scale of the packet evolution and will be discussed elsewhere. The movie linked to Fig. 4 shows the whole 20-cycle evolution.

*α*0 [18

18. H. R. Reiss and V. P. Krainov, “Approximation for a Coulomb-Volkov solution in strong fields,” Phys. Rev. A **50**, R911 (1994). [CrossRef]

19. P. Gross and H. Rabitz, “On the generality of optimal control theory for laser-induced control field design,” J. Chem. Phys.105 (1996). [CrossRef]

## References and links

1. | J. Mostowski and J. J. Sanchez-Mondragon, “Interaction of highly excited hydrogen atoms with a resonant oscillating field,” Opt. Commun. |

2. | M. Pont and M. Gavrila, “Stabilization of atomic hydrogen in superintence, hight-frequency laser fields of circular polarization,” Phys. Rev. Lett. |

3. | T. Uzer, E. A. Lee, D. Farrelly, and A. F. Brunello, “Synthesis of a classical atom: wavepacket analogs of the Trojan Asteroids,” Contemporary Physics |

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

5. | A. Buchleitner and D. Delande, “Nondispersive electronic wave packets in multiphoton processes,” Phys. Rev. Lett. |

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

7. | M. Kalinski, J. H. Eberly, J. A West, and C. R. Stroud Jr., “The Rutherford atom in quantum theory,” Phys. Rev. Lett. (to be published). Physica Scripta |

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

9. | I. Bialynicki-Birula and Z. Bialynicka-Birula, “Radiative decay of Trojan wave packets,” Phys. Rev. A |

10. | M. Kalinski and J. H. Eberly, “New states of Hydrogen in circularly polarized electromagnetic field,” Phys. Rev. Lett. |

11. | W. Pauli and M. Fiertz, Nuovo Cimento, 15, 167 (1933). |

12. | W. C. Henneberger, “Perturbation method for atoms in intense light beams,” Phys. Rev. Lett. |

13. | M. Gavrila and J. Z. Kaminski, “Free-free transitions in intense high-frequency laser fields,” Phys. Rev. Lett. |

14. | K. J. LaGattuta, “Laser effect in photoionization: Numerical solution for one-dimensional |

15. | K. J. LaGattuta, “Laser-assisted scattering from a one-dimensional |

16. | L. I. Schiff, |

17. | N. Meyer, L. Benet, C. Lipp, D. Trautmann, C. Jung, and T. H. Seliman, “Chaotic scattering off a rotating target,” J. Phys. A |

18. | H. R. Reiss and V. P. Krainov, “Approximation for a Coulomb-Volkov solution in strong fields,” Phys. Rev. A |

19. | P. Gross and H. Rabitz, “On the generality of optimal control theory for laser-induced control field design,” J. Chem. Phys.105 (1996). [CrossRef] |

**OCIS Codes**

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

(270.6620) Quantum optics : Strong-field processes

**ToC Category:**

Focus Issue: Quantum control of photons and matter

**History**

Original Manuscript: November 9, 2000

Published: January 15, 2001

**Citation**

Matt Kalinski, "Quantum localization in circularly polarized electromagnetic field in ultra-strong field limit," Opt. Express **8**, 112-117 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-2-112

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### References

- J. Mostowski, and J. J. Sanchez-Mondragon, "Interaction of highly excited hydrogen atoms with a resonant oscillating field," Opt. Commun. 29, 293 (1979). [CrossRef]
- M. Pont and M. Gavrila, "Stabilization of atomic hydrogen in superintence, hight-frequency laser fields of circular polarization," Phys. Rev. Lett. 65, 2362 (1990). [CrossRef] [PubMed]
- T. Uzer, E. A. Lee, D. Farrelly, and A. F. Brunello, "Synthesis of a classical atom: wavepacket analogs of the Trojan Asteroids," Contemporary Physics 41, 14 (2000). [CrossRef]
- J. H. Eberly, I. Bialynicki-Birula, and M. Kalinski, "Lagrange equilibrium points in celestial mechanics and nonspreading wave packets for strongly driven Rydberg electrons," Phys. Rev. Lett. 73, 1777 (1994). [CrossRef] [PubMed]
- A. Buchleitner and D. Delande, "Nondispersive electronic wave packets in multiphoton processes," Phys. Rev. Lett. 75, 1487 (1995). [CrossRef] [PubMed]
- J. Zakrzewski, D. Delande and A. Buchleitner, "Nonspreading electronic wave packets and conductance fluctuations," Phys. Rev. Lett. 75, 4015 (1995). [CrossRef] [PubMed]
- M. Kalinski, J. H. Eberly, J. A West, and C. R. Stroud, Jr., "The Rutherford atom in quantum theory," Phys. Rev. Lett. (to be published). Physica Scripta 20, 539 (1979).
- M. Kalinski and J. H. Eberly, "Guiding electron orbits with chirped light" Opt. Express 1, 216 (1997). http://www.opticsexpress.org/oearchive/source/2328.htm [CrossRef]
- I. Bialynicki-Birula, and Z. Bialynicka-Birula, "Radiative decay of Trojan wave packets," Phys. Rev. A 56, 3623 (1997). [CrossRef]
- M. Kalinski and J. H. Eberly, "New states of Hydrogen in circularly polarized electromagnetic field," Phys. Rev. Lett. 77, 2420 (1996). [CrossRef] [PubMed]
- W. Pauli and M. Fiertz, Nuovo Cimento, 15, 167 (1933).
- W. C. Henneberger, "Perturbation method for atoms in intense light beams," Phys. Rev. Lett. 21, 838 (1968). [CrossRef]
- M. Gavrila and J. Z. Kaminski, "Free-free transitions in intense high-frequency laser fields," Phys. Rev. Lett. 52, 613 (1984). [CrossRef]
- K. J. LaGattuta, "Laser effect in photoionization: Numerical solution for one-dimensional d potential," Phys. Rev. A 40, 683 (1989) [CrossRef] [PubMed]
- K. J. LaGattuta, "Laser-assisted scattering from a one-dimensional d -potential: An exact solution," Phys. Rev. A 49, 1745 (1994) [CrossRef] [PubMed]
- L. I. Schiff, Quantum Mechanics, (McGraw-Hill, Lisbon, 1968).
- N. Meyer, L. Benet, C. Lipp, D. Trautmann, C. Jung and T. H. Seliman, "Chaotic scattering off a rotating target," J. Phys. A 28, 2529 (1995). [CrossRef]
- H. R. Reiss and V. P. Krainov, "Approximation for a Coulomb-Volkov solution in strong fields," Phys. Rev. A 50, R911 (1994). [CrossRef]
- P. Gross and H. Rabitz, "On the generality of optimal control theory for laser-induced control field design," J. Chem. Phys. 105 (1996). [CrossRef]

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