Frequency Analysis of 3D Electronic <span class="aps-inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>1</mn><mi>/</mi><mi mathvariant="italic">r</mi></math></span> Dynamics: Tuning between Order and Chaos

Jan von Milczewski; David Farrelly; T. Uzer

doi:10.1103/PhysRevLett.78.1436

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Frequency Analysis of 3D Electronic $1 / r$ Dynamics: Tuning between Order and Chaos

Jan von Milczewski¹, David Farrelly², and T. Uzer¹

¹School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332-0430
²Department of Chemistry and Biochemistry, Utah State University, Logan, Utah 84322-0300

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Phys. Rev. Lett. 78, 1436 – Published 24 February, 1997

DOI: https://doi.org/10.1103/PhysRevLett.78.1436

Abstract

Frequency analysis is used to chart the global phase space structure for the most general case of an electron bound by Coulombic and harmonic potentials and perturbed by tilted electric and magnetic fields. Distinct order-chaos transitions are found together with an example of a wave function scarred by a 3D “exotic” periodic orbit, thereby providing compelling evidence for the power of frequency analysis in studying multidimensional chaos.

References (25)

R. C. Ashoori, Nature (London) 379, 413 (1996).
T. F. Gallagher, Rydberg Atoms (Cambridge University Press, Cambridge,1994).
Physics of Low-Dimensional Semiconductor Structures, P. Butcher, N. H. March, and M. P. Tosi (Plenum, New York,1993).
M. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer-Verlag, New York,1990).
D. L. Shepelyansky and A. D. Stone, Phys. Rev. Lett. 74, 2098 (1995).
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B. R. Johnson, J. O. Hirschfelder, and K.-H. Yang, Rev. Mod. Phys. 55, 122 (1983).
Suitable measures of the electric and magnetic field strengths in the absence of a dot confining potential and taking $ε = 1$ are $E_{0} {= e Z / a}_{0}^{2}$ and $B_{0} = ℏ {c / e a}_{0}^{2}$ where $a_{0} = ℏ^{2} / μ {Ze}^{2}$ where μ is the reduced mass. It is convenient to work with the dimensionless quantities ${F = E / E}_{0}$ and $γ {= B / B}_{0}$ . In a quantum dot effective atomic units are used [[1]].
M. M. Dignam and J. E. Sipe, Phys. Rev. B 45, 6819 (1992).
R. L. Greene and K. K. Bajaj, Phys. Rev. B 31, 913 (1985).
D. Farrelly, Phys. Lett. A 191, 265 (1994).
J. Laskar and P. Robutel, Nature (London) 361, 608 (1993).
J. von Milczewski, G. H. F. Diercksen, and T. Uzer, Phys. Rev. Lett. 76, 2890 (1996).
D. Farrelly, T. Uzer, P. E. Raines, J. P. Skelton, and J. A. Milligan, Phys. Rev. A 45, 4738 (1992).
J. von Milczewski, G. H. F. Diercksen, and T. Uzer, Phys. Rev. Lett. 73, 2428 (1994).
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[c1] R. C. Ashoori, Nature (London) 379, 413 (1996).

[c2] T. F. Gallagher, Rydberg Atoms (Cambridge University Press, Cambridge,1994).

[c3] Physics of Low-Dimensional Semiconductor Structures, P. Butcher, N. H. March, and M. P. Tosi (Plenum, New York,1993).

[c4] M. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer-Verlag, New York,1990).

[c5] D. L. Shepelyansky and A. D. Stone, Phys. Rev. Lett. 74, 2098 (1995).

[c6] A. J. Lichtenberg and M. A. Lieberman, Regular and Chaotic Dynamics (Springer-Verlag, New York,1992).

[c7] J. Laskar, Physica (Amsterdam) 67D, 257 (1993).

[c8] H. S. Dumas and J. Laskar, Phys. Rev. Lett. 70, 2975 (1993).

[c9] E. J. Heller, Phys. Rev. Lett. 53, 1515 (1984).

[c10] L. Marmet, H. Held, G. Raithel, J. A. Yeazell, and H. Walther, Phys. Rev. Lett. 72, 3779 (1994).

[c11] T. Pang and S. G. Louie, Phys. Rev. Lett. 65, 1635 (1990).

[c12] D. C. Reynolds and T. C. Collins, Excitons: Their Properties and Uses (Academic Press, London,1981).

[c13] T. M. Fromhold, P. B. Wilkinson, F. W. Sheard, L. Eaves, J. Miao, and G. Edwards, Phys. Rev. Lett. 75, 1142 (1995).

[c14] P. B. Wilkinson, T. M. Fromhold, L. Eaves, F. W. Sheard, N. Miura, and T. Takamatsu, Nature (London) 380, 608 (1996).

[c15] G. Muller, G. S. Boebinger, H. Mathur, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 75, 2875 (1995).

[c16] B. R. Johnson, J. O. Hirschfelder, and K.-H. Yang, Rev. Mod. Phys. 55, 122 (1983).

[c17] Suitable measures of the electric and magnetic field strengths in the absence of a dot confining potential and taking $ε = 1$ are $E_{0} {= e Z / a}_{0}^{2}$ and $B_{0} = ℏ {c / e a}_{0}^{2}$ where $a_{0} = ℏ^{2} / μ {Ze}^{2}$ where μ is the reduced mass. It is convenient to work with the dimensionless quantities ${F = E / E}_{0}$ and $γ {= B / B}_{0}$ . In a quantum dot effective atomic units are used [[1]].

[c18] M. M. Dignam and J. E. Sipe, Phys. Rev. B 45, 6819 (1992).

[c19] R. L. Greene and K. K. Bajaj, Phys. Rev. B 31, 913 (1985).

[c20] D. Farrelly, Phys. Lett. A 191, 265 (1994).

[c21] J. Laskar and P. Robutel, Nature (London) 361, 608 (1993).

[c22] J. von Milczewski, G. H. F. Diercksen, and T. Uzer, Phys. Rev. Lett. 76, 2890 (1996).

[c23] D. Farrelly, T. Uzer, P. E. Raines, J. P. Skelton, and J. A. Milligan, Phys. Rev. A 45, 4738 (1992).

[c24] J. von Milczewski, G. H. F. Diercksen, and T. Uzer, Phys. Rev. Lett. 73, 2428 (1994).

[c25] A. Deprit, Celest. Mech. 1, 12 (1969).

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Frequency Analysis of 3D Electronic $1 / r$ Dynamics: Tuning between Order and Chaos

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Frequency Analysis of 3D Electronic 1/r Dynamics: Tuning between Order and Chaos

Abstract

References (25)

Frequency Analysis of 3D Electronic $1 / r$ Dynamics: Tuning between Order and Chaos