# Rydberg atoms in a magnetic field and quantum chaos

Prof. Dominique Delande accepted the invitation on 24 July 2009 (self-imposed deadline: 24 December 2009).

**Short note and Related References
added by Scholarpedia Editor D.Shepelyansky in May 2020 **

The hydrogen and Rydberg atoms in a uniform magnetic field are studied analytically and numerically as real and physical examples of simple systems chaotic in the classical limit [1],[2],[3]. The quantum spectrum is shown to have a region of approximate integrability which breaks down with approach to the classical escape threshold. Classical dynamics depends only on the scaled energy given as the true energy divided by the third root of the square of the field strength. The classical transition from regular motion below the escape threshold to chaos near the escape threshold is accompanied by a corresponding transition in statistical properties of the short ranged quantum spectral fluctuations shown to be close to the results of Random matrix theory in agreement with the Bohigas-Giannoni-Schmit conjecture. Knowledge of the classical periodic orbits leads to a quantitative understanding of the low frequency properties of the quantum spectra as summarized in the Gutzwiller trace formula. These developments have led to a deeper understanding of the long known “quasi-Landau resonances” and other modulations in photoabsorption spectra alowing to describe experiments of Welge, Klepner et al. The detailed related references and reviews of this research field can be find at [4],[5].

## Related References

- D.Wintgen, H.Friedrich, "Regularity and irregularity in spectra of the magnetized Hydrogen atom", Phys. Rev. Lett. 57: 571 (1986)
- D.Delande, J.C.Gay, "Quantum chaos and statistical properties of energy Levels: numerical study of the Hydrogen atom in a magnetic field", Phys. Rev. Lett. 57: 2006 (1986)
- Chun-ho Iu, G.R.Welch, M.M.Kash, D.Klepner, D.Delande, J.C.Gay, "Diamagnetic Rydberg atom: confrontation of calculated and observed spectra", Phys. Rev. Lett. 66: 145 (1991)
- H.Friedrich, D.Wintgen, "The Hydgoren atom in a uniform magnetic field - an example of chaos", Physics Reports 183(2): 37 (1989)
- R.Blumel, W.P.Reinhardt, "Chaos in atomic physics", Cambridge University Press, Cambridge UK (1997)

## See also internal links

Bohigas-Giannoni-Schmit conjecture, Microwave billiards and quantum chaos, Quantum chaos, Random matrix theory, Shnirelman theorem