Correspondence of unstable periodic orbits and quasi-Landau modulations

We study the correlation between modulations in the photoabsorption cross sections of diamagnetic Rydberg atoms and classical periodic orbits in a completely chaotic regime. Extensive quantum-mechanical calculations of photoabsorption cross sections and classical calculations of periodic trajectories are presented. In accordance with the scaling property of the classical Hamiltonian, the photoabsorption cross sections are studied for fixed values of the scaled energy ɛ=E/${\gamma }^{2/3}$ (γ is the magnetic field strength parameter) as a function of ${\gamma }^{\mathrm{-}1/3}$. The modulations of the photoabsorption cross sections appear as prominent peaks in the Fourier-transformed cross sections and can be related to periodic classical orbits. The positions of the peaks are given quantitatively by the scaled action along the periodic orbits and the relative strengths of the peaks can be understood qualitatively by considering the geometry of the orbits and the transition involved and the stability of the classical orbits. The one-to-one correspondence between peaks in the Fourier-transformed cross sections and classical periodic orbits applies not only to the traditional quasi-Landau modulation associated with the classical orbit in the plane perpendicular to the field, but also to a large number of further modulations corresponding to topologically different series of periodic orbits. In the completely irregular region around the zero-field threshold, the unstable periodic classical orbits and the observable peaks in the modulated cross sections cannot in general be related to individual eigenstates of the quantum system.