Using standard tools of nonlinear dynamics we analyze recently discovered instabilities of radio-frequency charged-particle traps. In the cw-driven cylindrical Kingdon trap the instabilities occur at the two values eta*(3) =3.6130467...and eta*(4) =4.4311244...of the trap's control parameter eta. Analytical estimates based on the theory of Mathieu functions predict eta*(3) =pi square root of [(363-32 pi(2))/(66 pi square root of (6-48 pi(2))]=3.6923922...and eta*(4) = [(square root of pi)/2) x [(363-32 pi(2))/(square root of (1089+48 pi(2))-12 pi)](1/2) =4.4965466... The kicked Kingdon trap, an analytically solvable model, predicts eta*(3) = 1/3 square root of 105=3.4156502...and eta*(4) = square root of 17=4.1231056... We show that similar instabilities occur in the two-particle Paul trap and the cw-driven spherical Kingdon trap.

Analytical arguments and numerical simulations suggest that the shapes of 3D microwave ionization curves measured by Koch and collaborators (see P.M. Koch and K. A. H. van Leeuwen, Phys. Rep. 255, 289 (1995)) depend only weakly on the angular momentum of the atoms in the initial microcanonical ensemble, but strongly on the principal quantum number and the magnetic quantum number. Based on this insight, coupled with the computational power of a high-end 60-node Beowulf PC cluster, we present the first 3D quantum calculations of microwave ionization curves in the experimentally relevant parameter regime. 32.80.Rm,05.45.+b

We present a complete multiphoton classification of prethreshold structures in measured microwave ionization data of hydrogen Rydberg atoms. We show that fourth-order quasienergy perturbation theory reproduces the locations of prethreshold structures, thus proving that prethreshold structures are a perturbative effect. Additional prethreshold structures are predicted to occur at field and frequency values, to our knowledge, not yet explored experimentally.