Stability of the Rydberg atom in the crossed magnetic and electric fields

Abstract

ABSTRACT:The stability of equilibrium positions of the Rydberg atom exposed to the uniform crossed electric and magnetic fields is analyzed. The dynamics of the system is described by an autonomous Hamiltonian depending on parameters a and f . By the normalization of the quadratic part of the Hamiltonian expansion in the neighborhood of the equilibrium position it is proved that for any f < 0 and 1 2, the equilibrium solution of the equations of motion is stable in Liapunov sense, while for f > 0 and a < 1/2, there is a domain of instability in the plain of parameters Ofa bounded by the curve d 3 = 0. In the domain of linear stability, it is proved that there are two curves in the plane Ofa, where the resonance conditions of third (ω 1 = 2ω 2 ) and fourth (ω 1 = 3ω 2 ) order are fulfilled. Moreover, by the normalization of the third-and fourth-order terms in the Hamiltonian expansion it is proved that in the case of the third-order resonance, the equilibrium position is unstable for all f > 0 different from f = 0.111572 and f = 0.281144, for which the stability takes place. In the case of the fourth-order resonance, there are two intervals of parameters for which the equilibrium position is unstable.