Kepler map

We present a consecutive derivation of mapping equations of motion for the one-dimensional classical hydrogenic atom in a monochromatic field of any frequency. We analyze this map in the case of high and low relative frequency of the field and transition from regular to chaotic behavior. We show that the map at aphelion is suitable for investigation of transition to chaotic behavior also in the low frequency field and even for adiabatic ionization when the strength of the external field is comparable with the Coulomb field. Moreover, the approximate analytical criterion (taking into account the electron's energy increase by the influence of the field) yields a threshold field strength quite close to the numerical results. We reveal that transition from adiabatic to chaotic ionization takes place when the ratio of the field frequency to the electron Kepler frequency approximately equals 0.1. For the dynamics and ionization in a very low frequency field the Kepler map can be converted to a differential equation and solved analytically. The threshold field of the adiabatic ionization obtained from the map is only 1.5% lower than the exact field strength of static field ionization. 1 1. INTRODUCTION It is already the third decade when the highly excited hydrogen atom in a microwave field remains one of the simplest and most proper real system for experimental and theoretical investigation of classical and quantum chaos in the nonlinear systems strongly driven by external driving fields (see reviews [1–5] and references herein). For theoretical analysis of transition to stochastic behavior and ionization processes of atoms in microwave fields approximate mapping equations of motion, rather than differential equations, are most convenient. Such a two-dimensional map (for the scaled energy of the one-dimensional atom in a monochromatic field and for the relative phase of the field), later called Kepler map [3,13], has been obtained in Refs. [6,7] after an integration of equations of motion for one period of the intrinsic motion of the electron between two subsequent passings of the aphelion, the largest distance from the nucleus. This map greatly facilitates numerical investigation of dynamics and ionization process and allows even an analytical estimation of the threshold field strengths for the onset of chaos, the diffusion coefficient of the electron in energy space and other characteristics of the system [3–10]. Moreover, this map is closely related to the expressions of quasiclassical dipole matrix elements for high atomic states [11,12]. The …