PHYSICAL REVIEW A VOLUME 50, NUMBER 6 DECEMBER 1994 Classical stabilization of periodically kicked hydrogen atoms Giulio Casati, Italo Guarneri, and Giorgio Mantica Universi ta di Milano, sede di Corno, via Lucini 3, 22100 Corno, Italy (Received 6 April 1994; revised manuscript received 29 June 1994) We investigate the stability of a classical three-dimensional hydrogen atom subject to a periodic train of alternating 5-like pulses, or "kicks. " The intense-field stabilization effect, widely investigated for the case of monochromatically driven atoms, is numerically shown to occur in this purely classical model, and its dependence on various parameters is analyzed. Our results support the view that classical intense-field stabilization is determined by the stability of the motion in a direction transverse to the field polarization, and confirm previously established estimates for the onset of stabilization. PACS number(s): 31. 50. + w, 32.SO.Rm, 42. 50. Hz When a beam of hydrogen atoms prepared in some ini- tial state interacts for a certain time with a radiation field of given intensity and frequency, some of the atoms are ionized, the ionization probability being a function of the initial state, the interaction time, and the field intensity and frequency as well. Ceteris paribus, one would naively expect this function to be monotonic in the field intensity, i.e. , a stronger field should always produce a larger ion- ization. This expectation is contradicted by theoretical investigations that have shown that, on the contrary, on increasing the field strength above a critical value the hy- drogen atom becomes increasingly stable against field- induced ionization. This surprising efFect, known as intense-field stabilization (IFS), was originally predicted on purely quantal grounds [1], but has subsequently been found even in classical systems, first in simplified one- dimensional (1D) models using a smoothed Coulomb po- tential [4], and then in realistic 3D models [2,3]. A pure- ly classical approach to IFS has yielded a rough theoreti- cal estimate for the intensity border above which IFS should be expected [2]. Being thus clear that quantum IFS is paralleled by a quite similar classical phenomenon [5], the conclusion that the two phenomena have a com- mon origin seems inescapable. Although only the quan- turn phenomenon is physically relevant in the atomic domain, much may be learned from a careful investiga- tion of its classical counterpart. However, in spite of the growing attention attracted by IFS, the explanation for this phenomenon is not yet com- pletely clear, also, because theoretical analysis has to take into account a number of distinct dynamical features, the relative weight of which has not been fully assessed in determining IFS. A typical example is the so-called "dichotomy. " In a reference frame oscillating with the external field, the nu- cleus itself oscillates and produces an average Coulomb field quite similar to the field due to a charge continuous- ly distributed along its trajectory. With a smooth (e. g. , monochromatic) driving, the nucleus spends a large part of its time in the vicinity of the turning points of its tra- jectory; therefore, the effective charge distribution is highly nonhomogeneous and looks like a sort of dumbbell. At low field the combination of "dumbbell" and centrifugal potentials produces an effective potential well which has a single minimum; however, as the field increases the potential well is more and more strained, and above a certain field intensity it splits into a double- well potential which has two minima located close to the extrerna of the dumbbell. This metamorphosis of the average potential experienced by the electron in the mov- ing frame has attracted considerable attention, both in quantum [6] and in classical descriptions [3, 5]. Generally speaking, the precise role of this and other intriguing features is not clear. Our understanding of the purely classical Kepler problem with a periodic driving is still far from complete, and further analysis is needed. For example, it is often argued, as an intuitive explana- tion of classical IFS, that the problem has two integrable limits, given respectively by the unperturbed Kepler motion and the free-field motion; the former limit is at- tained on neglecting the external field, the latter on neglecting interaction with the nucleus. On increasing the field intensity, the second limit is approached, and this should explain why the motion becomes stable. This naive interpretation is not satisfactory, because the free-field motion is not bounded; it consists of oscilla- tions pius a drift that would lead to fast ionization in an overwhelming majority of cases. As a matter of fact, at very large field intensities the survival probability drops to zero (see Fig. 1). In addition, the atomic part of the Hamiltonian is a singular perturbation of the free-field Hamiltonian: it is never "small" compared to the latter, because its effect is always quite strong in the proximity of the nucleus. Therefore, in order to explain IFS, one has to understand why, in the stabilized regime, the elec- tron does not come close to the nucleus, even though it remains at all times confined in a bounded region. In this paper we present results for a model of a classi- cal 3D hydrogen atom subject to a periodic sequence of 5-like pulses of fixed strength and direction and alternat- ing sign. The motion of the electron under such a driving is conveniently described in a stroboscopic way, by iterat- ing a map that yields the evolution over one complete period of the external kicking field. In comparison with 1050-2947/94/50(6)/5018(7)/$06. 00 50 5018 1994 The American Physical Society