Abstract
The classical dynamics of the hydrogen atom in parallel static and microwave electric fields are analysed. This work is motivated by recent experiments on excited hydrogen atoms in such fields, which show enhanced resonant ionization at certain combinations of field strengths.By analysing the dynamics using an appropriate representation and averaging approximations, a simple picture of the ionization process is obtained. This shows how the resonant dynamics are controlled by a separatrix that develops and moves through phase space as the fields are switched on and provides necessary conditions for a dynamical resonance to affect the ionization probability. In addition, these methods yield a simple approximate Hamiltonian that facilitates quantal calculations.Using high-order perturbation theory, we obtain a series expansion for the position of the dynamical resonance and an estimate for its radius of convergence. Because, unusually, the resonance island moves through the phase space, the position of the dynamical resonance does not coincide precisely with the ionization maxima. Moreover, there are circumstances in which the field switch-on time dramatically affects the classical ionization probability; for long switch times, it reflects the shape of the incipient homoclinic tangle of the initial state, making it impossible to predict the resonance shape. Additionally, for a similar reason, the resonance ionization time can reflect the timescale of the motion near the separatrix, which is therefore much longer than conventional static field Stark ionization. All these effects are confirmed using accurate Monte Carlo calculations using the exact Hamiltonian.The dynamical structures producing these effects are present in the quantum dynamics; so we conclude that, for sufficiently large principal quantum numbers, the effects seen here will also be observed in the quantum dynamics.
Highlights
Using high-order perturbation theory, we obtain a series expansion for the position of the dynamical resonance and an estimate for its radius of convergence
In figure we show the classical ionization probabilities for the envelope 16-50-16, in which the j = 7–10, 12, and 15–19 resonances are clearly visible, but the j = 6, 11 and 14, marked by the arrows, are missing: other calculations show that the j = 5 resonance is missing and theory suggests that the j = 2–4 resonance islands are too narrow to affect the ionization probability, that is Ie < Iec − Ie(0), as discussed in the previous section
We relate this graph to the ionization curve in figure 12 by recalling that a dynamical resonance affects the ionization probability only if it can transport an orbit to a region Ie > Iec, where Iec is defined by Fcrit(Iec, Im) = Fμ + Fs(j), so the resonance island width must exceed the difference Iec − Ie(0)
Summary
The Hamiltonian for the hydrogen atom in parallel static and microwave electric fields, as in the experiments of Galvez et al (2000, 2005), has been derived by Leopold and Richards (1991) and, provided the field envelope λ(t) changes sufficiently slowly, is given by. Where μ is the atomic reduced mass, e the electron charge and λ(t) the envelope function describing the passage of the atom through the cavity. This Hamiltonian has azimuthal symmetry so that the z-component of angular momentum, Im, is conserved. For the particular experiments described by Galvez et al (2005), λ(t) has the 16-113-16 configuration, meaning that it rises monotonically from zero to unity in 16 field periods, remains constant for 113 periods and decreases monotonically to zero in 16 periods.
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