The effect of parallel static and microwave electric fields on excited   hydrogen atoms
  Motivated by recent experiments we analyse the classical dynamics of a hydrogen atom in parallel static and microwave electric fields. Using an appropriate representation and averaging approximations we show that resonant ionisation is controlled by a separatrix, and provide necessary conditions for a dynamical resonance to affect the ionisation probability.   The position of the dynamical resonance is computed using a high-order perturbation series, and estimate its radius of convergence. We show that the position of the dynamical resonance does not coincide precisely with the ionisation maxima, and that the field switch-on time can dramatically affect the ionisation signal which, for long switch times, reflects the shape of an incipient homoclinic. Similarly, the resonance ionisation time can reflect the time-scale of the separatrix motion, which is therefore longer than conventional static field Stark ionisation. We explain why these effects should be observed in the quantum dynamics.   PACs: 32.80.Rm, 33.40.+f, 34.10.+x, 05.45.Ac, 05.45.Mt 
D. Richards