Nonperturbative and perturbative treatments of parametric heating in atom traps

We study the quantum description of parametric heating in harmonic potentials both nonperturbatively and perturbatively, having in mind applications to atom traps. The first approach establishes an explicit connection between classical and quantum descriptions; it also gives analytic expressions for properties such as the width of fractional frequency parametric resonances. The second approach gives an alternative insight into the problem and can be directly extended to take into account nonlinear effects. This is especially important for shallow traps.