Quantum Fermi acceleration in the resonant gaps of a periodically driven one-dimensional potential box

We study numerically the quantum mechanics of a point particle in the one-dimensional potential box, whose boundary oscillates periodically according to the sawtooth driving law. We perform very accurate numerical calculations over up to about 500 periods. Unlike in smooth driving, this system admits classical Fermi acceleration, because the Kolmogorov–Arnold–Moser theorem does not apply, and surprisingly also admits quantum Fermi acceleration, but only in the extremely narrow resonant gaps located at the values of the driving parameter that corresponds to either one-photon transitions or multi-photon transitions. In the gaps the energy of the particle increases indefinitely, quadratically with time, as predicted by our quite general two-level theory derived for general periodic quantum systems, whilst outside the gaps it oscillates and displays beatings with long or very long periods which are theoretically unexplained, but very well and clearly manifested. (paper)