Quantum resonances and ratchets in free-falling frames

Quantum resonance (QR) is defined in the free-falling frame of the quantum kicked particle subjected to gravity. The general QR conditions are derived. They imply the rationality of the gravity parameter η, the kicking-period parameter τ (2π), and the quasimomentum β. Exact results are obtained concerning wave-packet evolution for arbitrary periodic kicking potentials in the case of integer τ (2π) (the main QRs). It is shown that a quantum ratchet generally arises in this case for resonant β. The noninertial nature of the free-falling frame affects the ratchet by effectively changing the kicking potential to one depending on (β,η). For a simple class of initial wave packets, it is explicitly shown that the ratchet characteristics are determined to a large extent by symmetry properties and by number-theoretical features of η.