Wave packet dynamics of an atomic ion in a Paul trap : Approximations and stability

Using numerical simulations of the time-dependent Schrödinger equation, we study the fullquantum dynamics of the motion of an atomic ion in a linear Paul trap. Such a trap is based on atime-varying, periodic electric ̄eld and hence corresponds to a time-dependent potential for theion, which we model exactly. We compare the center-of-mass motion with that obtained fromclassical equations of motion, as well as to results based on a time-independent e®ective po-tential. We also study the oscillations of the width of the ion's wave packet, including close tothe border between stable (bounded) and unstable (unbounded) trajectories. Our results con- ̄rm that the center-of-mass motion always follows the classical trajectory, that the width of thewave packet is bounded for trapping within the stability region, and therefore that the classicaltrapping criterion is fully applicable to quantum motion