Accelerated and Airy-Bloch oscillations

A quantum particle subjected to a constant force undergoes an accelerated motion following a parabolic path, which differs from the classical motion just because of wave packet spreading (quantum diffusion). However, when a periodic potential is added (such as in a crystal) the particle undergoes Bragg scattering and an oscillatory (rather than accelerated) motion is found, corresponding to the famous Bloch oscillations. Here we introduce an exactly-solvable quantum Hamiltonian model, corresponding to a generalized Wannier-Stark Hamiltonian $\hat{H}$, in which a quantum particle shows an intermediate dynamical behavior, namely an oscillatory motion superimposed to an accelerated one. Such a novel dynamical behavior is referred to as {\it accelerated Bloch oscillations}. Analytical expressions of the spectrum, improper eigenfunctions and propagator of the generalized Wannier-Stark Hamiltonian $\hat{H}$ are derived. Finally, it is shown that acceleration and quantum diffusion in the generalized Wannier-Stark Hamiltonian are prevented for Airy wave packets, which undergo a periodic breathing dynamics that can be referred to as {\it Airy-Bloch oscillations}.