Transmission through a potential barrier in an arbitrary quantum state

We investigate the scattering of a quantum-mechanical particle on an oscillating rectangular potential barrier. In the first step, we show the calculation for a classical potential revealing coherent, quantized energy exchange between the particle and potential, respectively, the likewise classical field that generates the latter. In order to explore the quantized energy transfer in more detail, the transition to a quantum field is performed and the Schr\"odinger equation for the composite quantum system consisting of particle and field is solved analytically as well. The result allows a physically very intuitive explanation of the underlying energy exchange processes and explicitly displays the occurring entanglement between particle and field due to their interaction, thus offering the possibility to use this combined system for quantum information experiments. Depending on the initial field state, incoherent or coherent energy exchange is observed. In the limit of high mean photon numbers the classical results can be fully retrieved.