Resonant tunnelling of a wavepacket in the sequential tunnelling model

The tunnelling of a wavepacket through a resonant tunnelling system is studied using the sequential tunnelling model. Fermi's Golden Rule cannot be applied to this process because the wavepacket is a coherent superposition of states which have different energies and in the Fermi Golden Rule it is assumed that contributions to the transition rate from states at different energies add incoherently. A time-dependent analysis shows that the occupancy of the resonant level increases as t², where t is the time after the wavepacket is incident on the barrier, for small values of t. If the tunnelling from the resonant level to the electrodes is ignored the occupancy of the resonant level is constant for large values of t which is expected because tunnelling from the wavepacket to the resonant level can only occur during the time that the wavepacket is in contact with the resonant tunnelling system. When a sequence of wavepackets is incident on the resonant tunnelling system the occupancy of the resonant level at large times reaches the value predicted by the Fermi Golden Rule.