Fast-forward scaling theory for phase imprinting on a BEC: creation of a wave packet with uniform momentum density and loading to Bloch states without disturbance

We study phase imprinting on Bose–Einstein condensates (BECs) with the fast-forward scaling theory revealing a nontrivial scaling property in quantum dynamics. We introduce a wave packet with uniform momentum density (WPUM) which has peculiar properties but is short-lived. The fast-forward scaling theory is applied to derive the driving potential for creation of the WPUMs in a predetermined time. Fast manipulation is essential for the creation of WPUMs because of the instability of the state. We also study loading of a BEC into a predetermined Bloch state in the lowest band from the ground state of a periodic potential. Controlled linear potential is not sufficient for creation of the Bloch state with large wavenumber because the change in the amplitude of the order parameter is not negligible. We derive the exact driving potential for creation of predetermined Bloch states using the obtained theory.