Fast-forward problem in quantum mechanics

We show the way to speed up the time evolution of a wave function (WF), i.e., to fast-forward the WF in microscopic and macroscopic quantum mechanics, by controlling the driving potential with resultant regulation of the additional phase of the WF, so that a target state is obtained in a shorter time. We first present a general framework of the fast-forwarding of a WF in quantum mechanics and provide an example of the fast-forward of a WF in two-dimensional (2D) free space. Then the framework of the fast-forward is extended to macroscopic quantum mechanics described by the nonlinear Schrödinger equation. We show the fast-forward of (i) transport of Bose-Einstein condensates trapped by a moving 2D harmonic potential and (ii) propagation of a soliton both in free space and through a potential barrier (: macroscopic quantum tunneling).