Fast-forward of adiabatic dynamics in quantum mechanics

Abstract

We propose a method to accelerate adiabatic dynamics of wave functions (WFs) in quantum mechanics to obtain a final adiabatic state except for the spatially uniform phase in any desired short time. In our previous work, acceleration of the dynamics of WFs was shown to obtain the final state in any short time by applying driving potential. We develop the previous theory of fast-forward to derive a driving potential for the fast-forward of adiabatic dynamics. A typical example is the fast-forward of adiabatic transport of a WF, which is the ideal transport in the sense that a stationary WF is transported to an aimed position in any desired short time without leaving any disturbance at the final time of the fast-forward. As other important examples, we show accelerated manipulations of WFs, such as their splitting and squeezing. The theory is also applicable to macroscopic quantum mechanics described by the nonlinear Schrödinger equation.

Fast frictionless expansion of an optical lattice

“…We propose to repeat the experiment [28] with the same trajectory but a different potential depth and a different frequency. If they change according to (9,10), then fast frictionless expansion can be realized. So far, we have studied 1-D optical lattice with variable spacing.…”

“…where dot is the time derivation with respect to the parameter t. The acceleration,Λ, is set to zero at initial and final times since the initial and final external frequencies are assumed to be zero (3,10). These conditions together with the conditions (2) guarantee the fast transitionless evolution.…”

Engineering fast and stable splitting of matter waves

“…The wave splitting via shortcuts avoids the final excitation and turns out to be signifficantly more stable than the adiabatic following with respect to the asymmetric perturbation. Specifically we shall use a simple inversion method: a streamlined version [6] of the fast-forward technique of Masuda and Nakamura [7] applied to GrossPitaievski (GP) or Schrödinger equations. We have previously found some obstacles to apply the invariantsbased method (at least using quadratic-in momentum invariants [6]) and the transitionless-driving algorythm [8] (because of difficulties to implement in practice the counter-diabatic terms).…”

“…Fast-forward approach.-The fast-forward method [6,7,9] may be used to generate external potentials to drive the matter wave from the initial single well to a final symmetric double well. The starting point of the streamlined version in [6] is the 3D time-dependent GP equation…”

“…Specifically we shall use a simple inversion method: a streamlined version [6] of the fast-forward technique of Masuda and Nakamura [7] applied to GrossPitaievski (GP) or Schrödinger equations. We have previously found some obstacles to apply the invariantsbased method (at least using quadratic-in momentum invariants [6]) and the transitionless-driving algorythm [8] (because of difficulties to implement in practice the counter-diabatic terms).Fast-forward approach.-The fast-forward method [6,7,9] may be used to generate external potentials to drive the matter wave from the initial single well to a final symmetric double well. The starting point of the streamlined version in [6] is the 3D time-dependent GP equation…”