Theory of Quantum Super Impulses

Abstract

A quantum impulse is a brief but strong perturbation that produces a sudden change in a wave function ψ(x). We develop a theory of quantum impulses, distinguishing between ordinary and super impulses. An ordinary impulse paints a phase onto ψ, while a super impulse—the main focus of this paper—deforms the wave function under an invertible map, μ:x→x^{′}. Borrowing tools from optimal-mass-transport theory and shortcuts to adiabaticity, we show how to design a super impulse that deforms a wave function under a desired map μ and we illustrate our results using solvable examples. We point out a strong connection between quantum and classical super impulses, expressed in terms of the path-integral formulation of quantum mechanics. We briefly discuss hybrid impulses, in which ordinary and super impulses are applied simultaneously. While our central results are derived for evolution under the time-dependent Schrödinger equation, they apply equally well to the time-dependent Gross-Pitaevskii equation and thus may be relevant for the manipulation of Bose-Einstein condensates.