Fast forward to the classical adiabatic invariant

TL;DR

This paper introduces a method to preserve the classical action invariant under non-adiabatic conditions by constructing a fast-forward potential, enabling control of trajectories and potential applications in quantum shortcuts.

Contribution

It presents a novel approach to maintain classical adiabatic invariants under rapid changes by designing a specific fast-forward potential energy function.

Findings

Successfully preserves classical action in non-adiabatic regimes

Constructs a local dynamical invariant constant along trajectories

Provides numerical simulations demonstrating the method

Abstract

We show how the classical action, an adiabatic invariant, can be preserved under non-adiabatic conditions. Specifically, for a time-dependent Hamiltonian $H = p^2/2m + U(q,t)$ in one degree of freedom, and for an arbitrary choice of action $I_0$, we construct a "fast-forward" potential energy function $V_{\rm FF}(q,t)$ that, when added to $H$, guides all trajectories with initial action $I_0$ to end with the same value of action. We use this result to construct a local dynamical invariant $J(q,p,t)$ whose value remains constant along these trajectories. We illustrate our results with numerical simulations. Finally, we sketch how our classical results may be used to design approximate quantum shortcuts to adiabaticity.