Shortcuts to adiabaticity for Lévy processes in harmonic traps.

Abstract

Lévy stochastic processes, with noise distributed according to a Lévy stable distribution, are ubiquitous in science. Focusing on the case of a particle trapped in an external harmonic potential, we address the problem of finding "shortcuts to adiabaticity": After the system is prepared in a given initial stationary state, we search for time-dependent protocols for the driving external potential, such that a given final state is reached in a given, finite time. These techniques, usually employed for stochastic processes with additive Gaussian noise, are typically based on a inverse-engineering approach allowing to find exact analytical solutions for the required protocol. We generalize the approach to the wider class of Lévy stochastic processes, both in the overdamped and (for pure translations) in the underdamped regime, by finding exact equationsfor the relevant characteristic functions in Fourier space.