Shortcuts To Adiabaticity for L\'evy processes in harmonic traps

Abstract

L\'evy stochastic processes, with noise distributed according to a L\'evy 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 used for stochastic processes with additive Gaussian noise, are typically based on a inverse-engineering approach. We generalise the approach to the wider class of L\'evy stochastic processes, both in the overdamped and in the underdamped regime, by finding exact equations for the relevant characteristic functions in Fourier space.