Optimal time evolution for Hermitian and non-Hermitian Hamiltonians

Abstract

Interest in optimal time evolution dates back to the end of the seventeenth century, when the famous brachistochrone problem was solved almost simultaneously by Newton, Leibniz, l'Hôpital, and Jacob and Johann Bernoulli. The word brachistochrone is derived from Greek and means shortest time (of flight). The classical brachistochrone problem is stated as follows: A bead slides down a frictionless wire from point A to point B in a homogeneous gravitational field. What is the shape of the wire that minimizes the time of flight of the bead? The solution to this problem is that the optimal (fastest) time evolution is achieved when the wire takes the shape of a cycloid, which is the curve that is traced out by a point on a wheel that is rollingon flat ground. © 2010 Springer-Verlag Berlin Heidelberg.