We say that a system exhibits monodromy if we take the system around a closed loop in its parameter space, and we find that the system does not come back to its original state. Many systems have this property: atoms in a trap, a hydrogen atom in crossed fields, and linear and quasilinear molecules.

A hydrogen atom in weak perpendicular and near-perpendicular electric and magnetic fields is one of the systems exhibiting quantum monodromy. Using perturbation theory, Sadovskii and Cushman predicted the presence of monodromy in perpendicular fields. It shows up as a defect in the lattice of quantum states. When the fields are tilted from perpendicular, these lattice defects undergo a series of bifurcations.

A newly discovered dynamical manifestation of monodromy can be illustrated by the behavior of atoms in a trap. Let us imagine a collection of noninteracting classical particles moving in a two-dimensional circular box with a hard reflecting wall, and with a cylindrically-symmetric potential energy barrier. Let us start all the particles moving on one line with angular momentum L = 0 , and with energy E < 0. Then let us impose additional smooth forces and torques on the particles so that [L(t),E(t)] move in a circle around the origin in the [L,E] plane. In other words, apply a torque to increase the angular momentum, then drive the particles to a higher energy (above the barrier), then reduce the angular momentum to a negative value, reduce the energy, and finally come back to the initial energy and angular momentum. Where in space do the particles end up? The answer is surprising.