Berry-like phases in structured atoms and molecules

Quantum mechanical phases arising from a periodically varying Hamiltonian are considered. These phases are derived from the eigenvalues of a stationary, “dressed” Hamiltonian that is able to treat internal atomic or molecular structure in addition to the time variation. In the limit of an adiabatic time variation, the usual Berry phase is recovered. For more rapid variation, nonadiabatic corrections to the Berry phase are recovered in perturbation theory, and their explicit dependence on internal structure emerges. Simple demonstrations of this formalism are given, to particles containing interacting spins, and to molecules in electric fields.

Figure 1

The axis of rotation with laboratory-fixed coordinates $\left\{\xi ,\eta ,\zeta \right\}$ as well as the field coordinates defined by $\left\{x,y,z\right\}$. The field direction rotates about the $\zeta$-axis with angular frequency ${\omega }_r$.

Figure 2

(Color online) The extra phase accumulated due to the rotation of the field. In the limit of very fast rotation, ${\omega }_r⪢{\omega }_L$, the system accumulates a phase of $4\pi$, which is unobservable. In this limit, the states are best represented by projections onto the axis of rotation. The various lines represent values of ${\theta }_r$ between $\pi /2$ (bottom line) and 0 (top line) in steps of $\pi /10$. As can be seen, when ${\theta }_r$ is zero, there is no measurable phase shift, since there is no enclosed solid angle.