Revealing Anisotropy in a Paul Trap Through Berry Phase

Abstract

When an ion confined in an anisotropic bidimensional Paul trap is subjected to a laser beam oriented along an arbitrary direction, the interaction between its electronic and vibrational degrees of freedom is described by a time-dependent Hamiltonian model as a consequence of the lack of symmetry. Appropriately choosing the laser frequency, the Hamiltonian model turns out to be sinusoidally oscillating at the difference between the proper frequencies of the center of mass of the ion. Thus, if the anisotropy of the trap is sufficiently small, the evolution of the system can be considered as adiabatic. In the context of this physical situation we have calculated the Berry phase acquired in a cycle by the instantaneous eigenstates of the trapped ion Hamiltonian. Suitably choosing the initial condition and a physical observable we succeed to forecast physical effects directly traceable back to the accumulated Berry phase. In particular we indeed bring to light that the mean value of the chosen observable after a cycle is the negative of that calculated at the same instant of time in the case of isotropic traps. This effect demonstrates that and how the Berry phase can be exploited to evidence the existence of a weak anisotropy in a Paul trap.