Berry phase in atom optics

The geometric phase manifests itself in many different phenomena of physics ranging from the polarization change in the propagation of light in fibers via the precession of a neutron in a magnetic field, to the quantum dynamics of dark states in an atom. Most recently, the geometric phase has also been used in topological quantum computing as realized with trapped ions . In the present thesis we propose a scheme to observe the geometric phase in the context of atom optics. The concept of the geometric phase arises in the context of a Hamiltonian dependent on the parameters which are slowly varying in time. When these variations are cyclic, that is the Hamiltonian returns to its initial form the instantaneous eigenstate will not necessarily regain its original value, but will pick up a phase. This phenomenon has been observed in experiments with polarized light, radio waves, molecules, etc. Several proposals were given for the observation of the geometric phase in atom interferometry. However, so far only the dependence on the atomic internal degrees of freedom was investigated. In the present thesis we extend this approach by taking into account atomic external degrees of freedom, that is the center-of-mass-motion of the atom. We consider the scattering of a two-level atom from a near-resonant standing light wave. Within the Raman-Nath approximation on the atomic center-of-mass motion, adiabatic turn-on and -off of the interaction together with the rotating wave approximation we obtain a condition for the cancellation of the dynamical phase and show that the scattering picture is determined only by the Berry phase dependent on the internal and external atomic degrees of freedom. Moreover, we propose a novel possibility to observe the Berry phase based on the atomic lens construction. This application of the Berry phase might even be useful in the realm of lithography with cold atoms