Non-abelian berry's phase effects and optical pumping of atoms

Jan Segert

R. R. Aldinger, A. B�hm, Mark E. Loewe

Jan Segert

The non-Abelian Berry’s phase effect for a family of operators H0+k⋅V is considered, where H0 is rotationally invariant, V is a vector operator, and k varies over the unit vectors in R3. The parameter space is the two-sphere. The time evolution in the adiabatic limit is given by a connnection on a fiber bundle over the two-sphere. All connections consistent with the rotational symmetry are classified, and the time evolution is explicitly calculated for a nondegenerate Hamiltonian, as well as for a Hamiltonian with a double degeneracy. In the nondegenerate case, the connection is uniquely determined by the symmetry. In the doubly degenerate case, the connection is in some instances not determined by the symmetry. The case of approximate degeneracy is also discussed. A possible experimental test of this effect using optical pumping is described elsewhere.

Ho-Meoyng Choi, Chueng‐Ryong Ji, Hyun Kyu Lee

We observe that the non-Abelian Berry’s potential can appear in the atomic system under the slowly rotating electric field (Stark effect). We calculate Berry’s phase using the perturbation theory in the weak external electric field. To elaborate the nature of non-Abelian character, we demonstrate the change of states during the adiabatic transportation for a particular set of paths.

Evgenii B. Aleksandrov, N. N. Yakobson

X. B. Wang, Koichi Matsumoto, Heng Fan +2 more

In a nondegenerate syste, the abelian Berry's phase will never cause transitions among the Hamiltonian's eigenstate. However, in a degenerate syatem, it is well known that the state transition can be caused by the non-abelian Berry phase. Actually, in a such a system, the phase factor is not always nonabelian. Even in the case that the phase is a nondiagonal matrix, it still can be an abelian phase and can cause the state transitions. We then propose a simple scheme to detect such an effect of the Berry phase to the degenerate states by using the optical entangled pair state. In this scheme, we need not control any external parameters during the quantum state evolution.