Population permutation group and phase operation in an anisotropic semiconductor quantum dot

We investigate the experimental realization of the population permutation group and the phase operation of a three-level system in an anisotropic quantum dot. A three-dimensional representation of the permutation group can be generated from the identity matrix and three off-diagonal operation matrices, which are the transformation matrices of the three-level system. The ultrafast optical methods to realize all the elements of the group are proposed. And the theoretical analysis coincides with the experimental observations. With the diagonal operations and the rotation operations along the equator of the qubit sphere, all the π-phase operations for the three states can be realized. These methods would be useful for the physical realization of population permutation group, multiple qubit rotation and Deutsch-Jozsa-like algorithm in a multilevel system.