Synthesis of arbitrary SU(3) transformations of atomic qutrits

Several scenarios are proposed and analyzed for engineering of arbitrary preselected SU(3) transformations of laser-driven atomic qutrits. Two of the most natural implementations of qutrits are considered, in which the three qutrit states are coupled to each other via two-photon transitions through either (i) a common state in a tripod linkage pattern or (ii) two upper states in an M-shaped linkage chain. The SU(3) transformation for the tripod qutrit can be realized by 3 Givens SU(2) rotations, which require 9 consecutive interaction steps. Alternatively, because under certain conditions the propagator of the tripod system reduces to the Householder reflection operator, any SU(3) transformation can be constructed physically by 3 Householder reflections, each of which is implemented in a single interaction step. As an example, the discrete Fourier transform can be synthesized by 7 consecutive interaction steps with Givens rotations or, alternatively, by only a single Householder reflection and a phase gate. For the M-qutrit, the propagator is given by coupled Householder reflections and it cannot be reduced to Givens rotations or independent Householder reflections. By using these coupled Householder reflections it is shown that an arbitrary SU(3) transformation of the M qutrit can be realized with just two fields in at most 3 interaction steps; the discrete Fourier transform, in particular, requires only 2 interaction steps.