  Manipulation of superpositions of discrete quantum states has a mathematical
counterpart in the motion of a unit-length statevector in an N-dimensional
Hilbert space. Any such statevector motion can be regarded as a succession of
two-dimensional rotations. But the desired statevector change can also be
treated as a succession of reflections, the generalization of Householder
transformations. In multidimensional Hilbert space such reflection sequences
offer more efficient procedures for statevector manipulation than do sequences
of rotations. We here show how such reflections can be designed for a system
with two degenerate levels - a generalization of the traditional two-state atom
- that allows the construction of propagators for angular momentum states. We
use the Morris-Shore transformation to express the propagator in terms of
Morris-Shore basis states and Cayley-Klein parameters, which allows us to
connect properties of laser pulses to Hilbert-space motion. Under suitable
conditions on the couplings and the common detuning, the propagators within
each set of degenerate states represent products of generalized Householder
reflections, with orthogonal vectors. We propose physical realizations of this
novel geometrical object with resonant, near-resonant and far-off-resonant
laser pulses. We give several examples of implementations in real atoms or
molecules.
