  Non-Abelian and non-adiabatic variants of Berry's geometric phase have been
pivotal in the recent advances in fault tolerant quantum computation gates,
while Berry's phase itself is at the heart of the study of topological phases
of matter. The geometrical and topological properties of the phase space of
spin$-1$ quantum states is richer than that of spin$-1/2$ quantum states and is
relatively unexplored. For instance, the spin vector of a spin-1 system, unlike
that of a spin$-1/2$ system, can lie anywhere on or inside the Bloch sphere
representing the phase space. Recently, a generalization of Berry's phase that
encapsulates the topology of spin-1 quantum states has been formulated in J.
Math. Phys., 59(6), 062105. This geometric phase includes loops that go inside
the Bloch sphere and is carried by the tensor of spin fluctuations, unlike
Berry's phase which is carried by the global phase of the quantum state.
Furthermore, due to a mathematical singularity at the center of the Bloch
sphere, the class of loops that pass through the center are called singular
loops and are significant because their geometric phase is non-Abelian. In
contrast with Berry's phase for spin$-1/2$ systems, whose properties come from
the topology of a sphere, the properties of singular loop geometric phases come
from the topology of the real projective plane $\mathbb{RP}^2$, which is more
non-trivial. Here we use coherent control of ultracold $^{87}$Rb atoms in an
optical trap to experimentally explore this geometric phase for singular loops
in a spin-1 quantum system.
