Singular Loops and their Non-Abelian Geometric Phases in Spin-1 Ultracold Atoms. (arXiv:1801.00586v1 [cond-mat.quant-gas])

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. Here we use ultracold atoms to study the unique properties of spin-1
geometric phase. 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. This suggests a generalization of Berry's phase to include closed paths
that go inside the Bloch sphere. Under this generalization, the special class
of loops that pass through the center, which we refer to as \textit{singular
loops}, are significant in two ways. First, their geometric phase is
non-Abelian and second, their geometrical properties are qualitatively
different from the nearby non-singular loops, making them akin to critical
points of a quantum phase transition. Here we use coherent control of ultracold
$^{87}$Rb atoms in an optical trap to experimentally explore the geometric
phase of singular loops in a spin-1 quantum system.