Geometry, topology and control of spin-1 quantum states

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Hebbe Madhusudhana, Bharath
Chapman, Michael S.
Kennedy, T. A. Brian
Etnyre, John B.
Sá de Melo, Carlos A. R.
Parker, Colin V.
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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 Rb atoms in an optical trap to experimentally explore the geometric phase of singular loops in a spin-1 quantum system.
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