Exploring Non-Abelian Geometric Phases in Spin-1 Ultracold Atoms

Abstract

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.

13 pages, 7 figures

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Cited by in corpus (10)

• Non-Abelian adiabatic geometric transformations in a cold Strontium gas
• Wilson loop and Wilczek-Zee phase from a non-Abelian gauge field
• Entangling quantum logic gates in neutral atoms via the microwave-driven spin-flip blockade
• Observation of spin-tensor induced topological phase transitions of triply degenerate points with a trapped ion
• Non-Abelian Geometric Phases Carried by the Spin Fluctuation Tensor
• An SU(2) geometric phase induced by periodically driven Raman process in an ultracold dilute Bose gas
• Spin-tensor Meissner currents of ultracold bosonic gas in an optical lattice
• Robust Weyl points in a 1D superlattice with transverse spin-orbit coupling
• An exactly solvable model for anyons with non-Abelian flux
• Inequivalence of Landau-Lifshitz and Landau-Lifshitz-Gilbert dynamics for a single quantum spin