Berry phase by description of the quantum systems in the Kustaanheimo-Stiefel bundle

Holger Lyre

Eamonn Cahill

The authors show how explicit use of the constraint equation that arises when the Kustaanheimo-Stiefel transformation (1965) is applied to the hydrogen atom in R 3 enables them to determine the wavefunctions for the constrained harmonic oscillator in R 4 with the correct restriction on the quantum numbers in a direct and natural manner. They then highlight and resolve a discrepancy between the R 4 oscillator wavefunctions produced by different authors by considering the measure to be used in R 4 and the operator ordering in the quantised Hamiltonian.

Gennadi Aleksandrovich Sardanashvily

We show that composite bundles Q→Σ→R, where Σ→R is the parameter bundle, provide the adequate mathematical description of mechanical systems with time-dependent parameters both in classical and quantum mechanics. In particular, the Berry phase phenomenon is described in terms of connections on composite Hilbert space bundles.

Pratul Bandyopadhyay

Jian-Te Wang, Jing-Xin Liu, Hai-Tao Ding +1 more

The Stiefel-Whitney insultor is a two-dimensional topological insulator protected by parity-time ($\mathcal{PT}$) symmetry. With a vanishing Chern number, the topology in this system is characterized by second Stiefel-Whitney class. We propose a feasible scheme to realize a four-band Stiefel-Whitney insultor with spin-orbit coupled ultracold atoms in an optical Raman lattice. Four selected spin states are coupled by carefully designed Raman lasers to generate the desired spin-orbit interactions with spacetime inversion symmetry. We map out a phase diagram with respect to the experimental parameters, where a large topological phase region exists. We further present two distinct detection methods to resolve the non-Abelian band topology, in both equilibrium and dynamical ways. The detection relies on the spin textures extracted from the time-of-flight imaging, showing the tomographic signatures in the ground states and long-time averaged patterns on certain submanifolds via a bulk-surface duality. Our work paves a realistic way to explore novel topology inside real Berry bundles with quantum matters.

R. R. Aldinger, A. B�hm, Mark E. Loewe