Non-Abelian quantum holonomy of hydrogenlike atoms

We study the Uhlmann holonomy [Rep. Math. Phys. 24, 229 (1986)] of quantum states for hydrogenlike atoms where the intrinsic spin and orbital angular momentum are coupled by the spin-orbit interaction and are subject to a slowly varying magnetic field. We show that the holonomy for the orbital angular momentum and spin subsystems is non-Abelian while the holonomy of the whole system is Abelian. Quantum entanglement in the states of the whole system is crucially related to the non-Abelian gauge structure of the subsystems. We analyze the phase of the Wilson loop variable associated with the Uhlmann holonomy and find a relation between the phase of the whole system and corresponding marginal phases. Based on the results for the model system, we provide evidence that the phase of the Wilson loop variable and the mixed-state geometric phase [E. Sjöqvist et al., Phys. Rev. Lett. 85, 2845 (2000).] are generally inequivalent.

Figure 1

Figure-8 loop on the parameter sphere ${\mathcal{S}}^2$ of magnetic field directions. The Abelian geometric phase of the whole system along this path is zero because the net enclosed area is zero. In contrast, the phases of the Wilson loop variable associated with the Uhlmann holonomy for $L$ and $S$ are in general nonzero.

Figure 2

$\xi$ defined in Eq. (33) as a function of ${\phi }_1-{\phi }_0$ for $l=3$, $\mu =\frac{3}{2}$, and the coupling strengths $g=3,13,37,50$. The corresponding phase ${\gamma }_{L;\pm }^{(l,\mu )}$ of the Wilson loop variable associated with the Uhlmann holonomy of the $L$ subsystem is given by $\arg \xi$.

Figure 3

$\zeta$ defined in Eq. (38) as a function of ${\phi }_1-{\phi }_0$ for $l=2$, $\mu =\frac{1}{2}$, and $g=3,20,50$. The difference $\Delta {\gamma }^{(l,\mu )}={\gamma }_{J;\pm }^{(l,\mu )}-{\gamma }_{L;\pm }^{(l,\mu )}-{\gamma }_{S;\pm }^{(l,\mu )}$ of the phases of the corresponding Wilson loop variables is given by $-\arg \zeta$.