Figure 1: Schematic illustration of spin geometric phases in quantum rings.
From: Control of the spin geometric phase in semiconductor quantum rings
(a) Quantum ring of radius r placed in a crown-like magnetic field with an angle θ with respect to the z axis. (b) In the electron’s rest frame, the textured field subtends a solid angle (blue) in a round trip around the ring. In adiabatic transport, the solid angle is proportional to the geometric phase. Spin precession around B is associated with the dynamical phase. (c) Rashba SO coupling induces an effective in-plane magnetic field BSO perpendicular to the electron momentum P. (d) In non-adiabatic transport, the effective magnetic field has an additional z component proportional to ω0pr/2. The geometric phase is proportional to the solid angle subtended by the total effective magnetic field. (e) The geometric phase in a Rashba ring can be controlled by adding an in-plane field 
. (f) The in-plane field modulates the geometric phase via changing the solid angle subtended by the total effective field.
