Berry Phase & Holonomy
Geometric Phase in Quantum Mechanics — Fiber Bundle Structure
Example System
Spin-½ in B field
Adiabatic 2-level
Photon polarization
Loop on Bloch Sphere
Loop polar angle θ_loop:
45°
Loop azimuthal steps:
60
Animation speed:
1×
▶ Start Loop
⏸ Pause / Resume
Berry Phase
Loop solid angle Ω
—
Berry phase γ_B
—
Phase (deg)
—
Holonomy
—
Step
0
Theory
γ_B = i∮⟨n|∇_R|n⟩·dR
For spin-½ on Bloch sphere:
γ_B = −½ Ω(C)
where Ω = solid angle of loop
Ω = 2π(1 − cos θ_loop)
Geometric (not dynamic):
independent of speed.
Physically: Aharonov-Bohm,
Pancharatnam polarization,
molecular conical intersections.