Adiabatic Theorem & Berry Phase

γ = i∮⟨n(R)|∇_R|n(R)⟩·dR — geometric phase from parallel transport on the Bloch sphere




Berry phase γ: -
Solid angle Ω: -
Adiabaticity: -

Bloch Sphere — Spin Direction

Spin Expectation Values vs Time

Berry Phase Accumulation

Adiabatic vs Diabatic: ⟨σz⟩ at end of cycle

Adiabatic theorem (Born-Fock 1928): If H(t) changes slowly enough, a system initially in the nth eigenstate remains in the nth instantaneous eigenstate. "Slowly" means Ω ≪ ω₀ = ΔE/ℏ, the level gap.

Berry phase (1984): Even in perfect adiabatic evolution, the state acquires a geometric phase γ_n = i∮⟨n(R)|∇_R|n(R)⟩·dR that depends only on the geometry of the path in parameter space — not on how fast it was traversed. For a spin-½ in a rotating field: γ = -Ω_solid/2 where Ω_solid is the solid angle enclosed.

Solid angle: For a cone of half-angle θ₀, Ω_solid = 2π(1-cos θ₀). The Berry phase γ = -π(1-cos θ₀). For θ₀=90° (equatorial circuit), γ = -π.

Diabatic limit: Fast rotation (Ω ≫ ω₀) → spin doesn't follow the field → Rabi oscillations. The transition from adiabatic to diabatic is governed by the Landau-Zener parameter.