Gapped adiabatic quantum evolution
For a fixed smooth finite-dimensional Hamiltonian protocol H(s), s=t/T, an initially isolated nondegenerate eigenstate with a nonzero gap throughout the path follows its instantaneous eigenspace in the sufficiently slow limit.
Key knowledge
For a fixed smooth finite-dimensional Hamiltonian protocol H(s), s=t/T, an initially isolated nondegenerate eigenstate with a nonzero gap throughout the path follows its instantaneous eigenspace in the sufficiently slow limit. It acquires dynamical phase −∫E_n dt/ħ and geometric phase i∫⟨n|ṅ⟩dt. Off-diagonal mixing is governed by ⟨m|ṅ⟩=⟨m|Ḣ|n⟩/(E_n−E_m).
Values and equations
|Ψ(t)⟩≈e^[−(i/ħ)∫E_n dt] e^(iγ_n)|n(t)⟩; γ_n=i∫⟨n|ṅ⟩dt; ⟨m|ṅ⟩=⟨m|Ḣ|n⟩/(E_n−E_m)
- H,E_n,E_m
- Hamiltonian and instantaneous energies J
- t,T
- Time and protocol duration s
- s,γ_n
- Dimensionless protocol parameter and phase
- ħ
- Reduced Planck constant J s
- |Ψ⟩,|n⟩,|m⟩
- Normalized actual/instantaneous states
- Ḣ
- Time derivative of Hamiltonian J s⁻¹
Conditions and scope
- Closed unitary system, normalized smooth instantaneous eigenvectors along the path, fixed smooth protocol and a gap uniformly bounded away from zero; adiabatic error must be controlled for the actual protocol and duration.
- The local dimensionless ratio ħ|⟨m|Ḣ|n⟩|/|E_n−E_m|² is a useful diagnostic, not a universal necessary-and-sufficient error guarantee. Gap closure, rapid resonant structure or degeneracies need their own analysis. Energy can change even when state population follows the eigenspace.
Sources · 1
References
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Evidence and review
Quantum Physics III: chapter 6, adiabatic approximation (2017 notes, 2018 course): §2 equations 2.12–2.17 and §3 equations 3.1–3.12, printed pages 5–8: instantaneous-state ansatz, exact basis couplings and phase, with explicit failure of an exact unqualified ansatz. §4.14–4.18 discuss scale diagnostics; excludes the heuristic ΔE≈H as a gauge/energy-zero independent theorem.
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