Adiabatic theorem

Remarks

Here $\varepsilon = 1/T$, where $T$ is the total time of the evolution in physical time $t = sT$ (with $\hbar = 1$), so $\varepsilon \to 0$ means a slow change of the Hamiltonian. The proof compares $U_\varepsilon$ with the adiabatic evolution, which maps $\operatorname{Ran} P(0)$ exactly onto $\operatorname{Ran} P(s)$. The difference is an oscillatory integral; one integration by parts, using the gap $g$ to invert the commutator with $H(s)$, gains the factor $\varepsilon$.

Example: let $\sigma_1(s) = \set{E(s)}$ be a simple eigenvalue, and prepare the system in its eigenstate at $s = 0$. At time $s$ it is found in the corresponding eigenstate of $H(s)$, up to a phase and an error $O(\varepsilon)$. The phase is the sum of a dynamical phase $-\varepsilon^{-1}\int_0^s E(u)\, du$ and a geometric phase.

The gap condition is necessary: when two eigenvalues cross or come close, the system can jump between them, as in Landau–Zener transitions.

Going deeper

For a student

The theorem is due in its first form to Born and Fock (1928), and in this operator form to Kato (1950). It is the quantum analogue of the classical adiabatic invariants. It underlies the Born–Oppenheimer approximation, where the slow nuclei play the role of the slowly varying parameter, and it is the source of the geometric phase. Adiabatic quantum computation also relies on it: one deforms a simple Hamiltonian into one whose ground state encodes the answer, and the running time is governed by the gap.