Inconsistency in the Application of the Adiabatic Theorem

The adiabatic theorem states that an initial eigenstate of a slowly varying Hamiltonian remains close to an instantaneous eigenstate of the Hamiltonian at a later time. We show that a perfunctory application of this statement is problematic if the change in eigenstate is significant, regardless of how closely the evolution satisfies the requirements of the adiabatic theorem. We also introduce an example of a two-level system with an exactly solvable evolution to demonstrate the inapplicability of the adiabatic approximation for a particular slowly varying Hamiltonian.

Figure 1

Fidelity $\mathcal{F}$ of Eq. (17) for ${\omega }_0=1{\mathrm{s}}^{-1}$ and $\tau =2\pi \times 10{\mathrm{s}}^{-1}$.