Necessary and sufficient condition for quantum adiabatic evolution by unitary control fields

Necessary and sufficient condition for quantum adiabatic evolution by unitary control fieldsThis work derives the necessary and sufficient condition for quantum adiabatic evolution, that is, when a system remains in an eigenstate of a Hamiltonian even if this Hamiltonian changes over time. It settles a problem that had been identified in the conditions for adiabatic quantum evolution that had been formulated over the last 50 years or so. With the finding that the widely used quantum adiabatic quantitative condition is not always valid, various new necessary or sufficient adiabatic conditions were proposed for replacing the traditional one. However, none of these conditions has been successfully shown to be both necessary and sufficient. Our work settles the problem by providing a simple condition with proofs for both its necessity and sufficiency, by using a gauge invariant formalism to extract all the nonadiabatic transitions. Counterintuitively, the condition reveals that quantum adiabatic evolution allows rapid changes and/or arbitrary numbers of energy crossings in the system Hamiltonian. New ways to achieve quantum adiabatic evolution by pulse sequences or fast varying fields are demonstrated.