Differential topology of adiabatically controlled quantum processes

It is shown that in a controlled adiabatic homotopy between two Hamiltonians, H ₀ and H ₁, the gap or “anti-crossing” phenomenon can be viewed as the development of cusps and swallow tails in the region of the complex plane where two critical value curves of the quadratic map associated with the numerical range of H ₀ + i H ₁ come close. The “near crossing” in the energy level plots happens to be a generic situation, in the sense that a crossing is a manifestation of the quadratic numerical range map being unstable in the sense of differential topology. The stable singularities that can develop are identified and it is shown that they could occur near the gap, making those singularities of paramount importance. Various applications, including the quantum random walk, are provided to illustrate this theory.