Superadiabatic evolution and adiabatic transition probability between two nondegenerate levels isolated in the spectrum

Abstract

The Schrödinger equation in the adiabatic limit when the Hamiltonian depends analytically on time and possesses for any fixed time two nondegenerate eigenvalues e1(t) and e2(t) bounded away from the rest of the spectrum is considered herein. An approximation of the evolution called superadiabatic evolution is constructed and studied. Then a solution of the equation which is asymptotically an eigenfunction of energy e1(t) when t→−∞ is considered. Using superadiabatic evolution, an explicit formula for the transition probability to the eigenstate of energy e2(t) when t→+∞, provided the two eigenvalues are sufficiently isolated in the spectrum, is derived. The end result is a decreasing exponential in the adiabaticity parameter times a geometrical prefactor.