Geometric amplitude factors in adiabatic quantum transitions

Abstract

Abstract The exponentially small probability of transition between two quantum states, induced by the slow change over infinite time of an analytic hamiltonian Ĥ = H(δt). Ŝ ( where δ is a small adiabatic parameter and Ŝ is the Vector spin -½ operator), contains an additional factor exp{ᴦg} of purely geometric origin (that is, independent of δ and ħ). For ᴦg to be non-zero, Ĥ must be complex hermitian rather than real symmetric, and the hamiltonian curve H(ז) must not lie in a plane through the origin nor be a helix identical (up to rigid motion) with its time reverse. An expression is given for ᴦg, as an integral from the real t axis to the complex time of degeneracy of the two states. Explicit examples are given. The geometric effect could be observed in experiments with spinning particles.