Geometric amplitude factors in adiabatic quantum transitions

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{ ᴦ g } of purely geometric origin (that is, independent of δ and ħ ). For ᴦ g 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 ᴦ g , 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.