Universal transition prefactors derived by superadiabatic renormalization

For time-dependent two-state quantum systems, the transition probability is exponentially small in the adiabatic parameter in , with the exponent determined by a transition point tc in the complex time plane. The authors study the in -independent prefactors associated with different sorts of transition point (which need not correspond to complex degeneracies of the adiabatic energy). Unlike previous approaches the method they use does not make use of special functions. It consists of applying first-order perturbation theory to the Schrodinger equation obtained by transforming to a series of 'superadiabatic' bases clinging ever more closely to the evolving state. If the original matrix elements share a leading singularity (t-tc)r, and their fractional deviation from this is (t-tc)s, the prefactor is 4 sin2(pi s/2(2r+s+2)). This is universal in the sense of being invariant under time reparametrization and quantum changes of frame.