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

Alain Joye (1) , Charles-Edouard Pfister (1)

1 EPFL - Ecole Polytechnique Fédérale de Lausanne

The Schriidinger equation in the adiabatic limit when the Hamiltonian depends analytically on time and possesses for any fixed time two nondegenerate eigen-values e,(t) and e,(f) 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 e,(t) when t- ,-co is considered. Using superadiabatic evolution, an explicit formula for the transition probability to the eigenstate of energy ez(t) when t+ + CO, 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.

Domaines: Mathématiques [math]; Physique mathématique [math-ph]

Alain Joye, Charles-Edouard Pfister. Superadiabatic evolution and adiabatic transition probability between two nondegenerate levels isolated in the spectrum. Journal of Mathematical Physics, 1993, 34 (2), ⟨10.1063/1.530255⟩. ⟨hal-01221129⟩