Adiabatic perturbation theory invariant under the unitary transformations

Abstract

We propose a new formalism of the adiabatic perturbation theory that is invariant under the unitary transformations. A strict distinction is made between the Hamiltonian operator and the instantaneous energy operator. The state of the system is expanded in terms of the eigenstates of the instantaneous energy operator. The expansion coefficients, which represent the transition amplitudes among instantaneous energy eigenstates, are invariant under the unitary transformations. A perturbative series, whose zeroth-order term is the adiabatic approximation invariant under the unitary transformations, is generated up to second order for the state of the system. A diagrammatic representation of the perturbative series is developed. The first-order nonadiabatic correction yields two first-order adiabatic conditions, of which the first one is the unitary-invariant version of the traditional adiabatic condition. A simple physical interpretation is given for it in terms of the physical power operator. The second-order nonadiabatic correction yields two second-order adiabatic conditions. The second first-order condition and the first second-order condition are the unitary-invariant versions of the supplemented conditions derived by Tong et al. [Phys. Rev. Lett. 98, 150402 (2007)]. The second second-order adiabatic condition provides an additional condition setting the bound on the total evolution time.