# Population control in nearly degenerate N-state atoms

An analytic description is presented for the dynamics of electronic population in nearly degenerate N-state atoms. The time-dependent Schrodinger equation, written in terms of the probability amplitudes, becomes analytically solvable in the degenerate limit, when all the states accessible by the system have the same energy. Two-state and three-state atoms, as well as possible generalizations on the arbitrary number of states, are considered in detail The degenerate state approximation allows one to gain additional control over the system by adjusting the parameters of a control field, including the field's frequency, intensity and the shape. The limitations on the frequency of the control field are much less restrictive than in other quantum control schemes, and based on the following conditions. To justify the use of N-state model, the frequency of the control field should be less than the resonant frequency of the transition from any of the active states to the nearest state outside the N-state manifold. At the same time, it should be much higher than the resonant frequency of the transitions between the active states (this enables the use of degenerate model for nearly degenerate states) These conditions define the structure of the electronic states of systems to which the degenerate states approximation can be applied: N states with approximately the same energy (active states), well isolated from all other states. We have identified some of the systems that have this kind of electronic structure, and used them to illustrate the method, as well as to estimate its accuracy and robustness Our overall goal is to generally determine conditions on both the external field and the transition matrix elements that lead to complete control of a population in a nearly degenerate N-state atom. The effects of the field parameters on the duration of the transferred state and the population leakage due to non-degeneracy are discussed. Additional requirements on the relative strength of the transition matrix elements are stated for three-state atoms. The conditions on the transition matrix and the external field are formulated conveniently in terms of quantum numbers. Different cases are considered where analytic solutions exist for many-state system. Detailed calculations are presented for transfer of the electrons in atomic hydrogen and lithium. The results of the calculations are compared with the exact numerical solutions of the coupled channel equations, obtained by direct integration using the standard 4th order Runge-Kutta algorithm