Using techniques of complex analysis in an algebraic approach, we solve the wave equation for a two-level atom interacting with a monochromatic light field exactly. A closed-form expression for the quasienergies is obtained, which shows that the Bloch-Siegert shift is always finite, regardless of whether the original or the shifted level spacing is an integral multiple of the driving frequency $\ensuremath{\omega}$. We also find that the wave functions, though finite when the original level spacing is an integral multiple of $\ensuremath{\omega}$, become divergent when the intensity-dependent shifted energy spacing is an integral multiple of the photon energy. This result provides an ab initio theoretical explanation for the occurrence of the Freeman resonances observed in above-threshold ionization experiments.
Algebraic methods are developed for deriving steady-state solutions of the Bloch equations for a system of
N atomic (molecular) levels resonantly coupled to strong coherent radiation fields in the rotating-wave approximation, and undergoing thermal (radiative and collisional) homogeneous relaxation. These methods are applied to a "ladder" model, in which the excitation and relaxation occur by one-level steps, with one field mode in resonance with each step. Explicit solution (in terms of coupling strengths, relaxation rates, and frequency detunings) is given for the three-level system, and some observations are made on systems of four or more levels. The process of solution of the
N2-dimensional set of Bloch equations involves matrices of dimension $N\ensuremath{-}1$ or less.
To produce electrons with well-defined spin via photoionization, realizing orbital-resolved ionization is the prerequisite due to spin-orbital coupling. In this paper, we numerically and experimentally demonstrate that the electrons emitted from two original energy degenerated orbitals (
j=3/2,m=1 and $j=3/2, m=\ensuremath{-}1$) of Xe atoms can be well separated in the energy domain through selectively achieving Freeman resonance during strong-field ionization using a circularly polarized laser field. This orbital resolved Freeman resonant ionization leads to strong spin sensitivity without the need of fine laser frequency and intensity tuning. This result enables the production of high-degree spin-polarized electrons by energy gating, which provides a unique degree of freedom for advancing many promising applications.