Simple vectorial model for the spectrum of a two-level atom in an intense low-frequency field

Abstract

An ansatz is proposed for the analytic solution of the Heisenberg equations of a two-level atom in a classical, linearly polarized and intense low-frequency field. According to this ansatz, the motion of the vector representing the state of the atom on the Bloch sphere consists of a fast precession around a direction, which in turn basculates slowly around an axis perpendicular to the field. The time evolution of the atomic state, which is obtained analytically by this vectorial model, is shown to compare quite satisfactorily with the exact numerical solution of the Heisenberg equations. The limits of validity of the ansatz are discussed. The spectrum of the light scattered by the two-level atom is evaluated analytically using the ansatz and it is shown to reproduce very precisely the exact spectrum evaluated numerically. It is shown that the slow component of the motion of the atomic Bloch vector yields the high-order harmonic part of the spectrum, whereas the fast precession gives rise to the hyper-Raman part. The appearance of the overall spectrum is discussed as a function of the various physical parameters of the model. The insight provided by the vectorial model is exploited to investigate the manifest universality of the main features of the overall spectrum and to suggest experiments to test the validity of the vectorial model.

Pair production at the focus of two equal and oppositely directed laser beams: The effect of the pulse shape

“…Finally, since some approximations have been done to obtain the final result Eq. (23) it is useful to compare it with the exact production probability obtained by numerically integrating the Schroedinger equation in the form (5) with the pulse shape (21) and by evaluating Eq. (7).…”

“…(7) after integrating numerically the Schroedinger equation in the form (5) with the pulse shape (21) and of the approximated asymptotic production probabilities (continuous curve) calculated by means of Eq. (23). The probabilities are plotted as functions of the peak laser electric field E expressed in unit of E 1 and the numerical values of all the parameters are those given below Eq.…”

High order harmonic generation: the role of the acceleration matrix elements and of the bound and continuum transitions

“…In particular it must be remarked that the two-level atom is appealing because it reproduces all the main experimental characteristics of the high order harmonic generation such as the presence of odd harmonics, plateau and cut-o and yields clear predictions that in part have been observed; it allows fast numerical calculations; it lends itself to analytical treatment [25][26][27][28] helping in understanding the physics of the nonlinear interaction and the interplay between the parameters and in nding suitable ranges of parameters that otherwise would escape numerical trials and casual attempts. However the two-level atom does not allow for ionization and does not support an energy continuum and it can be directly compared to more realistic models only with di culty.…”

Analytic expression for exact-ground-state energy based on an operator method for a class of anharmonic potentials

“…For convenience, let us set S in Eqs. (23,24) to unity so that the problem is reduce to the solution of R, T from the following equations…”

“…The study of anharmonic potential [18][19][20] has always been an exciting and interesting field due to its broad applications in quantum field theory [21,22], nuclear models, atomic and molecular physics [23][24][25], condensed matter physics [26,27], statistical physics and chemical physics [28]. Indeed, numerous numerical methods including renormalized strong coupling expansion, perturbation expansion, supersymmetric quantum mechanics, WKB, iteration based on the generalized Bloch equation, state-dependent diagonalization, Hill determinant method, phase-integral approach, iterative Bogoliubov transformations, eigenvalue moment method, perturbativevariation, and algebraic method have been proposed to investigate these anharmonic potentials [29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45].…”