Algebraic derivation of the iterative‐perturbative method as a global squeezing and displacement transformation

Abstract

ABSTRACT:We make use of iterative Bogoliubov transformations (IBT) and employ a procedure derived by Mundarain et al. to derive an approximate analytic expression for the evolution operator of a system consisting of a quartic oscillator coupled to a time-dependent electric field. As a result, this expression resumes the whole iterative process as a global squeezing and displacement transformation in which the compression coefficients and translation parameters are obtained explicitly, thus giving a complete analytic structure to the process that supports and complements the conventional IBT and the bilinear approximation (Echave, J., et al. J Chem Phys 1990, 92, 1188, which are mainly numerical and semiclassic approaches respectively.

Evaluation of state-to-state transitions and squeezing properties in molecules of the type H–X using the recently derived analytical IBT

“…Nevertheless, making use of a methodology proposed by Yuen [7] in which the author takes into account the optical properties of two-photon coherent states, Mundarain et al [8] were able to derive an analytical expression for the evolution operator and evaluate the transition probabilities between the states of a harmonic oscillator, neglecting the higher order terms. Since the validity of excluding that part of the Hamiltonian containing the operators that do not form a Lie algebra has been argued extensively, most recently Paz and Urdaneta [1,9] have followed the procedure employed by Mundarain together with the iterative process used by R ecamier and J auregui [10] for similar systems, to include higher order terms in the boson operators and study their relevance for the evaluation of state-to-state transition probabilities in a quartic onedimensional anharmonic oscillator interacting with a low intensity time dependent electric field. As a result the final analytical expression resumes the whole iterative procedure as a global squeezing and displacement transformation in which the compression coefficients and translation parameters were obtained explicitly.…”