Nonlinear response of a deformed oscillator

Abstract

We analize the response of a single anharmonic diatomic molecule to a monochromatic time dependent electric field E ω (t) and evaluate the temporal evolution of the dipole moment, phase space trajectories and several transition probabilities as a function of the intensity of the field. We define deformed boson operators A † = f (n)a † , A = af (n) in terms of the usual harmonic oscillator creation and annihilation operators a, a † ,n = a † a and choose the number operator function f (n) such that the energy spectrum of a harmonic oscillator-type Hamiltonian written in terms of the deformed operators yield an energy spectra similar to that of a Morse potential. TheoryMuch work has been devoted to the study of the optical linear and nonlinear response of harmonic systems to time dependent electric and magnetic fields either with classical [1] or quantum mechanical treatments [2,3]. Laser pulses with well controlled temporal characteristics of amplitude and frequency have a high potential for selective manipulation of the internal state of atoms and molecules [4,5,6]. Cooling techniques have allowed the trapping and manipulation of atoms by optical means, however, fluctuations of the electromagnetic fields used to trap or to modify the quantum state lead to decoherence in ion traps and limits the trap stability [7]. A study involving multiple photon excitation for a linearly driven anharmonic oscillator using 1

Markovian master equation for nonlinear systems

“…Our goal in this section is to depict the dynamics on phase space of what is called the nonlinear Morse-like oscillator [20] from the perspective of the temporal evolution of its associated coherent states. To this end we will make use of the phase-space description based upon the Wigner quasidistribution function, taking into account that the oscillator is viewed as an open quantum system whose evolution is governed by the master equation (34).…”

“…It is worth mentioning that the free evolution of such a system has already been analyzed in [22,23,25,26] in terms of its nonlinear coherent states. It was first found in [20] that by choosing the deformation function ( )…”

“…The wide applicability of this nonlinear algebraic formalism has been demonstrated in a manifold of circumstances. Among the most representative, we can quote the paper of Man'ko et al [6] about the Weyl-Wigner-Moyal representation for foscillators, in which the well-known Kerr-like nonlinearity of media was taken into account; the description of the centerof-mass motion of a laser-driven trapped ion [7][8][9]; generalizations of the Jaynes-Cummings model in which the interaction between a two or three-level atom and the radiation field is nonlinear in the field variables [10][11][12], as well as including Kerr type nonlinearities [13][14][15]; the relation between the deformation function of the f-deformed oscillator and the two-dimensional harmonic oscillator on the flat space and on the sphere [16]; and generalizations of coherent states for confining systems such as the symmetric Pöschl-Teller potentials [17,18] and the Morse potential [20][21][22][23][24][25][26].…”

Lie algebraic method applied to a pulsed anharmonic oscillator

“…Recently we analyzed the nonlinear response of a deformed oscillator to a time‐dependent electric field, using deformed creation and annihilation operators with a deformation function, such that the energy spectrum was similar to that of a Morse potential. The linear response found for the harmonic and the deformed cases was similar; however, when nonlinear terms in the expansion of the coordinate were incorporated, the differences increased significantly 1. Since the deformation function depends only on the number operator, the harmonic oscillator eigenstates are also eigenstates of the deformed oscillator; to evaluate the averages, we used the number representation.…”

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

“…On the other hand, in most of these calculations the cubic term has been neglected and only the quartic term has been retained, and the resulting iterative process is done straightforwardly. In those cases in which the cubic term is retained, it was common to solve the problem numerically employing methods such as the conventional IBT 4, 5 and recently others like the bilinear approximation 6, where the iterative result is achieved in one step.…”