Lie algebraic method applied to a pulsed anharmonic oscillator

Abstract

ABSTRACT:Using nonlinear coherent states defined as approximate eigenstates of a deformed annihilation operator, we evaluate the response to a classical pulsed electric field of a system supporting a finite number of bound states. We calculate the temporal evolution of the average value of several observables like the momentum and the diplacement coordinate.

Phase space picture of Morse-like coherent states based upon the Wigner function

“…[23] by means of an approximate version of the deformed displacement operator in which the number operator is replaced by an average value, i.e,n → α|n|α ≡n. Hence the disentanglement problem of the exponential (20) was circumvented by applying standard mathematical techniques, since in such a case the commutator between and † is a function ofn, which is obviously no longer an operator. However, the displacement operator thus obtained turned out to be approximately unitary depending on the value of the parameter α.…”

Nonlinear coherent states for nonlinear systems

“…where L n m is an associated Laguerre polynomial. Once a specific form for the function f (n) has been chosen, one can obtain the CSs pertinent to that deformed oscillator using equation ( 18) [28]. In [29], these states were used for the construction of even and odd combinations of Morse-like CSs with the finding that the even combination displays squeezing while the odd combination does not, in agreement with the conduct shown by the even combination of field CSs.…”

Markovian master equation for nonlinear systems

“…In writing the Hamiltonian in equation ( 21), the rotating-wave approximation was applied by dropping the anti resonant terms like b A ˆk and b A ˆk † † . Comparing the interaction Hamiltonian, equation (21) with that used in the pioneer work by Mancini [27], it can be noticed that their structure is similar but here H ˆInt is a linear function of the deformed oscillator operators A ând A ˆ † instead of a ˆand a ˆ †.…”

“…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].…”