Abstract
We show here that the model Hamiltonian used in our paper for an ion vibrating in a q-analog harmonic oscillator trap and interacting with a classical single-mode light field is indeed obtained by replacing the usual bosonic creation and annihilation operators of the harmonic trap model by their q-deformed counterparts. The approximations made in our paper amount to using, for the ion-laser interaction in a q-analog harmonic oscillator trap, the operator ${F}_{q}{=e}^{\ensuremath{-}(|\ensuremath{\epsilon}{|}^{2}/2)}{e}^{i\ensuremath{\epsilon}{A}^{\ifmmode\dagger\else\textdagger\fi{}}}{e}^{i\ensuremath{\epsilon}A},$ which is analogous to the corresponding operator for an ion in a harmonic oscillator trap that is ${F=e}^{\ensuremath{-}(|\ensuremath{\epsilon}{|}^{2}/2)}{e}^{i\ensuremath{\epsilon}{a}^{\ifmmode\dagger\else\textdagger\fi{}}}{e}^{i\ensuremath{\epsilon}a}.$ Here a and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$ are the usual creation and annihilation operators, whereas A and ${A}^{\ifmmode\dagger\else\textdagger\fi{}}$ are the q-deformed bosonic creation and annihilation operators. There is no problem with diagonalizing this operator using the basis states $|g,m〉$ and $|e,m〉,$ where m stands for the motional number state. In our paper we do not claim to have diagonalized the operator ${F}_{q}{=e}^{i\ensuremath{\epsilon}{(A}^{\ifmmode\dagger\else\textdagger\fi{}}+A)}$ for which the basis states $|g,m〉$ and $|e,m〉$ are not analytic vectors.
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