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arXiv:1910.12561

Reply to Comment on "A no-go result for the quantum damped harmonic oscillator"

Fabio Bagarello, Francesco Gargano, Federico Roccati

Published Oct 28, 2019• 0 citations
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Abstract

In a recent paper, \cite{deguchi}, Deguchi and Fujiwara claim that our results in \cite{BGR} are wrong, and compute what they claim is the square integrable vacuum of their annihilation operators. In this brief note, we show that their vacuum is indeed not a vacuum, and we try to explain what is behind their mistake. We also consider a very simple example clarifying the core of the problem.

DOI
10.48550/arxiv.1910.12561

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In a recent paper, \cite{deguchi}, Deguchi and Fujiwara claim that our results in \cite{BGR} are wrong, and compute what they claim is the square integrable vacuum of their annihilation operators. In this brief note, we show that their vacuum is indeed not a vacuum, and we try to explain what is behind their mistake. We also consider a very simple example clarifying the core of the problem.
In a recent paper, Hassoul et al.[1], the authors proposed an analysis of the quantum dynamics for general time-dependent three coupled oscillators through an approach based on their decouplement using the unitary transformation method. Thus, to diagonalize the transformed Hamiltonian, they introduce a new unitary operator corresponding to a three-dimensional rotation parameterized by Euler angles. Through this procedure, Hassoul et al. [1] claim that the coupled oscillatory subsystems are completely decoupled. This last approach is partly wrong. In this brief note, we show that their method is indeed not correct and we try to explain what truly lies behind their mistakes. We also propose a brief discussion on an alternative method that might achieve satisfactory results.
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〉|g,m〉∣g,m〉 and ∣e,m〉,|e,m〉,∣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〉|g,m〉∣g,m〉 and ∣e,m〉|e,m〉∣e,m〉 are not analytic vectors.