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arXiv:quant-ph/9903068 · Physical Review A

Reply to “Comment on ‘Single two-level ion in an anharmonic oscillator trap: Time evolution of theQfunction and population inversion’ ”

S. Shelly Sharma, Naresh Kumar Sharma, L. Zamick

Published Apr 1, 1999• 1 citation
arXivHTMLPDF

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.

DOI
10.1103/physreva.59.3138

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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 Fq=e\ensuremath−(∣\ensuremathϵ∣2/2)ei\ensuremathϵA\ifmmode†\else\textdagger\fiei\ensuremathϵA,{F}_{q}{=e}^{\ensuremath{-}(|\ensuremath{\epsilon}{|}^{2}/2)}{e}^{i\ensuremath{\epsilon}{A}^{\ifmmode\dagger\else\textdagger\fi{}}}{e}^{i\ensuremath{\epsilon}A},Fq​=e\ensuremath−(∣\ensuremathϵ∣2/2)ei\ensuremathϵA\ifmmode†\else\textdagger\fiei\ensuremathϵA, which is analogous to the corresponding operator for an ion in a harmonic oscillator trap that is F=e\ensuremath−(∣\ensuremathϵ∣2/2)ei\ensuremathϵa\ifmmode†\else\textdagger\fiei\ensuremathϵa.{F=e}^{\ensuremath{-}(|\ensuremath{\epsilon}{|}^{2}/2)}{e}^{i\ensuremath{\epsilon}{a}^{\ifmmode\dagger\else\textdagger\fi{}}}{e}^{i\ensuremath{\epsilon}a}.F=e\ensuremath−(∣\ensuremathϵ∣2/2)ei\ensuremathϵa\ifmmode†\else\textdagger\fiei\ensuremathϵa. Here a and a\ifmmode†\else\textdagger\fi{a}^{\ifmmode\dagger\else\textdagger\fi{}}a\ifmmode†\else\textdagger\fi are the usual creation and annihilation operators, whereas A and A\ifmmode†\else\textdagger\fi{A}^{\ifmmode\dagger\else\textdagger\fi{}}A\ifmmode†\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 Fq=ei\ensuremathϵ(A\ifmmode†\else\textdagger\fi+A){F}_{q}{=e}^{i\ensuremath{\epsilon}{(A}^{\ifmmode\dagger\else\textdagger\fi{}}+A)}Fq​=ei\ensuremathϵ(A\ifmmode†\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.
Recently, Sharma et al. [Phys. Rev. A 56, 694 (1997)] proposed a q-deformed oscillator trap model to investigate the time evolution of a single two-level ion in an anharmonic oscillator trap, with the anharmonicity being parametrized by the deformation parameter q. The model is obtained by simply replacing the usual bosonic creation and annihilation operators in the harmonic trap model by their q-deformed counterparts. In the present paper we have reexamined the validity of the q-deformed oscillator trap model, and our analysis indicates that such a naive generalization actually does not exist. This is accomplished by demonstrating that the eigenstates of this q-deformed oscillator trap model are not normalizable.
We investigate the time evolution of the population inversion as well as the QQQ function for center-of-mass motion for a two-level ion, interacting with single-mode laser light field, in an anharmonic-oscillator trap. The anharmonicities of the trap are quantified in terms of the deformation parameter of the qqq-analog trap. For $\ensuremath{\tau}=0.003$ ${(q=e}^{\ensuremath{\tau}})$ the collapses and revivals of population inversion become well-defined facilitating experimental observation, but for large $\ensuremath{\tau}\ensuremath{\sim}0.1$ the time dependence of population inversion is completely wiped out. The quasiprobability function shows an atypical behavior with a single peak at t=0t=0t=0 splitting up into as many as seven fragments (at t=130t=130t=130) during collapse and at a later time these segments come together to signal the revival of population inversion. A small degree of anharmonicity is seen to enhance the coherence of the ion-trap system, whereas for large $\ensuremath{\tau}$ ($\ensuremath{\sim}0.1$ in the present context) the coherence is completely lost.
Atom in a qqq-Analog Harmonic Oscillator Trap
We study the population inversion and Q-function of a two-level atom, interacting with single-mode laser light field, in a qqq-analog harmonic oscillator trap for increasing qqq. For τ=.003(q=eτ)τ=.003(q=e^τ)τ=.003(q=eτ) the collapses and revivals of population inversion become well defined facilitating experimental observation but for large τ∼0.1τ\sim 0.1τ∼0.1 the time dependence of population inversion is completely wiped out.
Fermions need to satisfy the Pauli exclusion principle: no two can be in the same state. This restriction is most compactly expressed in a second quantization formalism by the requirement that the creation and annihilation operators of the electrons satisfy anticommutation relations. The usual classical limit of quantum mechanics corresponds to creation and annihilation operators that satisfy commutation relations, as for a harmonic oscillator. We discuss a simple classical limit for Fermions. This limit is shown to correspond to an anharmonic oscillator, with just one bound excited state. The vibrational quantum number of this anharmonic oscillator, which is therefore limited to the range 0 to 1, is the classical analog of the quantum mechanical occupancy. This interpretation is also true for Bosons, except that they correspond to a harmonic oscillator so that the occupancy is from 0 up. The formalism is intended to be useful for simulating the behavior of highly correlated Fermionic systems, so the extension to many electron states is also discussed.