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Physical Review A

Single two-level ion in an anharmonic-oscillator trap: Time evolution of the Q function and population inversion

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

Published Jul 1, 1997• 15 citations
Publisher

Abstract

We investigate the time evolution of the population inversion as well as the $Q$ 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 $q$-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 $t=0$ splitting up into as many as seven fragments (at $t=130$) 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.

DOI
10.1103/physreva.56.694

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The Schrödinger cat state of trapped ions in harmonic and anharmonic oscillator traps

S. Shelly Sharma, Naresh Kumar Sharma

We examine the time evolution of a two-level ion interacting with a light field in a harmonic oscillator trap and in a trap with anharmonicities. The anharmonicities of the trap are quantified in terms of the deformation parameter τ characterizing the q -analogue of the harmonic oscillator trap. Initially the ion is prepared in a Schrödinger cat state. The entanglement of the centre-of-mass motional states and the internal degrees of freedom of the ion results in a characteristic collapse and revival pattern. We numerically calculate the population inversion I ( t ), quasi-probabilities Q ( t ) and partial mutual quantum entropy S ( P ) for the system as a function of time. Interestingly, small deformations of the trap enhance the contrast between population inversion collapse and revival peaks as compared to the zero-deformation case. For β = 3 and 4 (β determines the average number of trap quanta linked to centre-of-mass motion) the best collapse and revival sequence is obtained for τ = 0.0047 and 0.004, respectively. For large values of τ, decoherence sets are accompanied by loss of amplitude of population inversion, and for τ ~0.1 the collapse and revival phenomenon disappear. Each collapse or revival of population inversion is characterized by a peak in the S ( P ) versus t plot. During the transition from collapse to revival and vice-versa, we have a minimum mutual entropy value that is S ( P ) = 0. Successive revival peaks show a lowering of the local maximum point indicating a dissipative irreversible change in the ionic state. An improved definition of the collapse and revival pattern as the anharmonicity of the trapping potential increases is also reflected in the quasi-probability versus t plots.

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We study the population inversion and Q-function of a two-level atom, interacting with single-mode laser light field, in a $q$-analog harmonic oscillator trap for increasing $q$. For $τ=.003(q=e^τ)$ the collapses and revivals of population inversion become well defined facilitating experimental observation but for large $τ\sim 0.1$ the time dependence of population inversion is completely wiped out.

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

Chi‐Fai Lo, K.L. Liu

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.

Physical Review A•Apr 1, 1999•6 citations

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

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.

arXiv:quant-ph/9903068•Physical Review A•Apr 1, 1999•1 citation

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Following the path of Bužek et al. in [Phys. Rev. A 56 2352 1997], we investigate the full intensity-dependent interaction between a motional two-level ion which is trapped in a f-deformed oscillator (so possessing vibrational modes too) with a single-mode nonlinear quantized field in the Lamb-Dicke limit. After finding the explicit form of the entire ion-field state vector in a general framework, we particularly restrict ourselves to an appropriate nonlinearity function (describing intensity-dependent interaction) corresponding to the center-of-mass motion of a trapped ion which explicitly depends on the Lamb-Dicke parameter as well as another well-known nonlinearity function in the form n. Then, we evaluate the effects of two different initial quantized field states as well as the considered nonlinearity function (via varying the Lamb-Dicke parameter) on a few physical properties of the system. For this purpose, we pay attention to the amount of entanglement, mean number of vibrational quanta (phonons), population inversion of the two-level ion and Q-distribution function. Finally, by comparing our results with the previously considered “linear” one which is pointed in the above Ref., we establish that, by tuning the Lamb-Dicke parameter, one can adjust the mentioned nonclassical properties of the system appropriately.

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Quantum Collapses and Revivals in a Quantized Trap

C. A. Blockley, Daniel F. Walls, H. Risken

A single two-level atom, with quantized centre-of-mass motion, is constrained to move in a one-dimensional harmonic potential, while interacting with a single-mode classical travelling light field. When the atom's centre-of-mass motion is in a coherent state, we show that the atomic inversion exhibits collapses and revivals. Whereas in the Jaynes-Cummings model this behaviour occurs due to the discrete nature of the light field, in our case the behaviour is due to the discrete nature of the vibrational trap states. The Q -function for the external motion is also calculated and shown to break into two peaks in the collapse region. Finally the parameter ranges under which the collapses and revivals can be observed are discussed, as well as the possibility of an experiment.

Europhysics Letters (EPL)•Feb 1, 1992•254 citations
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 \ensuremathτ=0.003\ensuremath{\tau}=0.003\ensuremathτ=0.003 (q=e\ensuremathτ){(q=e}^{\ensuremath{\tau}})(q=e\ensuremathτ) the collapses and revivals of population inversion become well-defined facilitating experimental observation, but for large \ensuremathτ\ensuremath∼0.1\ensuremath{\tau}\ensuremath{\sim}0.1\ensuremathτ\ensuremath∼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 \ensuremathτ\ensuremath{\tau}\ensuremathτ (\ensuremath∼0.1\ensuremath{\sim}0.1\ensuremath∼0.1 in the present context) the coherence is completely lost.
We examine the time evolution of a two-level ion interacting with a light field in a harmonic oscillator trap and in a trap with anharmonicities. The anharmonicities of the trap are quantified in terms of the deformation parameter τ characterizing the q -analogue of the harmonic oscillator trap. Initially the ion is prepared in a Schrödinger cat state. The entanglement of the centre-of-mass motional states and the internal degrees of freedom of the ion results in a characteristic collapse and revival pattern. We numerically calculate the population inversion I ( t ), quasi-probabilities Q ( t ) and partial mutual quantum entropy S ( P ) for the system as a function of time. Interestingly, small deformations of the trap enhance the contrast between population inversion collapse and revival peaks as compared to the zero-deformation case. For β = 3 and 4 (β determines the average number of trap quanta linked to centre-of-mass motion) the best collapse and revival sequence is obtained for τ = 0.0047 and 0.004, respectively. For large values of τ, decoherence sets are accompanied by loss of amplitude of population inversion, and for τ ~0.1 the collapse and revival phenomenon disappear. Each collapse or revival of population inversion is characterized by a peak in the S ( P ) versus t plot. During the transition from collapse to revival and vice-versa, we have a minimum mutual entropy value that is S ( P ) = 0. Successive revival peaks show a lowering of the local maximum point indicating a dissipative irreversible change in the ionic state. An improved definition of the collapse and revival pattern as the anharmonicity of the trapping potential increases is also reflected in the quasi-probability versus t plots.
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.
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 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.
Following the path of Bužek et al. in [Phys. Rev. A 56 2352 1997], we investigate the full intensity-dependent interaction between a motional two-level ion which is trapped in a f-deformed oscillator (so possessing vibrational modes too) with a single-mode nonlinear quantized field in the Lamb-Dicke limit. After finding the explicit form of the entire ion-field state vector in a general framework, we particularly restrict ourselves to an appropriate nonlinearity function (describing intensity-dependent interaction) corresponding to the center-of-mass motion of a trapped ion which explicitly depends on the Lamb-Dicke parameter as well as another well-known nonlinearity function in the form n. Then, we evaluate the effects of two different initial quantized field states as well as the considered nonlinearity function (via varying the Lamb-Dicke parameter) on a few physical properties of the system. For this purpose, we pay attention to the amount of entanglement, mean number of vibrational quanta (phonons), population inversion of the two-level ion and Q-distribution function. Finally, by comparing our results with the previously considered “linear” one which is pointed in the above Ref., we establish that, by tuning the Lamb-Dicke parameter, one can adjust the mentioned nonclassical properties of the system appropriately.
A single two-level atom, with quantized centre-of-mass motion, is constrained to move in a one-dimensional harmonic potential, while interacting with a single-mode classical travelling light field. When the atom's centre-of-mass motion is in a coherent state, we show that the atomic inversion exhibits collapses and revivals. Whereas in the Jaynes-Cummings model this behaviour occurs due to the discrete nature of the light field, in our case the behaviour is due to the discrete nature of the vibrational trap states. The Q -function for the external motion is also calculated and shown to break into two peaks in the collapse region. Finally the parameter ranges under which the collapses and revivals can be observed are discussed, as well as the possibility of an experiment.