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arXiv:quant-ph/0011090 · Journal of Physics B Atomic Molecular and Optical Physics

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

S. Shelly Sharma, Naresh Kumar Sharma

Published Mar 26, 2002• 10 citations
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Abstract

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.

DOI
10.1088/0953-4075/35/7/303

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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.

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A single ion trapped in a harmonic potential and laser cooled at the node of a standing wave can be described in terms of the Jaynes-Cummings Hamiltonian known in quantum optics. Similar to nonclassical states of light we propose to prepare nonclassical states of the ion motion in a harmonic trap. In particular, several schemes for the preparation of Fock states are presented, and techniques for the preparation of squeezed states and coherent states of the ion motion are indicated. For an observation of these nonclassical states we propose to measure quantum collapse and revival in the population inversion of a single trapped ion.

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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.
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.
In this paper we use the quasi-mutual entropy, which does not depend on quantum channel, as a measure of information content of ionic state due to ion-laser interaction in a single trapped ion. It is found that the quasi-mutual entropy is a good measure of the entanglement degree of the ionic state at any time. We examine numerically the population inversion, the photon number distribution and degree of entanglement due to quasi-mutual entropy for a single trapped ion interacting with a laser held. During the transition from collapse to revival and vice-versa we have a minimum degree of entanglement value. Successive revival peaks show a lowering of the local maximum points indicating a dissipative irreversible change in the ionic state. We propose a generation of Bell-type states having a simple initial state preparation of a single trapped ion.
The time evolution of ionic internal states for trapped cold ions interacting with external laser in a harmonic oscillator potential, and a potential with a small nonlinear component varying as x6, modeled through a q-deformed harmonic oscillator, is briefly discussed. The entanglement of ionic internal states and center of mass vibrational states results in collapse and revival of population inversion. For deformation parameter, τ = 0.004(q = eτ), the population inversion shows an extremely well behaved collapse and revival pattern. Next a simple application where entanglement is used for implementing quantum logic gates with ion trap placed inside an optical cavity, is presented. It is found that faster ionic state manipulation times can be obtained by working beyond the Lamb-Dicke regime.
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.
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.
A single ion trapped in a harmonic potential and laser cooled at the node of a standing wave can be described in terms of the Jaynes-Cummings Hamiltonian known in quantum optics. Similar to nonclassical states of light we propose to prepare nonclassical states of the ion motion in a harmonic trap. In particular, several schemes for the preparation of Fock states are presented, and techniques for the preparation of squeezed states and coherent states of the ion motion are indicated. For an observation of these nonclassical states we propose to measure quantum collapse and revival in the population inversion of a single trapped ion.