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arXiv:quant-ph/0105060

Quantum mutual entropy for two-level ion in a q-analog trap

S. Shelly Sharma, Naresh Kumar Sharma

Published May 13, 2001• 0 citations
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Abstract

Quantum mutual entropy is used as a measure of information content of ionic state due to ion-laser interaction in a q-analog trap. The initial state of the system is a Schrodinger cat state. It is found that the partial mutual entropy is a good measure of the entanglement and purity of the ionic state at $t>0$.

DOI
10.48550/arxiv.quant-ph/0105060

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Quantum mutual entropy is used as a measure of information content of ionic state due to ion-laser interaction in a q-analog trap. The initial state of the system is a Schrodinger cat state. It is found that the partial mutual entropy is a good measure of the entanglement and purity of the ionic state at t>0t>0t>0.
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.
In the present communication we explore and develop the theory of entanglement measures. As a result we obtain a general criterion of a bipartite mixed states entanglement due to the quasi-mutual entropy for a single trapped two-level ion interacting with a laser field. This general expression of the entanglement degree is obtained with the physical significance and without the diagonal approximation. We indicate how the entanglement affected by different values of the Lamb-Dicke parameter of the initially generalized binomial state. Among the new features found, it is shown, for the large number of quanta, that the shape of entanglement could oscillate around the maximum value, which is related to the existence of a strong entanglement in the present system.
The time evolution of the atomic Wehrl entropy and long-lived entanglement generation using a single trapped ion interacting with a laser field are analyzed. Starting from the Heisenberg equation of motion, an exact solution of the system is obtained by indicating that there are some interesting features when a time-dependent modulating function is considered. We demonstrate that the long-living quantum entanglement can be obtained using the time-independent interaction when the field is initially in a pair cat states.
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.