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Journal of Modern Optics

Decay and survivability of entanglement in two-dimensional ionic motions

Mahmoud Abdel‐Aty

Published Nov 30, 2007• 3 citations
Publisher

Abstract

This contribution describes some new features of quantum entanglement in a single three-level trapped ion confined in a two-dimensional (2D) harmonic potential. The dynamical properties of quantum entanglement are studied in terms of the reduced-density linear entropy considering two specific initial states of the field. Allowing the instantaneous position of the centre-of-mass motion of the ion to be explicitly time dependent, it is shown that either sudden death of entanglement or survivability of quantum entanglement can be obtained with a specific choice of the initial state parameters. The difference in evolution picture corresponding to the multi-quanta processes is discussed.

DOI
10.1080/09500340701296499

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This contribution describes some new features of quantum entanglement in a single three-level trapped ion confined in a two-dimensional (2D) harmonic potential. The dynamical properties of quantum entanglement are studied in terms of the reduced-density linear entropy considering two specific initial states of the field. Allowing the instantaneous position of the centre-of-mass motion of the ion to be explicitly time dependent, it is shown that either sudden death of entanglement or survivability of quantum entanglement can be obtained with a specific choice of the initial state parameters. The difference in evolution picture corresponding to the multi-quanta processes is discussed.
he dynamical properties of quantum entanglement in a time-dependent three-level-trapped ion interacting with a laser field are studied in terms of the reduced-density linear entropy considering two specific initial states of the field. Allowing the instantaneous position of the center-of-mass motion of the ion to be explicitly time-dependent, it is shown that either sudden death of entanglement or survivability of quantum entanglement can be obtained with a specific choice of the initial state parameters. The difference in evolution picture corresponding to the multi-quanta processes is discussed.
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.
Trapped and laser-cooled ions are increasingly used for a variety of modern high-precision experiments, frequency standard applications and quantum information processing. Therefore, in this communication we present a comprehensive analysis of the pattern of information entropy arising in the time evolution of an ion interacting with a laser field. A general analytic approach is proposed for a three-level trapped-ion system in the presence of the time-dependent couplings. By working out an exact analytic solution, we conclusively analyse the general properties of the von Neumann entropy and quantum information entropy. It is shown that the information entropy is affected strongly by the time-dependent coupling and exhibits long time periodic oscillations. This feature attributed to the fact that in the time-dependent region Rabi oscillation is time dependent. Using parameters corresponding to a specific three-level ionic system, a single beryllium ion in a RF-(Paul) trap, we obtain illustrative examples of some novel aspects of this system in the dynamical evolution. Our results establish an explicit relation between the exact information entropy and the entanglement between the multi-level ion and the laser field. We show that different nonclassical effects arise in the dynamics of the ionic population inversion, depending on the initial states of the vibrational motion/field and on the values of Lamb–Dicke parameter η.
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 propose the use of atomic Wehrl entropy associated to the reduced atomic density operator as an entanglement indicator of bipartite systems. This is applied to a two-level system (one single harmonically trapped ion) by taking into account the linear center-of-mass-motional degree of freedom. Detailed analytical and explicit expressions are given, taking into account different configurations. The results show the important roles played by the laser phase and initial state setting in the evolution of the atomic Q-function, atomic Wehrl entropy and marginal atomic Q-function. Our procedure of using atomic Wehrl entropy may be applied to a system with Hilbert space of high dimension.