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Journal of Physics Conference Series

Comparing concurrence and negativity in time-dependent ionic-phononic system with fifteen dimensional density matrix

Rasim Dermez

Published Oct 1, 2016• 5 citations
Publisher

Abstract

It is solved a time-dependent Hamiltonian using a unitary transformation method which Λ( t ) type is used to engineer a cascade Ξ scheme interaction between the vibrational phonons and trapped three-level ion. Quantum entanglement is characterized by comparing concurrence and negativity of the time-dependent ionic-phononic system. In this quantum system, we obtain that the amount of concurrence can be tuned between 0 and 0.99 while the amount of negativity changes between 0 and 0.49.

DOI
10.1088/1742-6596/766/1/012012

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It is solved a time-dependent Hamiltonian using a unitary transformation method which Λ( t ) type is used to engineer a cascade Ξ scheme interaction between the vibrational phonons and trapped three-level ion. Quantum entanglement is characterized by comparing concurrence and negativity of the time-dependent ionic-phononic system. In this quantum system, we obtain that the amount of concurrence can be tuned between 0 and 0.99 while the amount of negativity changes between 0 and 0.49.
In this study, our goal is to obtain the entanglement dynamics of trapped three-level ion interaction two laser beams in beyond Lamb-Dicke parameters. Three values of LDP, η=0.09, η=0.2 and η=0.3 are given. We used the concurrence and the negativity to measure the amount of quantum entanglement created in the system. The interacting trapped ion led to the formation of phonons as a result of the coupling. In two quantum systems (ion-phonons), analytical formulas describing both these measurements are constructed. These formulas and probability coefficients include first order terms of final state vector. We report that long survival time of entanglement can be provided with two quantum measures. Negativity and concurrence maximum values are obtained N = 0.553 and for LDP = 0.3. As a similar, the other two values of LDP are determined and taken into account throughout this paper. For a more detailed understanding of entanglement measurement results, “contour plot” was preferred in Mathematica 8.
In this paper, we consider two phonons and three-level trapped ion in configuration forming Hilbert 12-space. The negativity and quantum correlations are revealed in trapped ion two phonon states system. Three values of LDP, = 0.006, =0.06 and =0.08 are given. The effects of the time-independent coupling in terms of the system, degree of quantum entanglement are investigated. Therefore, we have found the main optimal times for obtaining the high amount of entanglement with negativity.In this paper, we consider two phonons and three-level trapped ion in configuration forming Hilbert 12-space. The negativity and quantum correlations are revealed in trapped ion two phonon states system. Three values of LDP, = 0.006, =0.06 and =0.08 are given. The effects of the time-independent coupling in terms of the system, degree of quantum entanglement are investigated. Therefore, we have found the main optimal times for obtaining the high amount of entanglement with negativity.
The entangled states of three-level of trapped ion and two phonons (into coherent state) in Lambda configuration forming a Hilbert space of 12-dimensional (D) are analyzed. The concurrence is considered "such as a quantum measure" in trapped ion-coherent state system. Four values of Lamb-Dicke parameter (LDP), eta=0.005, 0.07, 0.08$ and $0.09 are investigated by the worths of concurrence. We demonstrate that as eta is increased, sudden birth of entangled states in the system is increased. Thus, establishing of entangled states can be tuned by LDP.
In this paper, the dynamical properties of a trapped three-level ion which interacts with a two-mode quantized field in the presence of a Kerr medium are studied. The ion-field coupling is also considered to be intensity-dependent (nonlinear interaction). In this respect, the Λ, V and ladder-type three-level ions are considered. Via solving the time-dependent Schörodinger equation, the state vectors of the corresponding ion-field systems are analytically obtained after choosing special initial conditions. In the continuation, the mean phonon number as a criterion of trapping energy, the population inversion as a criterion of exchanging energy (between the ion and the field) and the linear entropy as a criterion of entanglement between the trapped ion and the field are studied. To get more general insight about the interaction conditions, the effect of intensity-dependent ion-field coupling (nonlinearity regime) on the dynamics of system is investigated. Finally, by comparing the obtained results for the three configurations of trapped three-level ions with the previously considered two-level one, we conclude that the degree of entanglement for three-level ions is more than the two-level ion. Also, among the considered trapped ions, the ladder-type ion possesses the most amount of entanglement with the quantized field, whereas the Λ-type possesses the least. We also conclude that, the general temporal behaviour of the other considered quantities (mean phonon number and population inversion) are the same for all three types of trapped three-level ions.
A scheme to generate three qubit maximally entangled W-states, using three trapped ions interacting with the red sideband tuned single-mode field of a high finesse cavity, is proposed. For the cavity field initially prepared in a number state, the probability of generating the three-ion W-state is calculated. By using the ion-cavity coupling strengths achieved in experimental realizations, the interaction time needed for W-state generation is found to be of the order of 10 µs. It is found that for a fixed number of photons in the cavity the nature of entanglement of ionic internal states can be manipulated by an appropriate choice of the initial-state phonon number. The ionic qubits in the W-like state are found to be entangled to cavity photons. Analytical expressions for global negativity and partial K -way negativities ( K = 2 to 4) are obtained to study the evolution of entanglement distribution as a function of the interaction parameter. Reversible entanglement exchange between different entanglement modes is observed. For specific values of the interaction parameter, the three ions and photon–phonon system are found to have 4-partite entanglement, generated by 2-way and 3-way correlations.