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Adıyaman üniversitesi fen bilimleri dergisi

Analyzing of Quantum Entanglement with Concurrence in the Deep Lamb-Dicke Regime

Rasim Dermez

Published Jun 1, 2020• 1 citation
Publisher

Abstract

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.

DOI
10.37094/adyujsci.666654

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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, 0.08andandand0.09 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 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 work, the quantum entanglement and quantum linearity in a trapped ion–phonon system are theoretically explained in terms of the second-order terms of the Lamb–Dicke parameter, η . For the trapped ion–phonon system, the optimum times are 2 and 10 and the optimum probability amplitudes of ions are 0.4 and 0.58, respectively. With the optimum times and amplitudes, it is shown that as η grows, the quality of the entanglement in the trapped ion–phonon system increases. In particular, we have demonstrated the quantum linearity due to second-order terms of η .
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.
To achieve more accuracy in a system containing N trapped two-level ions interacting with a single-mode quantized field, in this paper we will consider the dipole–dipole interaction (DDI) between the trapped ions. In more detail, we first suggest an appropriate form of a quantized DDI Hamiltonian between two trapped ions that has arisen from the spontaneous emission of the excited one. Accordingly, we have considered the spontaneous emission as the origin of dissipation in our proposed model. Then, the case in which the trapped ions interact with a single-mode quantized field in the Lamb–Dicke regime as well as the first vibrational sideband is considered, in which the DDI between the trapped ions is also taken into account. Using the state vector, which is analytically obtained for the whole system under our consideration, we evaluate the amount of entanglement between the field and the two trapped ions by utilizing the linear entropy. We observe that the entanglement depends on the distance between the two trapped ions; i.e., decreasing the distance led to decrement in the amount of entanglement. In the continuation, we extend the proposed model at first to three (N=3) and then to more trapped ions (N>3), and the most general entangled state vectors related to the outlined systems are obtained, for which the same scenario (for two trapped ions) is repeated. Comparing the latter cases (N≥3) with the previous one (N=2), we conclude that the entanglement is sensitive to the number of interacting trapped ions in addition to the distances between them in such a way that by increasing the number of trapped ions, the degree of entanglement between them and the field is increased. These results help one to control the entanglement between the field and the trapped ions in a more accurate manner by tuning the distances between the trapped ions and their numbers.
We create entangled states of the spin and motion of a single 40Ca+ ion in a linear ion trap. We theoretically study and experimentally observe the behavior outside the Lamb-Dicke regime, where the trajectory in phase space is modified and the motional coherent states become squeezed. We directly observe the modification of the return time of the trajectory, and infer the squeezing. The mesoscopic entanglement is observed up to Deltaalpha=5.1 with coherence time 170 micros and mean phonon excitation n = 16.