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We exploit a scheme to obtain a long-lived entanglement using a driven central spin interacting with an antiferromagnetic spin bath. Our numerical results show the effects of different parameters on the population inversion and the entanglement dynamics in terms of the linear entropy. It is shown that the long-lived entanglement is an intriguing result corresponding to the collapse region of the atomic inversion. As illustration, we examine the long-time interaction of the entanglement under the resonance and off-resonance regimes.

One of the outstanding challenges for multi-particle quantum information processing is to accurately find the states of many particles in a scalable fashion [

Recently experimental interest has been increased in electronic spin systems, where the most prominent source of decoherence is thought to be electronic. Examples of these systems are superconducting quantum dots [

There is an elegant way of studying fundamental information inequality in term of the quantum relative entropy [

It is the purpose of this paper to give an analysis of the entanglement dynamics of a central spin interacting with an antiferromagnetic spin environment. The article is organized as follows: we display the Hamiltonian in Section 2. Then some dynamical aspects related to the population inversion and linear entropy are discussed in Section 3. Finally concluding remarks are drawn in Section 4.

The system, which is considered here, represents a central spin interacting with an antiferromagnetic spin environment [

where

where

where the connection between any atom and its nearest neighbors is described by the vector

The dynamical model used here is similar to the well-known spin-boson model discussed earlier [

Consider then the following initial state of the system

where

with a unity of the Boltzmann constant. We denote by

Following the standard procedure and assume that the density matrix of the antiferromagnetic bath is assumed to satisfy the Boltzmann distribution [

From Equation (5), we obtain a set of algebraic equations for the complex probability amplitudes of the quantum states which can be solved in the usual way [

where

It is also possible to compute the general solution of the system by considering more general initial states. Here, we have considered a system with a separable initial density matrix of the composed system, so that it makes sense how entanglement dynamics propagates. In the following section, we are interested in examining the relation between the long-lived entanglement and atomic inversion collapse.

In an effort to present a numerical characterization, we have performed some calculations of the linear entropy and atomic inversion quantities for a particular set of parameters, some of which can be considered as realistic, while some other parameters look perhaps too optimistic. However, dimensionless parameters are used and our results can be useful under different scenarios.

As an entanglement measure von Neumann entropy has been used when the system starts from a pure state and many generalizations have been proposed. Among them the Tsallis entropy [

where

It is pointed out in a straightforward way that such a quantity, the linear entropy of the reduced density matrix, can be used as a measure of the entanglement. Even if linear entropy is not additive in the usual sense [

Since the resulting series, Equations (6)-(8), cannot be analytically summed in a closed form, we evaluate them numerically. In

recognizing in which situations entanglement can be performed, we compute the atomic inversion in

At the end of this Section, we point out that the different values of the system parameters lead to the same one-to-one correspondence between the atomic inversion and the entanglement. One interesting problem in quantum information processing that exhibits connections between the collapse-revival phenomenon and long-lived entanglement, where the long collapse period of the atomic inversion has been shown at the same time of the steady-state entanglement known as the long-lived entanglement. In this equivalence the divergence of the collapse’s length depends on the system parameters. It is interesting to note that the basic features of entanglement in this model at different values of the detuning parameter turn up in the context of localization phenomena.

We have shown that long-distance steady state entanglement in a driven central spin interacting with antiferromagnetic spin bath systems can be coherently controlled through the tuning difference between the Larmor frequency and the magnetic field frequency. This entanglement is measured by analyzing the dynamical behavior of the linear entropy. We also found that there exist one-to-one correspondence between the long-lived entanglement and collapse regime of the atomic inversion. Surprisingly enough, using different values of the system parameters the steady state entanglement can be achieved whenever the off-resonant case is considered. The results presented here help to identify clearly which types of parameters can be used to obtain a long-lived entanglement. The entanglement dynamics become quite irregular as the coupling strength of the spin bath increases. This happens because of the competing interaction of the field between the atom and the spin bath. Since the interest in spin bath is quite relevant in relation to the so-called quantum non-demolition measurements hence, our results may be useful in that context.

I would like to acknowledge the support from Deanship for Scientific Research, University of Bahrain, project No. 19/2014.