Dzyaloshiniskii-Moriya (DM) interaction in three directions (Dx, Dy and Dz) is used to generate entangled network from partially entangled states in the presence of the spin-orbit coupling. The effect of the spin coupling on the entanglement between any two nodes of the network is investigated. The entanglement is quantified using Woottores concurrence method. It is shown that the entanglement decays as the coupling increases. For larger values of the spin coupling, the entanglement oscillates within finite bounds. For the initially entangled channels, the upper bound does not exceed its initial value, whereas the entanglement reaches its maximum value for the channels generated via indirect interaction.
Quantum Information Technology (QIT) promises faster and more secure means of data manipulation by making use of the quantum properties of matter [
The Dzyaloshiniskii-Moriya (DM) interaction is a natural phenomena discovered in 1960 by Moriya Dzyaloshinskii-Moriya (DM) as an antisymmetric and anisotropic exchange coupling between two spins) [
Metwally [
In this paper, we introduce the model and its evolution in Section 2. The entanglement between different nodes is quantified for different values of the spin-orbits coupling and the DM’s strength is discussed in Section 3. We then summarize our results and discussion in Section 4.
It is assumed that entangled network consists of four nodes is generated by using partial entangled states of Werner type [
Consider a partial entangled state generated by a source of the form
where
The initial state of the total system is given by
The evolution of the initial state (2) is given by,
where
where
The Hamiltonian describing the evolution of the system for a two-qubit spin-orbit chain with the DM interaction is switched on the x-axis can be written as
where k, l represent the nodes which are connected together via DM interaction with x-component of strength Dx; Jx, Jy and Jz are the x, y and z-component of the real coupling coefficients, respectively, and the
The final density operator of the network is given by
where
is the unitary operator which can be written explicitly in the basis
where
And the rest of the components
Similarly, the Hamiltonian describing the evolution of the system when the interaction is switched to the y-direction can be written as.
In this case, the unitary operator is given by
In matrix form, the unitary operator Equation (10) can be written as a matrix Equation (8) and the elements of the matrix are:
And the rest of the components
Similarly, the Hamiltonian describing the evolution of the system when the interaction is switched to the z-direction can be written as.
In this case, the unitary operator is given by
Is the unitary operator which used to connect between particle “2” and “3” when the interaction is switched on y-axis and defined by
In matrix form, the unitary operator Equation (13) can be written as in
Where the elements of the matrix are:
And the rest of the components
In this section, we quantify the entanglement between each pair of nodes. Practically, we consider the channels
where
The entanglement behavior (concurrence) of the entangled state between nodes “1” and “2”,
tween the second and third node, and consequently some correlations are lost. However, the upper and lower bounds will depend on the non-zero coupling when it is switched on. This behavior shows that the minimum bound of C for Jx ¹ 0 is always larger than that depicted for Jy ¹ 0 or Jz ¹ 0. On the other hand, the concurrence vanishes completely for Jy ¹ 0 or Jz ¹ 0, i.e., C = 0 as the scaled time increased without exceeding the initial upper limit [
The dynamics of concurrence, C, for the channel
We investigate the behavior of the entanglement for state
On the other hand, the oscillation increased if all the couplings have non-zero values.
The dynamics of the entanglement in the quantum network when the interaction is switched on y-direction is shown in Figures 5-7.
The effect of Jx, Jy and Jz in the dynamics of the entanglement over channel “12” is depicted in
The dynamics of the entanglement on channel “13” with the effect of the spin-orbit interaction when DM interaction is switched on the y-axis.
raction. It is clear that there is no entanglement at the initial stage during the interaction period. The entanglement appeared at t = 2.5 and reaches its maximum at t = 3.75 and then vanishes after t = 5. It remains zero there until t = 10 and the same behavior is repeated periodically. The effect of x-spin-orbit is depicted by the dotted black line which has the same effect as the z-spin-orbit interaction to which the generation of the entanglement started at t = 1, and the generation of entanglement without spin-orbit is started at t = 2.5. The maximum value of the entanglement is 0.4 at t = 3. It decreased to zero at t = 6 and increased again after t = 8 up to maximum at t = 12. The effect of the y-spin orbit interaction is represented by the dashed green line.
This figure shows the new feature is observed as the spin-orbit and the number of oscillations increased.
Figures 7(a)-(c) show the entanglement dynamics between node “1” and node “4”. In
The dotted green line shows no effect on the spin-orbit interaction in y-axis.
can see in the red and dashed green lines for Jx = Jy = 0.5 and Jy = Jz = 0.5, respectively. The dotted black line and dotted blue line are the combination of the spin-orbit interaction in xand z-direction which leads to the increasing in the entanglement oscillations. The effect of the spin-orbit affects the entanglement dynamics on channel “14” (see
The effect of the spin-orbit interaction on the dynamics of the entanglement for the channel “12”, “13” and “14” with DM interaction is switched on the z-axis is depicted in Figures 8-10.
of the spin-orbit interaction in one directional Jx, Jy and Jz. By comparison between
We discussed the effect of the spin-orbit (Heisenberg XYZ model) coupling to the entanglement between different nodes in the quantum network. It is shown that the entanglement decays for nonzero coupling. The phenomena of the entanglement sudden-death and sudden-birth appeared for larger coupling values. It shows that the coupling constant of the entangled channel initially has no effect on the upper bound of the entanglement. However, the lower bound of the entanglement does not vanish for non-zero couplings. The number of oscillations is increased as the coupling is increased. For entangled channels which are generated either via direct or indirect interactions, the concurrence and the number of oscillations are increased as the coupling is increased.
Finally, it is shown that the generated entangled channel between any two nodes via indirect interaction has a large degree of entanglement and the upper bound exceeds the initial entangled state. Therefore, one may generate maximum entangled state from the less entangled state by controlling the spin-orbit coupling. This means that terminals of the generated entangled network can be used to perform quantum information task with high efficiency.