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El-Shahat T M, Ismail M Kh. Some studies of the interaction between two two-level atoms and SU(1, 1) quantum systems. Chinese Physics B, 2018, 27(10): 100201  
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Some studies of the interaction between two two-level atoms and SU(1, 1) quantum systems
El-Shahat T M , Ismail M Kh
Faculty of Science, Al-Azhar University, Assiut 71524, Egypt

 

† Corresponding author. E-mail: momran484@yahoo.com

Abstract

Herein, we present an approach to look for the best phenomenon to measure quantum correlation. The system of two isolated qubits each interacting with a single-mode cavity was theoretically created to study the quantum correlation. Some of the phenomena, such as the quantum discord and concurrence, were generated through such a system. The influences of initial state purity, qubit motion, and detuning parameters were discussed for the phenomena. These parameters for a specific value show that the behavior of phenomena are analogous. It is interesting to mention that some values of detuning undergo a sudden death of phenomena, and the quantum discord still captures the qubits quantum correlation. We predict that the quantum discord may be a better measure of quantum correlation than concurrence.

PACS: 02.20.Bb
Keyword:an isolated atom with a cavity SU(1, 1);quantum discord concurrence
1. Introduction

In recent years, the main resource for tasks of quantum information processing was quantum correlations (QCs). One of the important kinds of QCs was the quantum entanglement. Entanglement played a main role in quantum information and computation,[1–7] quantum state teleportation,[8] and quantum cryptography.[9] Entanglement was observed for some theoretical works[10] and experimental researches.[11] Some of these works were mixed separable states and allowed us to show the quantum algorithms advantages over their classical analogies. The classical algorithms were observed for some systems without entanglement.[12,13]

Besides, another important type of QCs was firstly defined by Ollivir[14] called quantum discord (QD). QD was used to quantify all nonclassical correlation in a system. Relative to the definition of mutual information, QD explains that two classical equivalent ways turn out to be inequivalent in the quantum domain. It has been previously mentioned that QD is an independent one and a more general measure of QCs that may include entanglement. Quantum states with nonclassical correlations were provided by a larger region through QD, and a nonzero QD but not entanglement may be responsible for the efficiency of quantum computing.[15–17] Such observations were suggested that QD could be another asset for quantum information processing.

Recently, some works were studied for QD[18–20] but the analytical calculation of QD was comparatively difficult. On the other hand, the authors in Ref. [21] obtained the analytical formulas of QD for a family of two-qubit states. For more general quantum states, the authors in Refs. [22], [23], [24], and [25] derived some straightforward expressions to obtain QD, and concluded that QD, classic correlation, and entanglement were independent measures of correlations.

This paper is organized as follows. Section 2 gives the model and its solution. Section 3 is devoted to studying two types of measures to analyze quantum correlation. Section 4 gives the numerical results and discussion. Finally, we give our conclusions in Section 5.

2. Main system and its density operator

In the present work, we employ the interaction system of two identical moving qubits (A and B) or atoms with two single-mode SU(1, 1) cavities (a and b). We assume that the system of each qubit–cavity is isolated. This means that the atom “A” at a constant velocity only interacted with a field “a”, and the atom “B” is only interacted with a field “b”, see Fig. 1.

Fig. 1. (color online) Schematic structure for the interaction of each isolated atom with a cavity.

The time-depended Hamiltonian for such a system is given by

with ħ = 1. The system (1) contains some abbreviations: (i) , , and refer to the symmetric Lie group generators of SU(1, 1); (ii) is the annihilation operator of photon (atom), is the Pauli spin matrix, ωj is the atomic transition frequency, νj is the field frequency, λ refers to the coupling constant, j = a, b for cavity operators and j = A, B for atoms operators; (iii) g refers to the cavity shape function. Now, we rewrite system (1) as the following formula.

where Δj = ωj − νj is the detuning parameter of qubit–field interaction. Applying the transformation to the system (2), then the Hamiltonian in the new transformation becomes

where and . Now, we define another form of shape function as follows:

where vj is the atomic velocity, (p1 and p2) are the atomic motion parameters, and pj represents the cavity half-wave number with the length Lj. Here, assuming that ΔA = ΔB = Δ, La = Lb = L, pa = pb = p, and vA = vB = v = λL/π, then the new shape function takes the form: θ(t) = tp1 + p2 sin(pλt)/λp. The effect of qubit motion exists when p1 = 0 and p2 = 1, and it is absent when p1 = 1 and p2 = 0.

To compute the density operator analytically, which will employ searching for the best measure of quantum correlation, we start by calculating the time evolution of system (3) as follows:

where

Based on the previous assumption, the two qubits do not interact, the time evolution for each qubit can be expressed as follows:

In this work, it is assumed that the two qubits are initially in the Werner state

where I4 is the identity matrix, ρAB(0) is called the atom–atom density operator for zero time, and r is the purity of the initial state varying from 0 to 1. Also, the two cavity fields “a” and “b” are initially in the states ρa(0)=|ma, ka⟩⟨ka, ma| and ρb(0)=|mb, kb⟩⟨ kb, mb|. For r = 0, the state becomes a maximal state, while for r = 1 it becomes a Bell states. Consequently, the density evolution operator of the system can be expressed as follows:

We taking the trace over the field variables to obtain the reduced density matrix of two qubits. Relative to the computational basis of states |ee⟩, |eg⟩, |ge⟩, |gg⟩, the matrix ρAB(t) can be written as follows:

The elements of the matrix are given by

With

3. Quantum correlation

Now, to find the better measure of quantum correlation between the two moving and non-moving qubits of the present system, we study some of the phenomena, such as quantum discord and entanglement.

3.1. Entanglement phenomenon

Here, to investigate the entanglement of two qubits, we study the concurrence (C) measure,[26] which is considered as an effective approach to quantify entanglement and is defined as follows:

where λ1, λ2, λ3, and λ4 are the eigenvalues of the matrix, , and C ∈ [0,1]. The unentangled states referring to the concurrence for theirs are zero while maximally entangled states referring to it are one. Now, for the density matrix ρAB, we rewrite the definition (9) in the following formula

3.2. Quantum discord phenomenon

Quantum discord is considered to be one of the important measures of quantum correlation, which is defined as a difference between the total correlation measured by quantum mutual information and the classical correlation[27,28] as follows:

The symbol S(ρj) of Eq. (11) represents the Von Neumann entropy (j refers to either the total system or the subsystem) and ∑kpkS(ρk) is the quantum conditional entropy with ρk = TrB[(I ⊗ Bk) ρAB (I ⊗ Bk)]/pk, pk = TrAB[(I ⊗ Bk) ρAB (I ⊗ Bk)] and Bk is to a projectors complete set for measuring the subsystem B. Based on the previous work[27,28] and the definition of Shannon entropy Γ(x) = −x log2x − (1 − x)log2(1 − x), the definition (11) can be rewritten in the following form

where

with the eigenvalues

and

In the following, we employ Eqs. (10) and (12) to discuss the dynamics behavior of these phenomena.

3.3. Phenomena behavior and their discussion

The effect of parameter r on the time evolution of concurrence is shown in Fig. 2, where concurrence is plotted for r and scaled time λt in the absence of the qubit motion effect. If 0 ≤ r ≤ 0.32, one can note that C is zero while if 0.32 < r ≤ 1 and Δ/λ = 0, it is observed that C evolves periodically and the sudden death of C appears, see Fig. 2(a). We observed that C is randomly evolved and the entanglement sudden death phenomenon disappears for 0.32 < r ≤ 1 and Δ/λ = 10, see Fig. 2(b). For the off-resonance case the value of r is increased, the local maximum value of the concurrence is increased, and it tends to be 1 when r = 1.

Fig. 2. (color online) Concurrence against scaled time λt and parameter r in the absence of the qubit motion effect for the fixed value of the Bargmann index ka = kb = 1/2 and the nonnegative integer ma = mb = 1. Panel (a) for Δ/λ = 0 and panel (b) for Δ/λ = 10.

On the other hand, to explore the effects of r on the quantum discord, we plot in Fig. 3, the QD against λt and r. In the resonance case, one observes that the time behavior of QD is periodical with period nπ, n = 0, 1, 2, 3 …, see Fig. 3(a). Also, for the off-resonance case, in Fig. 3(b), the quantum discord is developed randomly and the periods of QD are early increased by increasing the purity of the initial state.

Fig. 3. (color online) The time evolution of quantum discord in the absence of the qubit motion effect with different values of r ∈ [0,1] for ka = kb = 1/2 and ma = mb = 1. Panel (a) for Δ/λ = 0 and panel (b) for Δ/λ = 10.

Now, we give fixed values for the parameter r to visualize the difference of QD from C. The time evolution of QD and C is shown in Fig. 4 for the fixed values of ka = kb = 1/2, ma = mb = 1, and different values of Δ/λ. One can observe that the detuning has a strong effect on these phenomena. It is noted that the amplitude of QD decreases during the time evolution when Δ/λ takes a suitable value, and a high amount of QD is obtained, see Fig. 4(c2). In addition, for Figs. 4(a1), 4(a2), 4(c1), and 4(c2), the evolutions of C in the resonance and off-resonance cases are always zero while those of QD are still nonzero for the region 0 ≤ r ≤ 0.32. Besides, for 0.32 < r ≤ 1 and Δ = 0, the concurrence phenomenon completely disappears and the sudden death of entanglement happens while the QD phenomenon remains nonzero with its corresponding fluctuations. Here, it is found that the amplitude of QD is smaller than the amplitude of C. Also, we find that the dynamics of QD is distinguished from C, where the behavior of C suddenly disappears during the evaluation while the behavior of QD still does not disappear. As a result, the absence of the entanglement does not indicate the absence of QD, see Figs. 4(b1) and 4(b2). So, the quantum discord still can capture the quantum correlation between the two qubits when C is zero. Here, for a suitable value of the detuning parameter, the amplitude of these phenomena decreases through the time evolution and obtains a high oscillation amount of QD and C, see Figs. 4(d1) and 4(d2). By increasing the values of r and detuning parameters more and more, we reach an important prediction: the amplitudes of these phenomena completely disappeared during the time evolution.

Fig. 4. (color online) The quantum discord and C versus λt in the absence of qubit motion influences for the resonance case in panels (a1), (a2), (b1), and (b2) and the off-resonance case in panels (c1), (c2), (d1), and (d2). Panels (a1) and (a2) for Δ/λ = 0, r = 0.3; Panels (b1) and (b2) for Δ/λ = 0, r = 0.6; Panels (c1) and (c2) for Δ/λ = 10, r = 0.3; Panels (d1) and (d2) for Δ/λ = 10, r = 0.6.

To visualize the influences of atomic motion and field-mode structure on quantum discord and concurrence C, the behavior of the phenomena with r = 1, ka = kb = 1/2, and ma = mb = 1 are displayed in Fig. 5. We observed that from Figs. 5(a1) and 5(a2), the time evolution of the phenomena behavior is similar to the above discussion and it exhibits the periodic dynamics with the same period as shown in the previous figures. So, the simulations of Figs. 5(a1) and 5(a2) are the same as those discussed above. In addition, by increasing the value of p the evolution periods of phenomena are decreased, as can be seen in Figs. 5(b1) and 5(b2).

Fig. 5. (color online) The behavior in the resonance case through the influences of atomic motion (p2) and field-mode structure (p). Panels (a1) and (a2) for Δ/λ = 0, r = 1, and p = 1; Panels (b1) and (b2) for Δ/λ = 0, r = 1, and p = 3.

In addition, in Fig. 6, these phenomena have been plotted as a contour versus time and r with the same conditions as given in Fig. 5. Also, it is concluded that the quantum discord still is obviously nonzero and the long-living quantum discord can be obtained.

Fig. 6. (color online) Contour plots of phenomena behavior in the resonance case. Panels (a) and (c) for Δ/λ = 0, p = 1; Panels (b) and (d) for Δ/λ = 0, p = 3.
4. Summary

In summary, some of the phenomena, such as the quantum discord and concurrence, have been demonstrated between two isolated moving qubits each separately interacting with a single-mode SU(1, 1) cavity. The effects of some parameters, such as the field-mode structure, purity of the initial state, atomic motion, and detuning have been studied for the current phenomena. These phenomena are strongly influenced due to these parameters. It is observed that in the absence of qubit motion and a specific r for the resonance and off-resonance cases the entanglement completely disappears while the quantum discord remains unchanged. Also, a sudden death via the detuning effect for these phenomena appears. Besides, in the presence of qubit motion and field-mode structure for r = 1, the phenomena exhibit the same behavior for the resonance and off-resonance cases. It is concluded that a minimum value of quantum discord is always greater than that of concurrence, and then when concurrence is zero the quantum discord can still capture the quantum correlation between the two qubits. Hence, the quantum discord is considered as a best measure of quantum correlation.

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