1. Introduction
Entanglement is a kind of quantum correlation that has been regarded as a valuable resource for quantum information processing [1] . However, it is was shown both theoretically [2] and experimentally [3] that some mixed separable states allow one to realize the advantages of quantum algorithms in comparison with their classical analogies and quantum nonlocality has been observed in the systems without entanglement [4] [5] . Such observations suggest us to conclude that the entanglement is not the only type of quantum correlation for quantum technology. A new but promising candidate is quantum discord first introduced by Ollivir and Zurek, which is used to quantify all nonclassical correlation in a system [6] . Quantum discord is built on the fact that two classical equivalent ways of defining the mutual information turn out to be inequivalent in the quantum domain. It is observed that quantum discord is a more general measure of quantum correlation that may include entanglement but is an independent one. Moreover, quantum discord provides a larger region of quantum states with nonclassical correlations, and a nonzero quantum discord but not entanglement may be responsible for the efficiency of a quantum computer [3] -[7] . Therefore, quantum discord could be a new resource for quantum information processing. Recently, numerous works have been devoted to the study of quantum discord [8] -[12] . Usually, quantum discord is somewhat difficult to calculate and we cannot obtain analytical solutions. But in [13] , the authors evaluated the quantum discord for a family of two-qubit states and obtained analytical formulas. For more general quantum states, such as the so-called 
states, the authors [14] -[21] derived explicit expressions for quantum discord and found that the quantum discord, entanglement and classic correlation were independent measures of correlations with no simple relative ordering. An extreme example is realized in the search of maximally discordant mixed states. It has revealed that states maximizing the quantum discord for given value of the classical correlations do not maximize the entanglement and are, in some cases, even separable [22] . Very recently a reliable and effective algorithm for the evaluation of the quantum discord of the general two-qubit states was presented, where the optimization problem for the quantum discord was expressed in a compact form [23] . In this work, we calculate and compare the time evolution of quantum discord and concurrence of two moving isolated atoms. This paper is organized as follows. In Section 2, we give the model and its solution. In Section 3, we give two types of measures including quantum discord and concurrence to analyze quantum correlation. Numerical results and discussion are given in Section 4. Finally we conclude in Section 5.
2. The Model and Its Solution
We consider two identical moving two-level atoms labeled 
and 
interact with two single-mode thermal cavity fields labeled 
and
. Each atom-cavity system is isolated (atom 
at a constant velocity interacting only with field 
and similarly for the atom 
and the field
). For simplicity, we suppose that these two atoms be coupled to the field with the same coupling constant
, and the field mode be resonant with the transition frequency of both atoms. The interaction Hamiltonian under the rotating wave approximation 
is
(1)
where 
is the creation (annihilation) for 
mode of the cavity field and
, 
are the raising and lowering operators respectively. 
denotes the mode structure of cavity field which can be defined specially as [24]
(2)
where, 
denotes the velocity of the 
atom and 
represents the half-wave number of cavity field with length
. We restrict our study for atomic in the 
-axis direction, so that only the 
-dependent field mode function needs to be taken into account. For simplicity we assume that
, 
and 
. Using standard technique, it can show that 
in Equation (1) gives rise to the following time evolution operator
(3)
where, 
and
. For a particular choice of the atomic motion velocity
, 
becomes
(4)
We assume that the two atoms are initially in the Werner state
(5)
Here, 
, 
is the 
identity matrix and 
varies from 0 to 1, and for 
the Werner becomes maximally states, while for 
they are the well known Bell states. The two cavity fields 
and 
are initially in single-mode thermal field states
(6)
The weight function 
and 
are,
(7)
where 
and 
are the mean photon numbers of the two thermal cavity fields a and b corresponding to the equilibrium cavity temperatures Ta and Tb respectively, 
is the Boltzmann constant and 
is the frequency of the field.
Using the evolution operator, we can generate the density operator of the system at time 
as
(8)
We can obtain the reduced density matrix for the two atoms at time t by tracing over the field variables. Under the standard basis 
the reduced density matrix 
can be obtained in the form
(9)
where the matrix elements are expressed as follow
(10)
with 
(the asterisk denotes complex conjugation) and all matrix elements are zero, 
and 
are given in Equation (7), and the time-dependent operations are expressed by
(11)
where 
is given in Equation (4).
3. Quantum Discord and Entanglement
In the following, we will investigate the quantum discord and entanglement between two moving atoms by using above results. In order to follow the entanglement of the two atoms, we choose the Wootters entanglement measure [25] , the concurrence C defined as
(12)
where the quantities 
are the square roots of the eigenvalues of the matrix
. The concurrence 
corresponds to an unentangled states and 
to a maximally entangled state.
On the other hand, quantum discord is based on the difference between the quantum mutual information and the classical correlation [26] . For a two-qubit quantum system, the total correlation is measured by their quantum mutual information
(13)
where 
and 
denote the reduced density matrix of 
and the density of the bipartite system respectively, and 
is the von Neumann entropy. Quantum discord, which quantifies the quantumness of correlation between 
and
, is then defined as the difference between the total correlation and classical correlation,
(14)
Here the classical correlation
, which quantifies the maximum information that one can obtain from the composite system by measurement on one of the subsystem, can be expressed as
(15)
where 
denotes a complete set of projective measurements performed locally on subsystem
, and
is the quantum conditional entropy with 
and
. By substituting from Equations (13) and (15) into Equation (14), one can obtain the quantum discord as
(16)
For a class of two-qubit states of qubits 
and
, i.e. the two-qubit 
states in standard basis 
,
(17)
It is easy to show that
(18)
where, 
being the four eigenvalues of 
and 
is the Shannon entropy.
According to Ref. [14] [27] , 
can be obtained as
(19)
where
For our system under consideration above, for the density matrix in Equation (9), one can obtain the explicit expressions of the concurrence 
and quantum discord 
in the form
(20)
(21)
where
(22)
(23)
with
(24)
where, the non zero elements 
are given in Equation (10).
Having obtained the results above, we are therefore in a position to discuss the dynamics behavior of the quantum discord and entanglement to the system consider in the following section.
4. Numerical Results and Discussion
Our goal in this article is to clarify what peculiar dynamics happened for quantum discord and entanglement (measured by concurrence) of the system under consideration as follow. In Figure 1, we plotted the concurrence 
as a function of scaled time 
and
. It is apparent that when 
the concurrence is zero,
while
, one can see that the concurrence evolves periodically and entanglement sudden death phenomenon occurs, as r increases the local maximum value of the concurrence increases and tends to 1 when r = 1. To study the effect of the atomic motion related to the field-mode structure parameter p in entanglement, we compare between Figure 1(a) 
and Figure 1(b)
, it is observed that when p increases the evolution period of the concurrence decreases and the time interval for zero entanglement becomes more and its wide becomes shorted. On the other hand, the quantum discord 
is plotted versus scaled time gt and r in Figure 2. The time behavior of quantum discord is periodical with period
, and its evolution periods are shorten with the increased parameter p. In order to compare between the quantum discord and concurrence, we plotted quantum discord and concurrence as a function of scaled time gt for different values of p and r in Figure 3. We have two main regions responding r: first region 0 ≤ r ≤ 0.33, in this region the concurrence
(a) (b)
Figure 2. Quantum discord QD plotted as a function of scaled time gt and r, for (a) 
and (b)
.
Figure 3. Concurrence C and quantum discord QD versus scaled time gt. The black solid and red dashed lines denote respectively QD and C.
always zero but the quantum discord remains non zero. Second region 0.33 < r ≤ 1, the concurrence vanishes periodically and entanglement sudden death happens, while the quantum discord remains non zero with its corresponding fluctuations. Also the frequencies of quantifiers in unit time are the same for concurrence and quantum discord. Whereas the amplitude of concurrence is greater than the amplitude of quantum discord. However, the dynamic of quantum discord is quite different, although entanglement suddenly disappears during the evaluation, the quantum discord remains non zero. It means that the absence of entanglement does not indicate the absence of quantum discord, so quantum discord still can capture the quantum correlation between the two atoms when the entanglement is zero.
5. Conclusion
We have investigated the quantum discord and entanglement between two isolated two-level atoms which are initially prepared in Werner state and each atom interacts with a thermal cavity field by considering the atomic motion and the field-mode structure. It is shown that the entanglement vanishes periodically and sudden death of entanglement appears, while the quantum discord remains non zero with its corresponding fluctuations. We observe that the atomic motion and the field-mode structure have the same effect on the behavior of quantum discord and entanglement. Although entanglement suddenly disappears during the evaluation, the quantum discord remains non zero. Quantum discord still can capture the quantum correlation between the two atoms when the entanglement is zero.