Entanglement between Two Tavis-Cummings Systems with N=2

We consider two non interacting two-atoms Tavis-Cummings systems and examine the entanglement among four two-state atoms. For our purpose, we assume that one atom from each system are entangled init ially and we obtain time-dependent concurrence among atoms. There are six pairwise concurrences among atoms and in this paper, we focus on remote atoms. We show that initially non-entangled atoms 2 A and 2 B becomes entangled by passing time, the process which may be interpreted as entanglement transfer between the cavities.


Introduction
Entanglement is one of the most fascinating aspect of quantum mechanics and is a defining future that makes fundamental d istinctions between quantum and classical Physics. Two quantum systems A and B are entangled when values of certain properties of system A are correlated with the same properties of system B [1]. Quantum entanglement has been viewed as an essential resources for quantum informat ion process, and a great deal of effort has been devoted to study and characterize the entanglement. Cavity quantum electrodynamics (QED) techniques has been recognized as a pro mising candidate for the physical realization of quantum informat ion processing [2].
In this paper we consider two non interacting two-ato ms Tavis-Cu mmings systems and examine the entanglement among four two-states atoms by obtaining the concurrences between each two atoms. The method which we have used is like the method that is used in ref [3]. For this purpose we first introduce the Tavis-Cu mmings model and then obtain the eigenvalues and eigenstates of the two-atoms Tavis-Cu mmings system in section 2. These eigenstates and eigenvalues are used for calculating the time evaluation of entanglement. We introduce the concept of concurrence for measuring the entanglement in section 3 and finally in section 4, we analy ze the entanglement among atoms in detail by calculating the reduced density matrice of each two two-levels ato ms and give some concludes in section 5.

The Tavis-Cummings Model
The Tavis-Cu mmings model (TCM ) describes the simp lest fundamental interaction between a single mode of the quantized electro magnetic field and a collection of N two-level ato ms under the rotating wave appro ximation approximation (RWA) condition [4] - [11]. The Hamiltonian of this model is written as (1) For the case of two-atom (N=2) Where a and + a are the annihilat ion and creation field operators. Each in itial state in this system is coupled to three states.
For examp le if the initial state is n e e i 2 1 = then this state is coupled with the states (3) Writing the Hamiltonian in these states, we obtain ( ) Then the eigenvalues are (5) and the correlated eigenstates are

Measure of Entanglement
To measure the entanglement, we need to introduce a convenient concept. We will adopt the Wooter's concurrence [13] as our measure: (7) where the quantities i λ are the eigenvalues in decreasing order of the matrix (8) where * ρ denotes the complex conjugation of ρ in the standard basis and y σ is the Pauli mat rix.
We may pay attention to this point that although the matrix ξ is not Hermit ian, the eigenvalues of this matrix are real and nonnegative. Reduction to a two-qubit form, will yield to a t wo-qubit mixed state always having the form: The concurrence for this matrix is easily found to be (10)

Entanglement Dynamics among
Atoms in Two Tavis Now, we are go ing to investigate the entanglement among atoms in this two Tavis-Cummings systems by calculating the concurrence among ato ms. The total init ial state has three parts: i) The entangled atoms: Th is in itial part can be written in terms of Bell states. We denote this part as follows: (11) ii) The two atoms which are not entangled initially. We can write this init ial part as below: (12) To make sure that these atoms are not entangled init ially, we must have 3 2 4 1 α α α α = (13) iii) The cavit ies Initial states: where n (m) is the number of photons in cavities A (B) at t=0.
Thus the total initial state is (15) which can be written as To simp lify our calcu lations, we suppose that the non entangled atoms are in itially in excited states ( ) and the nu mber of photons in two cavities are equal (n =m). By applying these conditions, the initial state becomes (17) To prepare for the time evolution we exp ress these states in terms of the eigenstates given in (6), and obtain (18) where (19) Thus, the initial state is (20) Since the time evolution of these eigenstates is specified we can transfer these time evolutions to the coefficients. Therefore (21) In order to take traces over individual ato ms or cavit ies, we need to revert to the bare bases and this leads to (22)       x    If we look at the in itial state in equation (19) we see that under the transformat ion the state remain unchanged.
That means at all times.
Similarly, we can find the exp licit exp ressions for and . The two cavit ies are not distinguishable, so we expect these C's to be the same and they are. The has the form The concurrence for this density matrix is [12] (35)

Conclusions
In this paper we considered two two-ato ms Tavis-Cu mmings systems. These two systems have no interaction together, but one atom fro m each system are entangled initially ( 1 The entanglement between 1 A and 1 B for times t>0 falls to zero and remains non-entangled for a period of t ime till becomes entangled again. We also see that the entangled intervals are smaller than non-entangled intervals. and solve the problem again. Or for the non-entangled atoms, which we assumed that were in their excited states initially, we can change this prescription and solve the problem again, but the methods for this new situation are similar with the method which we used before.