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Entanglement with phase decoherence

The system of an atom couples to two distinct optical cavities with phase decoherence is studied by making use of a dynamical algebraic method. We adopt the concurrence to characterize the entanglement between atom and cavities or between two optical cavities in the presence of the phase decoherence. It is found that the entanglement between atom and cavities can be controlled by adjusting the detuning parameter. Finally, we show that even if the atom is initially prepared in a maximally mixed state, it can also entangle the two mode cavity fields.


I. INTRODUCTION
Quantum entanglement was first introduced by Einstein, Podolsky and Rosen in their famous paper in 1935 [1]. It has been recently recognized that entanglement can be used as an important resource for quantum information processing [2]. Entanglement can exhibit the nature of a nonlocal correlation between quantum systems that have no classical interpretation. However, real quantum systems will unavoidably be influenced by surrounding environments. The interaction between the environment and quantum systems of interest can lead to decoherence. It is therefore of great importance to prevent or minimize the influence of environmental noise in the practical realization of quantum information processing. In order to prevent the effect of decoherence, several approaches have been proposed such as quantum error-correcting approach [3] or quantum error-avoiding approach [4,5].
Instead of attempting to shield the system from the environmental noise, Plenio and Huelge [6] use white noise to play a constructive role and generate the controllable entanglement by incoherent sources. Similar work on this aspect has also been considered by other authors [7]. In this paper, we investigate an atom couples to two distinct optical cavities with the phase decoherence and show how the entanglement between atom and cavities or between two optical cavities can be generated in the presence of the phase decoherence. In section II, we study the system with phase decoherence by making use of the dynamic algebraic method [8] and find the exact solution of the master equation for the system. The exact solution is then used to discuss the influence of the phase decoherence on the probability of occupation in ground state. In section III, we use the concurrence to characterize the entanglement between atom and cavities or between two optical cavities by means of the exact solution for the system. It is shown that the entanglement between atom and cavities can be controlled by adjusting the detuning parameter. Finally, we show that even if the atom is initially prepared in a maximally mixed state, it can also entangle the two mode cavity fields. A conclusion is given in Section IV.

II. SOLUTION OF AN ATOM COUPLES TO TWO DISTINCT OPTICAL CAVITIES WITH PHASE DECOHERENCE
We consider the situation that an atomic system is surrounded by two distinct optical cavities initially prepared in the vacuum state. The Hamiltonian for the system can be described by [6], where |e and |g are the excited state and the ground state of the two-level atom, ω 0 is atomic transition frequency, g a(b) is the coupling constant of the atom to cavity modes a(b), and a (a † ), b (b † ) are the annihilation (creation) operators of a mode of frequency ω a and b mode of frequency ω b , respectively. In Ref. [6], Plenio and Huelge use white noise as the actual driving force of the system and study numerically the entanglement between two optical cavities for the system in the resonant case. Here, we investigate analytically the entanglement between atom and cavities or between two optical cavities with phase decoherence by making use of the dynamical algebraical method. To reduce the complexity, we consider the case of ω a = ω b = ω. It is easy to verify that there exists two constants of motion in Hamiltonian (1), where g = g 2 a + g 2 b . It is easily proved that the operator K 1 and K 2 commute with Hamiltonian (1). We then introduce the operators as follows It can be shown that the operators S i (i = 0, ±) satisfy the following commutation relations where S 0 and S ± are the generators of the SU(2) algebra [9]. In terms of the SU(2) generators, we can rewrite Hamiltonian (1) as where ∆ = ω 0 − ω denotes detuning. In this paper, we consider the pure phase decoherence mechanism only. In this situation, the master equation governing the time evolution for the system under the Markovian approximation is given by [10] dρ dt where γ is the phase decoherence rate. Noted that the equation with the similar form has been proposed to describing the intrinsic decoherence [11]. The formal solution of the master equation (6) can be expressed as follows [8], where ρ(0) is the density operators of the initial atom-field system and M k (t) is defined by By means of the SU(2) dynamical algebraic structure, we obtain the explicit expression for the operator M k where We assume that the cavity fields are prepared initially in vacuum state |00 , and the atom is prepared in the excited state |e . The time evolution of ρ(t) can be written as, The |ϕ in Eq. (12) is a single-photon entangled state. Recently, much attention has been paid to investigate the preparation of the single-photon maximally entangled state [12]. It is noted that when the two coupling coefficients g a = g b , the state |ϕ is nothing but a single-photon maximally entangled state. We then show that if a projective measurement on the atom in the {|e , |g } basis is made, the atom will be projected on the ground state |g with the probability P g in the case of ∆ = 0, If the measurement result is |g , the two distinct cavity fields are in the single-photon maximally entangled state √ 2 2 (|10 + |01 ). In Fig.1, we plot the probability P g as the function of time t for different values of phase decoherence rate γ. It is shown that if the decoherence rate γ is zero, the two distinct cavity fields are in the maximally entangled single-photon state at the time t = π 2g with unit probability.

III. THE ENTANGLEMENT BETWEEN ATOM AND CAVITIES OR TWO OPTICAL CAVITIES
In order to quantify the degree of entanglement, several measures [13] of entanglement have been introduced for both pure and mixed quantum states. In this section, we adopt the concurrence to calculate the entanglement between atom and cavities or between two optical cavities with the phase decoherence. The concurrence related to the density operator ρ of a mixed state is defined by [14] C(ρ) = max{λ 1 where the λ i (i = 1, 2, 3, 4) are the square roots of the eigenvalues in decreasing order of magnitude of the "spin-flipped" density operator R where the asterisk indicates complex conjugation. The concurrence varies from C = 0 for an unentangled state to C = 1 for a maximally entangled state. We first investigate the quantum correlation between the atom and cavity modes. If we deal with the two cavity modes as system B, and the atom as system A, then ρ(t) in Eq.(11) can be thought of as the density operator of a two-qubit mixed state. In the basis |11 s ≡ |00 ⊗ |e , |10 s ≡ |00 ⊗ |g , |01 s ≡ |ϕ ⊗ |e , |00 s ≡ |ϕ ⊗ |g , the explicit expression of the concurrence C AB describing the entanglement between the system A and system B can be found to be, From Eq.(16), we can see that the detuning ∆ plays a key role in the quantum correlation between the atom and cavity modes. If the decoherence rate γ is not equal to zero, the concurrence C AB remains in the value 2g|∆|/Ω 2 in the limit t → ∞. In the strong coupling case, i.e., g a , g b ≫ ∆, the concurrence C AB of the stationary state ρ(∞) is approximately |∆|/(2g). On the other hand, in the large detuning limit, the concurrence C AB of the stationary state is approximately 2g/|∆|. In Fig.2, the concurrence C AB is plotted as a function of the time t and decoherence rate γ. We show the concurrence C AB as a function of the detuning parameter ∆ and the decoherence rate γ at a fixed time in Fig.3. In the limit t → ∞, concurrence C AB is plotted as a function of the detuning parameter ∆ in Fig.4. From Fig.4, we can see that C AB increases with the detuning ∆ parameter. This means that the entanglement between the atom and the cavity fields can be controlled by adjusting the detuning parameter. Now, we turn our discussion to the resonant case, i.e. ∆ = 0. In this case, C AB = | sin(2gt)| exp(−2g 2 γt). In Ref. [15], it has been proved that for any pure states of three qubits 1, 2 and 3, the entanglement is distributed following the inequality for the squared concurrence where C 1,(23) is the single-qubit concurrence defined as the concurrence between the qubit 1 and the rest of qubits (2,3). For any mixed states of three qubits 1, 2, and 3, there is analogous inequality for the squared concurrence as follows where C 2 min 1(23) is the minimum of average over all possible pure state decomposition of the three qubits mixed state [15].
In the present paper, we may expect that the pair entanglement between the atom and the a(b) mode cavity field is determined by the coupling coefficient g a (g b ). It is easy to prove that there exist the simple relations, and C a /C b = g a /g b , where C a (C b ) is the concurrence describing entanglement between the atom and the a(b) mode cavity field. Thus, our result is in agreement with that obtained in Ref. [15]. Next, we investigate the entanglement between light fields of two distinct cavities. By tracing out the degree of freedom of the atom in density matrix ρ(t) in Eq.(11), we obtain the reduced density matrix ρ B (t) describing the two light fields as follows, Then, the concurrence C B characterizing the entanglement of two light fields can be derived as From Eq.(21), we can see that the concurrence C B is equal to zero at time t = 2nπ/Ω, (n = 0, 1, 2...) in the case of γ = 0. At these specific time, the two cavity modes have no pair entanglement. However, in the case with γ = 0, the two cavity modes is always entangled for the time t > 0. In Fig.5, we plot the concurrence C B as the function of time t and damping rate γ. From Fig.5, we see that the entanglement between the two distinct light fields increases with the phase decoherence rate γ within the time range 2nπ/ √ ∆ 2 + 4g 2 ≤ t < (2n + 1 2 )π/ √ ∆ 2 + 4g 2 or (2n + 3 2 )π/ √ ∆ 2 + 4g 2 < t ≤ (2n + 2)π/ √ ∆ 2 + 4g 2 (n = 0, 1, 2, ...). The concurrence C B is displayed as a function of the phase decoherence rate γ for three different values of the detuning parameters at fixed time in Fig.6. The stationary state entanglement of the two cavity modes measured by concurrence is 4g a g b /(∆ 2 + 4g 2 ). This means that the stationary state entanglement achieves its maximal value 1/2 in the resonant case with g a = g b .
Finally, we discuss how much entanglement between the two mode cavity fields can be achieved if the initial atom is prepared in a thermal state and the cavity fields are prepared in the vacuum states. We assume that the initial atom is in the state ρ A (0) = δ|g g| + (1 − δ)|e e|, where 0 ≤ δ ≤ 1, and the cavity fields are still in the vacuum state |00 . Our calculation shows that C ′ AB = (1 − δ)C AB and C ′ B = (1 − δ)C B . This means that even if the initial atom is prepared in a maximally mixed state 1 2 |g g| + 1 2 |e e|, it can still entangle the two mode cavity fields. In this case, the concurrence C ′ B equals 1 4 in the steady state for ∆ = 0 and g a = g b .

IV. CONCLUSION
In this paper, we investigate analytically the entanglement between atom and cavities or between two optical cavities with phase decoherence by making use of the dynamic algebraic method. It is found that the entanglement between atom and cavities can be controlled by adjusting the detuning parameter. Finally, we show that even if the atom is initially in a maximally mixed state, it can also entangle two mode cavity fields initially prepared in vacuum state. The approach adopted here can be employed to investigate the entanglement between two optical cavities mediated by a two-level atom in those cases, in which the two mode cavity fields are initially prepared in another separable states.    2. The concurrence C AB as a function of the time t and the phase damping rate γ for g a = g b = 1 and ∆ = 5.

FIG.3
The concurrence C AB as a function of the detuning parameter ∆ and the phase damping rate γ for g a = g b = 1 and t = 10.

FIG.4
The concurrence C AB of the steady state as a function of the detuning parameter ∆ for g a = g b = 1 and γ = 0.1.

FIG.5
The concurrence C B as a function of the time t and the phase damping rate γ for g a = g b = 1 and ∆ = 0.

FIG.6
The concurrence C B as a function of the phase damping rate γ for various values of the detuning parameter: ∆ = 0 (Solid), ∆ = 1 (Dash) and ∆ = 2 (Dot) with t = 2 and g a = g b = 1.