Control the entanglement of two atoms in an optical cavity via white noise

Two two-level atoms within a leaky optical cavity is driven by two independent external optical white noise fields. We investigate how entanglement between two atoms arises in such a situation. The steady state entanglement of two atoms is also investigated. A stochastic-resonance-like behavior of entanglement is revealed. Finally, the Bell violation between atoms is discussed.


I. INTRODUCTION
Quantum entanglement plays a crucial role in quantum information and quantum computation [1]. 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 errorcorrecting approach [2] or quantum error-avoiding approach [3,4].
Instead of attempting to shield the system from the environmental noise, Plenio and Huelge [5] use white noise to play a constructive role and generate the controllable entanglement by incoherent sources. They showed that the noise-assisted entanglement exhibits the stochastic resonance behavior. Similar work on this aspect has also been considered by other authors [6][7][8]. In this paper, we study the quantum system in which two two-level atoms within a leaky optical cavity is driven by two independent external optical white noise fields. We investigate how entanglement between two atoms arises in such a situation. It is shown that white noise exhibits dual aspects, i.e., playing either a destructive or a constructive role in quantum information processing. Recently, Clark and Parkins [9] proposed a scheme to controllably entangle the internal states of two atoms trapped in a high-finesse optical cavity by employing quantum-reservoir engineering. By making use of laser and cavity fields to drive two separate Raman transitions between stable atomic ground states, the two atoms is effectively coupled to a squeezed reservoir. Phase-sensitive reservoir correlations leads to entanglement between the atoms. Different from their scheme, we will focus here on the problem of generating entanglement when only incoherent sources are available and show that controllable entanglement can arise in this situation. We show that, if two atoms are simultaneous driven by two independent white noise field with the same intensity, the entanglement between them is suppressed and eventually completely destroyed by the noise. However, in another case, in which only one atom is exposed in white noise field, the steady state entanglement of the two atoms is non-monotonic function of both the intensity of noise driving field and the spontaneous decay rate. A double resonance behavior emerges. Moreover, the threshold value of the spontaneous decay rate, below which there is not any steady state entanglement, increases with the intensity of noise field. This paper is organized as follows: In section II, we outline the experimental set up, in which two atoms are trapped in a optical cavity and driven by the thermal field. The spectral width of the thermal field is large compared to the linewidth of the atomic transition so that its effect is that of a white noise source. we model this system by a master equation and give a explicit analytical solution of the time evolution density matrix. In section III, based on the density matrix, we obtain the analytical expression of the concurrence characterizing the entanglement between two atoms. Both the entanglement during the time evolution and the steady state entanglement are inves-2 tigated. A conclusion is given in section IV.

II. THE MASTER EQUATION DESCRIBING TWO ATOMS TRAPPED IN A OPTICAL CAVITY AND DRIVEN BY NOISE FIELD
The system we consider here is two atom trapped in a optical cavity. The atoms are driven by two independent thermal fields and separated by a large enough distance that they feel no direct dipole-dipole interaction. The cavity has a field decay k and a frequency ω. The two eigenstates of the individual atom (|0 , |1 ) constitute the qubit states. The master equation for the total system density operator is (h = 1) where where a and a † are the annihilation and creation operators of the cavity field with frequency ω, and ω 0 is the transition frequency of the atoms and g is the atom-cavity coupling constant. The Liouvilleans L cav ρ and L at ρ are given by and where Γ (j) describes the coupling strength of the jth atom to the external fields and n (j) T Γ (j) is the transition rate due to the thermal field. The spectral width of the thermal field is large compared to the linewidth of the atomic transition so that its effect is that of a white noise source. Here, n (j) T can be interpreted as an effective photon number and that spontaneous decay of the atom out of the cavities is included in this scenario via the n (j) T + 1 term.
In the large detuning limit, i.e., ∆ = ω 0 − ω ≫ g √n + 1 withn being the mean photon number of the cavity field, there is no energy exchange between the atomic system and the cavity. We can obtain the effective Hamiltonian H e [10,11] The first and second terms describe the photon number dependent Stark shifts, and the third term describes the dipole coupling between the first and second atoms induced by the virtual photon process. When the cavity mode is initially in the vacuum state |0 , it will remain in the vacuum state throughout the procedure, the effective Hamiltonian reduces toH As the cavity mode will then never be populated, we can disregard the cavity mode in the following. Now, the master equation (1) can be reduced to where ρ s is the density matrix describing the subsystem containing only the two atoms. Firstly, we discuss the case with n (1) , two atoms are driven by two independent thermal fields with the same intensity. We assume that the atom 1 and atom 2 are initially in the pure product state |1 1 ⊗ |0 2 . Then, the explicit analytical solution of the master equation (7) can be obtained as follows, 4 In order to quantify the degree of entanglement, we adopt the concurrence C defined by Wooters [12]. The concurrence varies from C = 0 for an unentangled state to C = 1 for a maximally entangled state. For two qubits, in the "Standard" eigenbasis: |1 ≡ |11 , |2 ≡ |10 , |3 ≡ |01 , |4 ≡ |00 , the concurrence may be calculated explicitly from the following: 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 matrix operator R = ρ s (σ y ⊗ σ y )ρ * s (σ y ⊗ σ y ), where the asterisk indicates complex conjugation. It is straightforward to compute analytically the concurrence for the density matrix ρ s (t), and the concurrence C s (t) related to the density matrix ρ s (t) is obtained as follows where |x| gives the absolute value of x. In Fig.1, we have plotted the concurrence C s (t) as a function of the time t and the intensity of the thermal field n T with g 2 /∆ = 0.2 and Γ = 0.01. It is shown that the entanglement between two atoms decreases with n T , and there is not any entanglement arising between two atoms during the time evolution when n T is beyond a threshold value depending on the coupling constant Γ and effective Rabi frequency g 2 /∆. In Fig.2, the concurrence C s (t) is plotted as the function of the time t and the coupling constant Γ of the atoms and the external field (Note that Γ is equivalent to the spontaneous emission rate if n T = 0) with two different values of effective photon number n T of the thermal field. In the case with n T = 0 and g 2 /∆ = 0.2 (see Fig.2(a)), the entanglement of two atoms always arises during the time evolution even in the presence of atomic spontaneous emission. However, in the case with n T = 0.3 and g 2 /∆ = 0.2 (see Fig.2(b)), a threshold value of Γ is found, beyond which there is not any entanglement arising during time evolution. All of the above discussions indicate that the two equal intensity independent thermal fields suppress the entanglement generation. But, that is not the full aspects concerning the role of thermal field played in the entanglement of two atoms. In the following section, we consider the situation in which, only one of the atoms is driven by the external thermal field. A different aspect of the thermal field will be found.

III. THE STEADY STATE ENTANGLEMENT OF TWO ATOMS
In the above section, we have discussed the case, in which the two atoms are simultaneously driven by an external thermal field. there is not any steady state entanglement between two atoms in that situation. In this section, we consider the situation in which, only one of the atoms is driven by the external thermal field. The master equation is given by where η is the spontaneous emission rate of the atom 2. We assume that the two atoms are initially in the ground state |0 1 ⊗ |0 2 . The explicit analytical solution of the steady state of the master equation (12) can be obtained as follows, where Ω = g 2 /∆. The concurrence C st related to the steady state ρ st is obtained as follows In Fig.3, we have plotted the concurrence C st as a function of the spontaneous emission rate η and the intensity of the thermal field n T with g 2 /∆ = 0.2 and Γ = 0.1. It is shown that the steady state entanglement exhibit a double stochastic-resonance-like behavior, which is similar with the results in Ref. [5]. The double stochastic-resonance-like behavior also emerges in Fig.4, in which C st is depicted as a function of the spontaneous emission rate η and the coupling constant Γ with g 2 /∆ = 0.2 and n T = 2. In Fig.5, we plot the C st as a function of Ω and the intensity of the thermal field n T with Γ = 0.1 and η = 0.5. It is shown that the threshold value of n T is strongly dependent on the value of Ω.
6 From Eq.(15), we can find the threshold values of the parameters Ω, n T , Γ and η, beyond which there is not entanglement in the steady state. Some simply inequalities can be derived as follows The gray area in Fig.6 depicts the region where the steady state of the two atoms is entangled in the case with Γ = 0.1 and η = 0.5. In Fig.7, the concurrence C st is plotted as a function of the intensity of the thermal field n T with Γ = 0.01 and η = 0.5 for four different values of g 2 /∆. In Fig.8(b), we show how two atoms initially in various different product states would eventually evolve into the entangled steady state in the presence of the external noise driving one of the atoms. Otherwise, in the absence of the intense enough external noise, two atoms firstly become entangled due to the dipole coupling induced by the virtual photon process, then, they rapidly lose the entanglement, as shown in Fig.8(a).
Finally, we attempt to discuss the nonlocality of two atoms in the steady state. The nonlocal property of two atoms can be characterized by the maximal violation of Bell inequality. The most commonly discussed Bell inequality is the CHSH inequality [13,14]. The CHSH operator readŝ where a, a ′ , b, b ′ are unit vectors. In the above notation, the Bell inequality reads The maximal amount of Bell violation of a state ρ is given by [15] where λ andλ are the two largest eigenvalues of T ρ T † ρ . The matrix T ρ is determined completely by the correlation functions being a 3 × 3 matrix whose elements are (T ρ ) nm = Tr(ρσ n ⊗ σ m ). Here, σ 1 ≡ σ x , σ 2 ≡ σ y , and σ 3 ≡ σ z denote the usual Pauli matrices. We call the quantity B the maximal violation measure, which indicates the Bell violation when B > 2 and the maximal violation when B = 2 √ 2. For the density operator ρ st in Eq.(13) characterizing the steady state of two atoms, λ +λ can be written as follows Recently, Verstraete et al. investigated the relations between the violation of the CHSH inequality and the concurrence for systems of two qubits [16]. They showed that the maximal value of B for given concurrence C is 2 √ 1 + C 2 , which can be achieved by the pure states and some Bell diagonal states. If the given concurrence C is larger than √ 2 2 , the minimal value of B is 2 √ 2C, which can be achieved by the maximal entangled mixed state. Furthermore, the entangled two qubits state with the concurrence C ≤ √ 2 2 may not violate any CHSH inequality, even after all possible local filtering operations, except their Bell diagonal normal form does violate the CHSH inequalities [16]. So, it is not difficult to understand the following results. Our calculations show that not any violation of CHSH inequality will be found in the steady state even though the steady state is entangled. Moreover, the stochastic-resonance-like behavior can not be observed in the Bell violation of two atoms during the evolution, and the stronger the noise intensity, the more rapid the Bell violation disappears, which is shown in Fig.9.

V. CONCLUSSION
In this paper, we investigate the problem of generating entanglement when only incoherent sources are available and show that controllable entanglement can arise in this situation. We show that, if two atoms are simultaneous driven by two independent white noise field with the same intensity, the entanglement between them is suppressed and eventually completely destroyed by the noise. However, in another case, in which only one atom is exposed in white noise field, the steady state entanglement of the two atoms is non-monotonic function of both the intensity of noise driving field and the spontaneous decay rate. A double resonance behavior emerges. Moreover, the threshold value of the spontaneous decay rate, below which there is not any steady state entanglement, increases with the intensity of noise field.   Figure 3: the concurrence C st is plotted as a function of the spontaneous emission rate η and the intensity of the thermal field n T with g 2 /∆ = 0.2 and Γ = 0.1.