1. Introduction

Quantum entanglement plays a crucial role in performing quantum information processing [1, 2]. From a fundamental perspective, it is a nonclassical effect which is indispensable for understanding fundamental quantum physics. From a technological perspective, it is useful for enhanced measurement techniques and is the basic requirement for transferring quantum information between different nodes in a quantum network. The preparation and storage of the entanglement between distant quantum nodes leaves a great challenge due to environmental decoherence as well as the imperfection of quantum system itself [3].

Recently, the system of atom-cavity-fiber [4–6] has been attracted much attentions due to the application in long distance and large-scale quantum information processing, such as distributed quantum computation [7, 8], quantum entanglement preparation [9–14], and the quantum communication [15]. For these unitary-dynamics-based schemes, atomic spontaneous emission, cavity decay and the fiber loss are three main dissipative factors which would affect the practical efficiency of these schemes. Traditionally, dissipation is considered as a detrimental effect in quantum information processing, however, recent theories and experiments show an interesting fact that the dissipation can be used as resources for quantum computation and entanglement generation [16–47]. In contrast with the unitary-dynamics-based schemes, here the dissipation plays a positive role in the preparation process so that the entanglement is robust against decoherence. Furthermore, since the entangled state appears as a steady-state, it is unnecessary to introduce an additional unitary feedback mechanism to stabilize it. Another merit of this approach is that we do not require specifying the initial state and controlling the evolution time accurately. Particularly, In 1999, Plenio et al. and Cabrillo et al. proposed schemes to prepare entanglement via dissipation [16, 17]. Immediately afterward, several schemes were suggested to study the entanglement in dissipative quantum system. In 2011, Kastoryano et al. proposed a dissipative scheme for preparing a maximally entangled state of two -type atoms trapped in one optical cavity [18], whose results are better than that based on the unitary dynamics. Subsequently, Shen et al. generalized this scheme to the coupled cavity system [32, 34] and atom-cavity-fiber system [33]. In 2013, Reiter et al. [45] demonstrated that two transmon qubits can be driven into the steady Bell state by combining the effective two-photon process induced by microwave driving with the photon loss. In 2014, Zheng et al. proposed two schemes to prepare the maximal entanglement between two atoms coupled to a decaying resonator [35,47]. All of the previous schemes need to use four or more classical fields to drive two qubits simultaneously, and relative amplitudes and phases of the driving fields applied to different qubits should be accurately set. Moveover, the experimental demonstrations of dissipative preparation of entanglement has also been reported in ion traps [44, 49], superconducting circuits [52], and the collective spin degrees of freedom of large atomic ensembles [48].

In this paper, we propose a feasible scheme to prepare and stabilize a maximally entangled Bell state in separate optical cavities coupled through an optical fiber, where only one qubit is needed to be driven by two classical fields with well-chosen frequencies. With currently achievable experiment parameters, the numerical simulation demonstrates that the distributed steady-state entanglement can be obtained with high fidelity, purity and CHSH correction. Compared with previous schemes [47], the present one does not require simultaneous driving to both qubits, which is a basic requirement to perform state transfer and quantum gate operation between separate nodes of a quantum network. Different from the Ref. [35], the present one generalize the idea that two atoms are trapped in a single cavity to the case when two atoms trapped into two separate cavities connected through an optical fiber. It is significant for quantum networks since only one unilateral driving on the qubit in one node is required with classical fields.

The organization of the rest paper is as follows: In Sec. 2, we present the details of our model and show the master equation describing the dynamics of the open dissipative system in Lindblad form. In Sec. 3, we first expatiate the system’s dressed state subspace. Then, a steady Bell state is generated by using the classical laser fields to drive one qubit within the dressed space. Furthermore, we assess the performance of this scheme through numerically calculating the fidelity, purity and Clauser-Horne-Shimony-Holt (CHSH) correlation, respectively. In Sec. 4, we construct a quantum teleportation scheme with multiple nodes and calculate the variation of the fidelity of teleportation with the increasing of the node number n. Finally, we present the conclusions inferred from this paper in Sec. 5.

2. Theoretical model

We consider a atom-cavity-fiber coupling system consisting of two distant cavities connected by a single-transverse-mode optical fiber, as shown in Fig. 1. Each atom has ground state |g〉 and excited state |e〉 with the corresponding energies 0 and ω₀, respectively. The atomic transition |g〉 → |e〉 is coupled resonantly to the cavity mode with the coupling constant g, and the first atom driven by two off-resonance optical lasers with corresponding detuning Δ_k (k = 1, 2). In the short fiber limit Lν/(2πc) ≤ 1, where L is the length of the fiber and ν is the decay rate of the cavity field into a continuum of the fiber modes, only one fiber mode essentially interacts with the cavity modes. For simplicity, we assume the interaction between cavity mode and fiber mode is resonant. Thus, in a rotating frame, the Hamiltonian of the system could be written as (setting ħ = 1 throughout this paper) H = H₀ + H_c,f + H_a,c + H_cl,

 

Fig. 1 Experimental setup and level diagram of the atoms. Two atoms resonantly interact with quantized field, respectively. The first atom is driven by two classical fields. γ, κ, and β denote the atomic spontaneous emission rate, cavity decay rate and fiber loss rate, respectively.

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Dissipation, which can occur via the fiber loss, atomic spontaneous emission and cavity decay, is a requisite component in the current scheme. The states in two-excitation subspace would be transformed to the corresponding states in one-excitation and zero excitation subspace via dissipation. The dynamics of the open dissipative system in Lindblad form could be described by the master equation

3. Preparation of the distributed entanglement

3.1. Dressed states

The spectroscopy of the system is well described by the dressed states, i.e., the eigenstates of the Hamiltonian H₀ + H_c,f + H_a,c, as shown in Fig. 2. Here, we use them to see clearly the roles of H_cl after choosing appropriate laser driving and detuning. We define the excitation number operator of the total system $N_e=\sum _{i=1,2}\left({|e〉}_{ii}〈e|+a_i^{†}{a}_i\right)+b^{†}b$. In Table 1, the eigenstates and corresponding eigenvalues of zero and single excitation subspaces are shown with the notation

Under the weak excitation condition and if the initial state is in the zero excitation subspace, we can safely discard the subspace with excitation number greater than or equals two, and $|{\mathrm{\Phi }}_1^0〉$ is the maximal entanglement we want to prepare for the two distributed atoms.

 

Fig. 2 Schematic diagram for the dressed state of the atom-cavity-fiber coupling system.

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Table 1. The eigenstates and eigenenergies of the Hamiltonian H₀ + H_c,f + H_a,c within the zero and single excitation subspaces.

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Table 2. The eigenstates and eigenenergies of the Hamiltonian H₀ + H_c,f + H_a,c within the two-excitation subspace.

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3.2. Roles of the classical laser field

Under the dressed state picture, The Hamiltonian H_cl can be rewritten as

3.3. Preparation process

The processes for producing and stabilizing the Bell state are shown in Fig. 3, we here require the atomic spontaneous emission to be much slower than other dynamical processes. Besides, the generation of the steady-state is independent of the initial states. If the system is initially in the ground state |Φ₀〉, it will be driven by the classical field Ω₁ to one-excitation dressed state $|{\mathrm{\Phi }}_1^{-}〉$ and then to $|{\mathrm{\Phi }}_2^0〉$ and $|{\mathrm{\Phi }}_2^2〉$ by classical field Ω₂. The photon loss results in the decaying channel $|{\mathrm{\Phi }}_2^0〉\to |{\mathrm{\Phi }}_1^0〉=|{\varphi }_{+}〉|0,0,0〉$. The state $|{\mathrm{\Phi }}_1^0〉$ is decoupled from the qubit-resonator coupling and cavity decay and is unaffected by the drives due to off-resonant, so it is a steady-state. On the other hand, the state $|{\mathrm{\Phi }}_2^2〉$ decays to the one-excitation dressed state $|{\mathrm{\Phi }}_1^{\pm }〉$, repumped by the classical field Ω₁ to $|{\mathrm{\Phi }}_2^0〉$ and $|{\mathrm{\Phi }}_2^2〉$. With the coherent driving and dissipation processes continuing, the population of dressed state $|{\mathrm{\Phi }}_2^2〉$ will decline gradually until all of qubit population is driven to the Bell state $|{\mathrm{\Phi }}_1^0〉$. If the system is initially in |e₁, g₂〉 |0, 0, 0〉 or |g₁, e₂〉 |0, 0, 0〉, which can be regarded as a superposition of the one-excitation dressed states $|{\mathrm{\Phi }}_1^0〉$ and $|{\mathrm{\Phi }}_1^{\pm }〉$. As has been shown, the populations of $|{\mathrm{\Phi }}_1^{\pm }〉$ are finally transferred to the steady-state due to coherent driving and dissipation process. For initial state |e₁, e₂〉 |0, 0, 0〉, the strong qubit-resonator coupling results in the transfer of population to the states |ϕ₋〉 |1, 0, 0〉 and |g₁, g₂〉 |2, 0, 0〉. Due to dissipation, these two state continuously decay to |ϕ₋〉 |0, 0, 0〉 and |g₁, g₂〉 |1, 0, 0〉, respectively, the specific superposition of the dressed states $|{\mathrm{\Phi }}_1^{+}〉$ and $|{\mathrm{\Phi }}_1^{-}〉$, which will be driven to the steady-state finally.

 

Fig. 3 Processes for producing and stabilizing Bell state. The interaction between system and environment is characterized by the photon loss and atomic spontaneous emission with the rates κ and γ, respectively.

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3.4. Performance of the scheme

In this part, we present numerical simulation of the full dynamics of the whole system. The most common way to assess the quality of the steady-state is fidelity which is defined as F(t) =Tr[(|ϕ₊〉 〈ϕ₊| ⊗ I_c) ρ̂_t→∞] with |ϕ₊〉 being the desired state and ρ̂_t→∞ being the practical steady-state density matrix.

The premise of locality and realism implies some constraints on the statistics of two spatially separated particles, which is known as Bell inequalities [50]. On that basis, CHSH correlation [51]. The system state violates Bell inequality when the CHSH correlation rises above 2, and quantum mechanisms predicts CHSH correlation S(t) equals $2\sqrt{2}$ for the maximal violation limit. The CHSH correlation S(t) is defined as

We also introduce purity to characterize mixture degree of the target steady-state entanglement which is defined through reduced density operators as

To verify the feasibility of the scheme, we set |Φ₀〉 = |g₁, g₁〉 |0, 0, 0〉 as the initial state, and solve the master equation numerically in zero, one and two excitation subspace. In our scheme, the decay rate of cavity field modes need to be far larger than the spontaneous emission rate of qubits, i.e. γ ≪ κ, the coupling to the cavity modes should be much stronger than the coupling to the driving field, i.e. Ω₁, Ω₂ ≪ g, and the decay rate of cavity field modes and fiber mode should be comparable with weak driving strengths, i.e. κ, β ≃ Ω₁, Ω₂. With these assumptions, we can efficiently generate and protect the Bell state |ϕ₊〉. We choose a set of feasible experimental parameters in a recent circuit QED experiment [52]: χ/2π ≃ 6 MHz, κ/2π ≃ 1.7 MHz, T₁ ≃ 9 μs, where T₁ is the qubit energy relaxation time, and χ = g²/Δ, with Δ being the qubit-resonator detuning. It is reasonable to set Δ = 10g in this dispersive qubit-resonator interaction system, yielding g/2π ≃ 60MHz, κ₁ = κ₂ ≃ 2.8 × 10⁻²g, β = 1.5 × 10⁻²g and γ ≃ 2.72 × 10⁻⁴g. The optimized Rabi frequencies are taken as Ω₁ = 0.080g and Ω₂ = 0.035g. These parameters are feasible in experiment for a number of quantum optical and solid state systems, for instance: ion traps systems [44,49], superconducting circuit QED systems [52], and plasmonic systems [53–55]. Especially, for the future implementation of quantum information with cold atoms, since the qubits need to be confined within the waist of the cavity mode in cavity QED model, and this can be realized with cold atoms trapped in a far detuned optical dipole trap [56, 57] or optical lattice [58–60], which can be mapped into a cavity QED model as our scheme utilized.

With these parameters, in Fig. 4(a) we plot the evolutions of the experimental fidelity versus the steady-state $|{\mathrm{\Phi }}_1^0〉$ and initial state |Φ₀〉. The result shows that the desired state could be prepared with fidelity more than 90% when the evolution time equals 1500/g. In Fig. 4(b), we optimized the dissipative factors and Rabi frequencies of the drivings by taking γ = 1.75 × 10⁻⁵g, κ₁ = 1.6×10⁻²g, κ₂ = 2.8×10⁻²g, β = 0.67×10⁻²g, Ω₁ = 0.025g, Ω₂ = 0.045g, with these optimized parameters, the optimal fidelity for steady Bell state is about 97.24% when the evolution time equals 3500/g. It is shown that the evolution time needed to arrive at the steady-state increases with the decreasing of Ω₁ and Ω₂ and the corresponding fidelity decreases with the increasing of γ. This can be explained as follows, when Ω₁ and Ω₂ are decreased, the unitary dynamics becomes slower and the time to reach stabilized state increases. On the other hand, the transition from the Bell state to the ground state is strongly suppressed as γ decreases, thus the fidelity increases.

 

Fig. 4 The fidelity of states |Φ₀〉 and $|{\mathrm{\Phi }}_1^0〉$ versus the dimensionless parameter gt for the initial state |Φ₀〉 by solving the full master equation. In (a) the experimental parameters are chosen as Ω₁ = 0.080g, Ω₂ = 0.035g, ν = 20g. And the dissipative factors are chosen as γ/g = 2.72 × 10⁻⁴, κ₁ = κ₂ = 2.8 × 10⁻²g, β = 1.5 × 10⁻²g. The inset of (b) is plotted with the optimized parameters: Ω₁ = 0.025g, Ω₂ = 0.045g and ν = 20g. And the dissipative factors: γ = 1.75 × 10⁻⁵g, κ₁ = 1.6 × 10⁻²g, κ₂ = 2.4 × 10⁻²g and β = 0.67 × 10⁻²g.

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With the experimental parameters and optimized parameters mentioned above, we plot the CHSH correlation and the purity of the target steady-state as a function of the evolution time, as shown in Fig. 5. The Numerical simulation in Fig. 5(a) shows that the system is reasonably stabilized to the target steady-state, with the experimental CHSH correlation about 2.532 and the optimal CHSH correlation about 2.731, which are both clearly exceeding the maximum value of 2 allowed by the local hidden variable theories. One can see from Fig. 5(b) that the evolution curve of experimental purity and optimal purity exhibit a valleys in the regime 0< t <500/g. This is due to the fact that the coherent driving is dominant in the early stage of evolution, it leads to the system to be in a mixture of a variety of quantum states. With the increasing of evolution time, the competition between the coherent driving and dissipation reaches a balance, and the target steady-state can be stabilized with the experimental purity about 83% and the optimal purity about 94%.

 

Fig. 5 The CHSH correlation and the purity of the qubit steady-state as a function of the time with the experimental parameters(the bule solid curve) and optimized parameters(the red solid curve), which are chosen as the same as Fig. 4.

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Besides, to verify the robustness of the scheme for the driving amplitudes, we plot the fidelity and purity of the target steady-state as a function of the Rabi frequencies Ω₁ and Ω₂ in Fig. 6(a) and Fig. 6(b), respectively. The result shows that both the fidelity and purity are higher than 90% and 80% within a wide range of Rabi frequencies, demonstrating the scheme is insensitive to deviations of these control parameters. It is worth noting that we have introduced the non-local bosonic modes in Eq. (5) to simplify the dynamics of the system under the condition of ν = 20g, by which we may safely eliminate the bosonic modes c₁ and c₂ in that they are largely dispersive with the two qubits. In Fig. 6(c), the fidelity of the desired state versus ν and evolution time shows that the robustness of the scheme against the variations of the coupling strength between the cavity mode and the fiber mode, in which the fidelity can reach 88.47% even when the parameter is taken as ν = 7.85g. Taking the parameters γ = 2.72 × 10⁻⁴g, ν = 20g, Ω₁ = 0.080g and Ω₂ = 0.035g, the fidelity of the steady-state as a function of cavity leakage rate κ and fiber loss rate β is shown as the Fig. 6(d). Numerical simulation shows that the fidelity is insensitive to variation of the fiber loss rate β. This is due to the fact that for the large detuning case the two qubits exchange energy only with one bosonic mode c which is independent of the fiber mode, hence the dissipation dynamics is dominated by cavity leakage rate κ, while the effect of fiber loss rate β can be approximately neglected for the dissipation dynamics of the whole system under the condition β ≪ g.

 

Fig. 6 The fidelity (a) and purity (b) of the target steady-state as a function of the parameters Ω₁/g and Ω₂/g with the initial state |Φ₀〉 at the time 1×10⁴/g. The parameters are chosen as ν = 20g. (c) The fidelity of the desired state versus ν and evolution time with the initial state |Φ₀〉. The parameters are chosen as Ω₁ = 0.080g, Ω₂ = 0.035g. Figures (a), (b) and (c) are plotted with the dissipative factors γ = 2.72×10⁻⁴g, κ₁ = κ₂ = 2.8×10⁻²g, β = 1.5 × 10⁻²g. (d) The fidelity of the target steady-state as a function of cavity leakage rate κ and fiber loss rate β with the initial state |Φ₀〉 at the time 1×10⁴/g. The parameters are chosen as ν = 20g, Ω₁ = 0.080g, Ω₂ = 0.035g, and qubit spontaneous emission rate γ = 2.72 × 10⁻⁴g.

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4. Quantum teleportation based on distributed steady state entanglement

In this section, as a practical application of our dissipative entanglement preparation scheme in quantum communication, we construct a quantum teleportation setup with multiple nodes as shown in Fig. 7. Suppose that each node is initially prepared in the distributed entangled steady state |ϕ₊〉, the fidelity and purity can be higher than 90% and 80% under the condition of experimental parameters, respectively. The unknown quantum state (referred to as a) to be teleported in Alice’s hands is |φ〉_a = α|g〉_a + β|e〉_a, where α and β are unknown parameters with |α|² + |β|² = 1. By using the standard teleportation procedure [61, 62], Bob could deterministically recover the unknown state only by some local operations (I₂, σ_x2, σ_z2, σ_x2σ_z2 on atom₂. In the following the atomic state at Bob’s side as an unknown quantum state which can be teleported from the first node to the nth node by performing same operation. We numerically calculate the fidelity of teleportation within the first node by the formula

 

Fig. 7 Schematic diagram for implementation of quantum teleportation scheme with multiple nodes. The information of unknown qubit can be teleported from the first node to the nth node. The dashed box denotes the first node to teleport an unknown quantum state from Alice to Bob. The dotted boxes means that two qubits belong to the same participant. The grey box in the bottom left is a quantum circuit of teleportation for the first node. Here H represents a Hadamard operation, σ_x, σ_z are the Pauli operators representing local qubit-flip operation, and I is the identity operator.

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Fig. 8 Fidelity of teleportation scheme with multiple nodes as a function of node number n.

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5. Conclusion

In conclusion, we have put forward an efficient scheme to prepare the distributed two-atom maximal entanglement Bell state in the atom-cavity-fiber system via the cavity decay. In our scheme only one qubit needs to be driven by classical control fields and during the preparation process, there is no requirement of the phase difference between these two classical field. This is meaningful for the long distance quantum information processing tasks, such as quantum teleportation and quantum dense coding, which can be greatly simplified for the experimental implementation since the entanglement distribution process can be saved and only one qubit is required to be operated by the either of parties, this guarantees the absolute security of long distance quantum information processing tasks and makes the scheme more robust than that based on the unitary dynamics. With the experimental parameters the fidelity, purity and CHSH correction of the distributed steady-state entanglement could reach up to 91%, 83% and 2.532, respectively. And the distributed entanglement is robust against the fluctuations of the Rabi frequencies of the classical field. As a practical application, quantum teleportation scheme with multiple nodes can be constructed, we also discussed the variation of the fidelity of teleportation with the increasing of node number n. Our quantum teleportation scheme can be realized effectively under the current experimental conditions when the number of nodes not more than 3. We hope that our work may be useful for the distributed quantum information processing tasks in the near future.