1. Introduction

Quantum entanglement, as a fundamental aspect in quantum mechanics, has occupied a central place in modern research because of its promise of enormous utility in quantum computing [1–3], cryptography [2,4], etc. Generally speaking, the more particles that can be entangled, the more clearly nonclassical effects are exhibited and the more useful the states are for quantum applications. Typically, tripartite GHZ state, first proposed by Greenberger, Horne, and Zeilinger, provides a possibility to test quantum mechanics against local hidden theory without inequality [5] and has practical applications in e.g., quantum secrete sharing [6].

Recently, many theoretical schemes [7–10] have been proposed to generate multipartite GHZ states, and a series of experimental preparations [11–13] have already been realized. The physical systems utilized to these generations include superconducting circuits [10, 13], trapped ions [12], and cavity QED systems, etc.. It is well-known that cavity QED, where atoms interact with quantized electromagnetic fields inside a cavity, is a useful platform to demonstrate fundamental quantum mechanics laws and for the implementation of quantum information processing [14]. Specifically, numerous proposals with cavity QED have been made for entangling atoms either inside a cavity [15, 16] or trapped individually in different cavities [17–21]. For example, Pellizzari [22] proposed an approach to realize the reliable transfer of quantum information between two atoms in distant cavities connected by an optical fiber. Based on this proposal, various schemes [23–32] have been proposed to realize the quantum manipulations of the atoms trapped in different cavities connected via optical fibers. However, these models either require strong cavity-fiber coupling [28] (which is not easy to achieve in the usual experiment), or need many laser beams to implement the desirable manipulations [29] (this increases the experimental complication), or is sensitive to the decays of cavity fields and fiber modes [30] (which limits its scalability).

Here, by using quantum Zeno dynamics we propose an alternative approach to entangle the atoms trapped in the distant cavities connected by optical fibers. It is well-known that quantum Zeno effect is an interesting phenomenon in quantum mechanics. Due to this effect the quantum system remains in its initial state via frequently measurements. Facchi et al [33–35] showed that this effect does not necessarily freeze the dynamics. Instead, by frequently projecting onto a multidimensional subspace, the system could evolve away from its initial state, although it remains in the so-called “Zeno subspace” [33,36]. Moreover, without making use of projection operators and nonunitary dynamics, the quantum Zeno effect can also be expressed in terms of a continuous coupling between the system and detector [35]. Generally, the system and its continuously coupling detector can be governed by the total Hamiltonian H_K = H + KH_a, with H is for the quantum system investigated and the H_a describing the additional interaction with the detector, and K the coupling constant. In the limit K → ∞, the subsystem of interest is dominated by the evolution operator U(t) = lim_K→∞ exp(iKH_at)U_K(t) which takes the form U(t) = exp(−it ∑_n P_nHP_n) [35]. Here, P_n is the eigenprojection of H_a = ∑_n λ_nP_n corresponding to the eigenvalue λ_n. As a consequence, the system-detector can be described by the evolution operator U_K(t) ∼ exp(−iKH_at)U(t) = exp[−i∑_n(Kλ_nP_n + P_nHP_n)t]. This result is of great importance in view of practical applications of the quantum Zeno dynamics, such as to prepare various quantum states [37, 38] and to implement the quantum gates [39–42].

Compared with previous protocols [28–30] for generating GHZ states by selective absorption and emission of photons, adiabatic passages, and dispersive interactions between the atoms and cavities, etc., our approach possesses the following advantages: (i) the expected entanglement can be established by only one step operation, (ii) it is robust with respect to parameter imprecision and atomic and fibers’ dissipations, (iii) under the Zeno condition, the cavity fields are not excited really and thus is insensitive to the decays of the cavities, (iv) the generalization to N-atom entanglement is direct, and no matter how many atoms are involved, two laser beams are enough to implement the generations.

The paper is organized as follows. In section 2, we present our generic approach by using quantum Zeno dynamics to implement the GHZ entanglement of the atoms trapped in three fiber-connected cavities. The validity of the used quantum Zeno dynamics is analyzed by numerical method in detail. The direct generalization for entangling the N atoms trapped individually in N cavity is provided in Sec. 3. Finally, in Sec. 4 we discuss the feasibility of our proposal and give our conclusions.

2. Generation of GHZ state of atoms trapped in different cavities by quantum Zeno dynamics

We consider the physical configuration shown in the Fig. 1, three Λ-type atoms are trapped in three distant optical cavities coupled by two short optical fibers. Each atom has one excited state |e〉 and two dipole-transition forbidden ground states |0〉, |1〉. The first and third atomic transitions |1〉 ↔ |e〉 are driven resonantly by classical lasers with the couplings coefficient Ω₁ and Ω₃, respectively. The other atomic transition is resonantly coupled to the corresponding cavity mode with coupling constant g_i,r(l) (i = 1, 2, 3). The subscript r(l) denotes the right (left) circularly polarization. In the short fiber limit, (2Lv̄)/(2πc) ≪ 1, where L is the length of the fiber and v̄ is the decay rate of the cavity fields into a continuum of fiber modes [23]. In the interaction picture, the Hamiltonian of the whole system can be written as

 

Fig. 1 Zeno manipulations of atoms in spatially-separated cavities connected via optical fibers. Here, Ω₁ (Ω₃) is the classical field coupled to the first (third) atom, and b₁ and b₂ are the bosonic operators in fibers and couple to the corresponding cavity modes.

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Under the Zeno condition g,v ≫ Ω₁, Ω₃, the above Hilbert subspace is split into nine invariant Zeno subspaces [35]

It is clear that the validity of our scheme mainly relies on if the so-called Zeno condition g,v ≫ Ω₁, Ω₃ is satisfied robustly. Now, we discuss how the ratio Ω₃/g influences the fidelity F = |〈Ψ(τ)|φ(τ)〉|², with |φ(τ)〉 being the relevant final state evolved by the original H_total defined in Eq. (1). On the other hand, the ratio v/g is another important factor affecting the fidelity [23–25, 28]. Indeed, from Fig. 2 we see that the smaller the ratio Ω₃/g corresponds to the higher fidelity. We also note that, even at the relatively-low ratio v/g = 0.5, the fidelity is still high, i.e., above 97%. This is very important for the practical application of our scheme, as the large cavity-fiber coupling is not easy to be satisfied in the realistic experiments. Furthermore, in order to obtain high fidelity in moderate time, we can choose typically the ratios: Ω₃/g = 0.04 and v/g = 1.

 

Fig. 2 The influence of the ratios: Ω₃/g and v/g, on the fidelity of the prepared GHZ state.

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Our proposal is also robust for the imprecision of experimental parameters. Indeed, our previous discussions are based on certain ideal assumptions, e.g., g_i,r(l) ≡ g, v_i ≡ v. Practically, the coupling strengths g_i,r(l) depend on the positions of the atoms in the cavities, and thus certain deviations are unavoidable. To numerically consider these deviations, we typically set g_2,r(l) = g, g_1,r = g + δg₁ and g_3,l = g + δg₃ for simulations. In Fig. 3(a), the fidelity of the prepared state versus the variation δg₁ and δg₃ is plotted. It is seen that, even under a deviation |δg_1,3| = 10%g, the fidelity is still sufficiently high, e.g., larger than 94%. Similarly, we plot the fidelity versus the deviation of the cavity-fiber coupling (v) in Fig. 3(b), and find also that the great robustness against the errors in the selected parameters.

 

Fig. 3 The fidelity of the three-atom GHZ state versus various parameter errors: (a) δg₁ and δg₃; (b) δv₁ and δv₂.

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We now numerically verify that the above effective dynamics restricted in the subspace without exciting the cavity modes is also robust. In fact, considering all possible states of the system, the state at the time t reads |φ_total〉 = ∑_i c_i(t)|ϕ_i〉 within the subspace spanned by the basic state vectors in Eq. (4). The occupation probability for each state vector |ϕ_i〉 during the evolution is P_i(t) = |c_i(t)|², and satisfies the condition ∑_i P_i(t) = 1. By solving the Schrödinger equation, the variation of the occupation probability P_c(t) (≡ ∑_i=3,5,7,9 P_i(t)) is portrayed in Fig. 4 (red line). As shown in Fig. 4, the occupation probability of the cavity photonic state is less than 0.04. Therefore, our claim that the scheme is immune to the cavity decay seems justifiable. Meanwhile, Fig. 4 also describes the occupation probabilities P_f (t)(≡ ∑_i=4,8 P_i(t), green line) and P_e(t)(≡ ∑_i=2,6,10 P_i(t), black line) for those states, wherein the photon is in the fiber and one of the atoms in its excited state, respectively. Since P_e > P_f in Fig. 4, our protocol is more sensitive to atomic spontaneous emission than the fiber loss. To check this, let us investigate the evolution of the system governed by the following non-Hermitian Hamiltonian

 

Fig. 4 The population probabilities of cavity photonic state P_c, fiber photonic state P_f and the atomic excited state P_e versus the dimensionless parameter gt by exactly solving the evolution equation of the system without any approximation.

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Fig. 5 The influences of atomic spontaneous emission γ/g, cavity field decay κ_c/g and fiber photonic leakage κ_f/g on the fidelity of the triatomic GHZ state for the typical ratios: Ω₃/g = 0.04 and v/g = 1.

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3. Generalization to N-atom entanglement

We now generalize the present scheme to generate N-atom GHZ states. Let us consider the configuration shown in Fig. 6, where N atoms are individually trapped in N cavities connected by N − 1 short fibers. The level configuration of the atoms between two ends are chosen the same as that of the middle atom in the above 3-atom case. The Hamiltonian of the present system reads

 

Fig. 6 N atom trapped in distant cavities.

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Fig. 7 The occupation probabilities of the state |ϕ′₁〉 versus the dimensionless parameter gt under the total Hamiltonian (solid-line) and effective Hamiltonian (dotted-line).

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4. Discussions and conclusions

The experimental feasibility of the proposed scheme is briefly analyzed as follows. Practically, the atomic configuration involved in our scheme can be implemented with the ⁸⁷Rb atom, whose relevant atomic levels are shown in Fig. 8. Where the first (third) atom is coupled resonantly to an external π-polarized classical field and a π₊ (π₋) polarized photon modes of the cavity, and the middle atom is coupled resonantly to a π₊ and a π₋ polarized modes. Based on the above discussions, in order to obtain fidelity larger than 90%, one should keep the decay rate γ < 0.0055g. This condition can be satisfied in recent experiments [44, 45], wherein for an optical cavity with the wavelength about 850 nm the parameters are set as: g/2π = 750 MHz, γ/2π = 2.62 MHz, κ_c/2π = 3.5 MHz. Also, a near perfect fiber-cavity coupling with an efficiency larger than 99.9% can be realized using fiber-taper coupling to high-Q silica microspheres [46]. The fiber loss at the 852 nm wavelength is about 2.2 dB/km [47], which corresponds to the fiber decay rate 1.52 × 10⁵ Hz. With these parameters, it seems that the present entanglement-generation scheme with a high fidelity larger than 93% could be feasible with the present experimental technique. The duration of the Zeno pulses Ω₁ and Ω₃ utilized to generate the desirable three-atom GHZ state can be estimated as $\pi /N_1\sqrt{{\mathrm{\Omega }}_1^2+{\mathrm{\Omega }}_3^2}\approx 1.43\times {10}^{-8}\text{s}$, which is easily implemented for the present laser technology.

 

Fig. 8 Experimental atomic configuration used to generate tripartite GHZ state. Here, Ω₁ and Ω₃ are the classical fields, and π₊ and π₋ are the quantized cavity modes with different polarizations, respectively.

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In conclusion, we have proposed an approach to achieve three-atom GHZ states by taking advantage of quantum Zeno dynamics. In present proposal, only one step operation is required to complete the generation. The generation of the GHZ states is controllable, as the proposed quantum Zeno dynamics can be switched on/off by controlling the classical drivings of the atoms at two ends. We note also that, the generated GHZ state is no longer evolving, at least theoretically, after switching off the classical drivings of the atoms at two-side cavities. Additionally, the evolution of system is always kept in the subspace with null-excitation of cavity fields. So, the scheme is immune to the decay of cavity modes. Moreover, the scheme loosens the requirement of strong cavity-fiber coupling. We also show that the proposal can be easily generalized to prepared the N-atom entanglement without increasing the number of the applied classical fields.