Adiabatic passage for three-dimensional entanglement generation through quantum Zeno dynamics

Yan Liang; Shi-Lei Su; Qi-Cheng Wu; Xin Ji; Shou Zhang

doi:10.1364/OE.23.005064

1. Introduction

Quantum entanglement, an interesting and attractive phenomenon in quantum mechanics, plays a significant role not only in testing quantum nonlocality, but also in a variety of quantum information tasks [1–8], such as quantum computing [9–11], teleportation [4], cryptography [1], precision measurements [12] and so on. Recently, high-dimensional entanglement is becoming more and more important since they are more secure than qubit systems, especially in the aspect of quantum key distribution. Besides, it has been demonstrated that violations of local realism by two entangled high-dimensional systems are stronger than that by two-dimensional systems [13]. So a lot of works have been done in generation of high-dimensional entanglement [14–21]. Li and Huang deterministically generated a three-dimensional entanglement via quantum Zeno dynamics(QZD) [14]. Chen et al. proposed a scheme to prepare three-dimensional entanglement state between a single atom and a Bose-Einstein condensate (BEC) via stimulated Raman adiabatic passage (STIRAP) technique [15]. Wu et al. proposed a scheme for generating a multiparticle three-dimensional entanglement by appropriately adiabatic evolutions [16]. Shao et al. and Su et al. proposed respectively the dissipative schemes for generating three-dimensional entanglement [17–19]. In experiment, two schemes have been put forward to generate high-dimensional entanglement by the means of the spatial modes of the electromagnetic field carrying orbital angular momentum [20, 21].

In order to realize the entanglement generation or population transfer in a quantum system with time-dependent interacting field, many schemes have been put forward. Such as π pulses, composite pulses, rapid adiabatic passage(RAP), stimulated Raman adiabatic passage, and their variants [22–24]. STIRAP is widely used in time-dependent interacting field because of the robustness for variations in the experimental parameters. But it usually requires a relatively long interaction time, so that the decoherence would destroy the intended dynamics, and finally lead to an error result. Therefore, reducing the time of dynamics towards the perfect final outcome is necessary and perhaps the most effective method to essentially fight against the dissipation which comes from noise or losses accumulated during the operational processes. Rencently, various schemes have been explored theoretically and experimentally to construct shortcuts for adiabatic passage [25–32].

On the other hand, the quantum Zeno effect is an interesting phenomenon in quantum mechanics. It stems from general features of the Schrödinger equation that yield quadratic behavior of the survival probability at short time [33, 34]. The quantum Zeno effect, which has been tested in many experiments, is the inhibition of transitions between quantum states by frequent measurements [35,36]. Recent studies [37–39] show that a quantum Zeno evolution will evolve away from its initial state, but it remains in the Zeno subspace defined by the measurements [34, 37] via frequently projecting onto a multidimensional subspace. This is known as QZD. Suppose that a dynamical evolution of a system can be governed by the Hamiltonian H_K = H_obs + KH_meas, where H_obs is the Hamiltonian of the investigated quantum system and the H_meas is regarded as an additional interaction Hamiltonian performing the measurement, while K is a coupling constant. In the limit K → ∞, the system is governed by the evolution operator U(t) = exp[−it ∑_n(Kλ_nP_n + P_nH_obsP_n)], which is an important basis for our following work, with P_n is the eigenprojections of H_meas with eigenvalues λ_n(H_meas = ∑_n λ_nP_n).

In this paper, we present an effective scheme to construct an adiabatic passage for three-dimensional entanglement generation between atoms motivated by the space division of QZD. The atoms are individually trapped in distant optical cavities connected by a fiber. Compared with previous works, our scheme has the following advantages: First, the two atoms three-dimensional entanglement can be achieved in one step, which will effectively reduce the complexity for implementing the scheme in experiment. Second, our scheme is based on the resonant interaction so the evolution time is very short. Third, the scheme is very robust against the photons leakage and atoms decay since the system only evolves in the null-excitation subspace. Fourth, the scheme can be expanded to generate N-atom three-dimensional entanglement and high-dimensional entanglement. This paper is structured as follows: In Sec. 2, we construct the fundamental model and give the effective dynamics to generate three-dimensional entanglement of two spatially separated atoms. In Sec. 3, we analyze the robustness of this scheme via numerical simulation. In Sec. 4, we generalize this proposal to generate N-atom three-dimensional entanglement. Besides, in Sec. 5, we expand our scheme to generate high-dimensional entanglement between two distant atoms. The conclusion appears in Sec. 6.

2. Model and effective dynamics generation of two atoms three-dimensional entanglement

The schematic setup for generating three-dimensional entanglement of two atoms is shown in Fig. 1. We consider a cavity-fibre-cavity system, in which two atoms are trapped in the corresponding optical cavities connected by a fiber. Under the short fiber limit (lv)/(2πc) ≪ 1, only the resonant mode of the fiber will interact with the cavity mode [40], where l is the length of the fiber and v is the decay rate of the cavity field into a continuum of fiber modes. The corresponding level structures of atoms are shown in Fig. 2. Atom A has two excited states |e_L〉,|e_R〉, and five ground states |1〉, |R〉, |L〉, |g〉 and |0〉, while atom B is a five-level system with three ground states |R〉, |L〉 and |g〉, two excited states |e〉_L and |e〉_R. For atom A, the transitions |0〉 ↔ |e_R〉 and |1〉 ↔ |e_L〉 are driven by classical fields with the same Rabi frequency Ω_A(t). And the transitions |R〉 ↔ |e_R〉 and |L〉 ↔ |e_L〉 are resonantly driven by the corresponding cavity mode a_Aj with j-circular polarization and the coupling strength is g_Aj (j = L, R). For atom B, the transitions |R〉 ↔ |e_R〉 and |L〉 ↔ |e_L〉 are driven by classical fields with the same Rabi frequency Ω_B(t), and the transitions |g〉 ↔ |e_R〉 and |g〉 ↔ |e_L〉 are resonantly driven by the corresponding cavity mode a_Bj with j-circular polarization and the coupling strength is g_Bj (j = L, R). The whole Hamiltonian in the interaction picture can be written as (h̄ = 1):

H_{total} = H_{a - l} + H_{a - c - f},

H_{a - l} = Ω_{A} (t) ({| e_{L} 〉}_{A} 〈 1 | + {| e_{R} 〉}_{A} 〈 0 |) + Ω_{B} (t) ({| e_{L} 〉}_{B} 〈 L | + {〈 e_{R} |}_{B} 〈 R |) + H . c .,

\begin{array}{l} H_{a - c - f} & = & g_{A L} a_{A L} {| e_{L} 〉}_{A} 〈 L | + g_{A R} a_{A R} {| e_{R} 〉}_{A} 〈 R | + g_{B L} a_{B L} {| e_{L} 〉}_{B} 〈 g | + g_{B R} a_{B R} {| e_{R} 〉}_{B} 〈 g | \\ + η b_{L} (a_{A L}^{†} + a_{B L}^{†}) + η b_{R} (a_{A R}^{†} | a_{B R}^{†}) + H . c ., \end{array}

where η is the coupling strength between cavity mode and the fiber mode, b_R(L) is the annihilation operator for the fiber mode with R(L)-circular polarization, a_A(B)R(L) is the annihilation operator for the corresponding cavity field with R(L)-circular polarization, and g_A(B)R(L) is the coupling strength between the corresponding cavity mode and the trapped atom.

Fig. 1 The schematic setup for generating two atoms three-dimensional entanglement. The two atoms are trapped in two spatially separated optical cavities connected by a fiber.

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Fig. 2 The level configurations of atom A and B.

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In order to obtain the following two atoms three-dimensional entanglement:

| Ψ 〉 = \frac{1}{\sqrt{3}} ({| R 〉}_{A} {| R 〉}_{B} + {| L 〉}_{A} {| L 〉}_{B} + {| g 〉}_{A} {| g 〉}_{B}),

we assume atom A in the state

| Ψ_{A} 〉 = \frac{1}{\sqrt{3}} ({| 1 〉}_{A} + {| 0 〉}_{A} + {| g 〉}_{A}),

while atom B in the state |g〉_B, both the cavity modes and the fiber mode in vacuum state |0〉_AC|0〉_BC|0〉_f, and then demonstrate that with the help of QZD the atom state |0〉_A|g〉_B can be adiabatically evolved to |R〉_A|R〉_B, and |1〉_A|g〉_B can be adiabatically evolved to |L〉_A|L〉_B. It is easy to know that |g〉_A|g〉_B will remain unchange since there is no excitation for all the field modes at the beginning. For the initial state |0〉_A|g〉_B|0〉_AC|0〉_BC|0〉_f, the whole system evolves in the subspace spanned by

\begin{array}{l} | ϕ_{1} 〉 & = & {| 0 〉}_{A} {| g 〉}_{B} {| 0 〉}_{AC} {| 0 〉}_{BC} {| 0 〉}_{f}, \\ | ϕ_{2} 〉 & = & {| e_{R} 〉}_{A} {| g 〉}_{B} {| 0 〉}_{AC} {| 0 〉}_{BC} {| 0 〉}_{f}, \\ {| ϕ 〉}_{3} & = & {| R 〉}_{A} {| g 〉}_{B} {| 1_{R} 〉}_{AC} {| 0 〉}_{BC} {| 0 〉}_{f}, \\ | ϕ_{4} 〉 & = & | R_{A} 〉 {| g 〉}_{B} {| 0 〉}_{AC} {| 0 〉}_{BC} {| 1_{R} 〉}_{f}, \\ | ϕ_{5} 〉 & = & {| R 〉}_{A} {| g 〉}_{B} {| 0 〉}_{AC} {| 1_{R} 〉}_{BC} {| 0 〉}_{f}, \\ | ϕ_{6} 〉 & = & {| R 〉}_{A} {| e_{R} 〉}_{B} {| 0 〉}_{AC} {| 0 〉}_{BC} {| 0 〉}_{f}, \\ | ϕ_{7} 〉 & = & {| R 〉}_{A} {| R 〉}_{B} {| 0 〉}_{AC} {| 0 〉}_{BC} {| 0 〉}_{f} . \end{array}

Setting Ω_A(t), Ω_B(t) ≪ η, g_AR(L), g_BR(L), then both the condition H_a-c-f ≫ H_a-l and the Zeno condition K → ∞ can be satisfied (H_a-l and H_a-c-f correspond respectively to H_obs and KH_meas in the third paragraph of Sec. 1). By performing the unitary transformation U = e^−iH_a-c-ft under condition H_a-c-f ≫ H_a-l, the Hilbert subspace can be divided into five invariant Zeno subspaces [38, 39]:

\begin{array}{l} Γ_{P 1} & = & {| ϕ_{1} 〉, | ϕ_{7} 〉, | ψ_{1} 〉}, \\ Γ_{P 2} & = & {| ψ_{2} 〉}, & Γ_{P 3} & = & {| ψ_{3} 〉}, \\ Γ_{P 4} & = & {| ψ_{4} 〉}, & Γ_{P 5} & = & {| ψ_{5} 〉}, \end{array}

with the eigenvalues λ₁ = 0, λ₂ = −g, λ₃ = g,

λ_{4} = - \sqrt{g^{2} + 2 η^{2}} = - ε

, and

λ_{5} = \sqrt{g^{2} + 2 η^{2}} = ε

, where we assume g_AR(L) = g_BR(L) = g for simplicity. Here

\begin{array}{l} | ψ_{1} 〉 & = & \frac{1}{ε} (η | ϕ_{2} 〉 - g | ϕ_{4} 〉 + η | ϕ_{6} 〉), \\ | ψ_{2} 〉 & = & \frac{1}{2} (- | ϕ_{2} 〉 + | ϕ_{3} 〉 - | ϕ_{5} 〉 + | ϕ_{6} 〉), \\ | ψ_{3} 〉 & = & \frac{1}{2} (- | ϕ_{2} 〉 - | ϕ_{3} 〉 + | ϕ_{5} 〉 + | ϕ_{6} 〉), \\ | ψ_{4} 〉 & = & \frac{1}{2 ε} (g | ϕ_{2} 〉 - ε | ϕ_{3} 〉 + 2 η | ϕ_{4} 〉) - ε | ϕ_{5} 〉 + g | ϕ_{6} 〉, \\ | ψ_{5} 〉 & = & \frac{1}{2 ε} (g | ϕ_{2} 〉 + ε | ϕ_{3} 〉 + 2 η | ϕ_{4} 〉) + ε | ϕ_{5} 〉 + g | ϕ_{6} 〉, \end{array}

and the corresponding projection

P_{i}^{α} = | α 〉 〈 α |, (| α 〉 \in Γ_{P i}) .

Under the above condition, the system Hamiltonian can be rewritten as the following form [39]:

\begin{array}{l} H_{total} & ≃ & \sum_{i, α, β} (λ_{i} P_{i}^{α} + P_{i}^{α} H_{a - l} P_{i}^{β}) \\ = & - g | ψ_{2} 〉 〈 ψ_{2} | + g | ψ_{3} 〉 〈 ψ_{3} | - ε | ψ_{4} 〉 〈 ψ_{4} | + ε | ψ_{5} 〉 〈 ψ_{5} | \\ + \frac{1}{ε} η (Ω_{A} (t) | ψ_{1} 〉 | ϕ_{1} 〉 + Ω_{B} (t) | ψ_{1} 〉 | ϕ_{7} 〉 + H . c .) . \end{array}

When we choose the initial state |ϕ₁〉 = |0〉_A|g〉_B|0〉_AC|0〉_BC|0〉_f, the Hamiltonian H_total reduces to

H_{eff} = Ω_{A 1} (t) | ψ_{1} 〉 | ϕ_{1} 〉 + Ω_{B 1} (t) | ψ_{1} 〉 | ϕ_{7} 〉 + H . c .,

where

Ω_{A 1} (t) = \frac{1}{ε} η Ω_{A} (t)

and

Ω_{B 1} (t) = \frac{1}{ε} η Ω_{B} (t)

. By using adiabatic passage method, we can obtain the dark state of H_eff:

| ψ_{D 1} 〉 = \frac{1}{\sqrt{Ω_{A 1} {(t)}^{2} + Ω_{B 1} {(t)}^{2}}} (- Ω_{B 1} (t) | ϕ_{1} 〉 + Ω_{A 1} (t) | ϕ_{7} 〉) .

When the pulses shape satisfy

lim_{t \to - \infty} \frac{Ω_{A 1} (t)}{Ω_{B 1} (t)} = 0, lim_{t \to + \infty} \frac{Ω_{A 1} (t)}{Ω_{B 1} (t)} = \infty,

thus, based on the effective Hamiltonian (11), our proposal for population transfer from |ϕ₁〉 to |ϕ₇〉 can be achieved.

On the other hand, if the initial state is |1〉_A|g〉_B|0〉_AC|0〉_BC|0〉_f, the whole system evolves in the subspace spanned by

\begin{array}{l} | ϕ_{1}^{'} 〉 & = & {| 1 〉}_{A} {| g 〉}_{B} {| 0 〉}_{AC} {| 0 〉}_{BC} {| 0 〉}_{f}, \\ | ϕ_{2}^{'} 〉 & = & {| e_{L} 〉}_{A} {| g 〉}_{B} {| 0 〉}_{AC} {| 0 〉}_{BC} {| 0 〉}_{f}, \\ | ϕ_{3}^{'} 〉 & = & {| L 〉}_{A} {| g 〉}_{B} {| 1_{L} 〉}_{AC} {| 0 〉}_{BC} {| 0 〉}_{f}, \\ | ϕ_{4}^{'} 〉 & = & {| L 〉}_{A} {| g 〉}_{B} {| 0 〉}_{AC} {| 0 〉}_{BC} {| 1_{L} 〉}_{f}, \\ | ϕ_{5}^{'} 〉 & = & {| L 〉}_{A} {| g 〉}_{B} {| 0 〉}_{AC} {| 1_{L} 〉}_{BC} {| 0 〉}_{f}, \\ | ϕ_{6}^{'} 〉 & = & {| L 〉}_{A} {| e_{L} 〉}_{B} {| 0 〉}_{AC} {| 0 〉}_{BC} {| 0 〉}_{f}, \\ | ϕ_{7}^{'} 〉 & = & {| L 〉}_{A} {| L 〉}_{B} {| 0 〉}_{AC} {| 0 〉}_{BC} {| 0 〉}_{f} . \end{array}

In this situation, with the method mentioned above, we can easily obtain the effective Hamiltonian:

H_{eff}^{'} = Ω_{A 1} (t) | ψ_{1}^{'} 〉 〈 ϕ_{1}^{'} | + Ω_{B 1} (t) | ψ_{1}^{'} 〉 〈 ϕ_{7}^{'} | + H . c .,

where

| ψ_{1}^{'} 〉 = \frac{1}{ε} (η | ϕ_{2}^{'} 〉 - g | ϕ_{4}^{'} 〉 + η | ϕ_{6}^{'} 〉)

,

Ω_{A 1} (t) = \frac{1}{ε} η Ω_{A} (t)

and

Ω_{B 1} (t) = \frac{1}{ε} η Ω_{B} (t)

. By using adiabatic passage method, we can obtain the dark state of H′_eff:

| ψ_{D 2} 〉 = \frac{1}{\sqrt{Ω_{A 1} {(t)}^{2} + Ω_{B 1} {(t)}^{2}}} (- Ω_{B 1} (t) | ϕ_{1}^{'} 〉 + Ω_{A 1} (t) | ϕ_{7}^{'} 〉) .

When the pulses shape satisfy Eq. (13), the initial state |ϕ′₁〉 involves to |ϕ′₇〉 eventually.

If the initial state is |g〉_A|g〉_B|0〉_AC|0〉_BC|0〉_f, it will not change at all during the whole evolution.

Therefore, the initial state

| Ψ (0) 〉 = \frac{1}{\sqrt{3}} ({| 0 〉}_{A} + {| 1 〉}_{A} + {| g 〉}_{A}) {| g 〉}_{B} {| 0 〉}_{AC} {| 0 〉}_{BC} {| 0 〉}_{f}

of the compound system will evolve to the state

| Ψ_{final} 〉 = \frac{1}{\sqrt{3}} ({| R 〉}_{A} {| R 〉}_{B} + {| L 〉}_{A} {| L 〉}_{B} + {| g 〉}_{A} {| g 〉}_{B}) {| 0 〉}_{AC} {| 0 〉}_{BC} {| 0 〉}_{f},

which is a product state of the two atoms three-dimensional entanglement, the cavity modes vacuum state, and the fiber mode vacuum state.

3. Numerical analysis and the robustness of the scheme

In order to generate two atoms three-dimensional entanglement, the conditions of Eq. (13) should be satisfied in our scheme. For this reason, we can choose the pulses shape of the laser fields Ω_A(t) and Ω_B(t) in the original Hamiltonian H_total as:

Ω_{A} (t) = Ω_{0} {sin}^{4} [π (t - τ) / 31 t_{0}]

and

Ω_{B} (t) = Ω_{0} {sin}^{4} (π t / 31 t_{0}) .

Here Ω₀ is the pulse amplitude, τ being the time delay. Ω_A(t) and Ω_B(t) are two delayed but partially overlapping pulses, so the condition in Eq. (13) can be satisfied. Figure 3 shows the Rabi frequencies Ω_A(t) and Ω_B(t) versus Ω₀t with

t_{0} = Ω_{0}^{- 1}

and τ = 5.27t₀. The population curves of |ϕ₁〉 and |ϕ₇〉 versus Ω₀t are depicted in Fig. 4, where the blue lines correspond to the effective Hamiltonian of Eq. (11) and the red lines correspond to the initial Hamiltonian of Eq. (1). Here we choose g = 20Ω₀, η = 100g [46],

t_{0} = Ω_{0}^{- 1}

, τ= 5.27t₀, and Ω_A(t), Ω_B(t) are defined by Eqs. (19) and (20), respectively. With these parameters, the adiabatic condition and the Zeno condition can be satisfied well. From Fig. 4 we can see that the numerical results of the effective Hamiltonian Eq. (11) and those of the initial Hamiltonian Eq. (1) agree with each other reasonably well, and the population inverts completely when Ω₀t is over 25 no matter under the initial Hamiltonian or under the effective Hamiltonian. Therefore, our effective Hamiltonian is valid. Through the above processes, we can generate two atoms three-dimensional entanglement successfully.

Fig. 3 The time dependence of the laser fields Ω_A(t) corresponding to solid line and Ω_B(t) corresponding to dashed line with $t_{0} = Ω_{0}^{- 1}$ and τ = 5.27t₀.

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Fig. 4 Time evolutions of the populations of corresponding system states, controlled by the effective Hamiltonian of Eq. (11) (blue line) and the initial Hamiltonian of Eq. (1) (red line). The subgraph is the partially enlarged view.

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It is well known that whether a scheme is applicable for quantum information processing and quantum computing depends on the robustness against possible mechanisms of decoherence. To examine the robustness of our scheme described in the previous sections, we consider the effect of photon leakage and atom spontaneous decay. The corresponding master equation for the whole system density matrix ρ(t) has the following form:

\begin{array}{l} ρ (\dot{t}) & = & - i [H_{total}, ρ (t)] - \sum_{j = L, R} \frac{κ_{f}}{2} [b_{j}^{+} b_{j} ρ (t) - 2 b_{j} ρ (t) b_{j}^{+} + ρ (t) b_{j}^{+} b_{j}] \\ - \sum_{j = L, R} \sum_{i = A, B} \frac{κ_{j}}{2} [a_{i j}^{+} a_{i j} ρ (t) - 2 a_{i j} ρ (t) a_{i j}^{+} + ρ (t) a_{i j}^{+} a_{i j}] \\ - \frac{γ_{A}}{2} {\sum_{h = L, 1} [σ_{e_{L}, e_{L}}^{A} ρ (t) - 2 σ_{h, e_{L}}^{A} ρ (t) σ_{e_{L}, h}^{A} + ρ (t) σ_{e_{L}, e_{L}}^{A}] \\ + \sum_{k = R, 0} [σ_{e_{R}, e_{R}}^{A} ρ (t) - 2 σ_{k, e_{R}}^{A} ρ (t) σ_{e_{R}, k}^{A} + ρ (t) σ_{e_{R}, e_{R}}^{A}]} \\ - \sum_{j = L, R} \sum_{l = j, g} \frac{γ_{B}}{2} [σ_{e_{j}, e_{j}}^{B} ρ (t) - 2 σ_{l, e_{j}}^{B} ρ (t) σ_{e_{j}, l}^{B} + ρ (t) σ_{e_{j}}^{B} σ_{e_{j}}^{B}], \end{array}

where H_total is given by Eq. (1). κ_f and κ_R(L) are the photon leakage rates of the fiber mode and cavity mode R(L), respectively. γ_A(B) is the atom A(B) spontaneous emission rate from the excited state |e_R〉(|e_L〉) to the ground state |R〉(|L〉) and |g〉, respectively. σ_m,n = |m〉〈n|(m, n = 0, 1, L, R, g, e_L, e_R). For simplicity, we assume κ_f = κ_R = κ_L = κ/2, γ_A = γ_B = γ/4 and the initial condition ρ(0) = |Ψ₀〈〉Ψ₀|. In Fig. 5, the fidelity of the final two atoms three-dimensional entanglement is plotted versus the dimensionless parameters κ/g and γ/g by numerically solving the master Eq. (21). From Fig. 5 we can see that the fidelity of two atoms three-dimensional entanglement is higher than 96.5% even in the range of κ and γ close to g.

Fig. 5 The fidelity corresponding to the target state versus κ/g and γ/g, with g = 20Ω₀, η = 100g, $t_{0} = Ω_{0}^{- 1}$ and τ= 5.27t₀.

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4. Generation of N-atom three-dimensional entanglement

In this section, we will show how to deterministically generate the N-atom three-dimensional entanglement. The schematic setup is shown in Fig. 6, where N atoms are trapped in N spatially separated cavities connected by optical fibers. S₂, S₃,..., S_N are the optical switch devices which are used to control the interaction between the n-th cavity and the first cavity [16, 41, 42]. The atom in cavity 1 possesses the level structure as atom A presented in Fig. 2, and the atoms in the other (n = 2, 3,...,N) cavities are the same as atom B given in Fig. 2. By only turning on the n-th switch S_n (n = 2,3,..., N), and the others off, the interaction between the 1st atom and the n-th atom can be achieved. In this case, the interaction Hamiltonian has the following form (h̄ = 1):

H_{total}^{n} = H_{a - l}^{n} + H_{a - c - f}^{n},

H_{a - l}^{n} = Ω_{A} (t) ({| e_{L} 〉}_{1} 〈 1 | + {| e_{R} 〉}_{1} 〈 0 |) + Ω_{B} (t) ({| e_{L} 〉}_{n} 〈 L | + {| e_{R} 〉}_{n} 〈 R |) + H . c .,

\begin{array}{l} H_{a - c - f}^{n} & = & g_{1 L} a_{1 L} {| e_{L} 〉}_{1} 〈 L | + g_{1 R} a_{1 R} {| e_{R} 〉}_{1} 〈 R | + g_{n L} a_{n L} {| e_{L} 〉}_{n} 〈 g | \\ + g_{n R} a_{n R} {| e_{R} 〉}_{n} 〈 g | + η b_{L} (a_{1 L}^{†} + a_{n L}^{†}) + η b_{R} (a_{1 R}^{†} + a_{n R}^{†}) + H . c ., \end{array}

where n represents the atom trapped in the n-th cavity, a_nR(L) is annihilation operator corresponding to the n-th cavity with R(L)-circular polarization, and g_nR(L) is coupling strength between the corresponding cavity mode and the trapped atom.

Fig. 6 Schematic setup for generation of N-atom three-dimensional entanglement.

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In order to obtain N-atom three-dimensional entanglement

{| φ 〉}_{N} = \frac{1}{\sqrt{3}} ({| R 〉}_{1} {| R 〉}_{2} \dots {| R 〉}_{N} + {| L 〉}_{1} {| L 〉}_{2} \dots {| L 〉}_{N} + {| g 〉}_{1} {| g 〉}_{2} \dots {| g 〉}_{N}),

we assume g_nR = g_nL = g for simplicity, and the initial state of the compound system is:

{| φ 〉}_{1} = \frac{1}{\sqrt{3}} ({| 0 〉}_{1} + {| 1 〉}_{1} + {| g 〉}_{1}) {| g 〉}_{2} {| g 〉}_{3} \dots {| g 〉}_{N} {| 0 〉}_{1 C} {| 0 〉}_{2 C} \dots {| 0 〉}_{N C} {| 0 〉}_{f} .

(i) First, turn on S₂ and keep other switches off, then the first two atoms can be prepared in the state $| Ψ 〉 = \frac{1}{\sqrt{3}} ({| R 〉}_{1} | R_{2} 〉 + {| L 〉}_{1} {| L 〉}_{2} + {| g 〉}_{1} {| g 〉}_{2})$ with the method mentioned in Sec. 2.
(ii) Then, turn off S₂, and apply another pulse on atom 1 to drive the transitions |L〉₁ → |1〉₁ and |R〉₁ → |0〉₁, which leads the compound system to the state: ${| ϕ 〉}_{2} = \frac{1}{\sqrt{3}} ({| 0 〉}_{1} {| R 〉}_{2} + {| 1 〉}_{1} {| L 〉}_{2} + {| g 〉}_{1} {| g 〉}_{2}) {| g 〉}_{3} {| g 〉}_{4} \dots {| g 〉}_{N} {| 0 〉}_{1 C} {| 0 〉}_{2 C} \dots {| 0 〉}_{N C} {| 0 〉}_{f} .$
(iii) Subsequently, only S_n (n = 3,..., N) is turned on, thus the 1st atom can interact with the n-th atom. Next, perform the same operations as the first step successively to make the state in Eq. (27) evolve to $\begin{array}{l} {| φ 〉}_{3} & = & \frac{1}{\sqrt{3}} ({| R 〉}_{1} {| R 〉}_{2} \dots {| R 〉}_{n} + {| L 〉}_{1} {| L 〉}_{2} \dots {| L 〉}_{n} + {| g 〉}_{1} {| g 〉}_{2} \dots {| g 〉}_{n}) {| g 〉}_{n + 1} {| g 〉}_{n + 2} \\ \dots {| g 〉}_{N} {| 0 〉}_{1 C} {| 0 〉}_{2 C} \dots {| 0 〉}_{N C} {| 0 〉}_{f} . \end{array}$
(iv) Finally we can obtain the N-atom three-dimensional entanglement after repeating the steps (ii) and (iii) (N − 2) times. Thus, the state of the compound system finally evolves to $\begin{array}{l} {| φ 〉}_{4} & = & \frac{1}{\sqrt{3}} ({| R 〉}_{1} {| R 〉}_{2} \dots {| R 〉}_{N} + {| L 〉}_{1} {| L 〉}_{2} \dots {| L 〉}_{N} \\ + {| g 〉}_{1} {| g 〉}_{2} \dots {| g 〉}_{N}) {| 0 〉}_{1 C} {| 0 〉}_{2 C} \dots {| 0 〉}_{N C} {| 0 〉}_{f} . \end{array}$ That is a product state of N-atom three-dimensional entanglement, the cavity modes state, and the fiber modes state.

5. Generation of high-dimensional entanglement of two spatially separated atoms

We note that the scheme can also be expanded to generate high-dimensional entanglement of two spatially separated atoms. The potential atomic configurations is plotted in Fig. 7. We assume that the cavity supports 2N independent modes of photon fields. Then the Hamiltonian can be written as (h̄ = 1):

ℋ_{total} = ℋ_{a - l} + ℋ_{a - c - f},

ℋ_{a - l} = Ω_{A} (t) ({| e_{L} 〉}_{A} 〈 1 | + {| e_{R} 〉}_{A} 〈 0 |) + Ω_{B} (t) \sum_{i = 1}^{N} ({| e_{Li} 〉}_{B} 〈 L_{i} | + {| e_{Ri} 〉}_{B} 〈 R_{i} |) + H . c .,

\begin{array}{l} H_{a - c - f} & = & \sum_{i = 1}^{N} [(g_{ALi} a_{ALi} {| e_{L} 〉}_{A} 〈 L_{i} | + g_{ARi} a_{ARi} {| e_{R} 〉}_{A} 〈 R_{i} |) + g_{BLi} a_{BLi} {| e_{Li} 〉}_{B} 〈 g | \\ + g_{BRi} a_{BRi} {| e_{Ri} 〉}_{B} 〈 g | + η b_{Li} (a_{ALi}^{†} + a_{BLi}^{†}) + η b_{Ri} (a_{ARi}^{†} + a_{BRi}^{†}) + H . c .] . \end{array}

Assume the initial state of the whole system is |1〉_A|g〉_B|0₁0₂....0_N〉_AC|0₁0₂....0_N〉_BC|0₁0₂....0_N〉_f, then the system will evolve in the subsystem spanned by the vectors {|ζ₁〉, |ζ₂〉....|ζ_5N+2〉}:

\begin{array}{l} Γ = {| ζ_{1} 〉 & = & {| 1 〉}_{A} {| g 〉}_{B} {| 0_{1} 0_{2} \dots 0_{N} 〉}_{AC} {| 0_{1} 0_{2} \dots 0_{N} 〉}_{BC} {| 0_{1} 0_{2} \dots 0_{N} 〉}_{f}, \\ | ζ_{2} 〉 & = & {| e_{L} 〉}_{A} {| g 〉}_{B} {| 0_{1} 0_{2} \dots 0_{N} 〉}_{AC} {| 0_{1} 0_{2} \dots 0_{N} 〉}_{BC} {| 0_{1} 0_{2} \dots 0_{N} 〉}_{f}, \\ | ζ_{2 + i} 〉 & = & {| L_{i} 〉}_{A} {| g 〉}_{B} {| \dots 1 i \dots 〉}_{AC} {| 0_{1} 0_{2} \dots 0_{N} 〉}_{BC} {| 0_{1} 0_{2} \dots 0_{N} 〉}_{f}, \\ | ζ_{2 + N + i} 〉 & = & {| L_{i} 〉}_{A} {| g 〉}_{B} {| 0_{1} 0_{2} \dots 0_{N} 〉}_{AC} {| 0_{1} 0_{2} \dots 0_{N} 〉}_{BC} {| \dots 1 i \dots 〉}_{f}, \\ | ζ_{2 + 2 N + i} 〉 & = & {| L_{i} 〉}_{A} {| g 〉}_{B} {| 0_{1} 0_{2} \dots 0_{N} 〉}_{AC} {| \dots 1 i \dots 〉}_{BC} {| 0_{1} 0_{2} \dots 0_{N} 〉}_{f}, \\ | ζ_{2 + 3 N + i} 〉 & = & {| L_{i} 〉}_{A} {| e_{Li} 〉}_{B} {| 0_{1} 0_{2} \dots 0_{N} 〉}_{AC} {| 0_{1} 0_{2} \dots 0_{N} 〉}_{BC} {| 0_{1} 0_{2} \dots 0_{N} 〉}_{f}, \\ | ζ_{2 + 4 N + i} 〉 & = & {| L_{i} 〉}_{A} {| L_{i} 〉}_{B} {| 0_{1} 0_{2} \dots 0_{N} 〉}_{AC} {| 0_{1} 0_{2} \dots 0_{N} 〉}_{BC} {| 0_{1} 0_{2} \dots 0_{N} 〉}_{f} .}, \end{array}

where i = 1, 2, 3,...,N. With the prior procedures presented in Sec. 2, we can obtain the effective Hamiltonian:

ℋ_{eff} = \frac{1}{\sqrt{N + 1}} (Ω_{A 1} (t) | X_{1} 〉 〈 ζ_{1} | + Ω_{B 1} (t) \sum_{j = 4 N + 3}^{5 N + 2} | X_{1} 〉 〈 ζ_{j} | + H . c .),

with

Ω_{A 1} (t) = \frac{1}{ε} η Ω_{A} (t)

and

Ω_{B 1} (t) = \frac{1}{ε} η Ω_{B} (t)

, where

| X_{1} 〉 = \frac{1}{ε \sqrt{2 N + 1}} (η | ζ_{2} 〉 - g \sum_{j = N + 3}^{2 N + 2} | ζ_{j} 〉 + η \sum_{k = 3 N + 3}^{4 N + 2} | ζ_{k} 〉) .

With the help of adiabatic passage method, we can obtain the dark state of ℋ_eff:

| X_{D} 〉 = \frac{1}{\sqrt{(N + 1) (Ω_{A 1} {(t)}^{2} + Ω_{B 1} {(t)}^{2})}} (- Ω_{B 1} (t) | ζ_{1} 〉 + Ω_{A 1} (t) \sum_{j = 4 N + 3}^{5 N + 2} | ζ_{j} 〉) .

When Eq. (13) is satisfied, the initial state |1〉_A|g〉_B|0₁0₂....0_N〉_AC|0₁0₂....0_N〉_BC |0₁0₂....0_N〉_f of the whole system will finally evolve to

\begin{array}{l} | ν 〉 & = & \frac{1}{\sqrt{N}} \sum_{j = 4 N + 3}^{5 N + 2} | ζ_{j} 〉 \\ = & \frac{1}{\sqrt{N}} \sum_{m = 1}^{N} {| L_{m} 〉}_{A} {| L_{m} 〉}_{B} {| 0_{1} 0_{2} \dots 0_{N} 〉}_{AC} {| 0_{1} 0_{2} \dots 0_{N} 〉}_{BC} {| 0_{1} 0_{2} \dots 0_{N} 〉}_{f}, \end{array}

which is a N-dimensional maximally entanglement.

Fig. 7 The potential atomic levels configuration for atom A and B.

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For the initial state |0〉_A|g〉_B|0₁0₂....0_N〉_AC |0₁0₂....0_N〉_BC|0₁0₂....0_N〉_f, by using the similar way from Eqs. (33)–(37), this initial state finally evolves to

| ν_{1} 〉 = \frac{1}{\sqrt{N}} \sum_{k = 1}^{N} {| R_{k} 〉}_{A} {| R_{k} 〉}_{B} {| 0_{1} 0_{2} \dots 0_{N} 〉}_{AC} {| 0_{1} 0_{2} \dots 0_{N} 〉}_{BC} {| 0_{1} 0_{2} \dots 0_{N} 〉}_{f} .

On the other hand, the initial state |g〉_A|g〉_B|0₁0₂....0_N〉_AC|0₁0₂....0_N〉_BC|0₁0₂....0_N〉_f don’t participate in the evolution.

Therefore, if we choose the initial state of the combined system as

{| ν 〉}_{0} = \frac{1}{\sqrt{3}} ({| 1 〉}_{A} + {| 0 〉}_{A} + {| g 〉}_{A}) {| g 〉}_{B} {| 0_{1} 0_{2} \dots 0_{N} 〉}_{AC} {| 0_{1} 0_{2} \dots 0_{N} 〉}_{BC} {| 0_{1} 0_{2} \dots 0_{N} 〉}_{f},

after implementing all the operations mentioned above, the high-dimensional entanglement can be obtained as the following form:

\begin{array}{l} | β 〉 & = & \frac{1}{\sqrt{2 N + 1}} [\sum_{m = 1}^{N} {| L_{m} 〉}_{A} {| L_{m} 〉}_{B} + \sum_{k = 1}^{N} {| R_{k} 〉}_{A} {| R_{k} 〉}_{B} \\ + {| g 〉}_{A} {| g 〉}_{B}] {| 0_{1} 0_{2} \dots 0_{N} 〉}_{AC} {| 0_{1} 0_{2} \dots 0_{N} 〉}_{BC} {| 0_{1} 0_{2} \dots 0_{N} 〉}_{f} . \end{array}

6. Analysis and discussion

We now analyze the feasibility of the experiment for this scheme. The appropriate atomic level configuration can be obtained from the hyperfine structure of cold alkali-metal atoms [43–45]. In this paper we adopt the ¹³³Cs. 5S_1/2 ground level |F = 3, m = 2〉(|F = 3, m = −2〉) corresponds to |R〉(|L〉) and |F = 2, m = 1〉(|F = 2, m = −1〉) corresponds to |0〉(|1〉), respectively, while 5P_3/2 excited level |F = 3, m = 1〉(|F = 3, m = −1〉) corresponds to |e_R〉(|e_L〉). Other hyperfine levels in the ground-state manifold can be used as |g〉 for atom A. For atom B, the states |R〉, |L〉 and |g〉 correspond to |F = 2, m = −1〉, |F = 2, m = 1〉 and |F = 3, m = 0〉 of 5S_1/2 ground levels, respectively. And |e_R〉(|e_L〉) corresponds to |F = 3, m = −1〉(|F = 3, m = 1〉) of 5P_3/2 excited level. In experiments, the cavity QED parameters g = 2.5GHz, κ = 10MHz, and γ = 10MHz have been realized in [47, 48]. For such parameters, the fidelity of our scheme is larger than 99.0%, so our scheme is robust against both the cavity decay, the fiber loss and the atomic spontaneous radiation and may be very promising within current experiment technology.

In summary, we have proposed a promising scheme to generate three-dimensional entanglement with the help of QZD in the cavity-fiber-cavity system. Because the atoms are resonant interaction, so the speed of producing entanglement is very fast compared with the dispersive protocols [49, 50]. Meanwhile, the influence of various decoherence processes such as spontaneous emission and photon loss on the generation of entanglement is also investigated. Because during the whole process the system keeps in a Zeno subspace without exciting the cavity field and the fiber, and all the atoms are in the ground states, thus the scheme is robust against the cavity, fiber and atomic decay. Numerical results show the generation of entanglement can be achieved with a high fidelity. Besides, the scheme can be generalized to generate N-atom three-dimensional entanglement and high-dimensional entanglement. We hope our work will have a crucial role in promoting the development of quantum information science.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant Nos. 11464046 and 61465013.

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Adiabatic passage for three-dimensional entanglement generation through quantum Zeno dynamics

Abstract

1. Introduction

2. Model and effective dynamics generation of two atoms three-dimensional entanglement

3. Numerical analysis and the robustness of the scheme

4. Generation of N-atom three-dimensional entanglement

5. Generation of high-dimensional entanglement of two spatially separated atoms

6. Analysis and discussion

Acknowledgments

References and links

Cited By

Figures (7)

Equations (40)

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