Generation of multipartite continuous-variable entanglement via atomic spin wave: Heisenberg-Langevin approach

Xihua Yang; Jie Shang; Bolin Xue; Yuanyuan Zhou; Min Xiao

doi:10.1364/OE.22.012563

1. Introduction

Much attention has been paid to the generation of multipartite continuous-variable (CV) entanglement in recent years due to its wide applications, such as quantum computation, quantum communication, quantum cryptography, and quantum networks [1–3]. Apart from the routinely-used way of combining the entangled twin beams produced through spontaneous parametric down-conversion in nonlinear crystals to create entanglement among multiple beams by employing linear optical elements, i.e., polarizing beam splitters [4,5], an alternative promising avenue is the use of nonlinear processes [e.g., nondegenerate four-wave mixing (FWM) or Raman scattering] in an atomic ensemble to produce multi-entangled beams. These generated beams have the virtues of narrow bandwidth, long correlation time, and nondegenerate frequencies, which are quite necessary for connecting different physical systems at the nodes of quantum networks and quite suitable for quantum memory required in quantum communication [3]. Based on the seminal paper by Duan et al. [6], nondegenerate quantum correlated and entangled narrow-band photon pairs have been experimentally achieved with both hot atoms [7–9] and magneto-optical trap (MOT) [10,11] by using the electromagnetically induced transparency [12–15] (EIT)-based double-Λ-type system. Recently, Kolchin [16] and Glorieux et al. [17] have theoretically studied EIT-based paired photon generation with the consideration of Langevin noise terms by using the Heisenberg-Langevin method. Through multiorder coherent Raman scattering processes, the generation of multipartite entanglement in a far-off-resonance medium with a prepared coherence has been investigated [18]. Very recently, we have proposed a convenient and flexible way to create multicolor multipartite CV entanglement to any desired number and quantum entangler via an atomic spin wave pre-established by strong coupling and probe fields in the Λ-type EIT configuration [19, 20]. However, in a realistic atomic ensemble, there will be various dephasing processes, such as finite interaction time between atoms and light, and collision-induced atomic coherence decay, etc, which would degrade the degree of entanglement and have not been taken into account in [19, 20].

In this paper, based on the experimental observation of generating multi-field correlations and anti-correlations via atomic spin coherence in ⁸⁵Rb atomic system [21] and theoretical examination of generating multipartite CV entanglement and quantum entangler via atomic spin wave [19, 20], we discuss in detail the effects of the atomic density, Rabi frequencies of the scattering fields, and coherence decay rate of the lower doublet on the generation of nondegenerate multipartite CV entanglement via atomic spin wave in the Λ-type EIT configuration in an atomic ensemble. By using the Heisenberg-Langevin formalism, we demonstrate that increasing the atomic density and Rabi frequencies of the scattering fields, and decreasing the coherence decay rate of the lower doublet, would strengthen the degree of multipartite entanglement. This method provides a proof-of-principle demonstration of conveniently generating multicolor multipartite entangled narrow-band fields to any desired number via atomic spin wave in an atomic ensemble with high optical depth, which may have promising applications in quantum information processing and quantum networks.

2. Heisenberg-langevin equations

We consider a collection of atoms having the Λ-type configuration shown in Fig. 1(a), similar to the one studied in [19–21], where the relevant energy levels and the applied and generated laser fields form a quintuple-Λ-type system. Levels $| 1 〉$ , $| 2 〉$ , and $| 3 〉$ correspond, respectively, to the ground-state hyperfine levels 5S_1/2 (F = 3), 5S_1/2 (F = 2), and the excited state 5P_1/2 in D₁ line of ⁸⁵Rb atom with the ground-state hyperfine splitting of 3.036 GHz. We assume the strong probe field E_p (with frequency $ω_{p}$ and Rabi frequency $Ω_{p}$ ) is tuned to resonance with the transition $| 2 〉$ - $| 3 〉$ , whereas the strong coupling field E_c (with frequency $ω_{c}$ and Rabi frequency $Ω_{c}$ ) is tuned close to the resonant transition $| 1 〉$ - $| 3 〉$ with a small finite detuning Δ = ω_c-ω₃₁ with ω_ij (i ≠ j) being the atomic resonant frequency between levels i and j; thus, the two-photon detuning of the coupling and probe fields from the Raman transition $| 2 〉$ - $| 3 〉$ - $| 1 〉$ is also equal to Δ. A third mixing field E_m with frequency $ω_{m}$ , which can be generated from the coupling or probe field with an acousto-optic modulator, off-resonantly couples levels $| 2 〉$ (or $| 1 〉$ ) and $| 3 〉$ with the detuning Δ₃₍₄₎ = ω_m-ω₃₂₍₃₁₎. As discussed in [19–24], at high atomic density and high laser powers, two Stokes fields E₁ & E₃ (with frequencies $ω_{1}$ & $ω_{3}$ ) and two anti-Stokes fields E₂ & E₄ (with frequencies $ω_{2}$ & $ω_{4}$ ) can be simultaneously created through nondegenerate FWM processes with the coupling, probe, and mixing fields acting on both $| 1 〉$ - $| 3 〉$ and $| 2 〉$ - $| 3 〉$ transitions. Equivalently, the above simultaneously generated four FWM fields can be regarded as scattering the coupling, probe, and mixing fields off the atomic spin wave (S) pre-built by the strong coupling and probe fields in the Λ-type EIT configuration formed by levels $| 1 〉$ , $| 2 〉$ , and $| 3 〉$ , where the induced spin wave acts as a frequency converter with frequency equal to the separation between the lower doublet [19, 25]. The equivalent configuration is shown in Fig. 1(b), which can be readily generalized to multi-Λ-type system by applying more laser fields tuned to the vicinity of the transitions $| 1 〉$ - $| 3 〉$ and/or $| 2 〉$ - $| 3 〉$ to mix with the induced atomic spin wave.

Fig. 1 (a) The quintuple-Λ-type system of the D₁ transitions in ⁸⁵Rb atom coupled by the coupling (E_c), probe (E_p), and mixing (E_m) fields based on the configuration in Refs [19, 21], where E_p, E_c, and E_m fields all drive both $| 1 〉 - | 3 〉$ and $| 2 〉 - | 3 〉$ transitions, and the corresponding Stokes fields (E₁ and E₃), and anti-Stokes fields (E₂ and E₄) are generated through four FWM processes. (b) The equivalent configuration of (a) with the two lower states driven by the atomic spin wave S induced by the strong E_c and E_p fields in the Λ-type EIT configuration.

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With the Heisenberg-Langevin method, we first investigate the generation of quantum anti-correlated and entangled Stokes field E₁ and anti-Stokes field E₂ via atomic spin wave in a triple-Λ-type system by blocking the mixing field E_m. We use the equivalent configuration shown in Fig. 1(b) to treat the generated bipartite entanglement. The detunings of the probe field E_p (with the Rabi frequency $Ω_{2}$ ) and coupling field E_c (with the Rabi frequency $Ω_{1}$ ) from the resonant transitions $| 1 〉$ - $| 3 〉$ and $| 2 〉$ - $| 3 〉$ are denoted as Δ₂ = ω_p-ω₃₁ and Δ₁ = ω_c-ω₃₂, respectively. In the present scheme, we assume that the generated Stokes and anti-Stokes fields are very weak as compared to the scattering fields, thus, the scattering fields can be treated classically, whereas the Stokes field E₁ and anti-Stokes field E₂ can be treated quantum mechanically, and we assume that the Rabi frequencies of the scattering fields $Ω_{1}$ and $Ω_{2}$ are far smaller than their frequency detunings Δ₁ and Δ₂, so that the coupling between different scattering fields can be neglected. Also, we assume that the coupling and probe fields for creating atomic spin wave are substantially strong, so that the atomic spin wave is strong enough to ensure that different scattering fields have negligible influence on it. Under this condition, the Heisenberg-Langevin equations for describing the evolution of the collective atomic operators $σ_{12} (z, t)$ , $σ_{13} (z, t)$ , and $σ_{23} (z, t)$ can be written as [16,17,26]

{\dot{σ}}_{12} (z, t) = - (γ_{0} + i Δ) σ_{12} - i g_{1} a_{1} (z, t) σ_{23}^{+} + i g_{2} a_{2}^{+} (z, t) σ_{13} + i Ω_{1} σ_{13} - i Ω_{2} σ_{23}^{+} + F_{12} (z, t),

{\dot{σ}}_{13} (z, t) = - [(γ_{1} + γ_{2}) / 2 - i (Δ_{1} - Δ)] σ_{13} + i g_{1} a_{1} (z, t) (σ_{11} - σ_{33}) + i Ω_{1} σ_{12} + F_{13} (z, t),

{\dot{σ}}_{23} (z, t) = - [(γ_{1} + γ_{2}) / 2 - i (Δ_{2} + Δ)] σ_{23} + i g_{2} a_{2} (z, t) (σ_{22} - σ_{33}) + i Ω_{2} σ_{12}^{+} + F_{23} (z, t),

where

γ_{1}

and

γ_{2}

are the population decay rates from level 3 to levels 1 and 2 and

γ_{0}

the coherence decay rate between levels 1 and 2,

a_{1}

and

a_{2}

are the quantum operators of the generated Stokes E₁ and anti-Stokes E₂ fields,

g_{1 (2)} = μ_{13 (23)} \cdot ε_{1 (2)} / ℏ

is the atom-field coupling constant with

μ_{13 (23)}

as the dipole moment for the 1-3 (2-3) transition and $ε_{1 (2)} = \sqrt{ℏ ω_{1 (2)} / 2 ε_{0} V}$ as the electric field of a single Stokes (anti-Stokes) photon with V as the interaction volume with length L and beam radius r, and

F_{i j} (z, t)

are the collective atomic

δ

-correlated Langevin noise operators. In the similar analysis described in [16, 17, 26], under the assumption that the uniformly-distributed pencil-shaped atomic sample is optically thin in the transverse direction and the scattering fields propagate without depletion, the evolution of the annihilation

a_{1}

and creation

a_{2}^{+}

operators for the Stokes and anti-Stokes fields can be described by the coupled propagation equations

(\frac{\partial}{\partial t} + c \frac{\partial}{\partial z}) a_{1} (z, t) = i g_{1} N σ_{13},

(\frac{\partial}{\partial t} + c \frac{\partial}{\partial z}) a_{2}^{+} (z, t) = - i g_{2} N σ_{23}^{+},

where N is the total number of atoms in the quantum interaction volume. As done in [16, 17], we use the perturbation analysis to treat the interaction of the atoms with the fields. In the zeroth-order perturbation expansion, by semiclassically treating the interaction of the atoms with the strong coupling and probe fields in the Λ-type EIT configuration, we get the steady-state mean values of the atomic operators

σ_{11}^{(0)}

,

σ_{22}^{(0)}

,

σ_{33}^{(0)}

,

σ_{12}^{(0)}

,

σ_{13}^{(0)}

and

σ_{23}^{(0)}

. By Fourier transforming Heisenberg-Langevin equations for the atomic operators

σ_{12} (z, t)

,

σ_{13} (z, t)

, and

σ_{23} (z, t)

and substituting the zeroth-order solution into equation for the operator

σ_{12} (z, t)

, we can get the first-order solution

σ_{12}^{(1)} (z, ω)

. Then substituting

σ_{12}^{(1)} (z, ω)

and the zeroth-order solution for the other operators into the equations for

σ_{13} (z, t)

and

σ_{23} (z, t)

, the first-order solution

σ_{13}^{(1)} (z, ω)

, and

σ_{23}^{(1)} (z, ω)

can be obtained, which are expressed as

\begin{array}{l} σ_{13} = \frac{1}{γ_{13} + i (ω - Δ_{1} + Δ)} {[i g_{1} (σ_{11}^{(0)} - σ_{33}^{(0)}) + \frac{g_{1} Ω_{1} σ_{23}^{+ (0)}}{i (ω + Δ) + γ_{0}}] a_{1} - \frac{g_{2} Ω_{1} σ_{13}^{(0)}}{i (ω + Δ) + γ_{0}} a_{2}^{+} \\ + \frac{i Ω_{1} (i Ω_{1}^{} σ_{13}^{(0)} - i Ω_{2} σ_{23}^{+ (0)})}{i (ω + Δ) + γ_{0}} + \frac{i Ω_{1} F_{12}}{i (ω + Δ) + γ_{0}} + F_{13}}, \end{array}

\begin{array}{l} σ_{23}^{+} = \frac{1}{γ_{23} + i (ω + Δ_{2} + Δ)} {- \frac{g_{1} Ω_{2}^{} σ_{23}^{+ (0)}}{i (ω + Δ) + γ_{0}} a_{1} - [i g_{2} (σ_{22}^{(0)} - σ_{33}^{(0)}) - \frac{g_{2} Ω_{2}^{*} σ_{13}^{(0)}}{i (ω + Δ) + γ_{0}}] a_{2}^{+} \\ - \frac{i Ω_{2}^{} (i Ω_{1}^{} σ_{13}^{(0)} - i Ω_{2} σ_{23}^{+ (0)})}{i (ω + Δ) + γ_{0}} - \frac{i Ω_{2}^{} F_{12}}{i (ω + Δ) + γ_{0}} + F_{23}^{+}} . \end{array}

Note that in the above first-order analysis, as done in [16, 17], we only consider the steady-state mean values of the zeroth-order atomic operators (neglecting the noise operators) for the first-order solution. Substituting

σ_{13}^{(1)}

and

σ_{23}^{(1)}

into Fourier- transformed coupled propagation equations, the output of the annihilation operator

a_{1} (L, ω)

and creation operator

a_{2}^{+} (L, ω)

for the Stokes and anti-Stokes fields with respect to the Fourier frequency

ω

can be obtained, which is a linear combination of the input of the operators

a_{1} (0, ω)

and

a_{2}^{+} (0, ω)

and Langevin noise terms. In a similar way as that in [19], we use the criterion

V = {(Δ u)}^{2} + {(Δ v)}^{2} < 4

proposed in [27] to verify the entanglement of the generated Stokes and anti-Stokes fields, where

u = x_{1} + x_{2}

and

v = p_{1} - p_{2}

with

x_{j} =(a_{j} + a_{j}^{+})

and

p_{j} = - i (a_{j} - a_{j}^{+})

. In what follows, the relevant parameters are scaled with m and MHz, or m⁻¹ and MHz⁻¹, and set according to the experimental conditions in [20,21,28,29] as follows: atomic density

n_{0} {=10}^{16}

,

r =1 \times 10^{-4}

,

L =0 .02

,

γ_{1} = γ_{2} =3

,

γ_{0} =0 .1

,

ω_{12} =3036

,

Ω_{p} = Ω_{c} =600

,

Δ_{2} = - Δ_{1} - Δ =3036

,

Δ = - 150

, and

Δ_{3} =2900

.

3. Results and discussions

Figure 2 shows the evolution of V as a function of the Fourier frequency $ω$ with Δ = -ω (i.e., energy conservation is fulfilled). It can be seen that, in a moderate range (i.e., 0~840MHz) of the Fourier frequency $ω$ (that is, with the two-photon detuning Δ in the range of −840~0 MHz), V becomes less than 4, which sufficiently demonstrates the generation of genuine bipartite entanglement. The similar range of the negative two-photon detuning for optimizing the generation of two-field entanglement with FWM has been examined in [17,28,29]. In the following, we set Δ = -ω = −150 MHz, and investigate the effects of the atomic density, Rabi frequencies of the scattering fields, and coherence decay rate of the lower doublet on the generation of multipartite CV entanglement.

Fig. 2 The evolution of V as a function of the Fourier frequency $ω$ with $L =0 .02$ , $r =1 \times 10^{-4}$ , $n_{0} {=10}^{16}$ , $γ_{1} = γ_{2} =3$ , $γ_{0} =0 .1$ , $ω_{12} =3036$ , $Ω_{p} = Ω_{c} =600$ , $Ω_{1} = Ω_{2} =120$ , $Δ_{2} = - Δ_{1} - Δ =3036$ , and $Δ = - ω$ in corresponding units of m and MHz, or m⁻¹ and MHz⁻¹.

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The dependence of V on the atomic density is depicted in Fig. 3. It is obvious that, V, with the initial value of 4, decrease gradually with the increase of the atomic density, which indicates two entangled fields are produced. When n₀ is about larger than $4 \times 10^{15}$ /m³, V nearly becomes equal to zero; this means two perfectly squeezed fields can be achieved. Similar evolution of V with respect to the Rabi frequency of the scattering fields is observed, as displayed in Fig. 4, where when $Ω_{1}$ and $Ω_{2}$ is about larger than 40 MHz, V nearly reaches zero. Thus, in order to generate high degree of bipartite entanglement, the atomic density and Rabi frequency of the scattering fields should be high enough.

Fig. 3 The evolution of V as a function of the atomic density n₀ with Δ = -ω = −150 MHz, and the other parameters are the same as those in Fig. 2.

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Fig. 4 The evolution of V as a function of the Rabi frequency $Ω$ = $Ω_{1}$ = $Ω_{2}$ of the scattering field with Δ = -ω = −150 MHz, and the other parameters are the same as those in Fig. 2.

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As is well known, the coherence decay rate of the lower doublet plays a critical role in the generation of the bipartite entanglement in the Λ-type three-level system. This is clearly demonstrated in Fig. 5 by examining the dependence of V on the coherence decay rate $γ_{0}$ of the lower doublet. When the coherence decay rate is very small, V is almost equal to zero, and two nearly perfectly squeezed fields can be created. The increase of the coherence decay rate would increase the value of V, that is, the degree of bipartite entanglement would be weakened. When the coherence decay rate grows high enough, V becomes equal to 4 and no entanglement can be produced. In addition, in this Λ-type atomic system, the correlation time of the generated bipartite entanglement is determined by the coherence decay time between the two lower states, which, in practice, can be long ( $\sim m s$ or even $\sim s$ [30,31]), thereby having the virtue suitable for quantum memory required in quantum communication [6–9].

Fig. 5 The evolution of V as a function of the coherence decay rate $γ_{0}$ of the lower doublet with Δ = -ω = −150 MHz, and the other parameters are the same as those in Fig. 2.

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In [19], we did not take into account the case of the generated Stokes and anti-Stokes fields with equal coupling strength. In fact, the highest degree of bipartite entanglement is achieved with the two generated fields nearly having equal coupling strength. This is confirmed in Fig. 6 by varying the ratio the coupling coefficient g₂₃ to g₁₃. It is obvious that the larger or smaller g₂₃ with respect to g₁₃, the weaker the degree of bipartite entanglement, and when the two coupling coefficients g₂₃ and g₁₃ are nearly equal to each other, the minimum value of V (almost equal to zero) is obtained, i.e., the maximum bipartite entanglement is produced.

Fig. 6 The evolution of V as a function of the ratio of the coupling coefficients g₂₃/g₁₃ with Δ = -ω = −150 MHz, and the other parameters are the same as those in Fig. 2.

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As discussed in [19], the above idea for generating bipartite entanglement via atomic spin wave can be easily extended to create multipartite entanglement to any desired order when more scattering fields are applied. With the Heisenberg-Langevin formalism, we demonstrate this concept by realizing tripartite entanglement through scattering an additional mixing field E₃ off the atomic spin wave, as shown in Fig. 1(b). We consider the case of generating three fields E₁, E₂, and E₃. Also, we assume that the Rabi frequency $Ω_{3}$ of the mixing field is small as compared to its frequency detuning Δ₃ = ω_m-ω₃₁, so that the coupling between different scattering fields can be neglected. Under this condition, the evolution of the operator $σ_{12}^{}$ should include the terms from the interaction of the mixing field E_m and the correspondingly generated Stokes field E₃ with the atomic medium, whereas the evolution of the annihilation operator $a_{3}$ for the generated Stokes field E₃ can be described by the same coupled propagation equation as the annihilation operator $a_{1}$ for the Stokes field E₁. In a similar way, by Fourier transforming the Heisenberg-Langevin equations and coupled propagation equations, we can get the output of the operators $a_{1} (L, ω)$ , $a_{3} (L, ω)$ , and $a_{2}^{+} (L, ω)$ for the Stokes fields E₁ and E₃ and anti-Stokes field E₂ with respect to the Fourier frequency $ω$ after the interaction of the atoms and the fields. As done in [19], the tripartite entanglement of the generated fields E₁, E₂, and E₃ can be demonstrated according to the criterion proposed by van Lock-Furusawa (VLF) [32,33] with inequalities:

\begin{array}{l} V_{12} = V (x_{1} + x_{2}) + V (p_{1} - p_{2} + g_{3} p_{3}) < 4 \\ V_{13} = V (x_{1} - x_{3}) + V (p_{1} + g_{2} p_{2} + p_{3}) < 4 \\ V_{23} = V (x_{2} + x_{3}) + V (g_{1} p_{1} + p_{2} - p_{3}) < 4 \end{array}

where

V (A) = < A^{2} > - < A >^{2}

and g_i is an arbitrary real number. Following Ref [34], we set

g_{1} = \frac{- (< p_{1} p_{2} > - < p_{1} p_{3} >)}{< p_{1}^{2} >}

,

g_{2} = \frac{- (< p_{1} p_{2} > + < p_{2} p_{3} >)}{< p_{2}^{2} >}

, and

g_{3} = \frac{- (< p_{1} p_{3} > - < p_{2} p_{3} >)}{< p_{3}^{2} >}

. Satisfying any pair of these three inequalities is sufficient to demonstrate the creation of tripartite entanglement.

Figures 7(a)–7(c) display the evolutions of the VLF correlations as a function of the Fourier frequency $ω$ with Δ = -ω, Δ₃ = 2900 MHz, and Ω₁ = Ω₃ = 80 MHz. It can be seen that, in a moderate range of the Fourier frequency $ω$ (i.e., about 100~700 MHz), V₁₂, V₁₃, and V₂₃ are all smaller than 4, and the minimal values of V₁₂, V₁₃, and V₂₃ are, respectively, about 0.75, 2, and 0.75, which sufficiently demonstrates that the generated fields E₁, E₂, and E₃ are CV entangled with each other. Beyond the ranges, the entanglement among the generated three fields E₁, E₂, and E₃ would disappear. Further calculations show that the dependence of the tripartite entanglement on the atomic density, Rabi frequencies of the scattering fields, and coherence decay rate of the lower doublet exhibits the similar behaviors as those of the bipartite entanglement; that is, the increase of the atomic density and Rabi frequencies of the scattering fields and the decrease of the coherence decay rate of the lower doublet would strengthen the degree of tripartite entanglement. In the same way, we can demonstrate the generation of tripartite entanglement among E₁, E₂, and E₄ fields as well. This clearly indicates that the above concept for producing bipartite and tripartite entanglement via atomic spin wave can be easily extended to generate multipartite entanglement to any desired number with more scattering fields, tuned to the vicinity of the transitions $| 1 〉$ - $| 3 〉$ and/or $| 2 〉$ - $| 3 〉$ , mixing with the prebuilt atomic spin wave.

Fig. 7 The evolutions of the VLF correlations V₁₂ (a), V₁₃ (b), and V₂₃ (c) as a function of the Fourier frequency $ω$ with Δ = -ω, Δ₃ = 2900 MHz, and Ω₁ = Ω₃ = 80 MHz, and the other parameters are the same as those in Fig. 2.

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The generated multipartite quantum correlations, quantum anti-correlations, and entanglement can be intuitively understood in terms of the interaction between the laser fields and atomic medium. As seen in Fig. 1(a), all the Stokes fields and anti-Stokes fields are produced through FWM processes, where every Stokes (anti-Stokes) photon generation is obtained by absorbing one scattering and one coupling (probe) photons and emitting one probe (coupling) photon; equivalently, the generated Stokes (or anti-Stokes) field can be regarded as the result of frequency down- (up-) conversion process through mixing the scattering field with the atomic spin wave S prebuilt by the strong coupling and probe fields (as shown in Fig. 1(b)). Since the generation of a Stokes (or an anti-Stokes) photon is accompanied with the creation (or annihilation) of an atomic spin wave excitation, the up-converted frequency component (i.e., anti-Stokes field) is quantum anti-correlated with the down-converted frequency component (i.e., Stokes field), therefore strong multipartite entanglement can be achieved. In this sense, the prebuilt atomic spin wave acts as a quantum beam splitter, and in principle, by using the atomic spin wave excitation, multipartite entangled CV fields to any desired number can be created through applying more scattering fields, as long as the scattering fields are weak and the atomic spin wave is strong enough to ensure that different scattering fields have negligible influence on it, which can be realized by employing substantially strong probe and coupling fields.

4.Conclusion

In conclusion, by using the Heisenberg-Langevin method, we have theoretically studied the effects of the atomic density, Rabi frequencies of the scattering fields, and coherence decay rate of the lower doublet on the generation of nondegenerate multipartite CV entanglement. The theoretical results provide a proof-of-principle demonstration of generating nondegenerate entangled narrow-band multiple fields to any desired number with long correlation time in an atomic ensemble with high optical depth, and give valuable evidences for further studies on multipartite CV entanglement via atomic spin wave and subsequent applications in quantum information processing and quantum networks.

Acknowledgments

This work is supported by NBRPC (Nos. 2012CB921804 and 2011CBA00205), the National Natural Science Foundation of China (Nos. 11274225, 10974132, 50932003, and 11021403), and Innovation Program of Shanghai Municipal Education Commission (No. 10YZ10).

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Generation of multipartite continuous-variable entanglement via atomic spin wave: Heisenberg-Langevin approach

Abstract

1. Introduction

2. Heisenberg-langevin equations

3. Results and discussions

4.Conclusion

Acknowledgments

References and Links

Cited By

Figures (7)

Equations (8)

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