## Abstract

Two schemes for the preparation of weighted continuous variable cluster states with four atomic ensembles are proposed. In the first scheme, the four separated atomic ensembles inside a two-mode ring cavity are driven by pulse laser fields. The basic idea of the scheme is to transfer the ensemble bosonic modes into suitable linear combinations that can be prepared in a pure cluster state by a sequential application of the laser pulses with the aid of the cavity dissipation. In the second one, we take two separate two-mode cavities, each containing two atomic ensembles. The distant cavities are coupled by dissipation in a cascade way. It has been found that the mixed cluster state can be produced. These schemes may contribute towards implementing continuous variable quantum computation, quantum communication and networking based on atomic ensembles.

© 2012 OSA

## 1. Introduction

Cluster states, defined as a class of highly entangled multi-qubit states [1], have been proposed as a potential resource for performing one-way quantum computation [2]. In 2006, this concept was extended by Zhang and Braunstein [3] from qubits system to continuous-variable (CV) system. Subsequently, CV cluster states have attracted a lot of attention and have been extensively investigated due to their wide range of application for one-way quantum computation [4, 5], which provides a promising approach to fulfil the capabilities of quantum information processing. A number of theoretical and experimental schemes based on the linear optics have been proposed for generation of cluster states [6–13]. For example, Su *et al.* [6] and Furusawa *et al.* [8, 11] have reported the experimental generation of CV four-mode cluster states by use of single-mode squeezed light and a set of linear beam splitters. Menicucci *et al.* [13] have designed a compact experiment to produce an arbitrarily large CV cluster state using just one single-mode vacuum squeezer and one quantum nondemolition gate. On the other hand, Pfister *et al.* [9] have demonstrated, through mapping CV cluster-state graph onto two-mode squeezing graphs, that a desired CV cluster state can be produced from a single optical parametric oscillator (OPO) pumped by a multi-frequency laser beam. Recently, they have experimentally generated 15 quadripartite entangled cluster states in the optical frequency comb of a single OPO [10]. Moreover, by means of the linear optical cluster states prepared off-line, homodyne detection, and electronic feeding forward, a deterministically controlled-X operation have been designed [14,15]. The unconditional CV one-way quantum computation has been demonstrated experimentally by use of a linear CV cluster state with four entangled optical modes [16]. The CV square four-mode cluster state is proposed to realize the quantum teamwork [17] for unconditionally multiparty communication with Gaussian states.

Apart from the linear optical schemes, another interesting subject is the creation of entanglement among atomic ensembles [18–28]. Schemes involving atomic ensembles have many advantages over the linear optics schemes. For instance, collective effect of atoms increases the coupling strength of an interaction between light and matter. Moveover, the existence of long atomic ground-state coherence lifetimes offers a robust medium for realizing quantum memory. More importantly, this feature has also been adopted to create entanglement between two atomic ensembles by use of quantum reservoir engineering [29–32]. A scheme for the generation of entanglement between two hot atomic ensembles placed inside a high-*Q* ring cavity composed of two co-propagating modes has been proposed [21]. It has been found that under the interaction of suitable external classical laser fields, the cavity dissipative relaxation can be used to drive the two ensembles into long-lived two-mode squeezed vacuum state. Recently, Krauter *et al.* [26, 27] demonstrated that the dissipation induced entanglement between two atomic ensembles coupled to the environment composed of the vacuum modes of the electromagnetic field. This entanglement is achieved by dissipation engineered with laser and magnetic fields. It is obtained that the two atomic ensembles are kept entangled at room temperature for about 0.015s. The experimental results have been explained by Muschik *et al.* [25] who developed a scheme involving two-level systems and showed that steady-state entanglement can be generated by incoherent pumping if both the single-particle cooling and heating noises can be ignored. Krauter *et al.* [26, 27] suggested that a true steady-state entanglement between the two atomic ensembles may be created to place the atomic ensembles in a cavity. In Sec. 2, we present a proposal to produce the CV cluster states for four atomic ensembles inside a ring cavity. We will show that through sequentially sending two series of laser pulses with appropriate amplitudes and phases, the four atomic ensembles can be deterministically evolved into a four-mode weighted CV cluster state with the aid of the cavity dissipation, which is similar to those schemes created in the frequency comb of a single optical parametric oscillator pumped by multi-frequency laser beams [9, 10].

On the other hand, remote entanglement among distant atomic ensembles is an essential ingredient source for quantum networks based on the atomic ensembles [33, 34]. The coupled atom-cavity-fiber systems are considered as a basic building blocks toward scalable quantum network [35–39]. Schemes [40,41] have been suggested for entanglement engineering between single atomic ensembles in separated cavities through exchange photons mediated by optical fiber. However, the realization of the CV distant quantum teamwork [17] requires the entanglement among four distant atomic ensembles. Therefore, in Sec. 3, we will present a scheme for the entanglement generation among four atomic ensembles, which are located in two separate two-mode cavities coupled in a cascaded way. In each cavity, there are two atomic ensembles. We will show that the four atomic ensembles in two different nodes can be prepared in a mixed cluster states, which may be utilized in universal quantum networks [35–39] and quantum team-work [17].

In this paper, we present two schemes involving four atomic ensembles that could serve as potential sources for creation of four-mode cluster states. The paper is organized as follows. In Sec. 2, we describe the first scheme. We assume that four atomic ensembles are placed inside a single two-mode high-*Q* cavity. It has been found that with the aid of the cavity dissipation and sending two serials of appropriate laser pulses to interact with the atomic ensembles, the four atomic ensembles can be unconditionally prepared in a four-mode weighted cluster state. In Sec. 3, we discuss the entanglement generation among four atomic ensembles inside two separated two-mode cavities, each containing two atomic ensembles. The two cavities are coupled by optical fiber in a cascaded way. The time-dependent behavior of the four atomic ensembles in two different cavities is analyzed. It is found that the four atomic ensembles in two different nodes can be prepared in a mixed cluster states with the similar correlations as shown in the first scheme. Finally, we summarize our results in Sec. 4.

## 2. Creation of four-mode cluster states in a single high-Q ring cavity

In the first scheme, we assume that four atomic ensembles are placed inside a single two-mode high-*Q* ring cavity, as illustrated in Fig. 1. The ring cavity is composed of two modes *a* and *b* with frequencies *ω _{a}* and

*ω*. Actually in the ring cavity, the mode

_{b}*a*(

*b*) owns two degenerate mutually counter-propagating modes, to which the atomic ensembles are equally coupled. External pulse lasers which are used to drive the atomic ensembles couple to only the two propagating modes, as illustrated in Fig. 1(a). We assume that the atoms are homogeneously distributed inside the ensembles, only the scattering of the cavity field which co-propagates with the driving lasers occurs. This allows us to ignore the coupling of the atoms to the cavity modes which counter-propagate with the driving lasers. This ignorance can be fulfilled in trapped room-temperature atomic ensembles, in which fast atomic oscillations over the interaction time lead to a collectively enhanced coupling of the atoms to the modes which are collinear with the driving laser fields. Therefore, the ring cavity with frequencies

*ω*and

_{a}*ω*can work in the two-mode approximation for the case of trapped room-temperature atomic ensembles [42].

_{b}The four ensembles contain, respectively, *N*_{1}, *N*_{2}, *N*_{3} and *N*_{4} identical four-level atoms, as shown in Fig. 1(b). The *j*-th atom in the *n*-th ensemble is composed of two stable (not decaying) ground states |0* _{jn}*〉, |1

*〉, and two excited states |2*

_{jn}*〉 and |3*

_{jn}*〉. The parameters Ω*

_{jn}_{2n}and Ω

_{3n}are Rabi frequencies of external pulse lasers which are coupled to the transitions |1

*〉 → |2*

_{jn}*〉 and |0*

_{jn}*〉 → |3*

_{jn}*〉, respectively. The cavity modes couple to the transitions |1*

_{jn}*〉 → |3*

_{jn}*〉 and |0*

_{jn}*〉 →|2*

_{jn}*〉 with the coupling strength*

_{jn}*g*and

_{an}*g*, respectively.

_{bn}#### 2.1. Effective Hamiltonian

The Hamiltonian of the system, written in the rotating-wave approximation has the following form

where*〉 equal to zero in the ensembles*

_{jn}*n*= 1,2, and in the ensembles

*n*= 3,4 we have chosen the state |1

*〉 as the ground state with the energy equal to zero, and the energies of the other three states are correspondingly denoted as*

_{jn}*ω*

_{1},

*ω*

_{2}and

*ω*

_{3}.

_{2n}(

*x*) and Ω

_{jn}_{3n}(

*x*) are the position dependent Rabi frequencies of the driving laser fields, and

_{jn}*ϕ*

_{2n}and

*ϕ*

_{3n}are their phases, respectively.

*g*(

_{an}*x*) and

_{jn}*g*(

_{bn}*x*), are also dependent on the atomic position

_{jn}*x*. In what follows, we will assume all the wave numbers of the laser fields are nearly equal and denoted by

_{jn}*k*, then the plane traveling wave representation for the laser fields and the cavity modes, in which

We now make a unitary transformation and assume that the laser frequencies *ω*_{L2}, *ω*_{L3} satisfy the resonance condition *ω*_{L2} + *ω*_{1} = *ω*_{L3} – *ω*_{1}, and the detunings of the laser fields from the atomic transition frequencies Δ_{2n} = *ω*_{2} – (*ω*_{L2} + *ω*_{1}), Δ_{3n} = *ω*_{3} – *ω*_{L3} are much larger than the Rabi frequencies and the cavity coupling strengths Δ_{2n}, Δ_{3n} ≫ Ω_{2n}, Ω_{3n}, *g _{an}*,

*g*. In this limit, we can perform the adiabatic approximation to eliminate the atomic excited states and obtain an effective two-level Hamiltonian which takes the form after omitting constant energy terms

_{bn}*δ*=

_{a}*ω*–

_{a}*ω*

_{L3}+

*ω*

_{1}and

*δ*=

_{b}*ω*– ω

_{b}_{L3}+

*ω*

_{1}are the detunings of the cavity frequencies from the Raman coupling resonance,

*Ĵ*,

_{zn}*Ĵ*

_{±n}are collective atomic operators for the

*n*-th atomic ensemble, defined as

Taking into account of large detunings of the driving fields, we can assume that the excitation probability of the atoms to the states {|1* _{jn}*〉} in ensembles

*N*

_{1}and

*N*

_{2}, and the excitation probability of the atoms to the states {|0

*〉} in ensembles*

_{jn}*N*

_{3}and

*N*

_{4}are much smaller than the total number of atoms, i.e., $\u3008{c}_{n}^{\u2020}{c}_{n}\u3009\ll {N}_{n}$. In this case, the collective atomic operators in the Holstein-Primakoff representation [43] can be well approximated by

*J*= −

_{zn}*N*/2 and ${J}_{-n}=\sqrt{{N}_{n}}{c}_{n}$. Here, the operators

_{n}*c*and ${c}_{n}^{\u2020}$ obey the standard bosonic commutation relation, $\left[{c}_{n},{c}_{n}^{\u2020}\right]=1$. Then, if we choose the detunings such as

_{n}#### 2.2. Creation of four-mode weighted cluster states

Since four-mode CV cluster states are the simplest cluster states which play an important role in one-way quantum computation for continuous variable [16], we now proceed to present a procedure that may prepare four hot atoms ensembles coupled to two co-propagating modes of a ring cavity in a four-mode CV cluster state. The proposed procedure is based on the interaction of the four bosonic modes through the cavity modes suitably driven by the external laser fields and the damping of the cavity modes. Thus the dynamics of the atom-field coupling system is governed by the following master equation

in which characterizing the damping of the cavity modes*a*and

*b*, with rates

*κ*and

_{a}*κ*, respectively. For simplicity, we set

_{b}*κ*=

_{a}*κ*=

_{b}*κ*in the following. Here,

*D*[

*ϑ*]

*ρ*≡ 2

*ϑρϑ*

^{†}–

*ϑ*

^{†}

*ϑρ*–

*ρϑ*

^{†}

*ϑ*, with

*ϑ*=

*a*,

*b*.

We will demonstrate that the system evolving under the effective Hamiltonian (10) can decay into a stationary squeezed vacuum state *S*|0_{c1}, 0_{c2}, 0_{c3}, 0_{c4}〉, where *S* is a four-mode squeezing operator

*ε*is the squeezing parameter which is assumed to be real without loss of generality. In order to do it, we make a unitary transformation that transforms the field operators

*c*into new operators

_{n}*e*=

_{n}*Tc*

_{n}T^{†}, which are linear combinations of the operators of different atomic ensembles, i.e.

*S*

_{14}(

*λε*) and

*S*

_{23}(

*ε*/

*λ*) are standard two-mode squeezing operators [44], each involving only a pair of the bosonic modes, (

*e*

_{1},

*e*

_{4}) and (

*e*

_{2},

*e*

_{3}), respectively. Since the pairs of modes (

*e*

_{1},

*e*

_{4}) and (

*e*

_{2},

*e*

_{3}) are orthogonal to each other, the procedure of preparation of the modes in desired squeezed vacuum states can then be done in two independent steps.

*Step 1.* First a series of laser pulses with duration *T*_{1} are sent to drive the four atomic ensembles, whose Rabi frequencies and phases are specially chosen as: *ϕ*_{21} = *ϕ*_{23} = *ϕ*_{32} = *ϕ*_{34} = *π*, *ϕ*_{22} = *ϕ*_{24} = *ϕ*_{31} = *ϕ*_{33} = 0, *β*_{21} = *λβ*_{22}, *β*_{23} = *λβ*_{24}, *β*_{31} = *λβ*_{32}, *β*_{33} = *λβ*_{34}, *β*_{34} = *β*_{22} and *β*_{32} = *β*_{24}. With this choice of the Rabi frequencies and phases, the Hamiltonian (10) takes the form

After that we do the two-mode squeezing transformation for the mode *e*_{2} and *e*_{3} as *ρ̃*_{1} = *S*_{23}(*ε*/*λ*)*TρT*^{†}*S*_{23}(−*ε*/*λ*) with the squeezing degree
$\frac{\epsilon}{\lambda}=\frac{1}{2}\text{ln}\left(\frac{{\beta}_{34}-{\beta}_{32}}{{\beta}_{34}+{\beta}_{32}}\right)$, Eq. (17) becomes

We see that the Hamiltonian represents a simple system of two independent linear mixers, where the bosonic modes *e*_{2} and *e*_{3} linearly couple to the cavity modes *a* and *b*, respectively. The master equation of the transformed density operator takes the form

*T*

_{1}of the driving laser pulses is sufficiently long, for example,

*T*

_{1}∼ 1/

*κ*, the cavity dissipative relaxation will force the system to be prepared in a stationary state, in which all the modes

*a*,

*b*,

*e*

_{2}and

*e*

_{3}are in their vacuum states. Thus, as a result of the interaction given by the Hamiltonian (18), and after a sufficiently long evolution time, the four modes

*a*,

*e*

_{2},

*b*and

*e*

_{3}will be found in the vacuum state

*ρ*

_{e1,e4}(

*T*

_{1}) depends on the initial state but not important here. From Eq. (21) we can find that with the help of the cavity dissipation, the combined modes

*e*

_{2}and

*e*

_{3}have been prepared in a pure state through the couplings between modes

*a*and

*e*

_{2}, and

*b*and

*e*

_{3}. Because all the four combined modes

*e*are orthogonal to each other, we can leave the two modes

_{n}*e*

_{2}and

*e*

_{3}decoupled with the cavity modes and prepare the modes

*e*

_{1}and

*e*

_{4}in a pure vacuum state further. This can be realized by adjusting the parameters of the driving lasers so that the mode

*a*linearly coupled to

*e*

_{1}, and

*b*linearly coupled to

*e*

_{4}, and call the cavity dissipation again. Then we need another step to get the pure target entangled state.

*Step 2*. We now turn off the lasers driving the mode *e*_{2} and *e*_{3}, and sent another series of pulses of driving lasers in time interval *T*_{2} with the specifically chosen phases and the Rabi frequencies: *ϕ*_{2i} = *ϕ*_{3i} = 0 (*i* = 1, 2, 3, 4), *β*_{22} = *λβ*_{21}, *β*_{24} = *λβ*_{23}, *β*_{32} = *λβ*_{31}, *β*_{34} = *λβ*_{33}, *β*_{33} = *β*_{21} and *β*_{31} = *β*_{23}. In this case, the effective Hamiltonian (10) takes the form

If we now make the same unitary transformation as Eq. (14) and a two-mode squeezing transformation on the density operator *ρ*̃_{2} = *S*_{14}(*ελ*)*S*_{23}(*ε*/*λ*)*TρT*^{†}*S*_{23}(−*ε*/*λ*)*S*_{14}(−*ελ*), with the squeezing parameter as
$\lambda \epsilon =\frac{1}{2}\text{ln}\left(\frac{{\beta}_{33}+{\beta}_{31}}{{\beta}_{33}-{\beta}_{31}}\right)$, we obtain the master equation

*e*

_{1},

*e*

_{4}):

Following the similar discussion as mentioned in the first step, under the damping of the cavity modes, if the duration of the laser pulses is sufficiently long, the system will evolve to a pure vacuum state determined by the density operator *ρ̃*_{2} = |*ψ̃*〉 〈*ψ̃*|, where |*ψ̃*〉 = |0_{e1}, 0_{e2}, 0_{e3}, 0_{e4}, 0* _{a}*, 0

*〉 is the vacuum state of the transformed system.*

_{b}Performing the inverse transformation from *ρ̃* to *ρ* or from *ψ̃* to *ψ*, we find that the final stationary state of the system is prepared in the four-mode squeezed state

*S*is given in Eq. (13).

Thus, a pure entangled state of the modes of four atomic ensembles interacting with two cavity modes can deterministically prepared in only two independent steps. Recently, Midgley *et al.* [47] have proposed a procedure for creation of cluster states using a single multimode optical parametric oscillator (OPO). They have shown that with an appropriate nonlinear medium and a pump beam with the right frequency content, the comb of quantum modes will encode a large square lattice continuous-variable cluster state. However, only a mixed entangled CV state can be produced. In the procedure presented here, a pure entangled state can be generated.

We now demonstrate that the state |*ψ*〉 is an example of a continuous-variable cluster state. To show this, we introduce the quadrature amplitude
${X}_{j}=\left({c}_{j}+{c}_{j}^{\u2020}\right)/2$ and phase
${P}_{j}=-i\left({c}_{j}-{c}_{j}^{\u2020}\right)/2$ components of the four modes involved, and find that the variances of the linear combinations according to the definition of cluster states [8, 9], are

*V*< 1 (

_{j}*j*= 1, 2, 3, 4), which means that the variances can be lower than the limit of quantum vacuum fluctuations. Also, we can see that the variances tend to zero when the squeezing parameter

*ε*→ ∞. In the case of finite

*ε*, the variances are not equal to zero, but are still smaller than 1, indicating that the modes are in a entangled state. When we make the transformation (

*X*→

*P*,

*P*→ –

*X*) on modes 3 and 4 respectively, we then find that the state |

*ψ*〉 is an analog of a four-mode continuous-variable weighted cluster state in squared type.

In summary of this section, we have proposed a scheme for the deterministic generation of a pure four-mode cluster state of four atomic ensembles trapped inside a two-mode cavity. The cluster state can be generated only in two steps in the preparation process, simply by applying laser pulses with appropriate phases and amplitudes to drive the atomic ensembles. Although the discussion here only focuses on the creation of four-mode pure cluster states, the present method can be easily generalized to the creation of multimode pure cluster states based on many atomic ensembles trapped in a two-mode cavity. This could be done as follows. It is well known from the definition of CV 2*N*-mode weighted [9] or unweighted cluster states [8] that all the variances of 2*N* linear combinations of quadrature operators *X _{j}* and

*P*should be below the vacuum noise limit, i.e. should be squeezed. This means we can construct

_{j}*N*independent two-mode squeezed vacuum states in the combined bosonic representation similar to Eq. (14). Adjusting appropriate parameters of the laser pulses driving the atomic ensembles, with the help of the cavity dissipation and after

*N*steps of the preparation, we can get a pure 2

*N*-mode cluster state. Similarly, for the preparation of the 2

*N*+ 1-mode cluster states, we need construct

*N*independent two-mode squeezed vacuum states and one single-mode squeezed vacuum state in the combined bosonic representation, and

*N*+ 1 steps are required to obtain the steady-state 2

*N*+ 1-mode cluster state following the similar procedures as discussed above.

We should point out that our technique differs from that put forward by Polzik *et al.* [25], in which the entangling mechanism is due to the coupling to the *x*-polarized vacuum modes in the propagation direction *z* of the laser field, which are shared by both ensembles and provide the desired common environment. Actually, in their method, because only the *x*-polarized vacuum modes are utilized to drag the two combined bosonic modes of the two atomic ensembles into two-mode squeezed state, for example, the *e*_{2} and *e*_{3} modes as we discussed in step 1, the single-particle heating and cooling noises are unavoidable as explained in Ref. [25]. These noises prevent the two atomic ensembles from a pure entangled state and the entanglement can be only kept up to 0.015s. However, here as described by Eq. (18), the combined modes *e*_{2} and *e*_{3} interact with the two cavity modes *a* and *b* respectively, these two cavity modes can simultaneously drag the *e*_{2} and *e*_{3} modes into the pure two-mode squeezed vacuum state with the help of the cavity dissipation. Evidently, there exist no single-particle heating and cooling noises. Consequently, this pure entangled state can be kept for a long time. Because the *e _{j}* modes commute with each other, the preparation of the

*e*

_{1}and

*e*

_{4}modes in an another proper two-mode squeezed vacuum state in step 2 has no influence on the state prepared in the step 1. Thus the method we described here is more efficient and robust to prepare for the pure

*N*-mode cluster state.

Moreover, it is vital to achieve a large entanglement over a short time for preparation. In our scheme the entanglement can be created over times much shorter than that in the scheme proposed by Polzik *et al.* [25]. For example, in our scheme, the system evolves into a stationary state during the time 1/*κ*, where *κ* is the damping rate of the cavity modes. But in the scheme proposed by Polzik group, the damping rate is proportional to *g*^{2}/*κ* (where *g* ≪ *k*). Thus, the system would reach a steady state at time *κ*/*g*^{2}, which is much longer than 1/*κ*. Hence, in our scheme one could achieve a steady-state entanglement over times much shorter than that in the Polzik *et al.* scheme. This makes out our method more robust for the operation of quantum devices such as quantum repeaters.

## 3. Creation of a four-mode cluster state in cascaded cavities

In this section, we pay our attention to the preparation of the cluster state for four atomic ensembles located in two separate ring cavity. Here in the two distantly coupled cavities, each of them containing two atomic ensembles, as illustrated in Fig. 2. Atomic energy levels and coupling configurations of the lasers are the same as that in the first scheme, with one exception that in the present scheme the modes *a _{i}* and

*b*(

_{i}*i*= 1, 2) propagate in opposite directions, while they propagated in the same directions in the first scheme. The atomic ensembles are selectively coupled to counter-propagating degenerate cavity modes such that the ensembles

*N*

_{1}and

*N*

_{2}in the left cavity are coupled to modes

*a*

_{1}and

*b*

_{1}, whereas the ensembles

*N*

_{3}and

*N*

_{4}in the right cavity are coupled to modes

*a*

_{2}and

*b*

_{2}, respectively. External pulse lasers with the Rabi frequencies Ω

_{21}and Ω

_{22}, applied to the left cavity, couple to the clockwise mode

*a*

_{1}, while the lasers with the Rabi frequencies Ω

_{31}and Ω

_{32}couple to the anti-clockwise mode

*b*

_{1}. Correspondingly, in the right cavity, laser pulses of the Rabi frequencies Ω

_{23}and Ω

_{24}couple to the anti-clockwise mode

*a*

_{2}, while the laser pulses of the Rabi frequencies Ω

_{33}and Ω

_{34}couple to the clockwise mode

*b*

_{2}. As in the first scheme, the laser pulses interact dispersively with the atoms.

Note that the dissipative parts of the master equation contain coupling between the cavity modes arranged in the cascade way that the mode *a*_{2} couples to mode *a*_{1} from right cavity into left cavity, but the mode *b*_{1} couples to mode *b*_{2} from left cavity into right cavity. With this arrangement our system behaves as a cascade open system. Also, as in the first scheme, we consider room-temperature atomic ensembles, so that the approximation of taking only the modes propagating in the direction of laser pulses is reasonable.

#### 3.1. Time evolution of density matrix

We use dissipation of the cavity modes as the coupling between the cavities and arrange it in cascade way such that the mode *a*_{2} couples to the mode *a*_{1} and the mode *b*_{1} couples to the mode *b*_{2} with no feedback coupling. In this case, the effective Hamiltonian takes the form

The master equation for the density operator of the system is of the form [48]

*κ*is the damping rate of the cavity modes, and

*η*is the coupling efficiency between the modes. For perfect coupling,

*η*= 1, and

*η*< 1 for an imperfect coupling.

To examine how this system evolves, we make the unitary transformation of the collective mode operators as Eq. (14). Then, we find that under this transformation, the effective Hamiltonian takes the form

where*e*

_{2},

*e*

_{3}) of the modes is independent of the dynamics of the pair (

*e*

_{1},

*e*

_{4}). Here

*β*

_{1}and

*β*

_{2}are real and are determined by the following relations

*b*

_{1},

*e*

_{2}) and (

*a*

_{2},

*e*

_{4}), and two linear mixing processes happen within (

*b*

_{2},

*e*

_{3}) and (

*a*

_{1},

*e*

_{1}). The CV cluster entanglement can be built up with the four atomic ensembles via these processes with the help of the cavity coupling.

#### 3.2. Entanglement analysis of the system

We now proceed to solve the master equation (29)
and to determine if the entanglement could be created among the modes. We consider pairs
of the modes (*c*_{1}, *c*_{2}) and
(*c*_{3}, *c*_{4}) as two different
subsystems, and will consider if an entanglement could be created between them. For
simplicity, we assume that initially all the cavity and the atomic modes are in the vacuum
states.

To determine entanglement between the modes in different cavities, we will use the nonpositive partial transpose criterion [49, 50]. The criterion states that a four-mode Gaussian state is separable, if and only if

where*V*′ is an 8×8 covariance matrix expressed by use of the variables

*X*and

_{j}*P*(

_{j}*j*= 1, 2, 3, 4) in the phase space. The covariance matrix

*V′*can be easily obtained from solving the master equation (29) and using Eq. (14). Since we are just interested in the entanglement between the modes in different cavities, we group the two atomic ensembles in each cavity as one partition respectively. Consequently, the partial transposition Γ is equivalent to time reversal and corresponds in phase space to a sign change of the momentum variables, i.e.,

*X*→ Γ

^{T}*X*= (

^{T}*X*

_{1},

*P*

_{1},

*X*

_{2},

*P*

_{2},

*X*

_{3}, –

*P*

_{3},

*X*

_{4}, –

*P*

_{4}). The matrix Λ is a block diagonal matrix with the blocks given by the 2 × 2 matrix

The minimum negative eigenvalue of Γ*V*′Γ + *i*Λ/2, which determines the non-separability of the pair of modes (1,2) and (3,4), is found to be

*n*

_{3}= 〈

*e*

_{2}

*e*

_{3}〉= 〈

*e*

_{1}

*e*

_{4}〉= −

*d*

_{1}

*d*

_{2}are correlations between the modes, and can be obtained from the Eqs. (29)–(31)

*e*

_{2}and

*e*

_{3}. Due to the nondegenerate parametric amplification process between the modes

*e*

_{2}and

*b*

_{1}, photons generated in the mode

*b*

_{1}have a strong nonclassical correlation with the mode

*e*

_{2}. The photon in the mode

*b*

_{1}can be transferred to the right cavity and established a correlation with the mode

*b*

_{2}through the cascaded dissipative process. With the help of the laser pulses, linear mixing processes occur between mode

*b*

_{2}and mode

*e*

_{3}. As the result, an entanglement between the modes

*e*

_{2}and

*e*

_{3}can be created through the interaction between the cavity modes

*b*

_{1}and

*b*

_{2}. In the same manner, due to the nondegenerate parametric amplification process between modes

*e*

_{4}and

*a*

_{2}, photons generated in the mode

*a*

_{2}can be transferred through the dissipative process of spontaneous emission into the mode

*a*

_{1}of the left cavity. Then, linear mixing processes in the left cavity can be established between the modes

*a*

_{1}and

*e*

_{1}, resulting in an entanglement between the mode

*e*

_{4}and

*e*

_{1}. Next, using the relations between the modes

*e*and

_{j}*c*(

_{j}*j*= 1, 2, 3, 4), see Eq. (14), we can easily see that the entanglement between the modes (

*e*

_{1},

*e*

_{4}) and (

*e*

_{2},

*e*

_{3}) leads to entanglement between the pairs of the modes (

*c*

_{1},

*c*

_{2}) and (

*c*

_{3},

*c*

_{4}).

These nonclassical correlations lead to the same properties of square cluster states as that in the first scheme, so we can examine the occurrence of cluster states using the variances (26), that in the present case are found to be

Here,*V*< 1 indicates that there is a four-mode squeezing, and the obtained state is a weighted cluster state in squared-type, which is similar to obtain in the previous section. However, we should note here that in this scheme all the variances

_{j}*V*are equal, which is in contrast to that obtained in the first scheme. One can also find from Eq. (40) that the variance

_{j}*V*< 1 can be written as

_{j}Comparing Eqs. (41) with (39), we see that the conditions for squeezing and entanglement are not the same, indicating that it is easy to build entanglement among atomic ensembles than multi-mode squeezing for characterizing the property of cluster state. Only in the limit of
$\u3008{e}_{2}^{\u2020}{e}_{2}\u3009=\u3008{e}_{3}^{\u2020}{e}_{3}\u3009$ these two conditions merge, showing that the conditions for entanglement and squeezing are the same. However, the mode *e*_{2} interacts with the mode *b*_{1} in a parametric amplification way, the mode *e*_{3} is coupled to the mode *b*_{2} through linear mixing process. Thus,
$\u3008{e}_{2}^{\u2020}{e}_{2}\u3009\ne \u3008{e}_{3}^{\u2020}{e}_{3}\u3009$, which then results in a mixed state. The mixture of the obtained state can be characterized by the purity

*ρ*

^{2}is equal to 1, the state of system is a pure state, otherwise it is a mixed state. Although in the present scheme only a mixed cluster state can be obtained, this entangled state may serve as a useful quantum resource for multiparty communication schemes in the continuous-variable field [38, 39], such as remote information concentration, quantum secret sharing, and superactivation.

We would like to point out that the mixture of the obtained cluster state in this scheme comes from the incoherent weak coupling of the two pairs of the cavity modes. In optical setting, Pysher *et al.* [10] mention that the purity of a mixed cluster state can be increased by filtering the pump with a ”mode-cleaner” cavity. However, in the atomic setting, the purity of a mixed cluster state can be improved by coherently strong coupling between distant cavities mediated by short fiber with present technology. For instance, Serafini *et al.* [36] have proposed that realizing effective quantum gates between two atoms in distant cavities coupled by optical fibre is possible. The highly reliable swap and entangling gates [36] and the multiatom entangled states [40] are achievable with the present technology.

Figure 3 shows the time evolution of the negativity *E _{n}*, the variances

*V*, and the purity

_{j}*P*for different values of the ratio

*α*=

*β*

_{1}/

*β*

_{2}and for the case of ideal coupling between the cavities,

*η*= 1. It is seen that the negativity and the variances depend strongly on the ratio

*α*. For

*α*= 1, the variances

*V*have the same tendency in time as the negativity

_{j}*E*that they evolve towards the minimum value of

_{n}*V*= 0 and at the same time the negativity

_{j}*E*evolves towards its optimal negative value of

_{n}*E*= −0.5. However, when

_{n}*α*≠ 1, the variances

*V*have a different tendency from the negativity

_{j}*E*. The variances are reduced below the vacuum limit only in a restricted time range and develop to large positive values at long times Γ

_{n}_{1}

*t*≫ 1. This means that the state of the system is a cluster state only in a short time regime, and the time region where the system is in the cluster state decreases with increasing

*α*. It is interesting to note that created cluster state is a mixed one rather than a pure state. Figure 3 also shows that the state of the system is a mixed state except the initial state. Especially, when

*α*≠ 1, in the short time region we can get better entanglement and squeezing than those when

*α*= 1, but in the long time region the squeezing disappears and the entanglement becomes worse than that when

*α*= 1. The reason is that the modes

*e*

_{2}and

*e*

_{3}interact differently with the cavity modes. The mode

*e*

_{2}is coupled yo the cavity mode

*b*

_{1}with parametric amplification, whereas the mode

*e*

_{3}is coupled to the cavity mode

*b*

_{2}via linear mixing, which then results in different numbers of excitations in the modes, i.e. $\u3008{e}_{2}^{\u2020}{e}_{2}\u3009\ne \u3008{e}_{3}^{\u2020}{e}_{3}\u3009$. With the time evolution, the difference between $\u3008{e}_{2}^{\u2020}{e}_{2}\u3009$ and $\u3008{e}_{3}^{\u2020}{e}_{3}\u3009$ becomes large, which leads to the decrease of the purity. Also, the large difference $\u3008{e}_{2}^{\u2020}{e}_{2}\u3009-\u3008{e}_{3}^{\u2020}{e}_{3}\u3009$ makes the squeezing condition Eq. (41) hard to be satisfied, so the squeezing will disappear.

Figure 4 shows the effect of the coupling efficiency *η* of the cavity modes. We see that a decrease of the coupling efficiency leads to a rapid increase of the variances *V _{j}*. However, the entanglement is less sensitive to the coupling efficiency

*η*and decreases slowly with decreasing

*η*. Thus, the bosonic modes of the atomic ensembles can be entangled even if there is no squeezing. In other words, the more prefect the coupling between the cavity modes is, the better weighted cluster state in squared shape can be obtained.

## 4. Conclusions

To summarize, we have proposed two schemes for the preparation of entangled CV weighted cluster states with four atomic ensembles. In the first scheme, the four separated atomic ensembles are located inside a two-mode ring cavity driven by pulse laser fields. The basic idea of the scheme is to transfer the four ensemble bosonic modes into suitable linear combinations that can be prepared them in a pure cluster state by a sequential application of the laser pulses in two steps. In the second one, the four atomic ensembles are arranged in two separated two-mode cavities, each containing two atomic ensembles. The distant cavities are coupled by dissipation in a cascade way. We have analyzed the dynamical behavior of the squeezing and entanglement between the bosonic modes of the atomic ensembles located in different cavities. It has been found that the mixed weighted cluster state can be produced. Our paper describes a better way toward creating the four-mode pure CV cluster states by dissipation. This has the experimental advantage that the coupling by dissipation can create entanglement. Meantime, a CV cluster state is a universal quantum computing resource, and the ability to create a cluster state by the dissipative coupling of distant cavities may be utilized in quantum networks. Up to now, only the four-mode cluster CV state in optics setting has been realized. There is no successful generation of the four-mode CV cluster state among atomic ensembles in experiment. Thus we believe that our present results could be benefit for the CV quantum information based on atomic ensembles.

## Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant Nos. 60878004 and 11074087), the Natural Science Foundation of Hubei Province (Grant No. 2010CDA075), the Nature Science Foundation of Wuhan City (Grant No. 201150530149), and the National Basic Research Program of China(Grant No. 2012CB921602

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