## Creation of four-mode weighted cluster states with atomic ensembles in high-Q ring cavities |

Optics Express, Vol. 20, Issue 3, pp. 3176-3191 (2012)

http://dx.doi.org/10.1364/OE.20.003176

Acrobat PDF (1243 KB)

### 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

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*et al.*[6

6. X. Su, A. Tan, X. Jia, J. Zhang, C. Xie, and K. Peng, “Experimental preparation of quadripartite cluster and Greenberger-Horne-Zeilinger entangled states for continuous variables,” Phys. Rev. Lett. **98**, 070502 (2007). [CrossRef] [PubMed]

*et al.*[8

8. M. Yukawa, R. Ukai, P. van Loock, and A. Furusawa, “Experimental generation of four-mode continuous-variable cluster states,” Phys. Rev. A **78**, 012301 (2008). [CrossRef]

11. M. Gu, C. Weedbrook, N. C. Menicucci, T. C. Ralph, and P. van Loock, “Quantum computing with continuous-variable clusters,” Phys. Rev. A **79**, 062318 (2009). [CrossRef]

*et al.*[13

13. N. C. Menicucci, X. Ma, and T. C. Ralph, “Arbitrarily large continuous-variable cluster states from a single quantum nondemolition gate,” Phys. Rev. Lett. **104**, 250503 (2010). [CrossRef] [PubMed]

*et al.*[9

9. H. Zaidi, N. C. Menicucci, S. T. Flammia, R. Bloomer, M. Pysher, and O. Pfister, “Entangling the optical frequency comb: simultaneous generation of multiple 2 × 2 and 2 × 3 continuous-variable cluster states in a single optical parametric oscillator,” Laser Phys. **18**, 659 (2008). [CrossRef]

10. M. Pysher, Y. Miwa, R. Shahrokhshahi, R. Bloomer, and O. Pfister, “Parallel generation of quadripartite cluster entanglement in the optical frequency comb,” Phys. Rev. Lett. **107**, 030505 (2011). [CrossRef] [PubMed]

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16. R. Ukai, N. Iwata, Y. Shimokawa, S. C. Armstrong, A. Politi, J. Yoshikawa, P. van Loock, and A. Furusawa, “Demonstration of unconditional one-way quantum computations for continuous variables,” Phys. Rev. Lett. **106**, 240504 (2011). [CrossRef] [PubMed]

18. L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature (London) **414**, 413–418 (2001). [CrossRef]

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29. J. I. Cirac, A. S. Parkins, R. Blatt, and P. Zoller, “”Dark” squeezed states of the motion of a trapped ion,” Phys. Rev. Lett. **70**, 556 (1993). [CrossRef] [PubMed]

32. J. T. Barreiro, P. Schindler, O. Gühne, T. Monz, M. Chwalla, C. F. Roos, M. Hennrich, and R. Blatt,“Experimental multiparticle entanglement dynamics induced by decoherence,” Nat. Phys. **6**, 943–946 (2010). [CrossRef]

*Q*ring cavity composed of two co-propagating modes has been proposed [21

21. A. S. Parkins, E. Solano, and J. I. Cirac,“Unconditional two-mode squeezing of separated atomic ensembles,” Phys. Rev. Lett. **96**, 053602 (2006). [CrossRef] [PubMed]

*et al.*[26

26. H. Krauter, C. A. Muschik, K. Jensen, W. Wasilewski, J. M. Petersen, J. I. Cirac, and E. S. Polzik, “Entanglement generated by dissipation and steady state entanglement of two macroscopic objects,” Phys. Rev. Lett. **107**, 080503 (2011). [CrossRef] [PubMed]

*et al.*[25

25. C. A. Muschik, E. S. Polzik, and J. I. Cirac, “Dissipatively driven entanglement of two macroscopic atomic ensembles,” Phys. Rev. A **83**, 052312 (2011). [CrossRef]

*et al.*[26

26. H. Krauter, C. A. Muschik, K. Jensen, W. Wasilewski, J. M. Petersen, J. I. Cirac, and E. S. Polzik, “Entanglement generated by dissipation and steady state entanglement of two macroscopic objects,” Phys. Rev. Lett. **107**, 080503 (2011). [CrossRef] [PubMed]

9. H. Zaidi, N. C. Menicucci, S. T. Flammia, R. Bloomer, M. Pysher, and O. Pfister, “Entangling the optical frequency comb: simultaneous generation of multiple 2 × 2 and 2 × 3 continuous-variable cluster states in a single optical parametric oscillator,” Laser Phys. **18**, 659 (2008). [CrossRef]

10. M. Pysher, Y. Miwa, R. Shahrokhshahi, R. Bloomer, and O. Pfister, “Parallel generation of quadripartite cluster entanglement in the optical frequency comb,” Phys. Rev. Lett. **107**, 030505 (2011). [CrossRef] [PubMed]

33. H. J. Kimble, “The quantum internet,” Nature (London) **453**, 1023–1030 (2008). [CrossRef]

34. N. Sangouard, C. Simon, H. de Riedmatten, and N. Gisin, “Quantum repeaters based on atomic ensembles and linear optics,” Rev. Mod. Phys. **83**, 33–80 (2011). [CrossRef]

35. J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett. **78**, 3221 (1997). [CrossRef]

39. J. DiGuglielmo, A. Samblowski, B. Hage, C. Pineda, J. Eisert, and R. Schnabel, “Experimental unconditional preparation and detection of a continuous bound entangled state of light,” Phys. Rev. Lett. **107**, 240503 (2011). [CrossRef]

40. P.-B. Li and F.-L. Li, “Deterministic generation of multiparticle entanglement in a coupled cavity-fiber system,” Opt. Express **19**, 1207–1216 (2011). [CrossRef] [PubMed]

41. S. B. Zheng, Z. B. Yang, and Y. Xia, “Generation of two-mode squeezed states for two separated atomic ensembles via coupled cavities,” Phys. Rev. A **81**, 015804 (2010). [CrossRef]

35. J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett. **78**, 3221 (1997). [CrossRef]

39. J. DiGuglielmo, A. Samblowski, B. Hage, C. Pineda, J. Eisert, and R. Schnabel, “Experimental unconditional preparation and detection of a continuous bound entangled state of light,” Phys. Rev. Lett. **107**, 240503 (2011). [CrossRef]

*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

*Q*ring cavity, as illustrated in Fig. 1. The ring cavity is composed of two modes

*a*and

*b*with frequencies

*ω*and

_{a}*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}42. L.-M. Duan, J. I. Cirac, and P. Zoller, “Three-dimensional theory for interaction between atomic ensembles and free-space light,” Phys. Rev. A **66**, 023818 (2002). [CrossRef]

*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

*〉, |1*

_{jn}*〉, 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

*〉 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

*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, are the coupling strengths of the effective two-level system to the cavity modes, and

*Ĵ*,

_{zn}*Ĵ*

_{±n}are collective atomic operators for the

*n*-th atomic ensemble, defined as

*〉} in ensembles*

_{jn}*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.,

43. T. Holstein and H. Primakoff, “Field dependence of the intrinsic domain magnetization of a ferromagnet,” Phys. Rev. **58**, 1098–1113 (1940). [CrossRef]

*J*= −

_{zn}*N*/2 and

_{n}*c*and

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

16. R. Ukai, N. Iwata, Y. Shimokawa, S. C. Armstrong, A. Politi, J. Yoshikawa, P. van Loock, and A. Furusawa, “Demonstration of unconditional one-way quantum computations for continuous variables,” Phys. Rev. Lett. **106**, 240504 (2011). [CrossRef] [PubMed]

*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*.

*S*|0

_{c1}, 0

_{c2}, 0

_{c3}, 0

_{c4}〉, where

*S*is a four-mode squeezing operator and

*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. In terms of the transformed operators, the squeezing operator (13) takes the form where

*S*

_{14}(

*λε*) and

*S*

_{23}(

*ε*/

*λ*) are standard two-mode squeezing operators [44

44. J. S. Peng and G. X. Li, *Introduction to Modern Quantum Optics* (World Scientific, 1998). [CrossRef]

*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 Evidently, the special choice of the phases and the Rabi frequencies is used to match the transformation defined as Eq. (14). Consequently, the effective Hamiltonian (16) can then be written as

*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 The above equation shows that the system behaves as a set of damped and linearly coupled harmonic oscillators. In order to ensure that the system decays to a stable steady state, we calculate the eigenvalues of Eq. (densmatr1) and find [45

45. G.-X. Li, H.-T. Tan, and S.-P. Wu, “Motional entanglement for two trapped ions in cascaded optical cavities,” Phys. Rev. A **70**, 064301 (2004). [CrossRef]

46. G.-X. Li, “Generation of pure multipartite entangled vibrational states for ions trapped in a cavity,” Phys. Rev. A **74**, 055801 (2006). [CrossRef]

*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 where

*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

*ρ*̃

_{2}=

*S*

_{14}(

*ελ*)

*S*

_{23}(

*ε*/

*λ*)

*TρT*

^{†}

*S*

_{23}(−

*ε*/

*λ*)

*S*

_{14}(−

*ελ*), with the squeezing parameter as

*e*

_{1},

*e*

_{4}):

*ρ̃*

_{2}= |

*ψ̃*〉 〈

*ψ̃*|, where |

*ψ̃*〉 = |0

_{e1}, 0

_{e2}, 0

_{e3}, 0

_{e4}, 0

*, 0*

_{a}*〉 is the vacuum state of the transformed system.*

_{b}*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 where the squeezing operator

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

*et al.*[47

47. S. L. W. Midgley, M. K. Olsen, A. S. Bradley, and O. Pfister,“Analysis of a continuous-variable quadripartite cluster state from a single optical parametric oscillator,” Phys. Rev. A **82**, 053826 (2010). [CrossRef]

*ψ*〉 is an example of a continuous-variable cluster state. To show this, we introduce the quadrature amplitude

8. M. Yukawa, R. Ukai, P. van Loock, and A. Furusawa, “Experimental generation of four-mode continuous-variable cluster states,” Phys. Rev. A **78**, 012301 (2008). [CrossRef]

9. H. Zaidi, N. C. Menicucci, S. T. Flammia, R. Bloomer, M. Pysher, and O. Pfister, “Entangling the optical frequency comb: simultaneous generation of multiple 2 × 2 and 2 × 3 continuous-variable cluster states in a single optical parametric oscillator,” Laser Phys. **18**, 659 (2008). [CrossRef]

*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.

*N*-mode weighted [9

**18**, 659 (2008). [CrossRef]

8. M. Yukawa, R. Ukai, P. van Loock, and A. Furusawa, “Experimental generation of four-mode continuous-variable cluster states,” Phys. Rev. A **78**, 012301 (2008). [CrossRef]

*N*linear combinations of quadrature operators

*X*and

_{j}*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.

*et al.*[25

25. C. A. Muschik, E. S. Polzik, and J. I. Cirac, “Dissipatively driven entanglement of two macroscopic atomic ensembles,” Phys. Rev. A **83**, 052312 (2011). [CrossRef]

*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

*a*and

_{i}*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.

*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

*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 where

*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.

*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 with

*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

*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.

49. L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller,“Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. **84**, **2722** (2000). [CrossRef] [PubMed]

50. R. Simon, “Peres-Horodecki separability criterion for continuous variable systems,” Phys. Rev. Lett. **84**, 2726–2729 (2000). [CrossRef] [PubMed]

*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

*V*′Γ +

*i*Λ/2, which determines the non-separability of the pair of modes (1,2) and (3,4), is found to be where

*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}).

*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}*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,

*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

38. J. Zhang, “Continuous-variable multipartite unlockable bound entangled Gussian states,” Phys. Rev. A **83**, 052327 (2011). [CrossRef]

39. J. DiGuglielmo, A. Samblowski, B. Hage, C. Pineda, J. Eisert, and R. Schnabel, “Experimental unconditional preparation and detection of a continuous bound entangled state of light,” Phys. Rev. Lett. **107**, 240503 (2011). [CrossRef]

*et al.*[10

10. M. Pysher, Y. Miwa, R. Shahrokhshahi, R. Bloomer, and O. Pfister, “Parallel generation of quadripartite cluster entanglement in the optical frequency comb,” Phys. Rev. Lett. **107**, 030505 (2011). [CrossRef] [PubMed]

*et al.*[36

36. A. Serafini, S. Mancini, and S. Bose, “Distributed quantum computation via optical fibers,” Phys. Rev. Lett. **96**, 010503 (2006). [CrossRef] [PubMed]

36. A. Serafini, S. Mancini, and S. Bose, “Distributed quantum computation via optical fibers,” Phys. Rev. Lett. **96**, 010503 (2006). [CrossRef] [PubMed]

40. P.-B. Li and F.-L. Li, “Deterministic generation of multiparticle entanglement in a coupled cavity-fiber system,” Opt. Express **19**, 1207–1216 (2011). [CrossRef] [PubMed]

*E*, the variances

_{n}*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.

*η*of the cavity modes. We see that a decrease of the coupling efficiency leads to a rapid increase of the variances

*V*. However, the entanglement is less sensitive to the coupling efficiency

_{j}*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

## Acknowledgments

## References and links

1. | H. J. Briegel and R. Raussendorf, “Persistent entanglement in arrays of interacting particles,” Phys. Rev. Lett. |

2. | R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett. |

3. | J. Zhang and S. L. Braunstein, “Continuous-variable Gaussian analog of cluster states,” Phys. Rev. A |

4. | N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, “Universal quantum computation with continuous-variable cluster states,” Phys. Rev. Lett. |

5. | N. C. Menicucci, S. T. Flammia, and O. Pfister, “One-way quantum computing in the optical frequency comb,” Phys. Rev. Lett. |

6. | X. Su, A. Tan, X. Jia, J. Zhang, C. Xie, and K. Peng, “Experimental preparation of quadripartite cluster and Greenberger-Horne-Zeilinger entangled states for continuous variables,” Phys. Rev. Lett. |

7. | P. van Loock, C. Weedbrook, and M. Gu, “Building Gaussian cluster states by linear optics,” Phys. Rev. A |

8. | M. Yukawa, R. Ukai, P. van Loock, and A. Furusawa, “Experimental generation of four-mode continuous-variable cluster states,” Phys. Rev. A |

9. | H. Zaidi, N. C. Menicucci, S. T. Flammia, R. Bloomer, M. Pysher, and O. Pfister, “Entangling the optical frequency comb: simultaneous generation of multiple 2 × 2 and 2 × 3 continuous-variable cluster states in a single optical parametric oscillator,” Laser Phys. |

10. | M. Pysher, Y. Miwa, R. Shahrokhshahi, R. Bloomer, and O. Pfister, “Parallel generation of quadripartite cluster entanglement in the optical frequency comb,” Phys. Rev. Lett. |

11. | M. Gu, C. Weedbrook, N. C. Menicucci, T. C. Ralph, and P. van Loock, “Quantum computing with continuous-variable clusters,” Phys. Rev. A |

12. | N. C. Menicucci, S. T. Flammia, H. Zaidi, and O. Pfister,“Ultracompact generation of continuous-variable cluster states,” Phys. Rev. A |

13. | N. C. Menicucci, X. Ma, and T. C. Ralph, “Arbitrarily large continuous-variable cluster states from a single quantum nondemolition gate,” Phys. Rev. Lett. |

14. | A. Tan, C. Xie, and K. Peng, “Quantum logical gates with linear quadripartite cluster states of continuous variables,” Phys. Rev. A |

15. | Y. Wang, X. Su, H. Shen, A. Tan, C. Xie, and K. Peng, “Toward demonstrating controlled-X operation based on continuous-variable four-partite cluster states and quantum teleporters,” Phys. Rev. A |

16. | R. Ukai, N. Iwata, Y. Shimokawa, S. C. Armstrong, A. Politi, J. Yoshikawa, P. van Loock, and A. Furusawa, “Demonstration of unconditional one-way quantum computations for continuous variables,” Phys. Rev. Lett. |

17. | J. Zhang, G. Adesso, C. Xie, and K. Peng, “Quantum teamwork for unconditional multiparty communication with gaussian states,” Phys. Rev. A |

18. | L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature (London) |

19. | M. Ying Wu, G. Payne, E. W. Hagley, and L. Deng, “Preparation of multiparty entangled states using pairwise perfectly efficient single-probe photon four-wave mixing,” Phys. Rev. A |

20. | C. W. Chou, H. de Riedmatten, D. Felinto, S. V. Polyakov, S. J. van Enk, and H. J. Kimble, “Measurement-induced entanglement for excitation stored in remote atomic ensembles,” Nature (London) |

21. | A. S. Parkins, E. Solano, and J. I. Cirac,“Unconditional two-mode squeezing of separated atomic ensembles,” Phys. Rev. Lett. |

22. | K. Hammerer, M. Aspelmeyer, E. S. Polzik, and P. Zoller, “Establishing Einstein-Poldosky-Rosen channels between nanomechanics and atomic ensembles,” Phys. Rev. Lett. |

23. | K. Hammerer, A. S. Sørensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. |

24. | K. Jensen, W. Wasilewski, H. Krauter, T. Fernholz, B. M. Nielsen, M. Owari, M. B. Plenio, A. Serafini, M. M. Wolf, and E. S. Polzik, “Quantum memory for entangled continuous-variable states,” Nat. Phys. |

25. | C. A. Muschik, E. S. Polzik, and J. I. Cirac, “Dissipatively driven entanglement of two macroscopic atomic ensembles,” Phys. Rev. A |

26. | H. Krauter, C. A. Muschik, K. Jensen, W. Wasilewski, J. M. Petersen, J. I. Cirac, and E. S. Polzik, “Entanglement generated by dissipation and steady state entanglement of two macroscopic objects,” Phys. Rev. Lett. |

27. | C. A. Muschik, H. Krauter, K. Hammerer, and E. S. Polzik, “Quantum information at the Interface of light with mesoscopic objects,” arXiv:1105.2947 (2011). |

28. | D.-C. Li, C.-H. Yuan, Z.-L. Cao, and W.-P. Zhang, “Storage and retrieval of continuous-variable polarization-entangled cluster states in atomic ensembles,” Phys. Rev. A |

29. | J. I. Cirac, A. S. Parkins, R. Blatt, and P. Zoller, “”Dark” squeezed states of the motion of a trapped ion,” Phys. Rev. Lett. |

30. | S. Diehl, A. Micheli, A. Kantian, B. Kraus, H. P. Büchler, and P. Zoller,“Quantum states and phases in driven open quantum systems with cold atoms,” Nat. Phys. |

31. | F. Verstraete, M. M. Wolf, and J. I. Cirac,“Quantum computation and quantum-state engineering
driven by dissipation,” Nat. Phys. |

32. | J. T. Barreiro, P. Schindler, O. Gühne, T. Monz, M. Chwalla, C. F. Roos, M. Hennrich, and R. Blatt,“Experimental multiparticle entanglement dynamics induced by decoherence,” Nat. Phys. |

33. | H. J. Kimble, “The quantum internet,” Nature (London) |

34. | N. Sangouard, C. Simon, H. de Riedmatten, and N. Gisin, “Quantum repeaters based on atomic ensembles and linear optics,” Rev. Mod. Phys. |

35. | J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett. |

36. | A. Serafini, S. Mancini, and S. Bose, “Distributed quantum computation via optical fibers,” Phys. Rev. Lett. |

37. | X.Y. Lv̈, L. G. Si, X. Y. Hao, and X. X. Yang, “Achieving multipartite entanglement of distant atoms through selective photon emission and absorption processes,” Phys. Rev. A |

38. | J. Zhang, “Continuous-variable multipartite unlockable bound entangled Gussian states,” Phys. Rev. A |

39. | J. DiGuglielmo, A. Samblowski, B. Hage, C. Pineda, J. Eisert, and R. Schnabel, “Experimental unconditional preparation and detection of a continuous bound entangled state of light,” Phys. Rev. Lett. |

40. | P.-B. Li and F.-L. Li, “Deterministic generation of multiparticle entanglement in a coupled cavity-fiber system,” Opt. Express |

41. | S. B. Zheng, Z. B. Yang, and Y. Xia, “Generation of two-mode squeezed states for two separated atomic ensembles via coupled cavities,” Phys. Rev. A |

42. | L.-M. Duan, J. I. Cirac, and P. Zoller, “Three-dimensional theory for interaction between atomic ensembles and free-space light,” Phys. Rev. A |

43. | T. Holstein and H. Primakoff, “Field dependence of the intrinsic domain magnetization of a ferromagnet,” Phys. Rev. |

44. | J. S. Peng and G. X. Li, |

45. | G.-X. Li, H.-T. Tan, and S.-P. Wu, “Motional entanglement for two trapped ions in cascaded optical cavities,” Phys. Rev. A |

46. | G.-X. Li, “Generation of pure multipartite entangled vibrational states for ions trapped in a cavity,” Phys. Rev. A |

47. | S. L. W. Midgley, M. K. Olsen, A. S. Bradley, and O. Pfister,“Analysis of a continuous-variable quadripartite cluster state from a single optical parametric oscillator,” Phys. Rev. A |

48. | C. W. Gardiner and P. Zoller, |

49. | L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller,“Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. |

50. | R. Simon, “Peres-Horodecki separability criterion for continuous variable systems,” Phys. Rev. Lett. |

**OCIS Codes**

(270.0270) Quantum optics : Quantum optics

(270.6570) Quantum optics : Squeezed states

(270.5585) Quantum optics : Quantum information and processing

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: October 14, 2011

Revised Manuscript: December 23, 2011

Manuscript Accepted: January 20, 2012

Published: January 26, 2012

**Citation**

Li-hui Sun, Yan-qin Chen, and Gao-xiang Li, "Creation of four-mode weighted cluster states with atomic ensembles in high-Q ring cavities," Opt. Express **20**, 3176-3191 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-3-3176

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### References

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- J. T. Barreiro, P. Schindler, O. Gühne, T. Monz, M. Chwalla, C. F. Roos, M. Hennrich, and R. Blatt,“Experimental multiparticle entanglement dynamics induced by decoherence,” Nat. Phys.6, 943–946 (2010). [CrossRef]
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- J. Zhang, “Continuous-variable multipartite unlockable bound entangled Gussian states,” Phys. Rev. A83, 052327 (2011). [CrossRef]
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