OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 3 — Jan. 30, 2012
  • pp: 3176–3191
« Show journal navigation

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

Li-hui Sun, Yan-qin Chen, and Gao-xiang Li  »View Author Affiliations


Optics Express, Vol. 20, Issue 3, pp. 3176-3191 (2012)
http://dx.doi.org/10.1364/OE.20.003176


View Full Text Article

Acrobat PDF (1243 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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

1. H. J. Briegel and R. Raussendorf, “Persistent entanglement in arrays of interacting particles,” Phys. Rev. Lett. 86, 910 (2001). [CrossRef] [PubMed]

], have been proposed as a potential resource for performing one-way quantum computation [2

2. R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett. 86, 5188 (2001). [CrossRef] [PubMed]

]. In 2006, this concept was extended by Zhang and Braunstein [3

3. J. Zhang and S. L. Braunstein, “Continuous-variable Gaussian analog of cluster states,” Phys. Rev. A 73, 032318 (2006). [CrossRef]

] 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

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. 97, 110501 (2006). [CrossRef] [PubMed]

, 5

5. N. C. Menicucci, S. T. Flammia, and O. Pfister, “One-way quantum computing in the optical frequency comb,” Phys. Rev. Lett. 101, 130501 (2008). [CrossRef] [PubMed]

], 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

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]

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]

]. For example, Su 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]

] and Furusawa 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

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]

] 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

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]

] 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

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]

] 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

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]

]. 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

14. A. Tan, C. Xie, and K. Peng, “Quantum logical gates with linear quadripartite cluster states of continuous variables,” Phys. Rev. A 79, 042338 (2009). [CrossRef]

,15

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 81, 022311 (2010). [CrossRef]

]. 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

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]

]. The CV square four-mode cluster state is proposed to realize the quantum teamwork [17

17. J. Zhang, G. Adesso, C. Xie, and K. Peng, “Quantum teamwork for unconditional multiparty communication with gaussian states,” Phys. Rev. A 103, 070501 (2009).

] 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

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]

28

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 84, 022328 (2011). [CrossRef]

]. 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

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

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]

]. 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

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]

]. 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

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]

, 27

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).

] 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

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]

] 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

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]

, 27

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).

] 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

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

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]

].

On the other hand, remote entanglement among distant atomic ensembles is an essential ingredient source for quantum networks based on the atomic ensembles [33

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

, 34

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]

]. The coupled atom-cavity-fiber systems are considered as a basic building blocks toward scalable quantum network [35

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

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]

]. Schemes [40

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

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]

] 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

17. J. Zhang, G. Adesso, C. Xie, and K. Peng, “Quantum teamwork for unconditional multiparty communication with gaussian states,” Phys. Rev. A 103, 070501 (2009).

] 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

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

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]

] and quantum team-work [17

17. J. Zhang, G. Adesso, C. Xie, and K. Peng, “Quantum teamwork for unconditional multiparty communication with gaussian states,” Phys. Rev. A 103, 070501 (2009).

].

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 ωa and ωb. Actually in the ring cavity, the mode 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 ωa and ωb can work in the two-mode approximation for the case of trapped room-temperature atomic ensembles [42

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]

].

Fig. 1 (a) A scheme for creation of four-mode cluster states involving a single high-Q ring cavity and four atomic ensembles. The ensembles are driven by pulse lasers of suitable chosen Rabi frequencies and phases. (b) Atomic energy levels and coupling configurations of the lasers and the cavity modes for ensembles n = 1,2 (left) and n = 3,4 (right).

The four ensembles contain, respectively, N1, N2, N3 and N4 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 |0jn〉, |1jn〉, and two excited states |2jn〉 and |3jn〉. The parameters Ω2n and Ω3n are Rabi frequencies of external pulse lasers which are coupled to the transitions |1jn〉 → |2jn〉 and |0jn〉 → |3jn〉, respectively. The cavity modes couple to the transitions |1jn〉 → |3jn〉 and |0jn〉 →|2jn〉 with the coupling strength gan and gbn, respectively.

2.1. Effective Hamiltonian

The Hamiltonian of the system, written in the rotating-wave approximation has the following form
H=H0+HAL+HAC,
(1)
where
H0=ωaaa+ωbbb+n=12j=1Nn(ω1|1jn1jn|+ω2|2jn2jn|+ω3|3jn3jn|)+n=34j=1Nn(ω1|0jn0jn|+ω2|2jn2jn|+ω3|3jn3jn|)
(2)
is the free Hamiltonian of the cavity modes and the atoms. Here, we have set the energy of the ground state |0jn〉 equal to zero in the ensembles n = 1,2, and in the ensembles n = 3,4 we have chosen the state |1jn〉 as the ground state with the energy equal to zero, and the energies of the other three states are correspondingly denoted as ω1, ω2 and ω3.
HAL=12n=12j=1Nn{Ω2n(xjn)exp[i(ωL2tϕ2n)]|2jn1jn|+Ω3n(xjn)exp[i(ωL3tϕ3n)]|3jn0jn|+H.c.}+12n=34j=1Nn{Ω2n(xjn)exp[i(ωL2tϕ2n)]|2jn0jn|+Ω3n(xjn)exp[i(ωL3tϕ3n)]|3jn1jn|+H.c.}
(3)
is the interaction Hamiltonian between the atoms and the driving fields, here the parameters Ω2n(xjn) and Ω3n(xjn) are the position dependent Rabi frequencies of the driving laser fields, and ϕ2n and ϕ3n are their phases, respectively.
HAC=n=12j=1Nn[gan(xjn)|2jn0jn|a+gbn(xjn)|3jn1jn|b+H.c.]+n=34j=1Nn[gan(xjn)|2jn1jn|a+gbn(xjn)|3jn0jn|b+H.c.]
(4)
is the interaction Hamiltonian between the atoms and the two cavity modes. Here the coupling constants of the atomic transitions to the cavity fields, gan(xjn) and gbn(xjn), are also dependent on the atomic position xjn. In what follows, we will assume all the wave numbers of the laser fields are nearly equal and denoted by k, then the plane traveling wave representation for the laser fields and the cavity modes, in which
Ω2n(xjn)=Ω2nexp(ikxjn),Ω3n(xjn)=Ω3nexp(ikxjn),gan(xjn)=ganexp(ikxjn),gbn(xjn)=gbnexp(ikxjn).
(5)

We now make a unitary transformation and assume that the laser frequencies ωL2, ωL3 satisfy the resonance condition ωL2 + ω1 = ωL3ω1, and the detunings of the laser fields from the atomic transition frequencies Δ2n = ω2 – (ωL2 + ω1), Δ3n = ω3ωL3 are much larger than the Rabi frequencies and the cavity coupling strengths Δ2n, Δ3n ≫ Ω2n, Ω3n, gan, gbn. 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
Heff=δaaa+δbbb+n=12gan2Δ2n(12NnJzn)aa+n=34gan2Δ2n(12Nn+Jzn)aa+[n=12gbn2Δ3n(12Nn+Jzn)+n=34gbn2Δ3n(12NnJzn)]bb+[n=12β2nexp(iϕ2n)Jn+n=34β2nexp(iϕ2n)J+n]a+H.c.+[n=12β3nexp(iϕ3n)J+n+n=34β3nexp(iϕ3n)Jn]b+H.c.,
(6)
where δa = ωaωL3 + ω1 and δb = ωb – ωL3 + ω1 are the detunings of the cavity frequencies from the Raman coupling resonance,
β2n=NnΩ2ngan2Δ2n,β3n=NnΩ3ngbn2Δ3n,
(7)
are the coupling strengths of the effective two-level system to the cavity modes, and Ĵzn, Ĵ±n are collective atomic operators for the n-th atomic ensemble, defined as
Jzn=12j=1Nn(|1jn1jn||0jn0jn|),J+n=j=1Nn|1jn0jn|,Jn=j=1Nn|0jn1jn|.
(8)

Taking into account of large detunings of the driving fields, we can assume that the excitation probability of the atoms to the states {|1jn〉} in ensembles N1 and N2, and the excitation probability of the atoms to the states {|0jn〉} in ensembles N3 and N4 are much smaller than the total number of atoms, i.e., cncnNn. In this case, the collective atomic operators in the Holstein-Primakoff representation [43

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

] can be well approximated by Jzn = −Nn/2 and Jn=Nncn. Here, the operators cn and cn obey the standard bosonic commutation relation, [cn,cn]=1. Then, if we choose the detunings such as
δa+n=12gan2NnΔ2n=δb+n=34gbn2NnΔ3n=0,
(9)
we arrive to a simple form of the effective Hamiltonian
Heff=[n=12β2nexp(iϕ2n)cn+n=34β2nexp(iϕ2n)cn]a+H.c.+[n=12β3nexp(iϕ3n)cn+n=34β3nexp(iϕ3n)cn]b+H.c..
(10)

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

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]

], 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
ρ˙=i[Heff,ρ]+𝒧cρ,
(11)
in which
𝒧cρ=κaD[a]ρ+κbD[b]ρ,
(12)
characterizing the damping of the cavity modes a and b, with rates κa and κb, respectively. For simplicity, we set κa = κb = κ in the following. Here, D[ϑ]ρ ≡ 2ϑρϑϑϑρρϑϑ, 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|0c1, 0c2, 0c3, 0c4〉, where S is a four-mode squeezing operator
S=exp[ε(c1c4+c2c3+c2c4)H.c.],
(13)
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 cn into new operators en = TcnT, which are linear combinations of the operators of different atomic ensembles, i.e.
e1=c1+λc21+λ2,e2=λc1+c21+λ2,e3=λc3c41+λ2,e4=c3+λc41+λ2.
(14)
In terms of the transformed operators, the squeezing operator (13) takes the form
S=TST=exp(λεe1e4ελe2e3H.c.)=S14(λε)S23(ε/λ),
(15)
where λ=(5+1)/2, S14(λε) and S23(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]

], each involving only a pair of the bosonic modes, (e1, e4) and (e2, e3), respectively. Since the pairs of modes (e1, e4) and (e2, e3) 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 T1 are sent to drive the four atomic ensembles, whose Rabi frequencies and phases are specially chosen as: ϕ21 = ϕ23 = ϕ32 = ϕ34 = π, ϕ22 = ϕ24 = ϕ31 = ϕ33 = 0, β21 = λβ22, β23 = λβ24, β31 = λβ32, β33 = λβ34, β34 = β22 and β32 = β24. With this choice of the Rabi frequencies and phases, the Hamiltonian (10) takes the form
Heff1=a(λβ22c1+β22c2λβ24c3+β24c4)+b(λβ32c1β32c2+λβ34c3β34c4)+H.c.
(16)
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
THeff1T=(λ2+1)12[a(β22e2β24e3)+b(β32e2+β34e3)]+H.c.
(17)

We see that the Hamiltonian represents a simple system of two independent linear mixers, where the bosonic modes e2 and e3 linearly couple to the cavity modes a and b, respectively. The master equation of the transformed density operator takes the form
ddtρ˜1=i[H˜eff1,ρ˜1]+κD[a]ρ˜1+κD[b]ρ˜1,
(19)
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

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

]
η±=κ±[κ24(λ2+1)(β342β322)]12,
(20)
As long as 4(λ2+1)(β342β322)>κ2, both eigenvalues have negative real parts. Under this condition, if the duration T1 of the driving laser pulses is sufficiently long, for example, T1 ∼ 1/κ, the cavity dissipative relaxation will force the system to be prepared in a stationary state, in which all the modes a, b, e2 and e3 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, e2, b and e3 will be found in the vacuum state
ρ˜1=|0e2,0e3,0a,0b0e2,0e3,0a,0b|ρe1,e4(T1),
(21)
where ρe1,e4 (T1) 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 e2 and e3 have been prepared in a pure state through the couplings between modes a and e2, and b and e3. Because all the four combined modes en are orthogonal to each other, we can leave the two modes e2 and e3 decoupled with the cavity modes and prepare the modes e1 and e4 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 e1, and b linearly coupled to e4, 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 e2 and e3, and sent another series of pulses of driving lasers in time interval T2 with the specifically chosen phases and the Rabi frequencies: ϕ2i = ϕ3i = 0 (i = 1, 2, 3, 4), β22 = λβ21, β24 = λβ23, β32 = λβ31, β34 = λβ33, β33 = β21 and β31 = β23. In this case, the effective Hamiltonian (10) takes the form
Heff2=a(β21c1+λβ21c2+β23c3+λβ23c4)+b(β31c1+λβ31c2+β33c3+λβ33c4)+H.c.
(22)

If we now make the same unitary transformation as Eq. (14) and a two-mode squeezing transformation on the density operator ρ̃2 = S14(ελ)S23(ε/λ)TρTS23(−ε/λ)S14(−ελ), with the squeezing parameter as λε=12ln(β33+β31β33β31), we obtain the master equation
ddtρ˜2=i[H˜eff2,ρ˜2]+κD[a]ρ˜2+κD[b]ρ˜2,
(23)
with the Hamiltonian involving only the pair of the collective modes (e1, e4):
H˜eff2=[(λ2+1)(β212β232)]12(ae1+be4)+H.c.
(24)

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 ρ̃2 = |ψ̃〉 〈ψ̃|, where |ψ̃〉 = |0e1, 0e2, 0e3, 0e4, 0a, 0b〉 is the vacuum state of the transformed system.

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
|ψ=S|0c1,0c2,0c3,0c4,
(25)
where the squeezing operator 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

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]

] 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 Xj=(cj+cj)/2 and phase Pj=i(cjcj)/2 components of the four modes involved, and find that the variances of the linear combinations according to the definition of cluster states [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]

, 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]

], are
V1=V(P115P3+25P4),V2=V(X3+15X125X2),V3=V(P2+25P3+15P4),V4=V(X425X115X2),
(26)
with
V1=V2=1λ2+1(λ2eε/λ+eλε),V3=V4=1λ2+1(λ2eλε+eε/λ).
(27)
Obviously, we can find the variances can be smaller than one, Vj < 1 (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 (XP,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 2N-mode weighted [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]

] or unweighted cluster states [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]

] that all the variances of 2N linear combinations of quadrature operators Xj and Pj should be below the vacuum noise limit, i.e. should be squeezed. This means we can construct 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 2N-mode cluster state. Similarly, for the preparation of the 2N + 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 2N + 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

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]

], 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 e2 and e3 modes as we discussed in step 1, the single-particle heating and cooling noises are unavoidable as explained in Ref. [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]

]. 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 e2 and e3 interact with the two cavity modes a and b respectively, these two cavity modes can simultaneously drag the e2 and e3 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 ej modes commute with each other, the preparation of the e1 and e4 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

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]

]. 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 g2/κ (where gk). Thus, the system would reach a steady state at time κ/g2, 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 ai and bi (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 N1 and N2 in the left cavity are coupled to modes a1 and b1, whereas the ensembles N3 and N4 in the right cavity are coupled to modes a2 and b2, respectively. External pulse lasers with the Rabi frequencies Ω21 and Ω22, applied to the left cavity, couple to the clockwise mode a1, while the lasers with the Rabi frequencies Ω31 and Ω32 couple to the anti-clockwise mode b1. Correspondingly, in the right cavity, laser pulses of the Rabi frequencies Ω23 and Ω24 couple to the anti-clockwise mode a2, while the laser pulses of the Rabi frequencies Ω33 and Ω34 couple to the clockwise mode b2. As in the first scheme, the laser pulses interact dispersively with the atoms.

Fig. 2 The scheme for creation of cluster states with four atomic ensembles in two cascade ring cavities each containing two atomic ensembles. In the left cavity mode b1 propagates anticlockwise direction but a1 propagates clockwise direction. In the right cavity mode b2 propagates clockwise direction but a2 propagates anticlockwise direction.

Note that the dissipative parts of the master equation contain coupling between the cavity modes arranged in the cascade way that the mode a2 couples to mode a1 from right cavity into left cavity, but the mode b1 couples to mode b2 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 a2 couples to the mode a1 and the mode b1 couples to the mode b2 with no feedback coupling. In this case, the effective Hamiltonian takes the form
Heff=(n=12β2ncna1+n=34β2ncna2+H.c.)+(n=12β3ncnb1+n=34β3ncnb2+H.c.),
(28)
where a1, b1 and a2, b2 are the creation operators of the modes in the left cavity and the right cavity, respectively.

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

48. C. W. Gardiner and P. Zoller, Quantum Noise (Springer-Verlag, 2000).

]
ρ˙=i[H˜eff,ρ]+𝒧caρ+𝒧cbρ,
(29)
with
𝒧caρ=κj=12(2ajρajajajρρajaj)2ηκ([a1,a2ρ]+[ρa2,a1])𝒧cbρ=κj=12(2bjρbjbjbjρρbjbj)2ηκ([b2,b1ρ]+[ρb1,b2]),
(30)
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.

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
H˜eff=THeffT=H1+H2,
(31)
where
H1=β1b2e3+β2b1e2+H.c.,H2=β1a1e1+β2a2e4+H.c..
(32)
This shows that the dynamics of the pair (e2, e3) of the modes is independent of the dynamics of the pair (e1, e4). Here β1 and β2 are real and are determined by the following relations
β21=β34=β1cosθ,β22=β33=β1sinθ,β23=β32=β2cosθ,β24=β31=β2sinθ,
(33)
with θ=arccos[(1+λ2)12]. Obviously, we can see from Eq. (32) that the effective Hamiltonian describes two different kinds of interaction processes. Two nondegenerate parametric amplification processes occur within (b1, e2) and (a2, e4), and two linear mixing processes happen within (b2, e3) and (a1, e1). 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 (c1, c2) and (c3, c4) 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

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

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

]. The criterion states that a four-mode Gaussian state is separable, if and only if
ΓVΓ+i2Λ0,
(34)
where V′ is an 8×8 covariance matrix expressed by use of the variables Xj and Pj (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., XT → ΓXT = (X1, P1, X2, P2, X3, – P3, X4, – P4). The matrix Λ is a block diagonal matrix with the blocks given by the 2 × 2 matrix
σ=(0110).
(35)

The minimum negative eigenvalue of ΓV′Γ + iΛ/2, which determines the non-separability of the pair of modes (1,2) and (3,4), is found to be
En=n1+n2[(n1n2)2+4n32]122,
(36)
where n1=e2e2=e4e4=d121,n2=e3e3=e1e1=d22, are average numbers of photons in the modes, n3 = 〈e2e3〉= 〈e1e4〉= −d1d2 are correlations between the modes, and can be obtained from the Eqs. (29)(31)
d1=12Ω[s+exp(12st)sexp(12s+t)],d2=2κβ1β2ηβ12+β22[exp(12s+t)Ωexp(12st)Ω+exp(12st)Ωexp(12s+t)Ω],
(37)
with
s±=κ±Ω,s±=κ±Ω,Ω=κ2+4β22,Ω=κ24β12,
(38)
where we have assumed all the modes are initially in vacuum states. Now, it is easy to find from Eq. (36), that the atomic ensembles are entangled whenever
[e2e2e3e3]12<|e2e3|.
(39)
This shows that nonclassical correlations can be established between the collective modes e2 and e3. Due to the nondegenerate parametric amplification process between the modes e2 and b1, photons generated in the mode b1 have a strong nonclassical correlation with the mode e2. The photon in the mode b1 can be transferred to the right cavity and established a correlation with the mode b2 through the cascaded dissipative process. With the help of the laser pulses, linear mixing processes occur between mode b2 and mode e3. As the result, an entanglement between the modes e2 and e3 can be created through the interaction between the cavity modes b1 and b2. In the same manner, due to the nondegenerate parametric amplification process between modes e4 and a2, photons generated in the mode a2 can be transferred through the dissipative process of spontaneous emission into the mode a1 of the left cavity. Then, linear mixing processes in the left cavity can be established between the modes a1 and e1, resulting in an entanglement between the mode e4 and e1. Next, using the relations between the modes ej and cj (j = 1, 2, 3, 4), see Eq. (14), we can easily see that the entanglement between the modes (e1, e4) and (e2, e3) leads to entanglement between the pairs of the modes (c1, c2) and (c3, c4).

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 e2e2=e3e3 these two conditions merge, showing that the conditions for entanglement and squeezing are the same. However, the mode e2 interacts with the mode b1 in a parametric amplification way, the mode e3 is coupled to the mode b2 through linear mixing process. Thus, e2e2e3e3, which then results in a mixed state. The mixture of the obtained state can be characterized by the purity
Trρ2=[2(e2e2e3e3)+1]12.
(42)
When Trρ2 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

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]

], 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

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]

] 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

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

] 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

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

] and the multiatom entangled states [40

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]

] are achievable with the present technology.

Figure 3 shows the time evolution of the negativity En, the variances Vj, and the purity P for different values of the ratio α = β1/β2 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 Vj have the same tendency in time as the negativity En that they evolve towards the minimum value of Vj = 0 and at the same time the negativity En evolves towards its optimal negative value of En = −0.5. However, when α ≠ 1, the variances Vj have a different tendency from the negativity En. The variances are reduced below the vacuum limit only in a restricted time range and develop to large positive values at long times Γ1t ≫ 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 e2 and e3 interact differently with the cavity modes. The mode e2 is coupled yo the cavity mode b1 with parametric amplification, whereas the mode e3 is coupled to the cavity mode b2 via linear mixing, which then results in different numbers of excitations in the modes, i.e. e2e2e3e3. With the time evolution, the difference between e2e2 and e3e3 becomes large, which leads to the decrease of the purity. Also, the large difference e2e2e3e3 makes the squeezing condition Eq. (41) hard to be satisfied, so the squeezing will disappear.

Fig. 3 Time evolution of the negativity En, the variance Vj and the purity Trρ2 for η = 1,β2 = 0.1κ and β1 = αβ2 with α = 1 (solid line), α = 2 (dashed line), α = 4 (dotted line). The parameter Γ1 is defined as Γ1=β22/κ.

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 Vj. 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.

Fig. 4 Time evolution of the negativity En and the variance Vj for β1 = β2 = 0.1κ and different η: η = 1 (solid line), η = 0.8 (dashed line), η = 0.6 (dotted line).

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

References and links

1.

H. J. Briegel and R. Raussendorf, “Persistent entanglement in arrays of interacting particles,” Phys. Rev. Lett. 86, 910 (2001). [CrossRef] [PubMed]

2.

R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett. 86, 5188 (2001). [CrossRef] [PubMed]

3.

J. Zhang and S. L. Braunstein, “Continuous-variable Gaussian analog of cluster states,” Phys. Rev. A 73, 032318 (2006). [CrossRef]

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. 97, 110501 (2006). [CrossRef] [PubMed]

5.

N. C. Menicucci, S. T. Flammia, and O. Pfister, “One-way quantum computing in the optical frequency comb,” Phys. Rev. Lett. 101, 130501 (2008). [CrossRef] [PubMed]

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]

7.

P. van Loock, C. Weedbrook, and M. Gu, “Building Gaussian cluster states by linear optics,” Phys. Rev. A 76, 032321 (2007). [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]

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]

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]

12.

N. C. Menicucci, S. T. Flammia, H. Zaidi, and O. Pfister,“Ultracompact generation of continuous-variable cluster states,” Phys. Rev. A 76, 010302 (2007). [CrossRef]

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]

14.

A. Tan, C. Xie, and K. Peng, “Quantum logical gates with linear quadripartite cluster states of continuous variables,” Phys. Rev. A 79, 042338 (2009). [CrossRef]

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 81, 022311 (2010). [CrossRef]

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]

17.

J. Zhang, G. Adesso, C. Xie, and K. Peng, “Quantum teamwork for unconditional multiparty communication with gaussian states,” Phys. Rev. A 103, 070501 (2009).

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]

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 69, 063803 (2004). [CrossRef]

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) 438, 828–832 (2005). [CrossRef]

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]

22.

K. Hammerer, M. Aspelmeyer, E. S. Polzik, and P. Zoller, “Establishing Einstein-Poldosky-Rosen channels between nanomechanics and atomic ensembles,” Phys. Rev. Lett. 102, 020501 (2009). [CrossRef] [PubMed]

23.

K. Hammerer, A. S. Sørensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82, 1041–1093 (2010). [CrossRef]

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. 7, 13–16 (2011). [CrossRef]

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]

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]

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 84, 022328 (2011). [CrossRef]

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]

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. 4, 878–883 (2008). [CrossRef]

31.

F. Verstraete, M. M. Wolf, and J. I. Cirac,“Quantum computation and quantum-state engineering driven by dissipation,” Nat. Phys. 5, 633–636 (2009). [CrossRef]

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]

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]

36.

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

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 79, 052330 (2009). [CrossRef]

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]

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]

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]

43.

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

44.

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

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]

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]

48.

C. W. Gardiner and P. Zoller, Quantum Noise (Springer-Verlag, 2000).

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]

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


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. H. J. Briegel and R. Raussendorf, “Persistent entanglement in arrays of interacting particles,” Phys. Rev. Lett.86, 910 (2001). [CrossRef] [PubMed]
  2. R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett.86, 5188 (2001). [CrossRef] [PubMed]
  3. J. Zhang and S. L. Braunstein, “Continuous-variable Gaussian analog of cluster states,” Phys. Rev. A73, 032318 (2006). [CrossRef]
  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.97, 110501 (2006). [CrossRef] [PubMed]
  5. N. C. Menicucci, S. T. Flammia, and O. Pfister, “One-way quantum computing in the optical frequency comb,” Phys. Rev. Lett.101, 130501 (2008). [CrossRef] [PubMed]
  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]
  7. P. van Loock, C. Weedbrook, and M. Gu, “Building Gaussian cluster states by linear optics,” Phys. Rev. A76, 032321 (2007). [CrossRef]
  8. M. Yukawa, R. Ukai, P. van Loock, and A. Furusawa, “Experimental generation of four-mode continuous-variable cluster states,” Phys. Rev. A78, 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]
  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]
  11. M. Gu, C. Weedbrook, N. C. Menicucci, T. C. Ralph, and P. van Loock, “Quantum computing with continuous-variable clusters,” Phys. Rev. A79, 062318 (2009). [CrossRef]
  12. N. C. Menicucci, S. T. Flammia, H. Zaidi, and O. Pfister,“Ultracompact generation of continuous-variable cluster states,” Phys. Rev. A76, 010302 (2007). [CrossRef]
  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]
  14. A. Tan, C. Xie, and K. Peng, “Quantum logical gates with linear quadripartite cluster states of continuous variables,” Phys. Rev. A79, 042338 (2009). [CrossRef]
  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. A81, 022311 (2010). [CrossRef]
  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]
  17. J. Zhang, G. Adesso, C. Xie, and K. Peng, “Quantum teamwork for unconditional multiparty communication with gaussian states,” Phys. Rev. A103, 070501 (2009).
  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]
  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. A69, 063803 (2004). [CrossRef]
  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)438, 828–832 (2005). [CrossRef]
  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]
  22. K. Hammerer, M. Aspelmeyer, E. S. Polzik, and P. Zoller, “Establishing Einstein-Poldosky-Rosen channels between nanomechanics and atomic ensembles,” Phys. Rev. Lett.102, 020501 (2009). [CrossRef] [PubMed]
  23. K. Hammerer, A. S. Sørensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys.82, 1041–1093 (2010). [CrossRef]
  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.7, 13–16 (2011). [CrossRef]
  25. C. A. Muschik, E. S. Polzik, and J. I. Cirac, “Dissipatively driven entanglement of two macroscopic atomic ensembles,” Phys. Rev. A83, 052312 (2011). [CrossRef]
  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]
  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. A84, 022328 (2011). [CrossRef]
  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]
  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.4, 878–883 (2008). [CrossRef]
  31. F. Verstraete, M. M. Wolf, and J. I. Cirac,“Quantum computation and quantum-state engineering driven by dissipation,” Nat. Phys.5, 633–636 (2009). [CrossRef]
  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]
  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]
  36. A. Serafini, S. Mancini, and S. Bose, “Distributed quantum computation via optical fibers,” Phys. Rev. Lett.96, 010503 (2006). [CrossRef] [PubMed]
  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. A79, 052330 (2009). [CrossRef]
  38. J. Zhang, “Continuous-variable multipartite unlockable bound entangled Gussian states,” Phys. Rev. A83, 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]
  40. P.-B. Li and F.-L. Li, “Deterministic generation of multiparticle entanglement in a coupled cavity-fiber system,” Opt. Express19, 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. A81, 015804 (2010). [CrossRef]
  42. L.-M. Duan, J. I. Cirac, and P. Zoller, “Three-dimensional theory for interaction between atomic ensembles and free-space light,” Phys. Rev. A66, 023818 (2002). [CrossRef]
  43. T. Holstein and H. Primakoff, “Field dependence of the intrinsic domain magnetization of a ferromagnet,” Phys. Rev.58, 1098–1113 (1940). [CrossRef]
  44. J. S. Peng and G. X. Li, Introduction to Modern Quantum Optics (World Scientific, 1998). [CrossRef]
  45. G.-X. Li, H.-T. Tan, and S.-P. Wu, “Motional entanglement for two trapped ions in cascaded optical cavities,” Phys. Rev. A70, 064301 (2004). [CrossRef]
  46. G.-X. Li, “Generation of pure multipartite entangled vibrational states for ions trapped in a cavity,” Phys. Rev. A74, 055801 (2006). [CrossRef]
  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. A82, 053826 (2010). [CrossRef]
  48. C. W. Gardiner and P. Zoller, Quantum Noise (Springer-Verlag, 2000).
  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]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1 Fig. 2 Fig. 3
 
Fig. 4
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited