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

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    [CrossRef]
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    [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).
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  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).
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  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).
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  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).
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  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).
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    [CrossRef] [PubMed]

2011

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]

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]

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]

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]

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]

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).

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]

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]

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

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]

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]

2010

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]

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]

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]

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

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]

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]

2009

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]

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]

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

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]

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]

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]

2008

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]

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

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]

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]

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]

2007

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]

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

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]

2006

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

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]

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

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]

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

2005

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]

2004

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]

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]

2002

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]

2001

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]

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

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

2000

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]

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

1997

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]

1993

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]

1940

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

Adesso, G.

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

Armstrong, S. C.

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]

Aspelmeyer, M.

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]

Barreiro, J. T.

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]

Blatt, R.

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]

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]

Bloomer, R.

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]

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]

Bose, S.

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

Bradley, A. S.

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]

Braunstein, S. L.

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

Briegel, H. J.

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

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

Büchler, H. P.

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]

Cao, Z.-L.

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]

Chou, C. W.

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]

Chwalla, M.

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]

Cirac, J. I.

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]

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]

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]

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]

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]

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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).
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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).
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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).
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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).
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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).
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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).
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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).
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N. C. Menicucci, S. T. Flammia, and O. Pfister, “One-way quantum computing in the optical frequency comb,” Phys. Rev. Lett. 101, 130501 (2008).
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N. C. Menicucci, S. T. Flammia, H. Zaidi, and O. Pfister,“Ultracompact generation of continuous-variable cluster states,” Phys. Rev. A 76, 010302 (2007).
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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).
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L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller,“Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84,  2722 (2000).
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Gisin, N.

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).
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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).
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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).
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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).
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Hage, B.

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).
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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).
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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).

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

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).
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Holstein, T.

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

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]

Jensen, K.

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).
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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).
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Jia, X.

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).
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Kantian, A.

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

Kraus, B.

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).
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Krauter, H.

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).
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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).

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).
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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).
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Li, G. X.

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Lukin, M. D.

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]

Lv¨, X.Y.

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]

Ma, X.

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]

Mabuchi, H.

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).
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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]

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]

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]

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]

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]

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]

Micheli, A.

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).
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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]

Miwa, Y.

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]

Monz, T.

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]

Muschik, C. A.

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).

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]

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]

Nielsen, B. M.

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]

Nielsen, M. A.

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]

Olsen, M. K.

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]

Owari, M.

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).
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Parkins, A. S.

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]

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]

Payne, G.

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]

Peng, J. S.

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

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).
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A. Tan, C. Xie, and K. Peng, “Quantum logical gates with linear quadripartite cluster states of continuous variables,” Phys. Rev. A 79, 042338 (2009).
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J. Zhang, G. Adesso, C. Xie, and K. Peng, “Quantum teamwork for unconditional multiparty communication with gaussian states,” Phys. Rev. A 103, 070501 (2009).

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]

Petersen, J. M.

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]

Pfister, O.

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]

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]

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]

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]

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]

Pineda, C.

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]

Plenio, M. B.

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]

Politi, A.

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]

Polyakov, S. V.

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]

Polzik, E. S.

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]

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]

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]

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).

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

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]

Primakoff, H.

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

Pysher, M.

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]

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]

Ralph, T. C.

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]

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]

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]

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H. J. Briegel and R. Raussendorf, “Persistent entanglement in arrays of interacting particles,” Phys. Rev. Lett. 86, 910 (2001).
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R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett. 86, 5188 (2001).
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Roos, C. F.

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]

Samblowski, A.

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]

Sangouard, N.

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]

Schindler, P.

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]

Schnabel, R.

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]

Serafini, A.

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]

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

Shahrokhshahi, R.

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]

Shen, H.

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]

Shimokawa, Y.

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]

Si, L. G.

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]

Simon, C.

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]

Simon, R.

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

Solano, E.

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]

Sørensen, A. S.

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

Su, X.

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]

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]

Tan, A.

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]

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]

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]

Tan, H.-T.

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]

Ukai, R.

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]

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]

van Enk, S. J.

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]

van Loock, P.

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]

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]

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]

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

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]

Verstraete, F.

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]

Wang, Y.

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]

Wasilewski, W.

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]

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]

Weedbrook, C.

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]

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

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]

Wolf, M. M.

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]

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]

Wu, S.-P.

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]

Xia, Y.

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]

Xie, C.

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]

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

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]

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]

Yang, X. X.

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]

Yang, Z. B.

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]

Ying Wu, M.

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]

Yoshikawa, J.

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]

Yuan, C.-H.

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]

Yukawa, M.

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]

Zaidi, H.

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]

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]

Zhang, J.

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

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

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]

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

Zhang, W.-P.

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]

Zheng, S. B.

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]

Zoller, P.

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]

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]

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]

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]

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]

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]

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]

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

Laser Phys.

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]

Nat. Phys.

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]

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]

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]

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]

Nature (London)

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

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]

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]

Opt. Express

Phys. Rev.

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

Phys. Rev. A

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

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]

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]

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]

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

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]

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

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]

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]

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]

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]

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]

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]

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

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

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]

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]

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]

Phys. Rev. Lett.

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]

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]

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

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]

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]

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]

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

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

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]

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]

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]

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]

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]

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]

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]

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]

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

Rev. Mod. Phys.

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

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]

Other

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).

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

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

Cited By

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Figures (4)

Fig. 1
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).

Fig. 2
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.

Fig. 3
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 Γ 1 = β 2 2 / κ.

Fig. 4
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).

Equations (42)

Equations on this page are rendered with MathJax. Learn more.

H = H 0 + H A L + H A C ,
H 0 = ω a a a + ω b b b + n = 1 2 j = 1 N n ( ω 1 | 1 j n 1 j n | + ω 2 | 2 j n 2 j n | + ω 3 | 3 j n 3 j n | ) + n = 3 4 j = 1 N n ( ω 1 | 0 j n 0 j n | + ω 2 | 2 j n 2 j n | + ω 3 | 3 j n 3 j n | )
H A L = 1 2 n = 1 2 j = 1 N n { Ω 2 n ( x j n ) exp [ i ( ω L 2 t ϕ 2 n ) ] | 2 j n 1 j n | + Ω 3 n ( x j n ) exp [ i ( ω L 3 t ϕ 3 n ) ] | 3 j n 0 j n | + H.c. } + 1 2 n = 3 4 j = 1 N n { Ω 2 n ( x j n ) exp [ i ( ω L 2 t ϕ 2 n ) ] | 2 j n 0 j n | + Ω 3 n ( x j n ) exp [ i ( ω L 3 t ϕ 3 n ) ] | 3 j n 1 j n | + H.c. }
H A C = n = 1 2 j = 1 N n [ g a n ( x j n ) | 2 j n 0 j n | a + g b n ( x j n ) | 3 j n 1 j n | b + H.c. ] + n = 3 4 j = 1 N n [ g a n ( x j n ) | 2 j n 1 j n | a + g b n ( x j n ) | 3 j n 0 j n | b + H.c. ]
Ω 2 n ( x j n ) = Ω 2 n exp ( i k x j n ) , Ω 3 n ( x j n ) = Ω 3 n exp ( i k x j n ) , g a n ( x j n ) = g a n exp ( i k x j n ) , g b n ( x j n ) = g b n exp ( i k x j n ) .
H eff = δ a a a + δ b b b + n = 1 2 g a n 2 Δ 2 n ( 1 2 N n J z n ) a a + n = 3 4 g a n 2 Δ 2 n ( 1 2 N n + J z n ) a a + [ n = 1 2 g b n 2 Δ 3 n ( 1 2 N n + J z n ) + n = 3 4 g b n 2 Δ 3 n ( 1 2 N n J z n ) ] b b + [ n = 1 2 β 2 n exp ( i ϕ 2 n ) J n + n = 3 4 β 2 n exp ( i ϕ 2 n ) J + n ] a + H.c. + [ n = 1 2 β 3 n exp ( i ϕ 3 n ) J + n + n = 3 4 β 3 n exp ( i ϕ 3 n ) J n ] b + H.c. ,
β 2 n = N n Ω 2 n g a n 2 Δ 2 n , β 3 n = N n Ω 3 n g b n 2 Δ 3 n ,
J z n = 1 2 j = 1 N n ( | 1 j n 1 j n | | 0 j n 0 j n | ) , J + n = j = 1 N n | 1 j n 0 j n | , J n = j = 1 N n | 0 j n 1 j n | .
δ a + n = 1 2 g a n 2 N n Δ 2 n = δ b + n = 3 4 g b n 2 N n Δ 3 n = 0 ,
H eff = [ n = 1 2 β 2 n exp ( i ϕ 2 n ) c n + n = 3 4 β 2 n exp ( i ϕ 2 n ) c n ] a + H.c. + [ n = 1 2 β 3 n exp ( i ϕ 3 n ) c n + n = 3 4 β 3 n exp ( i ϕ 3 n ) c n ] b + H.c. .
ρ ˙ = i [ H eff , ρ ] + 𝒧 c ρ ,
𝒧 c ρ = κ a D [ a ] ρ + κ b D [ b ] ρ ,
S = exp [ ε ( c 1 c 4 + c 2 c 3 + c 2 c 4 ) H.c. ] ,
e 1 = c 1 + λ c 2 1 + λ 2 , e 2 = λ c 1 + c 2 1 + λ 2 , e 3 = λ c 3 c 4 1 + λ 2 , e 4 = c 3 + λ c 4 1 + λ 2 .
S = T S T = exp ( λ ε e 1 e 4 ε λ e 2 e 3 H.c. ) = S 14 ( λ ε ) S 23 ( ε / λ ) ,
H eff 1 = a ( λ β 22 c 1 + β 22 c 2 λ β 24 c 3 + β 24 c 4 ) + b ( λ β 32 c 1 β 32 c 2 + λ β 34 c 3 β 34 c 4 ) + H.c.
T H eff 1 T = ( λ 2 + 1 ) 1 2 [ a ( β 22 e 2 β 24 e 3 ) + b ( β 32 e 2 + β 34 e 3 ) ] + H.c.
H ˜ eff 1 = S 23 ( ε / λ ) T H eff 1 T S 23 ( ε / λ ) = [ ( λ 2 + 1 ) ( β 34 2 β 32 2 ) ] 1 2 ( a e 2 + b e 3 ) + H.c.
d d t ρ ˜ 1 = i [ H ˜ eff 1 , ρ ˜ 1 ] + κ D [ a ] ρ ˜ 1 + κ D [ b ] ρ ˜ 1 ,
η ± = κ ± [ κ 2 4 ( λ 2 + 1 ) ( β 34 2 β 32 2 ) ] 1 2 ,
ρ ˜ 1 = | 0 e 2 , 0 e 3 , 0 a , 0 b 0 e 2 , 0 e 3 , 0 a , 0 b | ρ e 1 , e 4 ( T 1 ) ,
H eff 2 = a ( β 21 c 1 + λ β 21 c 2 + β 23 c 3 + λ β 23 c 4 ) + b ( β 31 c 1 + λ β 31 c 2 + β 33 c 3 + λ β 33 c 4 ) + H.c.
d d t ρ ˜ 2 = i [ H ˜ eff 2 , ρ ˜ 2 ] + κ D [ a ] ρ ˜ 2 + κ D [ b ] ρ ˜ 2 ,
H ˜ eff 2 = [ ( λ 2 + 1 ) ( β 21 2 β 23 2 ) ] 1 2 ( a e 1 + b e 4 ) + H.c.
| ψ = S | 0 c 1 , 0 c 2 , 0 c 3 , 0 c 4 ,
V 1 = V ( P 1 1 5 P 3 + 2 5 P 4 ) , V 2 = V ( X 3 + 1 5 X 1 2 5 X 2 ) , V 3 = V ( P 2 + 2 5 P 3 + 1 5 P 4 ) , V 4 = V ( X 4 2 5 X 1 1 5 X 2 ) ,
V 1 = V 2 = 1 λ 2 + 1 ( λ 2 e ε / λ + e λ ε ) , V 3 = V 4 = 1 λ 2 + 1 ( λ 2 e λ ε + e ε / λ ) .
H eff = ( n = 1 2 β 2 n c n a 1 + n = 3 4 β 2 n c n a 2 + H.c. ) + ( n = 1 2 β 3 n c n b 1 + n = 3 4 β 3 n c n b 2 + H.c. ) ,
ρ ˙ = i [ H ˜ eff , ρ ] + 𝒧 c a ρ + 𝒧 c b ρ ,
𝒧 c a ρ = κ j = 1 2 ( 2 a j ρ a j a j a j ρ ρ a j a j ) 2 η κ ( [ a 1 , a 2 ρ ] + [ ρ a 2 , a 1 ] ) 𝒧 c b ρ = κ j = 1 2 ( 2 b j ρ b j b j b j ρ ρ b j b j ) 2 η κ ( [ b 2 , b 1 ρ ] + [ ρ b 1 , b 2 ] ) ,
H ˜ eff = T H eff T = H 1 + H 2 ,
H 1 = β 1 b 2 e 3 + β 2 b 1 e 2 + H.c. , H 2 = β 1 a 1 e 1 + β 2 a 2 e 4 + H.c..
β 21 = β 34 = β 1 cos θ , β 22 = β 33 = β 1 sin θ , β 23 = β 32 = β 2 cos θ , β 24 = β 31 = β 2 sin θ ,
Γ V Γ + i 2 Λ 0 ,
σ = ( 0 1 1 0 ) .
E n = n 1 + n 2 [ ( n 1 n 2 ) 2 + 4 n 3 2 ] 1 2 2 ,
d 1 = 1 2 Ω [ s + exp ( 1 2 s t ) s exp ( 1 2 s + t ) ] , d 2 = 2 κ β 1 β 2 η β 1 2 + β 2 2 [ exp ( 1 2 s + t ) Ω exp ( 1 2 s t ) Ω + exp ( 1 2 s t ) Ω exp ( 1 2 s + t ) Ω ] ,
s ± = κ ± Ω , s ± = κ ± Ω , Ω = κ 2 + 4 β 2 2 , Ω = κ 2 4 β 1 2 ,
[ e 2 e 2 e 3 e 3 ] 1 2 < | e 2 e 3 | .
V j = n 1 + n 2 2 | n 3 | + 1 , j = 1 , 2 , 3 , 4.
1 4 ( e 2 e 2 e 3 e 3 ) 2 + e 2 e 2 e 3 e 3 < | e 2 e 3 | 2 .
Tr ρ 2 = [ 2 ( e 2 e 2 e 3 e 3 ) + 1 ] 1 2 .

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