Abstract

We propose an experimentally feasible scheme to generate nonmaximal entanglement between two atomic ensembles. The degree of entanglement is readily tunable. The scheme involves laser manipulation of atomic ensembles; the setup includes adjustable quarter- and half-wave plates, a beam splitter, polarizing beam splitters, and single-photon detectors, and fits well the state of current experimental technology. Finally, we use the nonmaximally entangled state of ensembles to demonstrate quantum nonlocality by detecting the Clauser–Horne–Shimony–Holt inequality.

© 2004 Optical Society of America

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    [CrossRef] [PubMed]
  44. J. F. Roch, K. Vigneron, Ph. Grelu, A. Sinatra, J.-Ph. Poizat, and Ph. Grangier, “Quantum nondemolition measurements using cold trapped atoms,” Phys. Rev. Lett. 78, 634–637 (1997).
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2003 (2)

C. M. Savage and J. Ruostekoski, “Energetically stable particle-like Skyrmions in a trapped Bose-Einstein condensate,” Phys. Rev. Lett. 91, 010403 (2003).
[CrossRef]

P. Xue and G.-C. Guo, “Scheme for preparation of multipartite entanglement of atomic ensembles,” Phys. Rev. A 67, 034302 (2003).
[CrossRef]

2002 (3)

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, “Entangling many atomic ensembles through laser manipulation,” Phys. Rev. Lett. 88, 170402 (2002).
[CrossRef] [PubMed]

P. Xue, C.-F. Li, and G.-C. Guo, “Addendum to ‘Efficient quantum-key-distribution scheme with nonmaximally entangled states’,” Phys. Rev. A 65, 034302 (2002).
[CrossRef]

2001 (10)

P. Xue, C.-F. Li, and G.-C. Guo, “Efficient quantum-key-distribution scheme with nonmaximally entangled states,” Phys. Rev. A 64, 032305 (2001).
[CrossRef]

J.-W. Pan, M. Daniell, S. Gasparoni, G. Weihs, and A. Zeilinger, “Experimental demonstration of four-photon entanglement and high-fidelity teleportation,” Phys. Rev. Lett. 86, 4435–4438 (2001).
[CrossRef] [PubMed]

M. D. Lukin, M. Fleischhauer, R. Cote, L.-M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, “Dipole blockade and quantum information processing in mesoscopic atomic ensembles,” Phys. Rev. Lett. 87, 037901 (2001).
[CrossRef] [PubMed]

L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414, 413–416 (2001).
[CrossRef] [PubMed]

S. Potting, M. Cramer, and P. Meystre, “Momentum state engineering and control in Bose-Einstein condensates,” Phys. Rev. A 64, 063613 (2001).
[CrossRef]

A. Sorensen and K. Molmer, “Entanglement and extreme spin squeezing,” Phys. Rev. Lett. 86, 4431–4434 (2001).
[CrossRef] [PubMed]

B. Julsgaard, A. Kozhekin, and E. S. Polzik, “Experimental long-lived entanglement of two macroscopic objects,” Nature 413, 400–403 (2001).
[CrossRef] [PubMed]

D. F. Phillips, A. Fleischhauer, A. Mair, and R. L. Walsworth, “Storage of light in atomic vapor,” Phys. Rev. Lett. 86, 783–786 (2001).
[CrossRef] [PubMed]

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409, 490–493 (2001).
[CrossRef] [PubMed]

P. Xue, C.-F. Li, and G.-C. Guo, “Reducing the communication complexity with quantum entanglement,” Phys. Rev. A 64, 032304 (2001).
[CrossRef]

2000 (4)

A. Kuzmich, L. Mandel, and N. P. Bigelow, “Generation of spin squeezing via continuous quantum nondemolition measurement,” Phys. Rev. Lett. 85, 1594–1597 (2000).
[CrossRef] [PubMed]

L.-M. Duan, J. I. Cirac, P. Zoller, and E. S. Polzik, “Quantum communication between atomic ensembles using coherent light,” Phys. Rev. Lett. 85, 5643–5646 (2000).
[CrossRef]

A. Rauschenbeutel, G. Nogues, S. Osnaghi, P. Bertet, M. Brune, J.-M. Raimond, and S. Haroche, “Step-by-step engineered multiparticle entanglement,” Science 288, 2024–2028 (2000).
[CrossRef] [PubMed]

C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, M. Rowe, Q. A. Turchette, W. M. Itano, D. J. Wineland, and C. Monroe, “Experimental entanglement of four particles,” Nature 404, 256–259 (2000).
[CrossRef] [PubMed]

1999 (4)

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, “Ultrabright source of polarization-entangled photons,” Phys. Rev. A 60, R773–R776 (1999).
[CrossRef]

A. G. White, D. F. V. James, P. H. Eberhard, and P. G. Kwiat, “Nonmaximally entangled states: production, characterization, and utilization,” Phys. Rev. Lett. 83, 3103–3107 (1999).
[CrossRef]

D. Bouwmeester, J.-W. Pan, M. Daniell, H. Weinfurter, and A. Zeilinger, “Observation of three-photon Greenberger-Horne-Zeilinger entanglement,” Phys. Rev. Lett. 82, 1345–1349 (1999).
[CrossRef]

J. Hald, J. L. Sorensen, C. Schori, and E. S. Polzik, “Spin squeezed atoms: a macroscopic entangled ensemble created by light,” Phys. Rev. Lett. 83, 1319–1322 (1999).
[CrossRef]

1998 (2)

A. Kuzmich, N. P. Bigelow, and L. Mandel, “Atomic quantum nondemolition measurements and squeezing,” Europhys. Lett. 42, 481–486 (1998).
[CrossRef]

Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, “Deterministic entanglement of two trapped ions,” Phys. Rev. Lett. 81, 3631–3634 (1998).
[CrossRef]

1997 (5)

G. Digiuseppe, F. De Martini, and D. Boschi, “Experimental test of the violation of local realism in quantum mechanics without Bell inequalities,” Phys. Rev. A 56, 176–181 (1997).
[CrossRef]

J. F. Roch, K. Vigneron, Ph. Grelu, A. Sinatra, J.-Ph. Poizat, and Ph. Grangier, “Quantum nondemolition measurements using cold trapped atoms,” Phys. Rev. Lett. 78, 634–637 (1997).
[CrossRef]

E. Hagley, X. Maĺtre, G. Nogues, C. Wunderlich, M. Brune, J. M. Raimond, and S. Haroche, “Generation of Einstein-Podolsky-Rosen pairs of atoms,” Phys. Rev. Lett. 79, 1–5 (1997).
[CrossRef]

L. K. Grover, “Quantum mechanics helps in searching for a needle in a haystack,” Phys. Rev. Lett. 79, 325–328 (1997).
[CrossRef]

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575–579 (1997).
[CrossRef]

1996 (1)

A. K. Ekert and R. Jozsa, “Quantum computation and Shor’s factoring algorithm,” Rev. Mod. Phys. 68, 733–753 (1996).
[CrossRef]

1995 (3)

P. G. Kwiat, K. Mattle, H. Weinfurter, and A. Zeilinger, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[CrossRef] [PubMed]

A. Garuccio, “Hardy’s approach, Eberhard’s inequality, and supplementary assumptions,” Phys. Rev. A 52, 2535–2537 (1995).
[CrossRef] [PubMed]

J. R. Torgerson, D. Branning, C. H. Monken, and L. Mandel, “Experimental demonstration of the violation of local realism without Bell inequalities,” Phys. Lett. A 204, 323–328 (1995).
[CrossRef]

1993 (3)

L. Hardy, “Nonlocality for two particles without inequalities for almost all entangled states,” Phys. Rev. Lett. 71, 1665–1668 (1993).
[CrossRef] [PubMed]

P. H. Eberhard, “Background level and counter efficiencies required for a loophole-free Einstein-Podolsky-Rosen experiment,” Phys. Rev. A 47, R747–R750 (1993).
[CrossRef] [PubMed]

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[CrossRef] [PubMed]

1992 (1)

C. H. Bennett and S. J. Wiesner, “Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states,” Phys. Rev. Lett. 69, 2881–2884 (1992).
[CrossRef] [PubMed]

1991 (1)

A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67, 661–663 (1991).
[CrossRef] [PubMed]

1969 (1)

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
[CrossRef]

1964 (1)

J. S. Bell, “On the Einstein-Podolsky-Rosen paradox,” Physics 1, 195–200 (1964).

Appelbaum, I.

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, “Ultrabright source of polarization-entangled photons,” Phys. Rev. A 60, R773–R776 (1999).
[CrossRef]

Behroozi, C. H.

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409, 490–493 (2001).
[CrossRef] [PubMed]

Bell, J. S.

J. S. Bell, “On the Einstein-Podolsky-Rosen paradox,” Physics 1, 195–200 (1964).

Bennett, C. H.

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[CrossRef] [PubMed]

C. H. Bennett and S. J. Wiesner, “Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states,” Phys. Rev. Lett. 69, 2881–2884 (1992).
[CrossRef] [PubMed]

Bertet, P.

A. Rauschenbeutel, G. Nogues, S. Osnaghi, P. Bertet, M. Brune, J.-M. Raimond, and S. Haroche, “Step-by-step engineered multiparticle entanglement,” Science 288, 2024–2028 (2000).
[CrossRef] [PubMed]

Bigelow, N. P.

A. Kuzmich, L. Mandel, and N. P. Bigelow, “Generation of spin squeezing via continuous quantum nondemolition measurement,” Phys. Rev. Lett. 85, 1594–1597 (2000).
[CrossRef] [PubMed]

A. Kuzmich, N. P. Bigelow, and L. Mandel, “Atomic quantum nondemolition measurements and squeezing,” Europhys. Lett. 42, 481–486 (1998).
[CrossRef]

Boschi, D.

G. Digiuseppe, F. De Martini, and D. Boschi, “Experimental test of the violation of local realism in quantum mechanics without Bell inequalities,” Phys. Rev. A 56, 176–181 (1997).
[CrossRef]

Bouwmeester, D.

D. Bouwmeester, J.-W. Pan, M. Daniell, H. Weinfurter, and A. Zeilinger, “Observation of three-photon Greenberger-Horne-Zeilinger entanglement,” Phys. Rev. Lett. 82, 1345–1349 (1999).
[CrossRef]

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575–579 (1997).
[CrossRef]

Branning, D.

J. R. Torgerson, D. Branning, C. H. Monken, and L. Mandel, “Experimental demonstration of the violation of local realism without Bell inequalities,” Phys. Lett. A 204, 323–328 (1995).
[CrossRef]

Brassard, G.

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[CrossRef] [PubMed]

Brune, M.

A. Rauschenbeutel, G. Nogues, S. Osnaghi, P. Bertet, M. Brune, J.-M. Raimond, and S. Haroche, “Step-by-step engineered multiparticle entanglement,” Science 288, 2024–2028 (2000).
[CrossRef] [PubMed]

E. Hagley, X. Maĺtre, G. Nogues, C. Wunderlich, M. Brune, J. M. Raimond, and S. Haroche, “Generation of Einstein-Podolsky-Rosen pairs of atoms,” Phys. Rev. Lett. 79, 1–5 (1997).
[CrossRef]

Cirac, J. I.

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 414, 413–416 (2001).
[CrossRef] [PubMed]

M. D. Lukin, M. Fleischhauer, R. Cote, L.-M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, “Dipole blockade and quantum information processing in mesoscopic atomic ensembles,” Phys. Rev. Lett. 87, 037901 (2001).
[CrossRef] [PubMed]

L.-M. Duan, J. I. Cirac, P. Zoller, and E. S. Polzik, “Quantum communication between atomic ensembles using coherent light,” Phys. Rev. Lett. 85, 5643–5646 (2000).
[CrossRef]

Clauser, J. F.

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
[CrossRef]

Cote, R.

M. D. Lukin, M. Fleischhauer, R. Cote, L.-M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, “Dipole blockade and quantum information processing in mesoscopic atomic ensembles,” Phys. Rev. Lett. 87, 037901 (2001).
[CrossRef] [PubMed]

Cramer, M.

S. Potting, M. Cramer, and P. Meystre, “Momentum state engineering and control in Bose-Einstein condensates,” Phys. Rev. A 64, 063613 (2001).
[CrossRef]

Crépeau, C.

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[CrossRef] [PubMed]

Daniell, M.

J.-W. Pan, M. Daniell, S. Gasparoni, G. Weihs, and A. Zeilinger, “Experimental demonstration of four-photon entanglement and high-fidelity teleportation,” Phys. Rev. Lett. 86, 4435–4438 (2001).
[CrossRef] [PubMed]

D. Bouwmeester, J.-W. Pan, M. Daniell, H. Weinfurter, and A. Zeilinger, “Observation of three-photon Greenberger-Horne-Zeilinger entanglement,” Phys. Rev. Lett. 82, 1345–1349 (1999).
[CrossRef]

De Martini, F.

G. Digiuseppe, F. De Martini, and D. Boschi, “Experimental test of the violation of local realism in quantum mechanics without Bell inequalities,” Phys. Rev. A 56, 176–181 (1997).
[CrossRef]

Digiuseppe, G.

G. Digiuseppe, F. De Martini, and D. Boschi, “Experimental test of the violation of local realism in quantum mechanics without Bell inequalities,” Phys. Rev. A 56, 176–181 (1997).
[CrossRef]

Duan, L.-M.

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, “Entangling many atomic ensembles through laser manipulation,” Phys. Rev. Lett. 88, 170402 (2002).
[CrossRef] [PubMed]

M. D. Lukin, M. Fleischhauer, R. Cote, L.-M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, “Dipole blockade and quantum information processing in mesoscopic atomic ensembles,” Phys. Rev. Lett. 87, 037901 (2001).
[CrossRef] [PubMed]

L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414, 413–416 (2001).
[CrossRef] [PubMed]

L.-M. Duan, J. I. Cirac, P. Zoller, and E. S. Polzik, “Quantum communication between atomic ensembles using coherent light,” Phys. Rev. Lett. 85, 5643–5646 (2000).
[CrossRef]

Dutton, Z.

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409, 490–493 (2001).
[CrossRef] [PubMed]

Eberhard, P. H.

A. G. White, D. F. V. James, P. H. Eberhard, and P. G. Kwiat, “Nonmaximally entangled states: production, characterization, and utilization,” Phys. Rev. Lett. 83, 3103–3107 (1999).
[CrossRef]

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D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575–579 (1997).
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C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, M. Rowe, Q. A. Turchette, W. M. Itano, D. J. Wineland, and C. Monroe, “Experimental entanglement of four particles,” Nature 404, 256–259 (2000).
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J.-W. Pan, M. Daniell, S. Gasparoni, G. Weihs, and A. Zeilinger, “Experimental demonstration of four-photon entanglement and high-fidelity teleportation,” Phys. Rev. Lett. 86, 4435–4438 (2001).
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C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
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J. F. Roch, K. Vigneron, Ph. Grelu, A. Sinatra, J.-Ph. Poizat, and Ph. Grangier, “Quantum nondemolition measurements using cold trapped atoms,” Phys. Rev. Lett. 78, 634–637 (1997).
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S. Potting, M. Cramer, and P. Meystre, “Momentum state engineering and control in Bose-Einstein condensates,” Phys. Rev. A 64, 063613 (2001).
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E. Hagley, X. Maĺtre, G. Nogues, C. Wunderlich, M. Brune, J. M. Raimond, and S. Haroche, “Generation of Einstein-Podolsky-Rosen pairs of atoms,” Phys. Rev. Lett. 79, 1–5 (1997).
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A. Rauschenbeutel, G. Nogues, S. Osnaghi, P. Bertet, M. Brune, J.-M. Raimond, and S. Haroche, “Step-by-step engineered multiparticle entanglement,” Science 288, 2024–2028 (2000).
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C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, M. Rowe, Q. A. Turchette, W. M. Itano, D. J. Wineland, and C. Monroe, “Experimental entanglement of four particles,” Nature 404, 256–259 (2000).
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J. Hald, J. L. Sorensen, C. Schori, and E. S. Polzik, “Spin squeezed atoms: a macroscopic entangled ensemble created by light,” Phys. Rev. Lett. 83, 1319–1322 (1999).
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J. F. Roch, K. Vigneron, Ph. Grelu, A. Sinatra, J.-Ph. Poizat, and Ph. Grangier, “Quantum nondemolition measurements using cold trapped atoms,” Phys. Rev. Lett. 78, 634–637 (1997).
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J. Hald, J. L. Sorensen, C. Schori, and E. S. Polzik, “Spin squeezed atoms: a macroscopic entangled ensemble created by light,” Phys. Rev. Lett. 83, 1319–1322 (1999).
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C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, M. Rowe, Q. A. Turchette, W. M. Itano, D. J. Wineland, and C. Monroe, “Experimental entanglement of four particles,” Nature 404, 256–259 (2000).
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J. F. Roch, K. Vigneron, Ph. Grelu, A. Sinatra, J.-Ph. Poizat, and Ph. Grangier, “Quantum nondemolition measurements using cold trapped atoms,” Phys. Rev. Lett. 78, 634–637 (1997).
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P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, “Ultrabright source of polarization-entangled photons,” Phys. Rev. A 60, R773–R776 (1999).
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D. F. Phillips, A. Fleischhauer, A. Mair, and R. L. Walsworth, “Storage of light in atomic vapor,” Phys. Rev. Lett. 86, 783–786 (2001).
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J.-W. Pan, M. Daniell, S. Gasparoni, G. Weihs, and A. Zeilinger, “Experimental demonstration of four-photon entanglement and high-fidelity teleportation,” Phys. Rev. Lett. 86, 4435–4438 (2001).
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D. Bouwmeester, J.-W. Pan, M. Daniell, H. Weinfurter, and A. Zeilinger, “Observation of three-photon Greenberger-Horne-Zeilinger entanglement,” Phys. Rev. Lett. 82, 1345–1349 (1999).
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C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
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Figures (3)

Fig. 1
Fig. 1

Relevant Raman-type, Λ-level structure of the alkali atoms in the ensembles. A pair of metastable lower states |g and |s can be achieved by, for example, Zeeman sublevels of electronic ground states 5S1/2 87Rb atoms, and |e (5P1/2) is the excited state.

Fig. 2
Fig. 2

Schematic setup for entangling two ensembles E1 and E2 in the nonmaximally entangled state. The ensembles are illuminated by synchronized pumping laser pulses, and the forward-scattering Stokes pulses are collected after filtering by polarizers and Pockels cell to separate the pumping pulses and Stokes pulses. The dashed line represents the pumping laser pulses with the frequency ωpump and the solid line represents the Stokes pulses with the frequency ωrepump.

Fig. 3
Fig. 3

Schematic setup for the realization of the CHSH inequality detection. Two pairs of ensembles L1, R1 and L2, R2 have been prepared in an EPR-type of entangled state and a nonmaximally entangled state, respectively. The collective atomic excitations on each side are transferred to the optical excitations, which are read by the detectors. By choosing the relative phase shift ϕL and ϕR, we can measure the CHSH inequality.

Equations (23)

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s=(1/Na)i=1Na|gis|,
|φ=|vaca|vacp+pcs+a+|vaca|vacp+o(pc),
A=αa1+exp(iϕ12)βa2,
A=βa1+exp(iϕ12)αa2,
|ψ12=(αs1++exp(iϕ12)βs2+)|vac12,
|ψ12=(βs1++exp(iϕ12)αs2+)|vac12,
ρ12=1c+1(c|vac12vac|+|ψ12ψ|),
ρ12=1c+1(c|vac12vac|+|ψ12ψ|),
|ΨϕL1R1=(sL1++exp(iϕ)sR1+)/2|vacL1R1
ρL1R1=1c+1(c|vacL1R1vac|+|ΨϕL1R1Ψϕ|).
|ψPNE=(αsL2+sR1++βsL1+sR2+)|vacL1L2R1R2.
E(ϕL, ϕR)=PD1D3+PD2D4-PD1D4-PD2D3
E(ϕL, ϕR)=4α2β2 cos(ϕL-ϕR),
S=E(ϕL1, ϕR3)+E(ϕL1, ϕR2)+E(ϕL2, ϕR3)-E(ϕL2, ϕR2)=82α2β2,
ρ˙=-i[H, ρ]+12 ijNa(2dijρdij+-dij+dijρ-ρdij+dij)+12κ(2cρc+-c+cρ-ρc+c),
dg,j=γg|gje|,
ds,j=γs|sje|,
do,j=γo|oje|.
Hint=-NaΔ2Δ2+Γ2/2(Ωg*s+c++Ω*gsc)+i2κc+c,
c˙=-i[c, Hint]+κcin(t),
S˙+=-i[S+, Hint],
s˙+=κ2s+-iκcin(t),
ceff=0tδcout+(t)cout(t)dt=a+(tδ)a(tδ)+0tδcin+(t)cin(t)dt,

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