Abstract

Quantum steering is used to describe the “spooky action-at-a-distance” nonlocality raised in the Einstein-Podolsky-Rosen (EPR) paradox, which is important for understanding entanglement distribution and constructing quantum networks. Here, in this paper, we study an experimentally feasible scheme for generating quantum steering based on cascaded four-wave-mixing (FWM) processes in hot rubidium (Rb) vapor. Quantum steering, including bipartite steering and genuine tripartite steering among the output light fields, is theoretically analyzed. We find the corresponding gain regions in which the bipartite and tripartite steering exist. The results of bipartite steering can be used to establish a hierarchical steering model in which one beam can steer the other two beams in the whole gain region; however, the other two beams cannot steer the first beam simultaneously. Moreover, the other two beams cannot steer with each other in the whole gain region. More importantly, we investigate the gain dependence of the existence of the genuine tripartite steering and we find that the genuine tripartite steering exists in most of the whole gain region in the ideal case. Also we discuss the effect of losses on the genuine tripartite steering. Our results pave the way to experimental demonstration of quantum steering in cascaded FWM process.

© 2017 Optical Society of America

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  1. M. D. Reid, P. D. Drummond, E. G. Cavalcanti, W. P. Bowen, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “Colloquium: The Einstein-Podolsky-Rosen paradox: From concepts to applications,” Rev. Mod. Phys.,  81, 1727–1751 (2009).
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  3. D. H. Smith, G. Gillett, M. de Almeida, C. Branciard, A. Fedrizzi, T. J. Weinhold, A. Lita, B. Calkins, T. Gerrits, H. M. Wiseman, S. W. Nam, and A. G. White, “Conclusive quantum steering with superconducting transition-edge sensors,” Nature Commun. 3, 625 (2012).
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  4. B. Wittmann, S. Ramelow, F. Steinlechner, N. K. Langford, N. Brunner, H. Wiseman, R. Ursin, and A. Zeilinger, “A. Loophole-free Einstein-Podolsky-Rosen experiment via quantum steering,” New J. Phys. 14, 053030 (2012).
    [Crossref]
  5. A. Bennet, D. A. Evans, D. J. Saunders, C. Branciard, E. Cavalcanti, H. M. Wiseman, and G. J. Pryde, “Arbitrarily loss-tolerant Einstein-Podolsky-Rosen steering allowing a demonstration over 1 km of optical fiber with no detection loophole,” Phys. Rev. X 2, 031003 (2012).
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    [Crossref]
  10. M. Wang, Q. H. Gong, and Q. Y. He, “Collective multipartite Einstein-Podolsky-Rosen steering: more secure optical networks,” Opt. Lett. 39, 6703–6706 (2014).
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  11. M. Wang, Q. H. Gong, Z. Ficek, and Q. Y. He, “Efficient scheme for perfect collective Einstein-Podolsky-Rosen steering,” Sci. Rep. 5, 12346 (2015).
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  12. Y. Xiang, F. X. Sun, M. Wang, Q. H. Gong, and Q. Y. He, “Detection of genuine tripartite entanglement and steering in hybrid optomechanics,” Opt. Express 23, 30104–30117 (2015).
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  21. A. MacRae, T. Brannan, R. Achal, and A. I. Lvovsky, “Tomography of a high-purity narrowband photon from a transient atomic collective excitation,” Phys. Rev. Lett. 109, 033601 (2012).
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  36. J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777 (2012).
    [Crossref]
  37. R. Pooser and J. Jing, “Continuous variable cluster state generation over the optical spatial mode comb,” Phys. Rev. A 90, 043841 (2014).
    [Crossref]
  38. Y. Cai, J. Feng, H. Wang, G. Ferrini, X. Xu, J. Jing, and N. Treps, “Quantum-network generation based on four-wave mixing,” Phys. Rev. A 91, 013843 (2015).
    [Crossref]
  39. M. Jasperse, L. D. Turner, and R. E. Scholten, “Relative intensity squeezing by four-wave mixing with loss: an analytic model and experimental diagnostic,” Optics Express 19(4), 3765–3774 (2011).
    [Crossref] [PubMed]
  40. J. Zhang and S. L. Braunstein, “Contunuous-variable Gaussian analog of cluster state,” Phys. Rev. A 73, 032318 (2006).
    [Crossref]
  41. X. Deng, Y. Xiang, C. Tian, G. Adesso, Q. He, Q. Gong, X. Su, C. Xie, and K. Peng, “Demonstration of monogamy relations for Einstein-Podolsky-Rosen steering in Gaussian cluster states,” Phys. Rev. Lett 118, 230501 (2017).
    [Crossref] [PubMed]

2017 (1)

X. Deng, Y. Xiang, C. Tian, G. Adesso, Q. He, Q. Gong, X. Su, C. Xie, and K. Peng, “Demonstration of monogamy relations for Einstein-Podolsky-Rosen steering in Gaussian cluster states,” Phys. Rev. Lett 118, 230501 (2017).
[Crossref] [PubMed]

2016 (3)

2015 (6)

R. C. Pooser and B. Lawrie, “Ultrasensitive measurement of microcantilever displacement below the shot-noise limit,” Optica 2, 393 (2015).
[Crossref]

C. S. Embrey, M. T. Turnbull, P. G. Petrov, and V. Boyer, “Observation of localized multi-spatial-mode quadrature squeezing,” Phys. Rev. X 5, 031004 (2015).

Y. Cai, J. Feng, H. Wang, G. Ferrini, X. Xu, J. Jing, and N. Treps, “Quantum-network generation based on four-wave mixing,” Phys. Rev. A 91, 013843 (2015).
[Crossref]

M. Wang, Q. H. Gong, Z. Ficek, and Q. Y. He, “Efficient scheme for perfect collective Einstein-Podolsky-Rosen steering,” Sci. Rep. 5, 12346 (2015).
[Crossref] [PubMed]

Y. Xiang, F. X. Sun, M. Wang, Q. H. Gong, and Q. Y. He, “Detection of genuine tripartite entanglement and steering in hybrid optomechanics,” Opt. Express 23, 30104–30117 (2015).
[Crossref] [PubMed]

S. Armstrong, M. Wang, R. Y. Teh, Q. Gong, Q. Y. He, J. Janousek, H. A. Bachor, M. D. Reid, and P. K. Lam, “Multipartite Einstein-Podolsky-Rosen steering and genuine tripartite entanglement with optical networks,” Nat. Phys. 11, 167 (2015).
[Crossref]

2014 (6)

M. Wang, Q. H. Gong, and Q. Y. He, “Collective multipartite Einstein-Podolsky-Rosen steering: more secure optical networks,” Opt. Lett. 39, 6703–6706 (2014).
[Crossref] [PubMed]

Q. Y. He and Z. Ficek, “EPR paradox and quantum steering in a three-mode optomechanical system,” Physical Review A,  89(2), 74–79 (2014).
[Crossref]

Z. Qin, L. Cao, H. Wang, A. M. Marino, W. Zhang, and J. Jing, “Experimental generation of multiple quantum correlated beams from hot rubidium vapor,” Phys. Rev. Lett. 113, 023602 (2014).
[Crossref] [PubMed]

F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014).
[Crossref] [PubMed]

N. Otterstrom, R. C. Pooser, and B. J. Lawrie, “Nonlinear optical magnetometry with accessible in situ optical squeezing,” Opt. Lett. 39, 6533 (2014).
[Crossref] [PubMed]

R. Pooser and J. Jing, “Continuous variable cluster state generation over the optical spatial mode comb,” Phys. Rev. A 90, 043841 (2014).
[Crossref]

2013 (3)

N. V. Corzo, Quentin Glorieux, A. M. Marino, J. B. Clark, R. T. Glasser, and P. D. Lett., “Rotation of the noise ellipse for squeezed vacuum light generated via four-wave mixing,” Phys. Rev. A 88, 043836 (2013).
[Crossref]

Q. Y. He and M. D. Reid, “Genuine Multipartite Einstein-Podolsky-Rosen Steering,” Phys. Rev. Lett. 111, 250403 (2013).
[Crossref]

J. Kong, J. Jing, H. Wang, C. Liu, and W. Zhang, “Experimental investigation of the visibility dependence in a nonlinear interferometer using parametric amplifiers,” Appl. Phys. Lett. 102, 011130 (2013).
[Crossref]

2012 (8)

A. MacRae, T. Brannan, R. Achal, and A. I. Lvovsky, “Tomography of a high-purity narrowband photon from a transient atomic collective excitation,” Phys. Rev. Lett. 109, 033601 (2012).
[Crossref] [PubMed]

D. H. Smith, G. Gillett, M. de Almeida, C. Branciard, A. Fedrizzi, T. J. Weinhold, A. Lita, B. Calkins, T. Gerrits, H. M. Wiseman, S. W. Nam, and A. G. White, “Conclusive quantum steering with superconducting transition-edge sensors,” Nature Commun. 3, 625 (2012).
[Crossref]

B. Wittmann, S. Ramelow, F. Steinlechner, N. K. Langford, N. Brunner, H. Wiseman, R. Ursin, and A. Zeilinger, “A. Loophole-free Einstein-Podolsky-Rosen experiment via quantum steering,” New J. Phys. 14, 053030 (2012).
[Crossref]

A. Bennet, D. A. Evans, D. J. Saunders, C. Branciard, E. Cavalcanti, H. M. Wiseman, and G. J. Pryde, “Arbitrarily loss-tolerant Einstein-Podolsky-Rosen steering allowing a demonstration over 1 km of optical fiber with no detection loophole,” Phys. Rev. X 2, 031003 (2012).

R. T. Glasser, U. Vogl, and P. D. Lett, “Stimulated generation of superluminal light pulses via four-wave mixing,” Phy. Rev. Lett. 108, 173902 (2012).
[Crossref]

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777 (2012).
[Crossref]

N. V. Corzo, A. M. Marino, K. M. Jones, and P. D. Lett, “Noiseless optical amplifier operating on hundreds of spatial modes,” Phy. Rev. Lett. 109, 043602 (2012).
[Crossref]

C. Weedbrook, S. Pirandola, R. G. Patron, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621 (2012).
[Crossref]

2011 (2)

M. Jasperse, L. D. Turner, and R. E. Scholten, “Relative intensity squeezing by four-wave mixing with loss: an analytic model and experimental diagnostic,” Optics Express 19(4), 3765–3774 (2011).
[Crossref] [PubMed]

J. Jing, C. Liu, Z. Zhou, Z. Y. Ou, and W. Zhang, “Realization of a nonlinear interferometer with parametric amplifiers,” Appl. Phys. Lett. 99, 011110 (2011).
[Crossref]

2010 (1)

D. J. Saunders, S. J. Jones, H. M. Wiseman, and G. J. Pryde, “Experimental EPR-steering using Bell-local states,” Nature Phys.,  6, 845–849 (2010).
[Crossref]

2009 (3)

M. D. Reid, P. D. Drummond, E. G. Cavalcanti, W. P. Bowen, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “Colloquium: The Einstein-Podolsky-Rosen paradox: From concepts to applications,” Rev. Mod. Phys.,  81, 1727–1751 (2009).
[Crossref]

A. M. Marino, R. C. Pooser, V. Boyer, and P. D. Lett, “Tunable Delay of Einstein-Podolsky-Rosen Entanglement,” Nature 457, 859 (2009).
[Crossref] [PubMed]

Ryan M. Camacho, Praveen K. Vudyasetu, and John C. Howell, “Four-wave-mixing stopped light in hot atomic rubidium vapour,” Nature Photon. 3, 103–106 (2009).
[Crossref]

2008 (2)

V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, “Entangled images from Four-Wave Mixing,” Science 321, 544 (2008).
[Crossref] [PubMed]

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

2007 (1)

2006 (1)

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

2005 (1)

S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513 (2005).
[Crossref]

2003 (1)

M. D. Lukin, “Trapping and manipulating photon states in atomic ensembles,” Rev. Mod. Phys. 75, 457 (2003).
[Crossref]

2001 (1)

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 (2001).
[Crossref]

Achal, R.

A. MacRae, T. Brannan, R. Achal, and A. I. Lvovsky, “Tomography of a high-purity narrowband photon from a transient atomic collective excitation,” Phys. Rev. Lett. 109, 033601 (2012).
[Crossref] [PubMed]

Adesso, G.

X. Deng, Y. Xiang, C. Tian, G. Adesso, Q. He, Q. Gong, X. Su, C. Xie, and K. Peng, “Demonstration of monogamy relations for Einstein-Podolsky-Rosen steering in Gaussian cluster states,” Phys. Rev. Lett 118, 230501 (2017).
[Crossref] [PubMed]

Andersen, U. L.

M. D. Reid, P. D. Drummond, E. G. Cavalcanti, W. P. Bowen, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “Colloquium: The Einstein-Podolsky-Rosen paradox: From concepts to applications,” Rev. Mod. Phys.,  81, 1727–1751 (2009).
[Crossref]

Arimondo, E.

Armstrong, S.

S. Armstrong, M. Wang, R. Y. Teh, Q. Gong, Q. Y. He, J. Janousek, H. A. Bachor, M. D. Reid, and P. K. Lam, “Multipartite Einstein-Podolsky-Rosen steering and genuine tripartite entanglement with optical networks,” Nat. Phys. 11, 167 (2015).
[Crossref]

Bachor, H. A.

S. Armstrong, M. Wang, R. Y. Teh, Q. Gong, Q. Y. He, J. Janousek, H. A. Bachor, M. D. Reid, and P. K. Lam, “Multipartite Einstein-Podolsky-Rosen steering and genuine tripartite entanglement with optical networks,” Nat. Phys. 11, 167 (2015).
[Crossref]

M. D. Reid, P. D. Drummond, E. G. Cavalcanti, W. P. Bowen, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “Colloquium: The Einstein-Podolsky-Rosen paradox: From concepts to applications,” Rev. Mod. Phys.,  81, 1727–1751 (2009).
[Crossref]

Bennet, A.

A. Bennet, D. A. Evans, D. J. Saunders, C. Branciard, E. Cavalcanti, H. M. Wiseman, and G. J. Pryde, “Arbitrarily loss-tolerant Einstein-Podolsky-Rosen steering allowing a demonstration over 1 km of optical fiber with no detection loophole,” Phys. Rev. X 2, 031003 (2012).

Bowen, W. P.

M. D. Reid, P. D. Drummond, E. G. Cavalcanti, W. P. Bowen, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “Colloquium: The Einstein-Podolsky-Rosen paradox: From concepts to applications,” Rev. Mod. Phys.,  81, 1727–1751 (2009).
[Crossref]

Boyer, V.

C. S. Embrey, M. T. Turnbull, P. G. Petrov, and V. Boyer, “Observation of localized multi-spatial-mode quadrature squeezing,” Phys. Rev. X 5, 031004 (2015).

A. M. Marino, R. C. Pooser, V. Boyer, and P. D. Lett, “Tunable Delay of Einstein-Podolsky-Rosen Entanglement,” Nature 457, 859 (2009).
[Crossref] [PubMed]

V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, “Entangled images from Four-Wave Mixing,” Science 321, 544 (2008).
[Crossref] [PubMed]

C. F. McCormick, V. Boyer, E. Arimondo, and P. D. Lett, “Strong relative intensity squeezing by four-wave mixing in rubidium vapor,” Opt. Lett. 32, 178 (2007).
[Crossref]

Branciard, C.

D. H. Smith, G. Gillett, M. de Almeida, C. Branciard, A. Fedrizzi, T. J. Weinhold, A. Lita, B. Calkins, T. Gerrits, H. M. Wiseman, S. W. Nam, and A. G. White, “Conclusive quantum steering with superconducting transition-edge sensors,” Nature Commun. 3, 625 (2012).
[Crossref]

A. Bennet, D. A. Evans, D. J. Saunders, C. Branciard, E. Cavalcanti, H. M. Wiseman, and G. J. Pryde, “Arbitrarily loss-tolerant Einstein-Podolsky-Rosen steering allowing a demonstration over 1 km of optical fiber with no detection loophole,” Phys. Rev. X 2, 031003 (2012).

Brannan, T.

A. MacRae, T. Brannan, R. Achal, and A. I. Lvovsky, “Tomography of a high-purity narrowband photon from a transient atomic collective excitation,” Phys. Rev. Lett. 109, 033601 (2012).
[Crossref] [PubMed]

Braunstein, S. L.

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

S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513 (2005).
[Crossref]

Brunner, N.

B. Wittmann, S. Ramelow, F. Steinlechner, N. K. Langford, N. Brunner, H. Wiseman, R. Ursin, and A. Zeilinger, “A. Loophole-free Einstein-Podolsky-Rosen experiment via quantum steering,” New J. Phys. 14, 053030 (2012).
[Crossref]

Cai, Y.

Y. Cai, J. Feng, H. Wang, G. Ferrini, X. Xu, J. Jing, and N. Treps, “Quantum-network generation based on four-wave mixing,” Phys. Rev. A 91, 013843 (2015).
[Crossref]

Calkins, B.

D. H. Smith, G. Gillett, M. de Almeida, C. Branciard, A. Fedrizzi, T. J. Weinhold, A. Lita, B. Calkins, T. Gerrits, H. M. Wiseman, S. W. Nam, and A. G. White, “Conclusive quantum steering with superconducting transition-edge sensors,” Nature Commun. 3, 625 (2012).
[Crossref]

Camacho, Ryan M.

Ryan M. Camacho, Praveen K. Vudyasetu, and John C. Howell, “Four-wave-mixing stopped light in hot atomic rubidium vapour,” Nature Photon. 3, 103–106 (2009).
[Crossref]

Cao, L.

Z. Qin, L. Cao, H. Wang, A. M. Marino, W. Zhang, and J. Jing, “Experimental generation of multiple quantum correlated beams from hot rubidium vapor,” Phys. Rev. Lett. 113, 023602 (2014).
[Crossref] [PubMed]

Cavalcanti, E.

A. Bennet, D. A. Evans, D. J. Saunders, C. Branciard, E. Cavalcanti, H. M. Wiseman, and G. J. Pryde, “Arbitrarily loss-tolerant Einstein-Podolsky-Rosen steering allowing a demonstration over 1 km of optical fiber with no detection loophole,” Phys. Rev. X 2, 031003 (2012).

Cavalcanti, E. G.

M. D. Reid, P. D. Drummond, E. G. Cavalcanti, W. P. Bowen, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “Colloquium: The Einstein-Podolsky-Rosen paradox: From concepts to applications,” Rev. Mod. Phys.,  81, 1727–1751 (2009).
[Crossref]

Cerf, N. J.

C. Weedbrook, S. Pirandola, R. G. Patron, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621 (2012).
[Crossref]

Chen, Z. B.

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777 (2012).
[Crossref]

Cirac, J. I.

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 (2001).
[Crossref]

Clark, J. B.

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R. T. Glasser, U. Vogl, and P. D. Lett, “Stimulated generation of superluminal light pulses via four-wave mixing,” Phy. Rev. Lett. 108, 173902 (2012).
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Vudyasetu, Praveen K.

Ryan M. Camacho, Praveen K. Vudyasetu, and John C. Howell, “Four-wave-mixing stopped light in hot atomic rubidium vapour,” Nature Photon. 3, 103–106 (2009).
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Wang, H.

J. Xin, H. Wang, and J. Jing, “The effect of losses on the quantum-noise cancellation in the SU(1,1) interferometer,” Appl. Phys. Lett. 109, 051107 (2016).
[Crossref]

Y. Cai, J. Feng, H. Wang, G. Ferrini, X. Xu, J. Jing, and N. Treps, “Quantum-network generation based on four-wave mixing,” Phys. Rev. A 91, 013843 (2015).
[Crossref]

Z. Qin, L. Cao, H. Wang, A. M. Marino, W. Zhang, and J. Jing, “Experimental generation of multiple quantum correlated beams from hot rubidium vapor,” Phys. Rev. Lett. 113, 023602 (2014).
[Crossref] [PubMed]

J. Kong, J. Jing, H. Wang, C. Liu, and W. Zhang, “Experimental investigation of the visibility dependence in a nonlinear interferometer using parametric amplifiers,” Appl. Phys. Lett. 102, 011130 (2013).
[Crossref]

Wang, M.

S. Armstrong, M. Wang, R. Y. Teh, Q. Gong, Q. Y. He, J. Janousek, H. A. Bachor, M. D. Reid, and P. K. Lam, “Multipartite Einstein-Podolsky-Rosen steering and genuine tripartite entanglement with optical networks,” Nat. Phys. 11, 167 (2015).
[Crossref]

M. Wang, Q. H. Gong, Z. Ficek, and Q. Y. He, “Efficient scheme for perfect collective Einstein-Podolsky-Rosen steering,” Sci. Rep. 5, 12346 (2015).
[Crossref] [PubMed]

Y. Xiang, F. X. Sun, M. Wang, Q. H. Gong, and Q. Y. He, “Detection of genuine tripartite entanglement and steering in hybrid optomechanics,” Opt. Express 23, 30104–30117 (2015).
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M. Wang, Q. H. Gong, and Q. Y. He, “Collective multipartite Einstein-Podolsky-Rosen steering: more secure optical networks,” Opt. Lett. 39, 6703–6706 (2014).
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Weedbrook, C.

C. Weedbrook, S. Pirandola, R. G. Patron, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621 (2012).
[Crossref]

Weinfurter, H.

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777 (2012).
[Crossref]

Weinhold, T. J.

D. H. Smith, G. Gillett, M. de Almeida, C. Branciard, A. Fedrizzi, T. J. Weinhold, A. Lita, B. Calkins, T. Gerrits, H. M. Wiseman, S. W. Nam, and A. G. White, “Conclusive quantum steering with superconducting transition-edge sensors,” Nature Commun. 3, 625 (2012).
[Crossref]

White, A. G.

D. H. Smith, G. Gillett, M. de Almeida, C. Branciard, A. Fedrizzi, T. J. Weinhold, A. Lita, B. Calkins, T. Gerrits, H. M. Wiseman, S. W. Nam, and A. G. White, “Conclusive quantum steering with superconducting transition-edge sensors,” Nature Commun. 3, 625 (2012).
[Crossref]

Wiseman, H.

B. Wittmann, S. Ramelow, F. Steinlechner, N. K. Langford, N. Brunner, H. Wiseman, R. Ursin, and A. Zeilinger, “A. Loophole-free Einstein-Podolsky-Rosen experiment via quantum steering,” New J. Phys. 14, 053030 (2012).
[Crossref]

Wiseman, H. M.

A. Bennet, D. A. Evans, D. J. Saunders, C. Branciard, E. Cavalcanti, H. M. Wiseman, and G. J. Pryde, “Arbitrarily loss-tolerant Einstein-Podolsky-Rosen steering allowing a demonstration over 1 km of optical fiber with no detection loophole,” Phys. Rev. X 2, 031003 (2012).

D. H. Smith, G. Gillett, M. de Almeida, C. Branciard, A. Fedrizzi, T. J. Weinhold, A. Lita, B. Calkins, T. Gerrits, H. M. Wiseman, S. W. Nam, and A. G. White, “Conclusive quantum steering with superconducting transition-edge sensors,” Nature Commun. 3, 625 (2012).
[Crossref]

D. J. Saunders, S. J. Jones, H. M. Wiseman, and G. J. Pryde, “Experimental EPR-steering using Bell-local states,” Nature Phys.,  6, 845–849 (2010).
[Crossref]

Wittmann, B.

B. Wittmann, S. Ramelow, F. Steinlechner, N. K. Langford, N. Brunner, H. Wiseman, R. Ursin, and A. Zeilinger, “A. Loophole-free Einstein-Podolsky-Rosen experiment via quantum steering,” New J. Phys. 14, 053030 (2012).
[Crossref]

Xiang, Y.

X. Deng, Y. Xiang, C. Tian, G. Adesso, Q. He, Q. Gong, X. Su, C. Xie, and K. Peng, “Demonstration of monogamy relations for Einstein-Podolsky-Rosen steering in Gaussian cluster states,” Phys. Rev. Lett 118, 230501 (2017).
[Crossref] [PubMed]

Y. Xiang, F. X. Sun, M. Wang, Q. H. Gong, and Q. Y. He, “Detection of genuine tripartite entanglement and steering in hybrid optomechanics,” Opt. Express 23, 30104–30117 (2015).
[Crossref] [PubMed]

Xie, C.

X. Deng, Y. Xiang, C. Tian, G. Adesso, Q. He, Q. Gong, X. Su, C. Xie, and K. Peng, “Demonstration of monogamy relations for Einstein-Podolsky-Rosen steering in Gaussian cluster states,” Phys. Rev. Lett 118, 230501 (2017).
[Crossref] [PubMed]

Xin, J.

J. Xin, H. Wang, and J. Jing, “The effect of losses on the quantum-noise cancellation in the SU(1,1) interferometer,” Appl. Phys. Lett. 109, 051107 (2016).
[Crossref]

Xu, X.

Y. Cai, J. Feng, H. Wang, G. Ferrini, X. Xu, J. Jing, and N. Treps, “Quantum-network generation based on four-wave mixing,” Phys. Rev. A 91, 013843 (2015).
[Crossref]

Zeilinger, A.

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777 (2012).
[Crossref]

B. Wittmann, S. Ramelow, F. Steinlechner, N. K. Langford, N. Brunner, H. Wiseman, R. Ursin, and A. Zeilinger, “A. Loophole-free Einstein-Podolsky-Rosen experiment via quantum steering,” New J. Phys. 14, 053030 (2012).
[Crossref]

Zhang, J.

J. Zhang and S. L. Braunstein, “Contunuous-variable Gaussian analog of cluster state,” Phys. Rev. A 73, 032318 (2006).
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Zhang, W.

F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014).
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Z. Qin, L. Cao, H. Wang, A. M. Marino, W. Zhang, and J. Jing, “Experimental generation of multiple quantum correlated beams from hot rubidium vapor,” Phys. Rev. Lett. 113, 023602 (2014).
[Crossref] [PubMed]

J. Kong, J. Jing, H. Wang, C. Liu, and W. Zhang, “Experimental investigation of the visibility dependence in a nonlinear interferometer using parametric amplifiers,” Appl. Phys. Lett. 102, 011130 (2013).
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J. Jing, C. Liu, Z. Zhou, Z. Y. Ou, and W. Zhang, “Realization of a nonlinear interferometer with parametric amplifiers,” Appl. Phys. Lett. 99, 011110 (2011).
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Zhou, Z.

J. Jing, C. Liu, Z. Zhou, Z. Y. Ou, and W. Zhang, “Realization of a nonlinear interferometer with parametric amplifiers,” Appl. Phys. Lett. 99, 011110 (2011).
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Zoller, P.

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 (2001).
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Zukowski, M.

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777 (2012).
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Appl. Phys. Lett. (3)

J. Jing, C. Liu, Z. Zhou, Z. Y. Ou, and W. Zhang, “Realization of a nonlinear interferometer with parametric amplifiers,” Appl. Phys. Lett. 99, 011110 (2011).
[Crossref]

J. Kong, J. Jing, H. Wang, C. Liu, and W. Zhang, “Experimental investigation of the visibility dependence in a nonlinear interferometer using parametric amplifiers,” Appl. Phys. Lett. 102, 011130 (2013).
[Crossref]

J. Xin, H. Wang, and J. Jing, “The effect of losses on the quantum-noise cancellation in the SU(1,1) interferometer,” Appl. Phys. Lett. 109, 051107 (2016).
[Crossref]

Nat. Commun. (1)

F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014).
[Crossref] [PubMed]

Nat. Phys. (1)

S. Armstrong, M. Wang, R. Y. Teh, Q. Gong, Q. Y. He, J. Janousek, H. A. Bachor, M. D. Reid, and P. K. Lam, “Multipartite Einstein-Podolsky-Rosen steering and genuine tripartite entanglement with optical networks,” Nat. Phys. 11, 167 (2015).
[Crossref]

Nature (1)

A. M. Marino, R. C. Pooser, V. Boyer, and P. D. Lett, “Tunable Delay of Einstein-Podolsky-Rosen Entanglement,” Nature 457, 859 (2009).
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Nature (London) (2)

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 (2001).
[Crossref]

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

Nature Commun. (1)

D. H. Smith, G. Gillett, M. de Almeida, C. Branciard, A. Fedrizzi, T. J. Weinhold, A. Lita, B. Calkins, T. Gerrits, H. M. Wiseman, S. W. Nam, and A. G. White, “Conclusive quantum steering with superconducting transition-edge sensors,” Nature Commun. 3, 625 (2012).
[Crossref]

Nature Photon. (1)

Ryan M. Camacho, Praveen K. Vudyasetu, and John C. Howell, “Four-wave-mixing stopped light in hot atomic rubidium vapour,” Nature Photon. 3, 103–106 (2009).
[Crossref]

Nature Phys. (1)

D. J. Saunders, S. J. Jones, H. M. Wiseman, and G. J. Pryde, “Experimental EPR-steering using Bell-local states,” Nature Phys.,  6, 845–849 (2010).
[Crossref]

New J. Phys. (1)

B. Wittmann, S. Ramelow, F. Steinlechner, N. K. Langford, N. Brunner, H. Wiseman, R. Ursin, and A. Zeilinger, “A. Loophole-free Einstein-Podolsky-Rosen experiment via quantum steering,” New J. Phys. 14, 053030 (2012).
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Opt. Express (1)

Opt. Lett. (4)

Optica (2)

Optics Express (1)

M. Jasperse, L. D. Turner, and R. E. Scholten, “Relative intensity squeezing by four-wave mixing with loss: an analytic model and experimental diagnostic,” Optics Express 19(4), 3765–3774 (2011).
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Phy. Rev. Lett. (2)

R. T. Glasser, U. Vogl, and P. D. Lett, “Stimulated generation of superluminal light pulses via four-wave mixing,” Phy. Rev. Lett. 108, 173902 (2012).
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N. V. Corzo, A. M. Marino, K. M. Jones, and P. D. Lett, “Noiseless optical amplifier operating on hundreds of spatial modes,” Phy. Rev. Lett. 109, 043602 (2012).
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Phys. Rev. A (4)

R. Pooser and J. Jing, “Continuous variable cluster state generation over the optical spatial mode comb,” Phys. Rev. A 90, 043841 (2014).
[Crossref]

Y. Cai, J. Feng, H. Wang, G. Ferrini, X. Xu, J. Jing, and N. Treps, “Quantum-network generation based on four-wave mixing,” Phys. Rev. A 91, 013843 (2015).
[Crossref]

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

N. V. Corzo, Quentin Glorieux, A. M. Marino, J. B. Clark, R. T. Glasser, and P. D. Lett., “Rotation of the noise ellipse for squeezed vacuum light generated via four-wave mixing,” Phys. Rev. A 88, 043836 (2013).
[Crossref]

Phys. Rev. Lett (1)

X. Deng, Y. Xiang, C. Tian, G. Adesso, Q. He, Q. Gong, X. Su, C. Xie, and K. Peng, “Demonstration of monogamy relations for Einstein-Podolsky-Rosen steering in Gaussian cluster states,” Phys. Rev. Lett 118, 230501 (2017).
[Crossref] [PubMed]

Phys. Rev. Lett. (3)

Z. Qin, L. Cao, H. Wang, A. M. Marino, W. Zhang, and J. Jing, “Experimental generation of multiple quantum correlated beams from hot rubidium vapor,” Phys. Rev. Lett. 113, 023602 (2014).
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A. MacRae, T. Brannan, R. Achal, and A. I. Lvovsky, “Tomography of a high-purity narrowband photon from a transient atomic collective excitation,” Phys. Rev. Lett. 109, 033601 (2012).
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Q. Y. He and M. D. Reid, “Genuine Multipartite Einstein-Podolsky-Rosen Steering,” Phys. Rev. Lett. 111, 250403 (2013).
[Crossref]

Phys. Rev. X (2)

A. Bennet, D. A. Evans, D. J. Saunders, C. Branciard, E. Cavalcanti, H. M. Wiseman, and G. J. Pryde, “Arbitrarily loss-tolerant Einstein-Podolsky-Rosen steering allowing a demonstration over 1 km of optical fiber with no detection loophole,” Phys. Rev. X 2, 031003 (2012).

C. S. Embrey, M. T. Turnbull, P. G. Petrov, and V. Boyer, “Observation of localized multi-spatial-mode quadrature squeezing,” Phys. Rev. X 5, 031004 (2015).

Physical Review A (1)

Q. Y. He and Z. Ficek, “EPR paradox and quantum steering in a three-mode optomechanical system,” Physical Review A,  89(2), 74–79 (2014).
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Rev. Mod. Phys. (5)

M. D. Reid, P. D. Drummond, E. G. Cavalcanti, W. P. Bowen, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “Colloquium: The Einstein-Podolsky-Rosen paradox: From concepts to applications,” Rev. Mod. Phys.,  81, 1727–1751 (2009).
[Crossref]

C. Weedbrook, S. Pirandola, R. G. Patron, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621 (2012).
[Crossref]

M. D. Lukin, “Trapping and manipulating photon states in atomic ensembles,” Rev. Mod. Phys. 75, 457 (2003).
[Crossref]

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777 (2012).
[Crossref]

S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513 (2005).
[Crossref]

Sci. Rep. (1)

M. Wang, Q. H. Gong, Z. Ficek, and Q. Y. He, “Efficient scheme for perfect collective Einstein-Podolsky-Rosen steering,” Sci. Rep. 5, 12346 (2015).
[Crossref] [PubMed]

Science (1)

V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, “Entangled images from Four-Wave Mixing,” Science 321, 544 (2008).
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Other (1)

M. Kafatos, Bell’s Theorem, Quantum Theory and Conceptions of the Universe (Springer1989).
[Crossref]

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Figures (5)
Fig. 1
Fig. 1 Cascaded FWM processes in hot Rb vapor. (a) Double-λ energy level of Rb D1 line: Δ and δ stand for the one-photon detuning and the two-photon detuning respectively. The interaction strength depends strongly on the one-photon detuning Δ and the two-photon detuning δ. (b) The cascaded FWM scheme.

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Fig. 2
Fig. 2 The values of St12 (a), St21 (b), St13 (c), St31 (d), St23 (e) and St32 (f) through equations (2–5) vary with G1 and G2.

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Fig. 3
Fig. 3 The mutual hierarchical relations of the bipartite steering among the output beams Ô1, Ô2 and Ô3.

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Fig. 4
Fig. 4 The values of St123 (a), St213 (b), St312 (c) in equation (2, 6–8) and St123+St213+St312 (d) vary with G1 and G2.

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Fig. 5
Fig. 5 The value of St123 + St213 + St312 vary with G1 and G2 when we consider losses due to imperfect optical transmission and detection efficiency.

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Equations (18)

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

H ^ 1 = i ℏ ζ 1 ψ c 1 2 b ^ 1 † a ^ 1 † + h . c . ,
H ^ 2 = i ℏ ζ 2 ψ c 2 2 b ^ 2 † a ^ 2 † + h . c . .
a ^ 1 ( t ) = G 1 a ^ 0 + G 1 − 1 v ^ 0 † ,
b ^ 1 † ( t ) = G 1 v ^ 0 † + G 1 − 1 a ^ 0 .
a ^ 2 ( t ) = G 2 a ^ 1 + G 2 − 1 v ^ 0 ′ † ,
b ^ 2 † ( t ) = G 2 v ^ 0 ′ † + G 2 − 1 a ^ 1 .
( a ^ 2 ( t ) b ^ 2 † ( t ) b ^ 1 † ( t ) ) = ( G 1 G 2 ( G 1 − 1 ) G 2 G 2 − 1 G 1 ( G 1 − 1 ) ( G 1 − 1 ) ( G 2 − 1 ) G 2 G 1 − 1 G 1 0 ) ( a ^ 0 v ^ 0 † v ^ ′ 0 † ) .
X ^ i = 1 2 ( Q ^ i † + Q ^ i ) , Y ^ i = i 2 ( Q ^ i † − Q ^ i ) .
St i j = Δ inf ( X ^ i j ) Δ inf ( Y ^ i j ) ,
Δ inf ( X ^ i j ) = Δ ( X ^ i + g o p t , X ^ j X ^ j ) ,
Δ inf ( Y ^ i j ) = Δ ( Y ^ i + g o p t , Y ^ j Y ^ j ) .
St i j k = Δ inf ( X ^ i j k ) Δ inf ( Y ^ i j k ) .
Δ inf ( X ^ i j k ) = Δ ( X ^ i + g o p t , X ^ j X ^ j + g o p t , X ^ k X ^ k )
Δ inf ( Y ^ i j k ) = Δ ( Y ^ i + g o p t , Y ^ j Y ^ j + g o p t , X ^ k X ^ k )
a ^ 1 ( t ) → η 1 a ^ 1 + 1 − η 1 ν ^ 1 ,
b ^ 1 ( t ) → η 2 b ^ 2 + 1 − η 2 ν ^ 2 ,
a ^ 2 ( t ) → η 3 a ^ 2 + 1 − η 3 ν ^ 3 ,
b ^ 2 ( t ) → η 4 b ^ 2 + 1 − η 4 ν ^ 2 .

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