Abstract

We propose a novel scheme to generate the entanglement between two cavity optomechanical systems (COMSs) via a flying two-level atom. We derive the analytical expressions for the generated entangled states. We find that there exist two processes for generating entanglement: one is the entanglement transfer between the two phonon-modes, and the other is the entanglement swapping-like process among the two photon-modes and the two phonon-modes. We analyze these two kinds of phenomena, respectively, by adjusting the distance between the two COMSs. Then we discuss the verification of the generated entangled states of the two COMSs, and analyze the decoherence of the generated entangled states. Finally, we discuss the experimental feasibility of our proposal.

© 2017 Optical Society of America

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  1. R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865–942 (2009).
    [Crossref]
  2. C. Monroe, D. M. Meekhof, B. E. King, and D. J. Wineland, “A Schrodinger cat superposition state of an atom,” Science 272, 1131 (1996).
    [Crossref] [PubMed]
  3. M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. Van der Zouw, and A. Zeilinger, “Wave–particle duality of C60 molecules,” Nature 401, 680–682 (1999).
    [Crossref]
  4. S. Gerlich, S. Eibenberger, M. Tomandl, S. Nimmrichter, K. Hornberger, P. J. Fagan, J. Tüen, M. Mayor, and M. Arndt, “Quantum interference of large organic molecules,” Nat. Commun. 2, 263 (2011).
    [Crossref] [PubMed]
  5. D. P. DiVincenzo, “Quantum computation,” Science 270, 255 (1995).
    [Crossref]
  6. A. Ekert and R. Jozsa, “Shor’s quantum algorithm for factorising numbers,” Rev. Mod. Phys. 68, 733–753 (1996).
    [Crossref]
  7. L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414, 413–418 (2001).
    [Crossref] [PubMed]
  8. V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photon. 5, 222–229 (2011).
    [Crossref]
  9. A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67, 661 (1991).
    [Crossref] [PubMed]
  10. V. Scarani, S. Iblisdir, N. Gisin, and A. Acin, “Quantum cloning,” Rev. Mod. Phys. 77, 1225 (2005).
    [Crossref]
  11. P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337 (1995).
    [Crossref] [PubMed]
  12. E. Hagley, X. Maitre, 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 (1997).
    [Crossref]
  13. 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 (1998).
    [Crossref]
  14. M. Bayer, P. Hawrylak, K. Hinzer, S. Fafard, M. Korkusinski, Z. R. Wasilewski, and A. Forchel, “Coupling and entangling of quantum states in quantum dot molecules,” Science 291, 451–453 (2001).
    [Crossref] [PubMed]
  15. A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87, 123602 (2001).
    [Crossref] [PubMed]
  16. J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature 426, 264–267 (2003).
    [Crossref]
  17. Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Nonlocal entanglement concentration scheme for partially entangled multipartite systems with nonlinear optics,” Phys. Rev. A 77, 062325 (2008).
    [Crossref]
  18. S. Horoche and D. Kleppner, “Cavity quantum electrodynamics,” Phys. Today 42, 24 (1989).
    [Crossref]
  19. A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004).
    [Crossref] [PubMed]
  20. R. Blatt and D. Wineland, “Entangled states of trapped atomic ions,” Nature 453, 1008–1015 (2008).
    [Crossref] [PubMed]
  21. H. Häfner, W. Häsel, C. F. Roos, J. Benhelm, M. Chwalla, T. Köber, and O. Güne, “Scalable multiparticle entanglement of trapped ions,” Nature 438, 643–646 (2005).
    [Crossref]
  22. T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321, 1172–1176 (2008).
    [Crossref] [PubMed]
  23. P. Meystre, “A short walk through quantum optomechanics,” Ann. Phys. (Berlin) 525, 215 (2013).
    [Crossref]
  24. Z. R. Gong, H. Ian, Y. X. Liu, C. P. Sun, and F. Nori, “Effective Hamiltonian approach to the Kerr nonlinearity in an optomechanical system,” Phys. Rev. A 80, 065801 (2009).
    [Crossref]
  25. R. Ghobadi, A. R. Bahrampour, and C. Simon, “Quantum optomechanics in the bistable regime,” Phys. Rev. A 84, 033846 (2011).
    [Crossref]
  26. C. H. Dong, V. Fiore, M. C. Kuzyk, and H. L. Wang, “Optomechanical dark mode,” Science 338, 1609–1613 (2012).
    [Crossref] [PubMed]
  27. A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
    [Crossref] [PubMed]
  28. S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
    [Crossref] [PubMed]
  29. M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).
    [Crossref]
  30. J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3, 1196 (2012).
    [Crossref] [PubMed]
  31. S. Bose, K. Jacobs, and P. L. Knight, “Preparation of nonclassical states in cavities with a moving mirror,” Phys. Rev. A 56, 4175 (1997).
    [Crossref]
  32. W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards quantum superpositions of a mirror,” Phys. Rev. Lett. 91, 130401 (2010).
    [Crossref]
  33. S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401 (2002).
    [Crossref] [PubMed]
  34. F. Xue, Y. X. Liu, C. P. Sun, and F. Nori, “Two-mode squeezed states and entangled states of two mechanical resonators,” Phys. Rev. B 76, 064305 (2007).
    [Crossref]
  35. G. Vacanti, M. Paternostro, G. M. Palma, and V. Vedral, “Optomechanical to mechanical entanglement transformation,” New J. Phys. 10, 095014 (2008).
    [Crossref]
  36. S. Huang and G. S. Agarwal, “Entangling nanomechanical oscillators in a ring cavity by feeding squeezed light,” New J. Phys. 11, 103044 (2009).
    [Crossref]
  37. X. W. Xu, Y. J. Zhao, and Y. X. Liu, “Entangled-state engineering of vibrational modes in a multimembrane optomechanical system,” Phys. Rev. A 88, 022325 (2013).
    [Crossref]
  38. C. Joshi, J. Larson, M. Jonson, E. Andersson, and P. Öhberg, “Entanglement of distant optomechanical systems,” Phys. Rev. A 85, 033805 (2012).
    [Crossref]
  39. J. Zhang, K. Peng, and S. L. Braunstein, “Quantum-state transfer from light to macroscopic oscillators,” Phys. Rev. A 68, 013808 (2003).
    [Crossref]
  40. D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98, 030405 (2007).
    [Crossref] [PubMed]
  41. T. A. Palomaki, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, “Entangling mechanical motion with microwave fields,” Science 342, 710–713 (2013).
    [Crossref] [PubMed]
  42. J. Q. Liao, Q. Q. Wu, and F. Nori, “Entangling two macroscopic mechanical mirrors in a two-cavity optomechanical system,” Phys. Rev. A 89, 014302 (2014).
    [Crossref]
  43. M. Zukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, “Event-ready-detectors Bell experiment via entanglement swapping,” Phys. Rev. Lett. 71, 4287–4290 (1993).
    [Crossref] [PubMed]
  44. L. M. Kuang and Z. Lan, “Generation of atom-photon entangled states in atomic bose-einstein condensate via electromagnetically induced transparency,” Phys. Rev. A 68, 043606 (2004).
    [Crossref]
  45. C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge University Press, 2005), Chap. 8.
  46. J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
    [Crossref] [PubMed]
  47. J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707–712 (2010).
    [Crossref]
  48. S. J. Phoenix, “Wave-packet evolution in the damped oscillator,” Phys. Rev. A 41, 5132 (1990).
    [Crossref] [PubMed]

2014 (1)

J. Q. Liao, Q. Q. Wu, and F. Nori, “Entangling two macroscopic mechanical mirrors in a two-cavity optomechanical system,” Phys. Rev. A 89, 014302 (2014).
[Crossref]

2013 (4)

T. A. Palomaki, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, “Entangling mechanical motion with microwave fields,” Science 342, 710–713 (2013).
[Crossref] [PubMed]

X. W. Xu, Y. J. Zhao, and Y. X. Liu, “Entangled-state engineering of vibrational modes in a multimembrane optomechanical system,” Phys. Rev. A 88, 022325 (2013).
[Crossref]

P. Meystre, “A short walk through quantum optomechanics,” Ann. Phys. (Berlin) 525, 215 (2013).
[Crossref]

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).
[Crossref]

2012 (3)

J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3, 1196 (2012).
[Crossref] [PubMed]

C. H. Dong, V. Fiore, M. C. Kuzyk, and H. L. Wang, “Optomechanical dark mode,” Science 338, 1609–1613 (2012).
[Crossref] [PubMed]

C. Joshi, J. Larson, M. Jonson, E. Andersson, and P. Öhberg, “Entanglement of distant optomechanical systems,” Phys. Rev. A 85, 033805 (2012).
[Crossref]

2011 (4)

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

R. Ghobadi, A. R. Bahrampour, and C. Simon, “Quantum optomechanics in the bistable regime,” Phys. Rev. A 84, 033846 (2011).
[Crossref]

S. Gerlich, S. Eibenberger, M. Tomandl, S. Nimmrichter, K. Hornberger, P. J. Fagan, J. Tüen, M. Mayor, and M. Arndt, “Quantum interference of large organic molecules,” Nat. Commun. 2, 263 (2011).
[Crossref] [PubMed]

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photon. 5, 222–229 (2011).
[Crossref]

2010 (3)

W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards quantum superpositions of a mirror,” Phys. Rev. Lett. 91, 130401 (2010).
[Crossref]

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref] [PubMed]

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707–712 (2010).
[Crossref]

2009 (3)

S. Huang and G. S. Agarwal, “Entangling nanomechanical oscillators in a ring cavity by feeding squeezed light,” New J. Phys. 11, 103044 (2009).
[Crossref]

Z. R. Gong, H. Ian, Y. X. Liu, C. P. Sun, and F. Nori, “Effective Hamiltonian approach to the Kerr nonlinearity in an optomechanical system,” Phys. Rev. A 80, 065801 (2009).
[Crossref]

R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865–942 (2009).
[Crossref]

2008 (5)

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321, 1172–1176 (2008).
[Crossref] [PubMed]

Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Nonlocal entanglement concentration scheme for partially entangled multipartite systems with nonlinear optics,” Phys. Rev. A 77, 062325 (2008).
[Crossref]

R. Blatt and D. Wineland, “Entangled states of trapped atomic ions,” Nature 453, 1008–1015 (2008).
[Crossref] [PubMed]

G. Vacanti, M. Paternostro, G. M. Palma, and V. Vedral, “Optomechanical to mechanical entanglement transformation,” New J. Phys. 10, 095014 (2008).
[Crossref]

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref] [PubMed]

2007 (2)

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98, 030405 (2007).
[Crossref] [PubMed]

F. Xue, Y. X. Liu, C. P. Sun, and F. Nori, “Two-mode squeezed states and entangled states of two mechanical resonators,” Phys. Rev. B 76, 064305 (2007).
[Crossref]

2005 (2)

H. Häfner, W. Häsel, C. F. Roos, J. Benhelm, M. Chwalla, T. Köber, and O. Güne, “Scalable multiparticle entanglement of trapped ions,” Nature 438, 643–646 (2005).
[Crossref]

V. Scarani, S. Iblisdir, N. Gisin, and A. Acin, “Quantum cloning,” Rev. Mod. Phys. 77, 1225 (2005).
[Crossref]

2004 (2)

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004).
[Crossref] [PubMed]

L. M. Kuang and Z. Lan, “Generation of atom-photon entangled states in atomic bose-einstein condensate via electromagnetically induced transparency,” Phys. Rev. A 68, 043606 (2004).
[Crossref]

2003 (2)

J. Zhang, K. Peng, and S. L. Braunstein, “Quantum-state transfer from light to macroscopic oscillators,” Phys. Rev. A 68, 013808 (2003).
[Crossref]

J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature 426, 264–267 (2003).
[Crossref]

2002 (1)

S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401 (2002).
[Crossref] [PubMed]

2001 (3)

M. Bayer, P. Hawrylak, K. Hinzer, S. Fafard, M. Korkusinski, Z. R. Wasilewski, and A. Forchel, “Coupling and entangling of quantum states in quantum dot molecules,” Science 291, 451–453 (2001).
[Crossref] [PubMed]

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87, 123602 (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, 413–418 (2001).
[Crossref] [PubMed]

1999 (1)

M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. Van der Zouw, and A. Zeilinger, “Wave–particle duality of C60 molecules,” Nature 401, 680–682 (1999).
[Crossref]

1998 (1)

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 (1998).
[Crossref]

1997 (2)

E. Hagley, X. Maitre, 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 (1997).
[Crossref]

S. Bose, K. Jacobs, and P. L. Knight, “Preparation of nonclassical states in cavities with a moving mirror,” Phys. Rev. A 56, 4175 (1997).
[Crossref]

1996 (2)

A. Ekert and R. Jozsa, “Shor’s quantum algorithm for factorising numbers,” Rev. Mod. Phys. 68, 733–753 (1996).
[Crossref]

C. Monroe, D. M. Meekhof, B. E. King, and D. J. Wineland, “A Schrodinger cat superposition state of an atom,” Science 272, 1131 (1996).
[Crossref] [PubMed]

1995 (2)

D. P. DiVincenzo, “Quantum computation,” Science 270, 255 (1995).
[Crossref]

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

1993 (1)

M. Zukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, “Event-ready-detectors Bell experiment via entanglement swapping,” Phys. Rev. Lett. 71, 4287–4290 (1993).
[Crossref] [PubMed]

1991 (1)

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

1990 (1)

S. J. Phoenix, “Wave-packet evolution in the damped oscillator,” Phys. Rev. A 41, 5132 (1990).
[Crossref] [PubMed]

1989 (1)

S. Horoche and D. Kleppner, “Cavity quantum electrodynamics,” Phys. Today 42, 24 (1989).
[Crossref]

Abouraddy, A. F.

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87, 123602 (2001).
[Crossref] [PubMed]

Acin, A.

V. Scarani, S. Iblisdir, N. Gisin, and A. Acin, “Quantum cloning,” Rev. Mod. Phys. 77, 1225 (2005).
[Crossref]

Agarwal, G. S.

S. Huang and G. S. Agarwal, “Entangling nanomechanical oscillators in a ring cavity by feeding squeezed light,” New J. Phys. 11, 103044 (2009).
[Crossref]

Alegre, T. P. M.

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

Andersson, E.

C. Joshi, J. Larson, M. Jonson, E. Andersson, and P. Öhberg, “Entanglement of distant optomechanical systems,” Phys. Rev. A 85, 033805 (2012).
[Crossref]

Arcizet, O.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref] [PubMed]

Arndt, M.

S. Gerlich, S. Eibenberger, M. Tomandl, S. Nimmrichter, K. Hornberger, P. J. Fagan, J. Tüen, M. Mayor, and M. Arndt, “Quantum interference of large organic molecules,” Nat. Commun. 2, 263 (2011).
[Crossref] [PubMed]

M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. Van der Zouw, and A. Zeilinger, “Wave–particle duality of C60 molecules,” Nature 401, 680–682 (1999).
[Crossref]

Aspelmeyer, M.

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98, 030405 (2007).
[Crossref] [PubMed]

Bahrampour, A. R.

R. Ghobadi, A. R. Bahrampour, and C. Simon, “Quantum optomechanics in the bistable regime,” Phys. Rev. A 84, 033846 (2011).
[Crossref]

Bawaj, M.

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).
[Crossref]

Bayer, M.

M. Bayer, P. Hawrylak, K. Hinzer, S. Fafard, M. Korkusinski, Z. R. Wasilewski, and A. Forchel, “Coupling and entangling of quantum states in quantum dot molecules,” Science 291, 451–453 (2001).
[Crossref] [PubMed]

Benhelm, J.

H. Häfner, W. Häsel, C. F. Roos, J. Benhelm, M. Chwalla, T. Köber, and O. Güne, “Scalable multiparticle entanglement of trapped ions,” Nature 438, 643–646 (2005).
[Crossref]

Biancofiore, C.

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).
[Crossref]

Blais, A.

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004).
[Crossref] [PubMed]

Blatt, R.

R. Blatt and D. Wineland, “Entangled states of trapped atomic ions,” Nature 453, 1008–1015 (2008).
[Crossref] [PubMed]

Böhm, H. R.

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98, 030405 (2007).
[Crossref] [PubMed]

Bose, S.

S. Bose, K. Jacobs, and P. L. Knight, “Preparation of nonclassical states in cavities with a moving mirror,” Phys. Rev. A 56, 4175 (1997).
[Crossref]

Bouwmeester, D.

W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards quantum superpositions of a mirror,” Phys. Rev. Lett. 91, 130401 (2010).
[Crossref]

Branning, D.

J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature 426, 264–267 (2003).
[Crossref]

Braunstein, S. L.

J. Zhang, K. Peng, and S. L. Braunstein, “Quantum-state transfer from light to macroscopic oscillators,” Phys. Rev. A 68, 013808 (2003).
[Crossref]

Brune, M.

E. Hagley, X. Maitre, 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 (1997).
[Crossref]

Chan, J.

J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3, 1196 (2012).
[Crossref] [PubMed]

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

Chang, D.

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

Chwalla, M.

H. Häfner, W. Häsel, C. F. Roos, J. Benhelm, M. Chwalla, T. Köber, and O. Güne, “Scalable multiparticle entanglement of trapped ions,” Nature 438, 643–646 (2005).
[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 414, 413–418 (2001).
[Crossref] [PubMed]

Deléglise, S.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref] [PubMed]

Deng, F. G.

Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Nonlocal entanglement concentration scheme for partially entangled multipartite systems with nonlinear optics,” Phys. Rev. A 77, 062325 (2008).
[Crossref]

Di Giuseppe, G.

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).
[Crossref]

DiVincenzo, D. P.

D. P. DiVincenzo, “Quantum computation,” Science 270, 255 (1995).
[Crossref]

Dong, C. H.

C. H. Dong, V. Fiore, M. C. Kuzyk, and H. L. Wang, “Optomechanical dark mode,” Science 338, 1609–1613 (2012).
[Crossref] [PubMed]

Duan, L. M.

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

Eibenberger, S.

S. Gerlich, S. Eibenberger, M. Tomandl, S. Nimmrichter, K. Hornberger, P. J. Fagan, J. Tüen, M. Mayor, and M. Arndt, “Quantum interference of large organic molecules,” Nat. Commun. 2, 263 (2011).
[Crossref] [PubMed]

Eichenfield, M.

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

Ekert, A.

A. Ekert and R. Jozsa, “Shor’s quantum algorithm for factorising numbers,” Rev. Mod. Phys. 68, 733–753 (1996).
[Crossref]

Ekert, A. K.

M. Zukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, “Event-ready-detectors Bell experiment via entanglement swapping,” Phys. Rev. Lett. 71, 4287–4290 (1993).
[Crossref] [PubMed]

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

Fafard, S.

M. Bayer, P. Hawrylak, K. Hinzer, S. Fafard, M. Korkusinski, Z. R. Wasilewski, and A. Forchel, “Coupling and entangling of quantum states in quantum dot molecules,” Science 291, 451–453 (2001).
[Crossref] [PubMed]

Fagan, P. J.

S. Gerlich, S. Eibenberger, M. Tomandl, S. Nimmrichter, K. Hornberger, P. J. Fagan, J. Tüen, M. Mayor, and M. Arndt, “Quantum interference of large organic molecules,” Nat. Commun. 2, 263 (2011).
[Crossref] [PubMed]

Ferreira, A.

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98, 030405 (2007).
[Crossref] [PubMed]

Fiore, V.

C. H. Dong, V. Fiore, M. C. Kuzyk, and H. L. Wang, “Optomechanical dark mode,” Science 338, 1609–1613 (2012).
[Crossref] [PubMed]

Forchel, A.

M. Bayer, P. Hawrylak, K. Hinzer, S. Fafard, M. Korkusinski, Z. R. Wasilewski, and A. Forchel, “Coupling and entangling of quantum states in quantum dot molecules,” Science 291, 451–453 (2001).
[Crossref] [PubMed]

Frunzio, L.

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004).
[Crossref] [PubMed]

Galassi, M.

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).
[Crossref]

Gavartin, E.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref] [PubMed]

Gerlich, S.

S. Gerlich, S. Eibenberger, M. Tomandl, S. Nimmrichter, K. Hornberger, P. J. Fagan, J. Tüen, M. Mayor, and M. Arndt, “Quantum interference of large organic molecules,” Nat. Commun. 2, 263 (2011).
[Crossref] [PubMed]

Gerry, C. C.

C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge University Press, 2005), Chap. 8.

Ghobadi, R.

R. Ghobadi, A. R. Bahrampour, and C. Simon, “Quantum optomechanics in the bistable regime,” Phys. Rev. A 84, 033846 (2011).
[Crossref]

Gigan, S.

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98, 030405 (2007).
[Crossref] [PubMed]

Giovannetti, V.

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photon. 5, 222–229 (2011).
[Crossref]

S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401 (2002).
[Crossref] [PubMed]

Girvin, S. M.

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref] [PubMed]

Gisin, N.

V. Scarani, S. Iblisdir, N. Gisin, and A. Acin, “Quantum cloning,” Rev. Mod. Phys. 77, 1225 (2005).
[Crossref]

Gong, Z. R.

Z. R. Gong, H. Ian, Y. X. Liu, C. P. Sun, and F. Nori, “Effective Hamiltonian approach to the Kerr nonlinearity in an optomechanical system,” Phys. Rev. A 80, 065801 (2009).
[Crossref]

Guerreiro, A.

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98, 030405 (2007).
[Crossref] [PubMed]

Güne, O.

H. Häfner, W. Häsel, C. F. Roos, J. Benhelm, M. Chwalla, T. Köber, and O. Güne, “Scalable multiparticle entanglement of trapped ions,” Nature 438, 643–646 (2005).
[Crossref]

Häfner, H.

H. Häfner, W. Häsel, C. F. Roos, J. Benhelm, M. Chwalla, T. Köber, and O. Güne, “Scalable multiparticle entanglement of trapped ions,” Nature 438, 643–646 (2005).
[Crossref]

Hagley, E.

E. Hagley, X. Maitre, 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 (1997).
[Crossref]

Haroche, S.

E. Hagley, X. Maitre, 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 (1997).
[Crossref]

Harris, J. G.

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707–712 (2010).
[Crossref]

Harris, J. G. E.

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref] [PubMed]

Häsel, W.

H. Häfner, W. Häsel, C. F. Roos, J. Benhelm, M. Chwalla, T. Köber, and O. Güne, “Scalable multiparticle entanglement of trapped ions,” Nature 438, 643–646 (2005).
[Crossref]

Hawrylak, P.

M. Bayer, P. Hawrylak, K. Hinzer, S. Fafard, M. Korkusinski, Z. R. Wasilewski, and A. Forchel, “Coupling and entangling of quantum states in quantum dot molecules,” Science 291, 451–453 (2001).
[Crossref] [PubMed]

Hill, J. T.

J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3, 1196 (2012).
[Crossref] [PubMed]

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

Hinzer, K.

M. Bayer, P. Hawrylak, K. Hinzer, S. Fafard, M. Korkusinski, Z. R. Wasilewski, and A. Forchel, “Coupling and entangling of quantum states in quantum dot molecules,” Science 291, 451–453 (2001).
[Crossref] [PubMed]

Hornberger, K.

S. Gerlich, S. Eibenberger, M. Tomandl, S. Nimmrichter, K. Hornberger, P. J. Fagan, J. Tüen, M. Mayor, and M. Arndt, “Quantum interference of large organic molecules,” Nat. Commun. 2, 263 (2011).
[Crossref] [PubMed]

Horne, M. A.

M. Zukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, “Event-ready-detectors Bell experiment via entanglement swapping,” Phys. Rev. Lett. 71, 4287–4290 (1993).
[Crossref] [PubMed]

Horoche, S.

S. Horoche and D. Kleppner, “Cavity quantum electrodynamics,” Phys. Today 42, 24 (1989).
[Crossref]

Horodecki, K.

R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865–942 (2009).
[Crossref]

Horodecki, M.

R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865–942 (2009).
[Crossref]

Horodecki, P.

R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865–942 (2009).
[Crossref]

Horodecki, R.

R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865–942 (2009).
[Crossref]

Huang, R. S.

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004).
[Crossref] [PubMed]

Huang, S.

S. Huang and G. S. Agarwal, “Entangling nanomechanical oscillators in a ring cavity by feeding squeezed light,” New J. Phys. 11, 103044 (2009).
[Crossref]

Ian, H.

Z. R. Gong, H. Ian, Y. X. Liu, C. P. Sun, and F. Nori, “Effective Hamiltonian approach to the Kerr nonlinearity in an optomechanical system,” Phys. Rev. A 80, 065801 (2009).
[Crossref]

Iblisdir, S.

V. Scarani, S. Iblisdir, N. Gisin, and A. Acin, “Quantum cloning,” Rev. Mod. Phys. 77, 1225 (2005).
[Crossref]

Itano, W. M.

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 (1998).
[Crossref]

Jacobs, K.

S. Bose, K. Jacobs, and P. L. Knight, “Preparation of nonclassical states in cavities with a moving mirror,” Phys. Rev. A 56, 4175 (1997).
[Crossref]

Jayich, A. M.

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707–712 (2010).
[Crossref]

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref] [PubMed]

Jonson, M.

C. Joshi, J. Larson, M. Jonson, E. Andersson, and P. Öhberg, “Entanglement of distant optomechanical systems,” Phys. Rev. A 85, 033805 (2012).
[Crossref]

Joshi, C.

C. Joshi, J. Larson, M. Jonson, E. Andersson, and P. Öhberg, “Entanglement of distant optomechanical systems,” Phys. Rev. A 85, 033805 (2012).
[Crossref]

Jozsa, R.

A. Ekert and R. Jozsa, “Shor’s quantum algorithm for factorising numbers,” Rev. Mod. Phys. 68, 733–753 (1996).
[Crossref]

Karuza, M.

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).
[Crossref]

Keller, C.

M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. Van der Zouw, and A. Zeilinger, “Wave–particle duality of C60 molecules,” Nature 401, 680–682 (1999).
[Crossref]

King, B. E.

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 (1998).
[Crossref]

C. Monroe, D. M. Meekhof, B. E. King, and D. J. Wineland, “A Schrodinger cat superposition state of an atom,” Science 272, 1131 (1996).
[Crossref] [PubMed]

Kippenberg, T. J.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref] [PubMed]

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321, 1172–1176 (2008).
[Crossref] [PubMed]

Kleppner, D.

S. Horoche and D. Kleppner, “Cavity quantum electrodynamics,” Phys. Today 42, 24 (1989).
[Crossref]

Knight, P. L.

S. Bose, K. Jacobs, and P. L. Knight, “Preparation of nonclassical states in cavities with a moving mirror,” Phys. Rev. A 56, 4175 (1997).
[Crossref]

C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge University Press, 2005), Chap. 8.

Köber, T.

H. Häfner, W. Häsel, C. F. Roos, J. Benhelm, M. Chwalla, T. Köber, and O. Güne, “Scalable multiparticle entanglement of trapped ions,” Nature 438, 643–646 (2005).
[Crossref]

Korkusinski, M.

M. Bayer, P. Hawrylak, K. Hinzer, S. Fafard, M. Korkusinski, Z. R. Wasilewski, and A. Forchel, “Coupling and entangling of quantum states in quantum dot molecules,” Science 291, 451–453 (2001).
[Crossref] [PubMed]

Kuang, L. M.

L. M. Kuang and Z. Lan, “Generation of atom-photon entangled states in atomic bose-einstein condensate via electromagnetically induced transparency,” Phys. Rev. A 68, 043606 (2004).
[Crossref]

Kuzyk, M. C.

C. H. Dong, V. Fiore, M. C. Kuzyk, and H. L. Wang, “Optomechanical dark mode,” Science 338, 1609–1613 (2012).
[Crossref] [PubMed]

Kwiat, P. G.

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

Lan, Z.

L. M. Kuang and Z. Lan, “Generation of atom-photon entangled states in atomic bose-einstein condensate via electromagnetically induced transparency,” Phys. Rev. A 68, 043606 (2004).
[Crossref]

Larson, J.

C. Joshi, J. Larson, M. Jonson, E. Andersson, and P. Öhberg, “Entanglement of distant optomechanical systems,” Phys. Rev. A 85, 033805 (2012).
[Crossref]

Lehnert, K. W.

T. A. Palomaki, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, “Entangling mechanical motion with microwave fields,” Science 342, 710–713 (2013).
[Crossref] [PubMed]

Leibfried, D.

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 (1998).
[Crossref]

Liao, J. Q.

J. Q. Liao, Q. Q. Wu, and F. Nori, “Entangling two macroscopic mechanical mirrors in a two-cavity optomechanical system,” Phys. Rev. A 89, 014302 (2014).
[Crossref]

Lin, Q.

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

Liu, Y. X.

X. W. Xu, Y. J. Zhao, and Y. X. Liu, “Entangled-state engineering of vibrational modes in a multimembrane optomechanical system,” Phys. Rev. A 88, 022325 (2013).
[Crossref]

Z. R. Gong, H. Ian, Y. X. Liu, C. P. Sun, and F. Nori, “Effective Hamiltonian approach to the Kerr nonlinearity in an optomechanical system,” Phys. Rev. A 80, 065801 (2009).
[Crossref]

F. Xue, Y. X. Liu, C. P. Sun, and F. Nori, “Two-mode squeezed states and entangled states of two mechanical resonators,” Phys. Rev. B 76, 064305 (2007).
[Crossref]

Lloyd, S.

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photon. 5, 222–229 (2011).
[Crossref]

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 414, 413–418 (2001).
[Crossref] [PubMed]

Maccone, L.

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photon. 5, 222–229 (2011).
[Crossref]

Maitre, X.

E. Hagley, X. Maitre, 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 (1997).
[Crossref]

Majer, J.

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004).
[Crossref] [PubMed]

Mancini, S.

S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401 (2002).
[Crossref] [PubMed]

Marquardt, F.

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref] [PubMed]

Marshall, W.

W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards quantum superpositions of a mirror,” Phys. Rev. Lett. 91, 130401 (2010).
[Crossref]

Mattle, K.

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

Mayor, M.

S. Gerlich, S. Eibenberger, M. Tomandl, S. Nimmrichter, K. Hornberger, P. J. Fagan, J. Tüen, M. Mayor, and M. Arndt, “Quantum interference of large organic molecules,” Nat. Commun. 2, 263 (2011).
[Crossref] [PubMed]

Meekhof, D. M.

C. Monroe, D. M. Meekhof, B. E. King, and D. J. Wineland, “A Schrodinger cat superposition state of an atom,” Science 272, 1131 (1996).
[Crossref] [PubMed]

Meystre, P.

P. Meystre, “A short walk through quantum optomechanics,” Ann. Phys. (Berlin) 525, 215 (2013).
[Crossref]

Molinelli, C.

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).
[Crossref]

Monroe, C.

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 (1998).
[Crossref]

C. Monroe, D. M. Meekhof, B. E. King, and D. J. Wineland, “A Schrodinger cat superposition state of an atom,” Science 272, 1131 (1996).
[Crossref] [PubMed]

Myatt, C. J.

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 (1998).
[Crossref]

Nairz, O.

M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. Van der Zouw, and A. Zeilinger, “Wave–particle duality of C60 molecules,” Nature 401, 680–682 (1999).
[Crossref]

Natali, R.

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).
[Crossref]

Nimmrichter, S.

S. Gerlich, S. Eibenberger, M. Tomandl, S. Nimmrichter, K. Hornberger, P. J. Fagan, J. Tüen, M. Mayor, and M. Arndt, “Quantum interference of large organic molecules,” Nat. Commun. 2, 263 (2011).
[Crossref] [PubMed]

Nogues, G.

E. Hagley, X. Maitre, 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 (1997).
[Crossref]

Nori, F.

J. Q. Liao, Q. Q. Wu, and F. Nori, “Entangling two macroscopic mechanical mirrors in a two-cavity optomechanical system,” Phys. Rev. A 89, 014302 (2014).
[Crossref]

Z. R. Gong, H. Ian, Y. X. Liu, C. P. Sun, and F. Nori, “Effective Hamiltonian approach to the Kerr nonlinearity in an optomechanical system,” Phys. Rev. A 80, 065801 (2009).
[Crossref]

F. Xue, Y. X. Liu, C. P. Sun, and F. Nori, “Two-mode squeezed states and entangled states of two mechanical resonators,” Phys. Rev. B 76, 064305 (2007).
[Crossref]

O’Brien, J. L.

J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature 426, 264–267 (2003).
[Crossref]

Öhberg, P.

C. Joshi, J. Larson, M. Jonson, E. Andersson, and P. Öhberg, “Entanglement of distant optomechanical systems,” Phys. Rev. A 85, 033805 (2012).
[Crossref]

Painter, O.

J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3, 1196 (2012).
[Crossref] [PubMed]

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

Palma, G. M.

G. Vacanti, M. Paternostro, G. M. Palma, and V. Vedral, “Optomechanical to mechanical entanglement transformation,” New J. Phys. 10, 095014 (2008).
[Crossref]

Palomaki, T. A.

T. A. Palomaki, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, “Entangling mechanical motion with microwave fields,” Science 342, 710–713 (2013).
[Crossref] [PubMed]

Paternostro, M.

G. Vacanti, M. Paternostro, G. M. Palma, and V. Vedral, “Optomechanical to mechanical entanglement transformation,” New J. Phys. 10, 095014 (2008).
[Crossref]

Peng, K.

J. Zhang, K. Peng, and S. L. Braunstein, “Quantum-state transfer from light to macroscopic oscillators,” Phys. Rev. A 68, 013808 (2003).
[Crossref]

Penrose, R.

W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards quantum superpositions of a mirror,” Phys. Rev. Lett. 91, 130401 (2010).
[Crossref]

Phoenix, S. J.

S. J. Phoenix, “Wave-packet evolution in the damped oscillator,” Phys. Rev. A 41, 5132 (1990).
[Crossref] [PubMed]

Pryde, G. J.

J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature 426, 264–267 (2003).
[Crossref]

Raimond, J. M.

E. Hagley, X. Maitre, 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 (1997).
[Crossref]

Ralph, T. C.

J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature 426, 264–267 (2003).
[Crossref]

Rivière, R.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref] [PubMed]

Roos, C. F.

H. Häfner, W. Häsel, C. F. Roos, J. Benhelm, M. Chwalla, T. Köber, and O. Güne, “Scalable multiparticle entanglement of trapped ions,” Nature 438, 643–646 (2005).
[Crossref]

Safavi-Naeini, A. H.

J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3, 1196 (2012).
[Crossref] [PubMed]

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

Saleh, B. E. A.

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87, 123602 (2001).
[Crossref] [PubMed]

Sankey, J. C.

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707–712 (2010).
[Crossref]

Scarani, V.

V. Scarani, S. Iblisdir, N. Gisin, and A. Acin, “Quantum cloning,” Rev. Mod. Phys. 77, 1225 (2005).
[Crossref]

Schliesser, A.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref] [PubMed]

Schoelkopf, R. J.

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004).
[Crossref] [PubMed]

Schuster, D. I.

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004).
[Crossref] [PubMed]

Sergienko, A. V.

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87, 123602 (2001).
[Crossref] [PubMed]

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

Sheng, Y. B.

Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Nonlocal entanglement concentration scheme for partially entangled multipartite systems with nonlinear optics,” Phys. Rev. A 77, 062325 (2008).
[Crossref]

Shih, Y.

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

Simmonds, R. W.

T. A. Palomaki, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, “Entangling mechanical motion with microwave fields,” Science 342, 710–713 (2013).
[Crossref] [PubMed]

Simon, C.

R. Ghobadi, A. R. Bahrampour, and C. Simon, “Quantum optomechanics in the bistable regime,” Phys. Rev. A 84, 033846 (2011).
[Crossref]

W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards quantum superpositions of a mirror,” Phys. Rev. Lett. 91, 130401 (2010).
[Crossref]

Sun, C. P.

Z. R. Gong, H. Ian, Y. X. Liu, C. P. Sun, and F. Nori, “Effective Hamiltonian approach to the Kerr nonlinearity in an optomechanical system,” Phys. Rev. A 80, 065801 (2009).
[Crossref]

F. Xue, Y. X. Liu, C. P. Sun, and F. Nori, “Two-mode squeezed states and entangled states of two mechanical resonators,” Phys. Rev. B 76, 064305 (2007).
[Crossref]

Teich, M. C.

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87, 123602 (2001).
[Crossref] [PubMed]

Teufel, J. D.

T. A. Palomaki, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, “Entangling mechanical motion with microwave fields,” Science 342, 710–713 (2013).
[Crossref] [PubMed]

Thompson, J. D.

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref] [PubMed]

Tomandl, M.

S. Gerlich, S. Eibenberger, M. Tomandl, S. Nimmrichter, K. Hornberger, P. J. Fagan, J. Tüen, M. Mayor, and M. Arndt, “Quantum interference of large organic molecules,” Nat. Commun. 2, 263 (2011).
[Crossref] [PubMed]

Tombesi, P.

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).
[Crossref]

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98, 030405 (2007).
[Crossref] [PubMed]

S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401 (2002).
[Crossref] [PubMed]

Tüen, J.

S. Gerlich, S. Eibenberger, M. Tomandl, S. Nimmrichter, K. Hornberger, P. J. Fagan, J. Tüen, M. Mayor, and M. Arndt, “Quantum interference of large organic molecules,” Nat. Commun. 2, 263 (2011).
[Crossref] [PubMed]

Turchette, Q. A.

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 (1998).
[Crossref]

Vacanti, G.

G. Vacanti, M. Paternostro, G. M. Palma, and V. Vedral, “Optomechanical to mechanical entanglement transformation,” New J. Phys. 10, 095014 (2008).
[Crossref]

Vahala, K. J.

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321, 1172–1176 (2008).
[Crossref] [PubMed]

Van der Zouw, G.

M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. Van der Zouw, and A. Zeilinger, “Wave–particle duality of C60 molecules,” Nature 401, 680–682 (1999).
[Crossref]

Vedral, V.

G. Vacanti, M. Paternostro, G. M. Palma, and V. Vedral, “Optomechanical to mechanical entanglement transformation,” New J. Phys. 10, 095014 (2008).
[Crossref]

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98, 030405 (2007).
[Crossref] [PubMed]

Vitali, D.

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).
[Crossref]

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98, 030405 (2007).
[Crossref] [PubMed]

S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401 (2002).
[Crossref] [PubMed]

Vos-Andreae, J.

M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. Van der Zouw, and A. Zeilinger, “Wave–particle duality of C60 molecules,” Nature 401, 680–682 (1999).
[Crossref]

Wallraff, A.

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004).
[Crossref] [PubMed]

Wang, H. L.

C. H. Dong, V. Fiore, M. C. Kuzyk, and H. L. Wang, “Optomechanical dark mode,” Science 338, 1609–1613 (2012).
[Crossref] [PubMed]

Wasilewski, Z. R.

M. Bayer, P. Hawrylak, K. Hinzer, S. Fafard, M. Korkusinski, Z. R. Wasilewski, and A. Forchel, “Coupling and entangling of quantum states in quantum dot molecules,” Science 291, 451–453 (2001).
[Crossref] [PubMed]

Weinfurter, H.

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

Weis, S.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref] [PubMed]

White, A. G.

J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature 426, 264–267 (2003).
[Crossref]

Wineland, D.

R. Blatt and D. Wineland, “Entangled states of trapped atomic ions,” Nature 453, 1008–1015 (2008).
[Crossref] [PubMed]

Wineland, D. J.

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 (1998).
[Crossref]

C. Monroe, D. M. Meekhof, B. E. King, and D. J. Wineland, “A Schrodinger cat superposition state of an atom,” Science 272, 1131 (1996).
[Crossref] [PubMed]

Winger, M.

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

Wood, C. S.

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 (1998).
[Crossref]

Wu, Q. Q.

J. Q. Liao, Q. Q. Wu, and F. Nori, “Entangling two macroscopic mechanical mirrors in a two-cavity optomechanical system,” Phys. Rev. A 89, 014302 (2014).
[Crossref]

Wunderlich, C.

E. Hagley, X. Maitre, 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 (1997).
[Crossref]

Xu, X. W.

X. W. Xu, Y. J. Zhao, and Y. X. Liu, “Entangled-state engineering of vibrational modes in a multimembrane optomechanical system,” Phys. Rev. A 88, 022325 (2013).
[Crossref]

Xue, F.

F. Xue, Y. X. Liu, C. P. Sun, and F. Nori, “Two-mode squeezed states and entangled states of two mechanical resonators,” Phys. Rev. B 76, 064305 (2007).
[Crossref]

Yang, C.

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707–712 (2010).
[Crossref]

Zeilinger, A.

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98, 030405 (2007).
[Crossref] [PubMed]

M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. Van der Zouw, and A. Zeilinger, “Wave–particle duality of C60 molecules,” Nature 401, 680–682 (1999).
[Crossref]

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

M. Zukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, “Event-ready-detectors Bell experiment via entanglement swapping,” Phys. Rev. Lett. 71, 4287–4290 (1993).
[Crossref] [PubMed]

Zhang, J.

J. Zhang, K. Peng, and S. L. Braunstein, “Quantum-state transfer from light to macroscopic oscillators,” Phys. Rev. A 68, 013808 (2003).
[Crossref]

Zhao, Y. J.

X. W. Xu, Y. J. Zhao, and Y. X. Liu, “Entangled-state engineering of vibrational modes in a multimembrane optomechanical system,” Phys. Rev. A 88, 022325 (2013).
[Crossref]

Zhou, H. Y.

Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Nonlocal entanglement concentration scheme for partially entangled multipartite systems with nonlinear optics,” Phys. Rev. A 77, 062325 (2008).
[Crossref]

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 414, 413–418 (2001).
[Crossref] [PubMed]

Zukowski, M.

M. Zukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, “Event-ready-detectors Bell experiment via entanglement swapping,” Phys. Rev. Lett. 71, 4287–4290 (1993).
[Crossref] [PubMed]

Zwickl, B. M.

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707–712 (2010).
[Crossref]

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref] [PubMed]

Ann. Phys. (Berlin) (1)

P. Meystre, “A short walk through quantum optomechanics,” Ann. Phys. (Berlin) 525, 215 (2013).
[Crossref]

Nat. Commun. (2)

J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3, 1196 (2012).
[Crossref] [PubMed]

S. Gerlich, S. Eibenberger, M. Tomandl, S. Nimmrichter, K. Hornberger, P. J. Fagan, J. Tüen, M. Mayor, and M. Arndt, “Quantum interference of large organic molecules,” Nat. Commun. 2, 263 (2011).
[Crossref] [PubMed]

Nat. Photon. (1)

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photon. 5, 222–229 (2011).
[Crossref]

Nat. Phys. (1)

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707–712 (2010).
[Crossref]

Nature (8)

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref] [PubMed]

M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. Van der Zouw, and A. Zeilinger, “Wave–particle duality of C60 molecules,” Nature 401, 680–682 (1999).
[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, 413–418 (2001).
[Crossref] [PubMed]

J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature 426, 264–267 (2003).
[Crossref]

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004).
[Crossref] [PubMed]

R. Blatt and D. Wineland, “Entangled states of trapped atomic ions,” Nature 453, 1008–1015 (2008).
[Crossref] [PubMed]

H. Häfner, W. Häsel, C. F. Roos, J. Benhelm, M. Chwalla, T. Köber, and O. Güne, “Scalable multiparticle entanglement of trapped ions,” Nature 438, 643–646 (2005).
[Crossref]

New J. Phys. (2)

G. Vacanti, M. Paternostro, G. M. Palma, and V. Vedral, “Optomechanical to mechanical entanglement transformation,” New J. Phys. 10, 095014 (2008).
[Crossref]

S. Huang and G. S. Agarwal, “Entangling nanomechanical oscillators in a ring cavity by feeding squeezed light,” New J. Phys. 11, 103044 (2009).
[Crossref]

Phys. Rev. A (11)

X. W. Xu, Y. J. Zhao, and Y. X. Liu, “Entangled-state engineering of vibrational modes in a multimembrane optomechanical system,” Phys. Rev. A 88, 022325 (2013).
[Crossref]

C. Joshi, J. Larson, M. Jonson, E. Andersson, and P. Öhberg, “Entanglement of distant optomechanical systems,” Phys. Rev. A 85, 033805 (2012).
[Crossref]

J. Zhang, K. Peng, and S. L. Braunstein, “Quantum-state transfer from light to macroscopic oscillators,” Phys. Rev. A 68, 013808 (2003).
[Crossref]

J. Q. Liao, Q. Q. Wu, and F. Nori, “Entangling two macroscopic mechanical mirrors in a two-cavity optomechanical system,” Phys. Rev. A 89, 014302 (2014).
[Crossref]

L. M. Kuang and Z. Lan, “Generation of atom-photon entangled states in atomic bose-einstein condensate via electromagnetically induced transparency,” Phys. Rev. A 68, 043606 (2004).
[Crossref]

S. J. Phoenix, “Wave-packet evolution in the damped oscillator,” Phys. Rev. A 41, 5132 (1990).
[Crossref] [PubMed]

Z. R. Gong, H. Ian, Y. X. Liu, C. P. Sun, and F. Nori, “Effective Hamiltonian approach to the Kerr nonlinearity in an optomechanical system,” Phys. Rev. A 80, 065801 (2009).
[Crossref]

R. Ghobadi, A. R. Bahrampour, and C. Simon, “Quantum optomechanics in the bistable regime,” Phys. Rev. A 84, 033846 (2011).
[Crossref]

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).
[Crossref]

S. Bose, K. Jacobs, and P. L. Knight, “Preparation of nonclassical states in cavities with a moving mirror,” Phys. Rev. A 56, 4175 (1997).
[Crossref]

Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Nonlocal entanglement concentration scheme for partially entangled multipartite systems with nonlinear optics,” Phys. Rev. A 77, 062325 (2008).
[Crossref]

Phys. Rev. B (1)

F. Xue, Y. X. Liu, C. P. Sun, and F. Nori, “Two-mode squeezed states and entangled states of two mechanical resonators,” Phys. Rev. B 76, 064305 (2007).
[Crossref]

Phys. Rev. Lett. (9)

W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards quantum superpositions of a mirror,” Phys. Rev. Lett. 91, 130401 (2010).
[Crossref]

S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401 (2002).
[Crossref] [PubMed]

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87, 123602 (2001).
[Crossref] [PubMed]

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

E. Hagley, X. Maitre, 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 (1997).
[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 (1998).
[Crossref]

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

M. Zukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, “Event-ready-detectors Bell experiment via entanglement swapping,” Phys. Rev. Lett. 71, 4287–4290 (1993).
[Crossref] [PubMed]

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98, 030405 (2007).
[Crossref] [PubMed]

Phys. Today (1)

S. Horoche and D. Kleppner, “Cavity quantum electrodynamics,” Phys. Today 42, 24 (1989).
[Crossref]

Rev. Mod. Phys. (3)

V. Scarani, S. Iblisdir, N. Gisin, and A. Acin, “Quantum cloning,” Rev. Mod. Phys. 77, 1225 (2005).
[Crossref]

A. Ekert and R. Jozsa, “Shor’s quantum algorithm for factorising numbers,” Rev. Mod. Phys. 68, 733–753 (1996).
[Crossref]

R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865–942 (2009).
[Crossref]

Science (7)

C. Monroe, D. M. Meekhof, B. E. King, and D. J. Wineland, “A Schrodinger cat superposition state of an atom,” Science 272, 1131 (1996).
[Crossref] [PubMed]

D. P. DiVincenzo, “Quantum computation,” Science 270, 255 (1995).
[Crossref]

M. Bayer, P. Hawrylak, K. Hinzer, S. Fafard, M. Korkusinski, Z. R. Wasilewski, and A. Forchel, “Coupling and entangling of quantum states in quantum dot molecules,” Science 291, 451–453 (2001).
[Crossref] [PubMed]

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref] [PubMed]

C. H. Dong, V. Fiore, M. C. Kuzyk, and H. L. Wang, “Optomechanical dark mode,” Science 338, 1609–1613 (2012).
[Crossref] [PubMed]

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321, 1172–1176 (2008).
[Crossref] [PubMed]

T. A. Palomaki, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, “Entangling mechanical motion with microwave fields,” Science 342, 710–713 (2013).
[Crossref] [PubMed]

Other (1)

C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge University Press, 2005), Chap. 8.

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Figures (3)

Fig. 1
Fig. 1 Schematic diagram of the system. A two-level atom sequentially passes through two cavity optomechanical systems (COMSs), and its states are detected when it exits from the second COMS. In this way the two COMSs can be entangled.
Fig. 2
Fig. 2 The temporal sequence of entangling two optomechanical systems. U JC ( k ) and U OM ( k ) are the time evolution operators defined in the text. The insets above time points represent the position of the atom.
Fig. 3
Fig. 3 The variation of the concurrence C versus the parameter β.

Equations (41)

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H ( k ) = ω a 2 σ z + ω c a k a k + ω m b k b k i G ( a k σ + a k + σ ) g a k a k ( b k + b k ) ,
H int ( k ) = i d U k d t U k + U k H ( k ) U k = ω m b k b k i G ( a k σ + e i Δ t a k σ e i Δ t ) g a k a k ( b k + b k ) ,
H int ( k ) H JC ( k ) = ω m b k b k i G ( a k σ + a k σ ) ,
H int ( k ) = H OM ( k ) = ω m b k b k g a k a k ( b k + b k ) ,
| Ψ 0 = | e | 0 c 1 | α m 1 | 0 c 2 | α m 2 ,
| Ψ ( t 1 ) = U JC ( 1 ) ( τ 1 ) U OM ( 2 ) ( τ 1 ) | Ψ 0 = { sin ( G τ 1 ) | g | 1 c 1 + cos ( G τ 1 ) | e | 0 c 1 } | α γ ( τ 1 ) m 1 | 0 c 2 | α γ ( τ 1 ) m 2 ,
| Ψ ( t 1 ) = 1 2 ( | g | 1 c 1 + | e | 0 c 1 ) | α m 1 | 0 c 2 | α m 2 .
| Ψ ( t 2 ) = U OM ( 1 ) ( τ 2 ) U OM ( 2 ) ( τ 2 ) | Ψ ( t 1 ) = 1 2 { e i θ + ( τ 2 ) | g | 1 c 1 | ϕ + ( τ 2 ) m 1 + | e | 0 c 1 | α γ ( τ 2 ) m 1 } | 0 c 2 | α γ ( τ 2 ) m 2 ,
| Ψ T ( t 2 ) = U OM ( 1 ) ( T m 2 ) U OM ( 2 ) ( T m 2 ) | Ψ ( t 1 ) = 1 2 ( e i Λ | g | 1 c 1 | 2 β α m 1 + | e | 0 c 1 | α m 1 ) | 0 c 2 | α m 2 ,
| Ψ T ( t 3 ) = U OM ( 1 ) ( τ 3 ) U JC ( 2 ) ( τ 3 ) Ψ T ( t 2 ) = 1 2 { sin ( G τ 3 ) | g | 1 c 2 + cos ( G τ 3 ) | e | 0 c 2 } | 0 c 1 | α γ ( τ 3 ) m 1 | α γ ( τ 3 ) m 2 + | g 1 2 e i [ Λ + θ ( τ 3 ) ] | 1 c 1 | ξ ( τ 3 ) m 1 | 0 c 2 | α γ ( τ 3 ) m 2 ,
| Ψ T ( t 3 ) = | g { 1 2 e i [ Λ + 2 π β 2 ] | 1 c 1 | 2 β α m 1 | 0 c 2 | α m 2 + 1 2 | 1 c 1 | α m 2 | 0 c 2 | α m 1 } + 1 2 | e | 0 c 1 | α m 1 | 0 c 2 | α m 2 ,
| Ψ T ( t d ) = U OM ( 1 ) ( τ d ) U OM ( 2 ) ( τ d ) | ψ T ( t 3 ) = | g { 1 2 e i [ Λ + 2 π β 2 + θ ( τ d ) ] | 1 c 1 | ξ ( τ d ) m 1 | 0 c 2 | α γ ( τ d ) m 2 + 1 2 e i θ ( τ d ) | 0 c 2 | α γ ( τ d ) m 1 | 1 c 1 | ϕ ( τ d ) m 2 } + 1 2 | e | 0 c 1 | α γ ( τ d ) m 1 | 0 c 2 | α γ ( τ d ) m 2 ,
| ψ T ( T m 2 + t 3 ) = ( 2 3 e i 4 π β 2 | 1 c 1 | 0 c 2 | α m 2 + 2 3 e i Λ + | 0 c 1 | 1 c 2 | α + 2 β m 2 ) | α m 1 ,
| ψ T ( T m + t 3 ) = U OM ( 1 ) ( T M / 2 ) U OM ( 2 ) ( T m / 2 ) | ψ T ( T m 2 + t 3 ) = ( 2 3 e i Λ e i 4 π β 2 | 1 c 1 | 2 β α m 1 | 0 c 2 + 1 3 e i 2 π β 2 | 0 c 1 | α ) m 1 | 1 c 2 ) | α m 2 ,
| Ψ S ( t 2 ) = U OM ( 1 ) ( T m ) U OM ( 2 ) ( T m ) | Ψ ( t 1 ) = 1 2 ( e i 2 π β 2 | g | 1 c 1 + | e | 0 c 1 | α m 1 ) | 0 c 2 | α m 2 ,
| Ψ S ( t 3 ) = U OM ( 1 ) ( τ 3 ) U JC ( 2 ) ( τ 3 ) | Ψ S ( t 2 ) = 1 2 { cos ( G τ 3 ) | e | 0 c 2 + sin ( G τ 3 ) | g | 1 c 2 } | α γ ( τ 3 ) m 2 | 0 c 1 | α γ ( τ 3 ) m 1 + 1 2 e i [ 2 π β 2 + θ + ( τ 3 ) ] | g | 1 c 1 | ϕ + ( τ 3 ) m 1 | 0 c 2 | α γ ( τ 3 ) m 2 ,
| Ψ S ( t 3 ) = { | g ( 1 2 e i 4 π β 2 | 1 c 1 | 0 c 2 + 1 2 | 0 c 1 | 1 c 2 ) + 1 2 | e | 0 c 1 | 0 c 2 } | α m 1 | α m 2 ,
| Ψ S ( t d ) = U OM ( 1 ) ( τ d ) U OM ( 2 ) ( τ d ) | Ψ S ( t 3 ) = | g { 1 2 e i [ 4 π β 2 + θ + ( τ d ) ] | 1 c 1 | ϕ + ( τ d ) m 1 | 0 c 2 | α γ ( τ d ) m 2 + 1 2 e i θ + ( τ d ) | 0 c 1 | α γ ( τ d ) m 1 | 1 c 2 | ϕ + ( τ d ) m 2 } + 1 2 | e | 0 c 1 | α γ ( τ d ) m 1 | 0 c 2 | α γ ( τ d ) m 2 .
| ψ S ( T m / 2 + t 3 ) = ( 2 3 e i 6 π β 2 | 1 c 1 | 0 c 2 + 1 3 e i 2 π β 2 | 0 c 1 | 1 c 2 ) | α m 1 | α m 2 .
| ψ S ( T m + t 3 ) = U OM ( 1 ) ( T m / 2 ) U OM ( 2 ) ( T m / 2 ) | ψ S ( T m 2 + t 3 ) = 2 3 e i ( 6 π β 2 + Λ ) | 1 c 1 | 2 β α m 1 | 0 c 2 | α m 2 + 1 3 e i ( 2 π β 2 + Λ ) | 0 c 1 | α m 1 | 1 c 2 | 2 β α m 2 .
| ψ D = U JC ( 1 ) ( 2 T m ) U JC ( 2 ) ( 2 T m ) | g 1 | g 2 | ψ S ( T m + t 3 ) = ( 2 3 e i ( 6 π β 2 + Λ ) | e 1 | g 2 | 2 β α m 1 | α m 2 1 3 e i ( 2 π β 2 + Λ ) | g 1 | e 2 | α m 1 | 2 β α m 2 ) | 0 c 1 | 0 c 2 .
| ψ D = | g 1 | g 2 | D + + | e 1 | e 2 | D + | g 1 | e 2 | D + + | e 1 | g 2 | D
| D ± = ± ( 1 6 e i ( 6 π β 2 + Λ ) | 2 β α m 1 | α m 2 + 1 12 e i ( 2 π β 2 + Λ ) | α m 1 | 2 β α m 2 ) ,
| D ± = ± ( 1 6 e i ( 6 π β 2 + Λ ) | 2 β α m 1 | α m 2 1 12 e i ( 2 π β 2 + Λ ) | α m 1 | 2 β α m 2 ) ,
C = 2 𝒩 2 6 ( 1 e 4 β 2 ) ,
d ρ d t = i ω m k = 1 2 [ b k b k , ρ ] + γ m 2 k = 1 2 ( 2 b k ρ b k b k b k ρ ρ b k + b k ) ,
ρ ( t ) = 2 { 1 6 | ϑ ( t ) m 1 ϑ ( t ) | | ϑ ( t ) m 2 ϑ ( t ) | + 1 12 | ϑ ( t ) m 1 ϑ ( t ) | | ϑ ( t ) m 2 ϑ ( t ) | + W ( t ) [ 2 12 e i 4 π β 2 | ϑ ( t ) m 1 ϑ ( t ) | | ϑ ( t ) m 2 ϑ ( t ) | + 2 12 e i 4 π β 2 | ϑ ( t ) m 1 ϑ ( t ) | | ϑ ( t ) m 2 ϑ ( t ) | ] } ,
d ρ d t = i ω m k = 1 2 [ b k b k , ρ ] + k = 1 2 ( J k ρ + L k ρ ) ,
ρ ( t ) = exp [ k = 1 2 ( J k + L k + F k ) t ] ρ ( 0 ) ,
ρ ( t ) = lim N j = 1 N { exp [ k = 1 2 ( J k + L k + F k ) Δ t j ] } ρ ( 0 ) .
ρ ( t ) = lim N j = 1 N { exp [ k = 1 2 ( J k + L k ) Δ t j ] exp [ k = 1 2 F k Δ t j ] } ρ ( 0 ) .
exp [ ( J k + L k ) Δ t ] | 2 β α m k α | = ( α | | 2 β α m k ) Q | ( 2 β α ) B 1 m k α B 1 | ,
ρ ( t 1 ) = exp [ k = 1 2 ( J k + L k ) Δ t ] exp [ k = 1 2 F k Δ t ] ρ ( 0 ) = 2 { 1 6 | η ( Δ t ) m 1 η ( Δ t ) | | η ( Δ t ) m 2 η ( Δ t ) | + 1 12 | η ( Δ t ) m 1 η ( Δ t ) | | η ( Δ t ) m 2 η ( Δ t ) | + w 1 [ 2 12 e i 4 π β 2 | η ( Δ t ) m 1 η ( Δ t ) | | η ( Δ t ) m 2 η ( Δ t ) | + 2 12 e i 4 π β 2 | η ( Δ t ) m 1 η ( Δ t ) | | η ( Δ t ) m 2 η ( Δ t ) | ] } ,
w 1 = | η ( Δ t ) | | η ( Δ t ) | 2 Q = exp [ 4 β 2 ( 1 e γ m Δ t ) ] .
O 2 = O 3 = = O N = e i ω m Δ t ,
B 2 = B 3 = = B N = e γ m Δ t / 2 ,
w 2 = exp [ 4 β 2 ( 1 e γ m Δ t ) e γ m Δ t ] , w 3 = exp [ 4 β 2 ( 1 e γ m Δ t ) e 2 γ m Δ t ] , w N = exp [ 4 β 2 ( 1 e γ m Δ t ) e ( N 1 ) γ m Δ t ] ,
ρ ( t ) = 2 { 1 6 | λ ( t ) m 1 λ ( t ) | | λ ( t ) m 2 λ ( t ) | + 1 12 | λ ( t ) m 1 λ ( t ) | | λ ( t ) m 2 λ ( t ) | + w ( t ) [ 2 12 e i 4 π β 2 | λ ( t ) m 1 λ ( t ) | | λ ( t ) m 2 λ ( t ) | + 2 12 e i 4 π β 2 | λ ( t ) m 1 λ ( t ) | | λ ( t ) m 2 λ ( t ) | ] } ,
O = O 1 × O 2 × × O N = e i ω m t ,
B = B 1 × B 2 × × B N = e γ m t / 2 ,
w ( t ) = lim N w 1 ( t ) × w 2 ( t ) w N ( t ) = lim N exp [ 4 β 2 ( 1 e γ m Δ t ) ( 1 + e γ m Δ t + + e ( N 1 ) γ m Δ t ) ] = exp [ 4 β 2 ( 1 e γ m t ) ] .

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