Abstract

Output entanglement is a key element in quantum information processing. Here, we show how to obtain optimal entanglement between two filtered output fields in a three-mode optomechanical system. First, we obtain the key analytical expression of optimal time delay between the two filtered output fields, from which we can obtain the optimal coupling for output entanglement without time delay. In this case, our linearized analysis predicts that the entanglement saturates to an optimal value as the optomechanical coupling is increased. Furthermore, we obtain the optimal output entanglement with time delay. These results should be very helpful in conceiving new optomechanical schemes of quantum information processing with their efficiency depending critically on the degree of output entanglement.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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  1. S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77(2), 513–577 (2005).
    [Crossref]
  2. C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84(2), 621–669 (2012).
    [Crossref]
  3. B. Julsgaard, A. Kozhekin, and E. S. Polzik, “Experimental long-lived entanglement of two macroscopic objects,” Nature 413(6854), 400–403 (2001).
    [Crossref]
  4. H. Krauter, C. A. Muschik, K. Jensen, W. Wasilewski, J. M. Petersen, J. I. Cirac, and E. S. Polzik, “Entanglement generated by dissipation and steady state entanglement of two macroscopic objects,” Phys. Rev. Lett. 107(8), 080503 (2011).
    [Crossref]
  5. A. J. Berkley, H. Xu, R. C. Ramos, M. A. Gubrud, F. W. Strauch, P. R. Johnson, J. R. Anderson, A. J. Dragt, C. J. Lobb, and F. C. Wellstood, “Entangled macroscopic quantum states in two superconducting qubits,” Science 300(5625), 1548–1550 (2003).
    [Crossref]
  6. M. Neeley, R. C. Bialczak, M. Lenander, E. Lucero, M. Mariantoni, D. Sank, H. Wang, M. Weides, J. Wenner, Y. Yin, T. Yamamoto, A. N. Cleland, and J. M. Martinis, “Generation of three-qubit entangled states using superconducting phase qubits,” Nature 467(7315), 570–573 (2010).
    [Crossref]
  7. L. DiCarlo, M. Reed, L. Sun, B. L. Johnson, J. M. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Preparation and measurement of three-qubit entanglement in a superconducting circuit,” Nature 467(7315), 574–578 (2010).
    [Crossref]
  8. E. Flurin, N. Roch, F. Mallet, M. H. Devoret, and B. Huard, “Generating entangled microwave radiation over two transmission lines,” Phys. Rev. Lett. 109(18), 183901 (2012).
    [Crossref]
  9. T. A. Palomaki, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, “Entangling mechanical motion with microwave fields,” Science 342(6159), 710–713 (2013).
    [Crossref]
  10. S. Mancini, D. Vitali, and P. Tombesi, “Scheme for teleportation of quantum states onto a mechanical resonator,” Phys. Rev. Lett. 90(13), 137901 (2003).
    [Crossref]
  11. S. Pirandola, S. Mancini, D. Vitali, and P. Tombesi, “Continuous-variable entanglement and quantum-state teleportation between optical and macroscopic vibrational modes through radiation pressure,” Phys. Rev. A 68(6), 062317 (2003).
    [Crossref]
  12. S. Pirandola, S. Mancini, D. Vitali, and P. Tombesi, “Light reflection upon a movable mirror as a paradigm for continuous variable teleportation network,” J. Mod. Opt. 51(6-7), 901–912 (2004).
    [Crossref]
  13. S. Pirandola, D. Vitali, P. Tombesi, and S. Lloyd, “Macroscopic entanglement by entanglement swapping,” Phys. Rev. Lett. 97(15), 150403 (2006).
    [Crossref]
  14. S. Kiesewetter, Q. Y. He, P. D. Drummond, and M. D. Reid, “Scalable quantum simulation of pulsed entanglement and Einstein-Podolsky-Rosen steering in optomechanics,” Phys. Rev. A 90(4), 043805 (2014).
    [Crossref]
  15. M. Bhattacharya, P.-L. Giscard, and P. Meystre, “Entangling the rovibrational modes of a macroscopic mirror using radiation pressure,” Phys. Rev. A 77(3), 030303 (2008).
    [Crossref]
  16. R. X. Chen, L. T. Shen, Z. B. Yang, H. Z. Wu, and S. B. Zheng, “Enhancement of entanglement in distant mechanical vibrations via modulation in a coupled optomechanical system,” Phys. Rev. A 89(2), 023843 (2014).
    [Crossref]
  17. J. Q. Liao, Q. Q. Wu, and F. Nori, “Entangling two macroscopic mechanical mirrors in a two-cavity optomechanical system,” Phys. Rev. A 89(1), 014302 (2014).
    [Crossref]
  18. C. J. Yang, J. H. An, W. Yang, and Y. Li, “Generation of stable entanglement between two cavity mirrors by squeezed-reservoir engineering,” Phys. Rev. A 92(6), 062311 (2015).
    [Crossref]
  19. M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99(25), 250401 (2007).
    [Crossref]
  20. C. Wipf, T. Corbitt, Y. Chen, and N. Mavalvala, “Route to ponderomotive entanglement of light via optically trapped mirrors,” New J. Phys. 10(9), 095017 (2008).
    [Crossref]
  21. C. Genes, A. Mari, P. Tombesi, and D. Vitali, “Robust entanglement of a micromechanical resonator with output optical fields,” Phys. Rev. A 78(3), 032316 (2008).
    [Crossref]
  22. Sh. Barzanjeh, D. Vitali, P. Tombesi, and G. J. Milburn, “Entangling optical and microwave cavity modes by means of a nanomechanical resonator,” Phys. Rev. A 84(4), 042342 (2011).
    [Crossref]
  23. Sh. Barzanjeh, M. Abdi, G. J. Milburn, P. Tombesi, and D. Vitali, “Reversible optical-to-microwave quantum interface,” Phys. Rev. Lett. 109(13), 130503 (2012).
    [Crossref]
  24. Sh. Barzanjeh, S. Pirandola, and C. Weedbrook, “Continuous-variable dense coding by optomechanical cavities,” Phys. Rev. A 88(4), 042331 (2013).
    [Crossref]
  25. Y.-D. Wang and A. A. Clerk, “Reservoir-engineered entanglement in optomechanical systems,” Phys. Rev. Lett. 110(25), 253601 (2013).
    [Crossref]
  26. M. C. Kuzyk, S. J. van Enk, and H. Wang, “Generating robust optical entanglement in weak-coupling optomechanical systems,” Phys. Rev. A 88(6), 062341 (2013).
    [Crossref]
  27. 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(3), 030405 (2007).
    [Crossref]
  28. S. G. Hofer, W. Wieczorek, M. Aspelmeyer, and K. Hammerer, “Quantum entanglement and teleportation in pulsed cavity optomechanics,” Phys. Rev. A 84(5), 052327 (2011).
    [Crossref]
  29. U. Akram, W. Munro, K. Nemoto, and G. J. Milburn, “Photon-phonon entanglement in coupled optomechanical arrays,” Phys. Rev. A 86(4), 042306 (2012).
    [Crossref]
  30. K. Sinha, S. Y. Lin, and B. L. Hu, “Mirror-field entanglement in a microscopic model for quantum optomechanics,” Phys. Rev. A 92(2), 023852 (2015).
    [Crossref]
  31. J. Li, I. Moaddel Haghighi, N. Malossi, S. Zippilli, and D. Vitali, “Generation and detection of large and robust entanglement between two different mechanical resonators in cavity optomechanics,” New J. Phys. 17(10), 103037 (2015).
    [Crossref]
  32. J. Li, G. Li, S. Zippilli, D. Vitali, and T. Zhang, “Enhanced entanglement of two different mechanical resonators via coherent feedback,” Phys. Rev. A 95(4), 043819 (2017).
    [Crossref]
  33. M. Asjad, P. Tombesi, and D. Vitali, “Feedback control of two-mode output entanglement and steering in cavity optomechanics,” Phys. Rev. A 94(5), 052312 (2016).
    [Crossref]
  34. Q. Lin, B. He, R. Ghobadi, and C. Simon, “Fully quantum approach to optomechanical entanglement,” Phys. Rev. A 90(2), 022309 (2014).
    [Crossref]
  35. L. Tian, “Robust photon entanglement via quantum interference in optomechanical interfaces,” Phys. Rev. Lett. 110(23), 233602 (2013).
    [Crossref]
  36. Z. J. Deng, S. J. M. Habraken, and F. Marquardt, “Entanglement rate for Gaussian continuous variable beams,” New J. Phys. 18(6), 063022 (2016).
    [Crossref]
  37. Z. J. Deng, X. B. Yan, Y. D. Wang, and C. W. Wu, “Optimizing the output-photon entanglement in multimode optomechanical systems,” Phys. Rev. A 93(3), 033842 (2016).
    [Crossref]
  38. Y.-D. Wang, S. Chesi, and A. A. Clerk, “Bipartite and tripartite output entanglement in three-mode optomechanical systems,” Phys. Rev. A 91(1), 013807 (2015).
    [Crossref]
  39. H.-K. Li, X.-X. Ren, Y.-C. Liu, and Y.-F. Xiao, “Photon-photon interactions in a largely detuned optomechanical cavity,” Phys. Rev. A 88(5), 053850 (2013).
    [Crossref]
  40. F. X. Sun, D. Mao, Y. T. Dai, Z. Ficek, Q. Y. He, and Q. H. Gong, “Phase control of entanglement and quantum steering in a three-mode optomechanical system,” New J. Phys. 19(12), 123039 (2017).
    [Crossref]
  41. X. B. Yan, “Enhanced output entanglement with reservoir engineering,” Phys. Rev. A 96(5), 053831 (2017).
    [Crossref]
  42. C. Dong, V. Fiore, M. C. Kuzyk, and H. Wang, “Optomechanical dark mode,” Science 338(6114), 1609–1613 (2012).
    [Crossref]
  43. J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3(1), 1196 (2012).
    [Crossref]
  44. R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, “Bidirectional and efficient conversion between microwave and optical light,” Nat. Phys. 10(4), 321–326 (2014).
    [Crossref]
  45. S. Barzanjeh, E. S. Redchenko, M. Peruzzo, M. Wulf, D. P. Lewis, G. Arnold, and J. M. Fink, “Stationary entangled radiation from micromechanical motion,” Nature 570(7762), 480–483 (2019).
    [Crossref]
  46. E. X. DeJesus and C. Kaufman, “Routh-Hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations,” Phys. Rev. A 35(12), 5288–5290 (1987).
    [Crossref]
  47. C. Gardiner and P. Zoller, Quantum Noise, 3rd ed. (Springer, New York, 2004).
  48. G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65(3), 032314 (2002).
    [Crossref]
  49. M. B. Plenio, “Logarithmic negativity: A full entanglement monotone that is not convex,” Phys. Rev. Lett. 95(9), 090503 (2005).
    [Crossref]

2019 (1)

S. Barzanjeh, E. S. Redchenko, M. Peruzzo, M. Wulf, D. P. Lewis, G. Arnold, and J. M. Fink, “Stationary entangled radiation from micromechanical motion,” Nature 570(7762), 480–483 (2019).
[Crossref]

2017 (3)

F. X. Sun, D. Mao, Y. T. Dai, Z. Ficek, Q. Y. He, and Q. H. Gong, “Phase control of entanglement and quantum steering in a three-mode optomechanical system,” New J. Phys. 19(12), 123039 (2017).
[Crossref]

X. B. Yan, “Enhanced output entanglement with reservoir engineering,” Phys. Rev. A 96(5), 053831 (2017).
[Crossref]

J. Li, G. Li, S. Zippilli, D. Vitali, and T. Zhang, “Enhanced entanglement of two different mechanical resonators via coherent feedback,” Phys. Rev. A 95(4), 043819 (2017).
[Crossref]

2016 (3)

M. Asjad, P. Tombesi, and D. Vitali, “Feedback control of two-mode output entanglement and steering in cavity optomechanics,” Phys. Rev. A 94(5), 052312 (2016).
[Crossref]

Z. J. Deng, S. J. M. Habraken, and F. Marquardt, “Entanglement rate for Gaussian continuous variable beams,” New J. Phys. 18(6), 063022 (2016).
[Crossref]

Z. J. Deng, X. B. Yan, Y. D. Wang, and C. W. Wu, “Optimizing the output-photon entanglement in multimode optomechanical systems,” Phys. Rev. A 93(3), 033842 (2016).
[Crossref]

2015 (4)

Y.-D. Wang, S. Chesi, and A. A. Clerk, “Bipartite and tripartite output entanglement in three-mode optomechanical systems,” Phys. Rev. A 91(1), 013807 (2015).
[Crossref]

K. Sinha, S. Y. Lin, and B. L. Hu, “Mirror-field entanglement in a microscopic model for quantum optomechanics,” Phys. Rev. A 92(2), 023852 (2015).
[Crossref]

J. Li, I. Moaddel Haghighi, N. Malossi, S. Zippilli, and D. Vitali, “Generation and detection of large and robust entanglement between two different mechanical resonators in cavity optomechanics,” New J. Phys. 17(10), 103037 (2015).
[Crossref]

C. J. Yang, J. H. An, W. Yang, and Y. Li, “Generation of stable entanglement between two cavity mirrors by squeezed-reservoir engineering,” Phys. Rev. A 92(6), 062311 (2015).
[Crossref]

2014 (5)

Q. Lin, B. He, R. Ghobadi, and C. Simon, “Fully quantum approach to optomechanical entanglement,” Phys. Rev. A 90(2), 022309 (2014).
[Crossref]

S. Kiesewetter, Q. Y. He, P. D. Drummond, and M. D. Reid, “Scalable quantum simulation of pulsed entanglement and Einstein-Podolsky-Rosen steering in optomechanics,” Phys. Rev. A 90(4), 043805 (2014).
[Crossref]

R. X. Chen, L. T. Shen, Z. B. Yang, H. Z. Wu, and S. B. Zheng, “Enhancement of entanglement in distant mechanical vibrations via modulation in a coupled optomechanical system,” Phys. Rev. A 89(2), 023843 (2014).
[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(1), 014302 (2014).
[Crossref]

R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, “Bidirectional and efficient conversion between microwave and optical light,” Nat. Phys. 10(4), 321–326 (2014).
[Crossref]

2013 (6)

H.-K. Li, X.-X. Ren, Y.-C. Liu, and Y.-F. Xiao, “Photon-photon interactions in a largely detuned optomechanical cavity,” Phys. Rev. A 88(5), 053850 (2013).
[Crossref]

T. A. Palomaki, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, “Entangling mechanical motion with microwave fields,” Science 342(6159), 710–713 (2013).
[Crossref]

L. Tian, “Robust photon entanglement via quantum interference in optomechanical interfaces,” Phys. Rev. Lett. 110(23), 233602 (2013).
[Crossref]

Sh. Barzanjeh, S. Pirandola, and C. Weedbrook, “Continuous-variable dense coding by optomechanical cavities,” Phys. Rev. A 88(4), 042331 (2013).
[Crossref]

Y.-D. Wang and A. A. Clerk, “Reservoir-engineered entanglement in optomechanical systems,” Phys. Rev. Lett. 110(25), 253601 (2013).
[Crossref]

M. C. Kuzyk, S. J. van Enk, and H. Wang, “Generating robust optical entanglement in weak-coupling optomechanical systems,” Phys. Rev. A 88(6), 062341 (2013).
[Crossref]

2012 (6)

Sh. Barzanjeh, M. Abdi, G. J. Milburn, P. Tombesi, and D. Vitali, “Reversible optical-to-microwave quantum interface,” Phys. Rev. Lett. 109(13), 130503 (2012).
[Crossref]

U. Akram, W. Munro, K. Nemoto, and G. J. Milburn, “Photon-phonon entanglement in coupled optomechanical arrays,” Phys. Rev. A 86(4), 042306 (2012).
[Crossref]

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84(2), 621–669 (2012).
[Crossref]

E. Flurin, N. Roch, F. Mallet, M. H. Devoret, and B. Huard, “Generating entangled microwave radiation over two transmission lines,” Phys. Rev. Lett. 109(18), 183901 (2012).
[Crossref]

C. Dong, V. Fiore, M. C. Kuzyk, and H. Wang, “Optomechanical dark mode,” Science 338(6114), 1609–1613 (2012).
[Crossref]

J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3(1), 1196 (2012).
[Crossref]

2011 (3)

H. Krauter, C. A. Muschik, K. Jensen, W. Wasilewski, J. M. Petersen, J. I. Cirac, and E. S. Polzik, “Entanglement generated by dissipation and steady state entanglement of two macroscopic objects,” Phys. Rev. Lett. 107(8), 080503 (2011).
[Crossref]

S. G. Hofer, W. Wieczorek, M. Aspelmeyer, and K. Hammerer, “Quantum entanglement and teleportation in pulsed cavity optomechanics,” Phys. Rev. A 84(5), 052327 (2011).
[Crossref]

Sh. Barzanjeh, D. Vitali, P. Tombesi, and G. J. Milburn, “Entangling optical and microwave cavity modes by means of a nanomechanical resonator,” Phys. Rev. A 84(4), 042342 (2011).
[Crossref]

2010 (2)

M. Neeley, R. C. Bialczak, M. Lenander, E. Lucero, M. Mariantoni, D. Sank, H. Wang, M. Weides, J. Wenner, Y. Yin, T. Yamamoto, A. N. Cleland, and J. M. Martinis, “Generation of three-qubit entangled states using superconducting phase qubits,” Nature 467(7315), 570–573 (2010).
[Crossref]

L. DiCarlo, M. Reed, L. Sun, B. L. Johnson, J. M. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Preparation and measurement of three-qubit entanglement in a superconducting circuit,” Nature 467(7315), 574–578 (2010).
[Crossref]

2008 (3)

M. Bhattacharya, P.-L. Giscard, and P. Meystre, “Entangling the rovibrational modes of a macroscopic mirror using radiation pressure,” Phys. Rev. A 77(3), 030303 (2008).
[Crossref]

C. Wipf, T. Corbitt, Y. Chen, and N. Mavalvala, “Route to ponderomotive entanglement of light via optically trapped mirrors,” New J. Phys. 10(9), 095017 (2008).
[Crossref]

C. Genes, A. Mari, P. Tombesi, and D. Vitali, “Robust entanglement of a micromechanical resonator with output optical fields,” Phys. Rev. A 78(3), 032316 (2008).
[Crossref]

2007 (2)

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99(25), 250401 (2007).
[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(3), 030405 (2007).
[Crossref]

2006 (1)

S. Pirandola, D. Vitali, P. Tombesi, and S. Lloyd, “Macroscopic entanglement by entanglement swapping,” Phys. Rev. Lett. 97(15), 150403 (2006).
[Crossref]

2005 (2)

S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77(2), 513–577 (2005).
[Crossref]

M. B. Plenio, “Logarithmic negativity: A full entanglement monotone that is not convex,” Phys. Rev. Lett. 95(9), 090503 (2005).
[Crossref]

2004 (1)

S. Pirandola, S. Mancini, D. Vitali, and P. Tombesi, “Light reflection upon a movable mirror as a paradigm for continuous variable teleportation network,” J. Mod. Opt. 51(6-7), 901–912 (2004).
[Crossref]

2003 (3)

S. Mancini, D. Vitali, and P. Tombesi, “Scheme for teleportation of quantum states onto a mechanical resonator,” Phys. Rev. Lett. 90(13), 137901 (2003).
[Crossref]

S. Pirandola, S. Mancini, D. Vitali, and P. Tombesi, “Continuous-variable entanglement and quantum-state teleportation between optical and macroscopic vibrational modes through radiation pressure,” Phys. Rev. A 68(6), 062317 (2003).
[Crossref]

A. J. Berkley, H. Xu, R. C. Ramos, M. A. Gubrud, F. W. Strauch, P. R. Johnson, J. R. Anderson, A. J. Dragt, C. J. Lobb, and F. C. Wellstood, “Entangled macroscopic quantum states in two superconducting qubits,” Science 300(5625), 1548–1550 (2003).
[Crossref]

2002 (1)

G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65(3), 032314 (2002).
[Crossref]

2001 (1)

B. Julsgaard, A. Kozhekin, and E. S. Polzik, “Experimental long-lived entanglement of two macroscopic objects,” Nature 413(6854), 400–403 (2001).
[Crossref]

1987 (1)

E. X. DeJesus and C. Kaufman, “Routh-Hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations,” Phys. Rev. A 35(12), 5288–5290 (1987).
[Crossref]

Abdi, M.

Sh. Barzanjeh, M. Abdi, G. J. Milburn, P. Tombesi, and D. Vitali, “Reversible optical-to-microwave quantum interface,” Phys. Rev. Lett. 109(13), 130503 (2012).
[Crossref]

Akram, U.

U. Akram, W. Munro, K. Nemoto, and G. J. Milburn, “Photon-phonon entanglement in coupled optomechanical arrays,” Phys. Rev. A 86(4), 042306 (2012).
[Crossref]

An, J. H.

C. J. Yang, J. H. An, W. Yang, and Y. Li, “Generation of stable entanglement between two cavity mirrors by squeezed-reservoir engineering,” Phys. Rev. A 92(6), 062311 (2015).
[Crossref]

Anderson, J. R.

A. J. Berkley, H. Xu, R. C. Ramos, M. A. Gubrud, F. W. Strauch, P. R. Johnson, J. R. Anderson, A. J. Dragt, C. J. Lobb, and F. C. Wellstood, “Entangled macroscopic quantum states in two superconducting qubits,” Science 300(5625), 1548–1550 (2003).
[Crossref]

Andrews, R. W.

R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, “Bidirectional and efficient conversion between microwave and optical light,” Nat. Phys. 10(4), 321–326 (2014).
[Crossref]

Arnold, G.

S. Barzanjeh, E. S. Redchenko, M. Peruzzo, M. Wulf, D. P. Lewis, G. Arnold, and J. M. Fink, “Stationary entangled radiation from micromechanical motion,” Nature 570(7762), 480–483 (2019).
[Crossref]

Asjad, M.

M. Asjad, P. Tombesi, and D. Vitali, “Feedback control of two-mode output entanglement and steering in cavity optomechanics,” Phys. Rev. A 94(5), 052312 (2016).
[Crossref]

Aspelmeyer, M.

S. G. Hofer, W. Wieczorek, M. Aspelmeyer, and K. Hammerer, “Quantum entanglement and teleportation in pulsed cavity optomechanics,” Phys. Rev. A 84(5), 052327 (2011).
[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(3), 030405 (2007).
[Crossref]

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99(25), 250401 (2007).
[Crossref]

Barzanjeh, S.

S. Barzanjeh, E. S. Redchenko, M. Peruzzo, M. Wulf, D. P. Lewis, G. Arnold, and J. M. Fink, “Stationary entangled radiation from micromechanical motion,” Nature 570(7762), 480–483 (2019).
[Crossref]

Barzanjeh, Sh.

Sh. Barzanjeh, S. Pirandola, and C. Weedbrook, “Continuous-variable dense coding by optomechanical cavities,” Phys. Rev. A 88(4), 042331 (2013).
[Crossref]

Sh. Barzanjeh, M. Abdi, G. J. Milburn, P. Tombesi, and D. Vitali, “Reversible optical-to-microwave quantum interface,” Phys. Rev. Lett. 109(13), 130503 (2012).
[Crossref]

Sh. Barzanjeh, D. Vitali, P. Tombesi, and G. J. Milburn, “Entangling optical and microwave cavity modes by means of a nanomechanical resonator,” Phys. Rev. A 84(4), 042342 (2011).
[Crossref]

Berkley, A. J.

A. J. Berkley, H. Xu, R. C. Ramos, M. A. Gubrud, F. W. Strauch, P. R. Johnson, J. R. Anderson, A. J. Dragt, C. J. Lobb, and F. C. Wellstood, “Entangled macroscopic quantum states in two superconducting qubits,” Science 300(5625), 1548–1550 (2003).
[Crossref]

Bhattacharya, M.

M. Bhattacharya, P.-L. Giscard, and P. Meystre, “Entangling the rovibrational modes of a macroscopic mirror using radiation pressure,” Phys. Rev. A 77(3), 030303 (2008).
[Crossref]

Bialczak, R. C.

M. Neeley, R. C. Bialczak, M. Lenander, E. Lucero, M. Mariantoni, D. Sank, H. Wang, M. Weides, J. Wenner, Y. Yin, T. Yamamoto, A. N. Cleland, and J. M. Martinis, “Generation of three-qubit entangled states using superconducting phase qubits,” Nature 467(7315), 570–573 (2010).
[Crossref]

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(3), 030405 (2007).
[Crossref]

Braunstein, S. L.

S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77(2), 513–577 (2005).
[Crossref]

Brukner, C.

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99(25), 250401 (2007).
[Crossref]

Cerf, N. J.

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84(2), 621–669 (2012).
[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(1), 1196 (2012).
[Crossref]

Chen, R. X.

R. X. Chen, L. T. Shen, Z. B. Yang, H. Z. Wu, and S. B. Zheng, “Enhancement of entanglement in distant mechanical vibrations via modulation in a coupled optomechanical system,” Phys. Rev. A 89(2), 023843 (2014).
[Crossref]

Chen, Y.

C. Wipf, T. Corbitt, Y. Chen, and N. Mavalvala, “Route to ponderomotive entanglement of light via optically trapped mirrors,” New J. Phys. 10(9), 095017 (2008).
[Crossref]

Chesi, S.

Y.-D. Wang, S. Chesi, and A. A. Clerk, “Bipartite and tripartite output entanglement in three-mode optomechanical systems,” Phys. Rev. A 91(1), 013807 (2015).
[Crossref]

Chow, J. M.

L. DiCarlo, M. Reed, L. Sun, B. L. Johnson, J. M. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Preparation and measurement of three-qubit entanglement in a superconducting circuit,” Nature 467(7315), 574–578 (2010).
[Crossref]

Cicak, K.

R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, “Bidirectional and efficient conversion between microwave and optical light,” Nat. Phys. 10(4), 321–326 (2014).
[Crossref]

Cirac, J. I.

H. Krauter, C. A. Muschik, K. Jensen, W. Wasilewski, J. M. Petersen, J. I. Cirac, and E. S. Polzik, “Entanglement generated by dissipation and steady state entanglement of two macroscopic objects,” Phys. Rev. Lett. 107(8), 080503 (2011).
[Crossref]

Cleland, A. N.

M. Neeley, R. C. Bialczak, M. Lenander, E. Lucero, M. Mariantoni, D. Sank, H. Wang, M. Weides, J. Wenner, Y. Yin, T. Yamamoto, A. N. Cleland, and J. M. Martinis, “Generation of three-qubit entangled states using superconducting phase qubits,” Nature 467(7315), 570–573 (2010).
[Crossref]

Clerk, A. A.

Y.-D. Wang, S. Chesi, and A. A. Clerk, “Bipartite and tripartite output entanglement in three-mode optomechanical systems,” Phys. Rev. A 91(1), 013807 (2015).
[Crossref]

Y.-D. Wang and A. A. Clerk, “Reservoir-engineered entanglement in optomechanical systems,” Phys. Rev. Lett. 110(25), 253601 (2013).
[Crossref]

Corbitt, T.

C. Wipf, T. Corbitt, Y. Chen, and N. Mavalvala, “Route to ponderomotive entanglement of light via optically trapped mirrors,” New J. Phys. 10(9), 095017 (2008).
[Crossref]

Dai, Y. T.

F. X. Sun, D. Mao, Y. T. Dai, Z. Ficek, Q. Y. He, and Q. H. Gong, “Phase control of entanglement and quantum steering in a three-mode optomechanical system,” New J. Phys. 19(12), 123039 (2017).
[Crossref]

DeJesus, E. X.

E. X. DeJesus and C. Kaufman, “Routh-Hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations,” Phys. Rev. A 35(12), 5288–5290 (1987).
[Crossref]

Deng, Z. J.

Z. J. Deng, S. J. M. Habraken, and F. Marquardt, “Entanglement rate for Gaussian continuous variable beams,” New J. Phys. 18(6), 063022 (2016).
[Crossref]

Z. J. Deng, X. B. Yan, Y. D. Wang, and C. W. Wu, “Optimizing the output-photon entanglement in multimode optomechanical systems,” Phys. Rev. A 93(3), 033842 (2016).
[Crossref]

Devoret, M. H.

E. Flurin, N. Roch, F. Mallet, M. H. Devoret, and B. Huard, “Generating entangled microwave radiation over two transmission lines,” Phys. Rev. Lett. 109(18), 183901 (2012).
[Crossref]

L. DiCarlo, M. Reed, L. Sun, B. L. Johnson, J. M. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Preparation and measurement of three-qubit entanglement in a superconducting circuit,” Nature 467(7315), 574–578 (2010).
[Crossref]

DiCarlo, L.

L. DiCarlo, M. Reed, L. Sun, B. L. Johnson, J. M. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Preparation and measurement of three-qubit entanglement in a superconducting circuit,” Nature 467(7315), 574–578 (2010).
[Crossref]

Dong, C.

C. Dong, V. Fiore, M. C. Kuzyk, and H. Wang, “Optomechanical dark mode,” Science 338(6114), 1609–1613 (2012).
[Crossref]

Dragt, A. J.

A. J. Berkley, H. Xu, R. C. Ramos, M. A. Gubrud, F. W. Strauch, P. R. Johnson, J. R. Anderson, A. J. Dragt, C. J. Lobb, and F. C. Wellstood, “Entangled macroscopic quantum states in two superconducting qubits,” Science 300(5625), 1548–1550 (2003).
[Crossref]

Drummond, P. D.

S. Kiesewetter, Q. Y. He, P. D. Drummond, and M. D. Reid, “Scalable quantum simulation of pulsed entanglement and Einstein-Podolsky-Rosen steering in optomechanics,” Phys. Rev. A 90(4), 043805 (2014).
[Crossref]

Eisert, J.

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99(25), 250401 (2007).
[Crossref]

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(3), 030405 (2007).
[Crossref]

Ficek, Z.

F. X. Sun, D. Mao, Y. T. Dai, Z. Ficek, Q. Y. He, and Q. H. Gong, “Phase control of entanglement and quantum steering in a three-mode optomechanical system,” New J. Phys. 19(12), 123039 (2017).
[Crossref]

Fink, J. M.

S. Barzanjeh, E. S. Redchenko, M. Peruzzo, M. Wulf, D. P. Lewis, G. Arnold, and J. M. Fink, “Stationary entangled radiation from micromechanical motion,” Nature 570(7762), 480–483 (2019).
[Crossref]

Fiore, V.

C. Dong, V. Fiore, M. C. Kuzyk, and H. Wang, “Optomechanical dark mode,” Science 338(6114), 1609–1613 (2012).
[Crossref]

Flurin, E.

E. Flurin, N. Roch, F. Mallet, M. H. Devoret, and B. Huard, “Generating entangled microwave radiation over two transmission lines,” Phys. Rev. Lett. 109(18), 183901 (2012).
[Crossref]

Frunzio, L.

L. DiCarlo, M. Reed, L. Sun, B. L. Johnson, J. M. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Preparation and measurement of three-qubit entanglement in a superconducting circuit,” Nature 467(7315), 574–578 (2010).
[Crossref]

Gambetta, J. M.

L. DiCarlo, M. Reed, L. Sun, B. L. Johnson, J. M. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Preparation and measurement of three-qubit entanglement in a superconducting circuit,” Nature 467(7315), 574–578 (2010).
[Crossref]

García-Patrón, R.

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84(2), 621–669 (2012).
[Crossref]

Gardiner, C.

C. Gardiner and P. Zoller, Quantum Noise, 3rd ed. (Springer, New York, 2004).

Genes, C.

C. Genes, A. Mari, P. Tombesi, and D. Vitali, “Robust entanglement of a micromechanical resonator with output optical fields,” Phys. Rev. A 78(3), 032316 (2008).
[Crossref]

Ghobadi, R.

Q. Lin, B. He, R. Ghobadi, and C. Simon, “Fully quantum approach to optomechanical entanglement,” Phys. Rev. A 90(2), 022309 (2014).
[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(3), 030405 (2007).
[Crossref]

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99(25), 250401 (2007).
[Crossref]

Girvin, S. M.

L. DiCarlo, M. Reed, L. Sun, B. L. Johnson, J. M. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Preparation and measurement of three-qubit entanglement in a superconducting circuit,” Nature 467(7315), 574–578 (2010).
[Crossref]

Giscard, P.-L.

M. Bhattacharya, P.-L. Giscard, and P. Meystre, “Entangling the rovibrational modes of a macroscopic mirror using radiation pressure,” Phys. Rev. A 77(3), 030303 (2008).
[Crossref]

Gong, Q. H.

F. X. Sun, D. Mao, Y. T. Dai, Z. Ficek, Q. Y. He, and Q. H. Gong, “Phase control of entanglement and quantum steering in a three-mode optomechanical system,” New J. Phys. 19(12), 123039 (2017).
[Crossref]

Gubrud, M. A.

A. J. Berkley, H. Xu, R. C. Ramos, M. A. Gubrud, F. W. Strauch, P. R. Johnson, J. R. Anderson, A. J. Dragt, C. J. Lobb, and F. C. Wellstood, “Entangled macroscopic quantum states in two superconducting qubits,” Science 300(5625), 1548–1550 (2003).
[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(3), 030405 (2007).
[Crossref]

Habraken, S. J. M.

Z. J. Deng, S. J. M. Habraken, and F. Marquardt, “Entanglement rate for Gaussian continuous variable beams,” New J. Phys. 18(6), 063022 (2016).
[Crossref]

Hammerer, K.

S. G. Hofer, W. Wieczorek, M. Aspelmeyer, and K. Hammerer, “Quantum entanglement and teleportation in pulsed cavity optomechanics,” Phys. Rev. A 84(5), 052327 (2011).
[Crossref]

He, B.

Q. Lin, B. He, R. Ghobadi, and C. Simon, “Fully quantum approach to optomechanical entanglement,” Phys. Rev. A 90(2), 022309 (2014).
[Crossref]

He, Q. Y.

F. X. Sun, D. Mao, Y. T. Dai, Z. Ficek, Q. Y. He, and Q. H. Gong, “Phase control of entanglement and quantum steering in a three-mode optomechanical system,” New J. Phys. 19(12), 123039 (2017).
[Crossref]

S. Kiesewetter, Q. Y. He, P. D. Drummond, and M. D. Reid, “Scalable quantum simulation of pulsed entanglement and Einstein-Podolsky-Rosen steering in optomechanics,” Phys. Rev. A 90(4), 043805 (2014).
[Crossref]

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(1), 1196 (2012).
[Crossref]

Hofer, S. G.

S. G. Hofer, W. Wieczorek, M. Aspelmeyer, and K. Hammerer, “Quantum entanglement and teleportation in pulsed cavity optomechanics,” Phys. Rev. A 84(5), 052327 (2011).
[Crossref]

Hu, B. L.

K. Sinha, S. Y. Lin, and B. L. Hu, “Mirror-field entanglement in a microscopic model for quantum optomechanics,” Phys. Rev. A 92(2), 023852 (2015).
[Crossref]

Huard, B.

E. Flurin, N. Roch, F. Mallet, M. H. Devoret, and B. Huard, “Generating entangled microwave radiation over two transmission lines,” Phys. Rev. Lett. 109(18), 183901 (2012).
[Crossref]

Jensen, K.

H. Krauter, C. A. Muschik, K. Jensen, W. Wasilewski, J. M. Petersen, J. I. Cirac, and E. S. Polzik, “Entanglement generated by dissipation and steady state entanglement of two macroscopic objects,” Phys. Rev. Lett. 107(8), 080503 (2011).
[Crossref]

Johnson, B. L.

L. DiCarlo, M. Reed, L. Sun, B. L. Johnson, J. M. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Preparation and measurement of three-qubit entanglement in a superconducting circuit,” Nature 467(7315), 574–578 (2010).
[Crossref]

Johnson, P. R.

A. J. Berkley, H. Xu, R. C. Ramos, M. A. Gubrud, F. W. Strauch, P. R. Johnson, J. R. Anderson, A. J. Dragt, C. J. Lobb, and F. C. Wellstood, “Entangled macroscopic quantum states in two superconducting qubits,” Science 300(5625), 1548–1550 (2003).
[Crossref]

Julsgaard, B.

B. Julsgaard, A. Kozhekin, and E. S. Polzik, “Experimental long-lived entanglement of two macroscopic objects,” Nature 413(6854), 400–403 (2001).
[Crossref]

Kaufman, C.

E. X. DeJesus and C. Kaufman, “Routh-Hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations,” Phys. Rev. A 35(12), 5288–5290 (1987).
[Crossref]

Kiesewetter, S.

S. Kiesewetter, Q. Y. He, P. D. Drummond, and M. D. Reid, “Scalable quantum simulation of pulsed entanglement and Einstein-Podolsky-Rosen steering in optomechanics,” Phys. Rev. A 90(4), 043805 (2014).
[Crossref]

Kim, M. S.

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99(25), 250401 (2007).
[Crossref]

Kozhekin, A.

B. Julsgaard, A. Kozhekin, and E. S. Polzik, “Experimental long-lived entanglement of two macroscopic objects,” Nature 413(6854), 400–403 (2001).
[Crossref]

Krauter, H.

H. Krauter, C. A. Muschik, K. Jensen, W. Wasilewski, J. M. Petersen, J. I. Cirac, and E. S. Polzik, “Entanglement generated by dissipation and steady state entanglement of two macroscopic objects,” Phys. Rev. Lett. 107(8), 080503 (2011).
[Crossref]

Kuzyk, M. C.

M. C. Kuzyk, S. J. van Enk, and H. Wang, “Generating robust optical entanglement in weak-coupling optomechanical systems,” Phys. Rev. A 88(6), 062341 (2013).
[Crossref]

C. Dong, V. Fiore, M. C. Kuzyk, and H. Wang, “Optomechanical dark mode,” Science 338(6114), 1609–1613 (2012).
[Crossref]

Lehnert, K. W.

R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, “Bidirectional and efficient conversion between microwave and optical light,” Nat. Phys. 10(4), 321–326 (2014).
[Crossref]

T. A. Palomaki, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, “Entangling mechanical motion with microwave fields,” Science 342(6159), 710–713 (2013).
[Crossref]

Lenander, M.

M. Neeley, R. C. Bialczak, M. Lenander, E. Lucero, M. Mariantoni, D. Sank, H. Wang, M. Weides, J. Wenner, Y. Yin, T. Yamamoto, A. N. Cleland, and J. M. Martinis, “Generation of three-qubit entangled states using superconducting phase qubits,” Nature 467(7315), 570–573 (2010).
[Crossref]

Lewis, D. P.

S. Barzanjeh, E. S. Redchenko, M. Peruzzo, M. Wulf, D. P. Lewis, G. Arnold, and J. M. Fink, “Stationary entangled radiation from micromechanical motion,” Nature 570(7762), 480–483 (2019).
[Crossref]

Li, G.

J. Li, G. Li, S. Zippilli, D. Vitali, and T. Zhang, “Enhanced entanglement of two different mechanical resonators via coherent feedback,” Phys. Rev. A 95(4), 043819 (2017).
[Crossref]

Li, H.-K.

H.-K. Li, X.-X. Ren, Y.-C. Liu, and Y.-F. Xiao, “Photon-photon interactions in a largely detuned optomechanical cavity,” Phys. Rev. A 88(5), 053850 (2013).
[Crossref]

Li, J.

J. Li, G. Li, S. Zippilli, D. Vitali, and T. Zhang, “Enhanced entanglement of two different mechanical resonators via coherent feedback,” Phys. Rev. A 95(4), 043819 (2017).
[Crossref]

J. Li, I. Moaddel Haghighi, N. Malossi, S. Zippilli, and D. Vitali, “Generation and detection of large and robust entanglement between two different mechanical resonators in cavity optomechanics,” New J. Phys. 17(10), 103037 (2015).
[Crossref]

Li, Y.

C. J. Yang, J. H. An, W. Yang, and Y. Li, “Generation of stable entanglement between two cavity mirrors by squeezed-reservoir engineering,” Phys. Rev. A 92(6), 062311 (2015).
[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(1), 014302 (2014).
[Crossref]

Lin, Q.

Q. Lin, B. He, R. Ghobadi, and C. Simon, “Fully quantum approach to optomechanical entanglement,” Phys. Rev. A 90(2), 022309 (2014).
[Crossref]

Lin, S. Y.

K. Sinha, S. Y. Lin, and B. L. Hu, “Mirror-field entanglement in a microscopic model for quantum optomechanics,” Phys. Rev. A 92(2), 023852 (2015).
[Crossref]

Liu, Y.-C.

H.-K. Li, X.-X. Ren, Y.-C. Liu, and Y.-F. Xiao, “Photon-photon interactions in a largely detuned optomechanical cavity,” Phys. Rev. A 88(5), 053850 (2013).
[Crossref]

Lloyd, S.

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84(2), 621–669 (2012).
[Crossref]

S. Pirandola, D. Vitali, P. Tombesi, and S. Lloyd, “Macroscopic entanglement by entanglement swapping,” Phys. Rev. Lett. 97(15), 150403 (2006).
[Crossref]

Lobb, C. J.

A. J. Berkley, H. Xu, R. C. Ramos, M. A. Gubrud, F. W. Strauch, P. R. Johnson, J. R. Anderson, A. J. Dragt, C. J. Lobb, and F. C. Wellstood, “Entangled macroscopic quantum states in two superconducting qubits,” Science 300(5625), 1548–1550 (2003).
[Crossref]

Lucero, E.

M. Neeley, R. C. Bialczak, M. Lenander, E. Lucero, M. Mariantoni, D. Sank, H. Wang, M. Weides, J. Wenner, Y. Yin, T. Yamamoto, A. N. Cleland, and J. M. Martinis, “Generation of three-qubit entangled states using superconducting phase qubits,” Nature 467(7315), 570–573 (2010).
[Crossref]

Mallet, F.

E. Flurin, N. Roch, F. Mallet, M. H. Devoret, and B. Huard, “Generating entangled microwave radiation over two transmission lines,” Phys. Rev. Lett. 109(18), 183901 (2012).
[Crossref]

Malossi, N.

J. Li, I. Moaddel Haghighi, N. Malossi, S. Zippilli, and D. Vitali, “Generation and detection of large and robust entanglement between two different mechanical resonators in cavity optomechanics,” New J. Phys. 17(10), 103037 (2015).
[Crossref]

Mancini, S.

S. Pirandola, S. Mancini, D. Vitali, and P. Tombesi, “Light reflection upon a movable mirror as a paradigm for continuous variable teleportation network,” J. Mod. Opt. 51(6-7), 901–912 (2004).
[Crossref]

S. Pirandola, S. Mancini, D. Vitali, and P. Tombesi, “Continuous-variable entanglement and quantum-state teleportation between optical and macroscopic vibrational modes through radiation pressure,” Phys. Rev. A 68(6), 062317 (2003).
[Crossref]

S. Mancini, D. Vitali, and P. Tombesi, “Scheme for teleportation of quantum states onto a mechanical resonator,” Phys. Rev. Lett. 90(13), 137901 (2003).
[Crossref]

Mao, D.

F. X. Sun, D. Mao, Y. T. Dai, Z. Ficek, Q. Y. He, and Q. H. Gong, “Phase control of entanglement and quantum steering in a three-mode optomechanical system,” New J. Phys. 19(12), 123039 (2017).
[Crossref]

Mari, A.

C. Genes, A. Mari, P. Tombesi, and D. Vitali, “Robust entanglement of a micromechanical resonator with output optical fields,” Phys. Rev. A 78(3), 032316 (2008).
[Crossref]

Mariantoni, M.

M. Neeley, R. C. Bialczak, M. Lenander, E. Lucero, M. Mariantoni, D. Sank, H. Wang, M. Weides, J. Wenner, Y. Yin, T. Yamamoto, A. N. Cleland, and J. M. Martinis, “Generation of three-qubit entangled states using superconducting phase qubits,” Nature 467(7315), 570–573 (2010).
[Crossref]

Marquardt, F.

Z. J. Deng, S. J. M. Habraken, and F. Marquardt, “Entanglement rate for Gaussian continuous variable beams,” New J. Phys. 18(6), 063022 (2016).
[Crossref]

Martinis, J. M.

M. Neeley, R. C. Bialczak, M. Lenander, E. Lucero, M. Mariantoni, D. Sank, H. Wang, M. Weides, J. Wenner, Y. Yin, T. Yamamoto, A. N. Cleland, and J. M. Martinis, “Generation of three-qubit entangled states using superconducting phase qubits,” Nature 467(7315), 570–573 (2010).
[Crossref]

Mavalvala, N.

C. Wipf, T. Corbitt, Y. Chen, and N. Mavalvala, “Route to ponderomotive entanglement of light via optically trapped mirrors,” New J. Phys. 10(9), 095017 (2008).
[Crossref]

Meystre, P.

M. Bhattacharya, P.-L. Giscard, and P. Meystre, “Entangling the rovibrational modes of a macroscopic mirror using radiation pressure,” Phys. Rev. A 77(3), 030303 (2008).
[Crossref]

Milburn, G. J.

Sh. Barzanjeh, M. Abdi, G. J. Milburn, P. Tombesi, and D. Vitali, “Reversible optical-to-microwave quantum interface,” Phys. Rev. Lett. 109(13), 130503 (2012).
[Crossref]

U. Akram, W. Munro, K. Nemoto, and G. J. Milburn, “Photon-phonon entanglement in coupled optomechanical arrays,” Phys. Rev. A 86(4), 042306 (2012).
[Crossref]

Sh. Barzanjeh, D. Vitali, P. Tombesi, and G. J. Milburn, “Entangling optical and microwave cavity modes by means of a nanomechanical resonator,” Phys. Rev. A 84(4), 042342 (2011).
[Crossref]

Moaddel Haghighi, I.

J. Li, I. Moaddel Haghighi, N. Malossi, S. Zippilli, and D. Vitali, “Generation and detection of large and robust entanglement between two different mechanical resonators in cavity optomechanics,” New J. Phys. 17(10), 103037 (2015).
[Crossref]

Munro, W.

U. Akram, W. Munro, K. Nemoto, and G. J. Milburn, “Photon-phonon entanglement in coupled optomechanical arrays,” Phys. Rev. A 86(4), 042306 (2012).
[Crossref]

Muschik, C. A.

H. Krauter, C. A. Muschik, K. Jensen, W. Wasilewski, J. M. Petersen, J. I. Cirac, and E. S. Polzik, “Entanglement generated by dissipation and steady state entanglement of two macroscopic objects,” Phys. Rev. Lett. 107(8), 080503 (2011).
[Crossref]

Neeley, M.

M. Neeley, R. C. Bialczak, M. Lenander, E. Lucero, M. Mariantoni, D. Sank, H. Wang, M. Weides, J. Wenner, Y. Yin, T. Yamamoto, A. N. Cleland, and J. M. Martinis, “Generation of three-qubit entangled states using superconducting phase qubits,” Nature 467(7315), 570–573 (2010).
[Crossref]

Nemoto, K.

U. Akram, W. Munro, K. Nemoto, and G. J. Milburn, “Photon-phonon entanglement in coupled optomechanical arrays,” Phys. Rev. A 86(4), 042306 (2012).
[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(1), 014302 (2014).
[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(1), 1196 (2012).
[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(6159), 710–713 (2013).
[Crossref]

Paternostro, M.

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99(25), 250401 (2007).
[Crossref]

Peruzzo, M.

S. Barzanjeh, E. S. Redchenko, M. Peruzzo, M. Wulf, D. P. Lewis, G. Arnold, and J. M. Fink, “Stationary entangled radiation from micromechanical motion,” Nature 570(7762), 480–483 (2019).
[Crossref]

Petersen, J. M.

H. Krauter, C. A. Muschik, K. Jensen, W. Wasilewski, J. M. Petersen, J. I. Cirac, and E. S. Polzik, “Entanglement generated by dissipation and steady state entanglement of two macroscopic objects,” Phys. Rev. Lett. 107(8), 080503 (2011).
[Crossref]

Peterson, R. W.

R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, “Bidirectional and efficient conversion between microwave and optical light,” Nat. Phys. 10(4), 321–326 (2014).
[Crossref]

Pirandola, S.

Sh. Barzanjeh, S. Pirandola, and C. Weedbrook, “Continuous-variable dense coding by optomechanical cavities,” Phys. Rev. A 88(4), 042331 (2013).
[Crossref]

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84(2), 621–669 (2012).
[Crossref]

S. Pirandola, D. Vitali, P. Tombesi, and S. Lloyd, “Macroscopic entanglement by entanglement swapping,” Phys. Rev. Lett. 97(15), 150403 (2006).
[Crossref]

S. Pirandola, S. Mancini, D. Vitali, and P. Tombesi, “Light reflection upon a movable mirror as a paradigm for continuous variable teleportation network,” J. Mod. Opt. 51(6-7), 901–912 (2004).
[Crossref]

S. Pirandola, S. Mancini, D. Vitali, and P. Tombesi, “Continuous-variable entanglement and quantum-state teleportation between optical and macroscopic vibrational modes through radiation pressure,” Phys. Rev. A 68(6), 062317 (2003).
[Crossref]

Plenio, M. B.

M. B. Plenio, “Logarithmic negativity: A full entanglement monotone that is not convex,” Phys. Rev. Lett. 95(9), 090503 (2005).
[Crossref]

Polzik, E. S.

H. Krauter, C. A. Muschik, K. Jensen, W. Wasilewski, J. M. Petersen, J. I. Cirac, and E. S. Polzik, “Entanglement generated by dissipation and steady state entanglement of two macroscopic objects,” Phys. Rev. Lett. 107(8), 080503 (2011).
[Crossref]

B. Julsgaard, A. Kozhekin, and E. S. Polzik, “Experimental long-lived entanglement of two macroscopic objects,” Nature 413(6854), 400–403 (2001).
[Crossref]

Purdy, T. P.

R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, “Bidirectional and efficient conversion between microwave and optical light,” Nat. Phys. 10(4), 321–326 (2014).
[Crossref]

Ralph, T. C.

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84(2), 621–669 (2012).
[Crossref]

Ramos, R. C.

A. J. Berkley, H. Xu, R. C. Ramos, M. A. Gubrud, F. W. Strauch, P. R. Johnson, J. R. Anderson, A. J. Dragt, C. J. Lobb, and F. C. Wellstood, “Entangled macroscopic quantum states in two superconducting qubits,” Science 300(5625), 1548–1550 (2003).
[Crossref]

Redchenko, E. S.

S. Barzanjeh, E. S. Redchenko, M. Peruzzo, M. Wulf, D. P. Lewis, G. Arnold, and J. M. Fink, “Stationary entangled radiation from micromechanical motion,” Nature 570(7762), 480–483 (2019).
[Crossref]

Reed, M.

L. DiCarlo, M. Reed, L. Sun, B. L. Johnson, J. M. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Preparation and measurement of three-qubit entanglement in a superconducting circuit,” Nature 467(7315), 574–578 (2010).
[Crossref]

Regal, C. A.

R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, “Bidirectional and efficient conversion between microwave and optical light,” Nat. Phys. 10(4), 321–326 (2014).
[Crossref]

Reid, M. D.

S. Kiesewetter, Q. Y. He, P. D. Drummond, and M. D. Reid, “Scalable quantum simulation of pulsed entanglement and Einstein-Podolsky-Rosen steering in optomechanics,” Phys. Rev. A 90(4), 043805 (2014).
[Crossref]

Ren, X.-X.

H.-K. Li, X.-X. Ren, Y.-C. Liu, and Y.-F. Xiao, “Photon-photon interactions in a largely detuned optomechanical cavity,” Phys. Rev. A 88(5), 053850 (2013).
[Crossref]

Roch, N.

E. Flurin, N. Roch, F. Mallet, M. H. Devoret, and B. Huard, “Generating entangled microwave radiation over two transmission lines,” Phys. Rev. Lett. 109(18), 183901 (2012).
[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(1), 1196 (2012).
[Crossref]

Sank, D.

M. Neeley, R. C. Bialczak, M. Lenander, E. Lucero, M. Mariantoni, D. Sank, H. Wang, M. Weides, J. Wenner, Y. Yin, T. Yamamoto, A. N. Cleland, and J. M. Martinis, “Generation of three-qubit entangled states using superconducting phase qubits,” Nature 467(7315), 570–573 (2010).
[Crossref]

Schoelkopf, R. J.

L. DiCarlo, M. Reed, L. Sun, B. L. Johnson, J. M. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Preparation and measurement of three-qubit entanglement in a superconducting circuit,” Nature 467(7315), 574–578 (2010).
[Crossref]

Shapiro, J. H.

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84(2), 621–669 (2012).
[Crossref]

Shen, L. T.

R. X. Chen, L. T. Shen, Z. B. Yang, H. Z. Wu, and S. B. Zheng, “Enhancement of entanglement in distant mechanical vibrations via modulation in a coupled optomechanical system,” Phys. Rev. A 89(2), 023843 (2014).
[Crossref]

Simmonds, R. W.

R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, “Bidirectional and efficient conversion between microwave and optical light,” Nat. Phys. 10(4), 321–326 (2014).
[Crossref]

T. A. Palomaki, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, “Entangling mechanical motion with microwave fields,” Science 342(6159), 710–713 (2013).
[Crossref]

Simon, C.

Q. Lin, B. He, R. Ghobadi, and C. Simon, “Fully quantum approach to optomechanical entanglement,” Phys. Rev. A 90(2), 022309 (2014).
[Crossref]

Sinha, K.

K. Sinha, S. Y. Lin, and B. L. Hu, “Mirror-field entanglement in a microscopic model for quantum optomechanics,” Phys. Rev. A 92(2), 023852 (2015).
[Crossref]

Strauch, F. W.

A. J. Berkley, H. Xu, R. C. Ramos, M. A. Gubrud, F. W. Strauch, P. R. Johnson, J. R. Anderson, A. J. Dragt, C. J. Lobb, and F. C. Wellstood, “Entangled macroscopic quantum states in two superconducting qubits,” Science 300(5625), 1548–1550 (2003).
[Crossref]

Sun, F. X.

F. X. Sun, D. Mao, Y. T. Dai, Z. Ficek, Q. Y. He, and Q. H. Gong, “Phase control of entanglement and quantum steering in a three-mode optomechanical system,” New J. Phys. 19(12), 123039 (2017).
[Crossref]

Sun, L.

L. DiCarlo, M. Reed, L. Sun, B. L. Johnson, J. M. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Preparation and measurement of three-qubit entanglement in a superconducting circuit,” Nature 467(7315), 574–578 (2010).
[Crossref]

Teufel, J. D.

T. A. Palomaki, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, “Entangling mechanical motion with microwave fields,” Science 342(6159), 710–713 (2013).
[Crossref]

Tian, L.

L. Tian, “Robust photon entanglement via quantum interference in optomechanical interfaces,” Phys. Rev. Lett. 110(23), 233602 (2013).
[Crossref]

Tombesi, P.

M. Asjad, P. Tombesi, and D. Vitali, “Feedback control of two-mode output entanglement and steering in cavity optomechanics,” Phys. Rev. A 94(5), 052312 (2016).
[Crossref]

Sh. Barzanjeh, M. Abdi, G. J. Milburn, P. Tombesi, and D. Vitali, “Reversible optical-to-microwave quantum interface,” Phys. Rev. Lett. 109(13), 130503 (2012).
[Crossref]

Sh. Barzanjeh, D. Vitali, P. Tombesi, and G. J. Milburn, “Entangling optical and microwave cavity modes by means of a nanomechanical resonator,” Phys. Rev. A 84(4), 042342 (2011).
[Crossref]

C. Genes, A. Mari, P. Tombesi, and D. Vitali, “Robust entanglement of a micromechanical resonator with output optical fields,” Phys. Rev. A 78(3), 032316 (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(3), 030405 (2007).
[Crossref]

S. Pirandola, D. Vitali, P. Tombesi, and S. Lloyd, “Macroscopic entanglement by entanglement swapping,” Phys. Rev. Lett. 97(15), 150403 (2006).
[Crossref]

S. Pirandola, S. Mancini, D. Vitali, and P. Tombesi, “Light reflection upon a movable mirror as a paradigm for continuous variable teleportation network,” J. Mod. Opt. 51(6-7), 901–912 (2004).
[Crossref]

S. Mancini, D. Vitali, and P. Tombesi, “Scheme for teleportation of quantum states onto a mechanical resonator,” Phys. Rev. Lett. 90(13), 137901 (2003).
[Crossref]

S. Pirandola, S. Mancini, D. Vitali, and P. Tombesi, “Continuous-variable entanglement and quantum-state teleportation between optical and macroscopic vibrational modes through radiation pressure,” Phys. Rev. A 68(6), 062317 (2003).
[Crossref]

van Enk, S. J.

M. C. Kuzyk, S. J. van Enk, and H. Wang, “Generating robust optical entanglement in weak-coupling optomechanical systems,” Phys. Rev. A 88(6), 062341 (2013).
[Crossref]

van Loock, P.

S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77(2), 513–577 (2005).
[Crossref]

Vedral, V.

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(3), 030405 (2007).
[Crossref]

Vidal, G.

G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65(3), 032314 (2002).
[Crossref]

Vitali, D.

J. Li, G. Li, S. Zippilli, D. Vitali, and T. Zhang, “Enhanced entanglement of two different mechanical resonators via coherent feedback,” Phys. Rev. A 95(4), 043819 (2017).
[Crossref]

M. Asjad, P. Tombesi, and D. Vitali, “Feedback control of two-mode output entanglement and steering in cavity optomechanics,” Phys. Rev. A 94(5), 052312 (2016).
[Crossref]

J. Li, I. Moaddel Haghighi, N. Malossi, S. Zippilli, and D. Vitali, “Generation and detection of large and robust entanglement between two different mechanical resonators in cavity optomechanics,” New J. Phys. 17(10), 103037 (2015).
[Crossref]

Sh. Barzanjeh, M. Abdi, G. J. Milburn, P. Tombesi, and D. Vitali, “Reversible optical-to-microwave quantum interface,” Phys. Rev. Lett. 109(13), 130503 (2012).
[Crossref]

Sh. Barzanjeh, D. Vitali, P. Tombesi, and G. J. Milburn, “Entangling optical and microwave cavity modes by means of a nanomechanical resonator,” Phys. Rev. A 84(4), 042342 (2011).
[Crossref]

C. Genes, A. Mari, P. Tombesi, and D. Vitali, “Robust entanglement of a micromechanical resonator with output optical fields,” Phys. Rev. A 78(3), 032316 (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(3), 030405 (2007).
[Crossref]

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99(25), 250401 (2007).
[Crossref]

S. Pirandola, D. Vitali, P. Tombesi, and S. Lloyd, “Macroscopic entanglement by entanglement swapping,” Phys. Rev. Lett. 97(15), 150403 (2006).
[Crossref]

S. Pirandola, S. Mancini, D. Vitali, and P. Tombesi, “Light reflection upon a movable mirror as a paradigm for continuous variable teleportation network,” J. Mod. Opt. 51(6-7), 901–912 (2004).
[Crossref]

S. Pirandola, S. Mancini, D. Vitali, and P. Tombesi, “Continuous-variable entanglement and quantum-state teleportation between optical and macroscopic vibrational modes through radiation pressure,” Phys. Rev. A 68(6), 062317 (2003).
[Crossref]

S. Mancini, D. Vitali, and P. Tombesi, “Scheme for teleportation of quantum states onto a mechanical resonator,” Phys. Rev. Lett. 90(13), 137901 (2003).
[Crossref]

Wang, H.

M. C. Kuzyk, S. J. van Enk, and H. Wang, “Generating robust optical entanglement in weak-coupling optomechanical systems,” Phys. Rev. A 88(6), 062341 (2013).
[Crossref]

C. Dong, V. Fiore, M. C. Kuzyk, and H. Wang, “Optomechanical dark mode,” Science 338(6114), 1609–1613 (2012).
[Crossref]

M. Neeley, R. C. Bialczak, M. Lenander, E. Lucero, M. Mariantoni, D. Sank, H. Wang, M. Weides, J. Wenner, Y. Yin, T. Yamamoto, A. N. Cleland, and J. M. Martinis, “Generation of three-qubit entangled states using superconducting phase qubits,” Nature 467(7315), 570–573 (2010).
[Crossref]

Wang, Y. D.

Z. J. Deng, X. B. Yan, Y. D. Wang, and C. W. Wu, “Optimizing the output-photon entanglement in multimode optomechanical systems,” Phys. Rev. A 93(3), 033842 (2016).
[Crossref]

Wang, Y.-D.

Y.-D. Wang, S. Chesi, and A. A. Clerk, “Bipartite and tripartite output entanglement in three-mode optomechanical systems,” Phys. Rev. A 91(1), 013807 (2015).
[Crossref]

Y.-D. Wang and A. A. Clerk, “Reservoir-engineered entanglement in optomechanical systems,” Phys. Rev. Lett. 110(25), 253601 (2013).
[Crossref]

Wasilewski, W.

H. Krauter, C. A. Muschik, K. Jensen, W. Wasilewski, J. M. Petersen, J. I. Cirac, and E. S. Polzik, “Entanglement generated by dissipation and steady state entanglement of two macroscopic objects,” Phys. Rev. Lett. 107(8), 080503 (2011).
[Crossref]

Weedbrook, C.

Sh. Barzanjeh, S. Pirandola, and C. Weedbrook, “Continuous-variable dense coding by optomechanical cavities,” Phys. Rev. A 88(4), 042331 (2013).
[Crossref]

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84(2), 621–669 (2012).
[Crossref]

Weides, M.

M. Neeley, R. C. Bialczak, M. Lenander, E. Lucero, M. Mariantoni, D. Sank, H. Wang, M. Weides, J. Wenner, Y. Yin, T. Yamamoto, A. N. Cleland, and J. M. Martinis, “Generation of three-qubit entangled states using superconducting phase qubits,” Nature 467(7315), 570–573 (2010).
[Crossref]

Wellstood, F. C.

A. J. Berkley, H. Xu, R. C. Ramos, M. A. Gubrud, F. W. Strauch, P. R. Johnson, J. R. Anderson, A. J. Dragt, C. J. Lobb, and F. C. Wellstood, “Entangled macroscopic quantum states in two superconducting qubits,” Science 300(5625), 1548–1550 (2003).
[Crossref]

Wenner, J.

M. Neeley, R. C. Bialczak, M. Lenander, E. Lucero, M. Mariantoni, D. Sank, H. Wang, M. Weides, J. Wenner, Y. Yin, T. Yamamoto, A. N. Cleland, and J. M. Martinis, “Generation of three-qubit entangled states using superconducting phase qubits,” Nature 467(7315), 570–573 (2010).
[Crossref]

Werner, R. F.

G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65(3), 032314 (2002).
[Crossref]

Wieczorek, W.

S. G. Hofer, W. Wieczorek, M. Aspelmeyer, and K. Hammerer, “Quantum entanglement and teleportation in pulsed cavity optomechanics,” Phys. Rev. A 84(5), 052327 (2011).
[Crossref]

Wipf, C.

C. Wipf, T. Corbitt, Y. Chen, and N. Mavalvala, “Route to ponderomotive entanglement of light via optically trapped mirrors,” New J. Phys. 10(9), 095017 (2008).
[Crossref]

Wu, C. W.

Z. J. Deng, X. B. Yan, Y. D. Wang, and C. W. Wu, “Optimizing the output-photon entanglement in multimode optomechanical systems,” Phys. Rev. A 93(3), 033842 (2016).
[Crossref]

Wu, H. Z.

R. X. Chen, L. T. Shen, Z. B. Yang, H. Z. Wu, and S. B. Zheng, “Enhancement of entanglement in distant mechanical vibrations via modulation in a coupled optomechanical system,” Phys. Rev. A 89(2), 023843 (2014).
[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(1), 014302 (2014).
[Crossref]

Wulf, M.

S. Barzanjeh, E. S. Redchenko, M. Peruzzo, M. Wulf, D. P. Lewis, G. Arnold, and J. M. Fink, “Stationary entangled radiation from micromechanical motion,” Nature 570(7762), 480–483 (2019).
[Crossref]

Xiao, Y.-F.

H.-K. Li, X.-X. Ren, Y.-C. Liu, and Y.-F. Xiao, “Photon-photon interactions in a largely detuned optomechanical cavity,” Phys. Rev. A 88(5), 053850 (2013).
[Crossref]

Xu, H.

A. J. Berkley, H. Xu, R. C. Ramos, M. A. Gubrud, F. W. Strauch, P. R. Johnson, J. R. Anderson, A. J. Dragt, C. J. Lobb, and F. C. Wellstood, “Entangled macroscopic quantum states in two superconducting qubits,” Science 300(5625), 1548–1550 (2003).
[Crossref]

Yamamoto, T.

M. Neeley, R. C. Bialczak, M. Lenander, E. Lucero, M. Mariantoni, D. Sank, H. Wang, M. Weides, J. Wenner, Y. Yin, T. Yamamoto, A. N. Cleland, and J. M. Martinis, “Generation of three-qubit entangled states using superconducting phase qubits,” Nature 467(7315), 570–573 (2010).
[Crossref]

Yan, X. B.

X. B. Yan, “Enhanced output entanglement with reservoir engineering,” Phys. Rev. A 96(5), 053831 (2017).
[Crossref]

Z. J. Deng, X. B. Yan, Y. D. Wang, and C. W. Wu, “Optimizing the output-photon entanglement in multimode optomechanical systems,” Phys. Rev. A 93(3), 033842 (2016).
[Crossref]

Yang, C. J.

C. J. Yang, J. H. An, W. Yang, and Y. Li, “Generation of stable entanglement between two cavity mirrors by squeezed-reservoir engineering,” Phys. Rev. A 92(6), 062311 (2015).
[Crossref]

Yang, W.

C. J. Yang, J. H. An, W. Yang, and Y. Li, “Generation of stable entanglement between two cavity mirrors by squeezed-reservoir engineering,” Phys. Rev. A 92(6), 062311 (2015).
[Crossref]

Yang, Z. B.

R. X. Chen, L. T. Shen, Z. B. Yang, H. Z. Wu, and S. B. Zheng, “Enhancement of entanglement in distant mechanical vibrations via modulation in a coupled optomechanical system,” Phys. Rev. A 89(2), 023843 (2014).
[Crossref]

Yin, Y.

M. Neeley, R. C. Bialczak, M. Lenander, E. Lucero, M. Mariantoni, D. Sank, H. Wang, M. Weides, J. Wenner, Y. Yin, T. Yamamoto, A. N. Cleland, and J. M. Martinis, “Generation of three-qubit entangled states using superconducting phase qubits,” Nature 467(7315), 570–573 (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(3), 030405 (2007).
[Crossref]

Zhang, T.

J. Li, G. Li, S. Zippilli, D. Vitali, and T. Zhang, “Enhanced entanglement of two different mechanical resonators via coherent feedback,” Phys. Rev. A 95(4), 043819 (2017).
[Crossref]

Zheng, S. B.

R. X. Chen, L. T. Shen, Z. B. Yang, H. Z. Wu, and S. B. Zheng, “Enhancement of entanglement in distant mechanical vibrations via modulation in a coupled optomechanical system,” Phys. Rev. A 89(2), 023843 (2014).
[Crossref]

Zippilli, S.

J. Li, G. Li, S. Zippilli, D. Vitali, and T. Zhang, “Enhanced entanglement of two different mechanical resonators via coherent feedback,” Phys. Rev. A 95(4), 043819 (2017).
[Crossref]

J. Li, I. Moaddel Haghighi, N. Malossi, S. Zippilli, and D. Vitali, “Generation and detection of large and robust entanglement between two different mechanical resonators in cavity optomechanics,” New J. Phys. 17(10), 103037 (2015).
[Crossref]

Zoller, P.

C. Gardiner and P. Zoller, Quantum Noise, 3rd ed. (Springer, New York, 2004).

J. Mod. Opt. (1)

S. Pirandola, S. Mancini, D. Vitali, and P. Tombesi, “Light reflection upon a movable mirror as a paradigm for continuous variable teleportation network,” J. Mod. Opt. 51(6-7), 901–912 (2004).
[Crossref]

Nat. Commun. (1)

J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3(1), 1196 (2012).
[Crossref]

Nat. Phys. (1)

R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, “Bidirectional and efficient conversion between microwave and optical light,” Nat. Phys. 10(4), 321–326 (2014).
[Crossref]

Nature (4)

S. Barzanjeh, E. S. Redchenko, M. Peruzzo, M. Wulf, D. P. Lewis, G. Arnold, and J. M. Fink, “Stationary entangled radiation from micromechanical motion,” Nature 570(7762), 480–483 (2019).
[Crossref]

B. Julsgaard, A. Kozhekin, and E. S. Polzik, “Experimental long-lived entanglement of two macroscopic objects,” Nature 413(6854), 400–403 (2001).
[Crossref]

M. Neeley, R. C. Bialczak, M. Lenander, E. Lucero, M. Mariantoni, D. Sank, H. Wang, M. Weides, J. Wenner, Y. Yin, T. Yamamoto, A. N. Cleland, and J. M. Martinis, “Generation of three-qubit entangled states using superconducting phase qubits,” Nature 467(7315), 570–573 (2010).
[Crossref]

L. DiCarlo, M. Reed, L. Sun, B. L. Johnson, J. M. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Preparation and measurement of three-qubit entanglement in a superconducting circuit,” Nature 467(7315), 574–578 (2010).
[Crossref]

New J. Phys. (4)

C. Wipf, T. Corbitt, Y. Chen, and N. Mavalvala, “Route to ponderomotive entanglement of light via optically trapped mirrors,” New J. Phys. 10(9), 095017 (2008).
[Crossref]

J. Li, I. Moaddel Haghighi, N. Malossi, S. Zippilli, and D. Vitali, “Generation and detection of large and robust entanglement between two different mechanical resonators in cavity optomechanics,” New J. Phys. 17(10), 103037 (2015).
[Crossref]

Z. J. Deng, S. J. M. Habraken, and F. Marquardt, “Entanglement rate for Gaussian continuous variable beams,” New J. Phys. 18(6), 063022 (2016).
[Crossref]

F. X. Sun, D. Mao, Y. T. Dai, Z. Ficek, Q. Y. He, and Q. H. Gong, “Phase control of entanglement and quantum steering in a three-mode optomechanical system,” New J. Phys. 19(12), 123039 (2017).
[Crossref]

Phys. Rev. A (22)

X. B. Yan, “Enhanced output entanglement with reservoir engineering,” Phys. Rev. A 96(5), 053831 (2017).
[Crossref]

S. G. Hofer, W. Wieczorek, M. Aspelmeyer, and K. Hammerer, “Quantum entanglement and teleportation in pulsed cavity optomechanics,” Phys. Rev. A 84(5), 052327 (2011).
[Crossref]

U. Akram, W. Munro, K. Nemoto, and G. J. Milburn, “Photon-phonon entanglement in coupled optomechanical arrays,” Phys. Rev. A 86(4), 042306 (2012).
[Crossref]

K. Sinha, S. Y. Lin, and B. L. Hu, “Mirror-field entanglement in a microscopic model for quantum optomechanics,” Phys. Rev. A 92(2), 023852 (2015).
[Crossref]

E. X. DeJesus and C. Kaufman, “Routh-Hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations,” Phys. Rev. A 35(12), 5288–5290 (1987).
[Crossref]

G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65(3), 032314 (2002).
[Crossref]

Z. J. Deng, X. B. Yan, Y. D. Wang, and C. W. Wu, “Optimizing the output-photon entanglement in multimode optomechanical systems,” Phys. Rev. A 93(3), 033842 (2016).
[Crossref]

Y.-D. Wang, S. Chesi, and A. A. Clerk, “Bipartite and tripartite output entanglement in three-mode optomechanical systems,” Phys. Rev. A 91(1), 013807 (2015).
[Crossref]

H.-K. Li, X.-X. Ren, Y.-C. Liu, and Y.-F. Xiao, “Photon-photon interactions in a largely detuned optomechanical cavity,” Phys. Rev. A 88(5), 053850 (2013).
[Crossref]

J. Li, G. Li, S. Zippilli, D. Vitali, and T. Zhang, “Enhanced entanglement of two different mechanical resonators via coherent feedback,” Phys. Rev. A 95(4), 043819 (2017).
[Crossref]

M. Asjad, P. Tombesi, and D. Vitali, “Feedback control of two-mode output entanglement and steering in cavity optomechanics,” Phys. Rev. A 94(5), 052312 (2016).
[Crossref]

Q. Lin, B. He, R. Ghobadi, and C. Simon, “Fully quantum approach to optomechanical entanglement,” Phys. Rev. A 90(2), 022309 (2014).
[Crossref]

C. Genes, A. Mari, P. Tombesi, and D. Vitali, “Robust entanglement of a micromechanical resonator with output optical fields,” Phys. Rev. A 78(3), 032316 (2008).
[Crossref]

Sh. Barzanjeh, D. Vitali, P. Tombesi, and G. J. Milburn, “Entangling optical and microwave cavity modes by means of a nanomechanical resonator,” Phys. Rev. A 84(4), 042342 (2011).
[Crossref]

Sh. Barzanjeh, S. Pirandola, and C. Weedbrook, “Continuous-variable dense coding by optomechanical cavities,” Phys. Rev. A 88(4), 042331 (2013).
[Crossref]

M. C. Kuzyk, S. J. van Enk, and H. Wang, “Generating robust optical entanglement in weak-coupling optomechanical systems,” Phys. Rev. A 88(6), 062341 (2013).
[Crossref]

S. Pirandola, S. Mancini, D. Vitali, and P. Tombesi, “Continuous-variable entanglement and quantum-state teleportation between optical and macroscopic vibrational modes through radiation pressure,” Phys. Rev. A 68(6), 062317 (2003).
[Crossref]

S. Kiesewetter, Q. Y. He, P. D. Drummond, and M. D. Reid, “Scalable quantum simulation of pulsed entanglement and Einstein-Podolsky-Rosen steering in optomechanics,” Phys. Rev. A 90(4), 043805 (2014).
[Crossref]

M. Bhattacharya, P.-L. Giscard, and P. Meystre, “Entangling the rovibrational modes of a macroscopic mirror using radiation pressure,” Phys. Rev. A 77(3), 030303 (2008).
[Crossref]

R. X. Chen, L. T. Shen, Z. B. Yang, H. Z. Wu, and S. B. Zheng, “Enhancement of entanglement in distant mechanical vibrations via modulation in a coupled optomechanical system,” Phys. Rev. A 89(2), 023843 (2014).
[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(1), 014302 (2014).
[Crossref]

C. J. Yang, J. H. An, W. Yang, and Y. Li, “Generation of stable entanglement between two cavity mirrors by squeezed-reservoir engineering,” Phys. Rev. A 92(6), 062311 (2015).
[Crossref]

Phys. Rev. Lett. (10)

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99(25), 250401 (2007).
[Crossref]

S. Mancini, D. Vitali, and P. Tombesi, “Scheme for teleportation of quantum states onto a mechanical resonator,” Phys. Rev. Lett. 90(13), 137901 (2003).
[Crossref]

S. Pirandola, D. Vitali, P. Tombesi, and S. Lloyd, “Macroscopic entanglement by entanglement swapping,” Phys. Rev. Lett. 97(15), 150403 (2006).
[Crossref]

E. Flurin, N. Roch, F. Mallet, M. H. Devoret, and B. Huard, “Generating entangled microwave radiation over two transmission lines,” Phys. Rev. Lett. 109(18), 183901 (2012).
[Crossref]

H. Krauter, C. A. Muschik, K. Jensen, W. Wasilewski, J. M. Petersen, J. I. Cirac, and E. S. Polzik, “Entanglement generated by dissipation and steady state entanglement of two macroscopic objects,” Phys. Rev. Lett. 107(8), 080503 (2011).
[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(3), 030405 (2007).
[Crossref]

Y.-D. Wang and A. A. Clerk, “Reservoir-engineered entanglement in optomechanical systems,” Phys. Rev. Lett. 110(25), 253601 (2013).
[Crossref]

Sh. Barzanjeh, M. Abdi, G. J. Milburn, P. Tombesi, and D. Vitali, “Reversible optical-to-microwave quantum interface,” Phys. Rev. Lett. 109(13), 130503 (2012).
[Crossref]

L. Tian, “Robust photon entanglement via quantum interference in optomechanical interfaces,” Phys. Rev. Lett. 110(23), 233602 (2013).
[Crossref]

M. B. Plenio, “Logarithmic negativity: A full entanglement monotone that is not convex,” Phys. Rev. Lett. 95(9), 090503 (2005).
[Crossref]

Rev. Mod. Phys. (2)

S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77(2), 513–577 (2005).
[Crossref]

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84(2), 621–669 (2012).
[Crossref]

Science (3)

A. J. Berkley, H. Xu, R. C. Ramos, M. A. Gubrud, F. W. Strauch, P. R. Johnson, J. R. Anderson, A. J. Dragt, C. J. Lobb, and F. C. Wellstood, “Entangled macroscopic quantum states in two superconducting qubits,” Science 300(5625), 1548–1550 (2003).
[Crossref]

T. A. Palomaki, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, “Entangling mechanical motion with microwave fields,” Science 342(6159), 710–713 (2013).
[Crossref]

C. Dong, V. Fiore, M. C. Kuzyk, and H. Wang, “Optomechanical dark mode,” Science 338(6114), 1609–1613 (2012).
[Crossref]

Other (1)

C. Gardiner and P. Zoller, Quantum Noise, 3rd ed. (Springer, New York, 2004).

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Figures (5)

Fig. 1.
Fig. 1. (a) A three-mode optomechanical system with a mechanical resonator (MR) interacted with two cavities. Cavity 1 is driven by a red-detuned laser, while cavity 2 is driven by a blue-detuned laser. The entanglement between the filtered output fields of two cavities $\hat {D}^{\textrm {out}}_{1}$ and $\hat {D}^{\textrm {out}}_{2}$ can be generated. (b) Spectral position of cavity resonances $\omega _{1}$, $\omega _{2}$ and driving frequencies ($\omega _{1}-\omega _{m}$ and $\omega _{2}+\omega _{m}$).
Fig. 2.
Fig. 2. (a) The output entanglement $\textrm {E}_{\textrm {N}}(\omega =0)$ are plotted vs $G_{2}/G_{1}$ with large bandwidth $\sigma =\kappa /10$ (black line), $\sigma =\kappa$ (red line), and the two dashed lines are their corresponding ones with numerical optimal time delay. (b) The output entanglement $\textrm {E}_{\textrm {N}}(\omega =0)$ are plotted vs $G_{2}/G_{1}$ with small bandwidth $\sigma =\kappa /10^{4}$ (red line), $\sigma =2\kappa /10^{3}$ (yellow line), the boundary bandwidth $\sigma _{b}=\frac {\sqrt {3}\kappa ^{3}}{4G_{1}^{2}}$ (violet line), and their common line with numerical optimal time delay (blue dashed line). The other parameters are $\gamma =1, \kappa =10^{5}, G_{1}=10\kappa$.
Fig. 3.
Fig. 3. The optimal output entanglement $\textrm {E}^{\textrm {opt}}_{\textrm {N}}(\omega =0)$ vs $G_{1}/\kappa$ with optimal coupling Eq. (7) for bandwidth $\sigma =\kappa /10$ (black line), $\sigma =\kappa /2$ (blue line), $\sigma =\kappa$ (red line), and the saturation values are plotted according to Eq. (9) (green dashed lines). The other parameters are $\gamma =1, \kappa =10^{5}, G_{1}=10\kappa$.
Fig. 4.
Fig. 4. (a) The optimal time delay $\tau _{\textrm {opt}}$ are plotted vs $G_{2}/G_{1}$ according to Eq. (6) (blue solid line) and the numerical result (red dashed line). (b) The output entanglement $\textrm {E}_{\textrm {N}}(\omega =0)$ are plotted vs $G_{2}/G_{1}$ with the optimal time delay according to Eq. (6) (blue solid line) and the numerical result (red dashed line). And the optimal output entanglement (see the green dot) is plotted according to Eqs. (10) and (11). The parameters are $\gamma =1, \kappa =10^{5}, \sigma =\kappa , G_{1}=10\kappa$.
Fig. 5.
Fig. 5. The output entanglement $\textrm {E}_{\textrm {N}}(\omega )$ are plotted vs the normalized center frequency $\omega /\kappa$, with numerical optimal time delay and optimal coupling Eq. (10) (red solid line), with no time delay ($\tau =0$) and equal-coupling ($\textrm {G}_{1}=\textrm {G}_{2}$) (blue dashed line). The parameters are $\gamma =1, \kappa =10^{5}, \sigma =\kappa , G_{1}=10\kappa$.

Equations (11)

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H = ω m b ^ b ^ + i = 1 , 2 [ ω i a ^ i a ^ i + g i ( b ^ + b ^ ) a ^ i a ^ i ] .
H ^ int = G 1 b ^ d ^ 1 + G 2 b ^ d ^ 2 + H.c.
d d t b ^ = γ 2 b ^ i ( G 1 d ^ 1 + G 2 d ^ 2 ) γ b ^ in , d d t d ^ 1 = κ 1 2 d ^ 1 i G 1 b ^ κ 1 d ^ 1 in , d d t d ^ 2 = κ 2 2 d ^ 2 + i G 2 b ^ κ 2 d ^ 2 in , .
D ^ i out [ ω , σ , τ i ] = 1 σ ω σ 2 ω + σ 2 d ω e i ω τ i d ^ i out ( ω ) .
D ^ 1 D ^ 2 = ω σ 2 ω + σ 2 4 G 1 G 2 κ [ 2 G 1 2 ( κ 2 i Ω ) + ( κ + 2 i Ω ) ( 2 G 2 2 + 2 Ω 2 + i Ω κ ) ] e i Ω τ σ ( κ 2 + 4 Ω 2 ) [ ( 8 G 1 2 8 G 2 2 κ 2 ) Ω 2 4 ( G 1 2 G 2 2 ) 2 4 Ω 4 ] d Ω .
τ opt = 20 ( G 2 2 G 1 2 ) + 5 κ 2 + 3 σ 2 10 ( G 1 2 + G 2 2 ) κ .
G 2 opt = 1 2 4 G 1 2 κ 2 3 σ 2 5 .
G 2 opt = G 1 + G 1 σ 2 3 κ κ σ 2 3 4 .
E N sat = ln [ ( κ 2 + σ 2 ) ( 15 κ 2 σ + 4 σ 3 3 α β ) 2 9 κ 2 σ 2 α 2 ]
G 2 opt = G 1 ( α ( 15 κ 2 σ + 4 σ 3 3 α β ) 400 ( 6 σ 3 β ) ) 1 / 4
E N opt = ln [ α ( 2 σ β ) ( 15 κ 2 σ + 4 σ 3 3 α β ) 4800 G 1 4 σ 2 ] .