Abstract

We propose to realize the two-mode continuous-variable entanglement of microwave photons in an electro-optic system, consisting of superconducting microwave resonators and optical cavities that are filled with certain electro-optic media. The cascaded and parallel schemes realize such entanglement via coherent control on the dynamics of the system, while the dissipative dynamical scheme utilizes the reservoir-engineering approach and exploits the optical dissipation as a useful resource. We show that, for all the schemes, the amount of entanglement is determined by the ratio of the effective coupling strengths of the “beam-splitter” and “two-mode squeezing” interactions, instead of their amplitudes.

© 2017 Optical Society of America

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References

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  6. L. Tian, “Robust photon entanglement via quantum interference in optomechanical interfaces,” Phys. Rev. Lett. 110, 233602 (2013).
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    [Crossref] [PubMed]
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  35. T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: Back-action at the mesoscale,” Science 321, 1172–1176 (2008).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  40. J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471, 204–208 (2010).
    [Crossref]
  41. S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460, 724–727 (2009).
    [Crossref] [PubMed]
  42. C. Herzog, G. Poberaj, and P. Günter, “Electro-optic behavior of lithium niobate at cryogenic temperatures,” Opt. Commun. 281, 793–796 (2008).
    [Crossref]
  43. Y. Barad, Y. Lu, Z. Y. Cheng, S. E. Park, and Q. M. Zhang, “Composition, temperature, and crystal orientation dependence of the linear electro-optic properties of Pb(Zn1/3Nb2/3)O-3-PbTiO3 single crystals,” Appl. Phys. Lett. 77, 1247–1249 (2000).
    [Crossref]
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2017 (1)

S. Dambach, B. Kubala, and J. Ankerhold, “Generating entangled quantum microwaves in a josephson-photonics device,” New J. Phys. 19, 023027 (2017).
[Crossref]

2016 (5)

V. C. Vivoli, T. Barnea, C. Galland, and N. Sangouard, “Proposal for an optomechanical bell test,” Phys. Rev. Lett. 116, 070405 (2016).
[Crossref] [PubMed]

R. W. Peterson, T. P. Purdy, N. S. Kampel, R. W. Andrews, P. L. Yu, K. W. Lehnert, and C. A. Regal, “Laser cooling of a micromechanical membrane to the quantum backaction limit,” Phys. Rev. Lett. 116, 063601 (2016).
[Crossref] [PubMed]

M. Frimmer, J. Gieseler, and L. Novotny, “Cooling mechanical oscillators by coherent control,” Phys. Rev. Lett. 117, 163601 (2016).
[Crossref] [PubMed]

G. Brawley, M. Vanner, P. E. Larsen, S. Schmid, A. Boisen, and W. Bowen, “Nonlinear optomechanical measurement of mechanical motion,” Nat. Commun. 7, 10988 (2016).
[Crossref] [PubMed]

C. Javerzac-Galy, K. Plekhanov, N. R. Bernier, L. D. Toth, A. K. Feofanov, and T. J. Kippenberg, “On-chip microwave-to-optical quantum coherent converter based on a superconducting resonator coupled to an electro-optic microresonator,” Phys. Rev. A 94, 053815 (2016).
[Crossref]

2015 (2)

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

Z. Li, S. L. Ma, and F. L. Li, “Generation of broadband two-mode squeezed light in cascaded double-cavity optomechanical systems,” Phys. Rev. A 92, 023856 (2015).
[Crossref]

2014 (3)

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

S. L. Ma, Z. Li, A. P. Fang, P. B. Li, S. Y. Gao, and F. L. Li, “Controllable generation of two-mode-entangled states in two-resonator circuit QED with a single gap-tunable superconducting qubit,” Phys. Rev. A 90, 062342 (2014).
[Crossref]

Q. Y. He and Z. Ficek, “Einstein-Podolsky-Rosen paradox and quantum steering in a three-mode optomechanical system,” Phys. Rev. A 89, 022332 (2014).
[Crossref]

2013 (7)

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

C. P. Yang, Q. P. Su, S. B. Zheng, and S. Y. Han, “Generating entanglement between microwave photons and qubits in multiple cavities coupled by a superconducting qutrit,” Phys. Rev. A 87, 022320 (2013).
[Crossref]

Z. L. Xiang, S. Ashhab, J. Q. You, and F. Nori, “Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems,” Rev. Mod. Phys. 85, 623–653 (2013).
[Crossref]

P. B. Li, S. Y. Gao, and F. L. Li, “Robust continuous-variable entanglement of microwave photons with cavity electromechanics,” Phys. Rev. A 88, 043802 (2013).
[Crossref]

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

C. Jiang, H. Liu, Y. Cui, X. Li, G. Chen, and B. Chen, “Electromagnetically induced transparency and slow light in two-mode optomechanics,” Opt. Express 21, 12165–12173 (2013).
[Crossref] [PubMed]

G. L. Cheng, A. X. Chen, and W. X. Zhong, “Tripartite entanglement of microwave radiation via nonlinear parametric interactions enhanced by quantum interference in superconducting quantum circuits,” J. Opt. Soc. Am. B 30, 2875–2881 (2013).
[Crossref]

2012 (4)

W. C. Jiang, X. Lu, J. Zhang, and Q. Lin, “High-frequency silicon optomechanical oscillator with an ultralow threshold,” Opt. Express 20, 15991–15996 (2012).
[Crossref] [PubMed]

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

P. B. Li, S. Y. Gao, and F. L. Li, “Engineering two-mode entangled states between two superconducting resonators by dissipation,” Phys. Rev. A 86, 012318 (2012).
[Crossref]

E. P. Menzel, R. Di Candia, F. Deppe, P. Eder, L. Zhong, M. Ihmig, M. Haeberlein, A. Baust, E. Hoffmann, D. Ballester, K. Inomata, T. Yamamoto, Y. Nakamura, E. Solano, A. Marx, and R. Gross, “Path entanglement of continuous-variable quantum microwaves,” Phys. Rev. Lett. 109, 250502 (2012).
[Crossref]

2011 (4)

J. Q. Liao and C. K. Law, “Parametric generation of quadrature squeezing of mirrors in cavity optomechanics,” Phys. Rev. A 83, 033820 (2011).
[Crossref]

H. Wang, M. Mariantoni, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, T. Yamamoto, Y. Yin, J. Zhao, J. M. Martinis, and A. N. Cleland, “Deterministic entanglement of photons in two superconducting microwave resonators,” Phys. Rev. Lett. 106, 060401 (2011).
[Crossref] [PubMed]

J. Teufel, T. Donner, D. Li, K. Lehnert, and R. Simmonds, “Sideband cooling micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[Crossref] [PubMed]

S. 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, 042342 (2011).
[Crossref]

2010 (3)

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, and M. Weides, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[Crossref]

M. Tsang, “Cavity quantum electro-optics,” Phys. Rev. A 81, 063837 (2010).
[Crossref]

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471, 204–208 (2010).
[Crossref]

2009 (4)

S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460, 724–727 (2009).
[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]

A. Mari and J. Eisert, “Gently modulating optomechanical systems,” Phys. Rev. Lett. 103, 213603 (2009).
[Crossref]

X. Y. Lü, L. L. Zheng, P. Huang, J. Li, and X. X. Yang, “Adiabatic passage scheme for entanglement between two distant microwave cavities interacting with single-molecule magnets,” J. Opt. Soc. Am. B 26, 1162–1168 (2009).
[Crossref]

2008 (4)

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

A. Schliesser, R. Riviere, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. 4, 415–419 (2008).
[Crossref]

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

C. Herzog, G. Poberaj, and P. Günter, “Electro-optic behavior of lithium niobate at cryogenic temperatures,” Opt. Commun. 281, 793–796 (2008).
[Crossref]

2007 (2)

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki, “On fundamental quantum noises of whispering gallery mode electro-optic modulators,” Opt. Express 15, 17401–17409 (2007).
[Crossref] [PubMed]

P. Chen, X. G. Tu, S. P. Li, J. C. Li, W. Lin, H. Y. Chen, D. Y. Liu, J. Y. Kang, Y. H. Zuo, and L. Zhao, “Enhanced Pockels effect in GaN/AlxGa1-xN superlattice measured by polarization-maintaining fiber Mach-Zehnder interferometer,” Appl. Phys. Lett. 91, 1447 (2007).

2004 (1)

H. Nha and H. J. Carmichael, “Proposed test of quantum nonlocality for continuous variables,” Phys. Rev. Lett. 93, 020401 (2004).
[Crossref] [PubMed]

2000 (3)

Y. Barad, Y. Lu, Z. Y. Cheng, S. E. Park, and Q. M. Zhang, “Composition, temperature, and crystal orientation dependence of the linear electro-optic properties of Pb(Zn1/3Nb2/3)O-3-PbTiO3 single crystals,” Appl. Phys. Lett. 77, 1247–1249 (2000).
[Crossref]

L. M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000).
[Crossref] [PubMed]

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000).
[Crossref] [PubMed]

Abdi, M.

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

Allman, M. S.

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471, 204–208 (2010).
[Crossref]

An, J. H.

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

Andrews, R. W.

R. W. Peterson, T. P. Purdy, N. S. Kampel, R. W. Andrews, P. L. Yu, K. W. Lehnert, and C. A. Regal, “Laser cooling of a micromechanical membrane to the quantum backaction limit,” Phys. Rev. Lett. 116, 063601 (2016).
[Crossref] [PubMed]

Anetsberger, G.

A. Schliesser, R. Riviere, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. 4, 415–419 (2008).
[Crossref]

Ankerhold, J.

S. Dambach, B. Kubala, and J. Ankerhold, “Generating entangled quantum microwaves in a josephson-photonics device,” New J. Phys. 19, 023027 (2017).
[Crossref]

Ansmann, M.

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, and M. Weides, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[Crossref]

Arcizet, O.

A. Schliesser, R. Riviere, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. 4, 415–419 (2008).
[Crossref]

Ashhab, S.

Z. L. Xiang, S. Ashhab, J. Q. You, and F. Nori, “Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems,” Rev. Mod. Phys. 85, 623–653 (2013).
[Crossref]

Aspelmeyer, M.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460, 724–727 (2009).
[Crossref] [PubMed]

Ballester, D.

E. P. Menzel, R. Di Candia, F. Deppe, P. Eder, L. Zhong, M. Ihmig, M. Haeberlein, A. Baust, E. Hoffmann, D. Ballester, K. Inomata, T. Yamamoto, Y. Nakamura, E. Solano, A. Marx, and R. Gross, “Path entanglement of continuous-variable quantum microwaves,” Phys. Rev. Lett. 109, 250502 (2012).
[Crossref]

Barad, Y.

Y. Barad, Y. Lu, Z. Y. Cheng, S. E. Park, and Q. M. Zhang, “Composition, temperature, and crystal orientation dependence of the linear electro-optic properties of Pb(Zn1/3Nb2/3)O-3-PbTiO3 single crystals,” Appl. Phys. Lett. 77, 1247–1249 (2000).
[Crossref]

Barnea, T.

V. C. Vivoli, T. Barnea, C. Galland, and N. Sangouard, “Proposal for an optomechanical bell test,” Phys. Rev. Lett. 116, 070405 (2016).
[Crossref] [PubMed]

Barzanjeh, S.

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

S. 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, 042342 (2011).
[Crossref]

Baust, A.

E. P. Menzel, R. Di Candia, F. Deppe, P. Eder, L. Zhong, M. Ihmig, M. Haeberlein, A. Baust, E. Hoffmann, D. Ballester, K. Inomata, T. Yamamoto, Y. Nakamura, E. Solano, A. Marx, and R. Gross, “Path entanglement of continuous-variable quantum microwaves,” Phys. Rev. Lett. 109, 250502 (2012).
[Crossref]

Bernier, N. R.

C. Javerzac-Galy, K. Plekhanov, N. R. Bernier, L. D. Toth, A. K. Feofanov, and T. J. Kippenberg, “On-chip microwave-to-optical quantum coherent converter based on a superconducting resonator coupled to an electro-optic microresonator,” Phys. Rev. A 94, 053815 (2016).
[Crossref]

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Milburn, G. J.

S. Barzanjeh, M. Abdi, G. J. Milburn, P. Tombesi, and D. Vitali, “Reversible optical-to-microwave quantum interface,” Phys. Rev. Lett. 109, 130503 (2012).
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S. 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, 042342 (2011).
[Crossref]

Nakamura, Y.

E. P. Menzel, R. Di Candia, F. Deppe, P. Eder, L. Zhong, M. Ihmig, M. Haeberlein, A. Baust, E. Hoffmann, D. Ballester, K. Inomata, T. Yamamoto, Y. Nakamura, E. Solano, A. Marx, and R. Gross, “Path entanglement of continuous-variable quantum microwaves,” Phys. Rev. Lett. 109, 250502 (2012).
[Crossref]

Neeley, M.

H. Wang, M. Mariantoni, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, T. Yamamoto, Y. Yin, J. Zhao, J. M. Martinis, and A. N. Cleland, “Deterministic entanglement of photons in two superconducting microwave resonators,” Phys. Rev. Lett. 106, 060401 (2011).
[Crossref] [PubMed]

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, and M. Weides, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
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H. Nha and H. J. Carmichael, “Proposed test of quantum nonlocality for continuous variables,” Phys. Rev. Lett. 93, 020401 (2004).
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Nori, F.

Z. L. Xiang, S. Ashhab, J. Q. You, and F. Nori, “Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems,” Rev. Mod. Phys. 85, 623–653 (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).
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Novotny, L.

M. Frimmer, J. Gieseler, and L. Novotny, “Cooling mechanical oscillators by coherent control,” Phys. Rev. Lett. 117, 163601 (2016).
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H. Wang, M. Mariantoni, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, T. Yamamoto, Y. Yin, J. Zhao, J. M. Martinis, and A. N. Cleland, “Deterministic entanglement of photons in two superconducting microwave resonators,” Phys. Rev. Lett. 106, 060401 (2011).
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A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, and M. Weides, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[Crossref]

Park, S. E.

Y. Barad, Y. Lu, Z. Y. Cheng, S. E. Park, and Q. M. Zhang, “Composition, temperature, and crystal orientation dependence of the linear electro-optic properties of Pb(Zn1/3Nb2/3)O-3-PbTiO3 single crystals,” Appl. Phys. Lett. 77, 1247–1249 (2000).
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Peterson, R. W.

R. W. Peterson, T. P. Purdy, N. S. Kampel, R. W. Andrews, P. L. Yu, K. W. Lehnert, and C. A. Regal, “Laser cooling of a micromechanical membrane to the quantum backaction limit,” Phys. Rev. Lett. 116, 063601 (2016).
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Plekhanov, K.

C. Javerzac-Galy, K. Plekhanov, N. R. Bernier, L. D. Toth, A. K. Feofanov, and T. J. Kippenberg, “On-chip microwave-to-optical quantum coherent converter based on a superconducting resonator coupled to an electro-optic microresonator,” Phys. Rev. A 94, 053815 (2016).
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Poberaj, G.

C. Herzog, G. Poberaj, and P. Günter, “Electro-optic behavior of lithium niobate at cryogenic temperatures,” Opt. Commun. 281, 793–796 (2008).
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Purdy, T. P.

R. W. Peterson, T. P. Purdy, N. S. Kampel, R. W. Andrews, P. L. Yu, K. W. Lehnert, and C. A. Regal, “Laser cooling of a micromechanical membrane to the quantum backaction limit,” Phys. Rev. Lett. 116, 063601 (2016).
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Regal, C. A.

R. W. Peterson, T. P. Purdy, N. S. Kampel, R. W. Andrews, P. L. Yu, K. W. Lehnert, and C. A. Regal, “Laser cooling of a micromechanical membrane to the quantum backaction limit,” Phys. Rev. Lett. 116, 063601 (2016).
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Riviere, R.

A. Schliesser, R. Riviere, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. 4, 415–419 (2008).
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Sangouard, N.

V. C. Vivoli, T. Barnea, C. Galland, and N. Sangouard, “Proposal for an optomechanical bell test,” Phys. Rev. Lett. 116, 070405 (2016).
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Sank, D.

H. Wang, M. Mariantoni, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, T. Yamamoto, Y. Yin, J. Zhao, J. M. Martinis, and A. N. Cleland, “Deterministic entanglement of photons in two superconducting microwave resonators,” Phys. Rev. Lett. 106, 060401 (2011).
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A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, and M. Weides, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[Crossref]

Savchenkov, A. A.

Schliesser, A.

A. Schliesser, R. Riviere, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. 4, 415–419 (2008).
[Crossref]

Schmid, S.

G. Brawley, M. Vanner, P. E. Larsen, S. Schmid, A. Boisen, and W. Bowen, “Nonlinear optomechanical measurement of mechanical motion,” Nat. Commun. 7, 10988 (2016).
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M. O. Scully and M. S. Zubairy, Quantum optics (Cambridge University, Cambridge, 1997).
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Seidel, D.

Simmonds, R.

J. Teufel, T. Donner, D. Li, K. Lehnert, and R. Simmonds, “Sideband cooling micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
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Simmonds, R. W.

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471, 204–208 (2010).
[Crossref]

Simon, C.

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000).
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Sirois, A. J.

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471, 204–208 (2010).
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Solano, E.

E. P. Menzel, R. Di Candia, F. Deppe, P. Eder, L. Zhong, M. Ihmig, M. Haeberlein, A. Baust, E. Hoffmann, D. Ballester, K. Inomata, T. Yamamoto, Y. Nakamura, E. Solano, A. Marx, and R. Gross, “Path entanglement of continuous-variable quantum microwaves,” Phys. Rev. Lett. 109, 250502 (2012).
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Su, Q. P.

C. P. Yang, Q. P. Su, S. B. Zheng, and S. Y. Han, “Generating entanglement between microwave photons and qubits in multiple cavities coupled by a superconducting qutrit,” Phys. Rev. A 87, 022320 (2013).
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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]

Teufel, J.

J. Teufel, T. Donner, D. Li, K. Lehnert, and R. Simmonds, “Sideband cooling micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
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Teufel, J. D.

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471, 204–208 (2010).
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Tian, L.

L. Tian, “Robust photon entanglement via quantum interference in optomechanical interfaces,” Phys. Rev. Lett. 110, 233602 (2013).
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Tombesi, P.

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

S. 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, 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, 032316 (2008).
[Crossref]

Toth, L. D.

C. Javerzac-Galy, K. Plekhanov, N. R. Bernier, L. D. Toth, A. K. Feofanov, and T. J. Kippenberg, “On-chip microwave-to-optical quantum coherent converter based on a superconducting resonator coupled to an electro-optic microresonator,” Phys. Rev. A 94, 053815 (2016).
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Tsang, M.

M. Tsang, “Cavity quantum electro-optics,” Phys. Rev. A 81, 063837 (2010).
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Tu, X. G.

P. Chen, X. G. Tu, S. P. Li, J. C. Li, W. Lin, H. Y. Chen, D. Y. Liu, J. Y. Kang, Y. H. Zuo, and L. Zhao, “Enhanced Pockels effect in GaN/AlxGa1-xN superlattice measured by polarization-maintaining fiber Mach-Zehnder interferometer,” Appl. Phys. Lett. 91, 1447 (2007).

Vahala, K. J.

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: Back-action at the mesoscale,” Science 321, 1172–1176 (2008).
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Vanner, M.

G. Brawley, M. Vanner, P. E. Larsen, S. Schmid, A. Boisen, and W. Bowen, “Nonlinear optomechanical measurement of mechanical motion,” Nat. Commun. 7, 10988 (2016).
[Crossref] [PubMed]

Vanner, M. R.

S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460, 724–727 (2009).
[Crossref] [PubMed]

Vitali, D.

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

S. 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, 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, 032316 (2008).
[Crossref]

Vivoli, V. C.

V. C. Vivoli, T. Barnea, C. Galland, and N. Sangouard, “Proposal for an optomechanical bell test,” Phys. Rev. Lett. 116, 070405 (2016).
[Crossref] [PubMed]

Wang, H.

H. Wang, M. Mariantoni, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, T. Yamamoto, Y. Yin, J. Zhao, J. M. Martinis, and A. N. Cleland, “Deterministic entanglement of photons in two superconducting microwave resonators,” Phys. Rev. Lett. 106, 060401 (2011).
[Crossref] [PubMed]

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, and M. Weides, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
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Y. D. Wang and A. A. Clerk, “Reservoir-engineered entanglement in optomechanical systems,” Phys. Rev. Lett. 110, 253601 (2013).
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H. Wang, M. Mariantoni, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, T. Yamamoto, Y. Yin, J. Zhao, J. M. Martinis, and A. N. Cleland, “Deterministic entanglement of photons in two superconducting microwave resonators,” Phys. Rev. Lett. 106, 060401 (2011).
[Crossref] [PubMed]

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, and M. Weides, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[Crossref]

Weihs, G.

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000).
[Crossref] [PubMed]

Weinfurter, H.

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000).
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Wenner, J.

H. Wang, M. Mariantoni, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, T. Yamamoto, Y. Yin, J. Zhao, J. M. Martinis, and A. N. Cleland, “Deterministic entanglement of photons in two superconducting microwave resonators,” Phys. Rev. Lett. 106, 060401 (2011).
[Crossref] [PubMed]

Whittaker, J. D.

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471, 204–208 (2010).
[Crossref]

Xiang, Z. L.

Z. L. Xiang, S. Ashhab, J. Q. You, and F. Nori, “Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems,” Rev. Mod. Phys. 85, 623–653 (2013).
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Yamamoto, T.

E. P. Menzel, R. Di Candia, F. Deppe, P. Eder, L. Zhong, M. Ihmig, M. Haeberlein, A. Baust, E. Hoffmann, D. Ballester, K. Inomata, T. Yamamoto, Y. Nakamura, E. Solano, A. Marx, and R. Gross, “Path entanglement of continuous-variable quantum microwaves,” Phys. Rev. Lett. 109, 250502 (2012).
[Crossref]

H. Wang, M. Mariantoni, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, T. Yamamoto, Y. Yin, J. Zhao, J. M. Martinis, and A. N. Cleland, “Deterministic entanglement of photons in two superconducting microwave resonators,” Phys. Rev. Lett. 106, 060401 (2011).
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Yang, C. J.

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

Yang, C. P.

C. P. Yang, Q. P. Su, S. B. Zheng, and S. Y. Han, “Generating entanglement between microwave photons and qubits in multiple cavities coupled by a superconducting qutrit,” Phys. Rev. A 87, 022320 (2013).
[Crossref]

Yang, W. L.

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

Yang, X. X.

Yin, Y.

H. Wang, M. Mariantoni, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, T. Yamamoto, Y. Yin, J. Zhao, J. M. Martinis, and A. N. Cleland, “Deterministic entanglement of photons in two superconducting microwave resonators,” Phys. Rev. Lett. 106, 060401 (2011).
[Crossref] [PubMed]

You, J. Q.

Z. L. Xiang, S. Ashhab, J. Q. You, and F. Nori, “Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems,” Rev. Mod. Phys. 85, 623–653 (2013).
[Crossref]

Yu, P. L.

R. W. Peterson, T. P. Purdy, N. S. Kampel, R. W. Andrews, P. L. Yu, K. W. Lehnert, and C. A. Regal, “Laser cooling of a micromechanical membrane to the quantum backaction limit,” Phys. Rev. Lett. 116, 063601 (2016).
[Crossref] [PubMed]

Zeilinger, A.

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000).
[Crossref] [PubMed]

Zhang, J.

Zhang, Q. M.

Y. Barad, Y. Lu, Z. Y. Cheng, S. E. Park, and Q. M. Zhang, “Composition, temperature, and crystal orientation dependence of the linear electro-optic properties of Pb(Zn1/3Nb2/3)O-3-PbTiO3 single crystals,” Appl. Phys. Lett. 77, 1247–1249 (2000).
[Crossref]

Zhao, J.

H. Wang, M. Mariantoni, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, T. Yamamoto, Y. Yin, J. Zhao, J. M. Martinis, and A. N. Cleland, “Deterministic entanglement of photons in two superconducting microwave resonators,” Phys. Rev. Lett. 106, 060401 (2011).
[Crossref] [PubMed]

Zhao, L.

P. Chen, X. G. Tu, S. P. Li, J. C. Li, W. Lin, H. Y. Chen, D. Y. Liu, J. Y. Kang, Y. H. Zuo, and L. Zhao, “Enhanced Pockels effect in GaN/AlxGa1-xN superlattice measured by polarization-maintaining fiber Mach-Zehnder interferometer,” Appl. Phys. Lett. 91, 1447 (2007).

Zheng, L. L.

Zheng, S. B.

C. P. Yang, Q. P. Su, S. B. Zheng, and S. Y. Han, “Generating entanglement between microwave photons and qubits in multiple cavities coupled by a superconducting qutrit,” Phys. Rev. A 87, 022320 (2013).
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Zhong, L.

E. P. Menzel, R. Di Candia, F. Deppe, P. Eder, L. Zhong, M. Ihmig, M. Haeberlein, A. Baust, E. Hoffmann, D. Ballester, K. Inomata, T. Yamamoto, Y. Nakamura, E. Solano, A. Marx, and R. Gross, “Path entanglement of continuous-variable quantum microwaves,” Phys. Rev. Lett. 109, 250502 (2012).
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Zhong, W. X.

Zoller, P.

L. M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000).
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Zubairy, M. S.

M. O. Scully and M. S. Zubairy, Quantum optics (Cambridge University, Cambridge, 1997).
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Zuo, Y. H.

P. Chen, X. G. Tu, S. P. Li, J. C. Li, W. Lin, H. Y. Chen, D. Y. Liu, J. Y. Kang, Y. H. Zuo, and L. Zhao, “Enhanced Pockels effect in GaN/AlxGa1-xN superlattice measured by polarization-maintaining fiber Mach-Zehnder interferometer,” Appl. Phys. Lett. 91, 1447 (2007).

Appl. Phys. Lett. (2)

Y. Barad, Y. Lu, Z. Y. Cheng, S. E. Park, and Q. M. Zhang, “Composition, temperature, and crystal orientation dependence of the linear electro-optic properties of Pb(Zn1/3Nb2/3)O-3-PbTiO3 single crystals,” Appl. Phys. Lett. 77, 1247–1249 (2000).
[Crossref]

P. Chen, X. G. Tu, S. P. Li, J. C. Li, W. Lin, H. Y. Chen, D. Y. Liu, J. Y. Kang, Y. H. Zuo, and L. Zhao, “Enhanced Pockels effect in GaN/AlxGa1-xN superlattice measured by polarization-maintaining fiber Mach-Zehnder interferometer,” Appl. Phys. Lett. 91, 1447 (2007).

J. Opt. Soc. Am. B (2)

Nat. Commun. (1)

G. Brawley, M. Vanner, P. E. Larsen, S. Schmid, A. Boisen, and W. Bowen, “Nonlinear optomechanical measurement of mechanical motion,” Nat. Commun. 7, 10988 (2016).
[Crossref] [PubMed]

Nat. Phys. (1)

A. Schliesser, R. Riviere, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. 4, 415–419 (2008).
[Crossref]

Nature (4)

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, and M. Weides, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[Crossref]

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471, 204–208 (2010).
[Crossref]

S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460, 724–727 (2009).
[Crossref] [PubMed]

J. Teufel, T. Donner, D. Li, K. Lehnert, and R. Simmonds, “Sideband cooling micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
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New J. Phys. (1)

S. Dambach, B. Kubala, and J. Ankerhold, “Generating entangled quantum microwaves in a josephson-photonics device,” New J. Phys. 19, 023027 (2017).
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Opt. Commun. (1)

C. Herzog, G. Poberaj, and P. Günter, “Electro-optic behavior of lithium niobate at cryogenic temperatures,” Opt. Commun. 281, 793–796 (2008).
[Crossref]

Opt. Express (3)

Phys. Rev. A (13)

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).
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P. B. Li, S. Y. Gao, and F. L. Li, “Engineering two-mode entangled states between two superconducting resonators by dissipation,” Phys. Rev. A 86, 012318 (2012).
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C. Javerzac-Galy, K. Plekhanov, N. R. Bernier, L. D. Toth, A. K. Feofanov, and T. J. Kippenberg, “On-chip microwave-to-optical quantum coherent converter based on a superconducting resonator coupled to an electro-optic microresonator,” Phys. Rev. A 94, 053815 (2016).
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Q. Y. He and Z. Ficek, “Einstein-Podolsky-Rosen paradox and quantum steering in a three-mode optomechanical system,” Phys. Rev. A 89, 022332 (2014).
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J. Q. Liao and C. K. Law, “Parametric generation of quadrature squeezing of mirrors in cavity optomechanics,” Phys. Rev. A 83, 033820 (2011).
[Crossref]

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

M. Tsang, “Cavity quantum electro-optics,” Phys. Rev. A 81, 063837 (2010).
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Z. Li, S. L. Ma, and F. L. Li, “Generation of broadband two-mode squeezed light in cascaded double-cavity optomechanical systems,” Phys. Rev. A 92, 023856 (2015).
[Crossref]

C. Genes, A. Mari, P. Tombesi, and D. Vitali, “Robust entanglement of a micromechanical resonator with output optical fields,” Phys. Rev. A 78, 032316 (2008).
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Figures (7)

Fig. 1
Fig. 1

The setup of both the cascaded and parallel schemes. In the cascade scheme, the optical cavity is driven by a laser of frequency ωL1 or ωL2 for the different periods, while in the parallel scheme, we apply both of these driving lasers at the same time.

Fig. 2
Fig. 2

A diagram of the process of the cascaded scheme, and a1, b1, b2 are the annihilation operators for the optical mode and two microwave modes, respectively. When 0 < t < T1 or T2 < t < T1 + T2, LC1 and the optical cavity exchange quantum states with each other, and during T1 < t < T2, the optical cavity mode and the LC2 mode get entangled.

Fig. 3
Fig. 3

The time evolution of the fluctuating photon number for different values of the non-dimensional decay rate ki. Na1, Nb1 and Nb2 are photon numbers for optical cavity, LC1 and LC2, respectively. (a) r = 0.5, k0 = k1 = k2 = 0, (b) r = 0.5, k0 = k1 = k2 = 0.1. The initial condition for both (a) and (b) is Na1(0) = n0,th = 0, Nb1(0) = Nb2(0) = n1,th = n2,th = 0.1. In (a) the ideal case, at each instant , the photon number of the optical mode goes to zero and those numbers of LC1 and LC2 become equal. From (b), we can find that even for the optical cavity, the photon number can not be zero at steady state.

Fig. 4
Fig. 4

The total variance versus phase for different values of the parameters: (a) without dissipation, r = 1 − 10−3, n1,th = n2,th = nth, and nth = 0, 0.1, 1; (b) with dissipation, r = 1 − 10−3, nth = 0, k0 = k1 = k2 = k, and k = 0.001, 0.01, 0.1.

Fig. 5
Fig. 5

The setup for the dissipative dynamical scheme. The two optical cavities are filled with electro-optic media, which are modulated by both LC1 and LC2 via the electro-optic effect. LC1 is in the red-detuned regime with respect to the previous optical cavity and blue-detuned regime with respect to the second one. For LC2, the situation is just opposite.

Fig. 6
Fig. 6

Plot of the total variance V versus the scaled time τnew under different decay conditions, with the parameters r = 0.5, nb,th = 0.01, k0 = 10, together with the result for an ideal two-mode squeezed state. The effective decay rate of the subsystem formed by the two superconducting microwave resonators in Eq. (26) is γ = 2 ( 1 r 2 ) k 0 = 0.15.

Fig. 7
Fig. 7

Plot of the total variance V versus the scaled time τ2 under the above experimental parameters. We can see that the optimized scaled time is τ2 = 2.43, or equally T2 ≈ 1.3 μs.

Equations (35)

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H I , i C = g i a 1 a 1 ( b i + b i ) ,
g i = ω a 1 n 3 r 0 2 d ω b i 2 C i ,
H d C = i [ E 0 a 1 e i ω L 1 t E 0 * a 1 e i ω L 1 t ] ,
H 1 C = Δ a 1 a 1 + ω b 1 b 1 b 1 + ω b 2 b 2 b 2 g 1 a 1 a 1 ( b 1 + b 1 ) + i ( E 0 a 1 E 0 * a 1 ) ,
H 1 , eff C = Δ δ a 1 δ a 1 + ω b 1 b 1 b 1 + ω b 2 b 2 b 2 G 1 ( δ a 1 b 1 + δ a 1 b 1 ) .
[ δ a 1 ( τ 1 + τ 2 ) b 1 ( τ 1 + τ 2 ) b 2 ( τ 1 + τ 2 ) ] = M 1 M 2 M 1 [ δ a 1 ( 0 ) b 1 ( 0 ) b 2 ( 0 ) ] ,
M 1 = [ cos ( τ 1 ) i sin ( τ 1 ) 0 i sin ( τ 1 ) cos ( τ 1 ) 0 0 0 1 ] ,
M 2 = [ cosh [ r ( τ 2 τ 1 ) ] 0 i sinh [ r ( τ 2 τ 1 ) ] 0 1 0 i sinh [ r ( τ 2 τ 1 ) 0 cosh [ r ( τ 2 τ 1 ) ] ] .
M 1 M 2 M 1 = [ 1 0 0 0 cosh [ r ( τ 2 τ 1 ) ] sinh [ r ( τ 2 τ 1 ) ] 0 sinh [ r ( τ 2 τ 1 ) ] cosh [ r ( τ 2 τ 1 ) ] ] .
H = H 0 + H I + H d ,
H 0 = ω a 1 a 1 a 1 + i = 1 2 ω b i b i b i ,
H I = j = 1 2 g j a 1 a 1 ( b j + b j ) ,
H d = j = 1 2 ( 1 ) j E j ( a 1 e i ω L j t + a 1 e i ω L j t ) .
H = j = 1 2 g j a 1 a 1 ( b j e i ω b j t + b j e i ω b j t ) + j = 1 2 ( 1 ) j E j [ a 1 e ( 1 ) j 1 i Δ j t + a 1 e ( 1 ) j i Δ j t ] ,
U = T exp { i 0 t j = 1 2 ( 1 ) j E j [ a 1 e ( 1 ) j 1 i Δ j t + a 1 e ( 1 ) j i Δ j t ] d t } ,
H P = U ( H i t ) U = k = 1 2 g k { a 1 a 1 + j = 1 2 E j Δ j [ a ( e ( 1 ) j + 1 i Δ j t 1 ) + H . c . ] } × ( b k e i ω b k t + b k e i ω b k t ) .
H e f f P = g 1 E 1 Δ 1 ( a 1 b 1 + a 1 b 1 ) g 2 E 2 Δ 2 ( a 1 b 2 + a 1 b 2 ) .
d d τ [ a 1 b 1 b 2 ] = [ k 0 i i r i k 1 0 i r 0 k 2 ] [ a 1 b 1 b 2 ] + [ f 0 f 1 f 2 ] ,
f i ( τ ) f j ( τ ) R = 2 k i n i , t h δ i j δ ( τ τ ) ,
[ a 1 ( τ ) b 1 ( τ ) b 2 ( τ ) ] = M P [ a 1 ( 0 ) b 1 ( 0 ) b 2 ( 0 ) ] ,
M P = [ cos ( 1 r 2 τ ) i sin ( 1 r 2 τ ) 1 r 2 i r sin ( 1 r 2 τ ) 1 r 2 i sin ( 1 r 2 τ ) 1 r 2 cos ( 1 r 2 τ ) r 2 1 r 2 [ cos ( 1 r 2 τ ) 1 ] r 1 r 2 i r sin ( 1 r 2 τ ) 1 r 2 [ 1 cos ( 1 r 2 τ ) ] r 1 r 2 1 r 2 cos ( 1 r 2 τ ) 1 r 2 ] .
M P = [ 1 0 0 0 1 + r 2 1 r 2 2 r 1 r 2 0 2 r 1 r 2 1 + r 2 1 r 2 ] .
d d τ [ a 1 ( τ ) a 2 ( τ ) b 1 ( τ ) b 2 ( τ ) ] = [ k 0 0 i i r 0 k 0 i r r i i r 0 0 i r i 0 0 ] [ a 1 ( τ ) a 2 ( τ ) b 1 ( τ ) b 2 ( τ ) ] + [ f a 1 ( τ ) f a 2 ( τ ) 0 0 ] .
a 1 ( τ ) = i k 0 [ b 1 ( τ ) + r b 2 ( τ ) ] + f a 1 ( τ ) k 0 ,
a 2 ( τ ) = i k 0 [ r b 1 ( τ ) + b 2 ( τ ) ] + f a 2 ( τ ) k 0 .
d d τ [ b 1 ( τ ) b 2 ( τ ) ] = 1 k 0 [ r 2 1 0 0 r 2 1 ] [ b 1 ( τ ) b 2 ( τ ) ] + i k 0 [ f a 1 ( τ ) + r f a 2 ( τ ) r f a 1 ( τ ) f a 2 ( τ ) ] .
τ new = ( 1 r 2 ) τ k 0 , τ new = ( 1 r 2 ) τ k 0 ,
f ˜ a i ( τ new ) = f a i ( τ new ) 2 k 0 , i = 1 , 2 ,
S ˜ ( τ new , ς ) = exp [ ς ( f ˜ a 1 ( τ new ) f ˜ a 2 ( τ new ) f ˜ a 1 ( τ new ) f ˜ a 2 ( τ new ) ) ] ,
b 1 ( τ new ) = e τ new b 1 ( 0 ) + 2 i 0 τ new e ( τ new τ new ) S ˜ ( τ new ) f ˜ a 1 ( τ new ) S ˜ ( τ new ) d τ new ,
b 2 ( τ new ) = e τ new b 2 ( 0 ) 2 i 0 τ new e ( τ new τ new ) S ˜ ( τ new ) f ˜ a 2 ( τ new ) S ˜ ( τ new ) d τ new .
n ˜ cav , i = Γ Δ i 2 + ( Γ / 2 ) 2 P ω a 1 .
V = 2 ( 1 r ) / ( 1 + r ) + 2 α ( n t h , 1 + n t h , 2 + 2 ) 1 + 2 α ,
α = k i / γ , i = 1 , 2 ,
γ = 2 ( 1 r 2 ) / k 0 ..