Abstract

Schrödinger’s famous Gedankenexperiment has inspired multiple generations of physicists to think about apparent paradoxes that arise when the logic of quantum physics is applied to macroscopic objects. The development of quantum technologies enabled us to produce physical analogues of Schrödinger’s cats, such as superpositions of macroscopically distinct states as well as entangled states of microscopic and macroscopic entities. Here we take one step further and prepare an optical state which, in Schrödinger’s language, is equivalent to a superposition of two cats, one of which is dead and the other alive, but it is not known in which state each individual cat is. Specifically, the alive and dead states are, respectively, the displaced single photon and displaced vacuum (coherent state), with the magnitude of displacement being on a scale of 108 photons. These two states have significantly different photon statistics and are therefore macroscopically distinguishable.

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

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References

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  14. M. Ansmann, H. Wang, R. C. Bialczak, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, A. N. Cleland, and J. M. Martinis, “Violation of Bell’s inequality in Josephson phase qubits,” Nature 461, 504–506 (2009).
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  16. A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
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    [Crossref]
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    [Crossref]
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    [Crossref]
  24. P. Sekatski, N. Sangouard, and N. Gisin, “Size of quantum superpositions as measured with classical detectors,” Phys. Rev. A 89, 012116 (2014).
    [Crossref]
  25. S. A. Babichev, J. Appel, and A. I. Lvovsky, “Homodyne tomography characterization and nonlocality of a dual-mode optical qubit,” Phys. Rev. Lett. 92, 193601 (2004).
    [Crossref]
  26. M. G. A. Paris, “Displacement operator by beam splitter,” Phys. Rev. A 217, 78–80 (1996).
    [Crossref]
  27. A. I. Lvovsky and S. A. Babichev, “Synthesis and tomographic characterization of the displaced Fock state of light,” Phys. Rev. A 66, 011801 (2002).
    [Crossref]
  28. F. A. M. de Oliveira, M. S. Kim, P. L. Knight, and V. Bužek, “Properties of displaced number states,” Phys. Rev. A 41, 2645–2652 (1990).
    [Crossref]
  29. S. R. Huisman, N. Jain, S. A. Babichev, F. Vewinger, A. N. Zhang, S. H. Youn, and A. I. Lvovsky, “Instant single-photon Fock state tomography,” Opt. Lett. 34, 2739–2741 (2009).
    [Crossref]
  30. A. V. Masalov, A. Kuzhamuratov, and A. I. Lvovsky, “Noise spectra in balanced optical detectors based on transimpedance amplifiers,” Rev. Sci. Instrum. 88, 113109 (2017).
    [Crossref]
  31. J. Appel, D. Hoffman, E. Figueroa, and A. I. Lvovsky, “Electronic noise in optical homodyne tomography,” Phys. Rev. A 75, 035802 (2007).
    [Crossref]
  32. S. A. Babichev, B. Brezger, and A. I. Lvovsky, “Remote preparation of a single-mode photonic qubit by measuring field quadrature noise,” Phys. Rev. Lett. 92, 047903 (2004).
    [Crossref]
  33. A. I. Lvovsky, “Iterative maximum-likelihood reconstruction in quantum homodyne tomography,” J. Opt. B 6, S556–S559 (2004).
    [Crossref]
  34. M. Ho, O. Morin, J. D. Bancal, N. Gisin, N. Sangouard, and J. Laurat, “Witnessing single-photon entanglement with local homodyne measurement,” Phys. Rev. Lett. 110, 130401 (2013).
    [Crossref]
  35. P. Sekatski, N. Sangouard, M. Stobińska, F. Bussières, M. Afzelius, and N. Gisin, “Proposal for exploring macroscopic entanglement with a single photon and coherent states,” Phys. Rev. A 86, 060301 (2012).
    [Crossref]
  36. F. Marquardt, B. Abel, and J. von Delft, “Measuring the size of a quantum superposition of two-many body states,” Phys. Rev. A 78, 012109 (2008).
    [Crossref]
  37. C. W. Lee and H. Jeong, “Quantification of macroscopic quantum superpositions within phase space,” Phys. Rev. Lett. 106, 220401 (2011).
    [Crossref]
  38. R. Ghobadi, S. Kumar, B. Pepper, D. Bouwmeester, A. I. Lvovsky, and C. Simon, “Optomechanical micro-macro entanglement,” Phys. Rev. Lett. 112, 080503 (2014).
    [Crossref]

2018 (4)

F. Fröwis, P. Sekatski, W. Dür, N. Gisin, and N. Sangouard, “Macroscopic quantum states: measures, fragility, and implementations,” Rev. Mod. Phys. 90, 025004 (2018).
[Crossref]

M. Dabrowski, M. Mazelanik, M. Parniak, A. Leszczynski, M. Lipka, and W. Wasilewski, “Certification of high-dimensional entanglement and Einstein-Podolsky-Rosen steering with cold atomic quantum memory,” Phys. Rev. A 98, 042126 (2018).
[Crossref]

I. Marinković, A. Wallucks, R. Riedinger, S. Hong, M. Aspelmeyer, and S. Gröblacher, “Optomechanical bell test,” Phys. Rev. Lett. 121, 220404 (2018).
[Crossref]

X. Huang, E. Zeuthen, D. V. Vasilyev, Q. He, K. Hammerer, and E. S. Polzik, “Unconditional steady-state entanglement in macroscopic hybrid systems by coherent noise cancellation,” Phys. Rev. Lett. 121, 103602 (2018).
[Crossref]

2017 (2)

A. V. Masalov, A. Kuzhamuratov, and A. I. Lvovsky, “Noise spectra in balanced optical detectors based on transimpedance amplifiers,” Rev. Sci. Instrum. 88, 113109 (2017).
[Crossref]

P. Zarkeshian, C. Deshmukh, N. Sinclair, S. K. Goyal, G. H. Aguilar, P. Lefebvre, M. G. Puigibert, V. B. Verma, F. Marsili, M. D. Shaw, S. W. Nam, K. Heshami, D. Oblak, W. Tittel, and C. Simon, “Entanglement between more than two hundred macroscopic atomic ensembles in a solid,” Nat. Commun. 8, 906 (2017).
[Crossref]

2016 (2)

V. Vivoli, P. Sekatski, and N. Sangouard, “What does it take to detect entanglement with the human eye?” Optica 3, 473–476 (2016).
[Crossref]

A. Tiranov, J. Lavoie, P. C. Strassmann, N. Sangouard, M. Afzelius, F. Bussières, and N. Gisin, “Demonstration of light-matter micro-macro quantum correlations,” Phys. Rev. Lett. 116, 190502 (2016).
[Crossref]

2014 (4)

N. Behbood, M. Ciurana, G. Colangelo, M. Napolitano, G. Tóth, R. J. Sewell, and M. W. Mitchell, “Generation of macroscopic singlet states in a cold atomic ensemble,” Phys. Rev. Lett. 113, 093601 (2014).
[Crossref]

M. Arndt and K. Hornberger, “Testing the limits of quantum mechanical superpositions,” Nat. Phys. 10, 271–277 (2014).
[Crossref]

P. Sekatski, N. Sangouard, and N. Gisin, “Size of quantum superpositions as measured with classical detectors,” Phys. Rev. A 89, 012116 (2014).
[Crossref]

R. Ghobadi, S. Kumar, B. Pepper, D. Bouwmeester, A. I. Lvovsky, and C. Simon, “Optomechanical micro-macro entanglement,” Phys. Rev. Lett. 112, 080503 (2014).
[Crossref]

2013 (4)

N. Bruno, A. Martin, P. Sekatski, N. Sangouard, R. T. Thew, and N. Gisin, “Displacement of entanglement back and forth between the micro and macro domains,” Nat. Phys. 9, 545–548 (2013).
[Crossref]

A. I. Lvovsky, R. Ghobadi, A. Chandra, A. S. Prasad, and C. Simon, “Observation of micro-macro entanglement of light,” Nat. Phys. 9, 541–544 (2013).
[Crossref]

M. Ho, O. Morin, J. D. Bancal, N. Gisin, N. Sangouard, and J. Laurat, “Witnessing single-photon entanglement with local homodyne measurement,” Phys. Rev. Lett. 110, 130401 (2013).
[Crossref]

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

2012 (2)

T. Sh. Iskhakov, I. N. Agafonov, M. V. Chekhova, and G. Leuchs, “Polarization-entangled light pulses of 105 photons,” Phys. Rev. Lett. 109, 150502 (2012).
[Crossref]

P. Sekatski, N. Sangouard, M. Stobińska, F. Bussières, M. Afzelius, and N. Gisin, “Proposal for exploring macroscopic entanglement with a single photon and coherent states,” Phys. Rev. A 86, 060301 (2012).
[Crossref]

2011 (1)

C. W. Lee and H. Jeong, “Quantification of macroscopic quantum superpositions within phase space,” Phys. Rev. Lett. 106, 220401 (2011).
[Crossref]

2010 (1)

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

2009 (3)

M. Ansmann, H. Wang, R. C. Bialczak, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, A. N. Cleland, and J. M. Martinis, “Violation of Bell’s inequality in Josephson phase qubits,” Nature 461, 504–506 (2009).
[Crossref]

P. Sekatski, N. Brunner, C. Branciard, N. Gisin, and C. Simon, “Towards quantum experiments with human eyes as detectors based on cloning via stimulated emission,” Phys. Rev. Lett. 103, 113601 (2009).
[Crossref]

S. R. Huisman, N. Jain, S. A. Babichev, F. Vewinger, A. N. Zhang, S. H. Youn, and A. I. Lvovsky, “Instant single-photon Fock state tomography,” Opt. Lett. 34, 2739–2741 (2009).
[Crossref]

2008 (2)

F. Marquardt, B. Abel, and J. von Delft, “Measuring the size of a quantum superposition of two-many body states,” Phys. Rev. A 78, 012109 (2008).
[Crossref]

J. Estève, C. Gross, A. Weller, S. Giovanazzi, and M. K. Oberthaler, “Squeezing and entanglement in a Bose–Einstein condensate,” Nature 455, 1216–1219 (2008).
[Crossref]

2007 (1)

J. Appel, D. Hoffman, E. Figueroa, and A. I. Lvovsky, “Electronic noise in optical homodyne tomography,” Phys. Rev. A 75, 035802 (2007).
[Crossref]

2005 (1)

C. W. Chou, H. de Riedmatten, D. Felinto, S. V. Polyakov, S. J. van Enk, and H. J. Kimble, “Measurement-induced entanglement for excitation stored in remote atomic ensembles,” Nature 438, 828–832 (2005).
[Crossref]

2004 (3)

S. A. Babichev, B. Brezger, and A. I. Lvovsky, “Remote preparation of a single-mode photonic qubit by measuring field quadrature noise,” Phys. Rev. Lett. 92, 047903 (2004).
[Crossref]

A. I. Lvovsky, “Iterative maximum-likelihood reconstruction in quantum homodyne tomography,” J. Opt. B 6, S556–S559 (2004).
[Crossref]

S. A. Babichev, J. Appel, and A. I. Lvovsky, “Homodyne tomography characterization and nonlocality of a dual-mode optical qubit,” Phys. Rev. Lett. 92, 193601 (2004).
[Crossref]

2003 (1)

W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards quantum superpositions of a mirror,” Phys. Rev. Lett. 91, 130401 (2003).
[Crossref]

2002 (1)

A. I. Lvovsky and S. A. Babichev, “Synthesis and tomographic characterization of the displaced Fock state of light,” Phys. Rev. A 66, 011801 (2002).
[Crossref]

2001 (1)

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

2000 (1)

J. R. Friedman, V. Patel, W. Chen, S. K. Tolpygo, and J. E. Lukens, “Quantum superposition of distinct macroscopic states,” Nature 406, 43–46 (2000).
[Crossref]

1999 (1)

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

1996 (1)

M. G. A. Paris, “Displacement operator by beam splitter,” Phys. Rev. A 217, 78–80 (1996).
[Crossref]

1990 (1)

F. A. M. de Oliveira, M. S. Kim, P. L. Knight, and V. Bužek, “Properties of displaced number states,” Phys. Rev. A 41, 2645–2652 (1990).
[Crossref]

Abel, B.

F. Marquardt, B. Abel, and J. von Delft, “Measuring the size of a quantum superposition of two-many body states,” Phys. Rev. A 78, 012109 (2008).
[Crossref]

Afzelius, M.

A. Tiranov, J. Lavoie, P. C. Strassmann, N. Sangouard, M. Afzelius, F. Bussières, and N. Gisin, “Demonstration of light-matter micro-macro quantum correlations,” Phys. Rev. Lett. 116, 190502 (2016).
[Crossref]

P. Sekatski, N. Sangouard, M. Stobińska, F. Bussières, M. Afzelius, and N. Gisin, “Proposal for exploring macroscopic entanglement with a single photon and coherent states,” Phys. Rev. A 86, 060301 (2012).
[Crossref]

Agafonov, I. N.

T. Sh. Iskhakov, I. N. Agafonov, M. V. Chekhova, and G. Leuchs, “Polarization-entangled light pulses of 105 photons,” Phys. Rev. Lett. 109, 150502 (2012).
[Crossref]

Aguilar, G. H.

P. Zarkeshian, C. Deshmukh, N. Sinclair, S. K. Goyal, G. H. Aguilar, P. Lefebvre, M. G. Puigibert, V. B. Verma, F. Marsili, M. D. Shaw, S. W. Nam, K. Heshami, D. Oblak, W. Tittel, and C. Simon, “Entanglement between more than two hundred macroscopic atomic ensembles in a solid,” Nat. Commun. 8, 906 (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, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[Crossref]

M. Ansmann, H. Wang, R. C. Bialczak, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, A. N. Cleland, and J. M. Martinis, “Violation of Bell’s inequality in Josephson phase qubits,” Nature 461, 504–506 (2009).
[Crossref]

Appel, J.

J. Appel, D. Hoffman, E. Figueroa, and A. I. Lvovsky, “Electronic noise in optical homodyne tomography,” Phys. Rev. A 75, 035802 (2007).
[Crossref]

S. A. Babichev, J. Appel, and A. I. Lvovsky, “Homodyne tomography characterization and nonlocality of a dual-mode optical qubit,” Phys. Rev. Lett. 92, 193601 (2004).
[Crossref]

Arndt, M.

M. Arndt and K. Hornberger, “Testing the limits of quantum mechanical superpositions,” Nat. Phys. 10, 271–277 (2014).
[Crossref]

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

Aspelmeyer, M.

I. Marinković, A. Wallucks, R. Riedinger, S. Hong, M. Aspelmeyer, and S. Gröblacher, “Optomechanical bell test,” Phys. Rev. Lett. 121, 220404 (2018).
[Crossref]

Babichev, S. A.

S. R. Huisman, N. Jain, S. A. Babichev, F. Vewinger, A. N. Zhang, S. H. Youn, and A. I. Lvovsky, “Instant single-photon Fock state tomography,” Opt. Lett. 34, 2739–2741 (2009).
[Crossref]

S. A. Babichev, B. Brezger, and A. I. Lvovsky, “Remote preparation of a single-mode photonic qubit by measuring field quadrature noise,” Phys. Rev. Lett. 92, 047903 (2004).
[Crossref]

S. A. Babichev, J. Appel, and A. I. Lvovsky, “Homodyne tomography characterization and nonlocality of a dual-mode optical qubit,” Phys. Rev. Lett. 92, 193601 (2004).
[Crossref]

A. I. Lvovsky and S. A. Babichev, “Synthesis and tomographic characterization of the displaced Fock state of light,” Phys. Rev. A 66, 011801 (2002).
[Crossref]

Bancal, J. D.

M. Ho, O. Morin, J. D. Bancal, N. Gisin, N. Sangouard, and J. Laurat, “Witnessing single-photon entanglement with local homodyne measurement,” Phys. Rev. Lett. 110, 130401 (2013).
[Crossref]

Behbood, N.

N. Behbood, M. Ciurana, G. Colangelo, M. Napolitano, G. Tóth, R. J. Sewell, and M. W. Mitchell, “Generation of macroscopic singlet states in a cold atomic ensemble,” Phys. Rev. Lett. 113, 093601 (2014).
[Crossref]

Bellini, M.

N. Biagi, L. S. Costanzo, M. Bellini, and A. Zavatta, “Entangling macroscopic light states by delocalized photon addition,” arxiv.org/abs/1811.10466 (2018).

Biagi, N.

N. Biagi, L. S. Costanzo, M. Bellini, and A. Zavatta, “Entangling macroscopic light states by delocalized photon addition,” arxiv.org/abs/1811.10466 (2018).

Bialczak, R. C.

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

M. Ansmann, H. Wang, R. C. Bialczak, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, A. N. Cleland, and J. M. Martinis, “Violation of Bell’s inequality in Josephson phase qubits,” Nature 461, 504–506 (2009).
[Crossref]

Bouwmeester, D.

R. Ghobadi, S. Kumar, B. Pepper, D. Bouwmeester, A. I. Lvovsky, and C. Simon, “Optomechanical micro-macro entanglement,” Phys. Rev. Lett. 112, 080503 (2014).
[Crossref]

W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards quantum superpositions of a mirror,” Phys. Rev. Lett. 91, 130401 (2003).
[Crossref]

Branciard, C.

P. Sekatski, N. Brunner, C. Branciard, N. Gisin, and C. Simon, “Towards quantum experiments with human eyes as detectors based on cloning via stimulated emission,” Phys. Rev. Lett. 103, 113601 (2009).
[Crossref]

Brezger, B.

S. A. Babichev, B. Brezger, and A. I. Lvovsky, “Remote preparation of a single-mode photonic qubit by measuring field quadrature noise,” Phys. Rev. Lett. 92, 047903 (2004).
[Crossref]

Brunner, N.

P. Sekatski, N. Brunner, C. Branciard, N. Gisin, and C. Simon, “Towards quantum experiments with human eyes as detectors based on cloning via stimulated emission,” Phys. Rev. Lett. 103, 113601 (2009).
[Crossref]

Bruno, N.

N. Bruno, A. Martin, P. Sekatski, N. Sangouard, R. T. Thew, and N. Gisin, “Displacement of entanglement back and forth between the micro and macro domains,” Nat. Phys. 9, 545–548 (2013).
[Crossref]

Bussières, F.

A. Tiranov, J. Lavoie, P. C. Strassmann, N. Sangouard, M. Afzelius, F. Bussières, and N. Gisin, “Demonstration of light-matter micro-macro quantum correlations,” Phys. Rev. Lett. 116, 190502 (2016).
[Crossref]

P. Sekatski, N. Sangouard, M. Stobińska, F. Bussières, M. Afzelius, and N. Gisin, “Proposal for exploring macroscopic entanglement with a single photon and coherent states,” Phys. Rev. A 86, 060301 (2012).
[Crossref]

Bužek, V.

F. A. M. de Oliveira, M. S. Kim, P. L. Knight, and V. Bužek, “Properties of displaced number states,” Phys. Rev. A 41, 2645–2652 (1990).
[Crossref]

Chandra, A.

A. I. Lvovsky, R. Ghobadi, A. Chandra, A. S. Prasad, and C. Simon, “Observation of micro-macro entanglement of light,” Nat. Phys. 9, 541–544 (2013).
[Crossref]

Chekhova, M. V.

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

Polzik, E. S.

X. Huang, E. Zeuthen, D. V. Vasilyev, Q. He, K. Hammerer, and E. S. Polzik, “Unconditional steady-state entanglement in macroscopic hybrid systems by coherent noise cancellation,” Phys. Rev. Lett. 121, 103602 (2018).
[Crossref]

Prasad, A. S.

A. I. Lvovsky, R. Ghobadi, A. Chandra, A. S. Prasad, and C. Simon, “Observation of micro-macro entanglement of light,” Nat. Phys. 9, 541–544 (2013).
[Crossref]

Puigibert, M. G.

P. Zarkeshian, C. Deshmukh, N. Sinclair, S. K. Goyal, G. H. Aguilar, P. Lefebvre, M. G. Puigibert, V. B. Verma, F. Marsili, M. D. Shaw, S. W. Nam, K. Heshami, D. Oblak, W. Tittel, and C. Simon, “Entanglement between more than two hundred macroscopic atomic ensembles in a solid,” Nat. Commun. 8, 906 (2017).
[Crossref]

Riedinger, R.

I. Marinković, A. Wallucks, R. Riedinger, S. Hong, M. Aspelmeyer, and S. Gröblacher, “Optomechanical bell test,” Phys. Rev. Lett. 121, 220404 (2018).
[Crossref]

Sangouard, N.

F. Fröwis, P. Sekatski, W. Dür, N. Gisin, and N. Sangouard, “Macroscopic quantum states: measures, fragility, and implementations,” Rev. Mod. Phys. 90, 025004 (2018).
[Crossref]

V. Vivoli, P. Sekatski, and N. Sangouard, “What does it take to detect entanglement with the human eye?” Optica 3, 473–476 (2016).
[Crossref]

A. Tiranov, J. Lavoie, P. C. Strassmann, N. Sangouard, M. Afzelius, F. Bussières, and N. Gisin, “Demonstration of light-matter micro-macro quantum correlations,” Phys. Rev. Lett. 116, 190502 (2016).
[Crossref]

P. Sekatski, N. Sangouard, and N. Gisin, “Size of quantum superpositions as measured with classical detectors,” Phys. Rev. A 89, 012116 (2014).
[Crossref]

N. Bruno, A. Martin, P. Sekatski, N. Sangouard, R. T. Thew, and N. Gisin, “Displacement of entanglement back and forth between the micro and macro domains,” Nat. Phys. 9, 545–548 (2013).
[Crossref]

M. Ho, O. Morin, J. D. Bancal, N. Gisin, N. Sangouard, and J. Laurat, “Witnessing single-photon entanglement with local homodyne measurement,” Phys. Rev. Lett. 110, 130401 (2013).
[Crossref]

P. Sekatski, N. Sangouard, M. Stobińska, F. Bussières, M. Afzelius, and N. Gisin, “Proposal for exploring macroscopic entanglement with a single photon and coherent states,” Phys. Rev. A 86, 060301 (2012).
[Crossref]

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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, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[Crossref]

M. Ansmann, H. Wang, R. C. Bialczak, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, A. N. Cleland, and J. M. Martinis, “Violation of Bell’s inequality in Josephson phase qubits,” Nature 461, 504–506 (2009).
[Crossref]

Sekatski, P.

F. Fröwis, P. Sekatski, W. Dür, N. Gisin, and N. Sangouard, “Macroscopic quantum states: measures, fragility, and implementations,” Rev. Mod. Phys. 90, 025004 (2018).
[Crossref]

V. Vivoli, P. Sekatski, and N. Sangouard, “What does it take to detect entanglement with the human eye?” Optica 3, 473–476 (2016).
[Crossref]

P. Sekatski, N. Sangouard, and N. Gisin, “Size of quantum superpositions as measured with classical detectors,” Phys. Rev. A 89, 012116 (2014).
[Crossref]

N. Bruno, A. Martin, P. Sekatski, N. Sangouard, R. T. Thew, and N. Gisin, “Displacement of entanglement back and forth between the micro and macro domains,” Nat. Phys. 9, 545–548 (2013).
[Crossref]

P. Sekatski, N. Sangouard, M. Stobińska, F. Bussières, M. Afzelius, and N. Gisin, “Proposal for exploring macroscopic entanglement with a single photon and coherent states,” Phys. Rev. A 86, 060301 (2012).
[Crossref]

P. Sekatski, N. Brunner, C. Branciard, N. Gisin, and C. Simon, “Towards quantum experiments with human eyes as detectors based on cloning via stimulated emission,” Phys. Rev. Lett. 103, 113601 (2009).
[Crossref]

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N. Behbood, M. Ciurana, G. Colangelo, M. Napolitano, G. Tóth, R. J. Sewell, and M. W. Mitchell, “Generation of macroscopic singlet states in a cold atomic ensemble,” Phys. Rev. Lett. 113, 093601 (2014).
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P. Zarkeshian, C. Deshmukh, N. Sinclair, S. K. Goyal, G. H. Aguilar, P. Lefebvre, M. G. Puigibert, V. B. Verma, F. Marsili, M. D. Shaw, S. W. Nam, K. Heshami, D. Oblak, W. Tittel, and C. Simon, “Entanglement between more than two hundred macroscopic atomic ensembles in a solid,” Nat. Commun. 8, 906 (2017).
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T. A. Palomaki, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, “Entangling mechanical motion with microwave fields,” Science 342, 710–713 (2013).
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P. Zarkeshian, C. Deshmukh, N. Sinclair, S. K. Goyal, G. H. Aguilar, P. Lefebvre, M. G. Puigibert, V. B. Verma, F. Marsili, M. D. Shaw, S. W. Nam, K. Heshami, D. Oblak, W. Tittel, and C. Simon, “Entanglement between more than two hundred macroscopic atomic ensembles in a solid,” Nat. Commun. 8, 906 (2017).
[Crossref]

R. Ghobadi, S. Kumar, B. Pepper, D. Bouwmeester, A. I. Lvovsky, and C. Simon, “Optomechanical micro-macro entanglement,” Phys. Rev. Lett. 112, 080503 (2014).
[Crossref]

A. I. Lvovsky, R. Ghobadi, A. Chandra, A. S. Prasad, and C. Simon, “Observation of micro-macro entanglement of light,” Nat. Phys. 9, 541–544 (2013).
[Crossref]

P. Sekatski, N. Brunner, C. Branciard, N. Gisin, and C. Simon, “Towards quantum experiments with human eyes as detectors based on cloning via stimulated emission,” Phys. Rev. Lett. 103, 113601 (2009).
[Crossref]

W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards quantum superpositions of a mirror,” Phys. Rev. Lett. 91, 130401 (2003).
[Crossref]

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P. Zarkeshian, C. Deshmukh, N. Sinclair, S. K. Goyal, G. H. Aguilar, P. Lefebvre, M. G. Puigibert, V. B. Verma, F. Marsili, M. D. Shaw, S. W. Nam, K. Heshami, D. Oblak, W. Tittel, and C. Simon, “Entanglement between more than two hundred macroscopic atomic ensembles in a solid,” Nat. Commun. 8, 906 (2017).
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P. Sekatski, N. Sangouard, M. Stobińska, F. Bussières, M. Afzelius, and N. Gisin, “Proposal for exploring macroscopic entanglement with a single photon and coherent states,” Phys. Rev. A 86, 060301 (2012).
[Crossref]

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A. Tiranov, J. Lavoie, P. C. Strassmann, N. Sangouard, M. Afzelius, F. Bussières, and N. Gisin, “Demonstration of light-matter micro-macro quantum correlations,” Phys. Rev. Lett. 116, 190502 (2016).
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T. A. Palomaki, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, “Entangling mechanical motion with microwave fields,” Science 342, 710–713 (2013).
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N. Bruno, A. Martin, P. Sekatski, N. Sangouard, R. T. Thew, and N. Gisin, “Displacement of entanglement back and forth between the micro and macro domains,” Nat. Phys. 9, 545–548 (2013).
[Crossref]

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A. Tiranov, J. Lavoie, P. C. Strassmann, N. Sangouard, M. Afzelius, F. Bussières, and N. Gisin, “Demonstration of light-matter micro-macro quantum correlations,” Phys. Rev. Lett. 116, 190502 (2016).
[Crossref]

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P. Zarkeshian, C. Deshmukh, N. Sinclair, S. K. Goyal, G. H. Aguilar, P. Lefebvre, M. G. Puigibert, V. B. Verma, F. Marsili, M. D. Shaw, S. W. Nam, K. Heshami, D. Oblak, W. Tittel, and C. Simon, “Entanglement between more than two hundred macroscopic atomic ensembles in a solid,” Nat. Commun. 8, 906 (2017).
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J. R. Friedman, V. Patel, W. Chen, S. K. Tolpygo, and J. E. Lukens, “Quantum superposition of distinct macroscopic states,” Nature 406, 43–46 (2000).
[Crossref]

Tóth, G.

N. Behbood, M. Ciurana, G. Colangelo, M. Napolitano, G. Tóth, R. J. Sewell, and M. W. Mitchell, “Generation of macroscopic singlet states in a cold atomic ensemble,” Phys. Rev. Lett. 113, 093601 (2014).
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C. W. Chou, H. de Riedmatten, D. Felinto, S. V. Polyakov, S. J. van Enk, and H. J. Kimble, “Measurement-induced entanglement for excitation stored in remote atomic ensembles,” Nature 438, 828–832 (2005).
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X. Huang, E. Zeuthen, D. V. Vasilyev, Q. He, K. Hammerer, and E. S. Polzik, “Unconditional steady-state entanglement in macroscopic hybrid systems by coherent noise cancellation,” Phys. Rev. Lett. 121, 103602 (2018).
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P. Zarkeshian, C. Deshmukh, N. Sinclair, S. K. Goyal, G. H. Aguilar, P. Lefebvre, M. G. Puigibert, V. B. Verma, F. Marsili, M. D. Shaw, S. W. Nam, K. Heshami, D. Oblak, W. Tittel, and C. Simon, “Entanglement between more than two hundred macroscopic atomic ensembles in a solid,” Nat. Commun. 8, 906 (2017).
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Vivoli, V.

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F. Marquardt, B. Abel, and J. von Delft, “Measuring the size of a quantum superposition of two-many body states,” Phys. Rev. A 78, 012109 (2008).
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M. Arndt, O. Nairz, J. Vos-Andreae, C. K. Keller, G. Zouw, and A. Zeilinger, “Wave-particle duality of C60 molecules,” Nature 401, 680–682 (1999).
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I. Marinković, A. Wallucks, R. Riedinger, S. Hong, M. Aspelmeyer, and S. Gröblacher, “Optomechanical bell test,” Phys. Rev. Lett. 121, 220404 (2018).
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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, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[Crossref]

M. Ansmann, H. Wang, R. C. Bialczak, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, A. N. Cleland, and J. M. Martinis, “Violation of Bell’s inequality in Josephson phase qubits,” Nature 461, 504–506 (2009).
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M. Dabrowski, M. Mazelanik, M. Parniak, A. Leszczynski, M. Lipka, and W. Wasilewski, “Certification of high-dimensional entanglement and Einstein-Podolsky-Rosen steering with cold atomic quantum memory,” Phys. Rev. A 98, 042126 (2018).
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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, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
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M. Ansmann, H. Wang, R. C. Bialczak, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, A. N. Cleland, and J. M. Martinis, “Violation of Bell’s inequality in Josephson phase qubits,” Nature 461, 504–506 (2009).
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J. Estève, C. Gross, A. Weller, S. Giovanazzi, and M. K. Oberthaler, “Squeezing and entanglement in a Bose–Einstein condensate,” Nature 455, 1216–1219 (2008).
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Wenner, J.

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
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M. Ansmann, H. Wang, R. C. Bialczak, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, A. N. Cleland, and J. M. Martinis, “Violation of Bell’s inequality in Josephson phase qubits,” Nature 461, 504–506 (2009).
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Youn, S. H.

Zarkeshian, P.

P. Zarkeshian, C. Deshmukh, N. Sinclair, S. K. Goyal, G. H. Aguilar, P. Lefebvre, M. G. Puigibert, V. B. Verma, F. Marsili, M. D. Shaw, S. W. Nam, K. Heshami, D. Oblak, W. Tittel, and C. Simon, “Entanglement between more than two hundred macroscopic atomic ensembles in a solid,” Nat. Commun. 8, 906 (2017).
[Crossref]

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N. Biagi, L. S. Costanzo, M. Bellini, and A. Zavatta, “Entangling macroscopic light states by delocalized photon addition,” arxiv.org/abs/1811.10466 (2018).

Zeilinger, A.

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

Zeuthen, E.

X. Huang, E. Zeuthen, D. V. Vasilyev, Q. He, K. Hammerer, and E. S. Polzik, “Unconditional steady-state entanglement in macroscopic hybrid systems by coherent noise cancellation,” Phys. Rev. Lett. 121, 103602 (2018).
[Crossref]

Zhang, A. N.

Zouw, G.

M. Arndt, O. Nairz, J. Vos-Andreae, C. K. Keller, G. Zouw, and A. Zeilinger, “Wave-particle duality of C60 molecules,” Nature 401, 680–682 (1999).
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A. I. Lvovsky, “Iterative maximum-likelihood reconstruction in quantum homodyne tomography,” J. Opt. B 6, S556–S559 (2004).
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P. Zarkeshian, C. Deshmukh, N. Sinclair, S. K. Goyal, G. H. Aguilar, P. Lefebvre, M. G. Puigibert, V. B. Verma, F. Marsili, M. D. Shaw, S. W. Nam, K. Heshami, D. Oblak, W. Tittel, and C. Simon, “Entanglement between more than two hundred macroscopic atomic ensembles in a solid,” Nat. Commun. 8, 906 (2017).
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Nat. Phys. (3)

M. Arndt and K. Hornberger, “Testing the limits of quantum mechanical superpositions,” Nat. Phys. 10, 271–277 (2014).
[Crossref]

N. Bruno, A. Martin, P. Sekatski, N. Sangouard, R. T. Thew, and N. Gisin, “Displacement of entanglement back and forth between the micro and macro domains,” Nat. Phys. 9, 545–548 (2013).
[Crossref]

A. I. Lvovsky, R. Ghobadi, A. Chandra, A. S. Prasad, and C. Simon, “Observation of micro-macro entanglement of light,” Nat. Phys. 9, 541–544 (2013).
[Crossref]

Nature (7)

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

C. W. Chou, H. de Riedmatten, D. Felinto, S. V. Polyakov, S. J. van Enk, and H. J. Kimble, “Measurement-induced entanglement for excitation stored in remote atomic ensembles,” Nature 438, 828–832 (2005).
[Crossref]

J. Estève, C. Gross, A. Weller, S. Giovanazzi, and M. K. Oberthaler, “Squeezing and entanglement in a Bose–Einstein condensate,” Nature 455, 1216–1219 (2008).
[Crossref]

J. R. Friedman, V. Patel, W. Chen, S. K. Tolpygo, and J. E. Lukens, “Quantum superposition of distinct macroscopic states,” Nature 406, 43–46 (2000).
[Crossref]

M. Ansmann, H. Wang, R. C. Bialczak, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, A. N. Cleland, and J. M. Martinis, “Violation of Bell’s inequality in Josephson phase qubits,” Nature 461, 504–506 (2009).
[Crossref]

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

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
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Opt. Lett. (1)

Optica (1)

Phys. Rev. A (8)

M. Dabrowski, M. Mazelanik, M. Parniak, A. Leszczynski, M. Lipka, and W. Wasilewski, “Certification of high-dimensional entanglement and Einstein-Podolsky-Rosen steering with cold atomic quantum memory,” Phys. Rev. A 98, 042126 (2018).
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P. Sekatski, N. Sangouard, and N. Gisin, “Size of quantum superpositions as measured with classical detectors,” Phys. Rev. A 89, 012116 (2014).
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J. Appel, D. Hoffman, E. Figueroa, and A. I. Lvovsky, “Electronic noise in optical homodyne tomography,” Phys. Rev. A 75, 035802 (2007).
[Crossref]

P. Sekatski, N. Sangouard, M. Stobińska, F. Bussières, M. Afzelius, and N. Gisin, “Proposal for exploring macroscopic entanglement with a single photon and coherent states,” Phys. Rev. A 86, 060301 (2012).
[Crossref]

F. Marquardt, B. Abel, and J. von Delft, “Measuring the size of a quantum superposition of two-many body states,” Phys. Rev. A 78, 012109 (2008).
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Phys. Rev. Lett. (12)

C. W. Lee and H. Jeong, “Quantification of macroscopic quantum superpositions within phase space,” Phys. Rev. Lett. 106, 220401 (2011).
[Crossref]

R. Ghobadi, S. Kumar, B. Pepper, D. Bouwmeester, A. I. Lvovsky, and C. Simon, “Optomechanical micro-macro entanglement,” Phys. Rev. Lett. 112, 080503 (2014).
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S. A. Babichev, B. Brezger, and A. I. Lvovsky, “Remote preparation of a single-mode photonic qubit by measuring field quadrature noise,” Phys. Rev. Lett. 92, 047903 (2004).
[Crossref]

M. Ho, O. Morin, J. D. Bancal, N. Gisin, N. Sangouard, and J. Laurat, “Witnessing single-photon entanglement with local homodyne measurement,” Phys. Rev. Lett. 110, 130401 (2013).
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S. A. Babichev, J. Appel, and A. I. Lvovsky, “Homodyne tomography characterization and nonlocality of a dual-mode optical qubit,” Phys. Rev. Lett. 92, 193601 (2004).
[Crossref]

I. Marinković, A. Wallucks, R. Riedinger, S. Hong, M. Aspelmeyer, and S. Gröblacher, “Optomechanical bell test,” Phys. Rev. Lett. 121, 220404 (2018).
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T. Sh. Iskhakov, I. N. Agafonov, M. V. Chekhova, and G. Leuchs, “Polarization-entangled light pulses of 105 photons,” Phys. Rev. Lett. 109, 150502 (2012).
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P. Sekatski, N. Brunner, C. Branciard, N. Gisin, and C. Simon, “Towards quantum experiments with human eyes as detectors based on cloning via stimulated emission,” Phys. Rev. Lett. 103, 113601 (2009).
[Crossref]

W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards quantum superpositions of a mirror,” Phys. Rev. Lett. 91, 130401 (2003).
[Crossref]

A. Tiranov, J. Lavoie, P. C. Strassmann, N. Sangouard, M. Afzelius, F. Bussières, and N. Gisin, “Demonstration of light-matter micro-macro quantum correlations,” Phys. Rev. Lett. 116, 190502 (2016).
[Crossref]

N. Behbood, M. Ciurana, G. Colangelo, M. Napolitano, G. Tóth, R. J. Sewell, and M. W. Mitchell, “Generation of macroscopic singlet states in a cold atomic ensemble,” Phys. Rev. Lett. 113, 093601 (2014).
[Crossref]

X. Huang, E. Zeuthen, D. V. Vasilyev, Q. He, K. Hammerer, and E. S. Polzik, “Unconditional steady-state entanglement in macroscopic hybrid systems by coherent noise cancellation,” Phys. Rev. Lett. 121, 103602 (2018).
[Crossref]

Rev. Mod. Phys. (1)

F. Fröwis, P. Sekatski, W. Dür, N. Gisin, and N. Sangouard, “Macroscopic quantum states: measures, fragility, and implementations,” Rev. Mod. Phys. 90, 025004 (2018).
[Crossref]

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A. V. Masalov, A. Kuzhamuratov, and A. I. Lvovsky, “Noise spectra in balanced optical detectors based on transimpedance amplifiers,” Rev. Sci. Instrum. 88, 113109 (2017).
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Science (1)

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

Other (1)

N. Biagi, L. S. Costanzo, M. Bellini, and A. Zavatta, “Entangling macroscopic light states by delocalized photon addition,” arxiv.org/abs/1811.10466 (2018).

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Figures (3)

Fig. 1.
Fig. 1. Scheme of the experiment. (a) Preparation of the macro–macro entangled state consists in splitting a heralded single photon and applying the phase-space displacement in both modes. (b, c) The two ways to characterize the macro–macro entangled state. (b) Reverse displacement followed by quadrature measurements on a homodyne detector. (c) Photon number measurement using a balanced detector with a reference beam. (d) Full setup. Half-wave plates are denoted by HWP, quarter-wave plates by QWP, polarization beam splitters by PBS, calcite crystals by C, balanced detectors by BD, displacement fields by DF, and local oscillator/reference fields by LO. The spatial modes in which the macro–macro entanglement Eq. (2) is present are shown by purple lines.
Fig. 2.
Fig. 2. Photon statistics of Bob’s mode as a function of energy measurement results in Alice’s mode: mean (a), variance (b). The measurements in both channels are relative to the reference pulse. The data corresponding to the relative phase φ = 0 between Alice and Bob is depicted in red, and φ = π / 2 in green. Circles show the experimental data and solid lines are the theoretical curves for the appropriate efficiencies. Insets: Wigner functions and quadrature marginal distributions of Bob’s states illustrating why the mean energy of Bob’s state depends on Alice’s result for φ = 0 but not φ = π / 2 . (c) Histograms of Bob’s detected photon numbers conditioned on different results of Alice’s measurement, showing a high degree of single-shot distinguishability at a macroscopic level. Red solid line: theoretical prediction; blue dashed line: prediction for 100%-efficient detectors with the reference beam.
Fig. 3.
Fig. 3. Two-mode density matrix, reconstructed via homodyne tomography after reverse displacements in both channels. The vacuum component appears due to the nonideal quantum efficiency. Concurrence C ( ρ ) = 0.32 .

Equations (14)

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

| Ψ 0 A B = 1 2 ( | 0 A | 1 B + e i φ | 1 A | 0 B ) ,
| Ψ A B = 1 2 ( D ^ ( α ) | 0 A D ^ ( α ) | 1 B + e i φ D ^ ( α ) | 1 A D ^ ( α ) | 0 B ) .
Ξ 0 ( n ) = e α 2 / 2 α n n ! ; Ξ 1 ( n ) = e α 2 / 2 α n n ! ( n α α ) .
| Ξ 1 ( n ) Ξ 0 ( n ) | { 0 for | n α 2 | α for | n α 2 | α .
n A | Ψ A B = Ξ 1 ( n A ) D ^ ( α ) | 0 B + e i φ Ξ 0 ( n A ) D ^ ( α ) | 1 B ,
H ^ = 1 2 ω ( X ^ 2 + P ^ 2 ) = 1 2 ω ( X ^ 0 2 + 2 X 0 Δ X ^ + Δ X ^ 2 + Δ ^ P 2 ) ,
| Ψ A B = 1 2 2 [ D ^ ( α ) ( | 0 + | 1 ) A D ^ ( α ) ( | 0 + | 1 ) B + D ^ ( α ) ( | 0 | 1 ) A D ^ ( α ) ( | 0 | 1 ) B ] .
pr ( n A , n B ) = n A , n B | ρ ^ A B | n A , n B = η 2 | Ξ 0 ( n A ) Ξ 1 ( n B ) + e i φ Ξ 1 ( n A ) Ξ 0 ( n B ) | 2 + ( 1 η ) Ξ 0 2 ( n A ) Ξ 0 2 ( n B ) .
e α 2 α 2 n n ! e δ n 2 / 2 α 2 2 π α ,
pr ( n A , n B ) = e δ n A 2 + δ n B 2 2 α 2 2 π α 4 [ η 2 ( δ n A 2 + δ n B 2 + 2 cos φ δ n A δ n B ) + ( 1 η ) α 2 ] .
p r ( n A , n B ) = e n A 2 + n B 2 4 α 2 32 π α 4 [ η ( n A 2 + n B 2 + 2 cos φ n A n B ) + 4 ( 2 η ) α 2 ] .
n B = 4 α 2 n A η cos φ η ( n A 2 2 α 2 ) + 8 α 2 ;
Δ n B 2 = 2 α 2 ( 2 α 2 ( η + 4 ) + n A 2 η ) 2 α 2 ( 4 η ) + n A 2 η 16 α 4 n A 2 η 2 cos 2 φ ( 2 α 2 ( 4 η ) + n A 2 η ) 2 .
Δ n B 2 | n A = 0 Δ n B 2 | n A = 4 + η 4 η ,