Abstract

We propose a nonlocal scheme for preparing a distributed steady-state entanglement of two atoms trapped in separate optical cavities coupled through an optical fiber based on the combined effect of the unitary dynamics and dissipative process. In this scheme, only the qubit of one node is driven by an external classical field, while the other one does not need to be manipulated by an external field. This is meaningful for long distance quantum information processing tasks, and the experimental implementation is greatly simplified due to the unilateral manipulation on one node and the process of entanglement distribution can be avoided. This guarantees the absolute security of long distance quantum information processing tasks and makes the scheme more robust than that based on the unitary dynamics. We introduce the purity to characterize the mixture degree of the target steady-state. The steady entanglement can be obtained independent of the initial state. Furthermore, based on the dissipative entanglement preparation scheme, we construct a quantum teleportation setup with multiple nodes as a practical application, and the numerical simulation demonstrates the scheme can be realized effectively under the current experimental conditions..

© 2017 Optical Society of America

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    [Crossref] [PubMed]
  36. X. Y. Chen, L. T. Shen, Z. B. Yang, H. Z. Wu, and M. F. Chen, “Engineering W-type steady states for three atoms via dissipation in an optical cavity,” J. Opt. Soc. Am. B 29, 1535–1540 (2012).
    [Crossref]
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  57. T. Calarco, E. A. Hinds, D. Jaksch, J. Schmiedmayer, J. I. Cirac, and P. Zoller, “Quantum gates with neutral atoms: Controlling collisional interactions in time-dependent traps,” Phys. Rev. A 61, 022304 (2000).
    [Crossref]
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2015 (2)

S. L. Su, Q. Guo, H. F. Wang, and S. Zhang, “Simplified scheme for entanglement preparation with Rydberg pumping via dissipation,” Phys. Rev. A 92, 022328 (2015).
[Crossref]

S. L. Su, X. Q. Shao, Q. Guo, L. Y. Cheng, H.-F. Wang, and S. Zhang, “Preparation of entanglement between atoms in spatially separated cavities via fiber loss,” Eur. Phys. J. D 69, 123 (2015).
[Crossref]

2014 (4)

S. B. Zheng and L. T. Shen, “Generation and stabilization of maximal entanglement between two atomic qubits coupled to a decaying resonator,” J. Phys. B: At. Mol. Opt. Phys. 47, 055502 (2014).
[Crossref]

S. L. Su, X. Q. Shao, H. F. Wang, and S. Zhang, “Scheme for entanglement generation in an atom-cavity system via dissipation,” Phys. Rev. A 90, 054302 (2014).
[Crossref]

S. L. Su, X. Q. Shao, H. F. Wang, and S. Zhang, “Preparation of three-dimensional entanglement for distant atoms in coupled cavities via atomic spontaneous emission and cavity decay,” Sci. Rep. 4, 7566 (2014).
[Crossref] [PubMed]

L. T. Shen, X. Y. Chen, Z. B. Yang, H. Z. Wu, and S. B. Zheng, “Preparation of two-qubit steady entanglement through driving a single qubit,” Opt. Lett. 39, 6046 (2014).
[Crossref] [PubMed]

2013 (9)

A. Gonzalez-Tudela and D. Porras, “Mesoscopic entanglement induced by spontaneous emission in Solid-State quantum optics,” Phys. Rev. Lett. 110, 080502 (2013).
[Crossref] [PubMed]

S. Shankar, M. Hatridge, Z. Leghtas, K. M. Sliwa, A. Narla, U. Vool, S. M. Girvin, L Frunzio, M. Mirrahimi, and M. H. Devoret, “Autonomously stabilized entanglement between two superconducting quantum bits,” Nature 504, 419–422 (2013).
[Crossref] [PubMed]

E. G. Dalla Torre, J. Otterbach, E. Demler, V. Vuletic, and M. D. Lukin, “Dissipative preparation of spin squeezed atomic ensembles in a steady state,” Phys. Rev. Lett. 110, 120402 (2013).
[Crossref] [PubMed]

D. D. Bhaktavatsala Rao and K. Mølmer, “Dark entangled steady states of interacting rydberg atoms,” Phys. Rev. Lett. 111, 033606 (2013).
[Crossref]

A. W. Carr and M. Saffman, “Preparation of entangled and antiferromagnetic states by dissipative rydberg pumping,” Phys. Rev. Lett. 111, 033607 (2013).
[Crossref] [PubMed]

Y. Lin, J. P. Gaebler, F. Reiter, T. R. Tan, R. Bowler, A. S. Sørensen, D. Leibfried, and D. J. Wineland, “Dissipative production of a maximally entangled steady state of two quantum bits,” Nature 504, 415–418 (2013).
[Crossref] [PubMed]

F. Reiter, L. Tornberg, G. Johansson, and A. S. Sørensen, “Steady-state entanglement of two superconducting qubits engineered by dissipation,” Phys. Rev. A 88, 032317 (2013).
[Crossref]

R. Sweke, I. Sinayskiy, and F. Petruccione, “Dissipative preparation of large W states in optical cavities,” Phys. Rev. A 87, 042323 (2013).
[Crossref]

L. T. Shen, X. Y. Chen, Z. B. Yang, H. Z. Wu, and S. B. Zheng, “Cooling distant atoms into steady entanglement via coupled cavities,” Quantum Inf. Comput. 13, 281 (2013).

2012 (7)

L. T. Shen, X. Y. Chen, Z. B. Yang, H. Z. Wu, and S. B. Zheng, “Distributed entanglement induced by dissipative bosonic media,” Europhys. Lett. 99, 20003 (2012).
[Crossref]

F. Reiter, M. J. Kastoryano, and A. S. Sørensen, “Driving two atoms in an optical cavity into an entangled steady state using engineered decay,” New J. Phys. 14, 053022 (2012).
[Crossref]

M. Gullans, T. G. Tiecke, D. E. Chang, J. Feist, J. D. Thompson, J. I. Cirac, P. Zoller, and M. D. Lukin, “Nanoplasmonic lattices for ultracold atoms,” Phys. Rev. Lett. 109, 235309 (2012).
[Crossref]

P. B. Li, S. Y. Gao, H. R. Li, S. L. Ma, and F. L. Li, “Dissipative preparation of entangled states between two spatially separated nitrogen-vacancy centers,” Phys. Rev. A 85, 042306 (2012).
[Crossref]

W. A. Li and L. F. Wei, “Controllable entanglement preparations between atoms in spatially-separated cavities via quantum Zeno dynamics,” Opt. Express 20, 13440–13450 (2012).
[Crossref] [PubMed]

X. Y. Chen, L. T. Shen, Z. B. Yang, H. Z. Wu, and M. F. Chen, “Engineering W-type steady states for three atoms via dissipation in an optical cavity,” J. Opt. Soc. Am. B 29, 1535–1540 (2012).
[Crossref]

L. B. Chen, P. Shi, C. H. Zheng, and Y. J. Gu, “Generation of three-dimensional entangled state between a single atom and a Bose-Einstein condensate via adiabatic passage,” Opt. Express 20, 14547–14555 (2012).
[Crossref] [PubMed]

2011 (8)

J. Busch, S. De, S. S. Ivanov, B. T. Torosov, T. P. Spiller, and A. Beige, “Cooling atom-cavity systems into entangled states,” Phys. Rev. A 84, 022316 (2011).
[Crossref]

L. Memarzadeh and S. Mancini, “Stationary entanglement achievable by environment-induced chain links,” Phys. Rev. A 83, 042329 (2011)
[Crossref]

K. G. H. Vollbrecht, C. A. Muschik, and J. I. Cirac, “Entanglement distillation by dissipation and continuous quantum repeaters,” Phys. Rev. Lett. 107, 120502 (2011).
[Crossref] [PubMed]

M. J. Kastoryano, F. Reiter, and A. S. Sørensen, “Dissipative preparation of entanglement in optical cavities,” Phys. Rev. Lett. 106, 090502 (2011).
[Crossref] [PubMed]

L. T. Shen, X. Y. Chen, Z. B. Yang, H. Z. Wu, and S. B. Zheng, “Steady-state entanglement for distant atoms by dissipation in coupled cavities,” Phys. Rev. A 84, 064302 (2011).
[Crossref]

A. Gonzalez-Tudela, D. Martín-Cano, E. Moreno, L. MartínMoreno, C. Tejedor, and F. J. García-Vidal, “Entanglement of two qubits mediated by one-dimensional plasmonic waveguides,” Phys. Rev. Lett. 106, 020501 (2011).
[Crossref] [PubMed]

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, 080503 (2011).
[Crossref] [PubMed]

J. T. Barreiro, M. Müller, P. Schindler, D. Nigg, T. Monz, M. Chwalla, M. Hennrich, C. F. Roos, P. Zoller, and R. Blatt, “An open-system quantum simulator with trapped ions,” Nature 470, 486–491 (2011).
[Crossref] [PubMed]

2010 (1)

A. F. Alharbi and Z. Ficek, “Deterministic creation of stationary entangled states by dissipation,” Phys. Rev. A 82, 054103 (2010).
[Crossref]

2009 (3)

D. G. Angelakis, S. Bose, and S. Mancini, “Steady state entanglement between hybrid light-matter qubits,” Europhys. Lett. 85, 20007 (2009).
[Crossref]

F. Verstraete, M. M. Wolf, and J. I. Cirac, “Quantum computation, quantum state engineering, and quantum phase transitions driven by dissipation,” Nat. Phys. 5, 633 (2009).
[Crossref]

G. Vacanti and A. Beige, “Cooling atoms into entangled states,” New J. Phys. 11, 083008 (2009).
[Crossref]

2008 (3)

C. Horhammer and H. Buttner, “Environment-induced two-mode entanglement in quantum Brownian motion,” Phys. Rev. A 77, 042305 (2008).
[Crossref]

S. Diehl, A. Micheli, A. Kantian, B. Kraus, H. P. Büchler, and P. Zoller, “Quantum states and phases in driven open quantum systems with cold atoms,” Nat. Phys. 4, 878 (2008).
[Crossref]

X. Y. Lü, J. B. Liu, C. L. Ding, and J. H. Li, “Dispersive atom-field interaction scheme for three-dimensional entanglement between two spatially separated atoms,” Phys. Rev. A 78, 032305 (2008).
[Crossref]

2007 (3)

J. Song, Y. Xia, and H. S. Song, “Entangled state generation via adiabatic passage in two distant cavities,” J. Phys. B 40, 4503 (2007).
[Crossref]

J. Song, Y. Xia, H. S. Song, J. L. Guo, and J. Nie, “Quantum computation and entangled-state generation through adiabatic evolution in two distant cavities,” Europhys. Lett. 80, 60001 (2007).
[Crossref]

Z. Q. Yin and F. L. Li, “Multiatom and resonant interaction scheme for quantum state transfer and logical gates between two remote cavities via an optical fiber,” Phys. Rev. A 75, 012324 (2007).
[Crossref]

2006 (2)

A. Serafini, S. Mancini, and S. Bose, “Distributed quantum computation via optical fibers,” Phys. Rev. Lett. 96, 010503 (2006).
[Crossref] [PubMed]

F. Benatti and R. Floreanini, “Entangling oscillators through environment noise,” J. Phys. A 39, 2689 (2006).
[Crossref]

2003 (3)

F. Benatti, R. Floreanini, and M. Piani, “Environment induced entanglement in markovian dissipative dynamics,” Phys. Rev. Lett. 91, 070402 (2003).
[Crossref] [PubMed]

S. Clark, A. Peng, M. Gu, and S. Parkins, “unconditional preparation of entanglement between atoms in cascaded optical cavities,” Phys. Rev. Lett. 91, 177901 (2003).
[Crossref] [PubMed]

C. Simon and W. T. M. Irvine, “Robust long-distance entanglement and a loophole-free bell test with ions and photons,” Phys. Rev. Lett. 91, 110405 (2003).
[Crossref] [PubMed]

2002 (2)

E. Charron, E. Tiesinga, F. Mies, and C. Williams, “Optimizing a phase gate using quantum interference,” Phys. Rev. Lett. 88, 077901 (2002).
[Crossref] [PubMed]

D. Braun, “Creation of entanglement by interaction with a common heat bath,” Phys. Rev. Lett. 89, 277901 (2002).
[Crossref]

2000 (1)

T. Calarco, E. A. Hinds, D. Jaksch, J. Schmiedmayer, J. I. Cirac, and P. Zoller, “Quantum gates with neutral atoms: Controlling collisional interactions in time-dependent traps,” Phys. Rev. A 61, 022304 (2000).
[Crossref]

1999 (5)

D. Jakche, H.-J. Briegel, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Entanglement of atoms via Cold Controlled Collisions,” Phys. Rev. Lett. 82, 1975 (1999).
[Crossref]

G. K. Brennen, C. M. Caves, P. S. Jessen, and I. H. Deutsch, “Quantum logic gates in optical lattices,” Phys. Rev. Lett. 82, 1060 (1999).
[Crossref]

S. J. van Enk, H. J. Kimble, J. I. Cirac, and P. Zoller, “Quantum communication with dark photons,” Phys. Rev. A 59, 2659 (1999).
[Crossref]

M. B. Plenio, S. F. Huelga, A. Beige, and P. L. Knight, “Cavity-loss-induced generation of entangled atoms,” Phys. Rev. A 59, 2468 (1999).
[Crossref]

C. Cabrillo, J. I. Cirac, P. García-Fernández, and P. Zoller, “Creation of entangled states of distant atoms by interference,” Phys. Rev. A 59, 1025 (1999).
[Crossref]

1998 (2)

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold bosonic atoms in optical lattices,” Phys. Rev. Lett. 81, 3108 (1998).
[Crossref]

G. Brassard, S. L. Braunstein, and R. Cleve, “Teleportation as a quantum computation,” Physica D 120, 43 (1998).
[Crossref]

1997 (2)

J. I. Cirac, P. Zoller, H.J. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett. 78, 3221 (1997).
[Crossref]

T. Pellizzari, “Quantum networking with optical fibres,” Phys. Rev. Lett. 79, 5242 (1997).
[Crossref]

1993 (1)

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W.K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895 (1993).
[Crossref] [PubMed]

1969 (1)

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880 (1969).
[Crossref]

1964 (1)

J. S. Bell, “On the Einstein Podolsky Rosen paradox,” Physics (Long Island City, N. Y.) 1, 195 (1964).

1935 (2)

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777 (1935).
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E. Schrödinger, “Die gegenwärtige situation in der quantenmechanik,” Naturwissenschaften 23, 823 (1935).
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Alharbi, A. F.

A. F. Alharbi and Z. Ficek, “Deterministic creation of stationary entangled states by dissipation,” Phys. Rev. A 82, 054103 (2010).
[Crossref]

Angelakis, D. G.

D. G. Angelakis, S. Bose, and S. Mancini, “Steady state entanglement between hybrid light-matter qubits,” Europhys. Lett. 85, 20007 (2009).
[Crossref]

Barreiro, J. T.

J. T. Barreiro, M. Müller, P. Schindler, D. Nigg, T. Monz, M. Chwalla, M. Hennrich, C. F. Roos, P. Zoller, and R. Blatt, “An open-system quantum simulator with trapped ions,” Nature 470, 486–491 (2011).
[Crossref] [PubMed]

Beige, A.

J. Busch, S. De, S. S. Ivanov, B. T. Torosov, T. P. Spiller, and A. Beige, “Cooling atom-cavity systems into entangled states,” Phys. Rev. A 84, 022316 (2011).
[Crossref]

G. Vacanti and A. Beige, “Cooling atoms into entangled states,” New J. Phys. 11, 083008 (2009).
[Crossref]

M. B. Plenio, S. F. Huelga, A. Beige, and P. L. Knight, “Cavity-loss-induced generation of entangled atoms,” Phys. Rev. A 59, 2468 (1999).
[Crossref]

Bell, J. S.

J. S. Bell, “On the Einstein Podolsky Rosen paradox,” Physics (Long Island City, N. Y.) 1, 195 (1964).

Benatti, F.

F. Benatti and R. Floreanini, “Entangling oscillators through environment noise,” J. Phys. A 39, 2689 (2006).
[Crossref]

F. Benatti, R. Floreanini, and M. Piani, “Environment induced entanglement in markovian dissipative dynamics,” Phys. Rev. Lett. 91, 070402 (2003).
[Crossref] [PubMed]

Bennett, C. H.

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W.K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895 (1993).
[Crossref] [PubMed]

Bhaktavatsala Rao, D. D.

D. D. Bhaktavatsala Rao and K. Mølmer, “Dark entangled steady states of interacting rydberg atoms,” Phys. Rev. Lett. 111, 033606 (2013).
[Crossref]

Blatt, R.

J. T. Barreiro, M. Müller, P. Schindler, D. Nigg, T. Monz, M. Chwalla, M. Hennrich, C. F. Roos, P. Zoller, and R. Blatt, “An open-system quantum simulator with trapped ions,” Nature 470, 486–491 (2011).
[Crossref] [PubMed]

Bose, S.

D. G. Angelakis, S. Bose, and S. Mancini, “Steady state entanglement between hybrid light-matter qubits,” Europhys. Lett. 85, 20007 (2009).
[Crossref]

A. Serafini, S. Mancini, and S. Bose, “Distributed quantum computation via optical fibers,” Phys. Rev. Lett. 96, 010503 (2006).
[Crossref] [PubMed]

Bowler, R.

Y. Lin, J. P. Gaebler, F. Reiter, T. R. Tan, R. Bowler, A. S. Sørensen, D. Leibfried, and D. J. Wineland, “Dissipative production of a maximally entangled steady state of two quantum bits,” Nature 504, 415–418 (2013).
[Crossref] [PubMed]

Brassard, G.

G. Brassard, S. L. Braunstein, and R. Cleve, “Teleportation as a quantum computation,” Physica D 120, 43 (1998).
[Crossref]

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W.K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895 (1993).
[Crossref] [PubMed]

Braun, D.

D. Braun, “Creation of entanglement by interaction with a common heat bath,” Phys. Rev. Lett. 89, 277901 (2002).
[Crossref]

Braunstein, S. L.

G. Brassard, S. L. Braunstein, and R. Cleve, “Teleportation as a quantum computation,” Physica D 120, 43 (1998).
[Crossref]

Brennen, G. K.

G. K. Brennen, C. M. Caves, P. S. Jessen, and I. H. Deutsch, “Quantum logic gates in optical lattices,” Phys. Rev. Lett. 82, 1060 (1999).
[Crossref]

Briegel, H.-J.

D. Jakche, H.-J. Briegel, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Entanglement of atoms via Cold Controlled Collisions,” Phys. Rev. Lett. 82, 1975 (1999).
[Crossref]

Bruder, C.

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold bosonic atoms in optical lattices,” Phys. Rev. Lett. 81, 3108 (1998).
[Crossref]

Büchler, H. P.

S. Diehl, A. Micheli, A. Kantian, B. Kraus, H. P. Büchler, and P. Zoller, “Quantum states and phases in driven open quantum systems with cold atoms,” Nat. Phys. 4, 878 (2008).
[Crossref]

Busch, J.

J. Busch, S. De, S. S. Ivanov, B. T. Torosov, T. P. Spiller, and A. Beige, “Cooling atom-cavity systems into entangled states,” Phys. Rev. A 84, 022316 (2011).
[Crossref]

Buttner, H.

C. Horhammer and H. Buttner, “Environment-induced two-mode entanglement in quantum Brownian motion,” Phys. Rev. A 77, 042305 (2008).
[Crossref]

Cabrillo, C.

C. Cabrillo, J. I. Cirac, P. García-Fernández, and P. Zoller, “Creation of entangled states of distant atoms by interference,” Phys. Rev. A 59, 1025 (1999).
[Crossref]

Calarco, T.

T. Calarco, E. A. Hinds, D. Jaksch, J. Schmiedmayer, J. I. Cirac, and P. Zoller, “Quantum gates with neutral atoms: Controlling collisional interactions in time-dependent traps,” Phys. Rev. A 61, 022304 (2000).
[Crossref]

Carr, A. W.

A. W. Carr and M. Saffman, “Preparation of entangled and antiferromagnetic states by dissipative rydberg pumping,” Phys. Rev. Lett. 111, 033607 (2013).
[Crossref] [PubMed]

Caves, C. M.

G. K. Brennen, C. M. Caves, P. S. Jessen, and I. H. Deutsch, “Quantum logic gates in optical lattices,” Phys. Rev. Lett. 82, 1060 (1999).
[Crossref]

Chang, D. E.

M. Gullans, T. G. Tiecke, D. E. Chang, J. Feist, J. D. Thompson, J. I. Cirac, P. Zoller, and M. D. Lukin, “Nanoplasmonic lattices for ultracold atoms,” Phys. Rev. Lett. 109, 235309 (2012).
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Charron, E.

E. Charron, E. Tiesinga, F. Mies, and C. Williams, “Optimizing a phase gate using quantum interference,” Phys. Rev. Lett. 88, 077901 (2002).
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Chen, L. B.

Chen, M. F.

Chen, X. Y.

L. T. Shen, X. Y. Chen, Z. B. Yang, H. Z. Wu, and S. B. Zheng, “Preparation of two-qubit steady entanglement through driving a single qubit,” Opt. Lett. 39, 6046 (2014).
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L. T. Shen, X. Y. Chen, Z. B. Yang, H. Z. Wu, and S. B. Zheng, “Cooling distant atoms into steady entanglement via coupled cavities,” Quantum Inf. Comput. 13, 281 (2013).

X. Y. Chen, L. T. Shen, Z. B. Yang, H. Z. Wu, and M. F. Chen, “Engineering W-type steady states for three atoms via dissipation in an optical cavity,” J. Opt. Soc. Am. B 29, 1535–1540 (2012).
[Crossref]

L. T. Shen, X. Y. Chen, Z. B. Yang, H. Z. Wu, and S. B. Zheng, “Distributed entanglement induced by dissipative bosonic media,” Europhys. Lett. 99, 20003 (2012).
[Crossref]

L. T. Shen, X. Y. Chen, Z. B. Yang, H. Z. Wu, and S. B. Zheng, “Steady-state entanglement for distant atoms by dissipation in coupled cavities,” Phys. Rev. A 84, 064302 (2011).
[Crossref]

Cheng, L. Y.

S. L. Su, X. Q. Shao, Q. Guo, L. Y. Cheng, H.-F. Wang, and S. Zhang, “Preparation of entanglement between atoms in spatially separated cavities via fiber loss,” Eur. Phys. J. D 69, 123 (2015).
[Crossref]

Chwalla, M.

J. T. Barreiro, M. Müller, P. Schindler, D. Nigg, T. Monz, M. Chwalla, M. Hennrich, C. F. Roos, P. Zoller, and R. Blatt, “An open-system quantum simulator with trapped ions,” Nature 470, 486–491 (2011).
[Crossref] [PubMed]

Cirac, J. I.

M. Gullans, T. G. Tiecke, D. E. Chang, J. Feist, J. D. Thompson, J. I. Cirac, P. Zoller, and M. D. Lukin, “Nanoplasmonic lattices for ultracold atoms,” Phys. Rev. Lett. 109, 235309 (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, 080503 (2011).
[Crossref] [PubMed]

K. G. H. Vollbrecht, C. A. Muschik, and J. I. Cirac, “Entanglement distillation by dissipation and continuous quantum repeaters,” Phys. Rev. Lett. 107, 120502 (2011).
[Crossref] [PubMed]

F. Verstraete, M. M. Wolf, and J. I. Cirac, “Quantum computation, quantum state engineering, and quantum phase transitions driven by dissipation,” Nat. Phys. 5, 633 (2009).
[Crossref]

T. Calarco, E. A. Hinds, D. Jaksch, J. Schmiedmayer, J. I. Cirac, and P. Zoller, “Quantum gates with neutral atoms: Controlling collisional interactions in time-dependent traps,” Phys. Rev. A 61, 022304 (2000).
[Crossref]

D. Jakche, H.-J. Briegel, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Entanglement of atoms via Cold Controlled Collisions,” Phys. Rev. Lett. 82, 1975 (1999).
[Crossref]

C. Cabrillo, J. I. Cirac, P. García-Fernández, and P. Zoller, “Creation of entangled states of distant atoms by interference,” Phys. Rev. A 59, 1025 (1999).
[Crossref]

S. J. van Enk, H. J. Kimble, J. I. Cirac, and P. Zoller, “Quantum communication with dark photons,” Phys. Rev. A 59, 2659 (1999).
[Crossref]

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold bosonic atoms in optical lattices,” Phys. Rev. Lett. 81, 3108 (1998).
[Crossref]

J. I. Cirac, P. Zoller, H.J. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett. 78, 3221 (1997).
[Crossref]

Clark, S.

S. Clark, A. Peng, M. Gu, and S. Parkins, “unconditional preparation of entanglement between atoms in cascaded optical cavities,” Phys. Rev. Lett. 91, 177901 (2003).
[Crossref] [PubMed]

Clauser, J. F.

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880 (1969).
[Crossref]

Cleve, R.

G. Brassard, S. L. Braunstein, and R. Cleve, “Teleportation as a quantum computation,” Physica D 120, 43 (1998).
[Crossref]

Crépeau, C.

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W.K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895 (1993).
[Crossref] [PubMed]

Dalla Torre, E. G.

E. G. Dalla Torre, J. Otterbach, E. Demler, V. Vuletic, and M. D. Lukin, “Dissipative preparation of spin squeezed atomic ensembles in a steady state,” Phys. Rev. Lett. 110, 120402 (2013).
[Crossref] [PubMed]

De, S.

J. Busch, S. De, S. S. Ivanov, B. T. Torosov, T. P. Spiller, and A. Beige, “Cooling atom-cavity systems into entangled states,” Phys. Rev. A 84, 022316 (2011).
[Crossref]

Demler, E.

E. G. Dalla Torre, J. Otterbach, E. Demler, V. Vuletic, and M. D. Lukin, “Dissipative preparation of spin squeezed atomic ensembles in a steady state,” Phys. Rev. Lett. 110, 120402 (2013).
[Crossref] [PubMed]

Deutsch, I. H.

G. K. Brennen, C. M. Caves, P. S. Jessen, and I. H. Deutsch, “Quantum logic gates in optical lattices,” Phys. Rev. Lett. 82, 1060 (1999).
[Crossref]

Devoret, M. H.

S. Shankar, M. Hatridge, Z. Leghtas, K. M. Sliwa, A. Narla, U. Vool, S. M. Girvin, L Frunzio, M. Mirrahimi, and M. H. Devoret, “Autonomously stabilized entanglement between two superconducting quantum bits,” Nature 504, 419–422 (2013).
[Crossref] [PubMed]

Diehl, S.

S. Diehl, A. Micheli, A. Kantian, B. Kraus, H. P. Büchler, and P. Zoller, “Quantum states and phases in driven open quantum systems with cold atoms,” Nat. Phys. 4, 878 (2008).
[Crossref]

Ding, C. L.

X. Y. Lü, J. B. Liu, C. L. Ding, and J. H. Li, “Dispersive atom-field interaction scheme for three-dimensional entanglement between two spatially separated atoms,” Phys. Rev. A 78, 032305 (2008).
[Crossref]

Einstein, A.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777 (1935).
[Crossref]

Feist, J.

M. Gullans, T. G. Tiecke, D. E. Chang, J. Feist, J. D. Thompson, J. I. Cirac, P. Zoller, and M. D. Lukin, “Nanoplasmonic lattices for ultracold atoms,” Phys. Rev. Lett. 109, 235309 (2012).
[Crossref]

Ficek, Z.

A. F. Alharbi and Z. Ficek, “Deterministic creation of stationary entangled states by dissipation,” Phys. Rev. A 82, 054103 (2010).
[Crossref]

Floreanini, R.

F. Benatti and R. Floreanini, “Entangling oscillators through environment noise,” J. Phys. A 39, 2689 (2006).
[Crossref]

F. Benatti, R. Floreanini, and M. Piani, “Environment induced entanglement in markovian dissipative dynamics,” Phys. Rev. Lett. 91, 070402 (2003).
[Crossref] [PubMed]

Frunzio, L

S. Shankar, M. Hatridge, Z. Leghtas, K. M. Sliwa, A. Narla, U. Vool, S. M. Girvin, L Frunzio, M. Mirrahimi, and M. H. Devoret, “Autonomously stabilized entanglement between two superconducting quantum bits,” Nature 504, 419–422 (2013).
[Crossref] [PubMed]

Gaebler, J. P.

Y. Lin, J. P. Gaebler, F. Reiter, T. R. Tan, R. Bowler, A. S. Sørensen, D. Leibfried, and D. J. Wineland, “Dissipative production of a maximally entangled steady state of two quantum bits,” Nature 504, 415–418 (2013).
[Crossref] [PubMed]

Gao, S. Y.

P. B. Li, S. Y. Gao, H. R. Li, S. L. Ma, and F. L. Li, “Dissipative preparation of entangled states between two spatially separated nitrogen-vacancy centers,” Phys. Rev. A 85, 042306 (2012).
[Crossref]

García-Fernández, P.

C. Cabrillo, J. I. Cirac, P. García-Fernández, and P. Zoller, “Creation of entangled states of distant atoms by interference,” Phys. Rev. A 59, 1025 (1999).
[Crossref]

García-Vidal, F. J.

A. Gonzalez-Tudela, D. Martín-Cano, E. Moreno, L. MartínMoreno, C. Tejedor, and F. J. García-Vidal, “Entanglement of two qubits mediated by one-dimensional plasmonic waveguides,” Phys. Rev. Lett. 106, 020501 (2011).
[Crossref] [PubMed]

Gardiner, C. W.

D. Jakche, H.-J. Briegel, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Entanglement of atoms via Cold Controlled Collisions,” Phys. Rev. Lett. 82, 1975 (1999).
[Crossref]

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold bosonic atoms in optical lattices,” Phys. Rev. Lett. 81, 3108 (1998).
[Crossref]

Girvin, S. M.

S. Shankar, M. Hatridge, Z. Leghtas, K. M. Sliwa, A. Narla, U. Vool, S. M. Girvin, L Frunzio, M. Mirrahimi, and M. H. Devoret, “Autonomously stabilized entanglement between two superconducting quantum bits,” Nature 504, 419–422 (2013).
[Crossref] [PubMed]

Gonzalez-Tudela, A.

A. Gonzalez-Tudela and D. Porras, “Mesoscopic entanglement induced by spontaneous emission in Solid-State quantum optics,” Phys. Rev. Lett. 110, 080502 (2013).
[Crossref] [PubMed]

A. Gonzalez-Tudela, D. Martín-Cano, E. Moreno, L. MartínMoreno, C. Tejedor, and F. J. García-Vidal, “Entanglement of two qubits mediated by one-dimensional plasmonic waveguides,” Phys. Rev. Lett. 106, 020501 (2011).
[Crossref] [PubMed]

Gu, M.

S. Clark, A. Peng, M. Gu, and S. Parkins, “unconditional preparation of entanglement between atoms in cascaded optical cavities,” Phys. Rev. Lett. 91, 177901 (2003).
[Crossref] [PubMed]

Gu, Y. J.

Gullans, M.

M. Gullans, T. G. Tiecke, D. E. Chang, J. Feist, J. D. Thompson, J. I. Cirac, P. Zoller, and M. D. Lukin, “Nanoplasmonic lattices for ultracold atoms,” Phys. Rev. Lett. 109, 235309 (2012).
[Crossref]

Guo, J. L.

J. Song, Y. Xia, H. S. Song, J. L. Guo, and J. Nie, “Quantum computation and entangled-state generation through adiabatic evolution in two distant cavities,” Europhys. Lett. 80, 60001 (2007).
[Crossref]

Guo, Q.

S. L. Su, X. Q. Shao, Q. Guo, L. Y. Cheng, H.-F. Wang, and S. Zhang, “Preparation of entanglement between atoms in spatially separated cavities via fiber loss,” Eur. Phys. J. D 69, 123 (2015).
[Crossref]

S. L. Su, Q. Guo, H. F. Wang, and S. Zhang, “Simplified scheme for entanglement preparation with Rydberg pumping via dissipation,” Phys. Rev. A 92, 022328 (2015).
[Crossref]

Hatridge, M.

S. Shankar, M. Hatridge, Z. Leghtas, K. M. Sliwa, A. Narla, U. Vool, S. M. Girvin, L Frunzio, M. Mirrahimi, and M. H. Devoret, “Autonomously stabilized entanglement between two superconducting quantum bits,” Nature 504, 419–422 (2013).
[Crossref] [PubMed]

Hennrich, M.

J. T. Barreiro, M. Müller, P. Schindler, D. Nigg, T. Monz, M. Chwalla, M. Hennrich, C. F. Roos, P. Zoller, and R. Blatt, “An open-system quantum simulator with trapped ions,” Nature 470, 486–491 (2011).
[Crossref] [PubMed]

Hinds, E. A.

T. Calarco, E. A. Hinds, D. Jaksch, J. Schmiedmayer, J. I. Cirac, and P. Zoller, “Quantum gates with neutral atoms: Controlling collisional interactions in time-dependent traps,” Phys. Rev. A 61, 022304 (2000).
[Crossref]

Holt, R. A.

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880 (1969).
[Crossref]

Horhammer, C.

C. Horhammer and H. Buttner, “Environment-induced two-mode entanglement in quantum Brownian motion,” Phys. Rev. A 77, 042305 (2008).
[Crossref]

Horne, M. A.

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880 (1969).
[Crossref]

Huelga, S. F.

M. B. Plenio, S. F. Huelga, A. Beige, and P. L. Knight, “Cavity-loss-induced generation of entangled atoms,” Phys. Rev. A 59, 2468 (1999).
[Crossref]

Irvine, W. T. M.

C. Simon and W. T. M. Irvine, “Robust long-distance entanglement and a loophole-free bell test with ions and photons,” Phys. Rev. Lett. 91, 110405 (2003).
[Crossref] [PubMed]

Ivanov, S. S.

J. Busch, S. De, S. S. Ivanov, B. T. Torosov, T. P. Spiller, and A. Beige, “Cooling atom-cavity systems into entangled states,” Phys. Rev. A 84, 022316 (2011).
[Crossref]

Jakche, D.

D. Jakche, H.-J. Briegel, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Entanglement of atoms via Cold Controlled Collisions,” Phys. Rev. Lett. 82, 1975 (1999).
[Crossref]

Jaksch, D.

T. Calarco, E. A. Hinds, D. Jaksch, J. Schmiedmayer, J. I. Cirac, and P. Zoller, “Quantum gates with neutral atoms: Controlling collisional interactions in time-dependent traps,” Phys. Rev. A 61, 022304 (2000).
[Crossref]

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold bosonic atoms in optical lattices,” Phys. Rev. Lett. 81, 3108 (1998).
[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, 080503 (2011).
[Crossref] [PubMed]

Jessen, P. S.

G. K. Brennen, C. M. Caves, P. S. Jessen, and I. H. Deutsch, “Quantum logic gates in optical lattices,” Phys. Rev. Lett. 82, 1060 (1999).
[Crossref]

Johansson, G.

F. Reiter, L. Tornberg, G. Johansson, and A. S. Sørensen, “Steady-state entanglement of two superconducting qubits engineered by dissipation,” Phys. Rev. A 88, 032317 (2013).
[Crossref]

Jozsa, R.

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W.K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895 (1993).
[Crossref] [PubMed]

Kantian, A.

S. Diehl, A. Micheli, A. Kantian, B. Kraus, H. P. Büchler, and P. Zoller, “Quantum states and phases in driven open quantum systems with cold atoms,” Nat. Phys. 4, 878 (2008).
[Crossref]

Kastoryano, M. J.

F. Reiter, M. J. Kastoryano, and A. S. Sørensen, “Driving two atoms in an optical cavity into an entangled steady state using engineered decay,” New J. Phys. 14, 053022 (2012).
[Crossref]

M. J. Kastoryano, F. Reiter, and A. S. Sørensen, “Dissipative preparation of entanglement in optical cavities,” Phys. Rev. Lett. 106, 090502 (2011).
[Crossref] [PubMed]

Kimble, H. J.

S. J. van Enk, H. J. Kimble, J. I. Cirac, and P. Zoller, “Quantum communication with dark photons,” Phys. Rev. A 59, 2659 (1999).
[Crossref]

Kimble, H.J.

J. I. Cirac, P. Zoller, H.J. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett. 78, 3221 (1997).
[Crossref]

Knight, P. L.

M. B. Plenio, S. F. Huelga, A. Beige, and P. L. Knight, “Cavity-loss-induced generation of entangled atoms,” Phys. Rev. A 59, 2468 (1999).
[Crossref]

Kraus, B.

S. Diehl, A. Micheli, A. Kantian, B. Kraus, H. P. Büchler, and P. Zoller, “Quantum states and phases in driven open quantum systems with cold atoms,” Nat. Phys. 4, 878 (2008).
[Crossref]

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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, 080503 (2011).
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Figures (8)

Fig. 1
Fig. 1 Experimental setup and level diagram of the atoms. Two atoms resonantly interact with quantized field, respectively. The first atom is driven by two classical fields. γ, κ, and β denote the atomic spontaneous emission rate, cavity decay rate and fiber loss rate, respectively.

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Fig. 2
Fig. 2 Schematic diagram for the dressed state of the atom-cavity-fiber coupling system.

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Fig. 3
Fig. 3 Processes for producing and stabilizing Bell state. The interaction between system and environment is characterized by the photon loss and atomic spontaneous emission with the rates κ and γ, respectively.

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Fig. 4
Fig. 4 The fidelity of states |Φ0〉 and | Φ 1 0 〉 versus the dimensionless parameter gt for the initial state |Φ0〉 by solving the full master equation. In (a) the experimental parameters are chosen as Ω1 = 0.080g, Ω2 = 0.035g, ν = 20g. And the dissipative factors are chosen as γ/g = 2.72 × 10−4, κ1 = κ2 = 2.8 × 10−2g, β = 1.5 × 10−2g. The inset of (b) is plotted with the optimized parameters: Ω1 = 0.025g, Ω2 = 0.045g and ν = 20g. And the dissipative factors: γ = 1.75 × 10−5g, κ1 = 1.6 × 10−2g, κ2 = 2.4 × 10−2g and β = 0.67 × 10−2g.

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Fig. 5
Fig. 5 The CHSH correlation and the purity of the qubit steady-state as a function of the time with the experimental parameters(the bule solid curve) and optimized parameters(the red solid curve), which are chosen as the same as Fig. 4.

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Fig. 6
Fig. 6 The fidelity (a) and purity (b) of the target steady-state as a function of the parameters Ω1/g and Ω2/g with the initial state |Φ0〉 at the time 1×104/g. The parameters are chosen as ν = 20g. (c) The fidelity of the desired state versus ν and evolution time with the initial state |Φ0〉. The parameters are chosen as Ω1 = 0.080g, Ω2 = 0.035g. Figures (a), (b) and (c) are plotted with the dissipative factors γ = 2.72×10−4g, κ1 = κ2 = 2.8×10−2g, β = 1.5 × 10−2g. (d) The fidelity of the target steady-state as a function of cavity leakage rate κ and fiber loss rate β with the initial state |Φ0〉 at the time 1×104/g. The parameters are chosen as ν = 20g, Ω1 = 0.080g, Ω2 = 0.035g, and qubit spontaneous emission rate γ = 2.72 × 10−4g.

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Fig. 7
Fig. 7 Schematic diagram for implementation of quantum teleportation scheme with multiple nodes. The information of unknown qubit can be teleported from the first node to the nth node. The dashed box denotes the first node to teleport an unknown quantum state from Alice to Bob. The dotted boxes means that two qubits belong to the same participant. The grey box in the bottom left is a quantum circuit of teleportation for the first node. Here H represents a Hadamard operation, σx, σz are the Pauli operators representing local qubit-flip operation, and I is the identity operator.

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Fig. 8
Fig. 8 Fidelity of teleportation scheme with multiple nodes as a function of node number n.

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Tables (2)

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Table 1 The eigenstates and eigenenergies of the Hamiltonian H0 + Hc,f + Ha,c within the zero and single excitation subspaces.

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Table 2 The eigenstates and eigenenergies of the Hamiltonian H0 + Hc,f + Ha,c within the two-excitation subspace.

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Equations (15)

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H 0 = ∑ i = 1 , 2 ω 0 | e i 〉 〈 e i | + ∑ j = A , B ω a a j † a j + ω b b † b ,
H c , f = ν ( b a A † + b a B † ) + H . c . ,
H a , c = g ( S 1 † a A + S 2 † a B ) + H . c . ,
H cl = ∑ k = 1 , 2 Ω k e − i ω k t S 1 † + H . c . ,
c = 2 2 ( a A − a B ) , c 1 = 1 2 ( a A + a B + 2 b ) , c 2 = 1 2 ( a A + a B − 2 b ) ,
H a , c = 1 2 g ( e − i 2 ν t c 1 + e i 2 ν t c 2 + 2 c ) S 1 † + 1 2 g ( e − i 2 ν t c 1 + e i 2 ν t c 2 − 2 c ) S 2 † + H . c . ,
H a , c = 2 2 g ( S 1 † − S 2 † ) c + H . c . .
ρ ^ ˙ = i [ ρ ^ , H ] + 1 2 ∑ j [ 2 L ^ j ρ ^ L ^ j † − ( L ^ j † L ^ j ρ ^ + ρ ^ L ^ j † L ^ j ) ] ,
| Φ 0 〉 = | g 1 , g 2 〉 | 0 , 0 , 0 〉 ,                   | Φ 1 0 〉 = | ϕ + 〉 | 0 , 0 , 0 〉 , | Ψ 1 − 〉 = | g 1 , g 2 〉 | 0 , 0 , 1 〉 ,                   | Ψ 1 + 〉 = | g 1 , g 2 〉 | 0 , 1 , 0 〉 , | Φ 1 ± 〉 = 1 2 [ | ϕ − 〉 | 0 , 0 , 0 〉 ± | g 1 , g 2 〉 | 1 , 0 , 0 〉 ] ,
| Ψ 2 − 〉 = | ϕ + 〉 | 0 , 0 , 1 〉 ,                     | Ψ 2 + 〉 = | ϕ + 〉 | 0 , 1 , 0 〉 , | Ψ 2 , ± 1 〉 = 1 2 [ | ϕ − 〉 | 0 , 0 , 1 〉 ± | g 1 , g 2 〉 | 1 , 0 , 1 〉 ] , | Ψ 2 , ± 2 〉 = 1 2 [ | ϕ − 〉 | 0 , 1 , 0 〉 ± | g 1 , g 2 〉 | 1 , 1 , 0 〉 ] , | Φ 2 0 〉 = | ϕ + 〉 | 1 , 0 , 0 〉 ,             | Φ 2 1 〉 = | g 1 , g 2 〉 | 0 , 1 , 1 〉 , | Φ 2 2 〉 = 1 3 ( | g 1 , g 2 〉 | 2 , 0 , 0 〉 + 2 | e 1 , e 2 〉 ) | 0 , 0 , 0 〉 , | Φ 2 ± 〉 = 1 2 [ ( | ϕ − 〉 ) | 1 , 0 , 0 〉 ± 1 3 ( 2 | g 1 , g 2 〉 | 2 , 0 , 0 〉                             − | e 1 , e 2 〉 | 0 , 0 , 0 〉 ) ] .
H cl = ∑ k = 1 , 2 [ 1 2 Ω k e − i ω k t ( | Φ 1 − 〉 + | Φ 1 † 〉 ) 〈 Φ 0 | + 1 2 Ω k e − i ω k t | Φ 1 0 〉 〈 Φ 0 | + 1 2 Ω k e − i ω k t | Φ 2 − 〉 〈 Φ 1 − | + 1 2 Ω k e − i ω k t ( | Φ 2 , − 1 〉 + | Φ 2 , + 1 〉 ) 〈 Φ 1 − | + 1 2 Ω k e − i ω k t | Φ 2 + 〉 〈 Φ 1 + | + 1 2 Ω k e − i ω k t ( | Φ 2 , − 2 〉 + | Φ 2 , + 2 〉 ) 〈 Φ 1 + | − ( 1 6 Ω k e − i ω k t | Φ 2 2 〉 + 1 2 Ω k e − i ω k t | Φ 2 0 〉 ) ( 〈 Φ 1 − | + 〈 Φ 1 + | ) + 2 4 Ω k e − i ω k t ( | Φ 2 − 〉 + | Φ 2 + 〉 ) ( 〈 Φ 1 − | − 〈 Φ 1 + | ) + 1 3 Ω k e − i ω k t | Φ 2 2 〉 〈 Φ 1 0 | + H . c . ] .
S ( t ) = Tr [ ( 𝒪 CHSH ) ρ ( t ) ] ,
𝒪 CHSH = σ y , 1 ⊗ − σ y , 2 − σ x , 2 2 + σ x , 1 ⊗ − σ y , 2 − σ x , 2 2 + σ x , 1 ⊗ σ y , 2 − σ x , 2 2 − σ y , 1 ⊗ σ y , 2 − σ x , 2 2 .
𝒫 ( t ) = Tr [ ρ ^ ( t ) 2 ] .
F T = 〈 φ | 2 I 2   〈 01 | a , 1 ρ ^ T | 01 〉 a , 1 I 2 | φ 〉 2 + 〈 φ | 2 σ x 2   〈 00 | a , 1 ρ ^ T | 00 〉 a , 1     σ x 2 † | φ 〉 2 + 〈 φ | 2 σ z 2     〈 11 | a , 1 ρ ^ T | 11 〉 a , 1     σ z 2 † | φ 2 〉 + 〈 φ | 2 σ x 2 σ z 2     〈 10 | a , 1 ρ ^ T | 10 〉 a , 1     σ x 2 † σ z 2 † | φ 〉 2 = 0.9415 ,

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