Abstract

In this paper, a scheme for the generation of long-living entanglement between two distant Λ-type three-level atoms separately trapped in two dissipative cavities is proposed. In this scheme, two dissipative cavities are coupled to their own non-Markovian environments and two three-level atoms are driven by the classical fields. The entangled state between the two atoms is produced by performing Bell state measurement (BSM) on photons leaving the dissipative cavities. Using the time-dependent Schördinger equation, we obtain the analytical results for the evolution of the entanglement. It is revealed that, by manipulating the detunings of classical field, the long-living stationary entanglement between two atoms can be generated in the presence of dissipation.

© 2017 Optical Society of America

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    [Crossref]
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    [Crossref] [PubMed]
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  32. X. Zou and W. Mathis, “One-step implementation of maximally entangled states of many three-level atoms in microwave cavity QED,” Phys. Rev. A 70(3), 035802 (2004).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
  41. X. S. Ma, S. Zotter, J. Kofler, R. Ursin, T. Jennewein, Č. Brukner, and A. Zeilinger, “Experimental delayed-choice entanglement swapping,” Nature 8(6), 479–484 (2012).

2016 (5)

F. Galve, L. A. Pachón, and D. Zueco, “Ultrafast optimal sideband cooling under non-Markovian evolution,” Phys. Rev. Lett. 116(18), 183602 (2016).
[Crossref]

A. Nourmandipour, M. K. Tavassoly, and M. Rafiee, “Dynamics and protection of entanglement in n-qubit systems within Markovian and non-Markovian environments,” Phys. Rev. A 93(2), 022327 (2016).
[Crossref]

A. Nourmandipour and M. K. Tavassoly, “Entanglement swapping between dissipative systems,” Phys. Rev. A 94(2), 022339 (2016).
[Crossref]

A. Nourmandipour, M. K. Tavassoly, and S. Mancini, “The entangling power of a glocal dissipative map,” Quantum Inf. Comput. 16(11), 0969–0981 (2016).

J. Song, Z. J. Zhang, Y. Xia, X. D. Sun, and Y. Y. Jiang, “Fast coherent manipulation of quantum states in open systems,” Opt. Express 24(19), 21674–21683 (2016).
[Crossref] [PubMed]

2015 (3)

A. Nourmandipour and M. K. Tavassoly, “Dynamics and protecting of entanglement in two-level systems interacting with a dissipative cavity: the Gardiner-Collett approach,” J. Phys. B: At. Mol. Opt. Phys. 48(16), 165502 (2015).
[Crossref]

R. Fischer, I. Vidal, D. Gilboa, R. R. Correia, A. C. Ribeiro-Teixeira, S. D. Prado, and Y. Silberberg, “Light with tunable non-Markovian phase imprint,” Phys. Rev. Lett. 115(7), 073901 (2015).
[Crossref] [PubMed]

A. F. Estrada, L. A. Pachón, and L. A. Pachón, “Quantum limit for driven linear non-Markovian open-quantum-systems,” New J. Phys. 17(3), 033038 (2015).
[Crossref]

2014 (1)

J. Cerrillo and J. Cao, “Non-Markovian dynamical maps: numerical processing of open quantum trajectories,” Phys. Rev. Lett. 112(11), 110401 (2014).
[Crossref] [PubMed]

2012 (1)

X. S. Ma, S. Zotter, J. Kofler, R. Ursin, T. Jennewein, Č. Brukner, and A. Zeilinger, “Experimental delayed-choice entanglement swapping,” Nature 8(6), 479–484 (2012).

2010 (2)

J. F. Triana, A. F. Estrada, and L. A. Pachón, “Bringing entanglement to the high temperature limit,” Phys. Rev. Lett. 105(18), 180501 (2010).
[Crossref]

M. Dukalski and Y. M. Blanter, “Periodic revival of entanglement of two strongly driven qubits in a dissipative cavity,” Phys. Rev. A 82(5), 052330 (2010);
[Crossref]

2009 (3)

H. Tan, H. Xia H, and G. Li, “Interference-induced enhancement of field entanglement from an intracavity three-level V-type atom,” Phys. Rev. A 79(6), 063805 (2009).
[Crossref]

R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81(2), 865 (2009).
[Crossref]

W. Wieczorek, R. Krischek, N. Kiesel, P. Michelberger, G. Tóth, and H. Weinfurter, “Experimental entanglement of a six-photon symmetric Dicke state,” Phys. Rev. Lett. 103(2), 020504 (2009).
[Crossref] [PubMed]

2008 (1)

S. Y. Ye, Z. R. Zhong, and S. B. Zheng, “Deterministic generation of three-dimensional entanglement for two atoms separately trapped in two optical cavities,” Phys. Rev. A 77(1), 014303 (2008).
[Crossref]

2006 (4)

M. F. Santos, P. Milman, L. Davidovich, and N. Zagury, “Direct measurement of finite-time disentanglement induced by a reservoir,” Phys. Rev. B 73(4), 040305 (2006).
[Crossref]

T. Yu and J. H. Eberly, “Quantum open system theory: bipartite aspects,” Phys. Rev. Lett. 97(14), 140403 (2006).
[Crossref] [PubMed]

F. Benatti and R. Floreanini, “Entangling oscillators through environment noise,” J. Phys. A 39(11), 2689 (2006).
[Crossref]

S. Oh and J. Kim, “Entanglement between qubits induced by a common environment with a gap,” Phys. Rev. B 73(6), 062306 (2006).
[Crossref]

2005 (1)

Ö. Çakir, H.T. Dung, L. Knöll, and D. G. Welsch, “Generation of long-living entanglement between two separate three-level atoms,” Phys. Rev. A 71(3), 032326 (2005).
[Crossref]

2004 (4)

X. Zou and W. Mathis, “One-step implementation of maximally entangled states of many three-level atoms in microwave cavity QED,” Phys. Rev. A 70(3), 035802 (2004).
[Crossref]

T. Yu and J. H. Eberly, “Finite-time disentanglement via spontaneous emission,” Phys. Rev. Lett. 93(14), 140404 (2004).
[Crossref] [PubMed]

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431(7005), 162–167 (2004).
[Crossref] [PubMed]

J. Johnson, J. Canning, T. Kaneko, J. K. Pru, and J. L. Tilly, “Germline stem cells and follicular renewal in the postnatal mammalian ovary,” Nature 428(6979), 145–150 (2004).
[Crossref] [PubMed]

2003 (3)

T. Yu and J. H. Eberly, “Qubit disentanglement and decoherence via dephasing,” Phys. Rev. B 68(16), 165322 (2003).
[Crossref]

F. Benatti, R. Floreanini, and M. Piani, “Environment induced entanglement in Markovian dissipative dynamics,” Phys. Rev. Lett. 91(7), 070402 (2003).
[Crossref] [PubMed]

X. Zou, K. Pahlke, and W. Mathis, “Generation of an entangled state of two three-level atoms in cavity QED,” Phys. Rev. A 67(4), 044301 (2003).
[Crossref]

2002 (2)

D. Bruss and C. Macchiavello, “Optimal eavesdropping in cryptography with three-dimensional quantum states,” Phys. Rev. Lett. 88(12), 127901 (2002).
[Crossref] [PubMed]

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

2001 (3)

M. Bourennane, A. Karlsson, and G. Björk, “Quantum key distribution using multilevel encoding,” Phys. Rev. A 64(1), 012306 (2001).
[Crossref]

R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett. 86(22), 5188 (2001).
[Crossref] [PubMed]

G. R. Guthohriein, M. Keller, K. Hayasaka, W. Lange, and H. Walther, “A single ion as a nanoscopic probe of an optical field,” Nature 414(6859), 256–259 (2001).

2000 (3)

C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, and C. Monroe, “Experimental entanglement of four particles,” Nature 404(6775), 256–259 (2000).
[Crossref] [PubMed]

S. B. Zheng and G. C. Guo, “Efficient scheme for two-atom entanglement and quantum information processing in cavity QED,” Phys. Rev. Lett. 85(11), 2392 (2000).
[Crossref] [PubMed]

D. Kaszlikowski, P. Gnaciński, M. Żukowski, W. Miklaszewski, and A. Zeilinger, “Violations of local realism by two entangled N-dimensional systems are stronger than for two qubits,” Phys. Rev. Lett. 85(21), 4418 (2000).
[Crossref] [PubMed]

1998 (1)

Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, and D. J. Wineland, “Deterministic entanglement of two trapped ions,” Phys. Rev. Lett. 81(17), 3631 (1998).
[Crossref]

1996 (1)

K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, “Dense coding in experimental quantum communication,” Phys. Rev. Lett. 76(25), 4656 (1996).
[Crossref] [PubMed]

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(13), 1895 (1993).
[Crossref] [PubMed]

1991 (1)

K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67(6), 661 (1991).
[Crossref] [PubMed]

1981 (1)

A. Aspect, P. Grangier, and G. Roger, “Experimental tests of realistic local theories via Bell’s theorem,” Phys. Rev. Lett. 47(7), 460 (1981).
[Crossref]

Aspect, A.

A. Aspect, P. Grangier, and G. Roger, “Experimental tests of realistic local theories via Bell’s theorem,” Phys. Rev. Lett. 47(7), 460 (1981).
[Crossref]

Benatti, F.

F. Benatti and R. Floreanini, “Entangling oscillators through environment noise,” J. Phys. A 39(11), 2689 (2006).
[Crossref]

F. Benatti, R. Floreanini, and M. Piani, “Environment induced entanglement in Markovian dissipative dynamics,” Phys. Rev. Lett. 91(7), 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(13), 1895 (1993).
[Crossref] [PubMed]

Björk, G.

M. Bourennane, A. Karlsson, and G. Björk, “Quantum key distribution using multilevel encoding,” Phys. Rev. A 64(1), 012306 (2001).
[Crossref]

Blais, A.

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431(7005), 162–167 (2004).
[Crossref] [PubMed]

Blanter, Y. M.

M. Dukalski and Y. M. Blanter, “Periodic revival of entanglement of two strongly driven qubits in a dissipative cavity,” Phys. Rev. A 82(5), 052330 (2010);
[Crossref]

Bourennane, M.

M. Bourennane, A. Karlsson, and G. Björk, “Quantum key distribution using multilevel encoding,” Phys. Rev. A 64(1), 012306 (2001).
[Crossref]

Brassard, G.

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(13), 1895 (1993).
[Crossref] [PubMed]

Briegel, H. J.

R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett. 86(22), 5188 (2001).
[Crossref] [PubMed]

Brukner, C.

X. S. Ma, S. Zotter, J. Kofler, R. Ursin, T. Jennewein, Č. Brukner, and A. Zeilinger, “Experimental delayed-choice entanglement swapping,” Nature 8(6), 479–484 (2012).

Bruss, D.

D. Bruss and C. Macchiavello, “Optimal eavesdropping in cryptography with three-dimensional quantum states,” Phys. Rev. Lett. 88(12), 127901 (2002).
[Crossref] [PubMed]

Çakir, Ö.

Ö. Çakir, H.T. Dung, L. Knöll, and D. G. Welsch, “Generation of long-living entanglement between two separate three-level atoms,” Phys. Rev. A 71(3), 032326 (2005).
[Crossref]

Canning, J.

J. Johnson, J. Canning, T. Kaneko, J. K. Pru, and J. L. Tilly, “Germline stem cells and follicular renewal in the postnatal mammalian ovary,” Nature 428(6979), 145–150 (2004).
[Crossref] [PubMed]

Cao, J.

J. Cerrillo and J. Cao, “Non-Markovian dynamical maps: numerical processing of open quantum trajectories,” Phys. Rev. Lett. 112(11), 110401 (2014).
[Crossref] [PubMed]

Cerrillo, J.

J. Cerrillo and J. Cao, “Non-Markovian dynamical maps: numerical processing of open quantum trajectories,” Phys. Rev. Lett. 112(11), 110401 (2014).
[Crossref] [PubMed]

Correia, R. R.

R. Fischer, I. Vidal, D. Gilboa, R. R. Correia, A. C. Ribeiro-Teixeira, S. D. Prado, and Y. Silberberg, “Light with tunable non-Markovian phase imprint,” Phys. Rev. Lett. 115(7), 073901 (2015).
[Crossref] [PubMed]

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(13), 1895 (1993).
[Crossref] [PubMed]

Davidovich, L.

M. F. Santos, P. Milman, L. Davidovich, and N. Zagury, “Direct measurement of finite-time disentanglement induced by a reservoir,” Phys. Rev. B 73(4), 040305 (2006).
[Crossref]

Dukalski, M.

M. Dukalski and Y. M. Blanter, “Periodic revival of entanglement of two strongly driven qubits in a dissipative cavity,” Phys. Rev. A 82(5), 052330 (2010);
[Crossref]

Dung, H.T.

Ö. Çakir, H.T. Dung, L. Knöll, and D. G. Welsch, “Generation of long-living entanglement between two separate three-level atoms,” Phys. Rev. A 71(3), 032326 (2005).
[Crossref]

Eberly, J. H.

T. Yu and J. H. Eberly, “Quantum open system theory: bipartite aspects,” Phys. Rev. Lett. 97(14), 140403 (2006).
[Crossref] [PubMed]

T. Yu and J. H. Eberly, “Finite-time disentanglement via spontaneous emission,” Phys. Rev. Lett. 93(14), 140404 (2004).
[Crossref] [PubMed]

T. Yu and J. H. Eberly, “Qubit disentanglement and decoherence via dephasing,” Phys. Rev. B 68(16), 165322 (2003).
[Crossref]

Ekert, K.

K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67(6), 661 (1991).
[Crossref] [PubMed]

Estrada, A. F.

A. F. Estrada, L. A. Pachón, and L. A. Pachón, “Quantum limit for driven linear non-Markovian open-quantum-systems,” New J. Phys. 17(3), 033038 (2015).
[Crossref]

J. F. Triana, A. F. Estrada, and L. A. Pachón, “Bringing entanglement to the high temperature limit,” Phys. Rev. Lett. 105(18), 180501 (2010).
[Crossref]

Fischer, R.

R. Fischer, I. Vidal, D. Gilboa, R. R. Correia, A. C. Ribeiro-Teixeira, S. D. Prado, and Y. Silberberg, “Light with tunable non-Markovian phase imprint,” Phys. Rev. Lett. 115(7), 073901 (2015).
[Crossref] [PubMed]

Floreanini, R.

F. Benatti and R. Floreanini, “Entangling oscillators through environment noise,” J. Phys. A 39(11), 2689 (2006).
[Crossref]

F. Benatti, R. Floreanini, and M. Piani, “Environment induced entanglement in Markovian dissipative dynamics,” Phys. Rev. Lett. 91(7), 070402 (2003).
[Crossref] [PubMed]

Frunzio, L.

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431(7005), 162–167 (2004).
[Crossref] [PubMed]

Galve, F.

F. Galve, L. A. Pachón, and D. Zueco, “Ultrafast optimal sideband cooling under non-Markovian evolution,” Phys. Rev. Lett. 116(18), 183602 (2016).
[Crossref]

Gilboa, D.

R. Fischer, I. Vidal, D. Gilboa, R. R. Correia, A. C. Ribeiro-Teixeira, S. D. Prado, and Y. Silberberg, “Light with tunable non-Markovian phase imprint,” Phys. Rev. Lett. 115(7), 073901 (2015).
[Crossref] [PubMed]

Gnacinski, P.

D. Kaszlikowski, P. Gnaciński, M. Żukowski, W. Miklaszewski, and A. Zeilinger, “Violations of local realism by two entangled N-dimensional systems are stronger than for two qubits,” Phys. Rev. Lett. 85(21), 4418 (2000).
[Crossref] [PubMed]

Grangier, P.

A. Aspect, P. Grangier, and G. Roger, “Experimental tests of realistic local theories via Bell’s theorem,” Phys. Rev. Lett. 47(7), 460 (1981).
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S. B. Zheng and G. C. Guo, “Efficient scheme for two-atom entanglement and quantum information processing in cavity QED,” Phys. Rev. Lett. 85(11), 2392 (2000).
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G. R. Guthohriein, M. Keller, K. Hayasaka, W. Lange, and H. Walther, “A single ion as a nanoscopic probe of an optical field,” Nature 414(6859), 256–259 (2001).

Hayasaka, K.

G. R. Guthohriein, M. Keller, K. Hayasaka, W. Lange, and H. Walther, “A single ion as a nanoscopic probe of an optical field,” Nature 414(6859), 256–259 (2001).

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R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81(2), 865 (2009).
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A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431(7005), 162–167 (2004).
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Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, and D. J. Wineland, “Deterministic entanglement of two trapped ions,” Phys. Rev. Lett. 81(17), 3631 (1998).
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X. S. Ma, S. Zotter, J. Kofler, R. Ursin, T. Jennewein, Č. Brukner, and A. Zeilinger, “Experimental delayed-choice entanglement swapping,” Nature 8(6), 479–484 (2012).

Jiang, Y. Y.

Johnson, J.

J. Johnson, J. Canning, T. Kaneko, J. K. Pru, and J. L. Tilly, “Germline stem cells and follicular renewal in the postnatal mammalian ovary,” Nature 428(6979), 145–150 (2004).
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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(13), 1895 (1993).
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J. Johnson, J. Canning, T. Kaneko, J. K. Pru, and J. L. Tilly, “Germline stem cells and follicular renewal in the postnatal mammalian ovary,” Nature 428(6979), 145–150 (2004).
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D. Kaszlikowski, P. Gnaciński, M. Żukowski, W. Miklaszewski, and A. Zeilinger, “Violations of local realism by two entangled N-dimensional systems are stronger than for two qubits,” Phys. Rev. Lett. 85(21), 4418 (2000).
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Keller, M.

G. R. Guthohriein, M. Keller, K. Hayasaka, W. Lange, and H. Walther, “A single ion as a nanoscopic probe of an optical field,” Nature 414(6859), 256–259 (2001).

Kielpinski, D.

C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, and C. Monroe, “Experimental entanglement of four particles,” Nature 404(6775), 256–259 (2000).
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Kiesel, N.

W. Wieczorek, R. Krischek, N. Kiesel, P. Michelberger, G. Tóth, and H. Weinfurter, “Experimental entanglement of a six-photon symmetric Dicke state,” Phys. Rev. Lett. 103(2), 020504 (2009).
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Kim, J.

S. Oh and J. Kim, “Entanglement between qubits induced by a common environment with a gap,” Phys. Rev. B 73(6), 062306 (2006).
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C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, and C. Monroe, “Experimental entanglement of four particles,” Nature 404(6775), 256–259 (2000).
[Crossref] [PubMed]

Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, and D. J. Wineland, “Deterministic entanglement of two trapped ions,” Phys. Rev. Lett. 81(17), 3631 (1998).
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Ö. Çakir, H.T. Dung, L. Knöll, and D. G. Welsch, “Generation of long-living entanglement between two separate three-level atoms,” Phys. Rev. A 71(3), 032326 (2005).
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Kofler, J.

X. S. Ma, S. Zotter, J. Kofler, R. Ursin, T. Jennewein, Č. Brukner, and A. Zeilinger, “Experimental delayed-choice entanglement swapping,” Nature 8(6), 479–484 (2012).

Krischek, R.

W. Wieczorek, R. Krischek, N. Kiesel, P. Michelberger, G. Tóth, and H. Weinfurter, “Experimental entanglement of a six-photon symmetric Dicke state,” Phys. Rev. Lett. 103(2), 020504 (2009).
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Kwiat, P. G.

K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, “Dense coding in experimental quantum communication,” Phys. Rev. Lett. 76(25), 4656 (1996).
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Lange, W.

G. R. Guthohriein, M. Keller, K. Hayasaka, W. Lange, and H. Walther, “A single ion as a nanoscopic probe of an optical field,” Nature 414(6859), 256–259 (2001).

Langer, C.

C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, and C. Monroe, “Experimental entanglement of four particles,” Nature 404(6775), 256–259 (2000).
[Crossref] [PubMed]

Leibfried, D.

Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, and D. J. Wineland, “Deterministic entanglement of two trapped ions,” Phys. Rev. Lett. 81(17), 3631 (1998).
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Li, G.

H. Tan, H. Xia H, and G. Li, “Interference-induced enhancement of field entanglement from an intracavity three-level V-type atom,” Phys. Rev. A 79(6), 063805 (2009).
[Crossref]

Ma, X. S.

X. S. Ma, S. Zotter, J. Kofler, R. Ursin, T. Jennewein, Č. Brukner, and A. Zeilinger, “Experimental delayed-choice entanglement swapping,” Nature 8(6), 479–484 (2012).

Macchiavello, C.

D. Bruss and C. Macchiavello, “Optimal eavesdropping in cryptography with three-dimensional quantum states,” Phys. Rev. Lett. 88(12), 127901 (2002).
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A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431(7005), 162–167 (2004).
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Mancini, S.

A. Nourmandipour, M. K. Tavassoly, and S. Mancini, “The entangling power of a glocal dissipative map,” Quantum Inf. Comput. 16(11), 0969–0981 (2016).

Mathis, W.

X. Zou and W. Mathis, “One-step implementation of maximally entangled states of many three-level atoms in microwave cavity QED,” Phys. Rev. A 70(3), 035802 (2004).
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X. Zou, K. Pahlke, and W. Mathis, “Generation of an entangled state of two three-level atoms in cavity QED,” Phys. Rev. A 67(4), 044301 (2003).
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Mattle, K.

K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, “Dense coding in experimental quantum communication,” Phys. Rev. Lett. 76(25), 4656 (1996).
[Crossref] [PubMed]

Meyer, V.

C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, and C. Monroe, “Experimental entanglement of four particles,” Nature 404(6775), 256–259 (2000).
[Crossref] [PubMed]

Michelberger, P.

W. Wieczorek, R. Krischek, N. Kiesel, P. Michelberger, G. Tóth, and H. Weinfurter, “Experimental entanglement of a six-photon symmetric Dicke state,” Phys. Rev. Lett. 103(2), 020504 (2009).
[Crossref] [PubMed]

Miklaszewski, W.

D. Kaszlikowski, P. Gnaciński, M. Żukowski, W. Miklaszewski, and A. Zeilinger, “Violations of local realism by two entangled N-dimensional systems are stronger than for two qubits,” Phys. Rev. Lett. 85(21), 4418 (2000).
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Milman, P.

M. F. Santos, P. Milman, L. Davidovich, and N. Zagury, “Direct measurement of finite-time disentanglement induced by a reservoir,” Phys. Rev. B 73(4), 040305 (2006).
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Monroe, C.

C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, and C. Monroe, “Experimental entanglement of four particles,” Nature 404(6775), 256–259 (2000).
[Crossref] [PubMed]

Myatt, C. J.

C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, and C. Monroe, “Experimental entanglement of four particles,” Nature 404(6775), 256–259 (2000).
[Crossref] [PubMed]

Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, and D. J. Wineland, “Deterministic entanglement of two trapped ions,” Phys. Rev. Lett. 81(17), 3631 (1998).
[Crossref]

Nourmandipour, A.

A. Nourmandipour, M. K. Tavassoly, and M. Rafiee, “Dynamics and protection of entanglement in n-qubit systems within Markovian and non-Markovian environments,” Phys. Rev. A 93(2), 022327 (2016).
[Crossref]

A. Nourmandipour and M. K. Tavassoly, “Entanglement swapping between dissipative systems,” Phys. Rev. A 94(2), 022339 (2016).
[Crossref]

A. Nourmandipour, M. K. Tavassoly, and S. Mancini, “The entangling power of a glocal dissipative map,” Quantum Inf. Comput. 16(11), 0969–0981 (2016).

A. Nourmandipour and M. K. Tavassoly, “Dynamics and protecting of entanglement in two-level systems interacting with a dissipative cavity: the Gardiner-Collett approach,” J. Phys. B: At. Mol. Opt. Phys. 48(16), 165502 (2015).
[Crossref]

Oh, S.

S. Oh and J. Kim, “Entanglement between qubits induced by a common environment with a gap,” Phys. Rev. B 73(6), 062306 (2006).
[Crossref]

Pachón, L. A.

F. Galve, L. A. Pachón, and D. Zueco, “Ultrafast optimal sideband cooling under non-Markovian evolution,” Phys. Rev. Lett. 116(18), 183602 (2016).
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A. F. Estrada, L. A. Pachón, and L. A. Pachón, “Quantum limit for driven linear non-Markovian open-quantum-systems,” New J. Phys. 17(3), 033038 (2015).
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J. F. Triana, A. F. Estrada, and L. A. Pachón, “Bringing entanglement to the high temperature limit,” Phys. Rev. Lett. 105(18), 180501 (2010).
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X. Zou, K. Pahlke, and W. Mathis, “Generation of an entangled state of two three-level atoms in cavity QED,” Phys. Rev. A 67(4), 044301 (2003).
[Crossref]

Peres, A.

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(13), 1895 (1993).
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F. Benatti, R. Floreanini, and M. Piani, “Environment induced entanglement in Markovian dissipative dynamics,” Phys. Rev. Lett. 91(7), 070402 (2003).
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Prado, S. D.

R. Fischer, I. Vidal, D. Gilboa, R. R. Correia, A. C. Ribeiro-Teixeira, S. D. Prado, and Y. Silberberg, “Light with tunable non-Markovian phase imprint,” Phys. Rev. Lett. 115(7), 073901 (2015).
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Pru, J. K.

J. Johnson, J. Canning, T. Kaneko, J. K. Pru, and J. L. Tilly, “Germline stem cells and follicular renewal in the postnatal mammalian ovary,” Nature 428(6979), 145–150 (2004).
[Crossref] [PubMed]

Rafiee, M.

A. Nourmandipour, M. K. Tavassoly, and M. Rafiee, “Dynamics and protection of entanglement in n-qubit systems within Markovian and non-Markovian environments,” Phys. Rev. A 93(2), 022327 (2016).
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R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett. 86(22), 5188 (2001).
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R. Fischer, I. Vidal, D. Gilboa, R. R. Correia, A. C. Ribeiro-Teixeira, S. D. Prado, and Y. Silberberg, “Light with tunable non-Markovian phase imprint,” Phys. Rev. Lett. 115(7), 073901 (2015).
[Crossref] [PubMed]

Roger, G.

A. Aspect, P. Grangier, and G. Roger, “Experimental tests of realistic local theories via Bell’s theorem,” Phys. Rev. Lett. 47(7), 460 (1981).
[Crossref]

Sackett, C. A.

C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, and C. Monroe, “Experimental entanglement of four particles,” Nature 404(6775), 256–259 (2000).
[Crossref] [PubMed]

Santos, M. F.

M. F. Santos, P. Milman, L. Davidovich, and N. Zagury, “Direct measurement of finite-time disentanglement induced by a reservoir,” Phys. Rev. B 73(4), 040305 (2006).
[Crossref]

Schoelkopf, R. J.

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431(7005), 162–167 (2004).
[Crossref] [PubMed]

Schuster, D. I.

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431(7005), 162–167 (2004).
[Crossref] [PubMed]

Silberberg, Y.

R. Fischer, I. Vidal, D. Gilboa, R. R. Correia, A. C. Ribeiro-Teixeira, S. D. Prado, and Y. Silberberg, “Light with tunable non-Markovian phase imprint,” Phys. Rev. Lett. 115(7), 073901 (2015).
[Crossref] [PubMed]

Song, J.

Sun, X. D.

Tan, H.

H. Tan, H. Xia H, and G. Li, “Interference-induced enhancement of field entanglement from an intracavity three-level V-type atom,” Phys. Rev. A 79(6), 063805 (2009).
[Crossref]

Tavassoly, M. K.

A. Nourmandipour, M. K. Tavassoly, and S. Mancini, “The entangling power of a glocal dissipative map,” Quantum Inf. Comput. 16(11), 0969–0981 (2016).

A. Nourmandipour and M. K. Tavassoly, “Entanglement swapping between dissipative systems,” Phys. Rev. A 94(2), 022339 (2016).
[Crossref]

A. Nourmandipour, M. K. Tavassoly, and M. Rafiee, “Dynamics and protection of entanglement in n-qubit systems within Markovian and non-Markovian environments,” Phys. Rev. A 93(2), 022327 (2016).
[Crossref]

A. Nourmandipour and M. K. Tavassoly, “Dynamics and protecting of entanglement in two-level systems interacting with a dissipative cavity: the Gardiner-Collett approach,” J. Phys. B: At. Mol. Opt. Phys. 48(16), 165502 (2015).
[Crossref]

Tilly, J. L.

J. Johnson, J. Canning, T. Kaneko, J. K. Pru, and J. L. Tilly, “Germline stem cells and follicular renewal in the postnatal mammalian ovary,” Nature 428(6979), 145–150 (2004).
[Crossref] [PubMed]

Tóth, G.

W. Wieczorek, R. Krischek, N. Kiesel, P. Michelberger, G. Tóth, and H. Weinfurter, “Experimental entanglement of a six-photon symmetric Dicke state,” Phys. Rev. Lett. 103(2), 020504 (2009).
[Crossref] [PubMed]

Triana, J. F.

J. F. Triana, A. F. Estrada, and L. A. Pachón, “Bringing entanglement to the high temperature limit,” Phys. Rev. Lett. 105(18), 180501 (2010).
[Crossref]

Turchette, Q. A.

Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, and D. J. Wineland, “Deterministic entanglement of two trapped ions,” Phys. Rev. Lett. 81(17), 3631 (1998).
[Crossref]

Ursin, R.

X. S. Ma, S. Zotter, J. Kofler, R. Ursin, T. Jennewein, Č. Brukner, and A. Zeilinger, “Experimental delayed-choice entanglement swapping,” Nature 8(6), 479–484 (2012).

Vidal, G.

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

Vidal, I.

R. Fischer, I. Vidal, D. Gilboa, R. R. Correia, A. C. Ribeiro-Teixeira, S. D. Prado, and Y. Silberberg, “Light with tunable non-Markovian phase imprint,” Phys. Rev. Lett. 115(7), 073901 (2015).
[Crossref] [PubMed]

Wallraff, A.

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431(7005), 162–167 (2004).
[Crossref] [PubMed]

Walther, H.

G. R. Guthohriein, M. Keller, K. Hayasaka, W. Lange, and H. Walther, “A single ion as a nanoscopic probe of an optical field,” Nature 414(6859), 256–259 (2001).

Weinfurter, H.

W. Wieczorek, R. Krischek, N. Kiesel, P. Michelberger, G. Tóth, and H. Weinfurter, “Experimental entanglement of a six-photon symmetric Dicke state,” Phys. Rev. Lett. 103(2), 020504 (2009).
[Crossref] [PubMed]

K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, “Dense coding in experimental quantum communication,” Phys. Rev. Lett. 76(25), 4656 (1996).
[Crossref] [PubMed]

Welsch, D. G.

Ö. Çakir, H.T. Dung, L. Knöll, and D. G. Welsch, “Generation of long-living entanglement between two separate three-level atoms,” Phys. Rev. A 71(3), 032326 (2005).
[Crossref]

Werner, R. F.

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

Wieczorek, W.

W. Wieczorek, R. Krischek, N. Kiesel, P. Michelberger, G. Tóth, and H. Weinfurter, “Experimental entanglement of a six-photon symmetric Dicke state,” Phys. Rev. Lett. 103(2), 020504 (2009).
[Crossref] [PubMed]

Wineland, D. J.

Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, and D. J. Wineland, “Deterministic entanglement of two trapped ions,” Phys. Rev. Lett. 81(17), 3631 (1998).
[Crossref]

Wood, C. S.

Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, and D. J. Wineland, “Deterministic entanglement of two trapped ions,” Phys. Rev. Lett. 81(17), 3631 (1998).
[Crossref]

Wootters, W. K.

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(13), 1895 (1993).
[Crossref] [PubMed]

Xia, Y.

Xia H, H.

H. Tan, H. Xia H, and G. Li, “Interference-induced enhancement of field entanglement from an intracavity three-level V-type atom,” Phys. Rev. A 79(6), 063805 (2009).
[Crossref]

Ye, S. Y.

S. Y. Ye, Z. R. Zhong, and S. B. Zheng, “Deterministic generation of three-dimensional entanglement for two atoms separately trapped in two optical cavities,” Phys. Rev. A 77(1), 014303 (2008).
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Yu, T.

T. Yu and J. H. Eberly, “Quantum open system theory: bipartite aspects,” Phys. Rev. Lett. 97(14), 140403 (2006).
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T. Yu and J. H. Eberly, “Finite-time disentanglement via spontaneous emission,” Phys. Rev. Lett. 93(14), 140404 (2004).
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T. Yu and J. H. Eberly, “Qubit disentanglement and decoherence via dephasing,” Phys. Rev. B 68(16), 165322 (2003).
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Zagury, N.

M. F. Santos, P. Milman, L. Davidovich, and N. Zagury, “Direct measurement of finite-time disentanglement induced by a reservoir,” Phys. Rev. B 73(4), 040305 (2006).
[Crossref]

Zeilinger, A.

X. S. Ma, S. Zotter, J. Kofler, R. Ursin, T. Jennewein, Č. Brukner, and A. Zeilinger, “Experimental delayed-choice entanglement swapping,” Nature 8(6), 479–484 (2012).

D. Kaszlikowski, P. Gnaciński, M. Żukowski, W. Miklaszewski, and A. Zeilinger, “Violations of local realism by two entangled N-dimensional systems are stronger than for two qubits,” Phys. Rev. Lett. 85(21), 4418 (2000).
[Crossref] [PubMed]

K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, “Dense coding in experimental quantum communication,” Phys. Rev. Lett. 76(25), 4656 (1996).
[Crossref] [PubMed]

Zhang, Z. J.

Zheng, S. B.

S. Y. Ye, Z. R. Zhong, and S. B. Zheng, “Deterministic generation of three-dimensional entanglement for two atoms separately trapped in two optical cavities,” Phys. Rev. A 77(1), 014303 (2008).
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S. B. Zheng and G. C. Guo, “Efficient scheme for two-atom entanglement and quantum information processing in cavity QED,” Phys. Rev. Lett. 85(11), 2392 (2000).
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Zhong, Z. R.

S. Y. Ye, Z. R. Zhong, and S. B. Zheng, “Deterministic generation of three-dimensional entanglement for two atoms separately trapped in two optical cavities,” Phys. Rev. A 77(1), 014303 (2008).
[Crossref]

Zotter, S.

X. S. Ma, S. Zotter, J. Kofler, R. Ursin, T. Jennewein, Č. Brukner, and A. Zeilinger, “Experimental delayed-choice entanglement swapping,” Nature 8(6), 479–484 (2012).

Zou, X.

X. Zou and W. Mathis, “One-step implementation of maximally entangled states of many three-level atoms in microwave cavity QED,” Phys. Rev. A 70(3), 035802 (2004).
[Crossref]

X. Zou, K. Pahlke, and W. Mathis, “Generation of an entangled state of two three-level atoms in cavity QED,” Phys. Rev. A 67(4), 044301 (2003).
[Crossref]

Zueco, D.

F. Galve, L. A. Pachón, and D. Zueco, “Ultrafast optimal sideband cooling under non-Markovian evolution,” Phys. Rev. Lett. 116(18), 183602 (2016).
[Crossref]

Zukowski, M.

D. Kaszlikowski, P. Gnaciński, M. Żukowski, W. Miklaszewski, and A. Zeilinger, “Violations of local realism by two entangled N-dimensional systems are stronger than for two qubits,” Phys. Rev. Lett. 85(21), 4418 (2000).
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J. Phys. A (1)

F. Benatti and R. Floreanini, “Entangling oscillators through environment noise,” J. Phys. A 39(11), 2689 (2006).
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J. Phys. B: At. Mol. Opt. Phys. (1)

A. Nourmandipour and M. K. Tavassoly, “Dynamics and protecting of entanglement in two-level systems interacting with a dissipative cavity: the Gardiner-Collett approach,” J. Phys. B: At. Mol. Opt. Phys. 48(16), 165502 (2015).
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Nature (5)

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Figures (7)

Fig. 1
Fig. 1 Schematic representation of the setup. BSM is performing Bell state measurement on photons leaving the cavities.
Fig. 2
Fig. 2 The evolution of the average linear entropy as a function of the scaled time τ = κt for different detunings in non-Markovian environments: Δ = 0, Δ l = 0 (orange curve), Δ = 15κ, Δ l = 0 (green curve), Δ = 0, Δ l = −15κ (red curve), and Δ = 15κ, Δ l = −15κ (blue curve). Other parameters: g = Ω = 10κ.
Fig. 3
Fig. 3 (a) The evolution of the populations of the states |e〉, |f〉, and |g〉 as a function of the scaled time τ = κt for the initial state |f〉 : the population of the state |e〉 (blue curve), the population of the state |f〉 (red curve), and the population of the state |e〉 (green curve). (b) The evolution of the populations of the state |g〉 as a function of the scaled time τ = κt for different detunings: Δ = 0, Δ l = 0 (orange curve), Δ = 15κ, Δ l = 0 (green curve), Δ = 0, Δ l = −15κ (red curve), and Δ = 15κ, Δ l = −15κ (blue curve). Other common parameters: Δ = Δ l = 0, g = Ω = 10κ, and θ = φ = 0.
Fig. 4
Fig. 4 The linear entropy of the atom-field at the scaled time τ = 15κt, as a function of detunings Δ and Δ l for initial atomic state |f〉. Other parameters: g = Ω = 10κ.
Fig. 5
Fig. 5 The evolution of the linear entropy between the atom and the cavity field over the scaled time τ = gt for different detunings in Markovian and non-Markovian environments: Δ = Δ l = 0 (orange curve), Δ = 1.5g, Δ l = 0 (green curve), Δ = 0, Δ l = −1.5g (red curve), and Δ = 1.5g, Δ l = −1.5g (blue curve). The dashed and solid lines denote Markovian and non-Markovian environments, respectively. Other parameters: θ = φ = 0.
Fig. 6
Fig. 6 (a) The evolution of the average negativity as a function of the scaled time τ = κt for different detunings in non-Markovian environments: Δ = Δ l = 0 (orange curve), Δ = 15κ, Δ l = 0 (green curve), Δ = 0, Δ l = −15κ (red curve), and Δ = 15κ, Δ l = −15κ (blue curve). (b) Density matrix of the two atoms at τ = 1κt: Δ1 = Δ2 = 15κ, Δ l 1 = Δ l 2 = −15κ, and θ1 = θ2 = φ1 = φ2 = 0. Other common parameters: g = Ω = 10κ.
Fig. 7
Fig. 7 The evolution of the negativity between the two atoms as a function of the scaled time τ = κt for different initial atomic states in (a) non-Markovian environments and (b) Markovian environments: θ1 = θ2 = 0, φ1 = φ2 = 0 (blue curve), θ1 = π/2, θ2 = 0, φ1 = φ2 = 0 (red curve), θ1 = π/2, θ2 = π/4, φ1 = φ2 = 0 (green curve), and θ1 = π/2, θ2 = π/4, φ1 = π, φ2 = 0 (orange curve). Other parameters: (a) Δ = 15κ, Δ l = −15κ and g = Ω = 10κ. (b) Δ = 0.15κ, Δ l = −0.15κ and g = Ω = 0.1κ.

Equations (29)

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H c i = ω c i a i a i + 0 B i ( η ) B i ( η ) d η + 0 G i ( η ) [ a i B i ( η ) + H . c . ] d η ,
H c i = ω A i ( ω ) A i ( ω ) d ω .
a i = α i * ( ω ) A i ( ω ) d ω ,
α i ( ω ) = κ i / π ω ω c i + i κ i ,
H i = ω A i ( ω ) A i ( ω ) d ω + ω e i | e i e i | + ω f i | f i f i | + ω g i | g i g i | + g i [ α i * ( ω ) A i ( ω ) | e i g i | + H . c . ] d ω + Ω i [ | e i f i | e i ω l i t + H . c . ] ,
H I i = g [ α * ( ω ) A ( ω ) | e i g i | e i ( ω e ω g ω ) t + H . c . ] d ω + Ω [ | e i f i | e i Δ l t + H . c . ] ,
| ψ ( 0 ) i = [ cos ( θ i / 2 ) | f i + sin ( θ i / 2 ) e i φ i | g i ] | 0 i ,
| ψ ( t ) i = [ E i ( t ) | e i + F i ( t ) | f i + G i ( t ) | g i ] | 0 i + U i ( t , ω ) | g i | 1 ω i d ω ,
E ˙ i ( t ) = i g α * ( ω ) e i ( ω e ω g ω ) t U i ( ω , t ) d ω i Ω e i Δ l t F i ( t )
F ˙ i ( t ) = i Ω e i Δ l t E i ( t )
G ˙ i ( t ) = 0
U ˙ i ( t ) = i g α ( ω ) e i ( ω e ω g ω ) t E i ( t )
E ˙ i ( t ) = 0 t f ( t t 1 ) E i ( t 1 ) d t 1 Ω 2 0 t e i Δ l ( t t 2 ) E i ( t 2 ) d t 2 i Ω F i ( 0 ) e i Δ l t ,
J ( ω ) = g 2 | α ( ω ) | 2 = 1 π g 2 κ ( ω ω c ) 2 + κ 2 .
f ( t t 1 ) = g 2 e ( κ + i Δ ) ( t t 1 ) ,
E ˙ i ( t ) = g 2 0 t e ( κ + i Δ ) ( t t 2 ) E i ( t 1 ) d t 1 Ω 2 0 t e i Δ l ( t t 1 ) E i ( t 2 ) d t 2 i Ω F i ( 0 ) e i Δ l t .
E i ( t ) = F i ( 0 ) k = 1 k = 3 c k e s k t ,
S A ( θ , φ , t ) = 1 Tr ( ρ A 2 ) ,
ρ A = ( | E i ( t ) | 2 E i ( t ) F i * ( t ) E i ( t ) G i * ( t ) F i ( t ) E i * ( t ) | F i ( t ) | 2 F i ( t ) G i * ( t ) G i ( t ) E i * ( t ) G i ( t ) F i * ( t ) 1 | E i ( t ) | 2 | F i ( t ) | 2 )
S A av ( t ) = 1 4 π S A ( θ , φ , t ) sin ( θ ) d θ d φ .
| ψ ( t ) = | ψ 1 ( t ) | ψ 2 ( t ) .
| Ψ + = 1 2 ( | 0 1 | 1 2 | 1 1 | 0 2 ) ,
| ψ A A ( t ) = Ψ + | ψ ( t ) = 1 N ( t ) [ X 12 ( t ) | e , g X 21 ( t ) | g , e + Y 12 ( t ) | f , g Y 21 ( t ) | g , f + ( Z 12 ( t ) Z 21 ( t ) | g , g ) ]
N ( t ) = | X 12 ( t ) | 2 + | X 21 ( t ) | 2 + | Y 12 ( t ) | 2 + | Y 21 ( t ) | 2 + | Z 12 ( t ) Z 21 ( t ) | 2 .
X i j ( t ) = E i ( t ) U j ( ω , t ) Θ * ( ω ) d ω ,
Y i j ( t ) = F i ( t ) U j ( ω , t ) Θ * ( ω ) d ω ,
Z i j ( t ) = G i ( t ) U j ( ω , t ) Θ * ( ω ) d ω .
N ( ρ ( t ) ) = ρ T A 1 2 ,
N a v ( ρ ( t ) ) = 1 16 π 2 N ( ρ ( t ) ) k = 1 2 sin ( θ k ) d θ k d φ k .

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