Abstract

We develop a one-step scheme for generating multiparticle entangled states between two cold atomic clouds in distant cavities coupled by an optical fiber. We show that, through suitably choosing the intensities and detunings of the fields and precisely tuning the time evolution of the system, multiparticle entanglement between the separated atomic clouds can be engineered deterministically, in which quantum manipulations are insensitive to the states of the cavity and losses of the fiber. The experimental feasibility of this scheme is analyzed based on recent experimental advances in the realization of strong coupling between cold 87Rb clouds and fiber-based cavity. This scheme may open up promising perspectives for implementing quantum communication and networking with coupled cavities connected by optical fibers.

© 2011 OSA

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  1. J. S. Bell, “On the einstein-podolsky-rosen paradox,” Phys. 1, 195–200 (1964).
  2. D. M. Greenberger, M.A. Horne, A. Shimony, and A. Zeilinger, “Bell’s theorem without inequalities,” Am. J. Phys. 58, 1131–1143 (1990).
    [Crossref]
  3. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge2000).
  4. W. Dur, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,” Phys. Rev. A 62, 062314 (2000).
    [Crossref]
  5. H. J. Briegel and R. Raussendorf, “Persistent entanglement in arrays of interacting particles,” Phys. Rev. Lett. 86, 910–913 (2001).
    [Crossref] [PubMed]
  6. For a review see, K. Hammerer, A. S. Sorensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82, 1041–1093 (2010), and references therein.
    [Crossref]
  7. For a review see, L.-M. Duan and C. Monroe, “Colloquium: Quantum networks with trapped ions,” Rev. Mod. Phys. 82, 1209–1224 (2010), and references therein.
    [Crossref]
  8. R. Blatt and D. Wineland, “Entangled states of trapped atomic ions,” Nature (London)453, 1008–1015 (2008).
    [Crossref]
  9. For a review see, D. Jaksch and P. Zoller, “The cold atom Hubbard toolbox,” Ann. Phys. 315, 52–79 (2005), and references therein.
    [Crossref]
  10. H. J. Kimble, “Strong interactions of single atoms and photons in cavity QED,” Phys. Scr. T76, 127–137 (1998).
    [Crossref]
  11. H. Mabuchi and A. C. Doherty, “Cavity quantum electrodynamics: coherence in Context,” Science 298, 1372–1377 (2002).
    [Crossref] [PubMed]
  12. 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, 3221–3224 (1997).
    [Crossref]
  13. A. D. Boozer, A. Boca, R. Miller, T. E. Northup, and H. J. Kimble, “Reversible state transfer between light and a single trapped atom,” Phys. Rev. Lett. 98, 193601 (2007).
  14. E. Solano, G. S. Agarwal, and H. Walther, “Strong-driving-assisted multipartite entanglement in cavity QED,” Phys. Rev. Lett. 90, 027903 (2003).
    [Crossref] [PubMed]
  15. P.-B. Li, Y. Gu, Q.-H. Gong, and G.-C. Guo, “Quantum-information transfer in a coupled resonator waveguide,” Phys. Rev. A 79, 042339 (2009).
    [Crossref]
  16. F. Mei, M. Feng, Y.-F. Yu, and Z.-M. Zhang, “Scalable quantum information processing with atomic ensembles and flying photons,” Phys. Rev. A 80, 042319 (2009).
    [Crossref]
  17. P.-B. Li, Y. Gu, Q.-H. Gong, and G.-C. Guo, “Generation of two-mode entanglement between separated cavities,” J. Opt. Soc. Am. B 26, 189–193 (2009).
    [Crossref]
  18. S. Kang, Y. Choi, S. Lim, W. Kim, J.-R. Kim, J.-H. Lee, and K. An, “Continuous control of the coupling constant in an atom-cavity system by using elliptic polarization and magnetic sublevels,” Opt. Express. 18, 9286–9302 (2010).
    [Crossref] [PubMed]
  19. H. J. Kimble, “The quantum internet,” Nature (London) 453, 1023–1030 (2008).
    [Crossref]
  20. Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature (London) 450, 272–276 (2007).
    [Crossref]
  21. M. Trupke, E. A. Hinds, S. Eriksson, E. A. Curtis, Z. Moktadir, E. Kukharenka, and M. Kraft, “Microfabricated high-finesse optical cavity with open access and small volume,” Appl. Phys. Lett. 87, 211106 (2005).
    [Crossref]
  22. D. Hunger, T. Steinmetz, Y. Colombe, C. Deutsch, T. W. Hansch, and J. Reichel, “A fiber Fabry-Perot cavity with high finesse,” New J. Phys. 12065038 (2010).
    [Crossref]
  23. T. Pellizzari, “Quantum networking with optical fibres,” Phys. Rev. Lett. 79, 5242–5245 (1997).
    [Crossref]
  24. A. Serafini, S. Mancini, and S. Bose, “Distributed quantum computation via optical fibers,” Phys. Rev. Lett. 96, 010503 (2006).
    [Crossref] [PubMed]
  25. 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]
  26. P. Peng and F. L. Li, “Entangling two atoms in spatially separated cavities through both photon emission and absorption processes,” Phys. Rev. A 75, 062320 (2007).
    [Crossref]
  27. Y. L. Zhou, Y. M. Wang, L. M. Liang, and C. Z. Li, “Quantum state transfer between distant nodes of a quantum network via adiabatic passage,” Phys. Rev. A 79,044304 (2009).
    [Crossref]
  28. J. Busch and A. Beige, “Generating single-mode behavior in fiber-coupled optical cavities,” arXiv:1009.1011v2 (2010).
  29. X.-Y. Lu, P.-J. Song, J.-B. Liu, and X. Yang, “N-qubit W state of spatially separated single molecule magnets,” Opt. Express 17, 14298–14311 (2010).
    [Crossref]
  30. K. T. Kapale and J. P. Dowling, “Bootstrapping approach for generating maximally path-entangled photon states,” Phys. Rev. Lett. 99, 053602 (2007).
    [Crossref] [PubMed]
  31. T. Brandes, “Coherent and collective quantum optical effects in mesoscopic systems,” Phys. Rep. 408, 315–474 (2005).
    [Crossref]
  32. D. F. V. James, “Quantum computation with hot and cold ions: an assessment of proposed schemes,” Fortschr. Phys. 48, 823–837 (2000).
    [Crossref]
  33. A. Sørensen and K. Mølmer, “Entanglement and quantum computation with ions in thermal motion,” Phys. Rev. A 62, 022311 (2000).
    [Crossref]
  34. S. L. Zhu, Z. D. Wang, and P. Zanardi, “Geometric quantum computation and multiqubit entanglement with superconducting qubits inside a cavity,” Phys. Rev. Lett. 94, 100502 (2005).
    [Crossref] [PubMed]
  35. K. Mølmer and A. Sørensen, “Multiparticle entanglement of hot trapped ions,” Phys. Rev. Lett. 82, 1835–1838 (1999).
    [Crossref]
  36. K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys. 70, 1003–1025 (1997).
    [Crossref]
  37. I. E. Linington and N. V. Vitanov, “Decoherence-free preparation of Dicke states of trapped ions by collective stimulated Raman adiabatic passage,” Phys. Rev. A 77, 062327 (2008).
    [Crossref]
  38. R. Schack and T. A. Brun, “A C++ library using quantum trajectories to solve quantum master equations,” Comput. Phys. Commun. 102, 210–228 (1997).
    [Crossref]

2010 (5)

For a review see, K. Hammerer, A. S. Sorensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82, 1041–1093 (2010), and references therein.
[Crossref]

For a review see, L.-M. Duan and C. Monroe, “Colloquium: Quantum networks with trapped ions,” Rev. Mod. Phys. 82, 1209–1224 (2010), and references therein.
[Crossref]

S. Kang, Y. Choi, S. Lim, W. Kim, J.-R. Kim, J.-H. Lee, and K. An, “Continuous control of the coupling constant in an atom-cavity system by using elliptic polarization and magnetic sublevels,” Opt. Express. 18, 9286–9302 (2010).
[Crossref] [PubMed]

D. Hunger, T. Steinmetz, Y. Colombe, C. Deutsch, T. W. Hansch, and J. Reichel, “A fiber Fabry-Perot cavity with high finesse,” New J. Phys. 12065038 (2010).
[Crossref]

X.-Y. Lu, P.-J. Song, J.-B. Liu, and X. Yang, “N-qubit W state of spatially separated single molecule magnets,” Opt. Express 17, 14298–14311 (2010).
[Crossref]

2009 (4)

Y. L. Zhou, Y. M. Wang, L. M. Liang, and C. Z. Li, “Quantum state transfer between distant nodes of a quantum network via adiabatic passage,” Phys. Rev. A 79,044304 (2009).
[Crossref]

P.-B. Li, Y. Gu, Q.-H. Gong, and G.-C. Guo, “Quantum-information transfer in a coupled resonator waveguide,” Phys. Rev. A 79, 042339 (2009).
[Crossref]

F. Mei, M. Feng, Y.-F. Yu, and Z.-M. Zhang, “Scalable quantum information processing with atomic ensembles and flying photons,” Phys. Rev. A 80, 042319 (2009).
[Crossref]

P.-B. Li, Y. Gu, Q.-H. Gong, and G.-C. Guo, “Generation of two-mode entanglement between separated cavities,” J. Opt. Soc. Am. B 26, 189–193 (2009).
[Crossref]

2008 (2)

H. J. Kimble, “The quantum internet,” Nature (London) 453, 1023–1030 (2008).
[Crossref]

I. E. Linington and N. V. Vitanov, “Decoherence-free preparation of Dicke states of trapped ions by collective stimulated Raman adiabatic passage,” Phys. Rev. A 77, 062327 (2008).
[Crossref]

2007 (4)

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]

P. Peng and F. L. Li, “Entangling two atoms in spatially separated cavities through both photon emission and absorption processes,” Phys. Rev. A 75, 062320 (2007).
[Crossref]

K. T. Kapale and J. P. Dowling, “Bootstrapping approach for generating maximally path-entangled photon states,” Phys. Rev. Lett. 99, 053602 (2007).
[Crossref] [PubMed]

Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature (London) 450, 272–276 (2007).
[Crossref]

2006 (1)

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

2005 (4)

T. Brandes, “Coherent and collective quantum optical effects in mesoscopic systems,” Phys. Rep. 408, 315–474 (2005).
[Crossref]

S. L. Zhu, Z. D. Wang, and P. Zanardi, “Geometric quantum computation and multiqubit entanglement with superconducting qubits inside a cavity,” Phys. Rev. Lett. 94, 100502 (2005).
[Crossref] [PubMed]

M. Trupke, E. A. Hinds, S. Eriksson, E. A. Curtis, Z. Moktadir, E. Kukharenka, and M. Kraft, “Microfabricated high-finesse optical cavity with open access and small volume,” Appl. Phys. Lett. 87, 211106 (2005).
[Crossref]

For a review see, D. Jaksch and P. Zoller, “The cold atom Hubbard toolbox,” Ann. Phys. 315, 52–79 (2005), and references therein.
[Crossref]

2003 (1)

E. Solano, G. S. Agarwal, and H. Walther, “Strong-driving-assisted multipartite entanglement in cavity QED,” Phys. Rev. Lett. 90, 027903 (2003).
[Crossref] [PubMed]

2002 (1)

H. Mabuchi and A. C. Doherty, “Cavity quantum electrodynamics: coherence in Context,” Science 298, 1372–1377 (2002).
[Crossref] [PubMed]

2001 (1)

H. J. Briegel and R. Raussendorf, “Persistent entanglement in arrays of interacting particles,” Phys. Rev. Lett. 86, 910–913 (2001).
[Crossref] [PubMed]

2000 (3)

W. Dur, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,” Phys. Rev. A 62, 062314 (2000).
[Crossref]

D. F. V. James, “Quantum computation with hot and cold ions: an assessment of proposed schemes,” Fortschr. Phys. 48, 823–837 (2000).
[Crossref]

A. Sørensen and K. Mølmer, “Entanglement and quantum computation with ions in thermal motion,” Phys. Rev. A 62, 022311 (2000).
[Crossref]

1999 (1)

K. Mølmer and A. Sørensen, “Multiparticle entanglement of hot trapped ions,” Phys. Rev. Lett. 82, 1835–1838 (1999).
[Crossref]

1998 (1)

H. J. Kimble, “Strong interactions of single atoms and photons in cavity QED,” Phys. Scr. T76, 127–137 (1998).
[Crossref]

1997 (4)

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, 3221–3224 (1997).
[Crossref]

K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys. 70, 1003–1025 (1997).
[Crossref]

R. Schack and T. A. Brun, “A C++ library using quantum trajectories to solve quantum master equations,” Comput. Phys. Commun. 102, 210–228 (1997).
[Crossref]

T. Pellizzari, “Quantum networking with optical fibres,” Phys. Rev. Lett. 79, 5242–5245 (1997).
[Crossref]

1990 (1)

D. M. Greenberger, M.A. Horne, A. Shimony, and A. Zeilinger, “Bell’s theorem without inequalities,” Am. J. Phys. 58, 1131–1143 (1990).
[Crossref]

1964 (1)

J. S. Bell, “On the einstein-podolsky-rosen paradox,” Phys. 1, 195–200 (1964).

1936 (1)

A. D. Boozer, A. Boca, R. Miller, T. E. Northup, and H. J. Kimble, “Reversible state transfer between light and a single trapped atom,” Phys. Rev. Lett. 98, 193601 (2007).

Agarwal, G. S.

E. Solano, G. S. Agarwal, and H. Walther, “Strong-driving-assisted multipartite entanglement in cavity QED,” Phys. Rev. Lett. 90, 027903 (2003).
[Crossref] [PubMed]

An, K.

S. Kang, Y. Choi, S. Lim, W. Kim, J.-R. Kim, J.-H. Lee, and K. An, “Continuous control of the coupling constant in an atom-cavity system by using elliptic polarization and magnetic sublevels,” Opt. Express. 18, 9286–9302 (2010).
[Crossref] [PubMed]

Beige, A.

J. Busch and A. Beige, “Generating single-mode behavior in fiber-coupled optical cavities,” arXiv:1009.1011v2 (2010).

Bell, J. S.

J. S. Bell, “On the einstein-podolsky-rosen paradox,” Phys. 1, 195–200 (1964).

Bergmann, K.

K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys. 70, 1003–1025 (1997).
[Crossref]

Blatt, R.

R. Blatt and D. Wineland, “Entangled states of trapped atomic ions,” Nature (London)453, 1008–1015 (2008).
[Crossref]

Boca, A.

A. D. Boozer, A. Boca, R. Miller, T. E. Northup, and H. J. Kimble, “Reversible state transfer between light and a single trapped atom,” Phys. Rev. Lett. 98, 193601 (2007).

Boozer, A. D.

A. D. Boozer, A. Boca, R. Miller, T. E. Northup, and H. J. Kimble, “Reversible state transfer between light and a single trapped atom,” Phys. Rev. Lett. 98, 193601 (2007).

Bose, S.

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

Brandes, T.

T. Brandes, “Coherent and collective quantum optical effects in mesoscopic systems,” Phys. Rep. 408, 315–474 (2005).
[Crossref]

Briegel, H. J.

H. J. Briegel and R. Raussendorf, “Persistent entanglement in arrays of interacting particles,” Phys. Rev. Lett. 86, 910–913 (2001).
[Crossref] [PubMed]

Brun, T. A.

R. Schack and T. A. Brun, “A C++ library using quantum trajectories to solve quantum master equations,” Comput. Phys. Commun. 102, 210–228 (1997).
[Crossref]

Busch, J.

J. Busch and A. Beige, “Generating single-mode behavior in fiber-coupled optical cavities,” arXiv:1009.1011v2 (2010).

Choi, Y.

S. Kang, Y. Choi, S. Lim, W. Kim, J.-R. Kim, J.-H. Lee, and K. An, “Continuous control of the coupling constant in an atom-cavity system by using elliptic polarization and magnetic sublevels,” Opt. Express. 18, 9286–9302 (2010).
[Crossref] [PubMed]

Chuang, I. L.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge2000).

Cirac, J. I.

W. Dur, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,” Phys. Rev. A 62, 062314 (2000).
[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, 3221–3224 (1997).
[Crossref]

Colombe, Y.

D. Hunger, T. Steinmetz, Y. Colombe, C. Deutsch, T. W. Hansch, and J. Reichel, “A fiber Fabry-Perot cavity with high finesse,” New J. Phys. 12065038 (2010).
[Crossref]

Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature (London) 450, 272–276 (2007).
[Crossref]

Curtis, E. A.

M. Trupke, E. A. Hinds, S. Eriksson, E. A. Curtis, Z. Moktadir, E. Kukharenka, and M. Kraft, “Microfabricated high-finesse optical cavity with open access and small volume,” Appl. Phys. Lett. 87, 211106 (2005).
[Crossref]

Deutsch, C.

D. Hunger, T. Steinmetz, Y. Colombe, C. Deutsch, T. W. Hansch, and J. Reichel, “A fiber Fabry-Perot cavity with high finesse,” New J. Phys. 12065038 (2010).
[Crossref]

Doherty, A. C.

H. Mabuchi and A. C. Doherty, “Cavity quantum electrodynamics: coherence in Context,” Science 298, 1372–1377 (2002).
[Crossref] [PubMed]

Dowling, J. P.

K. T. Kapale and J. P. Dowling, “Bootstrapping approach for generating maximally path-entangled photon states,” Phys. Rev. Lett. 99, 053602 (2007).
[Crossref] [PubMed]

Duan, L.-M.

For a review see, L.-M. Duan and C. Monroe, “Colloquium: Quantum networks with trapped ions,” Rev. Mod. Phys. 82, 1209–1224 (2010), and references therein.
[Crossref]

Dubois, G.

Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature (London) 450, 272–276 (2007).
[Crossref]

Dur, W.

W. Dur, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,” Phys. Rev. A 62, 062314 (2000).
[Crossref]

Eriksson, S.

M. Trupke, E. A. Hinds, S. Eriksson, E. A. Curtis, Z. Moktadir, E. Kukharenka, and M. Kraft, “Microfabricated high-finesse optical cavity with open access and small volume,” Appl. Phys. Lett. 87, 211106 (2005).
[Crossref]

Feng, M.

F. Mei, M. Feng, Y.-F. Yu, and Z.-M. Zhang, “Scalable quantum information processing with atomic ensembles and flying photons,” Phys. Rev. A 80, 042319 (2009).
[Crossref]

Gong, Q.-H.

P.-B. Li, Y. Gu, Q.-H. Gong, and G.-C. Guo, “Quantum-information transfer in a coupled resonator waveguide,” Phys. Rev. A 79, 042339 (2009).
[Crossref]

P.-B. Li, Y. Gu, Q.-H. Gong, and G.-C. Guo, “Generation of two-mode entanglement between separated cavities,” J. Opt. Soc. Am. B 26, 189–193 (2009).
[Crossref]

Greenberger, D. M.

D. M. Greenberger, M.A. Horne, A. Shimony, and A. Zeilinger, “Bell’s theorem without inequalities,” Am. J. Phys. 58, 1131–1143 (1990).
[Crossref]

Gu, Y.

P.-B. Li, Y. Gu, Q.-H. Gong, and G.-C. Guo, “Quantum-information transfer in a coupled resonator waveguide,” Phys. Rev. A 79, 042339 (2009).
[Crossref]

P.-B. Li, Y. Gu, Q.-H. Gong, and G.-C. Guo, “Generation of two-mode entanglement between separated cavities,” J. Opt. Soc. Am. B 26, 189–193 (2009).
[Crossref]

Guo, G.-C.

P.-B. Li, Y. Gu, Q.-H. Gong, and G.-C. Guo, “Generation of two-mode entanglement between separated cavities,” J. Opt. Soc. Am. B 26, 189–193 (2009).
[Crossref]

P.-B. Li, Y. Gu, Q.-H. Gong, and G.-C. Guo, “Quantum-information transfer in a coupled resonator waveguide,” Phys. Rev. A 79, 042339 (2009).
[Crossref]

Hammerer, K.

For a review see, K. Hammerer, A. S. Sorensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82, 1041–1093 (2010), and references therein.
[Crossref]

Hansch, T. W.

D. Hunger, T. Steinmetz, Y. Colombe, C. Deutsch, T. W. Hansch, and J. Reichel, “A fiber Fabry-Perot cavity with high finesse,” New J. Phys. 12065038 (2010).
[Crossref]

Hinds, E. A.

M. Trupke, E. A. Hinds, S. Eriksson, E. A. Curtis, Z. Moktadir, E. Kukharenka, and M. Kraft, “Microfabricated high-finesse optical cavity with open access and small volume,” Appl. Phys. Lett. 87, 211106 (2005).
[Crossref]

Horne, M.A.

D. M. Greenberger, M.A. Horne, A. Shimony, and A. Zeilinger, “Bell’s theorem without inequalities,” Am. J. Phys. 58, 1131–1143 (1990).
[Crossref]

Hunger, D.

D. Hunger, T. Steinmetz, Y. Colombe, C. Deutsch, T. W. Hansch, and J. Reichel, “A fiber Fabry-Perot cavity with high finesse,” New J. Phys. 12065038 (2010).
[Crossref]

Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature (London) 450, 272–276 (2007).
[Crossref]

Jaksch, D.

For a review see, D. Jaksch and P. Zoller, “The cold atom Hubbard toolbox,” Ann. Phys. 315, 52–79 (2005), and references therein.
[Crossref]

James, D. F. V.

D. F. V. James, “Quantum computation with hot and cold ions: an assessment of proposed schemes,” Fortschr. Phys. 48, 823–837 (2000).
[Crossref]

Kang, S.

S. Kang, Y. Choi, S. Lim, W. Kim, J.-R. Kim, J.-H. Lee, and K. An, “Continuous control of the coupling constant in an atom-cavity system by using elliptic polarization and magnetic sublevels,” Opt. Express. 18, 9286–9302 (2010).
[Crossref] [PubMed]

Kapale, K. T.

K. T. Kapale and J. P. Dowling, “Bootstrapping approach for generating maximally path-entangled photon states,” Phys. Rev. Lett. 99, 053602 (2007).
[Crossref] [PubMed]

Kim, J.-R.

S. Kang, Y. Choi, S. Lim, W. Kim, J.-R. Kim, J.-H. Lee, and K. An, “Continuous control of the coupling constant in an atom-cavity system by using elliptic polarization and magnetic sublevels,” Opt. Express. 18, 9286–9302 (2010).
[Crossref] [PubMed]

Kim, W.

S. Kang, Y. Choi, S. Lim, W. Kim, J.-R. Kim, J.-H. Lee, and K. An, “Continuous control of the coupling constant in an atom-cavity system by using elliptic polarization and magnetic sublevels,” Opt. Express. 18, 9286–9302 (2010).
[Crossref] [PubMed]

Kimble, H. J.

H. J. Kimble, “The quantum internet,” Nature (London) 453, 1023–1030 (2008).
[Crossref]

H. J. Kimble, “Strong interactions of single atoms and photons in cavity QED,” Phys. Scr. T76, 127–137 (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, 3221–3224 (1997).
[Crossref]

A. D. Boozer, A. Boca, R. Miller, T. E. Northup, and H. J. Kimble, “Reversible state transfer between light and a single trapped atom,” Phys. Rev. Lett. 98, 193601 (2007).

Kraft, M.

M. Trupke, E. A. Hinds, S. Eriksson, E. A. Curtis, Z. Moktadir, E. Kukharenka, and M. Kraft, “Microfabricated high-finesse optical cavity with open access and small volume,” Appl. Phys. Lett. 87, 211106 (2005).
[Crossref]

Kukharenka, E.

M. Trupke, E. A. Hinds, S. Eriksson, E. A. Curtis, Z. Moktadir, E. Kukharenka, and M. Kraft, “Microfabricated high-finesse optical cavity with open access and small volume,” Appl. Phys. Lett. 87, 211106 (2005).
[Crossref]

Lee, J.-H.

S. Kang, Y. Choi, S. Lim, W. Kim, J.-R. Kim, J.-H. Lee, and K. An, “Continuous control of the coupling constant in an atom-cavity system by using elliptic polarization and magnetic sublevels,” Opt. Express. 18, 9286–9302 (2010).
[Crossref] [PubMed]

Li, C. Z.

Y. L. Zhou, Y. M. Wang, L. M. Liang, and C. Z. Li, “Quantum state transfer between distant nodes of a quantum network via adiabatic passage,” Phys. Rev. A 79,044304 (2009).
[Crossref]

Li, F. L.

P. Peng and F. L. Li, “Entangling two atoms in spatially separated cavities through both photon emission and absorption processes,” Phys. Rev. A 75, 062320 (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]

Li, P.-B.

P.-B. Li, Y. Gu, Q.-H. Gong, and G.-C. Guo, “Generation of two-mode entanglement between separated cavities,” J. Opt. Soc. Am. B 26, 189–193 (2009).
[Crossref]

P.-B. Li, Y. Gu, Q.-H. Gong, and G.-C. Guo, “Quantum-information transfer in a coupled resonator waveguide,” Phys. Rev. A 79, 042339 (2009).
[Crossref]

Liang, L. M.

Y. L. Zhou, Y. M. Wang, L. M. Liang, and C. Z. Li, “Quantum state transfer between distant nodes of a quantum network via adiabatic passage,” Phys. Rev. A 79,044304 (2009).
[Crossref]

Lim, S.

S. Kang, Y. Choi, S. Lim, W. Kim, J.-R. Kim, J.-H. Lee, and K. An, “Continuous control of the coupling constant in an atom-cavity system by using elliptic polarization and magnetic sublevels,” Opt. Express. 18, 9286–9302 (2010).
[Crossref] [PubMed]

Linington, I. E.

I. E. Linington and N. V. Vitanov, “Decoherence-free preparation of Dicke states of trapped ions by collective stimulated Raman adiabatic passage,” Phys. Rev. A 77, 062327 (2008).
[Crossref]

Linke, F.

Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature (London) 450, 272–276 (2007).
[Crossref]

Liu, J.-B.

Lu, X.-Y.

Mabuchi, H.

H. Mabuchi and A. C. Doherty, “Cavity quantum electrodynamics: coherence in Context,” Science 298, 1372–1377 (2002).
[Crossref] [PubMed]

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, 3221–3224 (1997).
[Crossref]

Mancini, S.

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

Mei, F.

F. Mei, M. Feng, Y.-F. Yu, and Z.-M. Zhang, “Scalable quantum information processing with atomic ensembles and flying photons,” Phys. Rev. A 80, 042319 (2009).
[Crossref]

Miller, R.

A. D. Boozer, A. Boca, R. Miller, T. E. Northup, and H. J. Kimble, “Reversible state transfer between light and a single trapped atom,” Phys. Rev. Lett. 98, 193601 (2007).

Moktadir, Z.

M. Trupke, E. A. Hinds, S. Eriksson, E. A. Curtis, Z. Moktadir, E. Kukharenka, and M. Kraft, “Microfabricated high-finesse optical cavity with open access and small volume,” Appl. Phys. Lett. 87, 211106 (2005).
[Crossref]

Mølmer, K.

A. Sørensen and K. Mølmer, “Entanglement and quantum computation with ions in thermal motion,” Phys. Rev. A 62, 022311 (2000).
[Crossref]

K. Mølmer and A. Sørensen, “Multiparticle entanglement of hot trapped ions,” Phys. Rev. Lett. 82, 1835–1838 (1999).
[Crossref]

Monroe, C.

For a review see, L.-M. Duan and C. Monroe, “Colloquium: Quantum networks with trapped ions,” Rev. Mod. Phys. 82, 1209–1224 (2010), and references therein.
[Crossref]

Nielsen, M. A.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge2000).

Northup, T. E.

A. D. Boozer, A. Boca, R. Miller, T. E. Northup, and H. J. Kimble, “Reversible state transfer between light and a single trapped atom,” Phys. Rev. Lett. 98, 193601 (2007).

Pellizzari, T.

T. Pellizzari, “Quantum networking with optical fibres,” Phys. Rev. Lett. 79, 5242–5245 (1997).
[Crossref]

Peng, P.

P. Peng and F. L. Li, “Entangling two atoms in spatially separated cavities through both photon emission and absorption processes,” Phys. Rev. A 75, 062320 (2007).
[Crossref]

Polzik, E. S.

For a review see, K. Hammerer, A. S. Sorensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82, 1041–1093 (2010), and references therein.
[Crossref]

Raussendorf, R.

H. J. Briegel and R. Raussendorf, “Persistent entanglement in arrays of interacting particles,” Phys. Rev. Lett. 86, 910–913 (2001).
[Crossref] [PubMed]

Reichel, J.

D. Hunger, T. Steinmetz, Y. Colombe, C. Deutsch, T. W. Hansch, and J. Reichel, “A fiber Fabry-Perot cavity with high finesse,” New J. Phys. 12065038 (2010).
[Crossref]

Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature (London) 450, 272–276 (2007).
[Crossref]

Schack, R.

R. Schack and T. A. Brun, “A C++ library using quantum trajectories to solve quantum master equations,” Comput. Phys. Commun. 102, 210–228 (1997).
[Crossref]

Serafini, A.

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

Shimony, A.

D. M. Greenberger, M.A. Horne, A. Shimony, and A. Zeilinger, “Bell’s theorem without inequalities,” Am. J. Phys. 58, 1131–1143 (1990).
[Crossref]

Shore, B. W.

K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys. 70, 1003–1025 (1997).
[Crossref]

Solano, E.

E. Solano, G. S. Agarwal, and H. Walther, “Strong-driving-assisted multipartite entanglement in cavity QED,” Phys. Rev. Lett. 90, 027903 (2003).
[Crossref] [PubMed]

Song, P.-J.

Sorensen, A. S.

For a review see, K. Hammerer, A. S. Sorensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82, 1041–1093 (2010), and references therein.
[Crossref]

Sørensen, A.

A. Sørensen and K. Mølmer, “Entanglement and quantum computation with ions in thermal motion,” Phys. Rev. A 62, 022311 (2000).
[Crossref]

K. Mølmer and A. Sørensen, “Multiparticle entanglement of hot trapped ions,” Phys. Rev. Lett. 82, 1835–1838 (1999).
[Crossref]

Steinmetz, T.

D. Hunger, T. Steinmetz, Y. Colombe, C. Deutsch, T. W. Hansch, and J. Reichel, “A fiber Fabry-Perot cavity with high finesse,” New J. Phys. 12065038 (2010).
[Crossref]

Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature (London) 450, 272–276 (2007).
[Crossref]

Theuer, H.

K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys. 70, 1003–1025 (1997).
[Crossref]

Trupke, M.

M. Trupke, E. A. Hinds, S. Eriksson, E. A. Curtis, Z. Moktadir, E. Kukharenka, and M. Kraft, “Microfabricated high-finesse optical cavity with open access and small volume,” Appl. Phys. Lett. 87, 211106 (2005).
[Crossref]

Vidal, G.

W. Dur, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,” Phys. Rev. A 62, 062314 (2000).
[Crossref]

Vitanov, N. V.

I. E. Linington and N. V. Vitanov, “Decoherence-free preparation of Dicke states of trapped ions by collective stimulated Raman adiabatic passage,” Phys. Rev. A 77, 062327 (2008).
[Crossref]

Walther, H.

E. Solano, G. S. Agarwal, and H. Walther, “Strong-driving-assisted multipartite entanglement in cavity QED,” Phys. Rev. Lett. 90, 027903 (2003).
[Crossref] [PubMed]

Wang, Y. M.

Y. L. Zhou, Y. M. Wang, L. M. Liang, and C. Z. Li, “Quantum state transfer between distant nodes of a quantum network via adiabatic passage,” Phys. Rev. A 79,044304 (2009).
[Crossref]

Wang, Z. D.

S. L. Zhu, Z. D. Wang, and P. Zanardi, “Geometric quantum computation and multiqubit entanglement with superconducting qubits inside a cavity,” Phys. Rev. Lett. 94, 100502 (2005).
[Crossref] [PubMed]

Wineland, D.

R. Blatt and D. Wineland, “Entangled states of trapped atomic ions,” Nature (London)453, 1008–1015 (2008).
[Crossref]

Yang, X.

Yin, Z. Q.

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]

Yu, Y.-F.

F. Mei, M. Feng, Y.-F. Yu, and Z.-M. Zhang, “Scalable quantum information processing with atomic ensembles and flying photons,” Phys. Rev. A 80, 042319 (2009).
[Crossref]

Zanardi, P.

S. L. Zhu, Z. D. Wang, and P. Zanardi, “Geometric quantum computation and multiqubit entanglement with superconducting qubits inside a cavity,” Phys. Rev. Lett. 94, 100502 (2005).
[Crossref] [PubMed]

Zeilinger, A.

D. M. Greenberger, M.A. Horne, A. Shimony, and A. Zeilinger, “Bell’s theorem without inequalities,” Am. J. Phys. 58, 1131–1143 (1990).
[Crossref]

Zhang, Z.-M.

F. Mei, M. Feng, Y.-F. Yu, and Z.-M. Zhang, “Scalable quantum information processing with atomic ensembles and flying photons,” Phys. Rev. A 80, 042319 (2009).
[Crossref]

Zhou, Y. L.

Y. L. Zhou, Y. M. Wang, L. M. Liang, and C. Z. Li, “Quantum state transfer between distant nodes of a quantum network via adiabatic passage,” Phys. Rev. A 79,044304 (2009).
[Crossref]

Zhu, S. L.

S. L. Zhu, Z. D. Wang, and P. Zanardi, “Geometric quantum computation and multiqubit entanglement with superconducting qubits inside a cavity,” Phys. Rev. Lett. 94, 100502 (2005).
[Crossref] [PubMed]

Zoller, P.

For a review see, D. Jaksch and P. Zoller, “The cold atom Hubbard toolbox,” Ann. Phys. 315, 52–79 (2005), and references therein.
[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, 3221–3224 (1997).
[Crossref]

Am. J. Phys. (1)

D. M. Greenberger, M.A. Horne, A. Shimony, and A. Zeilinger, “Bell’s theorem without inequalities,” Am. J. Phys. 58, 1131–1143 (1990).
[Crossref]

Ann. Phys. (1)

For a review see, D. Jaksch and P. Zoller, “The cold atom Hubbard toolbox,” Ann. Phys. 315, 52–79 (2005), and references therein.
[Crossref]

Appl. Phys. Lett. (1)

M. Trupke, E. A. Hinds, S. Eriksson, E. A. Curtis, Z. Moktadir, E. Kukharenka, and M. Kraft, “Microfabricated high-finesse optical cavity with open access and small volume,” Appl. Phys. Lett. 87, 211106 (2005).
[Crossref]

Comput. Phys. Commun. (1)

R. Schack and T. A. Brun, “A C++ library using quantum trajectories to solve quantum master equations,” Comput. Phys. Commun. 102, 210–228 (1997).
[Crossref]

Fortschr. Phys. (1)

D. F. V. James, “Quantum computation with hot and cold ions: an assessment of proposed schemes,” Fortschr. Phys. 48, 823–837 (2000).
[Crossref]

J. Opt. Soc. Am. B (1)

Nature (London) (2)

H. J. Kimble, “The quantum internet,” Nature (London) 453, 1023–1030 (2008).
[Crossref]

Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature (London) 450, 272–276 (2007).
[Crossref]

New J. Phys. (1)

D. Hunger, T. Steinmetz, Y. Colombe, C. Deutsch, T. W. Hansch, and J. Reichel, “A fiber Fabry-Perot cavity with high finesse,” New J. Phys. 12065038 (2010).
[Crossref]

Opt. Express (1)

Opt. Express. (1)

S. Kang, Y. Choi, S. Lim, W. Kim, J.-R. Kim, J.-H. Lee, and K. An, “Continuous control of the coupling constant in an atom-cavity system by using elliptic polarization and magnetic sublevels,” Opt. Express. 18, 9286–9302 (2010).
[Crossref] [PubMed]

Phys. (1)

J. S. Bell, “On the einstein-podolsky-rosen paradox,” Phys. 1, 195–200 (1964).

Phys. Rep. (1)

T. Brandes, “Coherent and collective quantum optical effects in mesoscopic systems,” Phys. Rep. 408, 315–474 (2005).
[Crossref]

Phys. Rev. A (8)

A. Sørensen and K. Mølmer, “Entanglement and quantum computation with ions in thermal motion,” Phys. Rev. A 62, 022311 (2000).
[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]

P. Peng and F. L. Li, “Entangling two atoms in spatially separated cavities through both photon emission and absorption processes,” Phys. Rev. A 75, 062320 (2007).
[Crossref]

Y. L. Zhou, Y. M. Wang, L. M. Liang, and C. Z. Li, “Quantum state transfer between distant nodes of a quantum network via adiabatic passage,” Phys. Rev. A 79,044304 (2009).
[Crossref]

P.-B. Li, Y. Gu, Q.-H. Gong, and G.-C. Guo, “Quantum-information transfer in a coupled resonator waveguide,” Phys. Rev. A 79, 042339 (2009).
[Crossref]

F. Mei, M. Feng, Y.-F. Yu, and Z.-M. Zhang, “Scalable quantum information processing with atomic ensembles and flying photons,” Phys. Rev. A 80, 042319 (2009).
[Crossref]

W. Dur, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,” Phys. Rev. A 62, 062314 (2000).
[Crossref]

I. E. Linington and N. V. Vitanov, “Decoherence-free preparation of Dicke states of trapped ions by collective stimulated Raman adiabatic passage,” Phys. Rev. A 77, 062327 (2008).
[Crossref]

Phys. Rev. Lett. (9)

H. J. Briegel and R. Raussendorf, “Persistent entanglement in arrays of interacting particles,” Phys. Rev. Lett. 86, 910–913 (2001).
[Crossref] [PubMed]

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, 3221–3224 (1997).
[Crossref]

A. D. Boozer, A. Boca, R. Miller, T. E. Northup, and H. J. Kimble, “Reversible state transfer between light and a single trapped atom,” Phys. Rev. Lett. 98, 193601 (2007).

E. Solano, G. S. Agarwal, and H. Walther, “Strong-driving-assisted multipartite entanglement in cavity QED,” Phys. Rev. Lett. 90, 027903 (2003).
[Crossref] [PubMed]

T. Pellizzari, “Quantum networking with optical fibres,” Phys. Rev. Lett. 79, 5242–5245 (1997).
[Crossref]

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

S. L. Zhu, Z. D. Wang, and P. Zanardi, “Geometric quantum computation and multiqubit entanglement with superconducting qubits inside a cavity,” Phys. Rev. Lett. 94, 100502 (2005).
[Crossref] [PubMed]

K. Mølmer and A. Sørensen, “Multiparticle entanglement of hot trapped ions,” Phys. Rev. Lett. 82, 1835–1838 (1999).
[Crossref]

K. T. Kapale and J. P. Dowling, “Bootstrapping approach for generating maximally path-entangled photon states,” Phys. Rev. Lett. 99, 053602 (2007).
[Crossref] [PubMed]

Phys. Scr. (1)

H. J. Kimble, “Strong interactions of single atoms and photons in cavity QED,” Phys. Scr. T76, 127–137 (1998).
[Crossref]

Rev. Mod. Phys. (3)

For a review see, K. Hammerer, A. S. Sorensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82, 1041–1093 (2010), and references therein.
[Crossref]

For a review see, L.-M. Duan and C. Monroe, “Colloquium: Quantum networks with trapped ions,” Rev. Mod. Phys. 82, 1209–1224 (2010), and references therein.
[Crossref]

K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys. 70, 1003–1025 (1997).
[Crossref]

Science (1)

H. Mabuchi and A. C. Doherty, “Cavity quantum electrodynamics: coherence in Context,” Science 298, 1372–1377 (2002).
[Crossref] [PubMed]

Other (3)

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge2000).

R. Blatt and D. Wineland, “Entangled states of trapped atomic ions,” Nature (London)453, 1008–1015 (2008).
[Crossref]

J. Busch and A. Beige, “Generating single-mode behavior in fiber-coupled optical cavities,” arXiv:1009.1011v2 (2010).

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Figures (2)

Fig. 1
Fig. 1

(a) Experimental setup. Two distant cold atomic gases are trapped in separate cavities connected by an optical fiber. The two cavity modes are coupled to the fiber mode with the coupling strength ν. (b) Atomic level structure with couplings to the cavity mode and driving laser fields.

Fig. 2
Fig. 2

Time evolution of the population and coherences of joint atomic ground states |000···000〉 and |111···111〉, as well as the fidelity. The first full curve (counted from above at δτ < 2) is the population of the joint ground state 000 |···000〉, the second one is the fidelity, the third one is population of the joint ground state |111···111〉, the last two curves are the imaginary and real part of the off-diagonal elements of the atomic density matrix, respectively. Results are displayed for different atomic numbers and cavity decay rates: (a) N = 2, κc = 0.1g0; (b) N = 2, κc = 0.5g0; (c) N = 5, κc = 0.1g0; (d) N = 5,κc = 0.5g0.

Equations (16)

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

1 = j = 1 , 2 { n = 1 N [ Ω 0 | e n j 0 n j | e i Δ 0 t + Ω 1 | e n j 0 n j | e i Δ 1 t + Ω 2 | e n j 1 n j | e i Δ 1 t + Ω 3 | e n j 1 n j | e i Δ 3 t + g 0 a ^ j | e n j 1 n j | e i ( Δ 0 δ ) t ] } + H . c . ,
1 = j = 1 , 2 { n = 1 N [ ( | Ω 0 | 2 Δ 0 + | Ω 1 | 2 Δ 1 ) | 0 n j 0 n j | + ( | Ω 2 | 2 Δ 1 + | Ω 3 | 2 Δ 3 + | g 0 | 2 Δ 2 a ^ j a ^ j ) | 1 n j 1 n j | + Ω 1 Ω 2 * Δ 1 | 1 n j 0 n j | + Ω 0 g 0 * Δ 0 a ^ j | 1 n j 0 n j | e i δ t ] } + H . c . ,
1 = j = 1 , 2 [ β S j + + β * S j ] + j = 1 , 2 [ Λ a j e i δ t S j + + Λ * a j e i δ t S j ]
c , f = ν b ( a ^ 1 + e i φ a ^ 2 ) + H . c . ,
= e i 0 t e i 0 t = j = 1 , 2 [ β S j + + β * S j ] + Λ * 2 [ e i 2 ν t c 1 + e i 2 ν t c 2 + 2 c ] S 1 e i δ t + Λ * 2 [ e i 2 ν t c 1 + e i 2 ν t c 2 2 c ] S 2 e i δ t + H . c .
e f f = { Λ * 2 [ e i 2 ν t c 1 + e i 2 ν t c 2 + 2 c ] e i δ t + Λ 2 [ e i 2 ν t c 1 + e i 2 ν t c 2 + 2 c ] e i δ t } × n = 1 N 1 2 ( | + n 1 + | | n 1 | ) + { Λ * 2 [ e i 2 ν t c 1 + e i 2 ν t c 2 2 c ] e i δ t + Λ 2 [ e i 2 ν t c 1 + e i 2 ν t c 2 2 c ] e i δ t } n = 1 N 1 2 ( | + n 2 + | | n 2 | ) .
eff = [ θ / 2 c e i δ t + θ * / 2 c e i δ t ] [ S x 1 S x 2 ] = Θ / 2 [ c e i δ t + i θ 0 + H . c . ] [ S x 1 S x 2 ] ,
𝒰 ( t ) = e i γ ( t ) [ S x 1 S x 2 ] 2 e [ α ( t ) c α * ( t ) c ] [ S x 1 S x 2 ] ,
𝒰 ( τ ) = e i λ τ [ S x 1 S x 2 ] 2 ,
| ψ f = M 1 , M 2 = N / 2 N / 2 c M 1 c M 2 ( 1 ) N / 2 M 2 e i λ ( M 1 M 2 ) 2 τ | N / 2 , M 1 x | N / 2 , M 2 x .
| ψ f = 1 2 M 1 , M 2 = N / 2 N / 2 c M 1 c M 2 ( 1 ) N / 2 M 2 [ e i π / 4 + e i π / 4 ( 1 ) M 1 M 2 ] | N / 2 , M 1 x | N / 2 , M 2 x = 1 2 [ e i π / 4 | N / 2 , N / 2 1 | N / 2 , N / 2 2 + e i π / 4 | N / 2 , N / 2 1 | N / 2 , N / 2 2 ] .
| ψ f = M 1 , M 2 = N / 2 N / 2 c M 1 c M 2 e i λ ( M 1 M 2 ) 2 τ | N / 2 , M 1 x | N / 2 , M 2 x
| ψ f = 1 2 M 1 , M 2 = N / 2 N / 2 c M 1 c M 2 [ e i π / 4 + e i π / 4 ( 1 ) M 1 M 2 ] | N / 2 , M 1 x | N / 2 , M 2 x = 1 2 [ e i π / 4 | N / 2 , N / 2 1 | N / 2 , N / 2 2 + [ e i π / 4 | N / 2 , N / 2 1 | N / 2 , N / 2 2 ] .
1 = j = 1 , 2 { n = 1 N [ Ω 1 Ω 2 Δ 1 e i ( k 1 k 2 ) r n j | 1 n j 0 n j | + Ω 0 g 0 * Δ 0 e i k 3 r n j a ^ j | 1 n j 0 n j | e i δ t ] } + H . c .
1 = j = 1 , 2 { n = 1 N [ Ω 1 Ω 2 * Δ 1 | 1 n j 0 n j | + Ω 0 g 0 * Δ 0 e i ( k 3 k 1 + k 2 ) r n j a ^ j | 1 n j 0 n j | e i δ t ] } + H . c .
d ρ d t = i [ , ρ ] + κ c j = 1 2 [ a ^ j ] ρ

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