Abstract

We propose a novel scheme to generate the entanglement between two cavity optomechanical systems (COMSs) via a flying two-level atom. We derive the analytical expressions for the generated entangled states. We find that there exist two processes for generating entanglement: one is the entanglement transfer between the two phonon-modes, and the other is the entanglement swapping-like process among the two photon-modes and the two phonon-modes. We analyze these two kinds of phenomena, respectively, by adjusting the distance between the two COMSs. Then we discuss the verification of the generated entangled states of the two COMSs, and analyze the decoherence of the generated entangled states. Finally, we discuss the experimental feasibility of our proposal.

© 2017 Optical Society of America

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  4. S. Gerlich, S. Eibenberger, M. Tomandl, S. Nimmrichter, K. Hornberger, P. J. Fagan, J. Tüen, M. Mayor, and M. Arndt, “Quantum interference of large organic molecules,” Nat. Commun. 2, 263 (2011).
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  28. S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
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  33. S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401 (2002).
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  34. F. Xue, Y. X. Liu, C. P. Sun, and F. Nori, “Two-mode squeezed states and entangled states of two mechanical resonators,” Phys. Rev. B 76, 064305 (2007).
    [Crossref]
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  36. S. Huang and G. S. Agarwal, “Entangling nanomechanical oscillators in a ring cavity by feeding squeezed light,” New J. Phys. 11, 103044 (2009).
    [Crossref]
  37. X. W. Xu, Y. J. Zhao, and Y. X. Liu, “Entangled-state engineering of vibrational modes in a multimembrane optomechanical system,” Phys. Rev. A 88, 022325 (2013).
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  38. C. Joshi, J. Larson, M. Jonson, E. Andersson, and P. Öhberg, “Entanglement of distant optomechanical systems,” Phys. Rev. A 85, 033805 (2012).
    [Crossref]
  39. J. Zhang, K. Peng, and S. L. Braunstein, “Quantum-state transfer from light to macroscopic oscillators,” Phys. Rev. A 68, 013808 (2003).
    [Crossref]
  40. D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98, 030405 (2007).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  42. J. Q. Liao, Q. Q. Wu, and F. Nori, “Entangling two macroscopic mechanical mirrors in a two-cavity optomechanical system,” Phys. Rev. A 89, 014302 (2014).
    [Crossref]
  43. M. Zukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, “Event-ready-detectors Bell experiment via entanglement swapping,” Phys. Rev. Lett. 71, 4287–4290 (1993).
    [Crossref] [PubMed]
  44. L. M. Kuang and Z. Lan, “Generation of atom-photon entangled states in atomic bose-einstein condensate via electromagnetically induced transparency,” Phys. Rev. A 68, 043606 (2004).
    [Crossref]
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    [Crossref] [PubMed]
  47. J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707–712 (2010).
    [Crossref]
  48. S. J. Phoenix, “Wave-packet evolution in the damped oscillator,” Phys. Rev. A 41, 5132 (1990).
    [Crossref] [PubMed]

2014 (1)

J. Q. Liao, Q. Q. Wu, and F. Nori, “Entangling two macroscopic mechanical mirrors in a two-cavity optomechanical system,” Phys. Rev. A 89, 014302 (2014).
[Crossref]

2013 (4)

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

X. W. Xu, Y. J. Zhao, and Y. X. Liu, “Entangled-state engineering of vibrational modes in a multimembrane optomechanical system,” Phys. Rev. A 88, 022325 (2013).
[Crossref]

P. Meystre, “A short walk through quantum optomechanics,” Ann. Phys. (Berlin) 525, 215 (2013).
[Crossref]

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).
[Crossref]

2012 (3)

J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3, 1196 (2012).
[Crossref] [PubMed]

C. H. Dong, V. Fiore, M. C. Kuzyk, and H. L. Wang, “Optomechanical dark mode,” Science 338, 1609–1613 (2012).
[Crossref] [PubMed]

C. Joshi, J. Larson, M. Jonson, E. Andersson, and P. Öhberg, “Entanglement of distant optomechanical systems,” Phys. Rev. A 85, 033805 (2012).
[Crossref]

2011 (4)

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

R. Ghobadi, A. R. Bahrampour, and C. Simon, “Quantum optomechanics in the bistable regime,” Phys. Rev. A 84, 033846 (2011).
[Crossref]

S. Gerlich, S. Eibenberger, M. Tomandl, S. Nimmrichter, K. Hornberger, P. J. Fagan, J. Tüen, M. Mayor, and M. Arndt, “Quantum interference of large organic molecules,” Nat. Commun. 2, 263 (2011).
[Crossref] [PubMed]

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photon. 5, 222–229 (2011).
[Crossref]

2010 (3)

W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards quantum superpositions of a mirror,” Phys. Rev. Lett. 91, 130401 (2010).
[Crossref]

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref] [PubMed]

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707–712 (2010).
[Crossref]

2009 (3)

S. Huang and G. S. Agarwal, “Entangling nanomechanical oscillators in a ring cavity by feeding squeezed light,” New J. Phys. 11, 103044 (2009).
[Crossref]

Z. R. Gong, H. Ian, Y. X. Liu, C. P. Sun, and F. Nori, “Effective Hamiltonian approach to the Kerr nonlinearity in an optomechanical system,” Phys. Rev. A 80, 065801 (2009).
[Crossref]

R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865–942 (2009).
[Crossref]

2008 (5)

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

Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Nonlocal entanglement concentration scheme for partially entangled multipartite systems with nonlinear optics,” Phys. Rev. A 77, 062325 (2008).
[Crossref]

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

G. Vacanti, M. Paternostro, G. M. Palma, and V. Vedral, “Optomechanical to mechanical entanglement transformation,” New J. Phys. 10, 095014 (2008).
[Crossref]

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref] [PubMed]

2007 (2)

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98, 030405 (2007).
[Crossref] [PubMed]

F. Xue, Y. X. Liu, C. P. Sun, and F. Nori, “Two-mode squeezed states and entangled states of two mechanical resonators,” Phys. Rev. B 76, 064305 (2007).
[Crossref]

2005 (2)

H. Häfner, W. Häsel, C. F. Roos, J. Benhelm, M. Chwalla, T. Köber, and O. Güne, “Scalable multiparticle entanglement of trapped ions,” Nature 438, 643–646 (2005).
[Crossref]

V. Scarani, S. Iblisdir, N. Gisin, and A. Acin, “Quantum cloning,” Rev. Mod. Phys. 77, 1225 (2005).
[Crossref]

2004 (2)

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, 162–167 (2004).
[Crossref] [PubMed]

L. M. Kuang and Z. Lan, “Generation of atom-photon entangled states in atomic bose-einstein condensate via electromagnetically induced transparency,” Phys. Rev. A 68, 043606 (2004).
[Crossref]

2003 (2)

J. Zhang, K. Peng, and S. L. Braunstein, “Quantum-state transfer from light to macroscopic oscillators,” Phys. Rev. A 68, 013808 (2003).
[Crossref]

J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature 426, 264–267 (2003).
[Crossref]

2002 (1)

S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401 (2002).
[Crossref] [PubMed]

2001 (3)

M. Bayer, P. Hawrylak, K. Hinzer, S. Fafard, M. Korkusinski, Z. R. Wasilewski, and A. Forchel, “Coupling and entangling of quantum states in quantum dot molecules,” Science 291, 451–453 (2001).
[Crossref] [PubMed]

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87, 123602 (2001).
[Crossref] [PubMed]

L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414, 413–418 (2001).
[Crossref] [PubMed]

1999 (1)

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

1998 (1)

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

1997 (2)

E. Hagley, X. Maitre, G. Nogues, C. Wunderlich, M. Brune, J. M. Raimond, and S. Haroche, “Generation of Einstein-Podolsky-Rosen pairs of atoms,” Phys. Rev. Lett. 79, 1 (1997).
[Crossref]

S. Bose, K. Jacobs, and P. L. Knight, “Preparation of nonclassical states in cavities with a moving mirror,” Phys. Rev. A 56, 4175 (1997).
[Crossref]

1996 (2)

A. Ekert and R. Jozsa, “Shor’s quantum algorithm for factorising numbers,” Rev. Mod. Phys. 68, 733–753 (1996).
[Crossref]

C. Monroe, D. M. Meekhof, B. E. King, and D. J. Wineland, “A Schrodinger cat superposition state of an atom,” Science 272, 1131 (1996).
[Crossref] [PubMed]

1995 (2)

D. P. DiVincenzo, “Quantum computation,” Science 270, 255 (1995).
[Crossref]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337 (1995).
[Crossref] [PubMed]

1993 (1)

M. Zukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, “Event-ready-detectors Bell experiment via entanglement swapping,” Phys. Rev. Lett. 71, 4287–4290 (1993).
[Crossref] [PubMed]

1991 (1)

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

1990 (1)

S. J. Phoenix, “Wave-packet evolution in the damped oscillator,” Phys. Rev. A 41, 5132 (1990).
[Crossref] [PubMed]

1989 (1)

S. Horoche and D. Kleppner, “Cavity quantum electrodynamics,” Phys. Today 42, 24 (1989).
[Crossref]

Abouraddy, A. F.

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87, 123602 (2001).
[Crossref] [PubMed]

Acin, A.

V. Scarani, S. Iblisdir, N. Gisin, and A. Acin, “Quantum cloning,” Rev. Mod. Phys. 77, 1225 (2005).
[Crossref]

Agarwal, G. S.

S. Huang and G. S. Agarwal, “Entangling nanomechanical oscillators in a ring cavity by feeding squeezed light,” New J. Phys. 11, 103044 (2009).
[Crossref]

Alegre, T. P. M.

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

Andersson, E.

C. Joshi, J. Larson, M. Jonson, E. Andersson, and P. Öhberg, “Entanglement of distant optomechanical systems,” Phys. Rev. A 85, 033805 (2012).
[Crossref]

Arcizet, O.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref] [PubMed]

Arndt, M.

S. Gerlich, S. Eibenberger, M. Tomandl, S. Nimmrichter, K. Hornberger, P. J. Fagan, J. Tüen, M. Mayor, and M. Arndt, “Quantum interference of large organic molecules,” Nat. Commun. 2, 263 (2011).
[Crossref] [PubMed]

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

Aspelmeyer, M.

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98, 030405 (2007).
[Crossref] [PubMed]

Bahrampour, A. R.

R. Ghobadi, A. R. Bahrampour, and C. Simon, “Quantum optomechanics in the bistable regime,” Phys. Rev. A 84, 033846 (2011).
[Crossref]

Bawaj, M.

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).
[Crossref]

Bayer, M.

M. Bayer, P. Hawrylak, K. Hinzer, S. Fafard, M. Korkusinski, Z. R. Wasilewski, and A. Forchel, “Coupling and entangling of quantum states in quantum dot molecules,” Science 291, 451–453 (2001).
[Crossref] [PubMed]

Benhelm, J.

H. Häfner, W. Häsel, C. F. Roos, J. Benhelm, M. Chwalla, T. Köber, and O. Güne, “Scalable multiparticle entanglement of trapped ions,” Nature 438, 643–646 (2005).
[Crossref]

Biancofiore, C.

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).
[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, 162–167 (2004).
[Crossref] [PubMed]

Blatt, R.

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

Böhm, H. R.

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98, 030405 (2007).
[Crossref] [PubMed]

Bose, S.

S. Bose, K. Jacobs, and P. L. Knight, “Preparation of nonclassical states in cavities with a moving mirror,” Phys. Rev. A 56, 4175 (1997).
[Crossref]

Bouwmeester, D.

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Figures (3)

Fig. 1
Fig. 1 Schematic diagram of the system. A two-level atom sequentially passes through two cavity optomechanical systems (COMSs), and its states are detected when it exits from the second COMS. In this way the two COMSs can be entangled.

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Fig. 2
Fig. 2 The temporal sequence of entangling two optomechanical systems. U JC ( k ) and U OM ( k ) are the time evolution operators defined in the text. The insets above time points represent the position of the atom.

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Fig. 3
Fig. 3 The variation of the concurrence C versus the parameter β.

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Equations (41)

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H ( k ) = ω a 2 σ z + ω c a k † a k + ω m b k † b k − i G ( a k σ + − a k + σ − ) − g a k † a k ( b k † + b k ) ,
H int ( k ) = i d U k † d t U k + U k † H ( k ) U k = ω m b k † b k − i G ( a k σ + e i Δ t − a k † σ − e − i Δ t ) − g a k † a k ( b k † + b k ) ,
H int ( k ) ≈ H JC ( k ) = ω m b k † b k − i G ( a k σ + − a k † σ − ) ,
H int ( k ) = H OM ( k ) = ω m b k † b k − g a k † a k ( b k † + b k ) ,
| Ψ 0 〉 = | e 〉 | 0 〉 c 1 | α 〉 m 1 | 0 〉 c 2 | α 〉 m 2 ,
| Ψ ( t 1 ) 〉 = U JC ( 1 ) ( τ 1 ) U OM ( 2 ) ( τ 1 ) | Ψ 0 〉 = { sin ( G τ 1 ) | g 〉 | 1 〉 c 1 + cos ( G τ 1 ) | e 〉 | 0 〉 c 1 } | α γ ( τ 1 ) 〉 m 1 | 0 〉 c 2 | α γ ( τ 1 ) 〉 m 2 ,
| Ψ ( t 1 ) 〉 = 1 2 ( | g 〉 | 1 〉 c 1 + | e 〉 | 0 〉 c 1 ) | α 〉 m 1 | 0 〉 c 2 | α 〉 m 2 .
| Ψ ( t 2 ) 〉 = U OM ( 1 ) ( τ 2 ) U OM ( 2 ) ( τ 2 ) | Ψ ( t 1 ) 〉 = 1 2 { e i θ + ( τ 2 ) | g 〉 | 1 〉 c 1 | ϕ + ( τ 2 ) 〉 m 1 + | e 〉 | 0 〉 c 1 | α γ ( τ 2 ) 〉 m 1 } | 0 〉 c 2 | α γ ( τ 2 ) 〉 m 2 ,
| Ψ T ( t 2 ) 〉 = U OM ( 1 ) ( T m 2 ) U OM ( 2 ) ( T m 2 ) | Ψ ( t 1 ) 〉 = 1 2 ( e i Λ − | g 〉 | 1 〉 c 1 | 2 β − α 〉 m 1 + | e 〉 | 0 〉 c 1 | − α 〉 m 1 ) | 0 〉 c 2 | − α 〉 m 2 ,
| Ψ T ( t 3 ) 〉 = U OM ( 1 ) ( τ 3 ) U JC ( 2 ) ( τ 3 ) Ψ T ( t 2 ) = 1 2 { sin ( G τ 3 ) | g 〉 | 1 〉 c 2 + cos ( G τ 3 ) | e 〉 | 0 〉 c 2 } | 0 〉 c 1 | − α γ ( τ 3 ) 〉 m 1 | − α γ ( τ 3 ) 〉 m 2 + | g 〉 1 2 e i [ Λ − + θ ′ ( τ 3 ) ] | 1 〉 c 1 | ξ ( τ 3 ) 〉 m 1 | 0 〉 c 2 | − α γ ( τ 3 ) 〉 m 2 ,
| Ψ T ( t 3 ) 〉 = | g 〉 { 1 2 e i [ Λ − + 2 π β 2 ] | 1 〉 c 1 | 2 β − α 〉 m 1 | 0 〉 c 2 | − α 〉 m 2 + 1 2 | 1 〉 c 1 | − α 〉 m 2 | 0 〉 c 2 | − α 〉 m 1 } + 1 2 | e 〉 | 0 〉 c 1 | − α 〉 m 1 | 0 〉 c 2 | − α 〉 m 2 ,
| Ψ T ( t d ) 〉 = U OM ( 1 ) ( τ d ) U OM ( 2 ) ( τ d ) | ψ T ( t 3 ) 〉 = | g 〉 { 1 2 e i [ Λ − + 2 π β 2 + θ ′ ( τ d ) ] | 1 〉 c 1 | ξ ( τ d ) 〉 m 1 | 0 〉 c 2 | − α γ ( τ d ) 〉 m 2 + 1 2 e i θ − ( τ d ) | 0 〉 c 2 | − α γ ( τ d ) 〉 m 1 | 1 〉 c 1 | ϕ − ( τ d ) 〉 m 2 } + 1 2 | e 〉 | 0 〉 c 1 | − α γ ( τ d ) 〉 m 1 | 0 〉 c 2 | − α γ ( τ d ) 〉 m 2 ,
| ψ T ( T m 2 + t 3 ) 〉 = ( 2 3 e i 4 π β 2 | 1 〉 c 1 | 0 〉 c 2 | α 〉 m 2 + 2 3 e i Λ + | 0 〉 c 1 | 1 〉 c 2 | α + 2 β 〉 m 2 ) | α 〉 m 1 ,
| ψ T ( T m + t 3 ) 〉 = U OM ( 1 ) ( T M / 2 ) U OM ( 2 ) ( T m / 2 ) | ψ T ( T m 2 + t 3 ) 〉 = ( 2 3 e i Λ − e i 4 π β 2 | 1 〉 c 1 | 2 β − α 〉 m 1 | 0 〉 c 2 + 1 3 e i 2 π β 2 | 0 〉 c 1 | − α ) 〉 m 1 | 1 c 2 〉 ) | − α 〉 m 2 ,
| Ψ S ( t 2 ) 〉 = U OM ( 1 ) ( T m ) U OM ( 2 ) ( T m ) | Ψ ( t 1 ) 〉 = 1 2 ( e i 2 π β 2 | g 〉 | 1 〉 c 1 + | e 〉 | 0 〉 c 1 | α 〉 m 1 ) | 0 〉 c 2 | α 〉 m 2 ,
| Ψ S ( t 3 ) 〉 = U OM ( 1 ) ( τ 3 ) U JC ( 2 ) ( τ 3 ) | Ψ S ( t 2 ) 〉 = 1 2 { cos ( G τ 3 ) | e 〉 | 0 〉 c 2 + sin ( G τ 3 ) | g 〉 | 1 〉 c 2 } | α γ ( τ 3 ) 〉 m 2 | 0 〉 c 1 | α γ ( τ 3 ) 〉 m 1 + 1 2 e i [ 2 π β 2 + θ + ( τ 3 ) ] | g 〉 | 1 〉 c 1 | ϕ + ( τ 3 ) 〉 m 1 | 0 〉 c 2 | α γ ( τ 3 ) 〉 m 2 ,
| Ψ S ( t 3 ) 〉 = { | g 〉 ( 1 2 e i 4 π β 2 | 1 〉 c 1 | 0 〉 c 2 + 1 2 | 0 〉 c 1 | 1 〉 c 2 ) + 1 2 | e 〉 | 0 〉 c 1 | 0 〉 c 2 } | α 〉 m 1 | α 〉 m 2 ,
| Ψ S ( t d ) 〉 = U OM ( 1 ) ( τ d ) U OM ( 2 ) ( τ d ) | Ψ S ( t 3 ) 〉 = | g 〉 { 1 2 e i [ 4 π β 2 + θ + ( τ d ) ] | 1 〉 c 1 | ϕ + ( τ d ) 〉 m 1 | 0 〉 c 2 | α γ ( τ d ) 〉 m 2 + 1 2 e i θ + ( τ d ) | 0 〉 c 1 | α γ ( τ d ) 〉 m 1 | 1 〉 c 2 | ϕ + ( τ d ) 〉 m 2 } + 1 2 | e 〉 | 0 〉 c 1 | α γ ( τ d ) 〉 m 1 | 0 〉 c 2 | α γ ( τ d ) 〉 m 2 .
| ψ S ( T m / 2 + t 3 ) 〉 = ( 2 3 e i 6 π β 2 | 1 〉 c 1 | 0 〉 c 2 + 1 3 e i 2 π β 2 | 0 〉 c 1 | 1 〉 c 2 ) | α 〉 m 1 | α 〉 m 2 .
| ψ S ( T m + t 3 ) 〉 = U OM ( 1 ) ( T m / 2 ) U OM ( 2 ) ( T m / 2 ) | ψ S ( T m 2 + t 3 ) 〉 = 2 3 e i ( 6 π β 2 + Λ − ) | 1 〉 c 1 | 2 β − α 〉 m 1 | 0 〉 c 2 | − α 〉 m 2 + 1 3 e i ( 2 π β 2 + Λ − ) | 0 〉 c 1 | − α 〉 m 1 | 1 〉 c 2 | 2 β − α 〉 m 2 .
| ψ D 〉 = U JC ( 1 ) ( 2 T m ) U JC ( 2 ) ( 2 T m ) | g 〉 1 | g 〉 2 | ψ S ( T m + t 3 ) 〉 = ( − 2 3 e i ( 6 π β 2 + Λ − ) | e 〉 1 | g 〉 2 | 2 β − α 〉 m 1 | − α 〉 m 2 − 1 3 e i ( 2 π β 2 + Λ − ) | g 1 〉 | e 〉 2 | − α 〉 m 1 | 2 β − α 〉 m 2 ) | 0 〉 c 1 | 0 〉 c 2 .
| ψ D ′ 〉 = | g 1 〉 | g 2 〉 | D + 〉 + | e 〉 1 | e 2 〉 | D − 〉 + | g 1 〉 | e 〉 2 | D + ′ 〉 + | e 〉 1 | g 〉 2 | D − ′ 〉
| D ± 〉 = ± ℳ ( 1 6 e i ( 6 π β 2 + Λ − ) | 2 β − α 〉 m 1 | − α 〉 m 2 + 1 12 e i ( 2 π β 2 + Λ − ) | − α 〉 m 1 | 2 β − α 〉 m 2 ) ,
| D ± ′ 〉 = ± ℳ ′ ( 1 6 e i ( 6 π β 2 + Λ − ) | 2 β − α 〉 m 1 | − α 〉 m 2 − 1 12 e i ( 2 π β 2 + Λ − ) | − α 〉 m 1 | 2 β − α 〉 m 2 ) ,
C = 2 𝒩 2 6 ( 1 − e − 4 β 2 ) ,
d ρ d t = − i ω m ∑ k = 1 2 [ b k † b k , ρ ] + γ m 2 ∑ k = 1 2 ( 2 b k ρ b k † − b k † b k ρ − ρ b k + b k ) ,
ρ ( t ) = ℳ 2 { 1 6 | ϑ ′ ( t ) 〉 m 1 〈 ϑ ′ ( t ) | ⊗ | ϑ ( t ) 〉 m 2 〈 ϑ ( t ) | + 1 12 | ϑ ( t ) 〉 m 1 〈 ϑ ( t ) | ⊗ | ϑ ′ ( t ) 〉 m 2 〈 ϑ ′ ( t ) | + W ( t ) [ 2 12 e i 4 π β 2 | ϑ ′ ( t ) 〉 m 1 〈 ϑ ( t ) | ⊗ | ϑ ( t ) 〉 m 2 〈 ϑ ′ ( t ) | + 2 12 e − i 4 π β 2 | ϑ ( t ) 〉 m 1 〈 ϑ ′ ( t ) | ⊗ | ϑ ′ ( t ) 〉 m 2 〈 ϑ ( t ) | ] } ,
d ρ d t = − i ω m ∑ k = 1 2 [ b k † b k , ρ ] + ∑ k = 1 2 ( J k ρ + L k ρ ) ,
ρ ( t ) = exp [ ∑ k = 1 2 ( J k + L k + F k ) t ] ρ ( 0 ) ,
ρ ( t ) = lim N → ∞ ∏ j = 1 N { exp [ ∑ k = 1 2 ( J k + L k + F k ) Δ t j ] } ρ ( 0 ) .
ρ ( t ) = lim N → ∞ ∏ j = 1 N { exp [ ∑ k = 1 2 ( J k + L k ) Δ t j ] exp [ ∑ k = 1 2 F k Δ t j ] } ρ ( 0 ) .
exp [ ( J k + L k ) Δ t ] | 2 β − α 〉 m k 〈 − α | = ( 〈 − α | | 2 β − α 〉 m k ) Q | ( 2 β − α ) B 1 〉 m k 〈 − α B 1 | ,
ρ ( t 1 ) = exp [ ∑ k = 1 2 ( J k + L k ) Δ t ] exp [ ∑ k = 1 2 F k Δ t ] ρ ( 0 ) = ℳ 2 { 1 6 | η ′ ( Δ t ) 〉 m 1 〈 η ′ ( Δ t ) | ⊗ | η ( Δ t ) 〉 m 2 〈 η ( Δ t ) | + 1 12 | η ( Δ t ) 〉 m 1 〈 η ( Δ t ) | ⊗ | η ′ ( Δ t ) 〉 m 2 〈 η ′ ( Δ t ) | + w 1 [ 2 12 e i 4 π β 2 | η ′ ( Δ t ) 〉 m 1 〈 η ( Δ t ) | ⊗ | η ( Δ t ) 〉 m 2 〈 η ′ ( Δ t ) | + 2 12 e − i 4 π β 2 | η ( Δ t ) 〉 m 1 〈 η ′ ( Δ t ) | ⊗ | η ′ ( Δ t ) 〉 m 2 〈 η ( Δ t ) | ] } ,
w 1 = | 〈 η ′ ( Δ t ) | | η ( Δ t ) 〉 | 2 Q = exp [ − 4 β 2 ( 1 − e − γ m Δ t ) ] .
O 2 = O 3 = ⋯ = O N = e i ω m Δ t ,
B 2 = B 3 = ⋯ = B N = e − γ m Δ t / 2 ,
w 2 = exp [ − 4 β 2 ( 1 − e − γ m Δ t ) e − γ m Δ t ] , w 3 = exp [ − 4 β 2 ( 1 − e − γ m Δ t ) e − 2 γ m Δ t ] , ⋯ w N = exp [ − 4 β 2 ( 1 − e − γ m Δ t ) e − ( N − 1 ) γ m Δ t ] ,
ρ ( t ) = ℳ 2 { 1 6 | λ ′ ( t ) 〉 m 1 〈 λ ′ ( t ) | ⊗ | λ ( t ) 〉 m 2 〈 λ ( t ) | + 1 12 | λ ( t ) 〉 m 1 〈 λ ( t ) | ⊗ | λ ′ ( t ) 〉 m 2 〈 λ ′ ( t ) | + w ( t ) [ 2 12 e i 4 π β 2 | λ ′ ( t ) 〉 m 1 〈 λ ( t ) | ⊗ | λ ( t ) 〉 m 2 〈 λ ′ ( t ) | + 2 12 e − i 4 π β 2 | λ ( t ) 〉 m 1 〈 λ ′ ( t ) | ⊗ | λ ′ ( t ) 〉 m 2 〈 λ ( t ) | ] } ,
O = O 1 × O 2 × ⋯ × O N = e i ω m t ,
B = B 1 × B 2 × ⋯ × B N = e − γ m t / 2 ,
w ( t ) = lim N → ∞ w 1 ( t ) × w 2 ( t ) ⋯ w N ( t ) = lim N → ∞ exp [ − 4 β 2 ( 1 − e − γ m Δ t ) ( 1 + e − γ m Δ t + ⋯ + e − ( N − 1 ) γ m Δ t ) ] = exp [ − 4 β 2 ( 1 − e − γ m t ) ] .

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