Abstract

We analyze a quantum teleportation protocol based on spatially and temporally resolved direct photodetection of the collective emission from a pair of atoms, one of which is entangled with a single mode of an optical cavity. We focus on the performance of the protocol as characterized by the fidelity of the teleported state and the success probability. We find that the fidelity approaches unity as the spacing of the atoms becomes much smaller than the emission wavelength with a success probability of 0.25. The fidelity remains above the classical limit of 23 for all atomic spacings with the ultimate limit of performance coming from the spatial resolution of the photodetection.

© 2010 Optical Society of America

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2009 (2)

R. Wagner, Jr., and J. P. Clemens, “Performance of a quantum teleportation protocol based on temporally resolved photodetection of collective spontaneous emission,” Phys. Rev. A 79, 042322 (2009).
[CrossRef]

R. Wagner, Jr., and J. P. Clemens, “Performance of a quantum teleportation protocol based on collective spontaneous emission,” J. Opt. Soc. Am. B 26, 541-548 (2009).
[CrossRef]

2008 (1)

2007 (2)

G. Chimczak and R. Tanas, “Improving fidelity in atomic state teleportation via cavity decay,” Phys. Rev. A 75, 022317 (2007).
[CrossRef]

A. Furusawa and N. Takei, “Quantum teleportation for continous variables and related quantum information processing,” Phys. Rep. 443, 97-119 (2007).
[CrossRef]

2006 (3)

J. F. Sherson, H. Krauter, R. K. Ollson, B. Julsgaard, K. Hammerer, I. Cirac, and E. S. Polzik, “Quantum teleportation between light and matter,” Nature 443, 557-560 (2006).
[CrossRef] [PubMed]

M. Yang and Z.-L. Cao, “Scheme for Bell-state-measurement-free quantum teleportation,” Int. J. Quantum Inf. 4, 341-346 (2006).
[CrossRef]

T. Di, F. Li, and M. S. Zubairy, “Preservation of nonclassicality in the continuous-variable quantum teleportation,” Opt. Commun. 260, 633-636 (2006).
[CrossRef]

2005 (4)

N. Takei, T. Aoki, S. Koike, K. I. Yoshino, K. Wakui, H. Yonezawa, T. Hiraoka, J. Mizuno, M. Takeoka, M. Ban, and A. Furusawa, “Experimental demonstration of quantum teleportation of a squeezed state,” Phys. Rev. A 72, 042304 (2005).
[CrossRef]

H. J. Carmichael, “Continuous variable teleportation of quantum fields,” Int. J. Quantum Inf. 3(Suppl. 1), 11-25 (2005).
[CrossRef]

Y. N. Chen, C. M. Li, D. S. Chuu, and T. Brandes, “Proposal for teleportation of charge qubits via super-radiance,” New J. Phys. 7, 172 (2005).
[CrossRef]

T. Di, A. Muthukrishnan, M. O. Scully, and M. S. Zubairy, “Quantum teleportation of an arbitrary superposition of atomic Dicke states,” Phys. Rev. A 71, 062308 (2005).
[CrossRef]

2004 (11)

M. Riebe, H. Haffner, C. F. Roos, W. Hansel, J. Benheim, G. P. T. Lancaster, T. W. Korber, C. Becher, F. Schmidt-Kaler, D. F. V. James, and R. Blatt, “Deterministic quantum teleportation with atoms,” Nature 429, 734-736 (2004).
[CrossRef] [PubMed]

M. D. Barrett, J. Chiaverini, T. Schaetz, J. Britton, W. M. Itano, J. D. Jost, E. Knill, C. Langer, D. Liebfried, R. Ozeri, and D. J. Wineland, “Deterministic quantum teleportation of atomic qubits,” Nature 429, 737-739 (2004).
[CrossRef] [PubMed]

D. Fattal, E. Diamanti, K. Inoue, and Y. Yamamoto, “Quantum teleportation with a quantum dot single photon source,” Phys. Rev. Lett. 92, 037904 (2004).
[CrossRef] [PubMed]

J. P. Clemens, L. Horvath, B. C. Sanders, and H. J. Carmichael, “Shot-to-shot fluctuations in the directed superradiant emission from extended atomic samples,” J. Opt. B: Quantum Semiclassical Opt. 6, S736-S741 (2004).
[CrossRef]

Y.-F. Huang, X.-F. Ren, Y.-S. Zhang, L.-M. Duan, and G.-C. Guo, “Experimental teleportation of a quantum controlled-NOT gate,” Phys. Rev. Lett. 93, 240501 (2004).
[CrossRef]

A. Gatti, I. V. Sokolov, M. I. Kolobov, and L. A. Lugiato, “Quantum fluctuations in holographic teleportation of optical images,” Eur. Phys. J. D 30, 123-135 (2004).
[CrossRef]

R. Ursin, T. Jennewein, M. Aspelmeyer, R. Kaltenbaek, M. Lindenthal, P. Walther, and A. Zeilinger, “Quantum teleportation across the Danube,” Nature 430, 849 (2004).
[CrossRef] [PubMed]

E. S. Guerra, “Teleportation of atomic states via cavity quantum electrodynamics,” Opt. Commun. 242, 541-549 (2004).
[CrossRef]

J. Cho and H.-W. Lee, “Quantum teleportation with atoms trapped in cavities,” Phys. Rev. A 70, 034305 (2004).
[CrossRef]

S.-B. Zheng, “Scheme for approximate conditional teleportation of an unknown atomic state without the Bell-state measurement,” Phys. Rev. A 69, 064302 (2004).
[CrossRef]

B. Yu, Z.-W. Zhou, Y. Zhang, G.-Y. Xiang, and G.-C. Guo, “Robust high-fidelity teleportation of an atomic state through the detection of cavity decay,” Phys. Rev. A 70, 014302 (2004).
[CrossRef]

2003 (8)

W. P. Bowen, N. Treps, B. C. Buchler, R. Schnabel, T. C. Ralph, H.-A. Bachor, T. Symul, and P. K. Lam, “Experimental investigation of continuous-variable quantum teleportation,” Phys. Rev. A 67, 032302 (2003).
[CrossRef]

T. C. Zhang, K. W. Goh, C. W. Chou, P. Lodahl, and H. J. Kimble, “Quantum teleportation of light beams,” Phys. Rev. A 67, 033802 (2003).
[CrossRef]

A. Dolinska, B. C. Buchler, W. P. Bowen, T. C. Ralph, and P. K. Lam, “Teleportation of continuous-variable polarization states,” Phys. Rev. A 68, 052308 (2003).
[CrossRef]

J. P. Clemens, L. Horvath, B. C. Sanders, and H. J. Carmichael, “Collective spontaneous emission from a line of atoms,” Phys. Rev. A 68, 023809 (2003).
[CrossRef]

L. Roa, A. Delgado, and I. Fuentes-Guridi, “Optimal conclusive teleportation of quantum states,” Phys. Rev. A 68, 022310 (2003).
[CrossRef]

S. D. Bartlett and W. J. Munro, “Quantum teleportation of optical quantum gates,” Phys. Rev. Lett. 90, 117901 (2003).
[CrossRef] [PubMed]

D. Kumar and P. N. Pandey, “Effect of noise on quantum teleportation,” Phys. Rev. A 68, 012317 (2003).
[CrossRef]

G. G. Carlo, G. Benenti, and G. Casati, “Teleportation in a noisy environment: a quantum trajectories approach,” Phys. Rev. Lett. 91, 257903 (2003).
[CrossRef]

2002 (6)

N.-K. Tran and O. Pfister, “Quantum teleportation with close-to-maximal entanglement from a beam splitter,” Phys. Rev. A 65, 052313 (2002).
[CrossRef]

S. Oh, S. Lee, and H. W. Lee, “Fidelity of quantum teleportation through noisy channels,” Phys. Rev. A 66, 022316 (2002).
[CrossRef]

J. P. Clemens and H. J. Carmichael, “Stochastic initiation of superradiance in a cavity: an approximation scheme within quantum trajectory theory,” Phys. Rev. A 65, 023815 (2002).
[CrossRef]

T. J. Johnson, S. D. Bartlett, and B. C. Sanders, “Continuous-variable quantum teleportation of entanglement,” Phys. Rev. A 66, 042326 (2002).
[CrossRef]

Y.-H. Kim, S. P. Kulik, and Y. Shih, “Quantum teleportation with a complete Bell state measurement,” J. Mod. Opt. 49, 221-236 (2002).
[CrossRef]

S. Giacomini, F. Sciarrino, E. Lombardi, and F. DeMartini, “Active teleportation of a quantum bit,” Phys. Rev. A 66, 030302(R) (2002).
[CrossRef]

2001 (4)

Y.-H. Kim, S. P. Kulik, and Y. Shih, “Quantum teleportation of a polarization state with a complete Bell state measurement,” Phys. Rev. Lett. 86, 1370-1373 (2001).
[CrossRef] [PubMed]

F. Grosshans and P. Grangier, “Quantum cloning and teleportation criteria for continuous quantum variables,” Phys. Rev. A 64, 010301(R) (2001).
[CrossRef]

X. Wang, “Quantum teleportation of entangled coherent states,” Phys. Rev. A 64, 022302 (2001).
[CrossRef]

S. F. Huelga, J. A. Vaccaro, A. Chefles, and M. B. Plenio, “Quantum remote control: teleportation of unitary operations,” Phys. Rev. A 63, 042303 (2001).
[CrossRef]

2000 (5)

J. A. Reina and N. F. Johnson, “Quantum teleportation in a solid-state system,” Phys. Rev. A 63, 012303 (2000).
[CrossRef]

H.-W. Lee and J. Kim, “Quantum teleportation and Bell's inequality using single-particle entanglement,” Phys. Rev. A 63, 012305 (2000).
[CrossRef]

H. J. Carmichael and K. Kim, “A quantum trajectory unraveling of the superradiance master equation,” Opt. Commun. 179, 417-427 (2000).
[CrossRef]

S. L. Braunstein, C. A. Fuchs, and H. J. Kimble, “Criteria for continuous-variable quantum teleportation,” J. Mod. Opt. 47, 267-278 (2000).

A. Kuzmich and E. S. Polzik, “Atomic quantum state teleportation and swapping,” Phys. Rev. Lett. 85, 5639-5642 (2000).
[CrossRef]

1999 (2)

G. J. Milburn and S. L. Braunstein, “Quantum teleportation with squeezed vacuum states,” Phys. Rev. A 60, 937-942 (1999).
[CrossRef]

S. Bose, P. L. Knight, M. B. Plenio, and V. Vedral, “Proposal for teleportation of an atomic state via cavity decay,” Phys. Rev. Lett. 83, 5158-5161 (1999).
[CrossRef]

1998 (6)

D. Bouwmeester, K. Mattle, J.-W. Pan, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Experimental quantum teleportation or arbitrary quantum states,” Appl. Phys. B 67, 749-752 (1998).
[CrossRef]

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Philos. Trans. R. Soc. London Ser. A 356, 1733-1737 (1998).
[CrossRef]

D. Boschi, S. Branca, F. DeMartini, L. Hardy, and S. Popescu, “Experimental realization of teleporting and unknown pure quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 80, 1121-1125 (1998).
[CrossRef]

M. A. Nielsen, E. Knill, and R. Laflamme, “Complete quantum teleportation using nuclear magnetic resonance,” Nature 396, 52-55 (1998).
[CrossRef]

S. L. Braunstein and H. J. Kimble, “Teleportation of continuous quantum variables,” Phys. Rev. Lett. 80, 869-872 (1998).
[CrossRef]

A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706-709 (1998).
[CrossRef] [PubMed]

1997 (1)

D. Bouwmeester, J.-W. Pan, K. Matttle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575-579 (1997).
[CrossRef]

1996 (2)

N. Gisin, “Nonlocality criteria for quantum teleportation,” Phys. Lett. A 210, 157-159 (1996).
[CrossRef]

R. G. DeVoe and R. G. Brewer, “Observation of superradiant and subradiant spontaneous emission of two trapped ions,” Phys. Rev. Lett. 76, 2049-2052 (1996).
[CrossRef] [PubMed]

1995 (2)

S. Massar and S. Popescu, “Optimal extraction of information from finite quantum ensembles,” Phys. Rev. Lett. 74, 1259-1262 (1995).
[CrossRef] [PubMed]

S. Popescu, “Bell's inequalities and density matrices: revealing “hidden” nonlocality,” Phys. Rev. Lett. 74, 2619-2622 (1995).
[CrossRef] [PubMed]

1994 (1)

L. Vaidman, “Teleportation of quantum states,” Phys. Rev. A 49, 1473-1476 (1994).
[CrossRef] [PubMed]

1993 (2)

C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895-1899 (1993).
[CrossRef] [PubMed]

U. Eichmann, J. C. Bergquist, J. J. Bollinger, J. M. Gilligan, W. M. Itano, D. J. Wineland, and M. G. Raizen, “Young's interference experiment with light scattered from two atoms,” Phys. Rev. Lett. 70, 2359-2362 (1993).
[CrossRef] [PubMed]

1970 (3)

R. H. Lehmberg, “Radiation from an N-atom system. II. spontaneous emission from a pair of atoms,” Phys. Rev. A 2, 889-896 (1970).
[CrossRef]

R. H. Lehmberg, “Radiation from an N-atom system. I. General formalism,” Phys. Rev. A 2, 883-888 (1970).
[CrossRef]

G. S. Agarwal, “Master-equation approach to spontaneous emission,” Phys. Rev. A 2, 2038-2046 (1970).
[CrossRef]

1969 (1)

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M. D. Barrett, J. Chiaverini, T. Schaetz, J. Britton, W. M. Itano, J. D. Jost, E. Knill, C. Langer, D. Liebfried, R. Ozeri, and D. J. Wineland, “Deterministic quantum teleportation of atomic qubits,” Nature 429, 737-739 (2004).
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C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895-1899 (1993).
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J. F. Sherson, H. Krauter, R. K. Ollson, B. Julsgaard, K. Hammerer, I. Cirac, and E. S. Polzik, “Quantum teleportation between light and matter,” Nature 443, 557-560 (2006).
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Figures (3)

Fig. 1
Fig. 1

Schematic diagram of the teleportation protocol. An atom prepared in the excited state moves through a resonant cavity in the vacuum state. After interacting with the cavity mode, the atom and cavity are entangled. This atom is then trapped in the vicinity of a second atom in an arbitrary, unknown state. The subsequent collective spontaneous emission is detected.

Fig. 2
Fig. 2

(a) Fidelity and (b) success probability for the combined photodetection strategy, minimized over the state to be teleported, for γ T equal to (i) 0.1, (ii) 1, (iii) 2, (iv) 5 .

Fig. 3
Fig. 3

Fidelity for γ T = 5 for (i) the temporally resolved, (ii) spatially resolved, and (iii) combined photodetection strategies.

Equations (36)

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| ψ c 1 = 1 2 ( | 0 e | 1 g ) ,
| ψ 2 = α | e + β | g .
| ψ c 12 = α 2 | 0 e e β 2 | 1 g g + 1 2 | + c | 12 + 1 2 | c | + 12 ,
| ± c = ± α | 1 + β | 0
| ± 12 = 1 2 ( | e g ± | g e )
ρ ̇ = i i , j = 1 2 Δ i j [ σ i + σ j , ρ ] + 1 2 i , j = 1 2 γ i j ( 2 σ j ρ σ i + σ i + σ j ρ ρ σ i + σ j ) ,
Δ i j = γ 3 4 { [ 1 ( d ̂ r ̂ i j ) 2 ] cos ξ i j ξ i j + [ 1 3 ( d ̂ r ̂ i j ) 2 ] ( sin ξ i j ξ i j 2 + cos ξ i j ξ i j 3 ) }
γ i j = γ 3 2 { [ 1 ( d ̂ r ̂ i j ) 2 ] sin ξ i j ξ i j + [ 1 3 ( d ̂ r ̂ i j ) 2 ] ( cos ξ i j ξ i j 2 sin ξ i j ξ i j 3 ) } ,
ξ i j = k 0 r i j = 2 π r i j λ 0 , r i j = r i r j ,
[ σ i + , σ j ] = δ i j σ i z , [ σ i ± , σ j z ] = 2 δ i j σ i ± .
J i = λ i j = 1 2 b j i σ j ,
ρ ̇ = i i , j = 1 2 Δ i j [ σ i + σ j , ρ ] + 1 2 i = 1 2 ( J i ρ J i J i J i ρ ρ J i J i ) .
S ( θ , ϕ ) = γ D ( θ , ϕ ) d Ω i = 1 2 e i k 0 R ̂ ( θ , ϕ ) r i σ i ,
D ( θ , ϕ ) = 3 8 π [ 1 ( d ̂ R ̂ ( θ , ϕ ) ) 2 ]
ρ ̇ = i i , j = 1 2 Δ i j [ σ i + σ j , ρ ] + 1 2 ( 2 S ( θ , ϕ ) ρ S ( θ , ϕ ) S ( θ , ϕ ) S ( θ , ϕ ) ρ ρ S ( θ , ϕ ) S ( θ , ϕ ) ) .
ρ ( t ) = REC P REC | ψ REC ( t ) ψ REC ( t ) | ,
ρ ̇ = i [ H , ρ ] + i = 1 n ( 2 O ̂ i ρ O ̂ i O ̂ i O ̂ i ρ ρ O ̂ i O ̂ i ) ,
| ψ REC ( t ) ¯ = B ( t t k ) O ̂ i k B ( t k t k 1 ) O ̂ i 2 B ( t 2 t 1 ) O ̂ i 1 B ( t 1 ) | ψ ( 0 ) ,
H B = H i i = 1 n O ̂ i O ̂ i ,
J 1 = γ + Γ 2 ( σ 1 + σ 2 ) ,
J 2 = γ Γ 2 ( σ 1 σ 2 ) ,
S ( θ ) = γ D ( θ ) sin θ d θ ( e i π s cos θ σ 1 + e i π s cos θ σ 2 ) ,
| ψ θ , t 1 = e i H B ( t t 1 ) S ( θ ) e i H B t 1 | ψ ( 0 ) = γ D ( θ ) sin θ d θ { α e γ t 1 e ( γ + Γ ) ( t t 1 ) 2 e i Δ ( t t 1 ) cos ξ | 0 | + 12 + i α e γ t 1 e ( γ Γ ) ( t t 1 ) 2 e i Δ ( t t 1 ) sin ξ | 0 | 12 + [ 1 2 e i Δ t 1 e ( γ + Γ ) t 1 2 cos ξ | c i 2 e i Δ t 1 e ( γ Γ ) t 1 2 sin ξ | + c ] | g g } ,
ρ c ¯ ( t , t 1 , θ ) = Tr 12 | ψ θ , t 1 ψ θ , t 1 | = γ D ( θ ) sin θ d θ { | α | 2 e 2 γ t 1 e ( γ + Γ ) ( t t 1 ) cos 2 ξ | 0 0 | + | α | 2 e 2 γ t 1 e ( γ Γ ) ( t t 1 ) sin 2 ξ | 0 0 | + 1 2 e ( γ + Γ ) t 1 cos 2 ξ | | + 1 2 e ( γ Γ ) t 1 sin 2 ξ | + + | + i 2 e 2 i Δ t 1 e γ t 1 sin ξ cos ξ | + | i 2 e 2 i Δ t 1 e γ t 1 sin ξ cos ξ | + | } ,
ρ c ¯ ( t , t 1 , θ ) = γ D ( θ ) sin θ d θ { | α | 2 e 2 γ t 1 e ( γ + Γ ) ( t t 1 ) cos 2 ξ | 0 0 | + | α | 2 e 2 γ t 1 e ( γ Γ ) ( t t 1 ) sin 2 ξ | 0 0 | + 1 2 e ( γ + Γ ) t 1 cos 2 ξ | + + | + 1 2 e ( γ Γ ) t 1 sin 2 ξ | | + i 2 e 2 i Δ t 1 e γ t 1 sin ξ cos ξ | + | i 2 e 2 i Δ t 1 e γ t 1 sin ξ cos ξ | + | } .
F u = 1 A + | ρ c ¯ ( t , t 1 , θ ) | + = 1 A γ D ( θ ) sin θ d θ { | α | 2 | β | 2 e 2 γ t 1 [ e ( γ + Γ ) ( t t 1 ) cos 2 ξ + e ( γ Γ ) ( t t 1 ) sin 2 ξ ] + ( | β | 2 | α | 2 ) 2 1 2 e ( γ + Γ ) t 1 cos 2 ξ + 1 2 e ( γ Γ ) t 1 sin 2 ξ + ( | β | 2 | α | 2 ) e γ t 1 sin ξ cos ξ sin ( 2 Δ t 1 ) } ,
F c = 1 A + | ρ c ¯ ( t , t 1 , θ ) | + = 1 A γ D ( θ ) sin θ d θ { | α | 2 | β | 2 e 2 γ t 1 [ e ( γ + Γ ) ( t t 1 ) cos 2 ξ + e ( γ Γ ) ( t t 1 ) sin 2 ξ ] + 1 2 e ( γ + Γ ) t 1 cos 2 ξ + ( | β | 2 | α | 2 ) 2 1 2 e ( γ Γ ) t 1 sin 2 ξ + ( | β | 2 | α | 2 ) e γ t 1 sin ξ cos ξ sin ( 2 Δ t 1 ) } ,
t 1 * = 1 2 Γ ln ( tan 2 ξ ) ,
ρ c ¯ ( T ) = 2 0 π 2 γ D ( θ ) sin θ d θ [ 0 t 1 * ρ c ¯ ( T , t 1 , θ ) d t 1 + t 1 * T ρ c ¯ ( T , t 1 , θ ) d t 1 ] = 2 0 π 2 γ D ( θ ) sin θ d θ { | 0 0 | | α | 2 [ cos 2 ξ e ( γ + Γ ) T 1 γ Γ [ 1 ζ ] + sin 2 ξ e ( γ Γ ) T 1 γ + Γ [ 1 ζ + ] + cos 2 ξ 1 γ Γ [ e ( γ + Γ ) T ζ e 2 γ T ] + sin 2 ξ 1 γ + Γ [ e ( γ Γ ) T ζ + e 2 γ T ] ] + | + + | [ 1 2 cos 2 ξ 1 γ + Γ [ 1 ζ + ] + 1 2 sin 2 ξ 1 γ Γ [ ζ e ( γ Γ ) T ] ] + | | [ 1 2 sin 2 ξ 1 γ Γ [ 1 ζ ] + 1 2 cos 2 ξ 1 γ + Γ [ ζ + e ( γ + Γ ) T ] ] + | + | i 2 sin ξ cos ξ [ 1 γ + 2 i Δ [ 1 e ( γ + 2 i Δ ) t 1 * ( θ ) ] 1 γ 2 i Δ [ e ( γ 2 i Δ ) t 1 * ( θ ) e ( γ 2 i Δ ) T ] ] | + | i 2 sin ξ cos ξ [ 1 γ 2 i Δ [ 1 e ( γ 2 i Δ ) t 1 * ( θ ) ] 1 γ + 2 i Δ [ e ( γ + 2 i Δ ) t 1 * ( θ ) e ( γ + 2 i Δ ) T ] ] } ,
I 1 = 2 0 π 2 γ D ( θ ) cos 2 ξ sin θ d θ ,
I 2 = 2 0 π 2 γ D ( θ ) sin 2 ξ sin θ d θ ,
I 3 = 2 0 π 2 γ D ( θ ) sin ξ cos ξ sin θ d θ ,
I 4 = 2 0 π 2 γ D ( θ ) cos 2 ξ e ( γ + Γ ) t 1 * ( θ ) sin θ d θ ,
I 5 = 2 0 π 2 γ D ( θ ) sin 2 ξ e ( γ Γ ) t 1 * ( θ ) sin θ d θ ,
N = Tr ρ c ¯ ( T ) = { | α | 2 I 1 1 γ Γ [ e ( γ + Γ ) T e 2 γ T ] + | α | 2 I 2 1 γ + Γ [ e ( γ Γ ) T e 2 γ T ] + 1 2 I 1 1 γ + Γ [ 1 e ( γ + Γ ) T ] + 1 2 I 2 1 γ Γ [ 1 e ( γ Γ ) T ] + ( | β | 2 | α | 2 ) I 3 2 ( γ 2 + 4 Δ 2 ) [ 4 Δ 4 Δ e γ T cos 2 Δ T + 2 γ e γ T sin 2 Δ T ] } ,
F = 1 N { | α | 2 | β | 2 I 1 1 γ Γ [ e ( γ + Γ ) T e 2 γ T ] + | α | 2 | β | 2 I 2 1 γ + Γ [ e ( γ Γ ) T e 2 γ T ] + 1 2 1 γ + Γ ( I 1 I 4 ) + 1 2 1 γ Γ ( I 5 I 2 e ( γ Γ ) T ) + ( | β | 2 | α | 2 ) 2 [ 1 2 1 γ Γ ( I 2 I 5 ) + 1 2 1 γ + Γ ( I 4 I 1 e ( γ + Γ ) t ) ] + ( | β | 2 | α | 2 ) I 3 2 ( γ 2 + 4 Δ 2 ) [ 4 Δ 4 Δ e γ T cos 2 Δ T + 2 γ e γ T sin 2 Δ T ] } .

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