Abstract

The twin-field quantum key distribution (TFQKD) protocol has garnered considerable attention in quantum communication because it overcomes the well-known fundamental limit of the secret key rate without quantum repeaters. In this study, we employ this scheme to demonstrate the long-distance distribution of entangled coherent state (ECS), which has not been addressed in the existing literature. We show a scheme for the distribution of ECS with a yield of $\sqrt {\eta }$, where η is the total efficiency of the whole transmission link. Compared to the cat-state based scheme, the success probability for fidelity is enhanced by 1.86 to 11.54 times in our new scheme, where the ECS is |ψECS(α = 2.0)〉 and the fixed fidelity (F) ranges from 0.99 to 0.75. The performance of our scheme in the presence of realistic on-off photon detector has also been investigated. Our work provides the application of TFQKD method toward continuous variable entanglement distribution and we believe that its application to other quantum information processing protocols are worth investigation in the near future.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (10)

Y. Israel, L. Cohen, X.-B. Song, J. Joo, H. S. Eisenberg, and Y. Silberberg, “Entangled coherent states created by mixing squeezed vacuum and coherent light,” Optica 6(6), 753–757 (2019).
[Crossref]

M. Curty, K. Azuma, and H.-K. Lo, “Simple security proof of twin-field type quantum key distribution protocol,” npj Quantum Information 5(1), 64 (2019).
[Crossref]

Z.-W. Yu, X.-L. Hu, C. Jiang, H. Xu, and X.-B. Wang, “Sending-or-not-sending twin-field quantum key distribution in practice,” Sci. Rep. 9(1), 3080 (2019).
[Crossref]

F. Grasselli, Á. Navarrete, and M. Curty, “Asymmetric twin-field quantum key distribution,” New J. Phys. 21(11), 113032 (2019).
[Crossref]

H.-L. Yin and Z.-B. Chen, “Coherent-state-based twin-field quantum key distribution,” Sci. Rep. 9(1), 14918 (2019).
[Crossref]

C. Cui, Z.-Q. Yin, R. Wang, W. Chen, S. Wang, G.-C. Guo, and Z.-F. Han, “Twin-field quantum key distribution without phase postselection,” Phys. Rev. Appl. 11(3), 034053 (2019).
[Crossref]

M. Minder, M. Pittaluga, G. L. Roberts, M. Lucamarini, J. F. Dynes, Z. L. Yuan, and A. J. Shields, “Experimental quantum key distribution beyond the repeaterless secret key capacity,” Nat. Photonics 13(5), 334–338 (2019).
[Crossref]

Y. Liu, Z.-W. Yu, W. Zhang, J.-Y. Guan, J.-P. Chen, C. Zhang, X.-L. Hu, H. Li, C. Jiang, J. Lin, T.-Y. Chen, L. You, Z. Wang, X.-B. Wang, Q. Zhang, and J.-W. Pan, “Experimental twin-field quantum key distribution through sending or not sending,” Phys. Rev. Lett. 123(10), 100505 (2019).
[Crossref]

X. Zhong, J. Hu, M. Curty, L. Qian, and H.-K. Lo, “Proof-of-principle experimental demonstration of twin-field type quantum key distribution,” Phys. Rev. Lett. 123(10), 100506 (2019).
[Crossref]

S. Wang, D.-Y. He, Z.-Q. Yin, F.-Y. Lu, C.-H. Cui, W. Chen, Z. Zhou, G.-C. Guo, and Z.-F. Han, “Beating the fundamental rate-distance limit in a proof-of-principle quantum key distribution system,” Phys. Rev. X 9(2), 021046 (2019).
[Crossref]

2018 (5)

X.-B. Wang, Z.-W. Yu, and X.-L. Hu, “Twin-field quantum key distribution with large misalignment error,” Phys. Rev. A 98(6), 062323 (2018).
[Crossref]

M. Huo, J. Qin, J. Cheng, Z. Yan, Z. Qin, X. Su, X. Jia, C. Xie, and K. Peng, “Deterministic quantum teleportation through fiber channels,” Sci. Adv. 4(10), eaas9401 (2018).
[Crossref]

J. Lin and N. Lütkenhaus, “Simple security analysis of phase-matching measurement-device-independent quantum key distribution,” Phys. Rev. A 98(4), 042332 (2018).
[Crossref]

M. Lucamarini, Z. L. Yuan, J. F. Dynes, and A. J. Shields, “Overcoming the rate-distance limit of quantum key distribution without quantum repeaters,” Nature 557(7705), 400–403 (2018).
[Crossref]

X. Ma, P. Zeng, and H. Zhou, “Phase-matching quantum key distribution,” Phys. Rev. X 8(3), 031043 (2018).
[Crossref]

2017 (1)

S. Pirandola, R. Laurenza, C. Ottaviani, and L. Banchi, “Fundamental limits of repeaterless quantum communications,” Nat. Commun. 8(1), 15043 (2017).
[Crossref]

2016 (1)

T. Liu, Q.-P. Su, S.-J. Xiong, J.-M. Liu, C.-P. Yang, and F. Nori, “Generation of a macroscopic entangled coherent state using quantum memories in circuit qed,” Sci. Rep. 6(1), 32004 (2016).
[Crossref]

2014 (1)

2013 (1)

Y. M. Zhang, X. W. Li, W. Yang, and G. R. Jin, “Quantum fisher information of entangled coherent states in the presence of photon loss,” Phys. Rev. A 88(4), 043832 (2013).
[Crossref]

2012 (3)

B. C. Sanders, “Review of entangled coherent states,” J. Phys. A: Math. Theor. 45(24), 244002 (2012).
[Crossref]

H.-K. Lo, M. Curty, and B. Qi, “Measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 108(13), 130503 (2012).
[Crossref]

V. D’Auria, O. Morin, C. Fabre, and J. Laurat, “Effect of the heralding detector properties on the conditional generation of single-photon states,” Eur. Phys. J. D 66(10), 249 (2012).
[Crossref]

2011 (1)

J. Joo, W. J. Munro, and T. P. Spiller, “Quantum metrology with entangled coherent states,” Phys. Rev. Lett. 107(8), 083601 (2011).
[Crossref]

2010 (1)

2009 (2)

C. C. Gerry, J. Mimih, and A. Benmoussa, “Maximally entangled coherent states and strong violations of bell-type inequalities,” Phys. Rev. A 80(2), 022111 (2009).
[Crossref]

A. Ourjoumtsev, F. Ferreyrol, R. Tualle-Brouri, and P. Grangier, “Preparation of non-local superpositions of quasi-classical light states,” Nat. Phys. 5(3), 189–192 (2009).
[Crossref]

2008 (1)

A. A. Semenov, A. V. Turchin, and H. V. Gomonay, “Detection of quantum light in the presence of noise,” Phys. Rev. A 78(5), 055803 (2008).
[Crossref]

2005 (2)

X.-B. Wang, “Beating the photon-number-splitting attack in practical quantum cryptography,” Phys. Rev. Lett. 94(23), 230503 (2005).
[Crossref]

H.-K. Lo, X. Ma, and K. Chen, “Decoy state quantum key distribution,” Phys. Rev. Lett. 94(23), 230504 (2005).
[Crossref]

2003 (3)

S. J. van Enk, “Entanglement capabilities in infinite dimensions: Multidimensional entangled coherent states,” Phys. Rev. Lett. 91(1), 017902 (2003).
[Crossref]

W.-Y. Hwang, “Quantum key distribution with high loss: Toward global secure communication,” Phys. Rev. Lett. 91(5), 057901 (2003).
[Crossref]

M. Paternostro, M. S. Kim, and B. S. Ham, “Generation of entangled coherent states via cross-phase-modulation in a double electromagnetically induced transparency regime,” Phys. Rev. A 67(2), 023811 (2003).
[Crossref]

2002 (1)

C. C. Gerry, A. Benmoussa, and R. A. Campos, “Nonlinear interferometer as a resource for maximally entangled photonic states: Application to interferometry,” Phys. Rev. A 66(1), 013804 (2002).
[Crossref]

2001 (3)

X. Wang, “Quantum teleportation of entangled coherent states,” Phys. Rev. A 64(2), 022302 (2001).
[Crossref]

H. Jeong, M. S. Kim, and J. Lee, “Quantum-information processing for a coherent superposition state via a mixedentangled coherent channel,” Phys. Rev. A 64(5), 052308 (2001).
[Crossref]

L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414(6862), 413–418 (2001).
[Crossref]

2000 (1)

N. Lütkenhaus, “Security against individual attacks for realistic quantum key distribution,” Phys. Rev. A 61(5), 052304 (2000).
[Crossref]

1993 (1)

L. Davidovich, A. Maali, M. Brune, J. M. Raimond, and S. Haroche, “Quantum switches and nonlocal microwave fields,” Phys. Rev. Lett. 71(15), 2360–2363 (1993).
[Crossref]

1992 (1)

B. C. Sanders, “Entangled coherent states,” Phys. Rev. A 45(9), 6811–6815 (1992).
[Crossref]

1967 (1)

Y. Aharonov and L. Susskind, “Charge superselection rule,” Phys. Rev. 155(5), 1428–1431 (1967).
[Crossref]

Aharonov, Y.

Y. Aharonov and L. Susskind, “Charge superselection rule,” Phys. Rev. 155(5), 1428–1431 (1967).
[Crossref]

Azuma, K.

M. Curty, K. Azuma, and H.-K. Lo, “Simple security proof of twin-field type quantum key distribution protocol,” npj Quantum Information 5(1), 64 (2019).
[Crossref]

Bai, Z.

Banchi, L.

S. Pirandola, R. Laurenza, C. Ottaviani, and L. Banchi, “Fundamental limits of repeaterless quantum communications,” Nat. Commun. 8(1), 15043 (2017).
[Crossref]

Benmoussa, A.

C. C. Gerry, J. Mimih, and A. Benmoussa, “Maximally entangled coherent states and strong violations of bell-type inequalities,” Phys. Rev. A 80(2), 022111 (2009).
[Crossref]

C. C. Gerry, A. Benmoussa, and R. A. Campos, “Nonlinear interferometer as a resource for maximally entangled photonic states: Application to interferometry,” Phys. Rev. A 66(1), 013804 (2002).
[Crossref]

Borregaard, J.

J. Borregaard, “Long-distance entanglement distribution using coherent states,” Ph.D. thesis, University of Copenhagen (2011).

Brune, M.

L. Davidovich, A. Maali, M. Brune, J. M. Raimond, and S. Haroche, “Quantum switches and nonlocal microwave fields,” Phys. Rev. Lett. 71(15), 2360–2363 (1993).
[Crossref]

Campos, R. A.

C. C. Gerry, A. Benmoussa, and R. A. Campos, “Nonlinear interferometer as a resource for maximally entangled photonic states: Application to interferometry,” Phys. Rev. A 66(1), 013804 (2002).
[Crossref]

Chen, J.-P.

Y. Liu, Z.-W. Yu, W. Zhang, J.-Y. Guan, J.-P. Chen, C. Zhang, X.-L. Hu, H. Li, C. Jiang, J. Lin, T.-Y. Chen, L. You, Z. Wang, X.-B. Wang, Q. Zhang, and J.-W. Pan, “Experimental twin-field quantum key distribution through sending or not sending,” Phys. Rev. Lett. 123(10), 100505 (2019).
[Crossref]

Chen, K.

H.-K. Lo, X. Ma, and K. Chen, “Decoy state quantum key distribution,” Phys. Rev. Lett. 94(23), 230504 (2005).
[Crossref]

Chen, T.-Y.

Y. Liu, Z.-W. Yu, W. Zhang, J.-Y. Guan, J.-P. Chen, C. Zhang, X.-L. Hu, H. Li, C. Jiang, J. Lin, T.-Y. Chen, L. You, Z. Wang, X.-B. Wang, Q. Zhang, and J.-W. Pan, “Experimental twin-field quantum key distribution through sending or not sending,” Phys. Rev. Lett. 123(10), 100505 (2019).
[Crossref]

Chen, W.

S. Wang, D.-Y. He, Z.-Q. Yin, F.-Y. Lu, C.-H. Cui, W. Chen, Z. Zhou, G.-C. Guo, and Z.-F. Han, “Beating the fundamental rate-distance limit in a proof-of-principle quantum key distribution system,” Phys. Rev. X 9(2), 021046 (2019).
[Crossref]

C. Cui, Z.-Q. Yin, R. Wang, W. Chen, S. Wang, G.-C. Guo, and Z.-F. Han, “Twin-field quantum key distribution without phase postselection,” Phys. Rev. Appl. 11(3), 034053 (2019).
[Crossref]

F.-Y. Lu, Z.-Q. Yin, R. Wang, G.-J. Fan-Yuan, S. Wang, D.-Y. He, W. Chen, W. Huang, B.-J. Xu, G.-C. Guo, and Z.-F. Han, “Practical issues of twin-field quantum key distribution,” arXiv: 1901.04264v3 (2019).

Chen, Z.-B.

H.-L. Yin and Z.-B. Chen, “Coherent-state-based twin-field quantum key distribution,” Sci. Rep. 9(1), 14918 (2019).
[Crossref]

Cheng, J.

M. Huo, J. Qin, J. Cheng, Z. Yan, Z. Qin, X. Su, X. Jia, C. Xie, and K. Peng, “Deterministic quantum teleportation through fiber channels,” Sci. Adv. 4(10), eaas9401 (2018).
[Crossref]

Cirac, J. I.

L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414(6862), 413–418 (2001).
[Crossref]

Cohen, L.

Cui, C.

C. Cui, Z.-Q. Yin, R. Wang, W. Chen, S. Wang, G.-C. Guo, and Z.-F. Han, “Twin-field quantum key distribution without phase postselection,” Phys. Rev. Appl. 11(3), 034053 (2019).
[Crossref]

Cui, C.-H.

S. Wang, D.-Y. He, Z.-Q. Yin, F.-Y. Lu, C.-H. Cui, W. Chen, Z. Zhou, G.-C. Guo, and Z.-F. Han, “Beating the fundamental rate-distance limit in a proof-of-principle quantum key distribution system,” Phys. Rev. X 9(2), 021046 (2019).
[Crossref]

Curty, M.

F. Grasselli, Á. Navarrete, and M. Curty, “Asymmetric twin-field quantum key distribution,” New J. Phys. 21(11), 113032 (2019).
[Crossref]

M. Curty, K. Azuma, and H.-K. Lo, “Simple security proof of twin-field type quantum key distribution protocol,” npj Quantum Information 5(1), 64 (2019).
[Crossref]

X. Zhong, J. Hu, M. Curty, L. Qian, and H.-K. Lo, “Proof-of-principle experimental demonstration of twin-field type quantum key distribution,” Phys. Rev. Lett. 123(10), 100506 (2019).
[Crossref]

H.-K. Lo, M. Curty, and B. Qi, “Measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 108(13), 130503 (2012).
[Crossref]

D’Auria, V.

V. D’Auria, O. Morin, C. Fabre, and J. Laurat, “Effect of the heralding detector properties on the conditional generation of single-photon states,” Eur. Phys. J. D 66(10), 249 (2012).
[Crossref]

Davidovich, L.

L. Davidovich, A. Maali, M. Brune, J. M. Raimond, and S. Haroche, “Quantum switches and nonlocal microwave fields,” Phys. Rev. Lett. 71(15), 2360–2363 (1993).
[Crossref]

Duan, L.-M.

L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414(6862), 413–418 (2001).
[Crossref]

Dynes, J. F.

M. Minder, M. Pittaluga, G. L. Roberts, M. Lucamarini, J. F. Dynes, Z. L. Yuan, and A. J. Shields, “Experimental quantum key distribution beyond the repeaterless secret key capacity,” Nat. Photonics 13(5), 334–338 (2019).
[Crossref]

M. Lucamarini, Z. L. Yuan, J. F. Dynes, and A. J. Shields, “Overcoming the rate-distance limit of quantum key distribution without quantum repeaters,” Nature 557(7705), 400–403 (2018).
[Crossref]

Eisenberg, H. S.

Fabre, C.

V. D’Auria, O. Morin, C. Fabre, and J. Laurat, “Effect of the heralding detector properties on the conditional generation of single-photon states,” Eur. Phys. J. D 66(10), 249 (2012).
[Crossref]

Fan-Yuan, G.-J.

F.-Y. Lu, Z.-Q. Yin, R. Wang, G.-J. Fan-Yuan, S. Wang, D.-Y. He, W. Chen, W. Huang, B.-J. Xu, G.-C. Guo, and Z.-F. Han, “Practical issues of twin-field quantum key distribution,” arXiv: 1901.04264v3 (2019).

Ferreyrol, F.

A. Ourjoumtsev, F. Ferreyrol, R. Tualle-Brouri, and P. Grangier, “Preparation of non-local superpositions of quasi-classical light states,” Nat. Phys. 5(3), 189–192 (2009).
[Crossref]

Gerry, C. C.

C. C. Gerry, J. Mimih, and A. Benmoussa, “Maximally entangled coherent states and strong violations of bell-type inequalities,” Phys. Rev. A 80(2), 022111 (2009).
[Crossref]

C. C. Gerry, A. Benmoussa, and R. A. Campos, “Nonlinear interferometer as a resource for maximally entangled photonic states: Application to interferometry,” Phys. Rev. A 66(1), 013804 (2002).
[Crossref]

Gisin, N.

Gomonay, H. V.

A. A. Semenov, A. V. Turchin, and H. V. Gomonay, “Detection of quantum light in the presence of noise,” Phys. Rev. A 78(5), 055803 (2008).
[Crossref]

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Figures (5)

Fig. 1.
Fig. 1. (a) Scheme for the direct transmission of ECS via two symmetric sides of a lossy channel. (b) Scheme for distributing ECS with Schrödinger$'$s cat state. (c) Distribution of ECS using TFQKD method. BS1 and BS2 are two beam splitters (BSs) with transmittance $\sqrt {\eta }$, which simulate the channel transmittance during halfway transmission. BS3 and BS4 are two BSs with transmittance $T$. BS5 is a $50:50$ BS for interference. $D_c$ and $D_d$ are two single photon detectors. The trash boxes denote the discarding operation.
Fig. 2.
Fig. 2. (a) Fidelity and (b) success probability for the distribution of ECS $|\psi _\mathrm {ECS}\rangle$ as a function of channel efficiency. (c) Tradeoff between fidelity and success probability for the schemes in Figs. 1(b) and (c). Curves are plotted according to the analytical results in Sec. 2 and Sec. 3 without any approximations. Other parameters: $\alpha =2.0$ and $T=0.95$.
Fig. 3.
Fig. 3. Yield as a function of transmission distance. The channel efficiency is assumed to be $0.20$ dB/km. Green dashed line represents the linear trend $Y=\eta$. Black solid line represents the square root trend $Y=\frac {1}{4}\sqrt {\eta }$. Other parameters: $\alpha =2.2$ and $T=0.98$.
Fig. 4.
Fig. 4. Fidelity and success probability when an inefficient on-off detector is used. Other parameters: $\alpha =2.5,T=0.98$.
Fig. 5.
Fig. 5. Yield of distribution of $|\psi _\mathrm {ECS}(\alpha )\rangle$ as a function of transmission distance. The channel efficiency is assumed to be $0.20$ dB/km. Green dashed line represents the linear trend $Y=\eta$. Black solid line represents the square root trend $Y=\frac {\tau }{4}\sqrt {\eta }$. Other parameters: $\alpha =2.5$, $T=0.98$, and (a) $\tau =0.90$ (b) $\tau =0.60$.

Equations (23)

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| ψ E C S ( α ) = 1 N ( α ) ( | α | α | α | α ) ,
U k ( t ) = exp [ ( a r c t a n ( 1 t ) / t ) ( a ^ k a ^ a ^ k a ^ ) ] ,
U ( t ) | α | β = | α t β 1 t | α 1 t + β t .
| ψ ( a ) = U A C ( η ) U B D ( η ) | ψ E C S ( α ) A B | 00 C D = 1 N ( | α η 1 4 , α 1 η , α η 1 4 , α 1 η A C B D           | α η 1 4 , α 1 η , α η 1 4 , α 1 η A C B D ) .
ρ ( a ) = T r C D | ψ ( a ) ψ ( b ) | = e 4 α 2 ( 1 η ) N ( α η 1 4 ) N ( α ) | ψ E C S ( α η 1 4 ) A B ψ E C S ( α η 1 4 ) |         + 1 e 4 α 2 ( 1 η ) N ( α ) | α η 1 4 , α η 1 4 α η 1 4 , α η 1 4 |         + 1 e 4 α 2 ( 1 η ) N ( α ) | α η 1 4 , α η 1 4 α η 1 4 , α η 1 4 | ,
| ψ ( b ) A C B D = 1 N c ( | T α , 1 T α A C + | T α , 1 T α A C ) ( | T α , 1 T α B D + | T α , 1 T α B D ) .
| ψ ( b ) A C E B D F = 1 N c × ( | T α , f 1 α , f 2 α , T α , f 1 α , f 2 α A C E B D F + | T α , f 1 α , f 2 α , T α , f 1 α , f 2 α A C E B D F + | T α , f 1 α , f 2 α , T α , f 1 α , f 2 α A C E B D F + | T α , f 1 α , f 2 α , T α , f 1 α , f 2 α A C E B D F ) ,
| ψ ( b ) A C E B D F = 1 N c × ( | T α , 0 , f 2 α , T α , 2 f 1 α , f 2 α A C E B D F + | T α , 2 f 1 α , f 2 α , T α , 0 , f 2 α A C E B D F + | T α , 2 f 1 α , f 2 α , T α , 0 , f 2 α A C E B D F + | T α , 0 , f 2 α , T α , 2 f 1 α , f 2 α A C E B D F ) .
P 10 ( b ) ρ 10 ( b ) = T r C D E F [ | ψ ( b ) A C E B D F ψ ( b ) | ( | 10 C D 10 | I E F ) ] .
ρ 10 ( b ) = e 4 f 2 2 α 2 N ( T α ) 2 2 e 4 ( f 1 2 1 ) α 2 | ψ E C S ( T α ) ψ E C S ( T α ) |               + 1 e 4 f 2 2 α 2 2 2 e 4 ( f 1 2 1 ) α 2 | T α , T α T α , T α |               + 1 e 4 f 2 2 α 2 2 2 e 4 ( f 1 2 1 ) α 2 | T α , T α T α , T α | .
P 10 ( b ) = 4 f 1 2 α 2 e 2 f 1 2 α 2 N c 2 ( 1 e 4 ( f 1 2 1 ) α 2 ) = η ( 1 T ) 4 α 2 e 2 f 1 2 α 2 N c 2 ( 1 e 4 ( f 1 2 1 ) α 2 ) .
| ψ h A C = 1 2 ( | α A | 0 C + | α A | 1 C ) ,
| ψ h D B = 1 2 ( | 0 D | α B + | 1 D | α B ) .
ρ A C = ρ B D = T r E [ U C E ( η ) | ψ h ψ h | | 0 E 0 | U C E ( η ) ] = 1 2 ( | ϕ ~ ϕ ~ | + ( 1 η ) | α , 0 α , 0 | ) ,
P 10 ( c ) ρ 10 ( c ) = T r C D [ U C D ( 1 2 ) ( ρ A C ρ D B ) U C D ( 1 2 ) | 10 C D 10 | ] .
ρ 10 ( c ) = N N + 2 2 η | ψ E C S ( α ) ψ E C S ( α ) | + 2 ( 1 η ) N + 2 ( 1 η ) | α , α α , α | ,
P 10 ( c ) = 1 4 ( 2 e 4 α 2 η ) η .
F ( a ) = e 2 α 2 ( 1 η 1 / 4 ) 2 ( 1 + e 4 α 2 ( 1 η ) ) ( 1 e 4 α 2 η 1 / 4 ) 2 2 ( 1 e 4 α 2 ) 2 , F ( b ) = e 2 ( 1 + T ) 2 α 2 ( 1 + e 4 T α 2 ) 2 ( 1 + e 4 ( 1 + T ) α 2 ( 1 + η ) ) 2 ( 1 e 4 α 2 ) ( 1 e 4 α 2 ( 1 + ( 1 + T ) η ) ) , F ( c ) = 1 e 4 α 2 1 + e 4 α 2 ( 2 + η ) .
Y ( a ) = c a 1 η + c a 2 η + O ( η 3 2 ) ,
Y ( b ) = c b 1 η + c b 2 η + O ( η 3 2 ) ,
Y ( c ) = 1 4 ( 1 e 4 α 2 ) η + O ( η 5 2 ) ,
c a 1 = 8 e 2 α 2 ( 1 + e 4 α 2 ) α 4 ( 1 + e 4 α 2 ) 2 , c a 2 = 16 e 2 α 2 α 6 ( 3 3 e 4 α 2 + 2 α 2 + 2 e 4 α 2 α 2 ) 3 ( 1 + e 4 α 2 ) 2 , c b 1 = e 6 α 2 4 T α 2 2 T α 2 ( e 4 T α 2 1 ) 2 ( e 4 α 2 + e 4 T α 2 ) ( 1 T ) α 2 2 ( e 4 α 2 1 ) 3 , c b 2 = e 6 α 2 4 T α 2 2 T α 2 ( e 4 T α 2 1 ) 2 ( e 4 T α 2 e 4 α 2 ) ( 1 T ) 2 α 4 ( e 4 α 2 1 ) 3 .
Y o n , o f f ( b ) = P o n , o f f ( b ) F o n , o f f ( b ) , Y o n , o f f ( c ) = P o n , o f f ( c ) F o n , o f f ( c ) .

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