Abstract

Collective atomic excitation can be realized by the Raman scattering. Such a photon-atom interface can form an SU(1,1)-typed atom-light hybrid interferometer, where the atomic Raman amplification processes take the place of the beam splitting elements in a traditional Mach-Zehnder interferometer. We numerically calculate the phase sensitivities and the signal-to-noise ratios (SNRs) of this interferometer with the method of homodyne detection and intensity detection, and give their differences of the optimal phase points to realize the best phase sensitivities and the maximal SNRs from these two detection methods. The difference of the effects of loss of light field and atomic decoherence on measure precision is analyzed.

© 2016 Optical Society of America

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References

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2016 (2)

B. P. Abbott and et al., (LIGO Scientific Collaboration and Virgo Collaboration), “Observation of gravitational waves from a binary black hole merger,” Phys. Rev. Lett. 116, 061102 (2016).
[Crossref]

S. A. Haine and W. Y. S. Lau, “Generation of atom-light entanglement in an optical cavity for quantum enhanced atom interferometry,” Phys. Rev. A 93, 023607 (2016).
[Crossref]

2015 (6)

H. Ma, D. Li, C.-H. Yuan, L. Q. Chen, Z. Y. Ou, and W. Zhang, “SU(1,1)-type light-atom-correlated interferometer,” Phys. Rev. A 92, 023847 (2015).
[Crossref]

B. Chen, C. Qiu, L. Q. Chen, K. Zhang, J. Guo, C.-H. Yuan, Z. Y. Ou, and W. Zhang, “Phase sensitive Raman process with correlated seeds,” Appl. Phys. Lett. 106, 111103 (2015).
[Crossref]

B. Chen, C. Qiu, S. Chen, J. Guo, L. Q. Chen, Z. Y. Ou, and W. Zhang, “Atom-light hybrid interferometer,” Phys. Rev. Lett. 115, 043602 (2015).
[Crossref] [PubMed]

R. D. Dobrzanski, M. Jarzyna, and J. Kolodyński, “Quantum limits in optical interferometry,” Prog. Opt. 60, 345 (2015).
[Crossref]

J. Peise, B. Lücke, L. Pezzè, F. Deuretzbacher, W. Ertmer, J. Arlt, A. Smerzi, L. Santos, and C. Klempt, “Interaction-free measurements by quantum Zeno stabilization of ultracold atoms,” Nat. Commun. 6, 6811 (2015).
[Crossref] [PubMed]

M. Gabbrielli, L. Pezzè, and A. Smerzi, “Spin-mixing interferometry with Bose-Einstein condensates,” Phys. Rev. Lett. 115, 163002 (2015).
[Crossref] [PubMed]

2014 (8)

F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014).
[Crossref]

W. Dür, M. Skotiniotis, F. Fröwis, and B. Kraus, “Improved quantum metrology using quantum error correction,” Phys. Rev. Lett. 112, 080801 (2014).
[Crossref]

E. M. Kessler, I. Lovchinsky, A. O. Sushkov, and M. D. Lukin, “Quantum error correction for metrology,” Phys. Rev. Lett. 112, 150802 (2014).
[Crossref] [PubMed]

S. Alipour, M. Mehboudi, and A. T. Rezakhani, “Quantum metrology in open systems: dissipative Cramér-Rao bound,” Phys. Rev. Lett. 112, 120405 (2014).
[Crossref]

G. Tóth and I. Apellaniz, “Quantum metrology from a quantum information science perspective,” J. Phys. A 47, 424006 (2014).
[Crossref]

S. S. Szigeti, B. Tonekaboni, W. Y. S. Lau, S. N. Hood, and S. A. Haine, “Squeezed-light-enhanced atom interferometry below the standard quantum limit,” Phys. Rev. A 90, 063630 (2014).
[Crossref]

Sh. Barzanjeh, D. P. DiVincenzo, and B. M. Terhal, “Dispersive qubit measurement by interferometry with parametric amplifiers,” Phys. Rev. B 90, 134515 (2014).
[Crossref]

D. Li, C.-H. Yuan, Z. Y. Ou, and W. Zhang, “The phase sensitivity of an SU(1,1) interferometer with coherent and squeezed-vacuum light,” New J. Phys. 16, 073020 (2014).
[Crossref]

2013 (5)

C.-H. Yuan, L. Q. Chen, Z. Y. Ou, and W. Zhang, “Correlation-enhanced phase-sensitive Raman scattering in atomic vapors,” Phys. Rev. A 87, 053835 (2013).
[Crossref]

S. A. Haine, “Information-recycling beam splitters for quantum enhanced atom interferometry,” Phys. Rev. Lett. 110, 053002 (2013).
[Crossref] [PubMed]

L. Li, Y. O. Dudin, and A. Kuzmich, “Entanglement between light and an optical atomic excitation,” Nature 498, 466 (2013).
[Crossref] [PubMed]

D. W. Berry, Michael J. W. Hall, and Howard M. Wiseman, “Stochastic Heisenberg limit: optimal estimation of a fluctuating phase,” Phys. Rev. Lett. 111, 113601 (2013).
[Crossref] [PubMed]

R. Chaves, J. B. Brask, M. Markiewicz, J. Kołodyński, and A. Acín, “Noisy metrology beyond the standard quantum limit,” Phys. Rev. Lett. 111, 120401 (2013).
[Crossref] [PubMed]

2012 (3)

R. Demkowicz-Dobrzanski, J. Kolodyński, and M. Gutǎ, “The elusive Heisenberg limit in quantum-enhanced metrology,” Nat. Commun. 3, 1063 (2012).
[Crossref] [PubMed]

Z. Y. Ou, “Enhancement of the phase-measurement sensitivity beyond the standard quantum limit by a nonlinear interferometer,” Phys. Rev. A 85, 023815 (2012).
[Crossref]

A. M. Marino, N. V. Corzo Trejo, and P. D. Lett, “Effect of losses on the performance of an SU(1,1) interferometer,” Phys. Rev. A 86, 023844 (2012).
[Crossref]

2011 (3)

B. M. Escher, R. L. de Matos Filho, and L. Davidovich, “General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology,” Nat. Phys. 7, 406 (2011).
[Crossref]

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

J. Jing, C. Liu, Z. Zhou, Z. Y. Ou, and W. Zhang, “Realization of a nonlinear interferometer with parametric amplifiers,” Appl. Phys. Lett. 99, 011110 (2011).
[Crossref]

2010 (6)

W. N. Plick, J. P. Dowling, and G. S. Agarwal, “Coherent-light-boosted sub-shot noise quantum interferometry,” New J. Phys. 12, 083014 (2010).
[Crossref]

C. Gross, T. Zibold, E. Nicklas, J. Estève, and M. K. Oberthaler, “Nonlinear atom interferometer surpasses classical precision limit,” Nature 464, 1165 (2010).
[Crossref] [PubMed]

M. Zwierz, C. A. Pérez-Delgado, and P. Kok, “General optimality of the Heisenberg limit for quantum metrology,” Phys. Rev. Lett. 105, 180402 (2010).
[Crossref]

K. Hammerer, A. S. Sørensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82, 1041 (2010).
[Crossref]

L. Q. Chen, G. W. Zhang, C.-L. Bian, C.-H. Yuan, Z. Y. Ou, and W. Zhang, “Observation of the Rabi oscillation of light driven by an atomic spin wave,” Phys. Rev. Lett. 105, 133603 (2010).
[Crossref]

C.-H. Yuan, L. Q. Chen, J. Jing, Z. Y. Ou, and W. Zhang, “Coherently enhanced Raman scattering in atomic vapor,” Phys. Rev. A 82, 013817 (2010).
[Crossref]

2009 (2)

L. Q. Chen, G. W. Zhang, C.-H. Yuan, J. Jing, Z. Y. Ou, and W. Zhang, “Enhanced Raman scattering by spatially distributed atomic coherence,” Appl. Phys. Lett. 95, 041115 (2009).
[Crossref]

R. D. Dobrzanski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Quantum phase estimation with lossy interferometers,” Phys. Rev. A 80, 013825 (2009).
[Crossref]

2008 (1)

J. P. Dowling, “Quantum optical metrology–the lowdown on high-N00N states,” Contemp. Phys. 49, 125 (2008).
[Crossref]

2006 (1)

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
[Crossref] [PubMed]

2004 (3)

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306, 1330 (2004).
[Crossref] [PubMed]

M. G. Raymer, “Quantum state entanglement and readout of collective atomic-ensemble modes and optical wave packets by stimulated Raman scattering,” J. Mod. Opt. 51, 1739 (2004).
[Crossref]

O. Steuernagel and S. Scheel, “Approaching the Heisenberg limit with two-mode squeezed states,” J. Opt. B: Quantum Semiclass. Opt. 6, S66 (2004).
[Crossref]

2002 (1)

H. Lee, P. Kok, and J. P. Dowling, “A quantum Rosetta stone for interferometry,” J. Mod. Opt. 49, 2325 (2002).
[Crossref]

2001 (1)

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 (2001).
[Crossref] [PubMed]

2000 (1)

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733 (2000).
[Crossref] [PubMed]

1998 (1)

T. Kim, O. Pfister, M. J. Holland, J. Noh, and J. L. Hall, “Influence of decorrelation on Heisenberg-limited interferometry with quantum correlated photons,” Phys. Rev. A 57, 4004 (1998).
[Crossref]

1997 (1)

B. C. Sanders, G. J. Milburn, and Z. Zhang, “Optimal quantum measurements for phase-shift estimation in optical interferometry,” J. Mod. Opt. 44, 1309 (1997).

1996 (1)

S. L. Braunstein, C. M. Caves, and G. J. Milburn, “Generalized uncertainty relations: theory, examples, and Lorentz invariance,” Ann. Phys. 247, 135 (1996).
[Crossref]

1995 (2)

R. Lynch, “The quantum phase problem: a critical review,” Phys. Rep. 256, 367 (1995).
[Crossref]

J. Jacobson, G. Björk, and Y. Yamamoto, “Quantum limit for the atom-light interferometer,” Appl. Phys. B 60, 187 (1995).
[Crossref]

1994 (2)

S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439 (1994).
[Crossref] [PubMed]

U. Leonhardt, “Quantum statistics of a two-mode SU(1,1) interferometer,” Phys. Rev. A 49, 1231 (1994).
[Crossref] [PubMed]

1990 (1)

A. Vourdas, “SU(2) and SU(1,1) phase states,” Phys. Rev. A 41, 1653 (1990).
[Crossref] [PubMed]

1987 (2)

M. Xiao, L. A. Wu, and H. J. Kimble, “Precision measurement beyond the short noise limit,” Phys. Rev. Lett. 59, 278 (1987).
[Crossref] [PubMed]

P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta, “Squeezed-light-enhanced polarization interferometer,” Phys. Rev. Lett. 59, 2153 (1987).
[Crossref] [PubMed]

1986 (1)

B. Yurke, S. L. McCall, and J. R. Klauder, “SU(2) and SU(1 1) interferometers,” Phys. Rev. A 33, 4033 (1986).
[Crossref]

1981 (1)

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693 (1981).
[Crossref]

Abbott, B. P.

B. P. Abbott and et al., (LIGO Scientific Collaboration and Virgo Collaboration), “Observation of gravitational waves from a binary black hole merger,” Phys. Rev. Lett. 116, 061102 (2016).
[Crossref]

Abrams, D. S.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733 (2000).
[Crossref] [PubMed]

Acín, A.

R. Chaves, J. B. Brask, M. Markiewicz, J. Kołodyński, and A. Acín, “Noisy metrology beyond the standard quantum limit,” Phys. Rev. Lett. 111, 120401 (2013).
[Crossref] [PubMed]

Agarwal, G. S.

W. N. Plick, J. P. Dowling, and G. S. Agarwal, “Coherent-light-boosted sub-shot noise quantum interferometry,” New J. Phys. 12, 083014 (2010).
[Crossref]

Alipour, S.

S. Alipour, M. Mehboudi, and A. T. Rezakhani, “Quantum metrology in open systems: dissipative Cramér-Rao bound,” Phys. Rev. Lett. 112, 120405 (2014).
[Crossref]

Anderson, B. E.

T. S. Horrom, B. E. Anderson, P. Gupta, and P. D. Lett, “SU(1,1) interferometry via four-wave mixing in Rb,” in 45th Winter Colloquium on the Physics of Quantum Electronics (PQE, 2015).

Apellaniz, I.

G. Tóth and I. Apellaniz, “Quantum metrology from a quantum information science perspective,” J. Phys. A 47, 424006 (2014).
[Crossref]

Arlt, J.

J. Peise, B. Lücke, L. Pezzè, F. Deuretzbacher, W. Ertmer, J. Arlt, A. Smerzi, L. Santos, and C. Klempt, “Interaction-free measurements by quantum Zeno stabilization of ultracold atoms,” Nat. Commun. 6, 6811 (2015).
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Figures (7)

Fig. 1
Fig. 1 (a) The intermode correlation between the Stokes field a ^ 1 and the atomic excitation b ^ 1 is generated by spontaneous Raman process. a ^ 0 is the initial input light field. b ^ 0 is in vacuum or an initial atomic collective excitation which can be prepared by another Raman process or electromagnetically induced transparency process. (b) During the delay time τ, the Stokes field a ^ 1 will be subject to the photon loss and evolute to a ^ 1 and the collective excitation b ^ 1 will undergo the collisional dephasing to b ^ 1 . A fictitious beam splitter (BS) is introduced to mimic the loss of photons into the environment. V ^ is the vacuum. (c) After the delay time τ, the light field a ^ 1 and its correlated atomic excitation b ^ 1 are used as initial seed for another enhanced Raman process. (d)–(f) The corresponding energy-level diagrams of different processes are shown.
Fig. 2
Fig. 2 The phase sensitivity ΔϕHD and the SNRHD versus the phase shift ϕ using the method of homodyne detection with (a) θα = π/2; (b) θα = 0. Parameters: g = 2, |α| = 10.
Fig. 3
Fig. 3 (a) Δna2, |⟨∂⟨na2⟩/∂ϕ⟩|, and the phase sensitivity ΔϕID; (b) ⟨na2⟩, Δna2 and the SNRID versus the phase shift ϕ using the method of intensity detection. Parameters: g = 2, |α| = 10.
Fig. 4
Fig. 4 (a) The optimal phase sensitivities Δϕ and (b) the maximal SNR versus the phase-sensing probe number nph. The optimal phase sensitivities ΔϕHD and ΔϕID are obtained at ϕ = 0 and ϕ = 0.062, respectively. The maximal SNRs are obtained at ϕ = 0 and θα = 0. Parameter: g = 2.
Fig. 5
Fig. 5 The linear correlation coefficients (a) Jx2; (b) Jn2 as a function of the phase shift ϕ for lossless case. Parameters: θ1 = 0, g = 2, |α| = 10.
Fig. 6
Fig. 6 The linear correlation coefficients Jx2 as a function of (a) the transmission rate T; (b) the collisional rate Γτ. Parameters: g = 2, |α| = 10, θα = π/2 and ϕ = 0.
Fig. 7
Fig. 7 The linear correlation coefficients Jn2 as a function of (a) the transmission rate T; (b) the collisional dephasing rate Γτ, where g = 2, |α| = 10, and θ2θ1 = π, and ϕ = 0.062.

Equations (42)

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a ^ ( t ) t = η A P b ^ ( t ) , b ^ ( t ) t = η A P a ^ ( t ) ,
a ^ ( t ) = u ( t ) a ^ ( 0 ) + v ( t ) b ^ ( 0 ) , b ^ ( t ) = u ( t ) b ^ ( 0 ) + v ( t ) a ^ ( 0 ) ,
a ^ 1 = T a ^ 1 ( t 1 ) e i ϕ + R V ^ ,
b ^ 1 = b ^ 1 ( t 1 ) e Γ τ + F ^ ,
a ^ 2 ( t 2 ) = U 1 a ^ 1 ( 0 ) + V 1 b ^ 1 ( 0 ) + R u 2 V ^ + v 2 F ^ ,
b ^ 2 ( t 2 ) = e i ϕ [ U 2 b ^ 1 ( 0 ) + V 2 a ^ 1 ( 0 ) ] + R v 2 V ^ + u 2 F ^ ,
U 1 = T u 1 u 2 e i ϕ + e Γ τ v 1 * v 2 , V 1 = T v 1 u 2 e i ϕ + e Γ τ u 1 * v 2 U 2 = e Γ τ u 1 u 2 e i ϕ + T v 1 * v 2 , V 2 = e Γ τ v 1 u 2 e i ϕ + T u 1 * v 2 .
Δ ϕ = ( Δ O ^ ) 2 1 / 2 | O ^ / ϕ | ,
SNR = O ^ ( Δ O ^ ) 2 1 / 2 .
Δ ϕ HD = ( Δ x ^ a 2 ) 2 1 / 2 N α cosh 2 g | sin ( ϕ + θ α ) | ,
SNR HD = N α [ cosh 2 g cos ( ϕ + θ α ) sinh 2 g cos ( θ α ) ] ( Δ x ^ a 2 ) 2 1 / 2 ,
( Δ x ^ a 2 ) 2 = 1 4 [ cosh 2 ( 2 g ) sinh 2 ( 2 g ) cos ϕ ] ,
Δ ϕ HD = 1 N a 1 2 cosh 2 g ,
SNR HD = 0 .
SNR HD = 2 N α ,
Δ ϕ ID = ( Δ n ^ a 2 ) 2 1 / 2 2 ( N α + 1 ) sinh 2 ( 2 g ) | sin ϕ | ,
SNR ID = 1 ( Δ n ^ a 2 ) 2 1 / 2 [ N α | cosh 2 g sinh 2 g e i ϕ | 2 + 1 2 sinh 2 ( 2 g ) ( 1 cos ϕ ) ] ,
( Δ n ^ a 2 ) 2 = N α | cosh 2 g sinh 2 g e i ϕ | 4 + 1 2 ( 1 + N α ) × sinh 2 ( 2 g ) | cosh 2 g sinh 2 g e i ϕ | 2 ( 1 cos ϕ ) .
( 1 + cos 2 ϕ ) 2 sinh 2 ( 2 g ) [ ( 2 N α + 1 ) cosh 2 ( 2 g ) + N α ] cos ϕ [ 4 N α ( 2 cosh 2 ( 2 g ) + sinh 4 ( 2 g ) ) + ( 1 + N α ) sinh 2 4 g ] = 0 .
x ^ a 2 = N α e Γ τ sinh 2 g cos θ α + T N α cosh 2 g cos ( ϕ + θ α ) ,
n ^ a 2 = ( T cosh 4 g + e 2 Γ τ sinh 4 g ) N α + sinh 2 g ( 1 e 2 Γ τ ) + 1 4 sinh 2 ( 2 g ) ( T + e 2 Γ τ ) 1 2 T e Γ τ ( N α + 1 ) sinh 2 ( 2 g ) cos ϕ ,
| x ^ a 2 ϕ | = T N cosh 2 g | sin ( ϕ + θ α ) | ,
| n ^ a ϕ | = 1 2 T e Γ τ ( N α + 1 ) sinh 2 ( 2 g ) | sin ( ϕ ) | .
( Δ x ^ a 2 ) 2 = 1 4 [ sinh 2 ( 2 g ) ( T 2 T e Γ τ cos ϕ ) + 2 e 2 Γ τ sinh 4 g + cosh ( 2 g ) ] ,
( Δ n ^ a 2 ) 2 = | U b | 4 N α + | U b V b | 2 ( 1 + N α ) + R cosh 2 g ( | U b | 2 N α + | V b | 2 ) + sinh 2 g [ | U b | 2 ( 1 + N α ) + R cosh 2 g ] ( 1 e 2 Γ τ ) ,
| U b | 2 = ( T cosh 2 g + e Γ τ sinh 2 g ) 2 2 T e Γ τ sinh 2 g cosh 2 g ( 1 + cos ϕ ) ,
| V b | 2 = 1 2 sinh 2 ( 2 g ) ( T + e 2 Γ τ 2 T e Γ τ cos ϕ ) ,
n ph = N α cosh ( 2 g ) + 2 sinh 2 g .
J ( A ^ , B ^ ) = c o v ( A ^ , B ^ ) ( Δ A ^ ) 2 1 / 2 ( Δ B ^ ) 2 1 / 2 ,
J x 1 ( x ^ a 1 , x ^ b 1 ) = cos θ 1 tanh ( 2 g ) ,
J y 1 ( y ^ a 1 , y ^ b 1 ) = cos θ 1 tanh ( 2 g ) ,
J n 1 ( n ^ a 1 , n ^ b 1 ) = ( 1 + 2 | a | 2 ) [ 4 coth 2 ( 2 g ) ( | a | 2 + | a | 4 ) + 1 ] 1 / 2 .
J x 2 ( x ^ a 2 , x ^ b 2 ) = c o v ( x ^ a 2 , x ^ b 2 ) ( Δ x ^ a 2 ) 2 1 / 2 ( Δ x ^ b 2 ) 2 1 / 2 ,
c o v ( x ^ a 2 , x ^ b 2 ) = 1 4 Re [ e i ϕ ( V 1 U 2 + U 1 V 2 ) + u 2 v 2 ( R + 1 e 2 Γ τ ) ] , ( Δ x ^ a 2 ) 2 = 1 4 [ | U 1 | 2 + | V 1 | 2 + R | u 2 | 2 + | v 2 | 2 ( 1 e 2 Γ τ ) ] , ( Δ x ^ b 2 ) 2 = 1 4 [ | U 2 | 2 + | V 2 | 2 + R | u 2 | 2 + | v 2 | 2 ( 1 e 2 Γ τ ) ] .
J n 2 ( n ^ a 2 , n ^ b 2 ) = c o v ( n ^ a 2 , n ^ b 2 ) ( Δ n ^ a 2 ) 2 1 / 2 ( Δ n ^ b 2 ) 2 1 / 2 ,
c o v ( n ^ a 2 , n ^ b 2 ) = | U 1 V 1 | 2 | α | 2 + ( 1 + | α | 2 ) Re [ U 1 * U 2 V 1 V 2 * ] + R [ Re [ e i ϕ U 2 V 1 u 2 * v 2 * ] + | α | 2 Re [ e i ϕ U 1 V 2 u 2 * v 2 * ] ] + [ R | u 2 v 2 | 2 + ( 1 + | α | 2 ) Re [ e i ϕ U 1 * V 2 * u 2 v 2 ] ] ( 1 e 2 Γ τ ) ,
( Δ n ^ a 2 ) 2 = | U 1 | 4 | α | 2 + | U 1 V 1 | 2 ( 1 + | α | 2 ) + R | u 2 | 2 ( | V 1 | 2 + | U 1 | 2 | α | 2 ) + | v 2 | 2 [ | U 1 | 2 ( | α | 2 + 1 ) + R | u 2 | 2 ] ( 1 e 2 Γ τ ) ,
( Δ n ^ b 2 ) 2 = | V 2 | 4 | α | 2 + | U 2 V 2 | 2 ( 1 + | α | 2 ) + R | v 2 | 2 ( | U 2 | 2 + | V 2 | 2 | α | 2 ) + | u 2 | 2 [ | V 2 | 2 ( 1 + | α | 2 ) + R | v 2 | 2 ] ( 1 e 2 Γ τ ) .
J x 2 ( x ^ a 2 , x ^ b 2 ) = 2 Re [ V U e i ϕ ] | U | 2 + | V | 2 = sinh ( 2 g ) cosh 2 ( 2 g ) sinh 2 ( 2 g ) cos ϕ [ cosh 2 g cos ( θ 1 + 3 ϕ ) + sinh 2 g cos ( θ 1 + ϕ ) cosh ( 2 g ) cos ( θ 1 + 2 ϕ ) ]
J n 2 ( n ^ a 2 , n ^ b 2 ) = | U V ( 1 + 2 | α | 2 ) | ( U V ) 1 / 2 = ( 1 + 2 | α | 2 ) × [ 4 [ 1 + sinh 2 ( 2 g ) ( 1 cos ϕ ) ] 2 ( | α | 2 + | α | 4 ) [ 1 + sinh 2 ( 2 g ) ( 1 cos ϕ ) ] 2 1 + 1 ] 1 / 2 ,
J x 2 ( x ^ a 2 , x ^ b 2 ) = tanh ( 4 g ) cos ( θ 1 ) ,
J n 2 ( n ^ a 2 , n ^ b 2 ) = 1 + 2 | α | 2 4 coth 2 ( 2 g 1 ) ( | α | 2 + | α | 4 ) + 1 = J n 1 ( n ^ a 1 , n ^ b 1 ) .

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