Abstract

We present a quantum model for electro-optic amplitude modulation, which is built upon quantum models of the main photonic components that constitute the modulator, that is, the guided-wave beamsplitter and the electro-optic phase modulator and accounts for all the different available modulator structures. General models are developed both for single and dual drive configurations and specific results are obtained for the most common configurations currently employed. Finally, the operation with two-photon input for the control of phase-modulated photons and the important topic of multicarrier modulation are also addressed.

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  1. L. Thylen, U. Westergren, P. Holmström, R. Schatz, and P. Jänes, “Recent developments in high-speed optical modulators,” in Optical Fiber Telecommunications V.A, I.P Kaminow, T. Li, and A.E. Willner, eds. (Academic Press, 2008), Chap. 7.
  2. G. P. Agrawal, Fiber-Optics Communications Systems (John Wiley & Sons, New York, 2002).
  3. C. H. Cox III, Analog Optical Links: Theory and Practice (Cambridge Univ. Press, 2004).
  4. M. Howerton, and W. K. Burns, “Broadband traveling wave modulators in LiNbO3,” in RF Photonic Technology in Optical Fiber Links, W.S. Chang, ed. (Cambridge Univ. Press, 2002), Chap. 5.
  5. G. E. Betts, “LiNbO3 external modulators and their use in high performance analog links,” in RF Photonic Technology in Optical Fiber Links(Cambridge Univ. Press, 2002), Chap. 4.
  6. A. Yariv, and P. Yeh, Photonics: Optical Electronics in Modern Communications (Oxford Univ. Press, 2006).
  7. N. Kashima, Passive optical Components for optical Fiber Transmission (Artech House, Boston, 1995).
  8. R. Syms, and J. Cozens, Optical Guided waves and Devices (McGraw-Hill, New York, 1992)
  9. B. E. A. Saleh, and M. C. Teich, Fundamentals of Photonics (John Wiley & Sons, New York, 1991)
  10. H. A. Bachor, and T. C. Ralph, A Guide to Experiments in Quantum Optics (Wiley-VCH, Weinheim, 2003).
  11. J. M. Mérolla, Y. Mazurenko, J.-P. Goedgebuer, L. Duraffourg, H. Porte, and W. T. Rhodes, “Quantum cryptographic device using single-photon phase modulation,” Phys. Rev. A 60(3), 1899–1905 (1999).
    [CrossRef]
  12. O. Guerreau, J.-M. Mérolla, A. Soujaeff, F. Patois, J. P. Goedgebuer, and F. J. Malassenet, “Long distance QKD transmission using single sideband detection scheme with WDM synchronization,” IEEE J. Sel. Top. Quantum Electron. 9(6), 1533–1540 (2003).
    [CrossRef]
  13. G. B. Xavier and J.-P. von der Weid, “Modulation schemes for frequency coded quantum key distribution,” Electron. Lett. 41(10), 607–608 (2005).
    [CrossRef]
  14. A. Ortigosa-Blanch and J. Capmany, “Subcarrier multiplexing optical quantum key distribution,” Phys. Rev. A 73(2), 024305 (2006).
    [CrossRef]
  15. P. Kolchin, C. Belthangady, S. Du, G. Y. Yin, and S. E. Harris, “Electro-optic modulation of single photons,” Phys. Rev. Lett. 101(10), 103601 (2008).
    [CrossRef] [PubMed]
  16. C. C. Gerry, and P. L. Knight, Introductory Quantum Optics (Cambridge Univ. Press, 2005).
  17. M. Fox, Quantum optics: An introduction (Oxford Univ. Press, 2006).
  18. J. C. Garrison, and R. Y. Chiao, Quantum Optics (Oxford Univ. Press, 2008).
  19. U. Leonhardt, “Quantum physics of simple optical instruments,” Rep. Prog. Phys. 66(7), 1207–1249 (2003).
    [CrossRef]
  20. P. Kumar and A. Prabhakar, “Evolution of quantum states in an electro-optic phase modulator,” IEEE J. Quantum Electron. 45(2), 149–156 (2009).
    [CrossRef]
  21. J. Capmany and C. R. Fernández-Pousa, “Quantum model for electro-optical phase modulation,” J. Opt. Soc. Am. B 27(6), A119–A129 (2010).
    [CrossRef]
  22. J. É. Capmany, A. Ortigosa-Blanch, J. É. Mora, A. Ruiz-Alba, W. Amaya, and A. Martinez, “Analysis of subcarrier multiplexed quantum key distribution systems: signal, intermodulation and quantum bit error rate,” IEEE J. Sel. Top. Quantum Electron. 15(6), 1607–1621 (2009).
    [CrossRef]

2010 (1)

2009 (2)

P. Kumar and A. Prabhakar, “Evolution of quantum states in an electro-optic phase modulator,” IEEE J. Quantum Electron. 45(2), 149–156 (2009).
[CrossRef]

J. É. Capmany, A. Ortigosa-Blanch, J. É. Mora, A. Ruiz-Alba, W. Amaya, and A. Martinez, “Analysis of subcarrier multiplexed quantum key distribution systems: signal, intermodulation and quantum bit error rate,” IEEE J. Sel. Top. Quantum Electron. 15(6), 1607–1621 (2009).
[CrossRef]

2008 (1)

P. Kolchin, C. Belthangady, S. Du, G. Y. Yin, and S. E. Harris, “Electro-optic modulation of single photons,” Phys. Rev. Lett. 101(10), 103601 (2008).
[CrossRef] [PubMed]

2006 (1)

A. Ortigosa-Blanch and J. Capmany, “Subcarrier multiplexing optical quantum key distribution,” Phys. Rev. A 73(2), 024305 (2006).
[CrossRef]

2005 (1)

G. B. Xavier and J.-P. von der Weid, “Modulation schemes for frequency coded quantum key distribution,” Electron. Lett. 41(10), 607–608 (2005).
[CrossRef]

2003 (2)

O. Guerreau, J.-M. Mérolla, A. Soujaeff, F. Patois, J. P. Goedgebuer, and F. J. Malassenet, “Long distance QKD transmission using single sideband detection scheme with WDM synchronization,” IEEE J. Sel. Top. Quantum Electron. 9(6), 1533–1540 (2003).
[CrossRef]

U. Leonhardt, “Quantum physics of simple optical instruments,” Rep. Prog. Phys. 66(7), 1207–1249 (2003).
[CrossRef]

1999 (1)

J. M. Mérolla, Y. Mazurenko, J.-P. Goedgebuer, L. Duraffourg, H. Porte, and W. T. Rhodes, “Quantum cryptographic device using single-photon phase modulation,” Phys. Rev. A 60(3), 1899–1905 (1999).
[CrossRef]

Amaya, W.

J. É. Capmany, A. Ortigosa-Blanch, J. É. Mora, A. Ruiz-Alba, W. Amaya, and A. Martinez, “Analysis of subcarrier multiplexed quantum key distribution systems: signal, intermodulation and quantum bit error rate,” IEEE J. Sel. Top. Quantum Electron. 15(6), 1607–1621 (2009).
[CrossRef]

Belthangady, C.

P. Kolchin, C. Belthangady, S. Du, G. Y. Yin, and S. E. Harris, “Electro-optic modulation of single photons,” Phys. Rev. Lett. 101(10), 103601 (2008).
[CrossRef] [PubMed]

Capmany, J.

J. Capmany and C. R. Fernández-Pousa, “Quantum model for electro-optical phase modulation,” J. Opt. Soc. Am. B 27(6), A119–A129 (2010).
[CrossRef]

A. Ortigosa-Blanch and J. Capmany, “Subcarrier multiplexing optical quantum key distribution,” Phys. Rev. A 73(2), 024305 (2006).
[CrossRef]

Capmany, J. É.

J. É. Capmany, A. Ortigosa-Blanch, J. É. Mora, A. Ruiz-Alba, W. Amaya, and A. Martinez, “Analysis of subcarrier multiplexed quantum key distribution systems: signal, intermodulation and quantum bit error rate,” IEEE J. Sel. Top. Quantum Electron. 15(6), 1607–1621 (2009).
[CrossRef]

Du, S.

P. Kolchin, C. Belthangady, S. Du, G. Y. Yin, and S. E. Harris, “Electro-optic modulation of single photons,” Phys. Rev. Lett. 101(10), 103601 (2008).
[CrossRef] [PubMed]

Duraffourg, L.

J. M. Mérolla, Y. Mazurenko, J.-P. Goedgebuer, L. Duraffourg, H. Porte, and W. T. Rhodes, “Quantum cryptographic device using single-photon phase modulation,” Phys. Rev. A 60(3), 1899–1905 (1999).
[CrossRef]

Fernández-Pousa, C. R.

Goedgebuer, J. P.

O. Guerreau, J.-M. Mérolla, A. Soujaeff, F. Patois, J. P. Goedgebuer, and F. J. Malassenet, “Long distance QKD transmission using single sideband detection scheme with WDM synchronization,” IEEE J. Sel. Top. Quantum Electron. 9(6), 1533–1540 (2003).
[CrossRef]

Goedgebuer, J.-P.

J. M. Mérolla, Y. Mazurenko, J.-P. Goedgebuer, L. Duraffourg, H. Porte, and W. T. Rhodes, “Quantum cryptographic device using single-photon phase modulation,” Phys. Rev. A 60(3), 1899–1905 (1999).
[CrossRef]

Guerreau, O.

O. Guerreau, J.-M. Mérolla, A. Soujaeff, F. Patois, J. P. Goedgebuer, and F. J. Malassenet, “Long distance QKD transmission using single sideband detection scheme with WDM synchronization,” IEEE J. Sel. Top. Quantum Electron. 9(6), 1533–1540 (2003).
[CrossRef]

Harris, S. E.

P. Kolchin, C. Belthangady, S. Du, G. Y. Yin, and S. E. Harris, “Electro-optic modulation of single photons,” Phys. Rev. Lett. 101(10), 103601 (2008).
[CrossRef] [PubMed]

Kolchin, P.

P. Kolchin, C. Belthangady, S. Du, G. Y. Yin, and S. E. Harris, “Electro-optic modulation of single photons,” Phys. Rev. Lett. 101(10), 103601 (2008).
[CrossRef] [PubMed]

Kumar, P.

P. Kumar and A. Prabhakar, “Evolution of quantum states in an electro-optic phase modulator,” IEEE J. Quantum Electron. 45(2), 149–156 (2009).
[CrossRef]

Leonhardt, U.

U. Leonhardt, “Quantum physics of simple optical instruments,” Rep. Prog. Phys. 66(7), 1207–1249 (2003).
[CrossRef]

Malassenet, F. J.

O. Guerreau, J.-M. Mérolla, A. Soujaeff, F. Patois, J. P. Goedgebuer, and F. J. Malassenet, “Long distance QKD transmission using single sideband detection scheme with WDM synchronization,” IEEE J. Sel. Top. Quantum Electron. 9(6), 1533–1540 (2003).
[CrossRef]

Martinez, A.

J. É. Capmany, A. Ortigosa-Blanch, J. É. Mora, A. Ruiz-Alba, W. Amaya, and A. Martinez, “Analysis of subcarrier multiplexed quantum key distribution systems: signal, intermodulation and quantum bit error rate,” IEEE J. Sel. Top. Quantum Electron. 15(6), 1607–1621 (2009).
[CrossRef]

Mazurenko, Y.

J. M. Mérolla, Y. Mazurenko, J.-P. Goedgebuer, L. Duraffourg, H. Porte, and W. T. Rhodes, “Quantum cryptographic device using single-photon phase modulation,” Phys. Rev. A 60(3), 1899–1905 (1999).
[CrossRef]

Mérolla, J. M.

J. M. Mérolla, Y. Mazurenko, J.-P. Goedgebuer, L. Duraffourg, H. Porte, and W. T. Rhodes, “Quantum cryptographic device using single-photon phase modulation,” Phys. Rev. A 60(3), 1899–1905 (1999).
[CrossRef]

Mérolla, J.-M.

O. Guerreau, J.-M. Mérolla, A. Soujaeff, F. Patois, J. P. Goedgebuer, and F. J. Malassenet, “Long distance QKD transmission using single sideband detection scheme with WDM synchronization,” IEEE J. Sel. Top. Quantum Electron. 9(6), 1533–1540 (2003).
[CrossRef]

Mora, J. É.

J. É. Capmany, A. Ortigosa-Blanch, J. É. Mora, A. Ruiz-Alba, W. Amaya, and A. Martinez, “Analysis of subcarrier multiplexed quantum key distribution systems: signal, intermodulation and quantum bit error rate,” IEEE J. Sel. Top. Quantum Electron. 15(6), 1607–1621 (2009).
[CrossRef]

Ortigosa-Blanch, A.

J. É. Capmany, A. Ortigosa-Blanch, J. É. Mora, A. Ruiz-Alba, W. Amaya, and A. Martinez, “Analysis of subcarrier multiplexed quantum key distribution systems: signal, intermodulation and quantum bit error rate,” IEEE J. Sel. Top. Quantum Electron. 15(6), 1607–1621 (2009).
[CrossRef]

A. Ortigosa-Blanch and J. Capmany, “Subcarrier multiplexing optical quantum key distribution,” Phys. Rev. A 73(2), 024305 (2006).
[CrossRef]

Patois, F.

O. Guerreau, J.-M. Mérolla, A. Soujaeff, F. Patois, J. P. Goedgebuer, and F. J. Malassenet, “Long distance QKD transmission using single sideband detection scheme with WDM synchronization,” IEEE J. Sel. Top. Quantum Electron. 9(6), 1533–1540 (2003).
[CrossRef]

Porte, H.

J. M. Mérolla, Y. Mazurenko, J.-P. Goedgebuer, L. Duraffourg, H. Porte, and W. T. Rhodes, “Quantum cryptographic device using single-photon phase modulation,” Phys. Rev. A 60(3), 1899–1905 (1999).
[CrossRef]

Prabhakar, A.

P. Kumar and A. Prabhakar, “Evolution of quantum states in an electro-optic phase modulator,” IEEE J. Quantum Electron. 45(2), 149–156 (2009).
[CrossRef]

Rhodes, W. T.

J. M. Mérolla, Y. Mazurenko, J.-P. Goedgebuer, L. Duraffourg, H. Porte, and W. T. Rhodes, “Quantum cryptographic device using single-photon phase modulation,” Phys. Rev. A 60(3), 1899–1905 (1999).
[CrossRef]

Ruiz-Alba, A.

J. É. Capmany, A. Ortigosa-Blanch, J. É. Mora, A. Ruiz-Alba, W. Amaya, and A. Martinez, “Analysis of subcarrier multiplexed quantum key distribution systems: signal, intermodulation and quantum bit error rate,” IEEE J. Sel. Top. Quantum Electron. 15(6), 1607–1621 (2009).
[CrossRef]

Soujaeff, A.

O. Guerreau, J.-M. Mérolla, A. Soujaeff, F. Patois, J. P. Goedgebuer, and F. J. Malassenet, “Long distance QKD transmission using single sideband detection scheme with WDM synchronization,” IEEE J. Sel. Top. Quantum Electron. 9(6), 1533–1540 (2003).
[CrossRef]

von der Weid, J.-P.

G. B. Xavier and J.-P. von der Weid, “Modulation schemes for frequency coded quantum key distribution,” Electron. Lett. 41(10), 607–608 (2005).
[CrossRef]

Xavier, G. B.

G. B. Xavier and J.-P. von der Weid, “Modulation schemes for frequency coded quantum key distribution,” Electron. Lett. 41(10), 607–608 (2005).
[CrossRef]

Yin, G. Y.

P. Kolchin, C. Belthangady, S. Du, G. Y. Yin, and S. E. Harris, “Electro-optic modulation of single photons,” Phys. Rev. Lett. 101(10), 103601 (2008).
[CrossRef] [PubMed]

Electron. Lett. (1)

G. B. Xavier and J.-P. von der Weid, “Modulation schemes for frequency coded quantum key distribution,” Electron. Lett. 41(10), 607–608 (2005).
[CrossRef]

IEEE J. Quantum Electron. (1)

P. Kumar and A. Prabhakar, “Evolution of quantum states in an electro-optic phase modulator,” IEEE J. Quantum Electron. 45(2), 149–156 (2009).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

J. É. Capmany, A. Ortigosa-Blanch, J. É. Mora, A. Ruiz-Alba, W. Amaya, and A. Martinez, “Analysis of subcarrier multiplexed quantum key distribution systems: signal, intermodulation and quantum bit error rate,” IEEE J. Sel. Top. Quantum Electron. 15(6), 1607–1621 (2009).
[CrossRef]

O. Guerreau, J.-M. Mérolla, A. Soujaeff, F. Patois, J. P. Goedgebuer, and F. J. Malassenet, “Long distance QKD transmission using single sideband detection scheme with WDM synchronization,” IEEE J. Sel. Top. Quantum Electron. 9(6), 1533–1540 (2003).
[CrossRef]

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

Phys. Rev. A (2)

J. M. Mérolla, Y. Mazurenko, J.-P. Goedgebuer, L. Duraffourg, H. Porte, and W. T. Rhodes, “Quantum cryptographic device using single-photon phase modulation,” Phys. Rev. A 60(3), 1899–1905 (1999).
[CrossRef]

A. Ortigosa-Blanch and J. Capmany, “Subcarrier multiplexing optical quantum key distribution,” Phys. Rev. A 73(2), 024305 (2006).
[CrossRef]

Phys. Rev. Lett. (1)

P. Kolchin, C. Belthangady, S. Du, G. Y. Yin, and S. E. Harris, “Electro-optic modulation of single photons,” Phys. Rev. Lett. 101(10), 103601 (2008).
[CrossRef] [PubMed]

Rep. Prog. Phys. (1)

U. Leonhardt, “Quantum physics of simple optical instruments,” Rep. Prog. Phys. 66(7), 1207–1249 (2003).
[CrossRef]

Other (13)

C. C. Gerry, and P. L. Knight, Introductory Quantum Optics (Cambridge Univ. Press, 2005).

M. Fox, Quantum optics: An introduction (Oxford Univ. Press, 2006).

J. C. Garrison, and R. Y. Chiao, Quantum Optics (Oxford Univ. Press, 2008).

L. Thylen, U. Westergren, P. Holmström, R. Schatz, and P. Jänes, “Recent developments in high-speed optical modulators,” in Optical Fiber Telecommunications V.A, I.P Kaminow, T. Li, and A.E. Willner, eds. (Academic Press, 2008), Chap. 7.

G. P. Agrawal, Fiber-Optics Communications Systems (John Wiley & Sons, New York, 2002).

C. H. Cox III, Analog Optical Links: Theory and Practice (Cambridge Univ. Press, 2004).

M. Howerton, and W. K. Burns, “Broadband traveling wave modulators in LiNbO3,” in RF Photonic Technology in Optical Fiber Links, W.S. Chang, ed. (Cambridge Univ. Press, 2002), Chap. 5.

G. E. Betts, “LiNbO3 external modulators and their use in high performance analog links,” in RF Photonic Technology in Optical Fiber Links(Cambridge Univ. Press, 2002), Chap. 4.

A. Yariv, and P. Yeh, Photonics: Optical Electronics in Modern Communications (Oxford Univ. Press, 2006).

N. Kashima, Passive optical Components for optical Fiber Transmission (Artech House, Boston, 1995).

R. Syms, and J. Cozens, Optical Guided waves and Devices (McGraw-Hill, New York, 1992)

B. E. A. Saleh, and M. C. Teich, Fundamentals of Photonics (John Wiley & Sons, New York, 1991)

H. A. Bachor, and T. C. Ralph, A Guide to Experiments in Quantum Optics (Wiley-VCH, Weinheim, 2003).

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Figures (5)

Fig. 1
Fig. 1

Different possible layouts for amplitude electro-optic modulators. After [5].

Fig. 2
Fig. 2

Quantum state labeling (upper) and single photon behavior (lower) of an optical beamsplitter.

Fig. 3
Fig. 3

Classical behavior of Y-branch guided-wave power splitter.

Fig. 4
Fig. 4

Typical configuration of a waveguide electro-optic phase modulator. (Lower) Black-box representation of the phase modulator under quantum regime.

Fig. 5
Fig. 5

Generic layout of an electro-optic amplitude modulator.

Equations (51)

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S ^ B S ( a ^ 1 a ^ 2 ) S ^ B S = ( t ' r ' r t ) ( a ^ 1 a ^ 2 ) ,
S ^ B S = exp ( j θ J ^ 1 ) = exp [ j θ ( a ^ 1 a ^ 2 + a ^ 2 a ^ 1 ) / 2 ] ,
t = t ' = cos ( θ / 2 ) r = r ' = j sin ( θ / 2 ) .
S ^ B S ( D ^ ( α ) 1 ) S ^ B S = D ^ ( α t ' ) D ^ ( α r ' ) S ^ B S ( 1 D ^ ( α ) ) S ^ B S = D ^ ( α r ) D ^ ( α t ) ,
S ^ B S ( | α 1 | β 2 ) = | α t ' + β r 1 ' | α r ' + β t 2 ' ,
t = t ' = 1 k     r = r ' = j k .
| t ' | 2 + | r ' | 2 = | t | 2 + | r | 2 = 1       r * t ' + r ' t * = 0.
t = t ' = 1 k       r ' = k = r ,
S ^ Y B ( a ^ 1 a ^ 2 ) S ^ Y B = ( 1 k k k 1 k ) ( a ^ 1 a ^ 2 ) .
S ^ Y B = exp ( j θ J ^ 2 ) = exp [ θ ( a ^ 1 a ^ 2 a ^ 2 a ^ 1 ) / 2 ] .
t = t ' = 1 k = cos ( θ / 2 )       r = r ' = k = sin ( θ / 2 ) .
S ^ P M a ^ n 0 S ^ P M = q = 1 C q ( q 0 ) a ^ q N r 0 .
C q ( q 0 ) = e j φ b ( j e j θ ) q q 0 [ J q q 0 ( m ) ( 1 ) q 0 J q + q 0 ( m ) ] e j φ b ( j e j θ ) q q 0 J q q 0 ( m ) ,
S ^ P M a ^ n 0 S ^ P M e j φ b s = 1 q o ( j e j θ ) s J s ( m ) a ^ n 0 + s N ,
S ^ P M = exp [ j ( χ T ^ N + χ * T ^ N + φ b N ^ p h ) ] ,
T ^ N = m = 1 a ^ m + N a ^ m         N ^ p h = T ^ 0 = m = 1 a ^ m a ^ m ,
S ^ E O M = S ^ B S o S ^ P M S ^ B S i = S ^ B S o ( S ^ P M 1 S ^ P M 2 ) S ^ B S i ,
S ^ E O M ( | 1 n | v a c ) = S ^ E O M ( a ^ n 1 ) | v a c | v a c = S ^ E O M ( a ^ n 1 ) S ^ E O M | v a c | v a c S ^ E O M ( | v a c | 1 n ) = S ^ E O M ( 1 a ^ n ) | v a c | v a c = S ^ E O M ( 1 a ^ n ) S ^ E O M | v a c | v a c ,
S ^ E O M ( a ^ n 1 ) S ^ E O M = t i ' t o ' ( S ^ P M 1 a ^ n S ^ P M 1 1 ) + r i ' r o ( S ^ P M 2 a ^ n S ^ P M 2 1 ) + t i ' r o ' ( 1 S ^ P M 1 a ^ n S ^ P M 1 ) + r i ' t o ( 1 S ^ P M 2 a ^ n S ^ P M 2 ) ,
S ^ E O M ( 1 a ^ n ) S ^ E O M = r i t o ' ( S ^ P M 1 a ^ n S ^ P M 1 1 ) + t i r o ( S ^ P M 2 a ^ n S ^ P M 2 1 ) + r i r o ' ( 1 S ^ P M 1 a ^ n S ^ P M 1 ) + t i t o ( 1 S ^ P M 2 a ^ n S ^ P M 2 ) ,
b ^ n 0 S ^ P M 1 a ^ n 0 S ^ P M 1 = q = 1 C q ( q 0 ) a ^ q N 1 r 1       c ^ n 0 S ^ P M 2 a ^ n 0 S ^ P M 2 = q = 1 C ¯ q ( q ¯ 0 ) a ^ q N 2 r 2 ,
S ^ E O M ( a ^ n 1 ) S ^ E O M = ( q = 1 [ t i ' t o ' C q ( q 0 ) + r i ' r o C ¯ q ( q 0 ) ] a ^ q N r ) 1 + 1 ( q = 1 [ t i ' r o ' C q ( q 0 ) + r i ' t o C ¯ q ( q 0 ) ] a ^ q N r ) ,
S ^ E O M ( 1 a ^ n ) S ^ E O M = ( q = 1 [ r i t o ' C q ( q 0 ) + t i r o C ¯ q ( q 0 ) ] a ^ q N r ) 1 + 1 ( q = 1 [ r i r o ' C q ( q 0 ) + t i t o C ¯ q ( q 0 ) ] a ^ q N r ) .
S ^ E O M ( | α n 0 n 0 | v a c ) = S ^ E O M ( D ^ n 0 ( α n 0 ) 1 ) S ^ E O M ( | v a c | v a c ) S ^ E O M ( | v a c | α n 0 n 0 ) = S ^ E O M ( 1 D ^ n o ( α n o ) ) S ^ E O M ( | v a c | v a c ) ,
S ^ E O M ( D ^ n 0 ( α n 0 ) 1 ) S ^ E O M = = [ q = 1 D ^ q N 1 r 1 ( α n 0 t i ' t o ' C q ( q 0 ) ) D ^ q N 2 r 2 ( α n o r i ' r o C ¯ q ( q ¯ 0 ) ) ] [ q = 1 D ^ q N 1 r 1 ( α n o t i ' r o ' C q ( q o ) ) D ^ q N 2 r 2 ( α n o r i ' t o C ¯ q ( q ¯ 0 ) ) ] ,
S ^ E O M ( 1 D ^ n 0 ( α n 0 ) ) S ^ E O M = [ q = 1 D ^ q N 1 r 1 ( α n o r i t o ' C q ( q 0 ) ) D ^ q N 2 r 2 ( α n o t i r o C ¯ q ( q ¯ 0 ) ) ] [ q = 1 D ^ q N 1 r 1 ( α n o r i r o ' C q ( q 0 ) ) D ^ q N 2 r 2 ( α n o t i t o C ¯ q ( q ¯ 0 ) ) ] .
S ^ E O M ( D ^ n 0 ( α n 0 ) 1 ) S ^ E O M = = [ q = 1 D ^ q N r [ α n 0 ( t i ' t o ' C q ( q 0 ) + r i ' r o C ¯ q ( q 0 ) ) ] ] [ q = 1 D ^ q N r [ α n o ( t i ' r o ' C q ( q 0 ) + r i ' t o C ¯ q ( q 0 ) ) ] ] ,
S ^ E O M ( 1 D ^ n 0 ( α n 0 ) ) S ^ E O M = = [ q = 1 D ^ q N r [ α n 0 ( r i t o ' C q ( q 0 ) + t i r o C ¯ q ( q 0 ) ) ] ] [ q = 1 D ^ q N r [ α n 0 ( r i r o ' C q ( q 0 ) + t i t o C ¯ q ( q 0 ) ) ] ] .
t o = t o ' = t i = t i ' = 1 / 2       r i ' = r o = r i = r o ' = 1 / 2 .
S ^ E O M ( a ^ n o 1 ) S ^ E O M = 1 2 [ q = 1 ( C q ( q 0 ) + C ¯ q ( q 0 ) ) a ^ q N r 1 ] + 1 2 [ 1 q = 1 ( C q ( q 0 ) + C ¯ q ( q 0 ) ) a ^ q N r ] ,
C q ( q 0 ) = e j φ b 1 ( j e j θ 1 ) q q 0 [ J q q 0 ( m 1 ) ( 1 ) q 0 J q + q 0 ( m 1 ) ] ç C ¯ q ( q 0 ) = e j φ b 2 ( j e j θ 2 ) q q 0 [ J q q 0 ( m 2 ) ( 1 ) q 0 J q + q 0 ( m 2 ) ] .
S ^ E O M | 1 n 0 | v a c = 1 2 [ q = 1 ( C q ( q 0 ) + C ¯ q ( q 0 ) ) | 1 q N r | v a c ] + 1 2 [ | v a c q = 1 ( C q ( q 0 ) + C ¯ q ( q 0 ) ) | 1 q N r ] .
C q ( q 0 ) = ( j ) q q 0 + 1 [ J q q 0 ( m ) ( 1 ) q 0 J q + q 0 ( m ) ] C ¯ q ( q 0 ) = ( j ) q q 0 + 1 [ J q q 0 ( m ) ( 1 ) q 0 J q + q 0 ( m ) ] = C q * ( q 0 ) ,
S ^ E O M ( a ^ n 0 1 ) S ^ E O M = q = 1 Re [ C q ( q 0 ) ]     a ^ q N r 1       +       1 q = 1 j Im [ C q ( q 0 ) ]     a ^ q N r .
C q ( q 0 ) = ( j ) q q 0 + 1 [ J q q 0 ( m ) ( 1 ) q 0 J q + q 0 ( m ) ] C ¯ q ( q 0 ) = ( j ) 2 q 2 q 0 [ J q q 0 ( m ) ( 1 ) q 0 J q + q 0 ( m ) ] .
C ¯ q 0 1 ( q 0 ) = C q 0 1 ( q 0 ) ,
S ^ E O M ( a ^ n 0 1 ) S ^ E O M = 1 2 [ ( S ^ P M 1 a ^ n 0 S ^ P M 1 + a ^ n 0 ) 1 ] + 1 2 [ 1 ( S ^ P M 1 a ^ n 0 S ^ P M 1 + a ^ n 0 ) ] =     = 1 2 [ ( q = 1 ( C q ( q 0 ) a ^ q N 1 r 1 ) + a ^ n 0 ) 1 ] + 1 2 [ 1 ( q = 1 ( C q ( q 0 ) a ^ q N 1 r 1 ) + a ^ n 0 ) ] ,
S ^ E O M | 1 n 0 | v a c = 1 2 [ ( ( C q 0 ( q 0 ) + 1 ) | 1 n 0 + q = 1 , q q o C q 0 ( q 0 ) | 1 q N r ) | v a c ] + + 1 2 [ | v a c ( ( C q 0 ( q 0 ) + 1 ) | 1 n 0 q = 1 , q q o C q ( q o ) | 1 q N r ) ] .
t o = t o ' = t i = t i ' = r i ' = r i = 1 / 2     r o = r o ' = j / 2 .
S ^ E O M ( a ^ n ´ 0 1 ) S ^ E O M = = 1 2 [ q = 1 ( C q ( q 0 ) + j C ¯ q ( q 0 ) ) a ^ q N r 1 ] + 1 2 [ 1 q = 1 ( j C q ( q 0 ) + C ¯ q ( q 0 ) ) a ^ q N r ] .
S ^ E O M ( a ^ n ´ 0 a ^ n 0 ) S ^ E O M = S ^ E O M ( a ^ n 0 1 ) S ^ E O M S ^ E O M ( 1 a ^ n 0 ) S ^ E O M = = t i ' t o ' r i t o ' ( b ^ n 0 b ^ n 0 1 ) + 2 t i ' t o ' r i r o ' ( b ^ n 0 b ^ n 0 ) + t i ' r o ' r i r o ' ( 1 b ^ n 0 b ^ n 0 ) + r i ' r o t i r o ( c ^ n 0 c ^ n 0 1 ) + + 2 r i ' r o t i t o ( c ^ n 0 c ^ n 0 ) + r i ' t o t i t o ( 1 c ^ n 0 c ^ n 0 ) + + [ t i t i ' + r i r i ' ] { r o t o ' ( b ^ n 0 c ^ n 0 1 ) + t o t o ' ( b ^ n 0 c ^ n 0 ) + r o r o ' ( c ^ n 0 b ^ n 0 ) + t o r o ' ( 1 b ^ n 0 c ^ n 0 ) } .
t o = t o ' = t i = t i ' = 1 / 2     r i ' = r i = r o = r o ' = j / 2 ,
S ^ E O M ( a ^ n ´ 0 a ^ n 0 ) S ^ E O M = i 4 ( b ^ n 0 b ^ n 0 1 ) 1 2 ( b ^ n 0 b ^ n 0 ) i 4 ( 1 b ^ n 0 b ^ n 0 ) i 4 ( c ^ n 0 c ^ n 0 1 ) 1 2 ( c ^ n 0 c ^ n 0 ) + i 4 ( 1 c ^ n 0 c ^ n 0 ) ,
S ^ E O M ( a ^ n ´ 0 a ^ n 0 ) S ^ E O M = e j Δ φ b 2 sin ( Δ φ b ) [ b ^ n 0 b ^ n 0 1 1 b ^ n 0 b ^ n 0 ] e j Δ φ b cos ( Δ φ b )     b ^ n 0 b ^ n 0 .
b ^ n 0 S ^ P M 1 a ^ n 0 S ^ P M 1 = e j φ b 1 a ^ n 0 + e j φ b 1 k = 1 M m 1 k [ j e j θ 1 k a ^ n 0 + N 1 k + j e j θ 1 k a ^ n 0 N 1 k ] c ^ n 0 S ^ P M 1 a ^ n 0 S ^ P M 1 = e j φ b 2 a ^ n 0 + e j φ b 2 k = 1 M m 2 k [ j e j θ 2 k a ^ n 0 + N 2 k + j e j θ 2 k a ^ n 0 N 2 k ] .
S ^ E O M ( D ^ n 0 ( α n 0 ) 1 ) S ^ E O M = A ^ B ^ A ^ = D ^ n 0 [ α n 0 ( t i ' t o ' e j φ b 1 + r i ' r o e j φ b 2 ) ] k = 1 M D ^ n 0 ± N 1 k [ t i ' t o ' m 1 k j α n 0 e j φ b 1 e ± j θ 1 k ] D ^ n 0 ± N 2 k [ r i ' r o m 2 k j α n 0 e j φ b 2 e ± j θ 2 k ] B ^ = D ^ n 0 [ α n 0 ( t i ' r o ' e j φ b 1 + r i ' t o e j φ b 2 ) ] k = 1 N D ^ n 0 ± N 1 k [ t i ' r o ' m 1 k j α n 0 e j φ b 1 e ± j θ 1 k ] D ^ n 0 ± N 2 k [ r i ' t o m 2 k j α n 0 e j φ b 2 e ± j θ 2 k ] ,
S ^ E O M ( 1 D ^ n 0 ( α n 0 ) ) S ^ E O M = E ^ F ^ E ^ = D ^ n 0 [ α n 0 ( r i t o ' e j φ b 1 + t i r o e j φ b 2 ) ] k = 1 M D ^ n 0 ± N 1 k [ r i t o ' m 1 k j α n 0 e j φ b 1 e ± j θ 1 k ] D ^ n 0 ± N 2 k [ t i r o m 2 k j α n 0 e j φ b 2 e ± j θ 2 k ] F ^ = D ^ n 0 [ α n 0 ( r i r o ' e j φ b 1 + t i t o e j φ b 2 ) ] k = 1 N D ^ n 0 ± N 1 k [ r i r o ' m 1 k j α n 0 e j φ b 1 e ± j θ 1 k ] D ^ n 0 ± N 2 k [ t i t o m 2 k j α n 0 e j φ b 2 e ± j θ 2 k ] .
A ^ = D ^ n 0 [ α n 0 ( t i ' t o ' e j φ b 1 + r i ' r o e j φ b 2 ) ] k = 1 M D ^ n 0 ± N k [ j α n o ( t i ' t o ' m 1 k e j φ b 1 e ± j θ 1 k + r i ' r o m 2 k e j φ b 2 e ± j θ 2 k ) ] B ^ = D ^ n 0 [ α n 0 ( t i ' r o ' e j φ b 1 + r i ' t o e j φ b 2 ) ] k = 1 M D ^ n 0 ± N k [ j α n o ( t i ' r o ' m 1 k e j φ b 1 e ± j θ 1 k + r i ' t o m 2 k e j φ b 2 e ± j θ 2 k ) ] E ^ = D ^ n 0 [ α n 0 ( r i t o ' e j φ b 1 + t i r o e j φ b 2 ) ] k = 1 M D ^ n 0 ± N k [ j α n o ( r i t o ' m 1 k e j φ b 1 e ± j θ 1 k + t i r o m 2 k e j φ b 2 e ± j θ 2 k ) ] F ^ = D ^ n 0 [ α n 0 ( r i r o ' e j φ b 1 + t i t o e j φ b 2 ) ] k = 1 M D ^ n 0 ± N k [ j α n o ( r i r o ' m 1 k e j φ b 1 e ± j θ 1 k + t i t o m 2 k e j φ b 2 e ± j θ 2 k ) ] .
| Ψ = S ^ E O M | α n 0 n 0 | v a c = S ^ E O M ( D ^ n 0 ( α n o ) 1 ) S ^ E O M | v a c | v a c = | Ψ 1 1 | Ψ 2 2 | Ψ 1 1 = | α n 0 ( t i ' t o ' e j φ b 1 + r i ' r o e j φ b 2 ) n 0 k = 1 M | j α n 0 ( t i ' t o ' m 1 k e j φ b 1 e ± j θ 1 k + r i ' r o m 2 k e j φ b 2 e ± j θ 2 k ) n o ± N k | Ψ 2 2 = | α n 0 ( t i ' r o ' e j φ b 1 + r i ' t o e j φ b 2 ) n 0 k = 1 M | j α n 0 ( t i ' r o ' m 1 k e j φ b 1 e ± j θ 1 k + r i ' t o m 2 k e j φ b 2 e ± j θ 2 k ) n o ± N k .
E ^ 1 + = j x ω ξ ( ω ) a ^ ω e j ω t 1       ξ ( ω ) = ω 2 ε o V ,
Ψ | E ^ 1 + + h c | Ψ = j x ξ ( ω 0 ) α n 0 ( t i ' t o ' e j φ b 1 + r i ' r o e j φ b 2 ) e j ω 0 t + + j x k = 1 M ξ ( ω 0 ± Ω k ) j α n 0 ( t i ' t o ' m 1 k e j φ b 1 e ± j θ 1 k + r i ' r o m 2 k e j φ b 2 e ± j θ 2 k ) e j ( ω 0 ± Ω k ) t + c c .

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