Abstract

We present a detailed analysis of the quantum description of electro-optical phase modulation. The results define a black-box type model for this device, which may be especially useful in the engineering steps leading to the design of complex quantum information systems incorporating one or more of these devices. By constructing an explicit representation of the phase modulation scattering operator, it is shown that an approach based entirely on its classical description leads to unphysical modes associated to non-positive frequencies. After modifying this operator, phase modulation is described, for the first time to the best of our knowledge, in terms of a unitary scattering operator Ŝ defined over positive-frequency modes. The modifications introduced by Ŝ in the process of sideband generation by phase modulation are shown to be insignificant when the radiation belongs to optical bands, thus being consistent with the classical description. Finally, the model is employed to characterize the important case of multitone radio- frequency modulation of an optical signal.

© 2010 Optical Society of America

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References

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  1. J. M. Merolla, 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, 1899–1905 (1999).
    [CrossRef]
  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, 1533–1540 (2003).
    [CrossRef]
  3. G. B. Xavier and J.-P. von der Weid, “Modulation schemes for frequency coded quantum key distribution,” Electron. Lett. 41, 607–608 (2005).
    [CrossRef]
  4. A. Ortigosa-Blanch and J. Capmany, “Subcarrier multiplexing optical quantum key distribution,” Phys. Rev. A 73, 024305 (2006).
    [CrossRef]
  5. P. Kolchin, C. Belthangady, S. Du, G. Y. Yin, and S. E. Harris, “Electro-optic modulation of single photons,” Phys. Rev. Lett. 101, 103601 (2008).
    [CrossRef] [PubMed]
  6. U. Leonhardt, “Quantum physics of simple optical instruments,” Rep. Prog. Phys. 66, 1207–1249 (2003).
    [CrossRef]
  7. M. Bloch, S. W. McLaughlin, J.-M. Merolla, and F. Patois, “Frequency-coded quantum key distribution,” Opt. Lett. 32, 201–203 (2007).
    [CrossRef]
  8. P. Kumar and A. Prabhakar, “Evolution of quantum states in an electro-optic phase modulator,” IEEE J. Quantum Electron. 45, 149–156 (2009).
    [CrossRef]
  9. L. Thylen, U. Westergren, P. Holmström, R. Schatz, and P. Jänes, “Recent developments in high-speed optical modulators,” in I.PKaminow, T.Li, and A.E.Willner, eds., in Optical Fiber Telecommunications V. A (Academic, 2008), Chap. 7.
    [CrossRef]
  10. J. Capmany, A. Ortigosa-Blanch, J. Mora, A. Ruiz, W. Amaya, and A. Martínez, “Analysis of subcarrier multiplexed quantum key distribution systems: signal, intermodulation and quantum bit error rate,” IEEE J. Sel. Top. Quantum Electron. 15, 1607–1621 (2009).
    [CrossRef]
  11. A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford Univ. Press, 2006) Chap. 9.
  12. A. B. Carson, P. B. Crilly, and J. C. Rutledge, Communication Systems, 4th ed. (McGraw Hill, 2002) Chap. 5.
  13. C. G. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge Univ. Press, 2005).

2009 (2)

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

J. Capmany, A. Ortigosa-Blanch, J. Mora, A. Ruiz, W. Amaya, and A. Martínez, “Analysis of subcarrier multiplexed quantum key distribution systems: signal, intermodulation and quantum bit error rate,” IEEE J. Sel. Top. Quantum Electron. 15, 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, 103601 (2008).
[CrossRef] [PubMed]

2007 (1)

M. Bloch, S. W. McLaughlin, J.-M. Merolla, and F. Patois, “Frequency-coded quantum key distribution,” Opt. Lett. 32, 201–203 (2007).
[CrossRef]

2006 (1)

A. Ortigosa-Blanch and J. Capmany, “Subcarrier multiplexing optical quantum key distribution,” Phys. Rev. A 73, 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, 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, 1533–1540 (2003).
[CrossRef]

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

1999 (1)

J. M. Merolla, 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, 1899–1905 (1999).
[CrossRef]

Amaya, W.

J. Capmany, A. Ortigosa-Blanch, J. Mora, A. Ruiz, W. Amaya, and A. Martínez, “Analysis of subcarrier multiplexed quantum key distribution systems: signal, intermodulation and quantum bit error rate,” IEEE J. Sel. Top. Quantum Electron. 15, 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, 103601 (2008).
[CrossRef] [PubMed]

Bloch, M.

M. Bloch, S. W. McLaughlin, J.-M. Merolla, and F. Patois, “Frequency-coded quantum key distribution,” Opt. Lett. 32, 201–203 (2007).
[CrossRef]

Capmany, J.

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

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

Carson, A. B.

A. B. Carson, P. B. Crilly, and J. C. Rutledge, Communication Systems, 4th ed. (McGraw Hill, 2002) Chap. 5.

Crilly, P. B.

A. B. Carson, P. B. Crilly, and J. C. Rutledge, Communication Systems, 4th ed. (McGraw Hill, 2002) Chap. 5.

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, 103601 (2008).
[CrossRef] [PubMed]

Duraffourg, L.

J. M. Merolla, 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, 1899–1905 (1999).
[CrossRef]

Gerry, C. G.

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

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, 1533–1540 (2003).
[CrossRef]

Goedgebuer, J.-P.

J. M. Merolla, 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, 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, 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, 103601 (2008).
[CrossRef] [PubMed]

Holmström, P.

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

Jänes, P.

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

Knight, P. L.

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

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, 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, 149–156 (2009).
[CrossRef]

Leonhardt, U.

U. Leonhardt, “Quantum physics of simple optical instruments,” Rep. Prog. Phys. 66, 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, 1533–1540 (2003).
[CrossRef]

Martínez, A.

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

Mazurenko, Y.

J. M. Merolla, 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, 1899–1905 (1999).
[CrossRef]

McLaughlin, S. W.

M. Bloch, S. W. McLaughlin, J.-M. Merolla, and F. Patois, “Frequency-coded quantum key distribution,” Opt. Lett. 32, 201–203 (2007).
[CrossRef]

Merolla, J. M.

J. M. Merolla, 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, 1899–1905 (1999).
[CrossRef]

Merolla, J.-M.

M. Bloch, S. W. McLaughlin, J.-M. Merolla, and F. Patois, “Frequency-coded quantum key distribution,” Opt. Lett. 32, 201–203 (2007).
[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, 1533–1540 (2003).
[CrossRef]

Mora, J.

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

Ortigosa-Blanch, A.

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

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

Patois, F.

M. Bloch, S. W. McLaughlin, J.-M. Merolla, and F. Patois, “Frequency-coded quantum key distribution,” Opt. Lett. 32, 201–203 (2007).
[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, 1533–1540 (2003).
[CrossRef]

Porte, H.

J. M. Merolla, 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, 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, 149–156 (2009).
[CrossRef]

Rhodes, W. T.

J. M. Merolla, 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, 1899–1905 (1999).
[CrossRef]

Ruiz, A.

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

Rutledge, J. C.

A. B. Carson, P. B. Crilly, and J. C. Rutledge, Communication Systems, 4th ed. (McGraw Hill, 2002) Chap. 5.

Schatz, R.

L. Thylen, U. Westergren, P. Holmström, R. Schatz, and P. Jänes, “Recent developments in high-speed optical modulators,” in I.PKaminow, T.Li, and A.E.Willner, eds., in Optical Fiber Telecommunications V. A (Academic, 2008), Chap. 7.
[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, 1533–1540 (2003).
[CrossRef]

Thylen, L.

L. Thylen, U. Westergren, P. Holmström, R. Schatz, and P. Jänes, “Recent developments in high-speed optical modulators,” in I.PKaminow, T.Li, and A.E.Willner, eds., in Optical Fiber Telecommunications V. A (Academic, 2008), Chap. 7.
[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, 607–608 (2005).
[CrossRef]

Westergren, U.

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

Xavier, G. B.

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

Yariv, A.

A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford Univ. Press, 2006) Chap. 9.

Yeh, P.

A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford Univ. Press, 2006) Chap. 9.

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, 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, 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, 149–156 (2009).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (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, 1533–1540 (2003).
[CrossRef]

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

Opt. Lett. (1)

M. Bloch, S. W. McLaughlin, J.-M. Merolla, and F. Patois, “Frequency-coded quantum key distribution,” Opt. Lett. 32, 201–203 (2007).
[CrossRef]

Phys. Rev. A (2)

J. M. Merolla, 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, 1899–1905 (1999).
[CrossRef]

A. Ortigosa-Blanch and J. Capmany, “Subcarrier multiplexing optical quantum key distribution,” Phys. Rev. A 73, 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, 103601 (2008).
[CrossRef] [PubMed]

Rep. Prog. Phys. (1)

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

Other (4)

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

A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford Univ. Press, 2006) Chap. 9.

A. B. Carson, P. B. Crilly, and J. C. Rutledge, Communication Systems, 4th ed. (McGraw Hill, 2002) Chap. 5.

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

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Figures (7)

Fig. 1
Fig. 1

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

Fig. 2
Fig. 2

Graphical representation of the process of coefficient construction for each resulting creation operator given by Eq. (24).

Fig. 3
Fig. 3

Simple diagram representing the possible transitions starting from an input photon present at a mode characterized by a frequency number n 0 and a modulating frequency given by a number N = 1 .

Fig. 4
Fig. 4

Auxiliary diagrams to compute the amplitude probability coefficient for the n 0 n 0 transition with N = 1 . Left: zero-order path. Upper right: Second-order paths. Lower right: Fourth-order paths.

Fig. 5
Fig. 5

Qualitative spectrum (the horizontal scale represents the mode number rather than the frequency) of a phase-modulated signal for an input single-mode state with mode number n 0 and N = 1 .

Fig. 6
Fig. 6

(Upper left) Allowed and forbidden transitions. (Upper right) A forbidden path. (Lower) Diagram showing that, for each forbidden path leading to a state characterized by n > 0 (lower left), there is another unique forbidden path (lower right) that leads to a state n < 0 built from the original one by transmitting it through the line corresponding to n = 0 and interchanging, from this point on, upward by downward transitions and vice versa.

Fig. 7
Fig. 7

Transition probabilities of phase-modulated one-photon states at modes n 0 = 6 (left) and n 0 = 10 (right). See the text.

Equations (52)

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Δ ϕ ( t ) = 2 π Δ η λ 0 l = π η 0 3 r λ 0 l d ( V DC + Δ V ( t ) ) = π V DC + Δ V ( t ) V π .
E in ( z , t ) = E 0 x ̂ e j ( ω o t k ( ω o ) z ) = E 0 x ̂ e j ω o ( t z v ) .
E out ( 0 , t ) = E in ( 0 , t ) e j Δ ϕ ( t ) = E 0 x ̂ e j ω o t e j φ b exp ( j π Δ V ( t ) V π ) ,
Δ V ( t ) = V m cos ( Ω t + θ ) ,
E out ( 0 , t ) = E in ( 0 , t ) e j Δ ϕ ( t ) = E 0 x ̂ e j ω o t e j φ b e j m cos ( Ω t + θ ) .
e j z cos ϑ = n = j n J n ( z ) e j n ϑ ,
E out ( z , t ) = E 0 x ̂ e j φ b n = ( j e j θ ) n J n ( m ) e j ( ω o + n Ω ) ( t z v ) = E 0 x ̂ n = C n e j ( ω o + n Ω ) ( t z v ) ,
C n = e j φ b ( j e j θ ) n J n ( m ) .
E in ( 0 , t ) = E out ( 0 , t ) e j φ b e j m cos ( Ω t + θ ) = E out ( 0 , t ) n = C n * e j n Ω t = E out ( 0 , t ) n = C n * e j n Ω t = E out ( 0 , t ) n = C ̃ n e j n Ω t ,
E in ( 0 , t ) = E 0 x ̂ e j ω o t n = α n e j n Ω t ,
E out ( z , t ) = E 0 x ̂ q , n = α n C q n e j ( ω o + q Ω ) ( t z v ) ,
| Ψ in = | α n 0 S ̂ | Ψ out = S ̂ | α n 0 = q = | C q α n 0 + q N .
| { α } q = | α q n 0 + q N S ̂ q = | k = C q k α k n 0 + q N .
| { α } = q = | α q n 0 + q N S ̂ 1 q = | k = C ̃ q k α k n 0 + q N = q = | k = C k q * α k n 0 + q N .
{ β } | S ̂ | { α } * = q n 0 + q N β q | k C q k α k n 0 + q N * = exp [ 1 2 q | s C q s α s | 2 1 2 q | β q | 2 + q s α s * C q s * β q ] = q exp [ 1 2 | n C ̃ q n β n | 2 1 2 | α q | 2 + s α q * C ̃ q s β s ] = q n 0 + q N α q | s C ̃ q s β s n 0 + q N = { α } | S ̂ 1 | { β } ,
S ̂ a ̂ n S ̂ = q = + C q a ̂ n + q N S ̂ a ̂ n S ̂ = q = + C q * a ̂ n + q N ,
S ̂ a ̂ n S ̂ | vac = S ̂ a ̂ n | vac = S ̂ | 1 n = q = C q a ̂ n + q N | vac = q = C q | 1 n + q N .
S ̂ S ̂ N ( χ , φ b ) = exp [ j G ̂ N ( χ , φ b ) ] = exp [ j ( χ T ̂ N + χ * T ̂ N + φ b N ̂ ph ) ] ,
S ̂ = exp ( j χ T ̂ N + j χ * T ̂ N ) exp ( j φ b N ̂ ph ) exp ( j Q ̂ ) exp ( j φ b N ̂ ph ) .
S ̂ a ̂ n S ̂ = e j Q ̑ e j φ b N ̂ ph a ̂ n e j φ b N ̂ ph e j Q ̑ = exp ( j a d Q ̑ ) exp ( j φ b a d N ̂ ph ) ( a n ) ,
exp ( j φ b a d N ̂ ph ) ( a ̂ n ) = p = 0 1 p ! ( j φ b ) p a d N ̂ ph ( p ) ( a ̂ n ) = p = 0 1 p ! ( j φ b ) p a ̂ n = e j φ b a ̂ n ,
exp ( j a d Q ̂ ) ( a ̂ n ) = p = 0 j p p ! a d Q ̂ ( p ) ( a ̂ n ) .
a d Q ̂ ( p ) ( a ̂ n ) = s = 0 p ( p s ) χ s ( χ * ) p s a ̂ n + ( 2 s p ) N
S ̂ a ̂ n S ̂ = e j φ b p = 0 j p p ! s = 0 p ( p s ) χ s ( χ * ) p s a ̂ n + ( 2 s p ) N .
S ̂ a ̂ n S ̂ = q = C q a ̂ n + q N
S ̂ a ̂ n S ̂ = e j φ b q = [ j q n = 0 ( 1 ) n ( 2 n + q ) ! ( q + 2 n q + n ) χ q + n ( χ * ) n ] a ̂ n + q N = e j φ b q = ( j m 2 e j θ ) q [ n = 0 ( 1 ) n ( 2 n + q ) ! ( q + 2 n q + n ) ( m 2 ) 2 n ] a ̂ n + q N .
J q ( m ) = ( m 2 ) q n = 0 ( 1 ) n ( m 2 ) 2 n n ! ( q + n ) ! ,
S ̂ a ̂ n S ̂ = q = e j φ b ( j e j θ ) q J q ( m ) a ̂ n + q N = q = C q a ̂ n + q N ,
A n 0 n 0 + 0 N = C 0 = 1 0 ! 2 | χ | 2 2 ! + 6 | χ | 4 4 ! + = n = 0 ( 1 ) n | χ | 2 n ( n ! ) 2 = J 0 ( 2 | χ | ) = J 0 ( m ) .
A n 0 n 0 + 1 N = C 1 = j e θ [ m 2 1 2 ( m 2 ) 3 + ] = j e j θ J 1 ( m ) .
A n 0 n 0 + q N = C q = s = 0 ( j χ ) q + s ( j χ * ) s ( q + 2 s ) ! ( q + 2 s ) ! ( q + s ) ! s ! = ( j e j θ ) q J q ( m ) ,
S ̂ S ̂ N ( χ , φ b ) = exp [ j G ̂ N ( χ , φ b ) ] = exp [ j ( χ T ̂ N + χ * T ̂ N + φ b N ̂ ph ) ] ,
T ̂ N = m = 1 a ̂ m + N a ̂ m T N = m = 1 a ̂ m a ̂ m + N N ̂ ph = T ̂ 0 = m = 1 a ̂ m a ̂ m ,
S ̂ a ̂ n o S ̂ = n > 0 C n ( n o ) a ̂ n
C 1 n 0 = ( j e j θ ) 1 n o J 1 n o ( m ) = ( j e j θ ) 1 ( j e j θ ) n o J 1 n o ( m ) ( j e j θ ) 1 ( j e j θ ) n o J 1 n o ( m ) = ( 1 ) ( j e j θ ) 1 n o J 1 n o ( m ) ,
C n n 0 = ( j e j θ ) n n o J n n o ( m ) ( 1 ) n ( j e j θ ) n n o J n n o ( m ) .
S ̂ a ̂ n o S ̂ = n = 1 e j φ b ( j e j θ ) n n o [ J n n o ( m ) ( 1 ) n J n n o ( m ) ] a ̂ n = n = 1 e j φ b ( j e j θ ) n n o [ J n n o ( m ) ( 1 ) n o J n + n o ( m ) ] a ̂ n = s = 1 n o e j φ b ( j e j θ ) s [ J s ( m ) ( 1 ) n o J s + 2 n o ( m ) ] a ̂ n o + s q = 1 D q , n o a ̂ q ,
S ̂ a ̂ n o S ̂ = s = 1 q o e j φ b ( j e j θ ) s [ J s ( m ) ( 1 ) q o J s + 2 q o ( m ) ] a ̂ n o + s N = q = 1 e j φ b ( j e j θ ) q q o [ J q q o ( m ) ( 1 ) q o J q + q o ( m ) ] a ̂ q N r 0 = q = 1 D q , q o a ̂ q N r 0 ,
q = 1 D p 0 , q D q , q 0 = q = 1 D q , p 0 * D q , q 0 = δ p 0 , q 0 .
S ( N , M ) ( φ , m 1 2 e j θ 1 , m 2 2 e j θ 2 ) = exp ( j φ N ̂ ph ) exp ( X ̂ + Y ̂ ) ,
X ̂ = j m 1 2 e j θ 1 T N + j m 1 2 e j θ 1 T N ,
Y ̂ = j m 2 2 e j θ 2 T M + j m 2 2 e j θ 2 T M .
exp ( X ̂ + Y ̂ ) = exp ( 1 2 [ X ̂ , Y ̂ ] + O ̂ ( m 3 ) ) exp ( X ̂ ) exp ( Y ̂ ) = exp ( 1 2 [ X ̂ , Y ̂ ] + O ̂ ( m 3 ) ) S ̂ N ( m 1 2 e j θ 1 ) S ̂ M ( m 2 2 e j θ 2 ) ,
S ( N , M ) ( φ , m 1 2 e j θ 1 , m 2 2 e j θ 2 ) = exp ( j φ N ̂ ph ) exp ( 1 2 [ X ̂ , Y ̂ ] + O ̂ ( m 3 ) ) S ̂ N ( m 1 2 e j θ 1 ) S ̂ M ( m 2 2 e j θ 2 ) .
S ( N , M ) a n o S ( N , M ) = exp [ j φ N ̂ ph ] exp [ 1 2 [ X ̂ , Y ̂ ] + O ̂ ( m 3 ) ] . S ̂ N S ̂ M a n o S ̂ M S ̂ N exp [ + 1 2 [ X ̂ , Y ̂ ] O ̂ ( m 3 ) ] exp [ j φ N ̂ ph ] .
S ̂ M a n o S ̂ M = q = 1 q o ( j e j θ ) q [ J q ( m 2 ) ( 1 ) q o J q + 2 q o ( m 2 ) ] a ̂ n o + q M = q = 1 q o D q ( q o , m 2 ) a ̂ n o + q M ,
S ̂ N S ̂ M a n S ̂ M S ̂ N = q = 1 q o D q ( q o , m 2 ) S ̂ N a ̂ n o + q M S ̂ N = { n o + q M = s o N u } = q = 1 q o D q ( q o , m 2 ) s = 1 s o D s ( s o , m 1 ) a ̂ n o + q M + s N = q = 1 q o s = 1 s o D q ( q o , m 2 ) D s ( s o , m 1 ) a ̂ n o + q M + s N ,
exp ( ± 1 2 [ X ̂ , Y ̂ ] + O ̂ ( m 3 ) ) I ̂ ± 1 2 [ X ̂ , Y ̂ ] + O ̂ ( m 3 )
S ( N , M ) a n o S ( N , M ) = e j φ b S ̂ N S ̂ M a n o S ̂ M S ̂ N + [ e j φ b S ̂ N S ̂ M a n o S ̂ M S ̂ N , Q ̂ ( m 2 ) ] ,
S ( N , M ) a n o S ( N , M ) e j φ b S ̂ N S ̂ M a n o S ̂ M S ̂ N = e j φ b D 0 ( q o , m 2 ) D 0 ( s o , m 1 ) a ̂ n o + e j φ b D 0 ( q o , m 2 ) D 1 ( s o , m 1 ) a ̂ n o + N + e j φ b D 0 ( q o , m 2 ) D 1 ( s o , m 1 ) a ̂ n o N + e j φ b D 1 ( q o , m 2 ) D 0 ( s o , m 1 ) a ̂ n o + M + e j φ b D 1 ( q o , m 2 ) D 0 ( s o , m 1 ) a ̂ n o M + .
S ( N 1 , N 2 , N k ) a n o S ( N 1 , N 2 , N k ) e j φ b q 1 = q 2 = q K = ( j e j θ 1 ) q 1 ( j e j θ K ) q K J q 1 ( m 1 ) J q K ( m K ) a ̂ n o + q 1 N 1 + q K N K = e j φ b ( r = 1 K J 0 ( m i ) ) a ̂ n o + e j φ b i = 1 K ( r = 1 r i K J 0 ( m i ) ) [ j e j θ i J 1 ( m i ) a ̂ n o + N i j e j θ i J 1 ( m i ) a ̂ n o N i ] +
S ( N 1 , N 2 , . N m ) a n o S ( N 1 , N 2 , . N m ) = e j φ b q 1 = q 2 = q k = ( j e j θ 1 ) q 1 ( j e j θ 2 ) q 2 ( j e j θ k ) q k ( m 1 q 1 q 1 ! ) ( m 2 q 2 q 2 ! ) ( m k q k q k ! ) a ̂ n o + q 1 N 1 + q 2 N 2 + q k N k = e j φ b a ̂ n o + e j φ b k = 1 k m k [ j e j θ k a ̂ n o + N k + j e j θ k a ̂ n o N k ] + Q ̂ ( m 2 ) .

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