Abstract

Quantization of optical solitons is discussed, using the nonlinear Fourier-transform (inverse-scattering) method. A quantum soliton is described in terms of two sets of conjugate variables such as the photon number and the phase, and the momentum and the center position. The theory of a soliton collision is extended to describe the quantum-nondemolition measurement of soliton photon number and momentum. These two variables are indeed quantum-nondemolition observables and can be measured by means of the phase and the center position of the probe soliton. It is demonstrated that the measurement error and backaction noise on the conjugate variable satisfy an uncertainty product.

© 1989 Optical Society of America

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  1. H. P. Yuen, Phys. Rev. A 13, 2226 (1976).
    [Crossref]
  2. H. P. Yuen and J. H. Shapiro, Opt. Lett. 4, 334 (1979).
    [Crossref] [PubMed]
  3. B. Yurke, Phys. Rev. A 32, 300 (1985); B. Yurke, P. Grangier, R. E. Slusher, and M. J. Potasek, Phys. Rev. A 35, 3586 (1987).
    [Crossref] [PubMed]
  4. R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
    [Crossref] [PubMed]
  5. M. W. Maeda, P. Kumar, and J. H. Shapiro, Opt. Lett. 12, 161 (1987); J. Opt. Soc. Am. B 4, 1501 (1987).
    [Crossref] [PubMed]
  6. R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
    [Crossref] [PubMed]
  7. L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986); M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, and H. J. Kimble, Phys. Rev. Lett. 59, 198 (1987).
    [Crossref] [PubMed]
  8. N. Imoto, H. A. Haus, and Y. Yamamoto, Phys. Rev. A 32, 2287 (1985).
    [Crossref] [PubMed]
  9. M. J. Potasek and B. Yurke, Phys. Rev. A 35, 3974 (1987).
    [Crossref] [PubMed]
  10. M. J. Potasek and B. Yurke, Phys. Rev. A 38, 1335 (1988).
    [Crossref] [PubMed]
  11. P. D. Drummond and S. J. Carter, J. Opt. Soc. Am. B 4, 1565 (1987).
    [Crossref]
  12. S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby, Phys. Rev. Lett. 58, 1841 (1987).
    [Crossref] [PubMed]
  13. R. M. Shelby, M. D. Levenson, and P. W. Bayer, Phys. Rev. B 31, 5244 (1985).
    [Crossref]
  14. V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).
  15. D. J. Kaup, J. Math. Phys. 16, 2036 (1975).
    [Crossref]
  16. M. Kitagawa and Y. Yamamoto, Phys. Rev. A 34, 3974 (1986).
    [Crossref] [PubMed]
  17. M. Wadati and M. Sakagami, J. Phys. Soc. Jpn. 53, 1933 (1984); M. Wadati, “Quantum inverse scattering method,” in Dynamical Problems in Soliton Systems, S. Takeno, ed. (Springer-Verlag, Berlin, 1985).
    [Crossref]
  18. H. A. Haus, Rev. Mod. Phys. 51, 331 (1979).
    [Crossref]
  19. Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers, I. Time dependent Hartree approximation; II. Exact solution,” Phys. Rev. A (to be published).

1988 (1)

M. J. Potasek and B. Yurke, Phys. Rev. A 38, 1335 (1988).
[Crossref] [PubMed]

1987 (4)

1986 (3)

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
[Crossref] [PubMed]

L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986); M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, and H. J. Kimble, Phys. Rev. Lett. 59, 198 (1987).
[Crossref] [PubMed]

M. Kitagawa and Y. Yamamoto, Phys. Rev. A 34, 3974 (1986).
[Crossref] [PubMed]

1985 (4)

R. M. Shelby, M. D. Levenson, and P. W. Bayer, Phys. Rev. B 31, 5244 (1985).
[Crossref]

N. Imoto, H. A. Haus, and Y. Yamamoto, Phys. Rev. A 32, 2287 (1985).
[Crossref] [PubMed]

B. Yurke, Phys. Rev. A 32, 300 (1985); B. Yurke, P. Grangier, R. E. Slusher, and M. J. Potasek, Phys. Rev. A 35, 3586 (1987).
[Crossref] [PubMed]

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
[Crossref] [PubMed]

1984 (1)

M. Wadati and M. Sakagami, J. Phys. Soc. Jpn. 53, 1933 (1984); M. Wadati, “Quantum inverse scattering method,” in Dynamical Problems in Soliton Systems, S. Takeno, ed. (Springer-Verlag, Berlin, 1985).
[Crossref]

1979 (2)

1976 (1)

H. P. Yuen, Phys. Rev. A 13, 2226 (1976).
[Crossref]

1975 (1)

D. J. Kaup, J. Math. Phys. 16, 2036 (1975).
[Crossref]

1972 (1)

V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).

Bayer, P. W.

R. M. Shelby, M. D. Levenson, and P. W. Bayer, Phys. Rev. B 31, 5244 (1985).
[Crossref]

Carter, S. J.

P. D. Drummond and S. J. Carter, J. Opt. Soc. Am. B 4, 1565 (1987).
[Crossref]

S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby, Phys. Rev. Lett. 58, 1841 (1987).
[Crossref] [PubMed]

DeVoe, R. G.

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
[Crossref] [PubMed]

Drummond, P. D.

S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby, Phys. Rev. Lett. 58, 1841 (1987).
[Crossref] [PubMed]

P. D. Drummond and S. J. Carter, J. Opt. Soc. Am. B 4, 1565 (1987).
[Crossref]

Hall, J. L.

L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986); M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, and H. J. Kimble, Phys. Rev. Lett. 59, 198 (1987).
[Crossref] [PubMed]

Haus, H. A.

N. Imoto, H. A. Haus, and Y. Yamamoto, Phys. Rev. A 32, 2287 (1985).
[Crossref] [PubMed]

H. A. Haus, Rev. Mod. Phys. 51, 331 (1979).
[Crossref]

Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers, I. Time dependent Hartree approximation; II. Exact solution,” Phys. Rev. A (to be published).

Hollberg, L. W.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
[Crossref] [PubMed]

Imoto, N.

N. Imoto, H. A. Haus, and Y. Yamamoto, Phys. Rev. A 32, 2287 (1985).
[Crossref] [PubMed]

Kaup, D. J.

D. J. Kaup, J. Math. Phys. 16, 2036 (1975).
[Crossref]

Kimble, H. J.

L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986); M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, and H. J. Kimble, Phys. Rev. Lett. 59, 198 (1987).
[Crossref] [PubMed]

Kitagawa, M.

M. Kitagawa and Y. Yamamoto, Phys. Rev. A 34, 3974 (1986).
[Crossref] [PubMed]

Kumar, P.

Lai, Y.

Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers, I. Time dependent Hartree approximation; II. Exact solution,” Phys. Rev. A (to be published).

Levenson, M. D.

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
[Crossref] [PubMed]

R. M. Shelby, M. D. Levenson, and P. W. Bayer, Phys. Rev. B 31, 5244 (1985).
[Crossref]

Maeda, M. W.

Mertz, J. C.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
[Crossref] [PubMed]

Perlmutter, S. H.

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
[Crossref] [PubMed]

Potasek, M. J.

M. J. Potasek and B. Yurke, Phys. Rev. A 38, 1335 (1988).
[Crossref] [PubMed]

M. J. Potasek and B. Yurke, Phys. Rev. A 35, 3974 (1987).
[Crossref] [PubMed]

Reid, M. D.

S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby, Phys. Rev. Lett. 58, 1841 (1987).
[Crossref] [PubMed]

Sakagami, M.

M. Wadati and M. Sakagami, J. Phys. Soc. Jpn. 53, 1933 (1984); M. Wadati, “Quantum inverse scattering method,” in Dynamical Problems in Soliton Systems, S. Takeno, ed. (Springer-Verlag, Berlin, 1985).
[Crossref]

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).

Shapiro, J. H.

Shelby, R. M.

S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby, Phys. Rev. Lett. 58, 1841 (1987).
[Crossref] [PubMed]

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
[Crossref] [PubMed]

R. M. Shelby, M. D. Levenson, and P. W. Bayer, Phys. Rev. B 31, 5244 (1985).
[Crossref]

Slusher, R. E.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
[Crossref] [PubMed]

Valley, J. F.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
[Crossref] [PubMed]

Wadati, M.

M. Wadati and M. Sakagami, J. Phys. Soc. Jpn. 53, 1933 (1984); M. Wadati, “Quantum inverse scattering method,” in Dynamical Problems in Soliton Systems, S. Takeno, ed. (Springer-Verlag, Berlin, 1985).
[Crossref]

Walls, D. F.

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
[Crossref] [PubMed]

Wu, H.

L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986); M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, and H. J. Kimble, Phys. Rev. Lett. 59, 198 (1987).
[Crossref] [PubMed]

Wu, L.-A.

L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986); M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, and H. J. Kimble, Phys. Rev. Lett. 59, 198 (1987).
[Crossref] [PubMed]

Yamamoto, Y.

M. Kitagawa and Y. Yamamoto, Phys. Rev. A 34, 3974 (1986).
[Crossref] [PubMed]

N. Imoto, H. A. Haus, and Y. Yamamoto, Phys. Rev. A 32, 2287 (1985).
[Crossref] [PubMed]

Yuen, H. P.

Yurke, B.

M. J. Potasek and B. Yurke, Phys. Rev. A 38, 1335 (1988).
[Crossref] [PubMed]

M. J. Potasek and B. Yurke, Phys. Rev. A 35, 3974 (1987).
[Crossref] [PubMed]

B. Yurke, Phys. Rev. A 32, 300 (1985); B. Yurke, P. Grangier, R. E. Slusher, and M. J. Potasek, Phys. Rev. A 35, 3586 (1987).
[Crossref] [PubMed]

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
[Crossref] [PubMed]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).

J. Math. Phys. (1)

D. J. Kaup, J. Math. Phys. 16, 2036 (1975).
[Crossref]

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

J. Phys. Soc. Jpn. (1)

M. Wadati and M. Sakagami, J. Phys. Soc. Jpn. 53, 1933 (1984); M. Wadati, “Quantum inverse scattering method,” in Dynamical Problems in Soliton Systems, S. Takeno, ed. (Springer-Verlag, Berlin, 1985).
[Crossref]

Opt. Lett. (2)

Phys. Rev. A (6)

B. Yurke, Phys. Rev. A 32, 300 (1985); B. Yurke, P. Grangier, R. E. Slusher, and M. J. Potasek, Phys. Rev. A 35, 3586 (1987).
[Crossref] [PubMed]

N. Imoto, H. A. Haus, and Y. Yamamoto, Phys. Rev. A 32, 2287 (1985).
[Crossref] [PubMed]

M. J. Potasek and B. Yurke, Phys. Rev. A 35, 3974 (1987).
[Crossref] [PubMed]

M. J. Potasek and B. Yurke, Phys. Rev. A 38, 1335 (1988).
[Crossref] [PubMed]

H. P. Yuen, Phys. Rev. A 13, 2226 (1976).
[Crossref]

M. Kitagawa and Y. Yamamoto, Phys. Rev. A 34, 3974 (1986).
[Crossref] [PubMed]

Phys. Rev. B (1)

R. M. Shelby, M. D. Levenson, and P. W. Bayer, Phys. Rev. B 31, 5244 (1985).
[Crossref]

Phys. Rev. Lett. (4)

S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby, Phys. Rev. Lett. 58, 1841 (1987).
[Crossref] [PubMed]

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
[Crossref] [PubMed]

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
[Crossref] [PubMed]

L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986); M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, and H. J. Kimble, Phys. Rev. Lett. 59, 198 (1987).
[Crossref] [PubMed]

Rev. Mod. Phys. (1)

H. A. Haus, Rev. Mod. Phys. 51, 331 (1979).
[Crossref]

Sov. Phys. JETP (1)

V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).

Other (1)

Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers, I. Time dependent Hartree approximation; II. Exact solution,” Phys. Rev. A (to be published).

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Figures (5)

Fig. 1
Fig. 1

(a) Quasi-probability densities of a squeezed state at input and output of a dispersive medium. (b) Diagram explaining the squeezing-angle rotation in the frequency domain.

Fig. 2
Fig. 2

(a) Quasi-probability densities of a coherent-state input and a squeezed output of a Kerr nonlinear medium. (b) Diagram explaining the squeezing in the frequency domain.

Fig. 3
Fig. 3

(a) Quasi-probability density of a number-state soliton |N〉. (b) The field amplitude of a number-state soliton.

Fig. 4
Fig. 4

(a) Quasi-probability density of a momentum-state soliton |p〉. (b) The field amplitude of a momentum-state soliton.

Fig. 5
Fig. 5

Setup for the quantum-nondemolition measurement for soliton photon number and momentum. A.O., acousto-optic.

Equations (119)

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k n L = 2 π n .
[ a ^ n , a ^ n ] = δ n n .
a ^ ( k ) = ( L 2 π ) 1 / 2 a ^ n = a ^ n Δ k ,
lim Δ k 0 [ a ^ ( k ) , a ^ ( k ) ] = lim Δ k 0 δ n n Δ k = δ ( k - k ) .
α k n = A exp [ - ( n Δ k ) 2 k 0 2 ] .
z a ^ ( z , k ) = i k a ^ ( z , k ) .
z a ^ ( z , k ) = i [ k ( ω 0 ) + ( ω - ω 0 ) d k d ω + 1 2 ( ω - ω 0 ) 2 d 2 k d ω 2 ] a ^ ( z , k ) ,
D ( d 2 k / d ω 2 ) / ( d k / d ω ) 2 ,
A ^ ( z , β ) = a ^ ( z , k ) exp [ i { k ( ω 0 ) - ( ω - ω 0 ) d k d ω } z ]
z A ^ ( z , β ) = i 2 β 2 D A ^ ( z , β ) .
β = ( d k / d ω ) ( ω - ω 0 )
[ A ( z , β ) , A ( z , β ) ] = δ ( β - β ) .
A ^ ( z , β ) = 1 2 π - α ^ ( z , x ) e i β x d x .
z α ^ ( z , x ) = - i 2 D 2 x 2 α ^ ( z , x ) .
i d d t A ^ = [ A ^ , H ^ ] ,
H ^ = - 2 v D β 2 A ( z , β ) A ^ ( z , β ) .
H ^ = - 2 v D [ x α ^ ( z , x ) ] [ x α ^ ( z , x ) ] d x
[ α ^ ( z , x ) , α ^ ( z , x ) ] = 1 2 π [ d β A ^ ( z , β ) e - i β x , d β A ^ ( z , β ) e i β x ] = 1 2 π d β d β δ ( β - β ) e - i β x e i β x = 1 2 π d β e - i β ( x - x ) = δ ( x - x ) ,
z α ^ ( z , x ) = i 2 κ α ^ ( z , x ) α ^ 2 ( z , x ) .
H ^ = - v κ 4 α ^ ( z , x ) 2 α ^ ( z , x ) 2 d x .
H ^ = - v κ 8 π A ^ ( z , β ) A ^ ( z , β ) × A ^ ( z , β ) A ^ ( z , β + β - β ) d β d β d β .
z A ^ ( z , β ) = - i κ 4 π d β d β A ^ ( z , β ) × A ^ ( z , β ) A ^ ( z , β + β - β ) .
H ^ = - 1 2 v D β 2 A ^ ( z , β ) A ^ ( z , β ) d β - v κ 8 π A ^ ( z , β ) × A ^ ( z , β ) A ^ ( z , β ) A ^ ( z , β + β - β ) d β d β d β .
z A ^ ( z , β ) = i 2 D β 2 A ^ ( z , β ) - i κ 4 π d β d β A ^ ( z , β ) × A ^ ( z , β ) A ^ ( z , β + β - β ) .
H ^ = - 2 v D [ x α ^ ( z , x ) ] [ x α ^ ( z , x ) ] d x - 4 v κ α ^ ( z , x ) 2 α ^ ( z , x ) 2 d x .
i z α ^ ( z , x ) = 1 2 D 2 x 2 α ^ ( z , x ) - 1 2 κ α ^ ( z , x ) α ^ 2 ( z , x ) .
v 1 x = - i ζ v 1 + ( κ 2 D ) 1 / 2 α ( 0 , x ) v 2 ,
v 2 x = i ζ v 2 - ( κ 2 D ) 1 / 2 α * ( 0 , x ) v 1 .
i v 1 z = [ D ζ 2 - κ 2 α * α ] v 1 + κ D ( i ζ α - 1 2 α x ) v 2 ,
i v 2 z = - κ D ( i ζ α * + 1 2 α * x ) v 1 - [ D ζ 2 - ( κ / 2 ) α * α ] v 2 ,
ρ ( ζ ) b ( ζ ) / a ( ζ ) ,
v 1 = α ( ζ ) e - i ζ x ,
v 2 = b ( ζ ) e i ζ x
ρ ( z , ξ ) = - ( κ 2 D ) 1 / 2 - A ( z , β = - 2 ξ ) exp ( - 2 i ξ x ) d x .
i z ρ j = - D ( 2 ζ j ) 2 ρ j
i z ρ ( ξ ) = - D ( 2 ξ ) 2 ρ ( ξ ) .
z a ( ξ ) = 0 ,
α ( z , x ) = 2 ( 2 D κ ) 1 / 2 η 1 sech [ 2 η 1 ( x - x 1 + 2 D ξ 1 z ) ] × exp [ - 2 i ξ 1 x - 2 i D ( ξ 1 2 - η 1 2 ) z + i ϕ 1 ] .
x 1 = 1 2 η 1 ln ( b 1 2 η 1 ) ,
ϕ 1 = - 2 arg b 1 .
N = - α * α d x ,
P = - i 2 - [ α * α x - α * x α ] d x ,
H = - [ v D 2 ( α x ) * ( α x ) - v κ 4 ( α * α ) 2 ] d x .
N = 4 D κ [ 2 j = 1 J η j + 1 2 π - d ξ ln ( 1 + ρ * ρ ) ] ,
P = 4 D κ [ - 4 j = 1 J ξ j η j - 1 π d ξ ξ ln ( 1 + ρ * ρ ) ] ,
H = 8 v D 2 κ [ j = 1 J ( 2 ξ j 2 η j - 2 3 η j 3 ) + 1 2 π d ξ ξ 2 ln ( 1 + ρ * ρ ) ] .
P j = ( - 2 ξ j ) .
b j = ρ a ζ | ζ = ζ j .
t ln b j = t b j b j = 2 i v D ζ j 2 ,
t ln b j 2 + i t arg ( b j ) = - 4 v D ξ j η j + i 2 v D ( ξ j 2 - η j 2 ) .
D t { ln b j 2 } = H p j
t p j = - H { ln b j 2 } = 0.
p j = - 2 ξ j / κ = P j / κ
q j = D ln b j 2 .
t { - arg ( b j ) } = - H N j
t N j = H { - arg ( b j ) } = 0 ,
N j = 8 D κ η j
ϕ j = - arg ( b j ) .
{ q i , p j } = δ i j
{ N i , ϕ j } = δ i j .
[ q ^ i , p ^ j ] = i δ i j
[ N ^ i , ϕ ^ j ] = i δ i j .
i t ρ ( ξ ) = - v D 2 ( 2 ξ ) 2 ρ ( ξ ) .
t ρ ( ξ ) 2 = 0
t arg [ ρ ( ξ ) ] = v D 2 ( 2 ξ ) 2 .
t { - arg [ ρ ( ξ ) ] } = - H N ( ξ )
t N ( ξ ) = H { - arg [ ρ ( ξ ) ] } = 0 ,
N ( ξ ) = 2 D Δ ξ π κ ln ( 1 + ρ ( ξ ) 2 )
ϕ ( ξ ) = - arg [ ρ ( ξ ) ] .
[ N ^ ( ξ ) , ϕ ^ ( ξ ) ] = i δ ( ξ - ξ ) .
N j = A j 1 2 + A j 2 2
ϕ j = arg ( A j 1 + i A j 2 ) .
[ A ^ i , A ^ j ] = δ i j
[ A ^ ( ξ ) , A ^ ( ξ ) ] = δ ( ξ - ξ ) .
Δ x 1 1 2 η 1 ,
Δ q 1 = 2 D Δ b 1 b 1 = 4 D η 1 Δ x 1 2 D .
Δ p 1 1 2 1 Δ q 1 4 D .
z diff = 1 4 D Δ ξ 1 η 1 .
z sp = π D η 1 2 2 D η 1 κ π .
z diff z sp N 4 π ,
Δ p 1 = 2 Δ ξ 1 / κ
Δ N 1 1 2 1 Δ ϕ 1 .
Δ ϕ rot = 2 D Δ η 1 2 z = 4 D η 1 2 Δ N 1 N 1 z = 4 π z z sp Δ N 1 N 1 .
Δ ϕ rot 1 2 1 Δ N 1 .
Δ z ph < z sp 8 π Δ N 1 2 N 1 = z sp 8 π .
ϕ 2 2 = - arg ( ζ 2 - ζ 1 ζ 2 - ζ 1 * ) = tan - 1 ( η 2 + η 1 ξ 2 - ξ 1 ) - tan - 1 ( η 2 - η 1 ξ 2 - ξ 1 ) .
tan ( ϕ 2 / 2 ) = 2 η 1 ( ξ 2 - ξ 1 ) ( η 2 2 - η 1 2 ) + ( ξ 2 - ξ 1 ) 2 .
ϕ ^ 2 1 = ϕ 2 1 + Δ ϕ ^ 2 1 ,
ξ ^ 2 1 = ξ 2 1 + Δ ξ ^ 2 1 ,
η ^ 2 1 = η 2 1 + Δ η ^ 2 1 .
Δ ϕ ^ 2 = 4 Δ η ^ 1 ( ξ 2 - ξ 1 ) × ( η 2 2 + η 1 2 ) + ( ξ 2 - ξ 1 ) 2 ( η 2 2 + η 1 2 ) 2 + 2 ( η 2 2 + η 1 2 ) ( ξ 2 - ξ 1 ) 2 + ( ξ 2 - ξ 1 ) 4 = F Δ N ^ 1 ,
F = κ 2 D ( ξ 2 - ξ 1 ) × ( η 2 2 + η 1 2 ) + ( ξ 2 - ξ 1 ) 2 ( η 2 2 - η 1 2 ) 2 + 2 ( η 2 2 + η 1 2 ) ( ξ 2 - ξ 1 ) 2 + ( ξ 2 - ξ 1 ) 4 .
F = κ 2 D ξ 2 - ξ 1 .
x 1 ( z ) - x 2 ( z ) = 2 ( ξ 1 - ξ 2 ) D .
x p 1 / η .
z eff = 1 2 D ξ 1 - ξ 2 η = 4 κ N ξ 1 - ξ 2 .
F = κ z eff x p .
x 2 = 1 η 2 ln | ζ 2 - ζ 1 ζ 2 - ζ 1 * | .
Δ x ^ 2 = - 4 η 1 ( ξ 2 - ξ 1 ) [ ( ξ 2 - ξ 1 ) 2 + ( η 2 + η 1 ) 2 ] [ ( ξ 2 - ξ 1 ) 2 + ( η 2 - η 1 ) 2 ] Δ ξ ^ 1 = G Δ P ^ 1 ,
G = 2 η 1 ( ξ 2 - ξ 1 ) [ ( ξ 2 - ξ 1 ) 2 + ( η 2 + η 1 ) 2 ] [ ( ξ 2 - ξ 1 ) 2 + ( η 2 - η 1 ) 2 ] .
G = 4 ( ξ 2 - ξ 1 ) κ D N .
Δ z eff z eff = Δ ξ 1 ξ 2 - ξ 1 .
Δ z eff z eff = Δ x 2 x p .
Δ x 2 = Δ ξ 1 ξ 2 - ξ 1 x p = 4 D κ N ( ξ 2 - ξ 1 ) Δ P 1 .
Δ N ^ 1 2 meas = Δ ϕ 2 2 fluct / F .
Δ ϕ ^ 1 2 backaction = F Δ N ^ 2 2 fluct .
Δ N 1 2 meas Δ ϕ ^ 1 2 backaction = Δ N ^ 2 2 fluct Δ ϕ ^ 2 2 fluct .
Δ N 2 2 fluct Δ ϕ 2 2 1 / 4
Δ P ^ 1 2 meas = Δ x ^ 2 2 fluct / G .
Δ x ^ 1 2 backaction = G Δ P ^ 2 2 fluct .
Δ P ^ 1 2 meas Δ x ^ 1 2 backaction = Δ P ^ 2 2 fluct Δ x ^ 2 2 fluct .
Δ ϕ ^ 2 = 2 cos 2 ( ϕ 2 / 2 ) Δ ( tan ϕ ^ 2 2 ) = 2 cos 2 ( ϕ 2 / 2 ) [ tan ( ϕ 2 / 2 ) η 1 Δ η ^ 1 + tan ( ϕ 2 / 2 ) ξ 1 Δ η ^ 1 ] = 1 ( η 2 2 - η 1 2 ) 2 + 2 ( η 2 2 + η 1 2 ) ( ξ 2 - ξ 1 ) 2 + ( ξ 2 - ξ 1 ) 4 × { [ ( η 2 2 + η 1 2 ) + ( ξ 2 - ξ 1 ) 2 ] 4 ( ξ 2 - ξ 1 ) Δ η ^ 1 + [ ( ξ 2 - ξ 1 ) 2 - ( η 2 2 - η 1 2 ) ] 4 η 1 Δ ξ ^ 1 } .
1 ξ 2 - ξ 1 = 2 v ( ω 2 - ω 1 ) .
1 η v Δ ω .
Δ ϕ ^ 2 = 4 ξ [ η 1 Δ ξ ^ 1 ξ + Δ η ^ 1 ] .
Δ ξ ^ 1 2 ( ξ / η 1 ) 2 Δ η ^ 1 2 = ξ 2 Δ N 1 2 meas N 1 2 .
Δ x ^ 2 = 1 η 2 { [ η 1 ln | ζ 1 - ζ 2 ζ 1 - ζ 2 * | ] Δ η ^ 1 + [ ξ 1 ln | ζ 1 - ζ 2 ζ 1 - ζ 2 * | ] Δ ξ ^ 1 } = 1 [ ( ξ 2 - ξ 1 ) 2 + ( η 2 - η 1 ) 2 ] [ ( ξ 2 - ξ 1 ) 2 + ( η 2 + η 1 ) 2 ] × { - 2 [ ( ξ 2 - ξ 1 ) 2 + ( η 2 2 - η 1 2 ) ] Δ η ^ 1 - 4 η 1 ( ξ 2 - ξ 1 ) Δ ξ ^ 1 } .
Δ x ^ 2 = - 4 ξ 2 [ 1 2 Δ η ^ 1 + η 1 Δ ξ ^ 1 ξ ] .
Δ η ^ 1 2 η 1 2 = Δ N ^ 1 2 N 1 2 4 Δ ξ ^ 1 2 ξ 2

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