Abstract

A linearized quantum theory of soliton squeezing and detection is presented. The linearization reduces the quantum problem to a classical one. The classical formulation provides physical insight. It is shown that a quantized soliton exhibits uncertainties in photon number and phase, position (time), and momentum (frequency). Detectors for the measurement of all four operators are discussed. The squeezing of the soliton in the fiber is analyzed. An optimal homodyne detector for detection of the squeezing is presented that suppresses the noise associated with the continuum and the uncertainties in position and momentum.

© 1990 Optical Society of America

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  1. S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby, Phys. Rev. Lett. 58, 1841 (1987).
    [CrossRef] [PubMed]
  2. P. D. Drummond and S. J. Carter, J. Opt. Soc. Am. B 4, 1565 (1987).
    [CrossRef]
  3. P. D. Drummond, S. J. Carter, and R. M. Shelby, Opt. Lett. 14, 373 (1989).
    [CrossRef] [PubMed]
  4. Y. Lai and H. A. Haus, Phys. Rev. A 40, 844 (1989).
    [CrossRef] [PubMed]
  5. Y. Lai and H. A. Haus, Phys. Rev. A 40, 1138 (1989).
  6. H. A. Haus, K. Watanabe, and Y. Yamamoto, J. Opt. Soc. Am. B 6, 1138 (1989).
    [CrossRef]
  7. J. Satsuma and N. Yajima, Suppl. Prog. Theor. Phys. 55, 284 (1974).
    [CrossRef]
  8. V. I. Karpman and V. V. Solovev, Physica 3D, 487 (1981).
  9. H. A. Haus and M. N. Islam, IEEE J. Quantum Electron. QE-21, 1172 (1985).
    [CrossRef]
  10. V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).
  11. A. Messiah, Quantum Mechanics, translated from the French by G. M. Temmer, (North-Holland, Amsterdam, 1961), Vol. 1, p. 301.
  12. B. Yurke, Phys. Rev. A 32, 311 (1985).
    [CrossRef] [PubMed]
  13. R. S. Bondurant, Phys. Rev. A 32, 2797 (1985).
    [CrossRef] [PubMed]

1989 (4)

1987 (2)

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]

1985 (3)

H. A. Haus and M. N. Islam, IEEE J. Quantum Electron. QE-21, 1172 (1985).
[CrossRef]

B. Yurke, Phys. Rev. A 32, 311 (1985).
[CrossRef] [PubMed]

R. S. Bondurant, Phys. Rev. A 32, 2797 (1985).
[CrossRef] [PubMed]

1981 (1)

V. I. Karpman and V. V. Solovev, Physica 3D, 487 (1981).

1974 (1)

J. Satsuma and N. Yajima, Suppl. Prog. Theor. Phys. 55, 284 (1974).
[CrossRef]

1972 (1)

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

Bondurant, R. S.

R. S. Bondurant, Phys. Rev. A 32, 2797 (1985).
[CrossRef] [PubMed]

Carter, S. J.

Drummond, P. D.

Haus, H. A.

Y. Lai and H. A. Haus, Phys. Rev. A 40, 844 (1989).
[CrossRef] [PubMed]

Y. Lai and H. A. Haus, Phys. Rev. A 40, 1138 (1989).

H. A. Haus, K. Watanabe, and Y. Yamamoto, J. Opt. Soc. Am. B 6, 1138 (1989).
[CrossRef]

H. A. Haus and M. N. Islam, IEEE J. Quantum Electron. QE-21, 1172 (1985).
[CrossRef]

Islam, M. N.

H. A. Haus and M. N. Islam, IEEE J. Quantum Electron. QE-21, 1172 (1985).
[CrossRef]

Karpman, V. I.

V. I. Karpman and V. V. Solovev, Physica 3D, 487 (1981).

Lai, Y.

Y. Lai and H. A. Haus, Phys. Rev. A 40, 1138 (1989).

Y. Lai and H. A. Haus, Phys. Rev. A 40, 844 (1989).
[CrossRef] [PubMed]

Messiah, A.

A. Messiah, Quantum Mechanics, translated from the French by G. M. Temmer, (North-Holland, Amsterdam, 1961), Vol. 1, p. 301.

Reid, M. D.

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

Satsuma, J.

J. Satsuma and N. Yajima, Suppl. Prog. Theor. Phys. 55, 284 (1974).
[CrossRef]

Shabat, A. B.

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

Shelby, R. M.

P. D. Drummond, S. J. Carter, and R. M. Shelby, Opt. Lett. 14, 373 (1989).
[CrossRef] [PubMed]

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

Solovev, V. V.

V. I. Karpman and V. V. Solovev, Physica 3D, 487 (1981).

Watanabe, K.

Yajima, N.

J. Satsuma and N. Yajima, Suppl. Prog. Theor. Phys. 55, 284 (1974).
[CrossRef]

Yamamoto, Y.

Yurke, B.

B. Yurke, Phys. Rev. A 32, 311 (1985).
[CrossRef] [PubMed]

Zakharov, V. E.

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

IEEE J. Quantum Electron. (1)

H. A. Haus and M. N. Islam, IEEE J. Quantum Electron. QE-21, 1172 (1985).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Lett. (1)

Phys. Rev. A (4)

Y. Lai and H. A. Haus, Phys. Rev. A 40, 844 (1989).
[CrossRef] [PubMed]

Y. Lai and H. A. Haus, Phys. Rev. A 40, 1138 (1989).

B. Yurke, Phys. Rev. A 32, 311 (1985).
[CrossRef] [PubMed]

R. S. Bondurant, Phys. Rev. A 32, 2797 (1985).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

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

Physica (1)

V. I. Karpman and V. V. Solovev, Physica 3D, 487 (1981).

Sov. Phys. JETP (1)

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

Suppl. Prog. Theor. Phys. (1)

J. Satsuma and N. Yajima, Suppl. Prog. Theor. Phys. 55, 284 (1974).
[CrossRef]

Other (1)

A. Messiah, Quantum Mechanics, translated from the French by G. M. Temmer, (North-Holland, Amsterdam, 1961), Vol. 1, p. 301.

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Figures (4)

Fig. 1
Fig. 1

Perturbation functions due to a change of (left to right, top to bottom) photon number, phase, momentum, and position. Figures are plotted with n0 = 100, |c| = 1.

Fig. 2
Fig. 2

Homodyne detection with a pulsed LO.

Fig. 3
Fig. 3

Optimum detection angle versus classical phase shift (propagation distance) under linearization approximation.

Fig. 4
Fig. 4

Optimal squeezing versus classical phase shift (propagation distance) under linearization approximation.

Tables (1)

Tables Icon

Table 1 Required LO Pulse Shapes for Measurement of Fluctuation Operators

Equations (72)

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i t ϕ ̂ ( x , t ) = 2 x 2 ϕ ̂ ( x , t ) + 2 c ϕ ̂ ( x , t ) ϕ ̂ ( x , t ) ϕ ̂ ( x , t ) .
[ ϕ ̂ ( x , t ) , ϕ ̂ ( x , t ) ] = δ ( x x ) ,
[ ϕ ̂ ( x , t ) , ϕ ̂ ( x , t ) ] = [ ϕ ̂ ( x , t ) , ϕ ̂ ( x , t ) ] = 0 .
ϕ 0 ( x , t ) = n 0 | c | 1 / 2 2 exp [ i n 0 2 | c | 2 4 t i p 0 2 t + i p 0 ( x x 0 ) + i θ 0 ] × sech [ n 0 | c | 2 ( x x 0 2 p 0 t ) ] .
ϕ ̂ ( x , t ) = ϕ 0 ( x , t ) + υ ̂ ( x , t ) ,
[ υ ̂ ( x , t ) , υ ̂ ( x , t ) ] = δ ( x x ) ,
[ υ ̂ ( x , t ) , υ ̂ ( x , t ) ] = [ υ ̂ ( x , t ) , υ ̂ ( x , t ) ] = 0 .
i t υ ̂ ( x , t ) = [ 2 x 2 + 4 | c | | ϕ 0 ( x , t ) | 2 ] υ ̂ ( x , t ) 2 | c | ϕ 0 ( x , t ) 2 υ ̂ ( x , t ) .
υ n ( x , t ) ϕ 0 ( x , t ) n 0 = [ 1 n 0 + i n 0 | c | 2 2 t | c | 2 x tanh ( n 0 | c | 2 x ) ] ϕ 0 ( x , t ) ,
υ θ ( x , t ) ϕ 0 ( x , t ) θ 0 = i ϕ 0 ( x , t ) ,
υ p ( x , t ) ϕ 0 ( x , t ) p 0 = [ i x + n 0 | c | t tanh ( n 0 | c | 2 x ) ] ϕ 0 ( x , t ) ,
υ x ( x , t ) ϕ 0 ( x , t ) x 0 = [ n 0 | c | 2 tanh ( n 0 | c | 2 x ) ] ϕ 0 ( x , t ) .
υ ̂ ( x , t ) = ϕ 0 ( x , t ) n 0 Δ n ̂ 0 + ϕ 0 ( x , t ) θ 0 Δ θ ̂ 0 + ϕ 0 ( x , t ) p 0 Δ p ̂ 0 + ϕ 0 ( x , t ) x 0 Δ x ̂ 0 + Δ υ ̂ c ( x , t ) .
t [ ( Re υ ) 2 + ( Im υ ) 2 ] d x = 0
t [ ( Re υ ) ( Re υ _ ) + ( Im υ ) ( Im υ _ ) ] d x = t Re υ _ * ( x , t ) υ ( x , t ) d x = 0 .
i t υ _ ( x , t ) = [ 2 x 2 + 4 | c | | ϕ 0 ( x , t ) | 2 ] υ _ ( x , t ) + 2 | c | ϕ 0 ( x , t ) 2 υ _ * ( x , t ) .
t Re υ _ ( x , t ) υ ( x , t ) d x = 0
t Re υ _ * ( x , t ) υ ( x , t ) d x = 0 .
υ _ ( x , t ) = i υ ( x , t ) .
υ _ n ( x , t ) = i [ 1 n 0 + i n 0 | c | 2 2 t | c | 2 x tanh ( n 0 | c | 2 x ) ] ϕ 0 ( x , t ) ,
υ _ θ ( x , t ) = ϕ 0 ( x , t ) ,
υ _ p ( x , t ) = [ x + i n 0 | c | t tanh ( n 0 | c | 2 x ) ] ϕ 0 ( x , t ) ,
υ _ x ( x , t ) = i [ n 0 | c | 2 tanh ( n 0 | c | 2 x ) ] ϕ 0 ( x , t ) .
[ υ _ j * ( x , t ) Δ υ ̂ c ( x , t ) + υ _ j ( x , t ) Δ υ ̂ c ( x , t ) ] d x = 0 , j = n , θ , p , x
Δ n ̂ 0 = [ υ _ θ * ( x , 0 ) υ ̂ ( x , 0 ) + υ _ θ ( x , 0 ) υ ̂ ( x , 0 ) ] d x ,
Δ θ ̂ 0 = { [ υ _ n * ( x , 0 ) υ ̂ ( x , 0 ) + υ _ n ( x , 0 ) υ ̂ ( x , 0 ) ] d x } ,
Δ p ̂ 0 = 1 n 0 { [ υ _ x * ( x , 0 ) υ ̂ ( x , 0 ) + υ _ x ( x , 0 ) υ ̂ ( x , 0 ) ] d x } ,
Δ x ̂ 0 = 1 n 0 { [ υ _ p * ( x , 0 ) υ ̂ ( x , 0 ) + υ _ p ( x , 0 ) υ ̂ ( x , 0 ) ] d x } .
[ Δ n ̂ 0 , Δ θ ̂ 0 ] = i ,
[ Δ x ̂ 0 , n 0 Δ p ̂ 0 ] = i ,
Δ n ̂ 0 2 = | υ _ θ ( x , 0 ) | 2 d x = n 0 ,
Δ θ ̂ 0 2 = | υ _ n ( x , 0 ) | 2 d x 0.6075 n 0 .
Δ n ̂ 0 2 × Δ θ ̂ 0 2 0.6075 > 0.25 .
Δ n ̂ ( t ) = υ _ θ * ( x , 0 ) exp ( i n 0 2 | c | 2 4 t ) υ ̂ ( x , t ) d x + h . c . ,
Δ θ ̂ ( t ) = υ _ n * ( x , 0 ) exp ( i n 0 2 | c | 2 4 t ) υ ̂ ( x , t ) d x + h . c . ,
Δ p ̂ ( t ) = 1 n 0 υ _ x * ( x , 0 ) exp ( i n 0 2 | c | 2 4 t ) υ ̂ ( x , t ) d x + h . c . ,
Δ x ̂ ( t ) = 1 n 0 υ _ p * ( x , 0 ) exp ( i n 0 2 | c | 2 4 t ) υ ̂ ( x , t ) d x + h . c . ,
Δ n ̂ ( t ) = Δ n ̂ 0 ,
Δ θ ̂ ( t ) = Δ θ ̂ 0 + n 0 | c | 2 2 Δ n ̂ 0 t ,
Δ p ̂ ( t ) = Δ p ̂ 0 ,
Δ x ̂ ( t ) = Δ x ̂ 0 + 2 Δ p ̂ 0 t .
Δ n ̂ ( t ) 2 n 0 â 1 ( t ) â ( t ) + â ( t ) 2 ,
n 0 Δ θ ̂ ( t ) â 2 ( t ) â ( t ) â ( t ) 2 i .
â ( t ) = μ â ( 0 ) + ν â ( 0 ) ,
μ 1 + i n 0 2 | c | 2 2 t ,
ν i n 0 2 | c | 2 2 t ,
â θ L ( t ) Re [ â ( t ) exp ( i θ L ) ] = Re { [ μ â ( 0 ) + ν â ( 0 ) ] exp ( i θ L ) } = Re [ μ exp ( i θ L ) + ν * exp ( i θ L ) ] â 1 ( 0 ) Im [ μ exp ( i θ L ) + ν * exp ( i θ L ) ] â 2 ( 0 ) ,
Var [ â θ L ( t ) ] = Re 2 [ μ ( t ) exp ( i θ L ) + ν * ( t ) exp ( i θ L ) ] â 1 2 ( 0 ) + Im 2 [ μ ( t ) exp ( i θ L ) + ν * ( t ) exp ( i θ L ) ] â 2 2 ( 0 ) ,
â 1 2 ( 0 ) = 1 / 4 ,
â 2 2 ( 0 ) 0.6075 .
M ̂ 1 ( t ) = 1 2 [ f L * ( x ) ϕ ̂ ( x , t ) d x + h . c . ] ,
f L ( x ) ϕ L ( x ) .
| f L ( x ) | 2 d x = 1 .
â θ L ( t ) = cos ( θ L ) â ( t ) + â ( t ) 2 + sin ( θ L ) â ( t ) â ( t ) 2 i = cos ( θ L ) 1 2 n 0 Δ n ̂ ( t ) + sin ( θ L ) n 0 Δ θ ̂ ( t ) .
â θ L ( t ) = 1 2 [ f * ( x ) υ ̂ ( x , t ) d x + h . c . ] ,
f ( x ) = [ cos ( θ L ) υ _ θ ( x , 0 ) n 0 + 2 sin ( θ L ) n 0 υ _ n ( x , 0 ) ] × exp ( i n 0 2 | c | 2 4 t ) .
f L ( x ) = f ( x ) / [ | f ( x ) | 2 d x ] 1 / 2 = f ( x ) / { [ cos 2 ( θ L ) + 4 × 0.6075 × sin 2 ( θ L ) ] } 1 / 2 .
Var ( θ L , t ) = Var [ â θ L ( t ) ] [ cos 2 ( θ L ) + 4 × 0.6075 × sin 2 ( θ L ) ] .
R ( t ) Var opt ( t ) Var opt ( 0 ) ,
Δ n ̂ ( t )
υ _ θ ( x , 0 ) exp ( i n 0 2 | c | 2 4 t )
Δ θ ̂ ( t )
υ _ n ( x , 0 ) exp ( i n 0 2 | c | 2 4 t )
Δ p ̂ ( t )
1 n 0 υ _ x ( x , 0 ) exp ( i n 0 2 | c | 2 4 t )
Δ x ̂ ( t )
1 n 0 υ _ p ( x , 0 ) exp ( i n 0 2 | c | 2 4 t )
Δ n ̂ ( t ) = υ _ θ * ( x , 0 ) exp ( i n 0 2 | c | 2 4 t ) υ ̂ ( x , t ) d x + h . c . = ( ϕ 0 ( x , 0 ) { [ υ n ( x , 0 ) + i n 0 | c | 2 2 t ϕ 0 ( x , 0 ) ] Δ n ̂ 0 + i ϕ 0 ( x , 0 ) Δ θ ̂ 0 + [ υ p ( x , 0 ) + n 0 | c | t tanh ( n 0 | c | 2 x ) ϕ 0 ( x , 0 ) ] Δ p ̂ 0 + υ x ( x , 0 ) Δ x ̂ 0 + Δ υ ̂ c ( x , t ) exp ( i n 0 2 | c | 2 4 t ) } d x ) + h . c . = 2 ϕ 0 ( x , 0 ) υ n ( x , 0 ) d x Δ ̂ n 0 = Δ n ̂ 0 .
υ n ( x , 0 ) ϕ 0 ( x , 0 ) d x = 1 / 2
Δ θ ̂ ( t ) = υ _ n * ( x , 0 ) exp ( i n 0 2 | c | 2 4 t ) υ ̂ ( x , t ) d x + h . c . = ( i υ n ( x , 0 ) { [ υ n ( x , 0 ) + i n 0 | c | 2 2 t ϕ 0 ( x , 0 ) ] Δ n ̂ 0 + i ϕ 0 ( x , 0 ) Δ θ ̂ 0 + [ υ p ( x , 0 ) + n 0 | c | t tanh ( n 0 | c | 2 x ) ϕ 0 ( x , 0 ) ] Δ p ̂ 0 + υ x ( x , 0 ) Δ x ̂ 0 + Δ υ ̂ c ( x , t ) exp ( i n 0 2 | c | 2 4 t ) } d x ) + h . c . = 2 [ n 0 | c | 2 2 t υ n ( x , 0 ) ϕ 0 ( x , 0 ) d x ] Δ n ̂ 0 + 2 [ υ n ( x , 0 ) ϕ 0 ( x , 0 ) d x ] Δ θ ̂ 0 = Δ θ ̂ 0 + n 0 | c | 2 2 Δ n ̂ 0 t .
D ̂ ( x , t ) = [ | ϕ L ( x ) + ϕ ̂ ( x ) | 2 | ϕ L ( x ) ϕ ̂ ( x ) | 2 2 ] h ( x ) = [ ϕ L * ( x ) ϕ ̂ ( x , t ) + h . c . ] h ( x ) .
M ̂ 1 ( t ) = [ ϕ L * ( x ) ϕ ̂ ( x , t ) + h . c . ] h ( x ) d x = [ h ( x ) d x ] [ ϕ L * ( x ) ϕ ̂ ( x , t ) d x + h . c . ] ϕ L * ( x ) ϕ ̂ ( x , t ) d x + h . c .

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