Abstract

We perform a calculation to determine how quantum mechanical fluctuations influence the propagation of a spatial soliton through a nonlinear material. To do so, we derive equations of motion for the linearized operators describing the deviation of the soliton position and transverse momentum from those of a corresponding classical solution to the nonlinear wave equation, and from these equations we determine the quantum uncertainty in the soliton position and transverse momentum. We find that under realistic laboratory conditions the quantum uncertainty in position is several orders of magnitude smaller the classical width of the soliton. This result suggests that the reliability of photonic devices based on spatial solitons is not compromised by quantum fluctuations.

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References

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  1. A pedagogical discussion of self-action effects including self-trapping and optical solitons is presented in Chapter 6 of R. W. Boyd, Nonlinear Optics (Academic, San Diego, 1992). See also E. M. Nagasako, R. W. Boyd, in Amazing Light, A volume dedicated to Charles Hard Townes on his 80th Birthday, edited by R. Y. Chiao (Springer, New York, 1996).
  2. G. A. Askar'yan, Sov. Phys. JETP 15, 1088 (1962).
  3. R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
    [CrossRef]
  4. V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).
  5. E. L. Dawes and J. H. Marburger, Phys. Rev. 179, 862 (1969); See also J. H. Marburger, Prog. Quantum Electron. 4, 35 (1975).
    [CrossRef]
  6. Y. Silberberg, Opt. Lett. 15, 1282 (1990).
    [CrossRef] [PubMed]
  7. J. S. Aitchison, Y. Silberberg, A. M. Weiner, D. E. Leaird, M. K. Oliver, J. L. Jackel, E. M. Vogel, and P. W. E. Smith, J. Opt. Soc. Am. B 8, 1290 (1991).
    [CrossRef]
  8. M. Shalaby and A. Barthelemy, Opt. Comm. 94, 341 (1992).
    [CrossRef]
  9. A. Villeneuve, J. S. Aitchison, J. U. Kang, P. G. Wigley, and G. I. Stegeman, Opt. Lett. 19, 761 (1994).
    [CrossRef] [PubMed]
  10. B. Luther-Davies and X. Yang, Opt. Lett. 17, 1755 (1992).
    [CrossRef] [PubMed]
  11. S. Blair, K. Wagner, and R. McLeod, Opt. Lett. 19, 1943 (1994).
    [CrossRef] [PubMed]
  12. E. M. Nagasako, R. W. Boyd, and G. S. Agarwal, Phys. Rev. A 55, 1412 (1997).
    [CrossRef]
  13. J. D. Gordon and H. A. Haus, Opt. Lett. 11, 665 (1986).
    [CrossRef] [PubMed]
  14. P. D. Drummond R. M. Shelby, S. R. Friberg, and Y. Yamomoto, Nature 365, 307 (1993).
    [CrossRef]
  15. H. A. Haus and M. N. Islam, IEEE J. Quantum Electron. QE-21, 1172 (1985).
    [CrossRef]
  16. H. A. Haus and Y. Lai, J. Opt. Soc. Am. B 7, 386 (1990).
    [CrossRef]
  17. Although c has the fixed value -1/2 using the present conventions, we retain c in our formulas for more ready comparisons of our results with those obtained using different normalization conventions.

Other (17)

A pedagogical discussion of self-action effects including self-trapping and optical solitons is presented in Chapter 6 of R. W. Boyd, Nonlinear Optics (Academic, San Diego, 1992). See also E. M. Nagasako, R. W. Boyd, in Amazing Light, A volume dedicated to Charles Hard Townes on his 80th Birthday, edited by R. Y. Chiao (Springer, New York, 1996).

G. A. Askar'yan, Sov. Phys. JETP 15, 1088 (1962).

R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

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

E. L. Dawes and J. H. Marburger, Phys. Rev. 179, 862 (1969); See also J. H. Marburger, Prog. Quantum Electron. 4, 35 (1975).
[CrossRef]

Y. Silberberg, Opt. Lett. 15, 1282 (1990).
[CrossRef] [PubMed]

J. S. Aitchison, Y. Silberberg, A. M. Weiner, D. E. Leaird, M. K. Oliver, J. L. Jackel, E. M. Vogel, and P. W. E. Smith, J. Opt. Soc. Am. B 8, 1290 (1991).
[CrossRef]

M. Shalaby and A. Barthelemy, Opt. Comm. 94, 341 (1992).
[CrossRef]

A. Villeneuve, J. S. Aitchison, J. U. Kang, P. G. Wigley, and G. I. Stegeman, Opt. Lett. 19, 761 (1994).
[CrossRef] [PubMed]

B. Luther-Davies and X. Yang, Opt. Lett. 17, 1755 (1992).
[CrossRef] [PubMed]

S. Blair, K. Wagner, and R. McLeod, Opt. Lett. 19, 1943 (1994).
[CrossRef] [PubMed]

E. M. Nagasako, R. W. Boyd, and G. S. Agarwal, Phys. Rev. A 55, 1412 (1997).
[CrossRef]

J. D. Gordon and H. A. Haus, Opt. Lett. 11, 665 (1986).
[CrossRef] [PubMed]

P. D. Drummond R. M. Shelby, S. R. Friberg, and Y. Yamomoto, Nature 365, 307 (1993).
[CrossRef]

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

H. A. Haus and Y. Lai, J. Opt. Soc. Am. B 7, 386 (1990).
[CrossRef]

Although c has the fixed value -1/2 using the present conventions, we retain c in our formulas for more ready comparisons of our results with those obtained using different normalization conventions.

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Equations (66)

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2 i k z A + 2 x 2 A + 2 k 2 n ̄ 2 n 0 A 2 A = 0 ,
E ( r , t ) = 1 2 x ̂ [ A ( x , z ) exp ( i β 0 z i ω 0 t ) + c . c . ] .
X 2 k x Z k z c 1 / 2
Φ ( + ) A n 0 / n ̄ 2 Φ ( ) ( Φ ( + ) ) * A * n 0 / n ̄ 2
i Z Φ ( + ) + 2 X 2 Φ ( + ) 2 c Φ ( + ) 2 Φ ( + ) = 0 .
Φ 0 ( + ) ( X , Z ) = N 0 c 1 / 2 2 exp [ i N 0 2 c 2 4 Z i p 0 2 Z + i p 0 ( X X 0 ) + i θ 0 ]
sech [ N 0 c 2 ( X X 0 2 p 0 Z ) ] ,
N 0 = Φ ( + ) ( X , Z ) 2 d X
Φ ̂ ( + ) ( X , Z ) = Φ ( + ) ( X , Z ) + Ψ ̂ ( + ) ( X , Z ) .
Ψ ̂ ( ) ( X , Z ) = Ψ ̂ ( + ) ( X , Z ) , Φ ̂ ( ) ( X , Z ) = Φ ̂ ( + ) ( X , Z ) ,
[ Ψ ̂ ( + ) ( X , Z ) , Ψ ̂ ( ) ( X , Z ) ] = S δ ( X X ) S = ħ c k 0 2 n 2 2 Δt L y .
i Z Ψ ̂ ( + ) + 2 X 2 Ψ ̂ ( + ) + 4 c Φ 0 ( + ) 2 Ψ ̂ ( + ) + 2 c ( Φ 0 ( + ) ) 2 Ψ ̂ ( ) = 0
Ψ ̂ ( + ) ( X , Z ) = Φ 0 ( + ) N 0 Δ N ̂ 0 ( + ) + Φ 0 ( + ) θ 0 Δ θ ̂ 0 ( + ) + Φ 0 ( + ) p 0 Δ p ̂ 0 ( + ) + Φ 0 ( + ) X 0 Δ X ̂ 0 ( + ) .
Ψ j ( + ) ( X , Z ) = Φ 0 ( + ) ( X , Z ) j 0 ,
Ψ N ( + ) ( X , Z ) = [ 1 N 0 + i N 0 c 2 2 Z c 2 X tanh ( N 0 c 2 2 X ) ] Φ 0 ( + ) ( X , Z ) ,
Ψ θ ( + ) ( X , Z ) = i Φ 0 ( + ) ( X , Z ) ,
Ψ p ( + ) ( X , Z ) = [ i X + N 0 c Z tanh ( N 0 c 2 X ) ] Φ 0 ( + ) ( X , Z ) ,
Ψ X ( + ) ( X , Z ) = [ N 0 c 2 tanh ( N 0 c 2 X ) ] Φ 0 ( + ) ( X , Z ) .
Δ N ̂ 0 = [ Ψ̲ θ ( ) ( X , 0 ) Ψ ̂ ( + ) ( X , 0 ) + Ψ̲ θ ( + ) ( X , 0 ) Ψ ̂ ( ) ( X , 0 ) ] d X ,
Δ θ ̂ 0 = [ Ψ̲ N ( ) ( X , 0 ) Ψ ̂ ( + ) ( X , 0 ) + Ψ̲ N ( + ) ( X , 0 ) Ψ ̂ ( ) ( X , 0 ) ] d X ,
Δ p ̂ 0 = 1 N 0 [ Ψ̲ X ( ) ( X , 0 ) Ψ ̂ ( + ) ( X , 0 ) + Ψ̲ X ( + ) ( X , 0 ) Ψ ̂ ( ) ( X , 0 ) ] d X ,
Δ X ̂ 0 = 1 N 0 [ Ψ̲ p ( ) ( X , 0 ) Ψ ̂ ( + ) ( X , 0 ) + Ψ̲ p ( + ) ( X , 0 ) Ψ ̂ ( ) ( X , 0 ) ] d X .
Δ N ̂ 0 = Δ N ̂ 0 ( + ) + Δ N ̂ 0 ( ) = Δ N ̂ 0 ( + ) + H . c .
[ Δ N ̂ 0 , Δ θ ̂ 0 ] = i S
[ Δ X ̂ 0 , N 0 Δ p ̂ 0 ] = i S .
Δ N ̂ ( Z ) = [ Ψ̲ θ ( ) ( X , 0 ) exp ( i N 0 2 c 2 Z 4 ) Ψ ̂ ( + ) ( X , Z )
+ Ψ̲ θ ( + ) ( X , 0 ) exp ( i N 0 2 c 2 Z 4 ) Ψ ̂ ( ) ( X , Z ) ] d X ,
Δ θ ̂ ( Z ) = [ Ψ̲ N ( ) ( X , 0 ) exp ( i N 0 2 c 2 Z 4 ) Ψ ̂ ( + ) ( X , Z )
+ Ψ̲ N ( + ) ( X , 0 ) exp ( i N 0 2 c 2 Z 4 ) Ψ ̂ ( ) ( X , Z ) ] d X ,
Δ p ̂ ( Z ) = 1 N 0 [ Ψ̲ X ( ) ( X , 0 )
exp ( i N 0 2 c 2 Z 4 ) Ψ ̂ ( + ) ( X , Z )
+ Ψ̲ X ( + ) ( X , 0 ) exp ( i N 0 2 c 2 Z 4 ) Ψ ̂ ( ) ( X , Z ) ] d X ,
Δ X ̂ ( Z ) = 1 N 0 [ Ψ̲ p ( ) ( X , 0 ) exp ( i N 0 2 c 2 Z 4 ) Ψ ̂ ( + ) ( X , Z )
+ Ψ̲ p ( + ) ( X , 0 ) exp ( i N 0 2 c 2 Z 4 ) Ψ ̂ ( ) ( X , Z ) ] d X .
d d Z Δ p ̂ = 0
d d Z Δ X ̂ = 2 Δ p ̂ ,
Δ p ̂ ( Z ) = Δ p ̂ 0
Δ X ̂ ( Z ) = Δ X ̂ 0 + 2 Δ p ̂ 0 Z .
Δ X ̂ 0 2 = π 2 S 3 N 0 3 c 2
Δ p ̂ 0 2 = N 0 c 2 S 12 .
Δ X 2 ( Z ) = π 2 S 3 N 0 3 c 2 + N 0 c 2 S 3 Z 2 .
A ( z ) = A peak sech ( x w ) exp ( i z 2 k w 2 )
A peak = 1 k w n 0 2 n ̄ 2 .
Δ n = 1 2 n ̄ 2 A peak 2
Δ n = n 0 2 k 2 w 2 .
N 0 = 2 k w .
Δ x 2 ( z ) = π 2 S ( k w ) w 2 12 + S 12 ( k w ) z 2 .
( Δ x ) rms w π 2 12 S k w .
i Ψ 1 ( + ) Z = L Ψ 1 ( + ) 2 c ( Φ 0 ( + ) ) 2 Ψ 1 ( )
i Ψ 2 ( + ) Z = L Ψ 2 ( + ) 2 c ( Φ 0 ( + ) ) 2 Ψ 2 ( )
L 2 X 2 + 4 c Φ 0 ( + ) 2 .
i ( Ψ 1 ( + ) Ψ 2 ( ) ) Z d X = [ Ψ 2 ( ) L Ψ 1 ( + ) + Ψ 1 ( + ) L Ψ 2 ( ) 2 c ( Φ 0 ( + ) ) 2 Ψ 1 ( ) Ψ 2 ( )
+ 2 c * ( Φ 0 ( ) ) 2 Ψ 2 ( + ) Ψ 1 ( + ) ] d X .
i Z ( Ψ 1 ( + ) Ψ 2 ( ) ) d X = [ 2 c ( Φ 0 ( + ) ) 2 Ψ 1 ( ) Ψ 2 ( )
+ 2 c * ( Φ 0 ( ) ) 2 Ψ 2 ( + ) Ψ 1 ( + ) ] d X .
Z [ Ψ 1 ( + ) ( i Ψ 2 ( ) ) d X + c . c . ] = 0 .
Ψ̢ i ( + ) ( X , Z ) = i Ψ i ( + ) ( X , Z ) ,
Z [ ( Ψ 2 ( + ) ( X , Z ) Ψ̢ 2 ( ) ( X , Z ) ) d X + c . c . ] = 0 .
Ψ i ( + ) ( X , Z ) = f i ( X ) exp ( i E i Z )
f 1 ( X ) f 2 * ( X ) d X = 0 ,
Δ p ̂ ( Z ) = 1 N 0 Ψ̲ X ( + ) ( X , Z ) * Ψ ̂ ( + ) ( X , Z ) d X
d d Z Δ p ̂ ( Z ) = 0 .
Ψ̢ p ( + ) ( X , Z ) * exp ( i N 0 2 c 2 Z 4 ) = Ψ̢ p ( + ) ( X , Z ) * 2 Z Ψ̢ X ( + ) ( X , Z ) *
Δ X ̂ ( Z ) = 1 N 0 Ψ̢ p ( + ) ( X , Z ) * Ψ ̂ ( + ) ( X , Z ) d X + c . c .
+ 2 Z 1 N 0 Ψ̢ X ( + ) ( X , Z ) * Ψ ̂ ( + ) ( X , Z ) d X + c . c . .
d d Z Δ X ̂ ( Z ) = 2 N 0 Ψ̢ X ( + ) ( X , Z ) * Ψ ̂ ( + ) ( X , Z ) d X + H . c . = 2 Δ p ̂ ( Z )

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