Abstract

The concept of particles of light is introduced by ascribing a mechanical degree of freedom to a radiational wave packet. Evolution of the position and momentum observables of such particle in a dispersive dielectric is studied. It is shown that an initial coherent state evolves into a dispersed state which is characterized by a reduction of quantum fluctuations below Standard Quantum Limit when a certain combination of the position and momentum observables is measured.

© 2001 Optical Society of America

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References

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  1. L. Mandel and E. Wolf, Optical coherence and quantum optics (Cambridge University Press, 1995).
  2. T. D. Newton and E. P. Wigner, “Localized states for elementary systems,” Rev. Mod. Phys. 21, 400 (1949).
    [Crossref]
  3. A. B. Matsko and V. V. Kozlov, “Second-quantized models for optical solitons in nonlinear fibers: Equal-time versus equal-space commutation relations,” Phys. Rev. A 62, 033811 (2000).
    [Crossref]
  4. B. Huttner and S. M. Barnett, “Quantization of the electromagnetic field in dielectrics”, Phys. Rev. A 46, 4306–4322 (1992).
    [Crossref] [PubMed]
  5. G. P. Agrawal, Chapter 3, Nonlinear fiber optics (Academic Press, 2001).
  6. C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693 (1981).
    [Crossref]
  7. H. A. Haus, Electromagnetic noise and quantum optical measurements (Springer-Verlag, 2000).
  8. I. A. Walmsley, “,” Opt. Lett. 17, 435–437 (1992).
    [Crossref] [PubMed]
  9. S. L. Braunstein and H. J. Kimble, “Teleportation of continuous quantum variables,” Phys. Rev. Lett. 80, 869–872 (1998).
    [Crossref]

2000 (1)

A. B. Matsko and V. V. Kozlov, “Second-quantized models for optical solitons in nonlinear fibers: Equal-time versus equal-space commutation relations,” Phys. Rev. A 62, 033811 (2000).
[Crossref]

1998 (1)

S. L. Braunstein and H. J. Kimble, “Teleportation of continuous quantum variables,” Phys. Rev. Lett. 80, 869–872 (1998).
[Crossref]

1992 (2)

B. Huttner and S. M. Barnett, “Quantization of the electromagnetic field in dielectrics”, Phys. Rev. A 46, 4306–4322 (1992).
[Crossref] [PubMed]

I. A. Walmsley, “,” Opt. Lett. 17, 435–437 (1992).
[Crossref] [PubMed]

1981 (1)

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693 (1981).
[Crossref]

1949 (1)

T. D. Newton and E. P. Wigner, “Localized states for elementary systems,” Rev. Mod. Phys. 21, 400 (1949).
[Crossref]

Agrawal, G. P.

G. P. Agrawal, Chapter 3, Nonlinear fiber optics (Academic Press, 2001).

Barnett, S. M.

B. Huttner and S. M. Barnett, “Quantization of the electromagnetic field in dielectrics”, Phys. Rev. A 46, 4306–4322 (1992).
[Crossref] [PubMed]

Braunstein, S. L.

S. L. Braunstein and H. J. Kimble, “Teleportation of continuous quantum variables,” Phys. Rev. Lett. 80, 869–872 (1998).
[Crossref]

Caves, C. M.

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693 (1981).
[Crossref]

Haus, H. A.

H. A. Haus, Electromagnetic noise and quantum optical measurements (Springer-Verlag, 2000).

Huttner, B.

B. Huttner and S. M. Barnett, “Quantization of the electromagnetic field in dielectrics”, Phys. Rev. A 46, 4306–4322 (1992).
[Crossref] [PubMed]

Kimble, H. J.

S. L. Braunstein and H. J. Kimble, “Teleportation of continuous quantum variables,” Phys. Rev. Lett. 80, 869–872 (1998).
[Crossref]

Kozlov, V. V.

A. B. Matsko and V. V. Kozlov, “Second-quantized models for optical solitons in nonlinear fibers: Equal-time versus equal-space commutation relations,” Phys. Rev. A 62, 033811 (2000).
[Crossref]

Mandel, L.

L. Mandel and E. Wolf, Optical coherence and quantum optics (Cambridge University Press, 1995).

Matsko, A. B.

A. B. Matsko and V. V. Kozlov, “Second-quantized models for optical solitons in nonlinear fibers: Equal-time versus equal-space commutation relations,” Phys. Rev. A 62, 033811 (2000).
[Crossref]

Newton, T. D.

T. D. Newton and E. P. Wigner, “Localized states for elementary systems,” Rev. Mod. Phys. 21, 400 (1949).
[Crossref]

Walmsley, I. A.

Wigner, E. P.

T. D. Newton and E. P. Wigner, “Localized states for elementary systems,” Rev. Mod. Phys. 21, 400 (1949).
[Crossref]

Wolf, E.

L. Mandel and E. Wolf, Optical coherence and quantum optics (Cambridge University Press, 1995).

Opt. Lett. (1)

Phys. Rev. A (2)

A. B. Matsko and V. V. Kozlov, “Second-quantized models for optical solitons in nonlinear fibers: Equal-time versus equal-space commutation relations,” Phys. Rev. A 62, 033811 (2000).
[Crossref]

B. Huttner and S. M. Barnett, “Quantization of the electromagnetic field in dielectrics”, Phys. Rev. A 46, 4306–4322 (1992).
[Crossref] [PubMed]

Phys. Rev. D (1)

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693 (1981).
[Crossref]

Phys. Rev. Lett. (1)

S. L. Braunstein and H. J. Kimble, “Teleportation of continuous quantum variables,” Phys. Rev. Lett. 80, 869–872 (1998).
[Crossref]

Rev. Mod. Phys. (1)

T. D. Newton and E. P. Wigner, “Localized states for elementary systems,” Rev. Mod. Phys. 21, 400 (1949).
[Crossref]

Other (3)

L. Mandel and E. Wolf, Optical coherence and quantum optics (Cambridge University Press, 1995).

G. P. Agrawal, Chapter 3, Nonlinear fiber optics (Academic Press, 2001).

H. A. Haus, Electromagnetic noise and quantum optical measurements (Springer-Verlag, 2000).

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Figures (1)

Fig. 1.
Fig. 1.

Sketch of the balanced homodyne detection for measuring position-momentum quadrature Eq. (20). A Gaussian pulse is splitted (BS1) into signal and local oscillator (LO). Both propagate through the same fiber (and thereby broaden similarly). Phase shift introduced by PS and shape transformation produced by “transformer” (e.g. by means of Fourier optics) shape the LO as Lq (τ)∝fq (τ), Eq. (20). Delay lines serve for spatial separating the signal and the LO.

Equations (20)

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[ E ̂ ( z , t ) , E ̂ ( z , t ) ] = c 4 π 0 S k e i [ k ( z z ) ω ( t t ) ] dk ,
E ̂ ( z , t ) = ( k 0 4 π 0 S ) 1 2 ϕ ̂ ( z , t ) e i ( k 0 z ω 0 t ) ,
[ ϕ ̂ ( z , t ) , ϕ ̂ ( z , t ) ] = δ ( t t ) .
X ̂ lim ε 0 t ϕ ̂ ( z , t ) ϕ ̂ ( z , t ) dt ( ε + N ̂ ) 1 ,
[ X ̂ , P ̂ ] = i ,
P ̂ i 2 ( ϕ ̂ t ϕ ̂ ϕ ̂ ϕ ̂ t ) dt ,
c ̂ ± ( z , ω ) z = ± ik ( ω ) c ̂ ± ( z , ω ) ± 2 Im [ k ( ω ) ] f ̂ ( z , ω ) ,
[ c ̂ ± ( z , ω ) , c ̂ ± ( z , ω ) ] = δ ( ω ω )
k ( ω ) = k 0 + k ( ω ω 0 ) + ( 1 2 ) k ( ω ω 0 ) 2 ,
E ̂ + ( z , t ) = i 4 π 0 S d ω [ k ( ω ) ] 1 2 c ̂ + ( z , ω ) e i ω t .
i z ϕ ̂ ( z , τ ) = k 2 2 τ 2 ϕ ̂ ( z , τ ) ,
X ̂ 1 n 0 ( τ ϕ * υ ̂ + τ ϕ υ ̂ ) d τ , P ̂ i ( ϕ * τ υ ̂ ϕ τ υ ̂ ) d τ ,
P ̂ ( z ) = P ̂ 0 , X ̂ ( z ) = X ̂ 0 + ( k z n 0 ) P ̂ 0 ,
[ X ̂ ( z ) , P ̂ ( z ) ] = i .
ϕ ( z , τ ) = n 0 π T p ( z ) exp [ τ 2 2 T p 2 ( z ) ] exp [ i τ 2 2 T p 2 ( z ) z Z D i 2 arctg z Z D ] ,
Δ P ̂ 2 coh ( in ) = ϕ ( 0 , τ ) τ 2 d τ = n 0 2 τ p 2 , Δ X ̂ 2 coh ( in ) = τ 2 ϕ ( 0 , τ ) 2 d τ n 0 2 = τ p 2 2 n 0 .
q ̂ = t c P ̂ cos ψ + t c 1 X ̂ sin ψ and q ̂ = t c P ̂ sin ψ + t c 1 X ̂ cos ψ ,
Δ q ̂ 2 min = 1 2 [ 1 + ( z 2 Z D ) 2 z 2 Z D ] 2 ,
Δ P ̂ 2 coh ( out ) = ϕ ( z , τ ) τ 2 d τ = n 0 2 τ p 2 , Δ X ̂ 2 coh ( out ) = τ 2 ϕ ( z , τ ) 2 d τ n 0 2 = T p 2 ( z ) 2 n 0 ,
q ̂ = [ f q * υ ̂ ( z , τ ) + H . c . ] d τ , f q it c ϕ ( z , τ ) τ cos ψ + τ ϕ ( z , τ ) t c n 0 sin ψ ,

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