Abstract

A symmetric nonlinear Mach–Zehnder interferometer containing a Kerr medium performs the operation of squeezing. The operation is performed by the interference of cotraveling waves, and the pump wave is removed. This mechanism is broadband. We analyze the operation of the interferometer and determine the degree of shot-noise reduction achieved in a balanced detector. A modification of the interferometer into a fiber ring reflector is described that accomplishes the squeezing and the pump separation.

© 1990 Optical Society of America

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References

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  1. R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
    [CrossRef] [PubMed]
  2. R. M. Shelby, M. D. Levenson, S H. Perlmutter, R. G. DeVoe, D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
    [CrossRef] [PubMed]
  3. L. W. Wu, H. J. Kimble, J. L. Hall, H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
    [CrossRef] [PubMed]
  4. M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, H. J. Kimble, Phys. Rev. Lett. 59, 198 (1987).
    [CrossRef] [PubMed]
  5. M. W. Maeda, P. Kumar, J. H. Shapiro, Opt. Lett. 12, 161 (1987);J. Opt. Soc. Am. B 4, 1501 (1987).
    [CrossRef] [PubMed]
  6. B. Yurke, P. Grangier, R. E. Slusher, M. J. Potasek, Phys. Rev. A 35, 3586 (1987).
    [CrossRef] [PubMed]
  7. M. Shirasaki, H. A. Haus, D L. Wong, in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1989), paper ThO1.
  8. M. J. LaGasse, D. Liu-Wong, J. G. Fujimoto, H. A. Haus, in Digest of the International Quantum Electronics Conference (Optical Society of America, Washington, D.C., 1988), paper TuF9,p. 280.
  9. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984).
  10. M. D. Levenson, R. M. Shelby, S. H. Perlmutter, Opt. Lett. 10, 514 (1985).
    [CrossRef] [PubMed]
  11. A. J. Akhiezer, V. B. Berestetski, Quantum Electrodynamics (Interscience, New York, 1965), p. 10.

1987

M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, H. J. Kimble, Phys. Rev. Lett. 59, 198 (1987).
[CrossRef] [PubMed]

M. W. Maeda, P. Kumar, J. H. Shapiro, Opt. Lett. 12, 161 (1987);J. Opt. Soc. Am. B 4, 1501 (1987).
[CrossRef] [PubMed]

B. Yurke, P. Grangier, R. E. Slusher, M. J. Potasek, Phys. Rev. A 35, 3586 (1987).
[CrossRef] [PubMed]

1986

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

L. W. Wu, H. J. Kimble, J. L. Hall, H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

1985

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

M. D. Levenson, R. M. Shelby, S. H. Perlmutter, Opt. Lett. 10, 514 (1985).
[CrossRef] [PubMed]

Akhiezer, A. J.

A. J. Akhiezer, V. B. Berestetski, Quantum Electrodynamics (Interscience, New York, 1965), p. 10.

Berestetski, V. B.

A. J. Akhiezer, V. B. Berestetski, Quantum Electrodynamics (Interscience, New York, 1965), p. 10.

Boyd, T. L.

M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, H. J. Kimble, Phys. Rev. Lett. 59, 198 (1987).
[CrossRef] [PubMed]

DeVoe, R. G.

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

Fujimoto, J. G.

M. J. LaGasse, D. Liu-Wong, J. G. Fujimoto, H. A. Haus, in Digest of the International Quantum Electronics Conference (Optical Society of America, Washington, D.C., 1988), paper TuF9,p. 280.

Grangier, P.

B. Yurke, P. Grangier, R. E. Slusher, M. J. Potasek, Phys. Rev. A 35, 3586 (1987).
[CrossRef] [PubMed]

Hall, J. L.

L. W. Wu, H. J. Kimble, J. L. Hall, H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

Haus, H. A.

M. J. LaGasse, D. Liu-Wong, J. G. Fujimoto, H. A. Haus, in Digest of the International Quantum Electronics Conference (Optical Society of America, Washington, D.C., 1988), paper TuF9,p. 280.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984).

M. Shirasaki, H. A. Haus, D L. Wong, in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1989), paper ThO1.

Hollberg, L. W.

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

Kimble, H. J.

M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, H. J. Kimble, Phys. Rev. Lett. 59, 198 (1987).
[CrossRef] [PubMed]

L. W. Wu, H. J. Kimble, J. L. Hall, H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

Kumar, P.

LaGasse, M. J.

M. J. LaGasse, D. Liu-Wong, J. G. Fujimoto, H. A. Haus, in Digest of the International Quantum Electronics Conference (Optical Society of America, Washington, D.C., 1988), paper TuF9,p. 280.

Levenson, M. D.

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

M. D. Levenson, R. M. Shelby, S. H. Perlmutter, Opt. Lett. 10, 514 (1985).
[CrossRef] [PubMed]

Liu-Wong, D.

M. J. LaGasse, D. Liu-Wong, J. G. Fujimoto, H. A. Haus, in Digest of the International Quantum Electronics Conference (Optical Society of America, Washington, D.C., 1988), paper TuF9,p. 280.

Maeda, M. W.

Mertz, J. C.

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

Orozco, L. A.

M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, H. J. Kimble, Phys. Rev. Lett. 59, 198 (1987).
[CrossRef] [PubMed]

Perlmutter, S H.

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

Perlmutter, S. H.

Potasek, M. J.

B. Yurke, P. Grangier, R. E. Slusher, M. J. Potasek, Phys. Rev. A 35, 3586 (1987).
[CrossRef] [PubMed]

Raizen, M. G.

M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, H. J. Kimble, Phys. Rev. Lett. 59, 198 (1987).
[CrossRef] [PubMed]

Shapiro, J. H.

Shelby, R. M.

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

M. D. Levenson, R. M. Shelby, S. H. Perlmutter, Opt. Lett. 10, 514 (1985).
[CrossRef] [PubMed]

Shirasaki, M.

M. Shirasaki, H. A. Haus, D L. Wong, in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1989), paper ThO1.

Slusher, R. E.

B. Yurke, P. Grangier, R. E. Slusher, M. J. Potasek, Phys. Rev. A 35, 3586 (1987).
[CrossRef] [PubMed]

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

Walls, D. F.

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

Wong, D L.

M. Shirasaki, H. A. Haus, D L. Wong, in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1989), paper ThO1.

Wu, H.

L. W. Wu, H. J. Kimble, J. L. Hall, H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

Wu, L. W.

L. W. Wu, H. J. Kimble, J. L. Hall, H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

Xiao, M.

M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, H. J. Kimble, Phys. Rev. Lett. 59, 198 (1987).
[CrossRef] [PubMed]

Yurke, B.

B. Yurke, P. Grangier, R. E. Slusher, M. J. Potasek, Phys. Rev. A 35, 3586 (1987).
[CrossRef] [PubMed]

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

Opt. Lett.

Phys. Rev. A

B. Yurke, P. Grangier, R. E. Slusher, M. J. Potasek, Phys. Rev. A 35, 3586 (1987).
[CrossRef] [PubMed]

Phys. Rev. Lett.

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

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

L. W. Wu, H. J. Kimble, J. L. Hall, H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, H. J. Kimble, Phys. Rev. Lett. 59, 198 (1987).
[CrossRef] [PubMed]

Other

M. Shirasaki, H. A. Haus, D L. Wong, in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1989), paper ThO1.

M. J. LaGasse, D. Liu-Wong, J. G. Fujimoto, H. A. Haus, in Digest of the International Quantum Electronics Conference (Optical Society of America, Washington, D.C., 1988), paper TuF9,p. 280.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984).

A. J. Akhiezer, V. B. Berestetski, Quantum Electrodynamics (Interscience, New York, 1965), p. 10.

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Figures (4)

Fig. 1
Fig. 1

Schematic of the squeezing interferometer.

Fig. 2
Fig. 2

Balanced-detector schematic.

Fig. 3
Fig. 3

Shot noise reduction as a function of peak phase.

Fig. 4
Fig. 4

Schematic of the experimental setup.

Equations (47)

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( z + 1 υ g t ) a ( z , t ) = i K a * ( z , t ) a ( z , t ) a ( z , t ) .
z a ( z , τ ) = i K a * ( z , τ ) a ( z , τ ) a ( z , τ ) .
τ = t z υ g ,
z a ( z , t ) = i K a * ( z , t ) a ( z , t ) a ( z , t ) .
1 2 π a ( z , t ) e i ω t d t a ( z , ω ) ,
1 2 π a ( z , ω ) e i ω t d ω a ( z , t ) ,
z a ( z , ω ) = i K d ω 1 d ω 2 a * ( z , ω 1 ) a ( z , ω 2 ) a ( z , ω + ω 1 ω 2 ) .
[ â ( z , ω ) , â ( z , ω ) ] = δ ( ω ω ) ,
[ â ( z , ω ) , â ( z , ω ) ] = [ â ( z , ω ) , â ( z , ω ) ] = 0.
z â ( z , ω ) = i K d ω 1 d ω 2 â ( z , ω 1 ) â ( z , ω 2 ) â ( z , ω + ω 1 ω 2 ) .
i d d z â ( z , ω ) = [ â ( z , ω ) , H ̂ ] ,
H ̂ = K 2 â ( z , ω ) â ( z , ω 1 ) â ( z , ω 2 ) â ( z , ω + ω 1 ω 2 ) d ω d ω 1 d ω 2 .
z â ( z , t ) = i K â ( z , t ) â ( z , t ) â ( z , t ) .
[ â ( z , t ) , â ( z , t ) ] = δ ( t t ) ,
[ â ( z , t ) , â ( z , t ) ] = [ â ( z , t ) , â ( z , t ) ] = 0 .
i d d z â ( z , t ) = [ â ( z , t ) , H ̂ ] ,
H ̂ = K 2 â ( z , t ) â ( z , t ) â ( z , t ) â ( z , t ) d t .
d d z â ( z , t ) = i K â ( z , t ) â ( z , t ) â ( z , t ) .
â ( l , t ) = exp ( i K l â â ) â ( 0 , t ) .
S = 1 2 [ 1 1 1 1 ] ,
ĉ = exp [ i κ ( â + b ) ( â + b ̂ ) 2 ] â + b ̂ 2 + exp [ i κ ( â b ) ( â b ̂ ) 2 ] â b ̂ 2 2 = exp [ i κ ( â + b ) ( â + b ̂ ) 2 ] + exp [ i κ ( â b ) ( â b ̂ ) 2 ] 2 â + exp [ i κ ( â + b ) ( â + b ̂ ) 2 ] exp [ i κ ( â b ) ( â b ̂ ) 2 ] 2 b ̂ .
exp [ i κ b ̂ b ̂ / 2 ]
ĉ = exp ( i κ 2 b ̂ b ̂ ) ( μ â + ν â ) ,
μ = ( 1 + i κ 2 | b | 2 ) ,
ν = i κ 2 b 2 .
Ĉ = exp ( i κ 2 b ̂ b ̂ ) b ,
d ̂ 1 = 1 2 ( ĉ + Ĉ ) ,
d ̂ 2 = 1 2 ( ĉ + Ĉ ) ,
Î ( t ) = e ( d ̂ 1 d ̂ 1 d ̂ 2 d ̂ 2 ) = e ( ĉ Ĉ + Ĉ ĉ ) .
Φ i i ( ω ) = 1 2 π d t d t 1 2 Î ( t ) Î ( t ) + Î ( t ) Î ( t ) exp [ i ω ( t t ) ] = 1 2 π d τ G ii ( τ ) e i ω τ .
G i i ( τ ) = d t 1 2 Î ( t ) Î ( t τ ) + Î ( t τ ) Î ( t )
τ = t t .
[ ĉ ( t ) , ĉ ( t ) ] = δ ( t t ) ,
G i i ( τ ) = e 2 | C | 2 δ ( τ ) .
Φ i i = e 2 π I 0 ,
ĉ = exp ( i κ 2 b ̂ b ̂ ) ( μ â + ν â ) ,
G i i ( τ ) = d t 1 2 Î ( t ) Î ( t ) + Î ( t ) Î ( t ) = e 2 d t [ | C ( t ) | 2 ( | μ | 2 + | ν | 2 ) + C ( t ) 2 μ * ν * + C * ( t ) 2 μ ν ] δ ( τ ) .
R = G i i ( τ ) μ , ν G i i ( τ ) μ = 1 , ν = 0 .
μ ( t ) = [ 1 + i Φ ( t ) ] = i 1 + Φ 2 ( t ) e i γ ,
ν ( t ) = i Φ ( t ) exp [ i arg ( b 2 ) ] ,
Φ ( t ) κ 2 | b ( t ) | 2 , γ = tan 1 1 Φ ( t ) .
G i i ( τ ) = e 2 δ ( τ ) d t | b ( t ) | 2 [ | μ | 2 + | ν | 2 2 | μ ν | ( γ ) ] = e 2 δ ( τ ) d t | b ( t ) | 2 [ 1 + Φ 2 + Φ 2 2 1 + Φ 2 Φ Φ 1 + Φ 2 ] = e 2 δ ( τ ) d t | b ( t ) | 2
C 2 ( t ) = b ( t ) 2 exp [ 2 i Φ ( t ) ] exp ( i γ 0 ) ,
γ 0 = tan 1 1 Φ 0 ,
G i i ( τ ) = e 2 δ ( τ ) d t | b ( t ) | 2 { 1 + 2 Φ 2 2 Φ 1 + Φ 2 cos ( γ γ 0 ) } = e 2 δ ( τ ) d t | b ( t ) | 2 { 1 + 2 Φ 2 2 Φ 1 + Φ 2 1 + Φ Φ 0 [ ( 1 + Φ 2 ) ( 1 + Φ 0 2 ) ] 1 / 2 } = e 2 δ ( τ ) d t | b ( t ) | 2 { 1 + 2 Φ 2 2 Φ ( 1 + Φ Φ 0 ) 1 + Φ 0 2 } = e 2 δ ( τ ) π τ 0 [ 1 + 2 Φ 0 2 3 2 Φ 0 2 3 2 Φ 0 1 + Φ 0 2 ] | b ( o ) | 2 ,
d t | b ( t ) | 2 = π τ 0 | b ( 0 ) | 2 .
R = d t | C ( t ) | 2 { 1 + 2 Φ 2 2 Φ 1 + Φ 2 } / d t | C ( t ) | 2 .

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