Abstract

An excess phase-to-amplitude conversion noise was observed in the interferometric cancellation for the coherent excitation of the squeezed output from an injection-locked laser. We show theoretically that this phase-to-amplitude conversion noise can be eliminated and that it is possible to generate squeezed vacuum states by the use of a semiconductor laser system.

© 1993 Optical Society of America

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References

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  1. W. H. Richardson, S. Machida, Y. Yamamoto, Phys. Rev. Lett. 66, 2867 (1991).
    [CrossRef] [PubMed]
  2. C. M. Caves, Phys. Rev. D 23, 1693 (1981).
    [CrossRef]
  3. Y. Lai, H. A. Haus, Y. Yamamoto, Opt. Lett. 16, 1517 (1991).
    [CrossRef] [PubMed]
  4. S. Inoue, S. Machida, Y. Yamamoto, H. Ohzu, “Squeezing in an injection-locked semiconductor laser,” Phys. Rev. A (to be published).
  5. L. Gillner, G. Björk, Y. Yamamoto, Phys. Rev. A 41, 5053 (1990).
    [CrossRef] [PubMed]
  6. H. A. Haus, Y. Yamamoto, Phys. Rev. A 29, 1261 (1984).
    [CrossRef]
  7. S. Machida, Y. Yamamoto, Opt. Lett. 14, 1045 (1989).
    [CrossRef] [PubMed]

1991 (2)

W. H. Richardson, S. Machida, Y. Yamamoto, Phys. Rev. Lett. 66, 2867 (1991).
[CrossRef] [PubMed]

Y. Lai, H. A. Haus, Y. Yamamoto, Opt. Lett. 16, 1517 (1991).
[CrossRef] [PubMed]

1990 (1)

L. Gillner, G. Björk, Y. Yamamoto, Phys. Rev. A 41, 5053 (1990).
[CrossRef] [PubMed]

1989 (1)

1984 (1)

H. A. Haus, Y. Yamamoto, Phys. Rev. A 29, 1261 (1984).
[CrossRef]

1981 (1)

C. M. Caves, Phys. Rev. D 23, 1693 (1981).
[CrossRef]

Björk, G.

L. Gillner, G. Björk, Y. Yamamoto, Phys. Rev. A 41, 5053 (1990).
[CrossRef] [PubMed]

Caves, C. M.

C. M. Caves, Phys. Rev. D 23, 1693 (1981).
[CrossRef]

Gillner, L.

L. Gillner, G. Björk, Y. Yamamoto, Phys. Rev. A 41, 5053 (1990).
[CrossRef] [PubMed]

Haus, H. A.

Inoue, S.

S. Inoue, S. Machida, Y. Yamamoto, H. Ohzu, “Squeezing in an injection-locked semiconductor laser,” Phys. Rev. A (to be published).

Lai, Y.

Machida, S.

W. H. Richardson, S. Machida, Y. Yamamoto, Phys. Rev. Lett. 66, 2867 (1991).
[CrossRef] [PubMed]

S. Machida, Y. Yamamoto, Opt. Lett. 14, 1045 (1989).
[CrossRef] [PubMed]

S. Inoue, S. Machida, Y. Yamamoto, H. Ohzu, “Squeezing in an injection-locked semiconductor laser,” Phys. Rev. A (to be published).

Ohzu, H.

S. Inoue, S. Machida, Y. Yamamoto, H. Ohzu, “Squeezing in an injection-locked semiconductor laser,” Phys. Rev. A (to be published).

Richardson, W. H.

W. H. Richardson, S. Machida, Y. Yamamoto, Phys. Rev. Lett. 66, 2867 (1991).
[CrossRef] [PubMed]

Yamamoto, Y.

Y. Lai, H. A. Haus, Y. Yamamoto, Opt. Lett. 16, 1517 (1991).
[CrossRef] [PubMed]

W. H. Richardson, S. Machida, Y. Yamamoto, Phys. Rev. Lett. 66, 2867 (1991).
[CrossRef] [PubMed]

L. Gillner, G. Björk, Y. Yamamoto, Phys. Rev. A 41, 5053 (1990).
[CrossRef] [PubMed]

S. Machida, Y. Yamamoto, Opt. Lett. 14, 1045 (1989).
[CrossRef] [PubMed]

H. A. Haus, Y. Yamamoto, Phys. Rev. A 29, 1261 (1984).
[CrossRef]

S. Inoue, S. Machida, Y. Yamamoto, H. Ohzu, “Squeezing in an injection-locked semiconductor laser,” Phys. Rev. A (to be published).

Opt. Lett. (2)

Phys. Rev. A (2)

L. Gillner, G. Björk, Y. Yamamoto, Phys. Rev. A 41, 5053 (1990).
[CrossRef] [PubMed]

H. A. Haus, Y. Yamamoto, Phys. Rev. A 29, 1261 (1984).
[CrossRef]

Phys. Rev. D (1)

C. M. Caves, Phys. Rev. D 23, 1693 (1981).
[CrossRef]

Phys. Rev. Lett. (1)

W. H. Richardson, S. Machida, Y. Yamamoto, Phys. Rev. Lett. 66, 2867 (1991).
[CrossRef] [PubMed]

Other (1)

S. Inoue, S. Machida, Y. Yamamoto, H. Ohzu, “Squeezing in an injection-locked semiconductor laser,” Phys. Rev. A (to be published).

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Figures (7)

Fig. 1
Fig. 1

Squeezed-vacuum-state generation by mixing an amplitude-squeezed state with a coherent state and its application to the Mach–Zehnder interferometer: FR, Faraday-rotator; PBS, polarization beam splitter; HTM, high-transmission mirror; BS, beam splitter.

Fig. 2
Fig. 2

Phase-to-amplitude conversion noise with the interference: open circle, quasi-probability density of the master-laser signal reflected by a high-transmission mirror; open ellipse, quasi-probability density of the transmitted injection-locked slave-laser signal; shaded ellipse, quasi-probability density of the combined signal.

Fig. 3
Fig. 3

The intensity noise spectral density with the interference as a function of the phase error δ: SQL, standard quantum limit. The dashed curve shows the noise level, which is obtained when the phase error δ is equal to zero.

Fig. 4
Fig. 4

The amplitude and phase noise spectral densities as a function of the locking bandwidth: SQL, standard quantum limit. (a) Normalized phase noise spectral density and (b) amplitude noise spectral density.

Fig. 5
Fig. 5

The experimental setup for the measurement of the intensity noise with the interference: HWP, half-wave plate; PBS, polarization beam splitter; NPBS, nonpolarization beam splitter; HTM, high-transmission mirror; PZT, piezo translator; LD, laser diode.

Fig. 6
Fig. 6

The intensity noise spectra with and without the interference. (a) Phase difference (πδ) ≃ 90°, (b) phase difference (πδ) ≃ 170 ± 5°, and (c) without the interference.

Fig. 7
Fig. 7

The intensity noise spectral densities at 22.8 MHz as a function of the locking bandwidth. Open circles, phase difference (πδ) ≃ 90°; open triangles, phase difference (πδ) = 170 ± 5°; open squares, without the interference; solid curves, theory (a) phase difference is 90° and (b) phase difference is 180°; SQL, standard quantum limit.

Tables (2)

Tables Icon

Table 1 Expressions for Coefficients Bi, Ci, and Di Defined in the Text

Tables Icon

Table 2 Expressions for Correlation Functions of Noise Operators

Equations (58)

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d A ˆ ( t ) d t = 1 2 [ ω Q + i 2 ( ω r ω ) ω μ 2 ( χ ˜ i i χ ˜ r ) ] A ˆ ( t ) + ( ω μ 2 χ ˜ i ) 1 / 2 f ˜ G ( t ) + ( ω Q 0 ) 1 / 2 f ˆ L ( t ) + ( ω Q e ) 1 / 2 [ F 0 + f ˆ ( t ) ] ,
1 Q = 1 Q e + 1 Q 0 .
ω μ 2 χ ˜ i = E ˜ c υ E ˜ υ c ,
d N ˜ c ( t ) d t = p N ˜ c ( t ) τ sp ( E ˜ c υ E ˜ υ c ) n ˆ ( t ) E ˜ c υ + Γ ˜ p ( t ) + Γ ˜ sp ( t ) + Γ ˜ ( t ) ,
H ˆ ( t ) = ( ω μ 2 χ ˜ i ) 1 / 2 f ˜ G ( t ) + ( ω Q 0 ) 1 / 2 f ˆ L ( t ) + ( ω Q e ) 1 / 2 f ˆ ( t ) ,
F ˜ c ( t ) = Γ ˜ p ( t ) + Γ ˜ sp ( t ) + Γ ˜ ( t ) .
N ˜ c ( t ) = N c 0 + Δ N ˜ c ( t ) ,
A ˆ ( t ) = [ A 0 + Δ A ˆ ( t ) ] exp { i [ ϕ 0 + Δ ϕ ˆ ( t ) ] } ,
n ˆ ( t ) = A ˆ ( t ) A ˆ ( t ) [ A 0 + Δ A ˆ ( t ) ] 2 A 0 2 + 2 A 0 Δ A ˆ ( t ) ,
χ ˜ i = χ ˜ i + d χ ˜ i d N c 0 Δ N ˜ c ,
χ ˜ r = χ ˜ r + d χ ˜ r d N c 0 Δ N ˜ c .
1 τ st = ω μ 2 A 0 2 d χ ˜ i d N c 0 ,
ω 0 = ω r + ω 2 μ 2 χ ˜ r ,
α = d χ ˜ r d N c 0 / d χ ˜ i d N c 0 ,
β = E ˜ c υ N c 0 / τ sp ,
n sp = E ˜ c υ E ˜ c υ E ˜ υ c ,
G = ω μ 2 χ ˜ i ,
1 τ p 0 = ω Q 0 , 1 τ p e = ω Q e , 1 τ p = ω Q .
d Δ N ˜ c ( t ) d t = A 1 Δ N ˜ c ( t ) + A 2 Δ A ˆ ( t ) + F ˜ c ( t ) ,
d Δ A ˆ ( t ) d t = A 3 Δ N ˜ c ( t ) A 5 Δ A ˆ ( t ) A 0 A 6 Δ ϕ ˆ ( t ) + H ˆ r ( t ) ,
d Δ ϕ ˆ ( t ) d t = A 4 Δ N ˜ c ( t ) + ( A 6 / A 0 ) Δ A ˆ ( t ) A 5 Δ ϕ ˆ ( t ) ( 1 / A 0 ) H ˆ i ( t ) ,
A 1 = 1 τ sp 1 τ st ,
A 2 = 2 A 0 G ,
A 3 = 1 2 A 0 τ st ,
A 4 = α 2 A 0 2 τ st ,
A 5 = F 0 A 0 ( 1 τ p e ) 1 / 2 cos ϕ 0 ,
A 6 = F 0 A 0 ( 1 τ p e ) 1 / 2 sin ϕ 0 .
H ˆ r = 1 / 2 ( H ˆ exp { i [ ϕ 0 + Δ ϕ ˆ ( t ) ] } + H ˆ exp { i [ ϕ 0 + Δ ϕ ˆ ( t ) ] } ) ,
H ˆ i = 1 / 2 i ( H ˆ exp { i [ ϕ 0 + Δ ϕ ˆ ( t ) ] } H ˆ exp { i [ ϕ 0 + Δ ϕ ˆ ( t ) ] } ) .
i Ω Δ N ˜ c ( Ω ) = A 1 Δ N ˜ c ( Ω ) + A 2 Δ A ˆ ( Ω ) + F ˜ c ( Ω ) ,
i Ω Δ A ˆ ( Ω ) = A 3 Δ N ˜ c ( Ω ) A 5 Δ A ˆ ( Ω ) A 0 A 6 Δ ϕ ˆ ( Ω ) + H ˆ r ( Ω ) ,
i Ω Δ ϕ ˆ ( Ω ) = A 4 Δ N ˜ c ( Ω ) + ( A 6 / A 0 ) Δ A ˆ ( Ω ) A 5 Δ ϕ ˆ ( Ω ) ( 1 / A 0 ) H ˆ i ( Ω ) .
Δ A ˆ ( 0 ) = ( 1 / B 1 ) [ B 2 F ˜ c ( 0 ) + B 3 H ˆ i ( 0 ) + B 4 H ˆ r ( 0 ) ] ,
Δ ϕ ˆ ( 0 ) = ( 1 / B 5 ) [ B 6 F ˜ c ( 0 ) + B 7 H ˆ i ( 0 ) + B 8 H ˆ r ( 0 ) ] .
r ˆ ( t ) = [ F 0 + f ˆ ( t ) ] + ( ω Q e ) 1 / 2 A ˆ ( t ) .
r ˆ ( t ) = [ r 0 + Δ r ˆ ( t ) ] exp { i [ ψ 0 + Δ ψ ˆ ( t ) ] } ,
C 4 r 0 Δ r ˆ ( t ) + C 3 [ Δ ϕ ˆ ( t ) Δ ψ ˆ ( t ) ] = f ˆ r ( t ) + C 1 Δ A ˆ ( t ) ,
C 3 r 0 Δ r ˆ ( t ) + C 4 [ Δ ϕ ˆ ( t ) Δ ψ ˆ ( t ) ] = f ˆ i ( t ) C 2 Δ A ˆ ( t ) F 0 Δ ϕ ˆ ( t ) ,
C 1 = ( 1 τ p e ) 1 / 2 cos ϕ 0 ,
C 2 = ( 1 τ p e ) 1 / 2 sin ϕ 0 ,
C 3 = A 0 C 2 ,
C 4 = A 0 C 1 F 0 ,
f ˆ r = 1 2 { f ˆ exp [ i Δ ϕ ˆ ( t ) ] + f ˆ exp [ i Δ ϕ ˆ ( t ) ] } ,
f ˆ i = 1 2 i { f ˆ exp [ i Δ ϕ ˆ ( t ) ] f ˆ exp [ i Δ ϕ ˆ ( t ) ] } .
Δ r ˆ ( 0 ) = C 5 [ C 6 Δ A ˆ ( 0 ) + C 7 Δ ϕ ˆ ( 0 ) + C 8 f ˆ r ( 0 ) + C 9 f ˆ i ( 0 ) ] ,
Δ ψ ¯ ( 0 ) = C 10 [ C 11 Δ A ˆ ( 0 ) + C 12 Δ ϕ ˆ ( 0 ) + C 3 f ˆ r ( 0 ) + C 4 f ˆ i ( 0 ) ] .
Δ r ˆ ( 0 ) = C 5 [ D 1 F ˜ c ( 0 ) + D 2 H ˆ i ( 0 ) + D 3 H ˆ r ( 0 ) + C 8 f ˆ r ( 0 ) + C 9 f ˆ i ( 0 ) ] ,
Δ ψ ˆ ( 0 ) = C 10 [ D 4 F ˜ c ( 0 ) + D 5 H ˆ i ( 0 ) + D 6 H ˆ r ( 0 ) + C 3 f ˆ r ( 0 ) + C 4 f ˆ i ( 0 ) ] .
Δ r ˆ ( 0 ) Δ r ˆ ( 0 ) = C 5 2 [ D 1 2 F ˜ c ( 0 ) F ˜ c ( 0 ) + D 2 2 H ˆ i ( 0 ) H ˆ i ( 0 ) + D 3 2 H ˆ r ( 0 ) H ˆ r ( 0 ) + C 8 2 f ˆ r ( 0 ) f ˆ r ( 0 ) + C 9 2 f ˆ i ( 0 ) f ˆ i ( 0 ) ] + 2 C 5 2 [ D 1 D 3 F ˜ c ( 0 ) H ˆ r ( 0 ) + D 2 C 8 H ˆ i ( 0 ) f ˆ r ( 0 ) + D 2 C 9 H ˆ i ( 0 ) f ˆ i ( 0 ) + D 3 C 8 H ˆ r ( 0 ) f ˆ r ( 0 ) + D 3 C 9 H ˆ r ( 0 ) f ˆ i ( 0 ) ] ,
Δ ψ ˆ ( 0 ) Δ ψ ˆ ( 0 ) = C 10 2 [ D 4 2 F ˜ c ( 0 ) F ˜ c ( 0 ) + D 5 2 H ˆ i ( 0 ) H ˆ i ( 0 ) + D 6 2 H ˆ r ( 0 ) H ˆ r ( 0 ) + C 3 2 f ˆ r ( 0 ) f ˆ r ( 0 ) + C 4 2 f ˆ i ( 0 ) f ˆ i ( 0 ) ] + 2 C 10 2 [ D 4 D 6 F ˜ c ( 0 ) H ˆ r ( 0 ) + D 5 C 3 H ˆ i ( 0 ) f ˆ r ( 0 ) + D 5 C 4 H ˆ i ( 0 ) f ˆ i ( 0 ) + D 6 C 3 H ˆ r ( 0 ) f ˆ r ( 0 ) + D 6 C 4 H ˆ r ( 0 ) f ˆ i ( 0 ) ] .
Δ r ˆ ( 0 ) Δ ψ ˆ ( 0 ) = C 5 C 10 [ D 1 D 4 F ˜ c ( 0 ) F ˜ c ( 0 ) + D 2 D 5 H ˆ i ( 0 ) H ˆ i ( 0 ) + D 3 D 6 H ˆ r ( 0 ) H ˆ r ( 0 ) + C 8 C 3 f ˆ r ( 0 ) f ˆ r ( 0 ) + C 9 C 4 f ˆ i ( 0 ) f ˆ i ( 0 ) ] + C 5 C 10 [ ( D 1 D 6 + D 3 D 4 ) F ˜ c ( 0 ) H ˆ r ( 0 ) + ( D 2 C 3 + C 8 D 5 ) H ˆ i ( 0 ) f ˆ r ( 0 ) + ( D 2 C 4 + C 9 D 5 ) H ˆ i ( 0 ) f ˆ i ( 0 ) + ( D 3 C 3 + C 8 D 5 ) H ˆ r ( 0 ) f ˆ r ( 0 ) + ( D 3 C 4 + C 9 D 6 ) H ˆ r ( 0 ) f ˆ i ( 0 ) ] .
I ˆ ( t ) = ( T ) 1 / 2 [ r 0 + Δ r ˆ ( t ) ] exp { i [ π δ + Δ ψ ˆ ( t ) ] } + ( 1 T ) 1 / 2 [ G 0 + g ˆ ( t ) ] ,
I ˆ a 1 ( t ) = ( T ) 1 / 2 r 0 cos δ + ( 1 T ) 1 / 2 G 0 ( T ) 1 / 2 Δ r ˆ ( t ) cos δ ( T ) 1 / 2 r 0 Δ ψ ˆ ( t ) sin δ + ( 1 T ) 1 / 2 Δ g ˆ r ( t ) ,
Δ g ˆ r ( t ) = ( 1 / 2 ) [ g ˆ ( t ) + g ˆ ( t ) ] .
I a 10 = ( T ) 1 / 2 r 0 cos δ + ( 1 T ) 1 / 2 G 0 .
Δ I ˆ a 1 ( t ) = ( T ) 1 / 2 Δ r ˆ ( t ) cos δ ( T ) 1 / 2 r 0 Δ ψ ˆ ( t ) sin δ + ( 1 T ) 1 / 2 Δ g ˆ r ( t ) .
Δ I ˆ a 1 ( 0 ) Δ I ˆ a 1 ( 0 ) = T cos 2 δ Δ r ˆ ( 0 ) Δ r ˆ ( 0 ) + 2 T r 0 cos δ sin δ Δ r ˆ ( 0 ) Δ ψ ˆ ( 0 ) + T r 0 2 sin 2 δ Δ ψ ˆ ( 0 ) Δ ψ ˆ ( 0 ) + ( 1 T ) Δ g ˆ r ( 0 ) Δ g ˆ r ( 0 ) ,
Δ g ˆ r ( 0 ) Δ g ˆ r ( 0 ) = 1 / 2 .

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