Abstract

We develop further the unified model for treating photon-counting and radiation-pressure fluctuations in the Michelson interferometer with input of squeezed vacuum state. The dependence of the quantum fluctuations on the phase of the input light is calculated. The analysis is restricted to a single-mode interferometer, but generalized in a way that includes both harmonic-oscillator and floating mirrors. We compare our results with those of other authors.

© 2002 Optical Society of America

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References

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  1. C. M. Caves, “Quantum-mechanical radiation-pressure fluctuations in an interferometer,” Phys. Rev. Lett. 45, 75–79 (1980).
    [CrossRef]
  2. C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
    [CrossRef]
  3. R. Loudon, “Quantum limit on the Michelson interferometer used for gravitational-wave detection,” Phys. Rev. Lett. 47, 815–818 (1981).
    [CrossRef]
  4. O. Assaf and Y. Ben-Aryeh, “Quantum mechanical noise in coherent-state and squeezed-state Michelson interferometers,” J. Opt. B: Quantum Semiclassical Opt. 4, 49–56 (2002).
    [CrossRef]
  5. A. Luis and L. L. Sanchez-Soto, “Mode transformation properties and quantum limits for a Fabry–Perot interferometer,” J. Mod. Opt. 38, 971–985 (1991).
    [CrossRef]
  6. A. Luis and L. L. Sanchez-Soto, “Breaking the standard quantum limit for interferometric measurements,” Opt. Commun. 89, 140–144 (1992).
    [CrossRef]
  7. H. P. Yuen, “Contractive states and standard quantum limit for monitoring free-mass positions,” Phys. Rev. Lett. 51, 719–722 (1983).
    [CrossRef]
  8. R. S. Bondurant and J. H. Shapiro, “Squeezed states in phase sensing interferometers,” Phys. Rev. D 30, 2548–2556 (1984).
    [CrossRef]
  9. W. G. Unruh, “Quantum noise in the interferometer detector,” in Quantum Optics, Experimental Gravitation and Measurement Theory, P. Meystre and M. O. Scully, eds. (Plenum, New York, 1983), pp. 647–660.
  10. D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, Berlin, 1994).
  11. H. J. Kimble, Yuri Levin, Andrey B. Matsko, Kip S. Thorne, and Sergey P. Vyatchanin, “Conversion of conventional gravitational-wave interferometers into quantum nondemo-lition interferometers by modifying their input and/or output optics,” Phys. Rev. D 65, 1–31 (2001); includes a list of references.
    [CrossRef]
  12. P. R. Saulson, Fundamentals of Interferometric Gravitational Wave Detectors (World Scientific, Singapore, 1994).
  13. K. S. Thorne, “Gravitational radiation,” in 300 Years of Gravitation, S. W. Hawking and W. Israel, eds. (Cambridge U. Press, Cambridge, UK, 1987) pp. 330–458.
  14. Y. Ben-Aryeh and M. Zahler, “The effects of the beam splitters on the quantum detection properties of the nonlinear Mach–Zender interferometer,” Opt. Commun. 85, 132–146 (1991).
    [CrossRef]

2002

O. Assaf and Y. Ben-Aryeh, “Quantum mechanical noise in coherent-state and squeezed-state Michelson interferometers,” J. Opt. B: Quantum Semiclassical Opt. 4, 49–56 (2002).
[CrossRef]

2001

H. J. Kimble, Yuri Levin, Andrey B. Matsko, Kip S. Thorne, and Sergey P. Vyatchanin, “Conversion of conventional gravitational-wave interferometers into quantum nondemo-lition interferometers by modifying their input and/or output optics,” Phys. Rev. D 65, 1–31 (2001); includes a list of references.
[CrossRef]

1992

A. Luis and L. L. Sanchez-Soto, “Breaking the standard quantum limit for interferometric measurements,” Opt. Commun. 89, 140–144 (1992).
[CrossRef]

1991

A. Luis and L. L. Sanchez-Soto, “Mode transformation properties and quantum limits for a Fabry–Perot interferometer,” J. Mod. Opt. 38, 971–985 (1991).
[CrossRef]

Y. Ben-Aryeh and M. Zahler, “The effects of the beam splitters on the quantum detection properties of the nonlinear Mach–Zender interferometer,” Opt. Commun. 85, 132–146 (1991).
[CrossRef]

1984

R. S. Bondurant and J. H. Shapiro, “Squeezed states in phase sensing interferometers,” Phys. Rev. D 30, 2548–2556 (1984).
[CrossRef]

1983

H. P. Yuen, “Contractive states and standard quantum limit for monitoring free-mass positions,” Phys. Rev. Lett. 51, 719–722 (1983).
[CrossRef]

1981

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
[CrossRef]

R. Loudon, “Quantum limit on the Michelson interferometer used for gravitational-wave detection,” Phys. Rev. Lett. 47, 815–818 (1981).
[CrossRef]

1980

C. M. Caves, “Quantum-mechanical radiation-pressure fluctuations in an interferometer,” Phys. Rev. Lett. 45, 75–79 (1980).
[CrossRef]

Assaf, O.

O. Assaf and Y. Ben-Aryeh, “Quantum mechanical noise in coherent-state and squeezed-state Michelson interferometers,” J. Opt. B: Quantum Semiclassical Opt. 4, 49–56 (2002).
[CrossRef]

Ben-Aryeh, Y.

O. Assaf and Y. Ben-Aryeh, “Quantum mechanical noise in coherent-state and squeezed-state Michelson interferometers,” J. Opt. B: Quantum Semiclassical Opt. 4, 49–56 (2002).
[CrossRef]

Y. Ben-Aryeh and M. Zahler, “The effects of the beam splitters on the quantum detection properties of the nonlinear Mach–Zender interferometer,” Opt. Commun. 85, 132–146 (1991).
[CrossRef]

Bondurant, R. S.

R. S. Bondurant and J. H. Shapiro, “Squeezed states in phase sensing interferometers,” Phys. Rev. D 30, 2548–2556 (1984).
[CrossRef]

Caves, C. M.

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
[CrossRef]

C. M. Caves, “Quantum-mechanical radiation-pressure fluctuations in an interferometer,” Phys. Rev. Lett. 45, 75–79 (1980).
[CrossRef]

Kimble, H. J.

H. J. Kimble, Yuri Levin, Andrey B. Matsko, Kip S. Thorne, and Sergey P. Vyatchanin, “Conversion of conventional gravitational-wave interferometers into quantum nondemo-lition interferometers by modifying their input and/or output optics,” Phys. Rev. D 65, 1–31 (2001); includes a list of references.
[CrossRef]

Levin, Yuri

H. J. Kimble, Yuri Levin, Andrey B. Matsko, Kip S. Thorne, and Sergey P. Vyatchanin, “Conversion of conventional gravitational-wave interferometers into quantum nondemo-lition interferometers by modifying their input and/or output optics,” Phys. Rev. D 65, 1–31 (2001); includes a list of references.
[CrossRef]

Loudon, R.

R. Loudon, “Quantum limit on the Michelson interferometer used for gravitational-wave detection,” Phys. Rev. Lett. 47, 815–818 (1981).
[CrossRef]

Luis, A.

A. Luis and L. L. Sanchez-Soto, “Breaking the standard quantum limit for interferometric measurements,” Opt. Commun. 89, 140–144 (1992).
[CrossRef]

A. Luis and L. L. Sanchez-Soto, “Mode transformation properties and quantum limits for a Fabry–Perot interferometer,” J. Mod. Opt. 38, 971–985 (1991).
[CrossRef]

Matsko, Andrey B.

H. J. Kimble, Yuri Levin, Andrey B. Matsko, Kip S. Thorne, and Sergey P. Vyatchanin, “Conversion of conventional gravitational-wave interferometers into quantum nondemo-lition interferometers by modifying their input and/or output optics,” Phys. Rev. D 65, 1–31 (2001); includes a list of references.
[CrossRef]

Sanchez-Soto, L. L.

A. Luis and L. L. Sanchez-Soto, “Breaking the standard quantum limit for interferometric measurements,” Opt. Commun. 89, 140–144 (1992).
[CrossRef]

A. Luis and L. L. Sanchez-Soto, “Mode transformation properties and quantum limits for a Fabry–Perot interferometer,” J. Mod. Opt. 38, 971–985 (1991).
[CrossRef]

Shapiro, J. H.

R. S. Bondurant and J. H. Shapiro, “Squeezed states in phase sensing interferometers,” Phys. Rev. D 30, 2548–2556 (1984).
[CrossRef]

Thorne, Kip S.

H. J. Kimble, Yuri Levin, Andrey B. Matsko, Kip S. Thorne, and Sergey P. Vyatchanin, “Conversion of conventional gravitational-wave interferometers into quantum nondemo-lition interferometers by modifying their input and/or output optics,” Phys. Rev. D 65, 1–31 (2001); includes a list of references.
[CrossRef]

Vyatchanin, Sergey P.

H. J. Kimble, Yuri Levin, Andrey B. Matsko, Kip S. Thorne, and Sergey P. Vyatchanin, “Conversion of conventional gravitational-wave interferometers into quantum nondemo-lition interferometers by modifying their input and/or output optics,” Phys. Rev. D 65, 1–31 (2001); includes a list of references.
[CrossRef]

Yuen, H. P.

H. P. Yuen, “Contractive states and standard quantum limit for monitoring free-mass positions,” Phys. Rev. Lett. 51, 719–722 (1983).
[CrossRef]

Zahler, M.

Y. Ben-Aryeh and M. Zahler, “The effects of the beam splitters on the quantum detection properties of the nonlinear Mach–Zender interferometer,” Opt. Commun. 85, 132–146 (1991).
[CrossRef]

J. Mod. Opt.

A. Luis and L. L. Sanchez-Soto, “Mode transformation properties and quantum limits for a Fabry–Perot interferometer,” J. Mod. Opt. 38, 971–985 (1991).
[CrossRef]

J. Opt. B: Quantum Semiclassical Opt.

O. Assaf and Y. Ben-Aryeh, “Quantum mechanical noise in coherent-state and squeezed-state Michelson interferometers,” J. Opt. B: Quantum Semiclassical Opt. 4, 49–56 (2002).
[CrossRef]

Opt. Commun.

A. Luis and L. L. Sanchez-Soto, “Breaking the standard quantum limit for interferometric measurements,” Opt. Commun. 89, 140–144 (1992).
[CrossRef]

Y. Ben-Aryeh and M. Zahler, “The effects of the beam splitters on the quantum detection properties of the nonlinear Mach–Zender interferometer,” Opt. Commun. 85, 132–146 (1991).
[CrossRef]

Phys. Rev. D

H. J. Kimble, Yuri Levin, Andrey B. Matsko, Kip S. Thorne, and Sergey P. Vyatchanin, “Conversion of conventional gravitational-wave interferometers into quantum nondemo-lition interferometers by modifying their input and/or output optics,” Phys. Rev. D 65, 1–31 (2001); includes a list of references.
[CrossRef]

R. S. Bondurant and J. H. Shapiro, “Squeezed states in phase sensing interferometers,” Phys. Rev. D 30, 2548–2556 (1984).
[CrossRef]

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
[CrossRef]

Phys. Rev. Lett.

R. Loudon, “Quantum limit on the Michelson interferometer used for gravitational-wave detection,” Phys. Rev. Lett. 47, 815–818 (1981).
[CrossRef]

C. M. Caves, “Quantum-mechanical radiation-pressure fluctuations in an interferometer,” Phys. Rev. Lett. 45, 75–79 (1980).
[CrossRef]

H. P. Yuen, “Contractive states and standard quantum limit for monitoring free-mass positions,” Phys. Rev. Lett. 51, 719–722 (1983).
[CrossRef]

Other

W. G. Unruh, “Quantum noise in the interferometer detector,” in Quantum Optics, Experimental Gravitation and Measurement Theory, P. Meystre and M. O. Scully, eds. (Plenum, New York, 1983), pp. 647–660.

D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, Berlin, 1994).

P. R. Saulson, Fundamentals of Interferometric Gravitational Wave Detectors (World Scientific, Singapore, 1994).

K. S. Thorne, “Gravitational radiation,” in 300 Years of Gravitation, S. W. Hawking and W. Israel, eds. (Cambridge U. Press, Cambridge, UK, 1987) pp. 330–458.

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Figures (2)

Fig. 1
Fig. 1

Uncertainty ΔZ resulting from PC and radiation-pressure fluctuations is described as a function of the phase ξ using the parameters |α|2=1019, C=10-18. The wave vector k corresponds to a wavelength λ of 500 nm. The horizontal dotted line represents the result for a coherent-state interferometer while the dotted–dashed and solid curves represent the results for a squeezed-state interferometer with squeezing amplitudes r=2 and r=4, respectively. The condition sin[k(Z2-Z1)]=1 is assumed.

Fig. 2
Fig. 2

Same as in Fig. 1 but with a change of the radiation-pressure constant to C=10-19.

Equations (26)

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aˆ1=12(aˆ+bˆ),aˆ2=12(aˆ-bˆ).
dˆ=12[exp(ikZ1+iC1aˆ1aˆ1)aˆ1+exp(ikZ2+iC2aˆ2aˆ2)aˆ2],
eˆ=12[exp(ikZ1+iC1aˆ1aˆ1)aˆ1-exp(ikZ2+iC2aˆ2aˆ2)aˆ2],
dˆdˆ=12 (aˆ1aˆ1+aˆ2aˆ2+{aˆ1 exp[ik(Z2-Z1)+iC(aˆ2aˆ2-aˆ1aˆ1)]aˆ2+H.C.}),
exp[iC(aˆ2aˆ2-aˆ1aˆ1)]1-iC(aˆbˆ+bˆaˆ)-C2/2(aˆbˆ+bˆaˆ)2.
dˆdˆ-eˆeˆ=cos[k(Z2-Z1)](|α|2+2C|α|2 sinh(r)cosh(r)sin(ξ)+C2/2{2|α|4 sinh(r)cosh(r)cos(ξ)-|α|4[2 sinh2(r)+1]-|α|2 sinh2(r)}),
δdˆdˆ-eˆeˆ=-k sin[k(Z2-Z1)]|α|2δZ.
(dˆdˆ-eˆeˆ)2=D+E sin(ξ)+F cos(ξ),
D=[1+2 sinh2(r)](|α|2+C2|α|6),
E=-4C sinh(r)cosh(r)|α|4,
F=(2|α|2-2C2|α|6)sinh(r)cosh(r).
cos(ξ)=-FE2+F2,sin(ξ)=-EE2+F2,
(dˆdˆ-eˆeˆ)2min=exp(-2r)(|α|2+C2|α|6).
dˆdˆ-eˆeˆ)2=|α|2 exp(-2r)+|α|6C2 exp(2r).
(dˆdˆ-eˆeˆ)21/2=δdˆdˆ-eˆeˆ,
(ΔZ)min=exp(-r)[|α|2+C2|α|6]1/2k-1(|α|2)-1.
ΔZ=[D+E sin(ξ)+F cos(ξ)]1/2k-1(|α|2)-1.
(ΔZˆ)min=exp(-r)k-12C,
(ΔZ)ξ=π=k-1[exp(-2r)|α|-2+exp(2r)C2|α|2]1/2.
kZˆi=kZi+Ciaˆiaˆi,i=1, 2
kΔZˆi=Ciaˆiaˆi=Cinˆi.
QiΔZˆi=2wbcnˆiτ,
Ci=2bwkcτ 1Qi.
ΔZ=Δpmτ
2wbcnˆ=mΔZˆτ=mτ Cnˆk,
Cfloat=2wbτkmc.

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