Abstract

We consider the problem of the measurement of very small displacements in the transverse plane of an optical image with a split photodetector. We show that the standard quantum limit for such a measurement, which is equal to the diffraction limit divided by the square root of the number of photons used in the measurement, cannot be overcome by use of ordinary single-mode squeezed light. We give the form of possible multimode nonclassical states of light, enabling us to enhance by orders of magnitude the resolution of such a measurement beyond the standard quantum limit.

© 2000 Optical Society of America

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  1. For reviews, see, for example, L. Lugiato, A. Gatti, and H. Wiedemann, Les Houches Session LXIII (North-Holland, Amsterdam, 1997), p. 431; M. Kolobov, “The spatial behavior of quantum noise,” Rev. Mod. Physics (to be published).
  2. Rayleigh, Philos. Mag. 261, 403, 477 (1879); Collected Optics Papers of Lord Rayleigh (Optical Society of America, Washington, D.C., 1994), part A, p. 117.
  3. C. K. Rushforth and R. W. Harris, J. Opt. Soc. Am. 58, 539 (1968).
  4. For a survey, see, for example, A. J. den Dekker and A. van den Bos, J. Opt. Soc. Am. A 14, 547 (1997).
    [CrossRef]
  5. J. Goodman, Statistical Optics (Wiley, New York, 1985), p. 85.
  6. M. Kolobov and I. Sokolov, Sov. Phys. JETP 63, 1105 (1986).
  7. M. Kolobov and I. Sokolov, Phys. Lett. A 140, 101 (1989).
    [CrossRef]
  8. C. Putman, B. De Grooth, N. Van Hulst, and J. Greve, J. Appl. Phys. 72, 6 (1992).
    [CrossRef]
  9. D. Fournier, A. Boccara, N. Amer, and R. Gerlach, Appl. Phys. Lett. 37, 519 (1980).
    [CrossRef]
  10. S. Kamimura, Appl. Opt. 26, 3425 (1987).
    [CrossRef] [PubMed]
  11. M. Levenson, W. Richardson, and S. Perlmutter, Opt. Lett. 14, 779 (1990).
    [CrossRef]
  12. J.-P. Poizat, T.-J. Chang, O. Ripoll, and P. Grangier, J. Opt. Soc. Am. B 15, 1757 (1998).
    [CrossRef]
  13. S. Machida, Y. Yamamoto, and Y. Itaya, Phys. Rev. Lett. 58, 1000 (1987); F. Marin, A. Bramati, E. Giacobino, T.-C. Zhang, J.-P. Poizat, J.-F. Roch, and P. Grangier, Phys. Rev. Lett. 75, 4606 (1995).
    [CrossRef] [PubMed]
  14. Actually a perfectly squeezed vacuum is excluded by our linearization procedure, as it has infinite noise on the other quadrature.
  15. I. Marzoli, A. Gatti, and L. Lugiato, Phys. Rev. Lett. 78, 2092 (1997).
    [CrossRef]

1998 (1)

1997 (2)

1992 (1)

C. Putman, B. De Grooth, N. Van Hulst, and J. Greve, J. Appl. Phys. 72, 6 (1992).
[CrossRef]

1990 (1)

1989 (1)

M. Kolobov and I. Sokolov, Phys. Lett. A 140, 101 (1989).
[CrossRef]

1987 (2)

S. Kamimura, Appl. Opt. 26, 3425 (1987).
[CrossRef] [PubMed]

S. Machida, Y. Yamamoto, and Y. Itaya, Phys. Rev. Lett. 58, 1000 (1987); F. Marin, A. Bramati, E. Giacobino, T.-C. Zhang, J.-P. Poizat, J.-F. Roch, and P. Grangier, Phys. Rev. Lett. 75, 4606 (1995).
[CrossRef] [PubMed]

1986 (1)

M. Kolobov and I. Sokolov, Sov. Phys. JETP 63, 1105 (1986).

1980 (1)

D. Fournier, A. Boccara, N. Amer, and R. Gerlach, Appl. Phys. Lett. 37, 519 (1980).
[CrossRef]

1968 (1)

1879 (1)

Rayleigh, Philos. Mag. 261, 403, 477 (1879); Collected Optics Papers of Lord Rayleigh (Optical Society of America, Washington, D.C., 1994), part A, p. 117.

Amer, N.

D. Fournier, A. Boccara, N. Amer, and R. Gerlach, Appl. Phys. Lett. 37, 519 (1980).
[CrossRef]

Boccara, A.

D. Fournier, A. Boccara, N. Amer, and R. Gerlach, Appl. Phys. Lett. 37, 519 (1980).
[CrossRef]

Chang, T.-J.

De Grooth, B.

C. Putman, B. De Grooth, N. Van Hulst, and J. Greve, J. Appl. Phys. 72, 6 (1992).
[CrossRef]

den Dekker, A. J.

Fournier, D.

D. Fournier, A. Boccara, N. Amer, and R. Gerlach, Appl. Phys. Lett. 37, 519 (1980).
[CrossRef]

Gatti, A.

I. Marzoli, A. Gatti, and L. Lugiato, Phys. Rev. Lett. 78, 2092 (1997).
[CrossRef]

For reviews, see, for example, L. Lugiato, A. Gatti, and H. Wiedemann, Les Houches Session LXIII (North-Holland, Amsterdam, 1997), p. 431; M. Kolobov, “The spatial behavior of quantum noise,” Rev. Mod. Physics (to be published).

Gerlach, R.

D. Fournier, A. Boccara, N. Amer, and R. Gerlach, Appl. Phys. Lett. 37, 519 (1980).
[CrossRef]

Goodman, J.

J. Goodman, Statistical Optics (Wiley, New York, 1985), p. 85.

Grangier, P.

Greve, J.

C. Putman, B. De Grooth, N. Van Hulst, and J. Greve, J. Appl. Phys. 72, 6 (1992).
[CrossRef]

Harris, R. W.

Itaya, Y.

S. Machida, Y. Yamamoto, and Y. Itaya, Phys. Rev. Lett. 58, 1000 (1987); F. Marin, A. Bramati, E. Giacobino, T.-C. Zhang, J.-P. Poizat, J.-F. Roch, and P. Grangier, Phys. Rev. Lett. 75, 4606 (1995).
[CrossRef] [PubMed]

Kamimura, S.

Kolobov, M.

M. Kolobov and I. Sokolov, Phys. Lett. A 140, 101 (1989).
[CrossRef]

M. Kolobov and I. Sokolov, Sov. Phys. JETP 63, 1105 (1986).

Levenson, M.

Lugiato, L.

I. Marzoli, A. Gatti, and L. Lugiato, Phys. Rev. Lett. 78, 2092 (1997).
[CrossRef]

For reviews, see, for example, L. Lugiato, A. Gatti, and H. Wiedemann, Les Houches Session LXIII (North-Holland, Amsterdam, 1997), p. 431; M. Kolobov, “The spatial behavior of quantum noise,” Rev. Mod. Physics (to be published).

Machida, S.

S. Machida, Y. Yamamoto, and Y. Itaya, Phys. Rev. Lett. 58, 1000 (1987); F. Marin, A. Bramati, E. Giacobino, T.-C. Zhang, J.-P. Poizat, J.-F. Roch, and P. Grangier, Phys. Rev. Lett. 75, 4606 (1995).
[CrossRef] [PubMed]

Marzoli, I.

I. Marzoli, A. Gatti, and L. Lugiato, Phys. Rev. Lett. 78, 2092 (1997).
[CrossRef]

Perlmutter, S.

Poizat, J.-P.

Putman, C.

C. Putman, B. De Grooth, N. Van Hulst, and J. Greve, J. Appl. Phys. 72, 6 (1992).
[CrossRef]

Rayleigh,

Rayleigh, Philos. Mag. 261, 403, 477 (1879); Collected Optics Papers of Lord Rayleigh (Optical Society of America, Washington, D.C., 1994), part A, p. 117.

Richardson, W.

Ripoll, O.

Rushforth, C. K.

Sokolov, I.

M. Kolobov and I. Sokolov, Phys. Lett. A 140, 101 (1989).
[CrossRef]

M. Kolobov and I. Sokolov, Sov. Phys. JETP 63, 1105 (1986).

van den Bos, A.

Van Hulst, N.

C. Putman, B. De Grooth, N. Van Hulst, and J. Greve, J. Appl. Phys. 72, 6 (1992).
[CrossRef]

Wiedemann, H.

For reviews, see, for example, L. Lugiato, A. Gatti, and H. Wiedemann, Les Houches Session LXIII (North-Holland, Amsterdam, 1997), p. 431; M. Kolobov, “The spatial behavior of quantum noise,” Rev. Mod. Physics (to be published).

Yamamoto, Y.

S. Machida, Y. Yamamoto, and Y. Itaya, Phys. Rev. Lett. 58, 1000 (1987); F. Marin, A. Bramati, E. Giacobino, T.-C. Zhang, J.-P. Poizat, J.-F. Roch, and P. Grangier, Phys. Rev. Lett. 75, 4606 (1995).
[CrossRef] [PubMed]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

D. Fournier, A. Boccara, N. Amer, and R. Gerlach, Appl. Phys. Lett. 37, 519 (1980).
[CrossRef]

J. Appl. Phys. (1)

C. Putman, B. De Grooth, N. Van Hulst, and J. Greve, J. Appl. Phys. 72, 6 (1992).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

Opt. Lett. (1)

Philos. Mag. (1)

Rayleigh, Philos. Mag. 261, 403, 477 (1879); Collected Optics Papers of Lord Rayleigh (Optical Society of America, Washington, D.C., 1994), part A, p. 117.

Phys. Lett. A (1)

M. Kolobov and I. Sokolov, Phys. Lett. A 140, 101 (1989).
[CrossRef]

Phys. Rev. Lett. (2)

S. Machida, Y. Yamamoto, and Y. Itaya, Phys. Rev. Lett. 58, 1000 (1987); F. Marin, A. Bramati, E. Giacobino, T.-C. Zhang, J.-P. Poizat, J.-F. Roch, and P. Grangier, Phys. Rev. Lett. 75, 4606 (1995).
[CrossRef] [PubMed]

I. Marzoli, A. Gatti, and L. Lugiato, Phys. Rev. Lett. 78, 2092 (1997).
[CrossRef]

Sov. Phys. JETP (1)

M. Kolobov and I. Sokolov, Sov. Phys. JETP 63, 1105 (1986).

Other (3)

For reviews, see, for example, L. Lugiato, A. Gatti, and H. Wiedemann, Les Houches Session LXIII (North-Holland, Amsterdam, 1997), p. 431; M. Kolobov, “The spatial behavior of quantum noise,” Rev. Mod. Physics (to be published).

J. Goodman, Statistical Optics (Wiley, New York, 1985), p. 85.

Actually a perfectly squeezed vacuum is excluded by our linearization procedure, as it has infinite noise on the other quadrature.

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Equations (8)

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NANB-NANB=NSASB+NANBNtot2×ΔNtot2-Ntot,
Dsql=ΔNtot,
Δ=NtotN-DD=0=Ntot2-+dyI0,y
ΔNA2NA-1=NANtotΔNtot2Ntot-1.
ΔNA2=NA+i=1,2Cai+aiAi2+CaiaiAi*2+2Ca1a2+A2A1*+2Ca1a2A1*A2*+c.c.,
Ai=SAuix,yE+x,ydxdy.
Ai=x>0uiE+dxdy-x<0uiE+dxdy.
ΔNA-NB2=Ntot1+4Δpo2-1×x>0uouedxdy2,

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