Abstract

Experiments to generate squeezed states of light are described for a collection of two-level atoms within a high-finesse cavity. The investigation is conducted in a regime for which the weak-field coupling of atoms to the cavity mode produces a splitting in the normal mode structure of the atom–field system that is large compared with the atomic linewidth. Reductions in photocurrent noise of 30% (−1.55 dB) below the noise level set by the vacuum state of the field are observed in a balanced homodyne detector. A degree of squeezing of approximately 50% is inferred for the field state in the absence of propagation and detection losses. The observed spectrum of squeezing extends over a very broad range of frequencies (~±75 MHz), with the frequency of best squeezing corresponding to an offset from the optical carrier given by the normal mode splitting.

© 1987 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  29. H. P. Yuen, V. W. S. Chan, Opt. Lett. 8, 177 (1983); errata, Opt. Lett. 8, 345 (1983); B. L. Schumaker, Opt. Lett. 9, 189 (1984).
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  31. L.-A. Wu, M. Xiao, H. J. Kimble, J. Opt. Soc. Am. B 4, 1465 (1987).
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  32. L. Mandel, E. Wolf, J. Opt. Soc. Am. 65, 413 (1975).
    [CrossRef]
  33. H. J. Kimble, D. E. Grant, A. T. Rosenberger, P. D. Drummond, in Laser Physics, J. D. Harvey, D. F. Walls, eds., Vol. 182 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1983), p. 14.
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  34. P. D. Drummond, IEEE J. Quantum Electron. QE-17, 301 (1981); Department of Physics, University of Auckland, Private Bag, Auckland 1, New Zealand (personal communication).
    [CrossRef]
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    [CrossRef]
  37. M. Xiao, H. J. Kimble, H. J. Carmichael, Phys. Rev. A 35, 3832 (1987).
    [CrossRef] [PubMed]

1987 (6)

D. A. Holm, M. Sargent, Phys. Rev. A 35, 2150 (1987).
[CrossRef] [PubMed]

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

L. A. Orozco, H. J. Kimble, A. T. Rosenberger, Opt. Commun. 62, 54 (1987); Phys. Rev. A (to be published).
[CrossRef]

G. V. Varada, M. S. Kumar, G. S. Agarwal, Opt. Commun. 62, 328 (1987).
[CrossRef]

L.-A. Wu, M. Xiao, H. J. Kimble, J. Opt. Soc. Am. B 4, 1465 (1987).
[CrossRef]

M. Xiao, H. J. Kimble, H. J. Carmichael, Phys. Rev. A 35, 3832 (1987).
[CrossRef] [PubMed]

1986 (2)

R. J. Brecha, L. A. Orozco, M. G. Raizen, Min Xiao, H. J. Kimble, J. Opt. Soc. Am. B 3, P238 (1986).

L.-A. Wu, H. J. Kimble, J. O. Hall, H. Wu, Phys. Rev. Lett. 57, 2520 (1986); L.-A. Wu, M. Xiao, H. J. Kimble, J. Opt. Soc. Am. B 4, 1465 (1987).
[CrossRef] [PubMed]

1985 (10)

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985); R. E. Slusher, B. Yurke, P. Grangier, A. La Porta, D. F. Walls, M. Reid, J. Opt. Soc. Am. B 4, 1453 (1987).
[CrossRef] [PubMed]

H.-I. Yoo, J. H. Eberly, Phys. Rep. 118(5), 239 (1985).
[CrossRef]

G. S. Agarwal, J. Opt. Soc. Am. B 2, 480 (1985).
[CrossRef]

H. J. Carmichael, Phys. Rev. A 33, 3262 (1985).
[CrossRef]

J. Seke, J. Opt. Soc. Am. B 2, 1687 (1985); Phys. Rev. A 33, 739 (1986).
[CrossRef]

D. Meschede, H. Walther, G. Müller, Phys. Rev. Lett. 54, 551 (1985); G. Rempe, H. Walther, N. Klein, Phys. Rev. Lett. 58, 353 (1987).
[CrossRef] [PubMed]

A. Heidmann, J. M. Raimond, S. Reynaud, Phys. Rev. Lett. 54, 326 (1985); A. Heidmann, J. M. Raimond, S. Reynaud, N. Zagury, Opt. Commun. 54, 54 (1985).
[CrossRef] [PubMed]

M. D. Reid, D. F. Walls, Phys. Rev. A 32, 396 (1985); Phys. Rev. A 34, 4929 (1986).
[CrossRef] [PubMed]

J. H. Shapiro, IEEE J. Quantum Electron. QE-21, 237 (1985).
[CrossRef]

M. J. Collet, D. F. Walls, Phys. Rev. A 32, 2887 (1985).
[CrossRef]

1984 (2)

M. J. Collet, C. W. Gardiner, Phys. Rev. A 30, 1386 (1984); C. W. Gardiner, M. J. Collett, Phys. Rev. A 31, 3761 (1985).
[CrossRef] [PubMed]

B. Yurke, Phys. Rev. A. 29, 408 (1984).
[CrossRef]

1983 (5)

Y. Kaluzny, P. Goy, M. Gross, J. M. Raimond, S. Haroche, Phys. Rev. Lett. 51, 1175 (1983).
[CrossRef]

A. T. Rosenberger, L. A. Orozco, H. J. Kimble, Phys. Rev. A 28, 2569 (1983).
[CrossRef]

H. J. Carmichael, D. F. Walls, P. D. Drummond, S. S. Hassan, Phys. Rev. A 27, 3112 (1983).
[CrossRef]

H. P. Yuen, V. W. S. Chan, Opt. Lett. 8, 177 (1983); errata, Opt. Lett. 8, 345 (1983); B. L. Schumaker, Opt. Lett. 9, 189 (1984).
[CrossRef] [PubMed]

L. A. Lugiato, M. Milani, Z. Phys. B 50, 171 (1983).
[CrossRef]

1982 (1)

L. A. Lugiato, G. Strini, Opt. Commun. 41, 67 (1982).
[CrossRef]

1981 (2)

P. D. Drummond, D. F. Walls, Phys. Rev. A 23, 2563 (1981).
[CrossRef]

P. D. Drummond, IEEE J. Quantum Electron. QE-17, 301 (1981); Department of Physics, University of Auckland, Private Bag, Auckland 1, New Zealand (personal communication).
[CrossRef]

1980 (1)

P. D. Drummond, D. F. Walls, J. Phys. A 13, 725 (1980).
[CrossRef]

1975 (1)

Agarwal, G. S.

G. V. Varada, M. S. Kumar, G. S. Agarwal, Opt. Commun. 62, 328 (1987).
[CrossRef]

G. S. Agarwal, J. Opt. Soc. Am. B 2, 480 (1985).
[CrossRef]

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]

H. J. Kimble, M. G. Raizen, L. A. Orozco, Min Xiao, T. L. Boyd, in Fundamentals of Quantum Optics II, F. Ehlotzky, ed. (Springer Verlag, Berlin, 1987).

Brecha, R. J.

R. J. Brecha, L. A. Orozco, M. G. Raizen, Min Xiao, H. J. Kimble, J. Opt. Soc. Am. B 3, P238 (1986).

Carmichael, H. J.

M. Xiao, H. J. Kimble, H. J. Carmichael, Phys. Rev. A 35, 3832 (1987).
[CrossRef] [PubMed]

H. J. Carmichael, Phys. Rev. A 33, 3262 (1985).
[CrossRef]

H. J. Carmichael, D. F. Walls, P. D. Drummond, S. S. Hassan, Phys. Rev. A 27, 3112 (1983).
[CrossRef]

H. J. Carmichael, Quantum Statistical Methods in Quantum Optics (Springer-Verlag, Berlin, to be published).

Chan, V. W. S.

Collet, M. J.

M. J. Collet, D. F. Walls, Phys. Rev. A 32, 2887 (1985).
[CrossRef]

M. J. Collet, C. W. Gardiner, Phys. Rev. A 30, 1386 (1984); C. W. Gardiner, M. J. Collett, Phys. Rev. A 31, 3761 (1985).
[CrossRef] [PubMed]

Drummond, P. D.

H. J. Carmichael, D. F. Walls, P. D. Drummond, S. S. Hassan, Phys. Rev. A 27, 3112 (1983).
[CrossRef]

P. D. Drummond, D. F. Walls, Phys. Rev. A 23, 2563 (1981).
[CrossRef]

P. D. Drummond, IEEE J. Quantum Electron. QE-17, 301 (1981); Department of Physics, University of Auckland, Private Bag, Auckland 1, New Zealand (personal communication).
[CrossRef]

P. D. Drummond, D. F. Walls, J. Phys. A 13, 725 (1980).
[CrossRef]

H. J. Kimble, D. E. Grant, A. T. Rosenberger, P. D. Drummond, in Laser Physics, J. D. Harvey, D. F. Walls, eds., Vol. 182 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1983), p. 14.
[CrossRef]

Eberly, J. H.

H.-I. Yoo, J. H. Eberly, Phys. Rep. 118(5), 239 (1985).
[CrossRef]

Gardiner, C. W.

M. J. Collet, C. W. Gardiner, Phys. Rev. A 30, 1386 (1984); C. W. Gardiner, M. J. Collett, Phys. Rev. A 31, 3761 (1985).
[CrossRef] [PubMed]

Goy, P.

Y. Kaluzny, P. Goy, M. Gross, J. M. Raimond, S. Haroche, Phys. Rev. Lett. 51, 1175 (1983).
[CrossRef]

Grant, D. E.

H. J. Kimble, D. E. Grant, A. T. Rosenberger, P. D. Drummond, in Laser Physics, J. D. Harvey, D. F. Walls, eds., Vol. 182 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1983), p. 14.
[CrossRef]

Gross, M.

Y. Kaluzny, P. Goy, M. Gross, J. M. Raimond, S. Haroche, Phys. Rev. Lett. 51, 1175 (1983).
[CrossRef]

Hall, J. O.

L.-A. Wu, H. J. Kimble, J. O. Hall, H. Wu, Phys. Rev. Lett. 57, 2520 (1986); L.-A. Wu, M. Xiao, H. J. Kimble, J. Opt. Soc. Am. B 4, 1465 (1987).
[CrossRef] [PubMed]

Haroche, S.

Y. Kaluzny, P. Goy, M. Gross, J. M. Raimond, S. Haroche, Phys. Rev. Lett. 51, 1175 (1983).
[CrossRef]

Hassan, S. S.

H. J. Carmichael, D. F. Walls, P. D. Drummond, S. S. Hassan, Phys. Rev. A 27, 3112 (1983).
[CrossRef]

Heidmann, A.

A. Heidmann, J. M. Raimond, S. Reynaud, Phys. Rev. Lett. 54, 326 (1985); A. Heidmann, J. M. Raimond, S. Reynaud, N. Zagury, Opt. Commun. 54, 54 (1985).
[CrossRef] [PubMed]

Hollberg, L. W.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985); R. E. Slusher, B. Yurke, P. Grangier, A. La Porta, D. F. Walls, M. Reid, J. Opt. Soc. Am. B 4, 1453 (1987).
[CrossRef] [PubMed]

Holm, D. A.

D. A. Holm, M. Sargent, Phys. Rev. A 35, 2150 (1987).
[CrossRef] [PubMed]

Kaluzny, Y.

Y. Kaluzny, P. Goy, M. Gross, J. M. Raimond, S. Haroche, Phys. Rev. Lett. 51, 1175 (1983).
[CrossRef]

Kimble, H. J.

L. A. Orozco, H. J. Kimble, A. T. Rosenberger, Opt. Commun. 62, 54 (1987); Phys. Rev. A (to be published).
[CrossRef]

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

L.-A. Wu, M. Xiao, H. J. Kimble, J. Opt. Soc. Am. B 4, 1465 (1987).
[CrossRef]

M. Xiao, H. J. Kimble, H. J. Carmichael, Phys. Rev. A 35, 3832 (1987).
[CrossRef] [PubMed]

L.-A. Wu, H. J. Kimble, J. O. Hall, H. Wu, Phys. Rev. Lett. 57, 2520 (1986); L.-A. Wu, M. Xiao, H. J. Kimble, J. Opt. Soc. Am. B 4, 1465 (1987).
[CrossRef] [PubMed]

R. J. Brecha, L. A. Orozco, M. G. Raizen, Min Xiao, H. J. Kimble, J. Opt. Soc. Am. B 3, P238 (1986).

A. T. Rosenberger, L. A. Orozco, H. J. Kimble, Phys. Rev. A 28, 2569 (1983).
[CrossRef]

M. G. Raizen, L. A. Orozco, H. J. Kimble, “Squeezed states in a coupled system” (submitted to Phys. Rev. A.)

H. J. Kimble, M. G. Raizen, L. A. Orozco, Min Xiao, T. L. Boyd, in Fundamentals of Quantum Optics II, F. Ehlotzky, ed. (Springer Verlag, Berlin, 1987).

H. J. Kimble, D. E. Grant, A. T. Rosenberger, P. D. Drummond, in Laser Physics, J. D. Harvey, D. F. Walls, eds., Vol. 182 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1983), p. 14.
[CrossRef]

Kumar, M. S.

G. V. Varada, M. S. Kumar, G. S. Agarwal, Opt. Commun. 62, 328 (1987).
[CrossRef]

Lane, A.

M. D. Reid, A. Lane, D. F. Walls, in Quantum Optics IV, J. D. Harvey, D. F. Walls, eds. (Springer-Verlag, Berlin, 1986), p. 31.
[CrossRef]

Lugiato, L. A.

L. A. Lugiato, M. Milani, Z. Phys. B 50, 171 (1983).
[CrossRef]

L. A. Lugiato, G. Strini, Opt. Commun. 41, 67 (1982).
[CrossRef]

L. A. Lugiato, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1984), Vol. 21, p. 69.
[CrossRef]

Mandel, L.

L. Mandel, E. Wolf, J. Opt. Soc. Am. 65, 413 (1975).
[CrossRef]

L. Mandel, in Optics in Four Dimension—1980, M. A. Machado, L. M. Narducci, eds., AIP Conference Proceedings No. 65 (American Institute of Physics, New York, 1981).

Mertz, J. C.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985); R. E. Slusher, B. Yurke, P. Grangier, A. La Porta, D. F. Walls, M. Reid, J. Opt. Soc. Am. B 4, 1453 (1987).
[CrossRef] [PubMed]

Meschede, D.

D. Meschede, H. Walther, G. Müller, Phys. Rev. Lett. 54, 551 (1985); G. Rempe, H. Walther, N. Klein, Phys. Rev. Lett. 58, 353 (1987).
[CrossRef] [PubMed]

Milani, M.

L. A. Lugiato, M. Milani, Z. Phys. B 50, 171 (1983).
[CrossRef]

Müller, G.

D. Meschede, H. Walther, G. Müller, Phys. Rev. Lett. 54, 551 (1985); G. Rempe, H. Walther, N. Klein, Phys. Rev. Lett. 58, 353 (1987).
[CrossRef] [PubMed]

Orozco, L. A.

L. A. Orozco, H. J. Kimble, A. T. Rosenberger, Opt. Commun. 62, 54 (1987); Phys. Rev. A (to be published).
[CrossRef]

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

R. J. Brecha, L. A. Orozco, M. G. Raizen, Min Xiao, H. J. Kimble, J. Opt. Soc. Am. B 3, P238 (1986).

A. T. Rosenberger, L. A. Orozco, H. J. Kimble, Phys. Rev. A 28, 2569 (1983).
[CrossRef]

M. G. Raizen, L. A. Orozco, H. J. Kimble, “Squeezed states in a coupled system” (submitted to Phys. Rev. A.)

H. J. Kimble, M. G. Raizen, L. A. Orozco, Min Xiao, T. L. Boyd, in Fundamentals of Quantum Optics II, F. Ehlotzky, ed. (Springer Verlag, Berlin, 1987).

Raimond, J. M.

A. Heidmann, J. M. Raimond, S. Reynaud, Phys. Rev. Lett. 54, 326 (1985); A. Heidmann, J. M. Raimond, S. Reynaud, N. Zagury, Opt. Commun. 54, 54 (1985).
[CrossRef] [PubMed]

Y. Kaluzny, P. Goy, M. Gross, J. M. Raimond, S. Haroche, Phys. Rev. Lett. 51, 1175 (1983).
[CrossRef]

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]

R. J. Brecha, L. A. Orozco, M. G. Raizen, Min Xiao, H. J. Kimble, J. Opt. Soc. Am. B 3, P238 (1986).

M. G. Raizen, L. A. Orozco, H. J. Kimble, “Squeezed states in a coupled system” (submitted to Phys. Rev. A.)

H. J. Kimble, M. G. Raizen, L. A. Orozco, Min Xiao, T. L. Boyd, in Fundamentals of Quantum Optics II, F. Ehlotzky, ed. (Springer Verlag, Berlin, 1987).

Reid, M. D.

M. D. Reid, D. F. Walls, Phys. Rev. A 32, 396 (1985); Phys. Rev. A 34, 4929 (1986).
[CrossRef] [PubMed]

M. D. Reid, A. Lane, D. F. Walls, in Quantum Optics IV, J. D. Harvey, D. F. Walls, eds. (Springer-Verlag, Berlin, 1986), p. 31.
[CrossRef]

Reynaud, S.

A. Heidmann, J. M. Raimond, S. Reynaud, Phys. Rev. Lett. 54, 326 (1985); A. Heidmann, J. M. Raimond, S. Reynaud, N. Zagury, Opt. Commun. 54, 54 (1985).
[CrossRef] [PubMed]

Rosenberger, A. T.

L. A. Orozco, H. J. Kimble, A. T. Rosenberger, Opt. Commun. 62, 54 (1987); Phys. Rev. A (to be published).
[CrossRef]

A. T. Rosenberger, L. A. Orozco, H. J. Kimble, Phys. Rev. A 28, 2569 (1983).
[CrossRef]

H. J. Kimble, D. E. Grant, A. T. Rosenberger, P. D. Drummond, in Laser Physics, J. D. Harvey, D. F. Walls, eds., Vol. 182 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1983), p. 14.
[CrossRef]

Sargent, M.

D. A. Holm, M. Sargent, Phys. Rev. A 35, 2150 (1987).
[CrossRef] [PubMed]

Seke, J.

Shapiro, J. H.

J. H. Shapiro, IEEE J. Quantum Electron. QE-21, 237 (1985).
[CrossRef]

Slusher, R. E.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985); R. E. Slusher, B. Yurke, P. Grangier, A. La Porta, D. F. Walls, M. Reid, J. Opt. Soc. Am. B 4, 1453 (1987).
[CrossRef] [PubMed]

Strini, G.

L. A. Lugiato, G. Strini, Opt. Commun. 41, 67 (1982).
[CrossRef]

Valley, J. F.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985); R. E. Slusher, B. Yurke, P. Grangier, A. La Porta, D. F. Walls, M. Reid, J. Opt. Soc. Am. B 4, 1453 (1987).
[CrossRef] [PubMed]

Varada, G. V.

G. V. Varada, M. S. Kumar, G. S. Agarwal, Opt. Commun. 62, 328 (1987).
[CrossRef]

Walls, D. F.

M. D. Reid, D. F. Walls, Phys. Rev. A 32, 396 (1985); Phys. Rev. A 34, 4929 (1986).
[CrossRef] [PubMed]

M. J. Collet, D. F. Walls, Phys. Rev. A 32, 2887 (1985).
[CrossRef]

H. J. Carmichael, D. F. Walls, P. D. Drummond, S. S. Hassan, Phys. Rev. A 27, 3112 (1983).
[CrossRef]

P. D. Drummond, D. F. Walls, Phys. Rev. A 23, 2563 (1981).
[CrossRef]

P. D. Drummond, D. F. Walls, J. Phys. A 13, 725 (1980).
[CrossRef]

M. D. Reid, A. Lane, D. F. Walls, in Quantum Optics IV, J. D. Harvey, D. F. Walls, eds. (Springer-Verlag, Berlin, 1986), p. 31.
[CrossRef]

Walther, H.

D. Meschede, H. Walther, G. Müller, Phys. Rev. Lett. 54, 551 (1985); G. Rempe, H. Walther, N. Klein, Phys. Rev. Lett. 58, 353 (1987).
[CrossRef] [PubMed]

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

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[CrossRef] [PubMed]

L.-A. Wu, H. J. Kimble, J. O. Hall, H. Wu, Phys. Rev. Lett. 57, 2520 (1986); L.-A. Wu, M. Xiao, H. J. Kimble, J. Opt. Soc. Am. B 4, 1465 (1987).
[CrossRef] [PubMed]

M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, H. J. Kimble, Phys. Rev. Lett. 59, 198 (1987).
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Figures (12)

Fig. 1
Fig. 1

Spectrum of squeezing S versus offset frequency ν = Ω / g N, with g N = γ μ C. Trace (i) μ = 5.4, C = 28, Δ = 7.3, θ = −0.9, x = 7.0; trace (ii) μ = 13.5, C = 52, Δ = −14.6, θ = 0.86, x = 16.4; trace (iii) μ = 50, C = 200, Δ = 60.0, θ = −1.0, x = 70.

Fig. 2
Fig. 2

Dependence of the minimum value of the spectrum of squeezing S on atomic cooperativity C for fixed μ. Note that, at any point on a curve, the values of (Δ, θ, Ω, x) were chosen for optimum squeezing.

Fig. 3
Fig. 3

Diagram of dressed states for N ≫ 1 atoms coupled to a single mode of a cavity. The three groups of states (0, 1, 2) correspond to zero, one and two excitations in the system. The degeneracy of the noninteracting system is lifted, in the presence of coupling, to produce splittings characterized by the frequency g N. The figure is drawn for Δ = θ = 0. The solid lines represent absorption of coherent pump photons, while the wavy lines in a and b depict two possible return paths to the ground state (following Varada et al.28).

Fig. 4
Fig. 4

Diagram of the elements of the experiment for generating and detecting squeezed states. BS1, BS2, BS3, beam splitters; EO1, EO2, electro-optic modulators; P1, P2, P3, P4, P5, polarizers; AO1, AO2, acousto-optic modulators; Q1, Q2, Q3, quarter-wave plates; H1, half-wave plate; S1, S2, S3, shutters; M1, M2, M3, mirrors; SD1, SD2, photodiodes; PZT1–PZT3, piezoelectric transducers.

Fig. 5
Fig. 5

Oscilloscope trace (Pi versus Pt) of the steady-state characteristics of absorptive optical bistability for case b. From the ratio R of input switching intensities Y1, Y2 (R = Y2/Y1) in traces such as this one, we determine the cooperativity parameter C. In this trace the input switching powers corresponding to (Y1 and Y2) are P1 = 87 ± 10 μW and P2 = 103 ± 12 μW, respectively.

Fig. 6
Fig. 6

A, Oscilloscope trace of the transmitted intensity Pt as a function of cavity detuning θ in the presence of atomic detuning Δ. The peak transmission from the cavity is 2.3 ± 0.3 μW and a scale for the cavity detuning is given. Note that the hysteresis associated with the scanning of the PZT was not corrected and accounts for a broadening of about 12 MHz, which is a small fraction of the detuning range between turning points shown in the figure. B, Theoretical prediction for the transmitted intensity Pt as a function of cavity detuning θ with atomic detuning Δ.

Fig. 7
Fig. 7

Relative level Φ(ϕ, Ω) of fluctuations of the difference photocurrent (i1i2) on a logarithmic scale as a function of analysis frequency Ω/2π. The LO phase ϕ is being swept at a rate somewhat higher than the rate at which Ω is varied. The data are for configuration b with C = 20 ± 5, Δ = 7 ± 1. Trace (i) Signal input blocked to define the vacuum level (the sloping character of this trace is due to the high-frequency falloff of the response of the photodiode–amplifier configuration). Trace (ii) Phase-sensitive fluctuations with signal beam present drop below the vacuum level. Traces such as this one over several ranges of Ω help us to search the parameter space and to find the region of best squeezing. The time taken for this trace is 0.2 sec; the sharp features near 130 and 160 MHz are generated by the flyback of the piezoelectric transducer.

Fig. 8
Fig. 8

Relative noise level Φ(ϕ, Ω) of fluctuations of the difference photocurrent (i1i2) versus LO phase ϕ at fixed analysis frequency Ω/2π = 124 MHz and bandwidth B = 300 kHz (postdetection video bandwidth, Δf = 106 Hz) for configuration a. Trace (i) Signal input blocked to define vacuum level (multiple trace average). Trace (ii) Phase-sensitive fluctuations with signal beam present drop below the vacuum level. Operating conditions C = 28 ± 3, μ = 5.4 ± 0.4, Δ = 7.0 ± 1.0, θ = −2 ± 1, x = 1 (within a factor of 3), Pi = 77 ± 10 μW. The time taken for the trace is 0.5 sec. The signal was not corrected for the electronic noise contribution.

Fig. 9
Fig. 9

Dependence of rms noise voltage V(ϕ) on LO phase ϕ at fixed analysis frequency Ω/2π = 280 MHz and bandwidth of 300 kHz (postdetection video bandwidth, 159 Hz) for case b. C = 45 ± 6, Δ = −14.6 ± 0.2. Trace (i) is the vacuum-plus-amplifier noise, trace (ii) is the squeezed beam signal, and trace (iii) is the amplifier noise level alone obtained with no light on the photodiodes. The sharp feature at the right side of the trace is due to the flyback of the piezoelectric used to scan the phase of the LO. Note that the ordinate is a linear scale in noise voltage. The time taken for the trace is 0.2 sec.

Fig. 10
Fig. 10

Relative noise level Φ(ϕ, Ω) of fluctuations of the difference photocurrent (i1i2) versus LO phase ϕ at fixed analysis frequency Ω/2π = 280 MHz and bandwidth of 300 kHz (postdetection video bandwidth, 159 Hz) for case b. Trace (i) Signal input blocked to define vacuum level. Trace (ii) Phase-sensitive fluctuations with signal beam present drop below the vacuum level. Operating conditions C = 52 ± 8, μ = 13.5 ± 0.5, Δ = −14.6 ± 0.2, x = 25 ± 5, Pi = 400 ± 100 μW. The time taken for the trace is 0.2 sec.

Fig. 11
Fig. 11

Histogram of frequency of occurrence of a given noise level around the minima of Φ(ϕ, Ω) obtained from many traces with the same parameters as in Fig. 10. The upper abcissa is the actual recorded noise reduction below the vacuum noise level, whereas the lower abcissa is the noise reduction after correction is made for the nonzero noise level of the amplifier. The asymmetry of the distribution presumably arises because there is some largest noise reduction produced by the system. For constant control parameters, the statistical fluctuations in photocurrent would give rise to a symmetric distribution, with HWHM indicated by the right-hand side of the figure. Since the control parameters are not constant, optimum squeezing is not recorded at each noise minimum, thereby skewing the distribution on the left-hand side.

Fig. 12
Fig. 12

Relative noise level Φ(ϕ, Ω) of fluctuations of the difference photocurrent (i1i2) versus LO phase ϕ at fixed analysis frequency Ω/2π = 280 MHz and bandwidth of 300 kHz (postdetection video bandwidth, 159 Hz), with the source of Doppler broadening corrected. Trace (i) Signal input blocked to define vacuum level. Trace (ii) Phase-sensitive fluctuations with signal beam present drop below the vacuum level for C = 52 ± 10, μ = 13.5 ± 0.5, and Δ = −14.6 ± 0.2.

Equations (4)

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H ^ = H ^ 0 + H ^ a + H ^ c , H ^ 0 = ( ω a / 2 ) J ^ z + ( ω c ) a ^ a ^ + g [ i J ^ - a ^ + H . C . ] .
A ( ϕ , Ω ) = - d τ e - i Ω τ : z ^ ( t , ϕ ) , z ^ ( t + τ , ϕ ) : .
S ( ϕ , Ω ) = 2 κ A ( ϕ , Ω ) .
R - ( Ω ) = 1 + ρ T 0 α η 2 S - ( Ω ) .

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