Abstract

We study the photon-number distribution in squeezed states of a single-mode radiation field. A U(1)-invariant squeezing criterion is compared and contrasted with a more restrictive criterion, with the help of suggestive geometric representations. The U(1) invariance of the photon-number distribution in a squeezed coherent state, with arbitrary complex squeeze and displacement parameters, is explicitly demonstrated. The behavior of the photon-number distribution for a representative value of the displacement and various values of the squeeze parameter is numerically investigated. A new kind of giant oscillation riding as an envelope over more rapid oscillations in this distribution is demonstrated.

© 1993 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. F. Walls, Nature (London) 306, 141 (1983), and references therein.
    [Crossref]
  2. R. Loudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987), and references therein.
    [Crossref]
  3. M. C. Teich and B. E. A. Saleh, Quantum Opt. 1, 153 (1989), and references therein.
    [Crossref]
  4. R. Loudon and P. L. Knight, eds., special issue on squeezed light, J. Mod. Opt. 34(6/7) (1987).
  5. H. J. Kimble and D. F. Walls, eds., feature on squeezed states of the electromagnetic field, J. Opt. Soc. Am. B 4(10), (1987).
    [Crossref]
  6. D. Stoler, Phys. Rev. D 1, 3217 (1970); Phys. Rev. D 4, 1925 (1971).
    [Crossref]
  7. H. P. Yuen, Phys. Rev. A 13, 2226 (1976).
    [Crossref]
  8. J. N. Hollenhorst, Phys. Rev. D 19, 1669 (1979).
    [Crossref]
  9. C. M. Caves, Phys. Rev. D 23, 1693 (1981).
    [Crossref]
  10. M. M. Nieto, in Frontiers of Nonequilibrium Statistical Physics, G. T. Moore and M. O. Scully, eds. (Plenum, New York, 1986); p. 287.
    [Crossref]
  11. R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
    [Crossref] [PubMed]
  12. R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
    [Crossref] [PubMed]
  13. L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
    [Crossref] [PubMed]
  14. M. W. Maeda, P. Kumar, and J. H. Shapiro, Opt. Lett. 3, 161 (1986).
  15. S. Machida, Y. Yamamoto, and Y. Itaya, Phys. Rev. Lett. 58, 1000 (1987).
    [Crossref] [PubMed]
  16. A. Heidmann, R. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Gamy, Phys. Rev. Lett. 59, 2555 (1987).
    [Crossref] [PubMed]
  17. R. Moschovich, B. Yurke, P. G. Kaminsky, A. D. Smith, A. H. Silver, R. W. Simon, and M. V. Schneider, Phys. Rev. Lett. 65, 1419 (1990).
    [Crossref]
  18. W. Schleich and J. A. Wheeler, J. Opt. Soc. Am. B 4, 1715 (1987).
    [Crossref]
  19. W. Schleich and J. A. Wheeler, Nature (London) 326, 574 (1987); in The Physics of Phase Space, Y. S. Kim and W. W. Zachary, eds. (Springer, New York, 1987); W. Schleich, J. A. Wheeler, and D. F. Walls, Phys. Rev. A 38, 1177 (1987).
    [Crossref]
  20. M. Hillery, Phys. Lett. A 111, 409 (1985); Phys. Rev. A 35, 725 (1987).
    [Crossref]
  21. R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 36, 3868 (1987); Phys. Rev. A 37, 3028 (1988); R. Simon, in Symmetries in Science II, B. Gruber and R. Lenczewski, eds. (Plenum, New York, 1986), p. 471; N. Mukunda, Curr. Sci. 59, 1135 (1990).
    [Crossref] [PubMed]
  22. See, for instance, T. F. Jordan, Linear Operators in Quantum Mechanics (Wiley, New York, 1974); G. Lion and M. Vergne, The Weil Representation, Maslov Index, and Theta Series (Birkhauser, Basel, 1980).
  23. H. Bacry, in Group Theoretical Methods in Physics, W. W. Zachary, ed. (World Scientific, Singapore, 1984), p. 215.
  24. There is in addition invariance under a scaling semigroup of Sp(2, ℛ), but this is not of interest here.
  25. This is the same as writing âexp(-iϕ)=[x^(ϕ)+ip^(ϕ)]2 and requiring that x^ (ϕ) be H squeezed for some value of ϕ. See, for instance, G. S. Agarwal and K. Tara, Phys. Rev. A 43, 492 (1991).
    [Crossref] [PubMed]
  26. G. S. Agarwal and G. Adam, Phys. Rev. A 38, 750 (1988).
    [Crossref] [PubMed]
  27. G. S. Agarwal and G. Adam, Phys. Rev. A 39, 6259 (1989).
    [Crossref] [PubMed]
  28. J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, Phys. Rev. Lett. 44, 1323 (1980).
    [Crossref]
  29. H-I. Yoo, J. J. Sanchez-Mondragon, and J. H. Eberly, J. Phys. A 4, 1383 (1981).
    [Crossref]
  30. R. R. Puri and G. S. Agarwal, Phys. Rev. A 35, 3433 (1987).
    [Crossref] [PubMed]
  31. M. Satyanarayana, P. Rice, R. Vyas, and H. J. Carmichael, J. Opt. Soc. Am. B 6, 228 (1989), and references therein.
    [Crossref]

1991 (1)

This is the same as writing âexp(-iϕ)=[x^(ϕ)+ip^(ϕ)]2 and requiring that x^ (ϕ) be H squeezed for some value of ϕ. See, for instance, G. S. Agarwal and K. Tara, Phys. Rev. A 43, 492 (1991).
[Crossref] [PubMed]

1990 (1)

R. Moschovich, B. Yurke, P. G. Kaminsky, A. D. Smith, A. H. Silver, R. W. Simon, and M. V. Schneider, Phys. Rev. Lett. 65, 1419 (1990).
[Crossref]

1989 (3)

M. C. Teich and B. E. A. Saleh, Quantum Opt. 1, 153 (1989), and references therein.
[Crossref]

G. S. Agarwal and G. Adam, Phys. Rev. A 39, 6259 (1989).
[Crossref] [PubMed]

M. Satyanarayana, P. Rice, R. Vyas, and H. J. Carmichael, J. Opt. Soc. Am. B 6, 228 (1989), and references therein.
[Crossref]

1988 (1)

G. S. Agarwal and G. Adam, Phys. Rev. A 38, 750 (1988).
[Crossref] [PubMed]

1987 (9)

S. Machida, Y. Yamamoto, and Y. Itaya, Phys. Rev. Lett. 58, 1000 (1987).
[Crossref] [PubMed]

A. Heidmann, R. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Gamy, Phys. Rev. Lett. 59, 2555 (1987).
[Crossref] [PubMed]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 36, 3868 (1987); Phys. Rev. A 37, 3028 (1988); R. Simon, in Symmetries in Science II, B. Gruber and R. Lenczewski, eds. (Plenum, New York, 1986), p. 471; N. Mukunda, Curr. Sci. 59, 1135 (1990).
[Crossref] [PubMed]

R. R. Puri and G. S. Agarwal, Phys. Rev. A 35, 3433 (1987).
[Crossref] [PubMed]

R. Loudon and P. L. Knight, eds., special issue on squeezed light, J. Mod. Opt. 34(6/7) (1987).

H. J. Kimble and D. F. Walls, eds., feature on squeezed states of the electromagnetic field, J. Opt. Soc. Am. B 4(10), (1987).
[Crossref]

R. Loudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987), and references therein.
[Crossref]

W. Schleich and J. A. Wheeler, J. Opt. Soc. Am. B 4, 1715 (1987).
[Crossref]

W. Schleich and J. A. Wheeler, Nature (London) 326, 574 (1987); in The Physics of Phase Space, Y. S. Kim and W. W. Zachary, eds. (Springer, New York, 1987); W. Schleich, J. A. Wheeler, and D. F. Walls, Phys. Rev. A 38, 1177 (1987).
[Crossref]

1986 (3)

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

L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[Crossref] [PubMed]

M. W. Maeda, P. Kumar, and J. H. Shapiro, Opt. Lett. 3, 161 (1986).

1985 (2)

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

M. Hillery, Phys. Lett. A 111, 409 (1985); Phys. Rev. A 35, 725 (1987).
[Crossref]

1983 (1)

D. F. Walls, Nature (London) 306, 141 (1983), and references therein.
[Crossref]

1981 (2)

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

H-I. Yoo, J. J. Sanchez-Mondragon, and J. H. Eberly, J. Phys. A 4, 1383 (1981).
[Crossref]

1980 (1)

J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, Phys. Rev. Lett. 44, 1323 (1980).
[Crossref]

1979 (1)

J. N. Hollenhorst, Phys. Rev. D 19, 1669 (1979).
[Crossref]

1976 (1)

H. P. Yuen, Phys. Rev. A 13, 2226 (1976).
[Crossref]

1970 (1)

D. Stoler, Phys. Rev. D 1, 3217 (1970); Phys. Rev. D 4, 1925 (1971).
[Crossref]

Adam, G.

G. S. Agarwal and G. Adam, Phys. Rev. A 39, 6259 (1989).
[Crossref] [PubMed]

G. S. Agarwal and G. Adam, Phys. Rev. A 38, 750 (1988).
[Crossref] [PubMed]

Agarwal, G. S.

This is the same as writing âexp(-iϕ)=[x^(ϕ)+ip^(ϕ)]2 and requiring that x^ (ϕ) be H squeezed for some value of ϕ. See, for instance, G. S. Agarwal and K. Tara, Phys. Rev. A 43, 492 (1991).
[Crossref] [PubMed]

G. S. Agarwal and G. Adam, Phys. Rev. A 39, 6259 (1989).
[Crossref] [PubMed]

G. S. Agarwal and G. Adam, Phys. Rev. A 38, 750 (1988).
[Crossref] [PubMed]

R. R. Puri and G. S. Agarwal, Phys. Rev. A 35, 3433 (1987).
[Crossref] [PubMed]

Bacry, H.

H. Bacry, in Group Theoretical Methods in Physics, W. W. Zachary, ed. (World Scientific, Singapore, 1984), p. 215.

Carmichael, H. J.

Caves, C. M.

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

DeVoe, R. G.

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

Eberly, J. H.

H-I. Yoo, J. J. Sanchez-Mondragon, and J. H. Eberly, J. Phys. A 4, 1383 (1981).
[Crossref]

J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, Phys. Rev. Lett. 44, 1323 (1980).
[Crossref]

Fabre, C.

A. Heidmann, R. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Gamy, Phys. Rev. Lett. 59, 2555 (1987).
[Crossref] [PubMed]

Gamy, G.

A. Heidmann, R. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Gamy, Phys. Rev. Lett. 59, 2555 (1987).
[Crossref] [PubMed]

Giacobino, E.

A. Heidmann, R. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Gamy, Phys. Rev. Lett. 59, 2555 (1987).
[Crossref] [PubMed]

Hall, J. L.

L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[Crossref] [PubMed]

Heidmann, A.

A. Heidmann, R. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Gamy, Phys. Rev. Lett. 59, 2555 (1987).
[Crossref] [PubMed]

Hillery, M.

M. Hillery, Phys. Lett. A 111, 409 (1985); Phys. Rev. A 35, 725 (1987).
[Crossref]

Hollberg, L. W.

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

Hollenhorst, J. N.

J. N. Hollenhorst, Phys. Rev. D 19, 1669 (1979).
[Crossref]

Horowicz, R.

A. Heidmann, R. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Gamy, Phys. Rev. Lett. 59, 2555 (1987).
[Crossref] [PubMed]

Itaya, Y.

S. Machida, Y. Yamamoto, and Y. Itaya, Phys. Rev. Lett. 58, 1000 (1987).
[Crossref] [PubMed]

Jordan, T. F.

See, for instance, T. F. Jordan, Linear Operators in Quantum Mechanics (Wiley, New York, 1974); G. Lion and M. Vergne, The Weil Representation, Maslov Index, and Theta Series (Birkhauser, Basel, 1980).

Kaminsky, P. G.

R. Moschovich, B. Yurke, P. G. Kaminsky, A. D. Smith, A. H. Silver, R. W. Simon, and M. V. Schneider, Phys. Rev. Lett. 65, 1419 (1990).
[Crossref]

Kimble, H. J.

L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[Crossref] [PubMed]

Knight, P. L.

R. Loudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987), and references therein.
[Crossref]

Kumar, P.

M. W. Maeda, P. Kumar, and J. H. Shapiro, Opt. Lett. 3, 161 (1986).

Levenson, M. D.

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

Loudon, R.

R. Loudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987), and references therein.
[Crossref]

Machida, S.

S. Machida, Y. Yamamoto, and Y. Itaya, Phys. Rev. Lett. 58, 1000 (1987).
[Crossref] [PubMed]

Maeda, M. W.

M. W. Maeda, P. Kumar, and J. H. Shapiro, Opt. Lett. 3, 161 (1986).

Mertz, J. C.

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

Moschovich, R.

R. Moschovich, B. Yurke, P. G. Kaminsky, A. D. Smith, A. H. Silver, R. W. Simon, and M. V. Schneider, Phys. Rev. Lett. 65, 1419 (1990).
[Crossref]

Mukunda, N.

R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 36, 3868 (1987); Phys. Rev. A 37, 3028 (1988); R. Simon, in Symmetries in Science II, B. Gruber and R. Lenczewski, eds. (Plenum, New York, 1986), p. 471; N. Mukunda, Curr. Sci. 59, 1135 (1990).
[Crossref] [PubMed]

Narozhny, N. B.

J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, Phys. Rev. Lett. 44, 1323 (1980).
[Crossref]

Nieto, M. M.

M. M. Nieto, in Frontiers of Nonequilibrium Statistical Physics, G. T. Moore and M. O. Scully, eds. (Plenum, New York, 1986); p. 287.
[Crossref]

Perlmutter, S. H.

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

Puri, R. R.

R. R. Puri and G. S. Agarwal, Phys. Rev. A 35, 3433 (1987).
[Crossref] [PubMed]

Reynaud, S.

A. Heidmann, R. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Gamy, Phys. Rev. Lett. 59, 2555 (1987).
[Crossref] [PubMed]

Rice, P.

Saleh, B. E. A.

M. C. Teich and B. E. A. Saleh, Quantum Opt. 1, 153 (1989), and references therein.
[Crossref]

Sanchez-Mondragon, J. J.

H-I. Yoo, J. J. Sanchez-Mondragon, and J. H. Eberly, J. Phys. A 4, 1383 (1981).
[Crossref]

J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, Phys. Rev. Lett. 44, 1323 (1980).
[Crossref]

Satyanarayana, M.

Schleich, W.

W. Schleich and J. A. Wheeler, J. Opt. Soc. Am. B 4, 1715 (1987).
[Crossref]

W. Schleich and J. A. Wheeler, Nature (London) 326, 574 (1987); in The Physics of Phase Space, Y. S. Kim and W. W. Zachary, eds. (Springer, New York, 1987); W. Schleich, J. A. Wheeler, and D. F. Walls, Phys. Rev. A 38, 1177 (1987).
[Crossref]

Schneider, M. V.

R. Moschovich, B. Yurke, P. G. Kaminsky, A. D. Smith, A. H. Silver, R. W. Simon, and M. V. Schneider, Phys. Rev. Lett. 65, 1419 (1990).
[Crossref]

Shapiro, J. H.

M. W. Maeda, P. Kumar, and J. H. Shapiro, Opt. Lett. 3, 161 (1986).

Shelby, R. M.

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

Silver, A. H.

R. Moschovich, B. Yurke, P. G. Kaminsky, A. D. Smith, A. H. Silver, R. W. Simon, and M. V. Schneider, Phys. Rev. Lett. 65, 1419 (1990).
[Crossref]

Simon, R.

R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 36, 3868 (1987); Phys. Rev. A 37, 3028 (1988); R. Simon, in Symmetries in Science II, B. Gruber and R. Lenczewski, eds. (Plenum, New York, 1986), p. 471; N. Mukunda, Curr. Sci. 59, 1135 (1990).
[Crossref] [PubMed]

Simon, R. W.

R. Moschovich, B. Yurke, P. G. Kaminsky, A. D. Smith, A. H. Silver, R. W. Simon, and M. V. Schneider, Phys. Rev. Lett. 65, 1419 (1990).
[Crossref]

Slusher, R. E.

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

Smith, A. D.

R. Moschovich, B. Yurke, P. G. Kaminsky, A. D. Smith, A. H. Silver, R. W. Simon, and M. V. Schneider, Phys. Rev. Lett. 65, 1419 (1990).
[Crossref]

Stoler, D.

D. Stoler, Phys. Rev. D 1, 3217 (1970); Phys. Rev. D 4, 1925 (1971).
[Crossref]

Sudarshan, E. C. G.

R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 36, 3868 (1987); Phys. Rev. A 37, 3028 (1988); R. Simon, in Symmetries in Science II, B. Gruber and R. Lenczewski, eds. (Plenum, New York, 1986), p. 471; N. Mukunda, Curr. Sci. 59, 1135 (1990).
[Crossref] [PubMed]

Tara, K.

This is the same as writing âexp(-iϕ)=[x^(ϕ)+ip^(ϕ)]2 and requiring that x^ (ϕ) be H squeezed for some value of ϕ. See, for instance, G. S. Agarwal and K. Tara, Phys. Rev. A 43, 492 (1991).
[Crossref] [PubMed]

Teich, M. C.

M. C. Teich and B. E. A. Saleh, Quantum Opt. 1, 153 (1989), and references therein.
[Crossref]

Valley, J. F.

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

Vyas, R.

Walls, D. F.

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

D. F. Walls, Nature (London) 306, 141 (1983), and references therein.
[Crossref]

Wheeler, J. A.

W. Schleich and J. A. Wheeler, J. Opt. Soc. Am. B 4, 1715 (1987).
[Crossref]

W. Schleich and J. A. Wheeler, Nature (London) 326, 574 (1987); in The Physics of Phase Space, Y. S. Kim and W. W. Zachary, eds. (Springer, New York, 1987); W. Schleich, J. A. Wheeler, and D. F. Walls, Phys. Rev. A 38, 1177 (1987).
[Crossref]

Wu, H.

L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[Crossref] [PubMed]

Wu, L.-A.

L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[Crossref] [PubMed]

Yamamoto, Y.

S. Machida, Y. Yamamoto, and Y. Itaya, Phys. Rev. Lett. 58, 1000 (1987).
[Crossref] [PubMed]

Yoo, H-I.

H-I. Yoo, J. J. Sanchez-Mondragon, and J. H. Eberly, J. Phys. A 4, 1383 (1981).
[Crossref]

Yuen, H. P.

H. P. Yuen, Phys. Rev. A 13, 2226 (1976).
[Crossref]

Yurke, B.

R. Moschovich, B. Yurke, P. G. Kaminsky, A. D. Smith, A. H. Silver, R. W. Simon, and M. V. Schneider, Phys. Rev. Lett. 65, 1419 (1990).
[Crossref]

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

J. Mod. Opt. (2)

R. Loudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987), and references therein.
[Crossref]

R. Loudon and P. L. Knight, eds., special issue on squeezed light, J. Mod. Opt. 34(6/7) (1987).

J. Opt. Soc. Am. B (3)

J. Phys. A (1)

H-I. Yoo, J. J. Sanchez-Mondragon, and J. H. Eberly, J. Phys. A 4, 1383 (1981).
[Crossref]

Nature (London) (2)

D. F. Walls, Nature (London) 306, 141 (1983), and references therein.
[Crossref]

W. Schleich and J. A. Wheeler, Nature (London) 326, 574 (1987); in The Physics of Phase Space, Y. S. Kim and W. W. Zachary, eds. (Springer, New York, 1987); W. Schleich, J. A. Wheeler, and D. F. Walls, Phys. Rev. A 38, 1177 (1987).
[Crossref]

Opt. Lett. (1)

M. W. Maeda, P. Kumar, and J. H. Shapiro, Opt. Lett. 3, 161 (1986).

Phys. Lett. A (1)

M. Hillery, Phys. Lett. A 111, 409 (1985); Phys. Rev. A 35, 725 (1987).
[Crossref]

Phys. Rev. A (6)

R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 36, 3868 (1987); Phys. Rev. A 37, 3028 (1988); R. Simon, in Symmetries in Science II, B. Gruber and R. Lenczewski, eds. (Plenum, New York, 1986), p. 471; N. Mukunda, Curr. Sci. 59, 1135 (1990).
[Crossref] [PubMed]

H. P. Yuen, Phys. Rev. A 13, 2226 (1976).
[Crossref]

R. R. Puri and G. S. Agarwal, Phys. Rev. A 35, 3433 (1987).
[Crossref] [PubMed]

This is the same as writing âexp(-iϕ)=[x^(ϕ)+ip^(ϕ)]2 and requiring that x^ (ϕ) be H squeezed for some value of ϕ. See, for instance, G. S. Agarwal and K. Tara, Phys. Rev. A 43, 492 (1991).
[Crossref] [PubMed]

G. S. Agarwal and G. Adam, Phys. Rev. A 38, 750 (1988).
[Crossref] [PubMed]

G. S. Agarwal and G. Adam, Phys. Rev. A 39, 6259 (1989).
[Crossref] [PubMed]

Phys. Rev. D (3)

J. N. Hollenhorst, Phys. Rev. D 19, 1669 (1979).
[Crossref]

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

D. Stoler, Phys. Rev. D 1, 3217 (1970); Phys. Rev. D 4, 1925 (1971).
[Crossref]

Phys. Rev. Lett. (7)

S. Machida, Y. Yamamoto, and Y. Itaya, Phys. Rev. Lett. 58, 1000 (1987).
[Crossref] [PubMed]

A. Heidmann, R. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Gamy, Phys. Rev. Lett. 59, 2555 (1987).
[Crossref] [PubMed]

R. Moschovich, B. Yurke, P. G. Kaminsky, A. D. Smith, A. H. Silver, R. W. Simon, and M. V. Schneider, Phys. Rev. Lett. 65, 1419 (1990).
[Crossref]

J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, Phys. Rev. Lett. 44, 1323 (1980).
[Crossref]

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

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

L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[Crossref] [PubMed]

Quantum Opt. (1)

M. C. Teich and B. E. A. Saleh, Quantum Opt. 1, 153 (1989), and references therein.
[Crossref]

Other (4)

M. M. Nieto, in Frontiers of Nonequilibrium Statistical Physics, G. T. Moore and M. O. Scully, eds. (Plenum, New York, 1986); p. 287.
[Crossref]

See, for instance, T. F. Jordan, Linear Operators in Quantum Mechanics (Wiley, New York, 1974); G. Lion and M. Vergne, The Weil Representation, Maslov Index, and Theta Series (Birkhauser, Basel, 1980).

H. Bacry, in Group Theoretical Methods in Physics, W. W. Zachary, ed. (World Scientific, Singapore, 1984), p. 215.

There is in addition invariance under a scaling semigroup of Sp(2, ℛ), but this is not of interest here.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Representing variance matrices in Minkowski space 2,1. The entire hyperboloid represents the set of all S-minimum-uncertainty states corresponding to general V. The hyperbola BOA, in the x0x2 plane, represents the set of all H minimum-uncertainty states corresponding to diagonal V. The circle passing through A and B in a plane perpendicular to the x0 axis represents typical free evolution.

Fig. 2
Fig. 2

Phase-space representation of means and variance. Note that under time evolution the ellipse changes orientation only and not intrinsic shape and size. It also rotates at the same rate and sense as its center, leading to constancy of angle χ.

Fig. 3
Fig. 3

Plots of photon-number distribution W(n) for various phase angles φ in the range of 0° to 90°, displaying the emergence of the giant oscillations. The squeeze parameters s = 201 throughout. The magnitude of the phase-space displacement, (x02 + p02)1/2, has been fixed at 7 2, for easy comparison with Ref. 18.

Fig. 4
Fig. 4

Showing W(n) for values of s = 21, 31, 41, 51, 101, 251. The parameters φ and (x02 + p02)1/2 are fixed at 88° and 7 2, respectively.

Tables (1)

Tables Icon

Table 1 Comparison of Uncertainty Principles, Minimum-Uncertainty States, Squeezing and Invariance Criteria

Equations (54)

Equations on this page are rendered with MathJax. Learn more.

[ x ^ , p ^ ] = i .
x 0 = ψ x ^ ψ ,             p 0 = ψ p ^ ψ .
V = [ ( Δ x ) 2 Δ ( x p ) Δ ( x p ) ( Δ p ) 2 ] , ( Δ x ) 2 = ψ ( x ^ - x 0 ) 2 ψ = ψ x ^ 2 ψ - x 0 2 , Δ ( x , p ) = ½ ψ { x ^ - x 0 , p ^ - p 0 } ψ = ½ ψ { x ^ , p ^ } ψ - x 0 p 0 , ( Δ p ) 2 = ψ ( p ^ - p 0 ) 2 ψ = ψ p ^ 2 ψ - p 0 2 .
Δ x Δ p 1 / 2.
det V ( Δ x ) 2 ( Δ p ) 2 - [ Δ ( x p ) ] 2 1 / 4.
Δ x Δ p { 1 4 + [ Δ ( x p ) ] 2 } 1 / 2 .
S S p ( 2 , R ) : S = [ a b b d ] ,             a , b , c , d real ,             a d - b c = 1 ;
( x ^ p ^ ) = S ( x ^ p ^ ) , [ x ^ , p ^ ] = i .
x ^ = a x ^ + b p ^ = U ( S ) - 1 x ^ U ( S ) , p ^ = c x ^ + d p ^ = U ( S ) - 1 p ^ U ( S ) , U ( S ) U ( S ) = ± U ( S S ) .
( x 0 p 0 ) = S ( x 0 p 0 ) ,
V = S V S T .
either             Δ x < 1 / 2
or             Δ p < 1 / 2 .
S = i σ 2 = [ 0 1 - 1 0 ] Sp ( 2 , R ) .
1 / 2 { Tr V - [ ( Tr V ) 2 - 4 det V ] 1 / 2 } < 1 / 2.
U ( 1 ) = { R ( θ ) = [ cos θ sin θ - sin θ cos θ ] | 0 θ < 2 π } Sp ( 2 , R ) ,
H 0 = 1 / 2 ( x ^ 2 + p ^ 2 )
exp ( - i t H 0 ) = U [ ( t ) ] , ( t ) = [ cos t sin t - sin t cos t ] U ( 1 ) Sp ( 2 , R ) ; U [ ( t ) ] - 1 ( x ^ p ^ ) U [ ( t ) ] = ( t ) ( x ^ p ^ ) .
ψ = exp ( - i t H 0 ) ψ , V = V ( t ) = ( t ) V ( t ) T .
x ^ θ 0 ( t ) = x ^ θ 0 cos t - p ^ θ 0 sin t = 1 2 [ a ^ exp ( - i θ 0 ) exp ( i t ) + a ^ exp ( i θ 0 ) exp ( - i t ) ] ,
Δ x Δ p = 1 / 2 Δ ( x , p ) = 0 det V = 1 / 4.
Δ x Δ p = 1 / 2 H squeezed unless Δ x = Δ p = 1 / 2 ;
det V = 1 / 4 S squeezed unless V = ( 1 / 2 ) 1 .
ψ u ( x ) = ( π u ) - 1 / 4 exp ( - x 2 2 u ) ,             u > 0 , V = 1 2 [ u 0 0 u - 1 ] , H squeezed except when u = 1.
ψ u , v ( x ) = ( π u ) - 1 / 4 exp [ - ( 1 - i v ) x 2 2 u ] ,             u > 0 , v real ; V = 1 2 [ u v v w ] ,             det V = 1 4 ; w = ( 1 + v 2 ) u - 1 ; S squeezed except when u = w = 1 ,             v = 0.
V = 1 2 [ u v v w ] , x V = ψ V ( x ) = ( π u ) - 1 / 4 exp [ - ( 1 - i v ) x 2 2 u ] .
x 0 , p 0 ; V exp [ i ( p 0 x ^ - x 0 p ^ ) ] V , x x 0 , p 0 ; V ψ x 0 , p 0 , V ( x ) = exp ( - i 2 x 0 p 0 ) ( π u ) - 1 / 4 × exp [ i p 0 x - ( 1 - i v ) ( x - x 0 ) 2 2 u ] .
exp ( - i t H 0 ) V U [ ( t ) ] V = exp [ i φ ( V , t ) ] V ( t ) , V ( t ) = ( t ) V ( t ) T .
V ( t ) = 1 2 [ u ( t ) v ( t ) v ( t ) w ( t ) ] , [ u ( t ) - w ( t ) 2 v ( t ) ] = ( 2 t ) ( u - w 2 v ) , u ( t ) + w ( t ) = u + w .
V = 1 2 [ u v v w ] x = ( x 0 , x 1 , x 2 ) 2 , 1 , x 0 = 1 2 ( u + w ) , x 1 = v , x 2 = ½ ( u - w ) .
u w - v 2 = 1 ( x 0 ) 2 - ( x 1 ) 2 - ( x 2 ) 2 = 1.
V = 1 2 [ u v v w ] at t = 0 V ( t 0 ) = 1 2 [ u 0 0 0 u 0 - 1 ] at t = t 0 , u 0 = ½ { u + w + sign v [ ( u + w ) 2 - 4 ] 1 / 2 } , w 0 = 1 u 0 = ½ { u + w - sign v [ ( u + w ) 2 - 4 ] 1 / 2 } ; u - w + 2 i v = exp ( 2 i t 0 ) ( u 0 - w 0 ) = exp ( 2 i t 0 ) sign v [ ( u - w ) 2 + 4 v 2 ] 1 / 2 } , 0 < t 0 < π / 2.
v u - w > 0 initial point in first or third quadrant of 1 - 2 plane , 0 < t 0 < π / 4 ; v u - w < 0 initial point in second or fourth quadrant of 1 - 2 plane , π / 4 < t 0 < π / 2.
exp ( - i t H 0 ) x 0 , p 0 ; V U [ ( t ) ] x 0 , p 0 ; V = exp [ i φ ( V , t ) ] x 0 ( t ) , p 0 ( t ) ; V ( t ) , [ x 0 ( t ) p 0 ( t ) ] = ( t ) ( x 0 p 0 ) .
E : ( x - x 0 , p - p 0 ) V - 1 ( x - x 0 p - p 0 ) = 1.
E ( t ) : [ x - x 0 ( t ) , p - p 0 ( t ) ] V ( t ) - 1 ( x - x 0 ( t ) p - p 0 ( t ) ) = 1.
( x p ) E ( t ) ( x p ) E ( t ) .
E : semimajor axis = Ω + , semiminor axis = Ω - , eccentricity = 1 - 4 Ω - 2 , area = π / 2.
x n = ψ n ( x ) = π - 1 / 4 n ! 2 n exp ( - 1 2 x 2 ) H n ( x ) ,             n = 0 , 1 , ,
λ n = n x 0 , p 0 ; V = - d x ψ n ( x ) * ψ x 0 , p 0 , V ( x ) , = ( π n ! 2 n ) - 1 / 2 u - 1 / 4 exp [ - i 2 x 0 p 0 - ( 1 - i v ) x 0 2 2 u ] μ n , μ n = - d x H n ( x ) × exp [ - x 2 2 + i p 0 x - ( 1 - i v ) ( x 2 - 2 x x 0 ) 2 u ] .
- d x H n ( x ) exp ( - ρ x 2 + σ x ) = ( π ρ ) 1 / 2 exp ( σ 2 4 ρ ) ( ρ - 1 ρ ) n / 2 H n ( σ 2 ρ ( ρ - 1 ) ) ,
ρ = 1 + u - i v 2 u , σ = x 0 + i ( u p 0 - v x 0 ) u .
λ n = ( n ! 2 n - 1 ) - 1 / 2 u 1 / 4 1 + u - i v ( 1 - u - i v 1 + u - i v ) n / 2 × exp [ - i 2 x 0 p 0 - ( 1 - i v ) x 0 2 2 u + [ x 0 + i ( u p 0 - v x 0 ) ] 2 2 u ( 1 + u - i v ) ] H n ( ξ ) , ξ = x 0 + i ( u p 0 - v x 0 ) [ ( 1 - u - i v ) ( 1 + u - i v ) ] 1 / 2 .
W n ( x 0 , p 0 ; V ) = λ n 2 = 1 n ! 2 n - 1 u ( 1 + u ) 2 + v 2 [ ( 1 - u ) 2 + v 2 ( 1 + u ) 2 + v 2 ] n / 2 × exp { - u ( 1 + u ) 2 + v 2 [ x 0 2 + p 0 2 + [ ( ( 1 + u ) 2 + v 2 ) / u ] 1 / 2 × [ ( ( 1 - u ) 2 + v 2 ) u ] 1 / 2 ξ 2 ] } H n ( ξ ) 2 .
u + w = U ( 1 )             invariant ( 1 ± u ) 2 + v 2 u = 1 + u 2 + v 2 u ± 2 = u + w ± 2 = U ( 1 )             invariant ,
2 V + σ 2 = [ u v - i v + i w ] .
2 V + σ 2 = ( u + w ) η η ,
η = ( η 1 η 2 ) = 1 u ( u + w ) ( - i u 1 - i v ) .
V ( t ) = ( t ) V ( t ) T 2 V ( t ) + σ 2 = ( t ) ( 2 V + σ 2 ) ( t ) T η ( t ) η ( t ) = ( t ) η η ( t ) T .
[ x 0 ( t ) p 0 ( t ) ] = ( t ) ( x 0 p 0 ) , η ( t ) = ( phase factor ) ( t ) η .
ξ = η 2 x 0 - η 1 p 0 η T η ,
V = 1 2 [ u 0 0 1 / u ] ,             0 < u 1 ; W n ( x 0 , p 0 ; V ) = 1 n ! 2 n - 1 u 1 / 2 ( 1 - u ) n ( 1 + u ) n + 1 × exp [ - u ( x 0 2 + p 0 2 ) + ( 1 - u 2 ) ξ 2 ( 1 + u ) 2 ] × H n ( ξ ) 2 , ξ = ( x 0 + i u p 0 ) / ( 1 - u 2 ) 1 / 2 .
x 0 + i p 0 = e i φ ( x 0 2 + p 0 2 ) 1 / 2 .
ξ = ( x 0 2 + p 0 2 1 - u 2 ) 1 / 2 ( cos φ + i u sin φ ) .

Metrics