Abstract

We investigate the quantum statistics of three time-dependent coupled oscillators in the presence of multiphoton processes. The system is connected with the two-atom multiphoton Tavis–Cummings model. The solution of the Heisenberg equations of the motion is obtained in a compact form. We assume that the modes are initially prepared in coherent states, and we discuss nonclassical phenomena (squeezing and sub-Poissonian behavior). Further, we examine the joint quasi-distribution functions as well as photon-number distribution and its factorial moments. The system has shown that the nonclassical effect is apparent in compound modes (1,3) and (2,3). Moreover, the superstructure phenomenon is observed when the photon transition is increased.

© 2006 Optical Society of America

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    [CrossRef]
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  5. D. F. Walls and R. Barakat, "Quantum-mechanical amplification and frequency conversion with a trilinear Hamiltonian," Phys. Rev. A 1, 446-453 (1970).
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  6. S. Carusotto, "Time evolution in stimulated Compton scattering," Phys. Rev. A 39, 1848-1859 (1989).
    [CrossRef] [PubMed]
  7. A. Bandilla, G. Drobny, and I. Jex, "Nondegenerate parametric interactions and nonclassical effects," Phys. Rev. A 53, 507-516 (1996).
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  29. M. S. Abdalla, F. A. A. El-Orany, and J. Perina, "SU(2) and SU(1,1) squeezing of interacting radiation modes," Acta Phys. Slov. 50, 613-616 (2000).
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  31. F. A. A. El-Orany, J. Perina, and M. S. Abdalla, "Evolution of Fock states in three mixed harmonic oscillators: quantum statistics," Int. J. Mod. Phys. B 15, 2125-2146 (2001).
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  37. K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, "Dense Coding in Experimental Quantum Communication," Phys. Rev. Lett. 76, 4656-4659 (1996).
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  38. D. S. Naik, C. G. Peterson, A. G. White, A. J. Berglund, and P. G. Kwiat, "Entangled state quantum cryptography: eavesdropping on the Ekert protocol," Phys. Rev. Lett. 84, 4733-4736 (2000).
    [CrossRef] [PubMed]
  39. W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, "Quantum cryptography using entangled photons in energy-time Bell states," Phys. Rev. Lett. 84, 4737-4740 (2000).
    [CrossRef] [PubMed]
  40. D. Bouwmeester, J.-W. Pan, M. Daniell, H. Weinfurter, and A. Zeilinger, "Observation of three-photon Greenberger-Horne-Zeilinger entanglement," Phys. Rev. Lett. 82, 1345-1349 (1999).
    [CrossRef]
  41. J. W. Pan, M. Daniell, S. Gasparoni, G. Weihs, and A. Zeilinger, "Experimental demonstration of four-photon entanglement and high-fidelity teleportation," Phys. Rev. Lett. 86, 4435-4438 (2001).
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  44. J. Katriel and D. G. Hummer, "Analytic solutions for three- and four-wave mixing via generalised Bose operators," J. Phys. A 14, 1211-1224 (1981).
    [CrossRef]
  45. C. M. Caves, "Quantum-mechanical noise in an interferometer," Phys. Rev. D 23, 1693-1708 (1981).
    [CrossRef]
  46. C. K. Hong and L. Mandel, "Higher-order squeezing of a quantum field," Phys. Rev. Lett. 54, 323-325 (1985).
    [CrossRef] [PubMed]
  47. M. Hillery, "Squeezing of the square of the field amplitude in second harmonic generation," Opt. Commun. 62, 135-138 (1987).
    [CrossRef]
  48. M. J. Collett and C. W. Gardiner, "Squeezing of intracavity and traveling-wave light fields produced in parametric amplification," Phys. Rev. A 30, 1386-1391 (1984).
    [CrossRef]
  49. M. J. Collett and R. Loudon, "Output properties of parametric amplifier in cavities," J. Opt. Soc. Am. B 4, 1525-1533 (1987).
    [CrossRef]
  50. C. K. Hong and L. Mandel, "Generation of higher-order squeezing of quantum electromagnetic fields," Phys. Rev. A 32, 974-982 (1985).
    [CrossRef] [PubMed]
  51. A. Luks, V. Perinová, and J. Perina, "Principal squeezing of vacuum fluctuations," Opt. Commun. 67, 149-151 (1988).
    [CrossRef]
  52. H. J.Kimble and D. P. Walls, "Squeezed states of the electromagnetic field, " J. Opt. Soc. Am. B 4, (1987).
  53. A. Luks, V. Perinová, and Z. Hradil, "Principal squeezing," Acta Phys. Pol. A 74, 713-721 (1988).
  54. R. Tanas, A. Miranowicz, and S. Kielich, "Squeezing and its graphical representations in the anharmonic oscillator model," Phys. Rev. A 43, 4014-4021 (1991).
    [CrossRef] [PubMed]
  55. V. Perinová, "Quantum statistics of quadratic optical parametric processes with intense coherent or stochastic pumping," Opt. Acta 28, 747-768 (1981).
    [CrossRef]
  56. V. Perinová and J. Perina, "Quantum statistics of quadratic optical parametric processes with intense coherent or stochastic pumping," Opt. Acta 28, 769-793 (1981).
    [CrossRef]
  57. B. R. Mollow and R. J. Glauber, "Quantum theory of parametric amplification. II," Phys. Rev. 160, 1097-1108 (1967).
    [CrossRef]
  58. M. Kozierowski and S. M. Chumakov, "Revivals in the off-resonant Jaynes-Cummings model with a Kerr medium," Acta Phys. Slov. 47, 301-310 (1997).
  59. M. Kozierowski, "Second-order collapses and revivals," Acta Phys. Slov. 48, 281-292 (1998).
  60. M. Kozierowski, "Thermal and squeezed vacuum Jaynes-Cummings models with a Kerr medium," J. Mol. Spectrosc. 48, 773-781 (2001).

2002 (2)

M. Abdel-Aty, M. S. Abdalla, and A.-S. F. Obada, "Entropy squeezing of a two-mode multiphoton Jaynes-Cummings model in the presence of a nonlinear medium," J. Opt. B 4, 134-142 (2002).
[CrossRef]

M. Abdel-Aty, M. S. Abdalla, and A.-S. F. Obada, "Entropy and phase properties of isotropic coupled oscillators interacting with a single atom: one- and two-photon processes," J. Opt. B 4, S133-S141 (2002).
[CrossRef]

2001 (7)

M. Abdel-Aty, M. S. Abdalla, and A.-S. F. Obada, "Quantum information and entropy squeezing of a two-level atom with a non-linear medium," J. Phys. A 34, 9129-9141 (2001).
[CrossRef]

F. A. A. El-Orany, J. Perina, and M. S. Abdalla, "Statistical properties of three quantized interacting oscillators," Phys. Scr. 63, 128-140 (2001).
[CrossRef]

F. A. A. El-Orany, J. Perina, and M. S. Abdalla, "Evolution of Fock states in three mixed harmonic oscillators: quantum statistics," Int. J. Mod. Phys. B 15, 2125-2146 (2001).
[CrossRef]

M. S. Abdalla, F. A. A. El-Orany, and J. Perina, "Statistical properties of a solvable three-Boson squeeze operator model," Eur. Phys. J. D 13, 423-438 (2001).
[CrossRef]

F. A. A. El-Orany, J. Perina, and M. S. Abdalla, "Generation of squeezed light in a nonlinear asymmetric directional coupler," J. Opt. B 3, 67-75 (2001).
[CrossRef]

J. W. Pan, M. Daniell, S. Gasparoni, G. Weihs, and A. Zeilinger, "Experimental demonstration of four-photon entanglement and high-fidelity teleportation," Phys. Rev. Lett. 86, 4435-4438 (2001).
[CrossRef] [PubMed]

M. Kozierowski, "Thermal and squeezed vacuum Jaynes-Cummings models with a Kerr medium," J. Mol. Spectrosc. 48, 773-781 (2001).

2000 (3)

M. S. Abdalla, F. A. A. El-Orany, and J. Perina, "SU(2) and SU(1,1) squeezing of interacting radiation modes," Acta Phys. Slov. 50, 613-616 (2000).

D. S. Naik, C. G. Peterson, A. G. White, A. J. Berglund, and P. G. Kwiat, "Entangled state quantum cryptography: eavesdropping on the Ekert protocol," Phys. Rev. Lett. 84, 4733-4736 (2000).
[CrossRef] [PubMed]

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, "Quantum cryptography using entangled photons in energy-time Bell states," Phys. Rev. Lett. 84, 4737-4740 (2000).
[CrossRef] [PubMed]

1999 (2)

D. Bouwmeester, J.-W. Pan, M. Daniell, H. Weinfurter, and A. Zeilinger, "Observation of three-photon Greenberger-Horne-Zeilinger entanglement," Phys. Rev. Lett. 82, 1345-1349 (1999).
[CrossRef]

M. S. Abdalla, F. A. A. El-Orany, and J. Perina, "Quantum statistical properties of nondegenerate optical parametric symmetric coupler," J. Phys. A 32, 3457-3483(1999).
[CrossRef]

1998 (1)

M. Kozierowski, "Second-order collapses and revivals," Acta Phys. Slov. 48, 281-292 (1998).

1997 (2)

M. Kozierowski and S. M. Chumakov, "Revivals in the off-resonant Jaynes-Cummings model with a Kerr medium," Acta Phys. Slov. 47, 301-310 (1997).

D. Bouwmeester, Jian-Weipan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, "Experimental quantum teleportation," Nature 390, 575-579 (1997).
[CrossRef]

1996 (2)

K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, "Dense Coding in Experimental Quantum Communication," Phys. Rev. Lett. 76, 4656-4659 (1996).
[CrossRef] [PubMed]

A. Bandilla, G. Drobny, and I. Jex, "Nondegenerate parametric interactions and nonclassical effects," Phys. Rev. A 53, 507-516 (1996).
[CrossRef] [PubMed]

1995 (1)

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, "New high-intensity source of polarization-entangled photon pairs," Phys. Rev. Lett. 75, 4337-4341 (1995).
[CrossRef] [PubMed]

1993 (1)

A. S. Shumovsky and B. Tanatar, "Quantum-statistical properties of a Raman-type model," Phys. Rev. A 48, 4735-4741 (1993).
[CrossRef] [PubMed]

1991 (1)

R. Tanas, A. Miranowicz, and S. Kielich, "Squeezing and its graphical representations in the anharmonic oscillator model," Phys. Rev. A 43, 4014-4021 (1991).
[CrossRef] [PubMed]

1990 (1)

1989 (1)

S. Carusotto, "Time evolution in stimulated Compton scattering," Phys. Rev. A 39, 1848-1859 (1989).
[CrossRef] [PubMed]

1988 (2)

A. Luks, V. Perinová, and J. Perina, "Principal squeezing of vacuum fluctuations," Opt. Commun. 67, 149-151 (1988).
[CrossRef]

A. Luks, V. Perinová, and Z. Hradil, "Principal squeezing," Acta Phys. Pol. A 74, 713-721 (1988).

1987 (3)

H. J.Kimble and D. P. Walls, "Squeezed states of the electromagnetic field, " J. Opt. Soc. Am. B 4, (1987).

M. J. Collett and R. Loudon, "Output properties of parametric amplifier in cavities," J. Opt. Soc. Am. B 4, 1525-1533 (1987).
[CrossRef]

M. Hillery, "Squeezing of the square of the field amplitude in second harmonic generation," Opt. Commun. 62, 135-138 (1987).
[CrossRef]

1985 (2)

C. K. Hong and L. Mandel, "Higher-order squeezing of a quantum field," Phys. Rev. Lett. 54, 323-325 (1985).
[CrossRef] [PubMed]

C. K. Hong and L. Mandel, "Generation of higher-order squeezing of quantum electromagnetic fields," Phys. Rev. A 32, 974-982 (1985).
[CrossRef] [PubMed]

1984 (1)

M. J. Collett and C. W. Gardiner, "Squeezing of intracavity and traveling-wave light fields produced in parametric amplification," Phys. Rev. A 30, 1386-1391 (1984).
[CrossRef]

1982 (2)

L. A. Lugiato and G. Strini, "Fluctuations in multiphoton optical bistability," Opt. Commun. 41, 447-449 (1982).
[CrossRef]

L. A. Lugiato and G. Strini, "On the squeezing obtainable in parametric oscillators and bistable absorption," Opt. Commun. 41, 67-70 (1982).
[CrossRef]

1981 (7)

D. F. Walls and P. Zoller, "Reduced quantum fluctuations in resonance fluorescence," Phys. Rev. Lett. 47, 709-711 (1981).
[CrossRef]

G. J. Milburn and D. F. Walls, "Production of squeezed states in a degenerate parametric amplifier," Opt. Commun. 39, 401-404 (1981).
[CrossRef]

W. J. Mielniczuk and J. Chrostowski, "Incoherent effects in the optical up-conversion process," Phys. Rev. A 23, 1382-1389 (1981).
[CrossRef]

J. Katriel and D. G. Hummer, "Analytic solutions for three- and four-wave mixing via generalised Bose operators," J. Phys. A 14, 1211-1224 (1981).
[CrossRef]

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

V. Perinová, "Quantum statistics of quadratic optical parametric processes with intense coherent or stochastic pumping," Opt. Acta 28, 747-768 (1981).
[CrossRef]

V. Perinová and J. Perina, "Quantum statistics of quadratic optical parametric processes with intense coherent or stochastic pumping," Opt. Acta 28, 769-793 (1981).
[CrossRef]

1979 (1)

1976 (1)

H. P. Yuen, "Two-photon coherent states of the radiation field," Phys. Rev. A 13, 2226-2243 (1976).
[CrossRef]

1970 (1)

D. F. Walls and R. Barakat, "Quantum-mechanical amplification and frequency conversion with a trilinear Hamiltonian," Phys. Rev. A 1, 446-453 (1970).
[CrossRef]

1969 (3)

E. A. Mishkin and D. F. Walls, "Quantum statistics of three interacting boson field modes," Phys. Rev. 185, 1618-1628 (1969).
[CrossRef]

J. Tucker and D. F. Walls, "Quantum theory of parametric frequency conversion," Ann. Phys. (N.Y.) 52, 1-15 (1969).
[CrossRef]

M. Tavis and F. W. Cummings, "Approximate solutions for an N-molecule-radiation-field Hamiltonian," Phys. Rev. 188, 692-695 (1969).
[CrossRef]

1968 (2)

M. Tavis and F. W. Cummings, "Exact solution for an N-molecule—radiation-field Hamiltonian," Phys. Rev. 170, 379-384 (1968).
[CrossRef]

R. Graham, "Photon statistics of the optical parametric oscillator including the threshold region," Z. Phys. 211, 469-482 (1968).
[CrossRef]

1967 (2)

B. R. Mollow and R. J. Glauber, "Quantum theory of parametric amplification. I," Phys. Rev. 160, 1076-1096 (1967).
[CrossRef]

B. R. Mollow and R. J. Glauber, "Quantum theory of parametric amplification. II," Phys. Rev. 160, 1097-1108 (1967).
[CrossRef]

1965 (3)

J. A. Giordmaine and R. C. Miller, "Tunable coherent parametric oscillation in LiNbO3 at optical frequencies," Phys. Rev. Lett. 14, 973-976 (1965).
[CrossRef]

E. T. Jaynes and F. W. Cummings, "Comparison of quantum and semiclassical radiation theories with application to the beam maser," Proc. IEEE 51, 89-109 (1965).
[CrossRef]

F. W. Cummings, "Stimulated emission of radiation in a single mode," Phys. Rev. 140, A1051-A1056 (1965).
[CrossRef]

1961 (1)

W. H. Louisell, A. Yariv, and A. E. Siegman, "Quantum fluctuations and noise in parametric processes," Phys. Rev. 124, 1646-1654 (1961).
[CrossRef]

1940 (1)

T. Holstein and H. Primakoff, "Field dependence of the intrinsic domain magnetization of a ferromagnet," Phys. Rev. 58, 1098-1113 (1940).
[CrossRef]

Abdalla, M. S.

M. Abdel-Aty, M. S. Abdalla, and A.-S. F. Obada, "Entropy squeezing of a two-mode multiphoton Jaynes-Cummings model in the presence of a nonlinear medium," J. Opt. B 4, 134-142 (2002).
[CrossRef]

M. Abdel-Aty, M. S. Abdalla, and A.-S. F. Obada, "Entropy and phase properties of isotropic coupled oscillators interacting with a single atom: one- and two-photon processes," J. Opt. B 4, S133-S141 (2002).
[CrossRef]

F. A. A. El-Orany, J. Perina, and M. S. Abdalla, "Evolution of Fock states in three mixed harmonic oscillators: quantum statistics," Int. J. Mod. Phys. B 15, 2125-2146 (2001).
[CrossRef]

M. S. Abdalla, F. A. A. El-Orany, and J. Perina, "Statistical properties of a solvable three-Boson squeeze operator model," Eur. Phys. J. D 13, 423-438 (2001).
[CrossRef]

F. A. A. El-Orany, J. Perina, and M. S. Abdalla, "Statistical properties of three quantized interacting oscillators," Phys. Scr. 63, 128-140 (2001).
[CrossRef]

M. Abdel-Aty, M. S. Abdalla, and A.-S. F. Obada, "Quantum information and entropy squeezing of a two-level atom with a non-linear medium," J. Phys. A 34, 9129-9141 (2001).
[CrossRef]

F. A. A. El-Orany, J. Perina, and M. S. Abdalla, "Generation of squeezed light in a nonlinear asymmetric directional coupler," J. Opt. B 3, 67-75 (2001).
[CrossRef]

M. S. Abdalla, F. A. A. El-Orany, and J. Perina, "SU(2) and SU(1,1) squeezing of interacting radiation modes," Acta Phys. Slov. 50, 613-616 (2000).

M. S. Abdalla, F. A. A. El-Orany, and J. Perina, "Quantum statistical properties of nondegenerate optical parametric symmetric coupler," J. Phys. A 32, 3457-3483(1999).
[CrossRef]

Abdel-Aty, M.

M. Abdel-Aty, M. S. Abdalla, and A.-S. F. Obada, "Entropy squeezing of a two-mode multiphoton Jaynes-Cummings model in the presence of a nonlinear medium," J. Opt. B 4, 134-142 (2002).
[CrossRef]

M. Abdel-Aty, M. S. Abdalla, and A.-S. F. Obada, "Entropy and phase properties of isotropic coupled oscillators interacting with a single atom: one- and two-photon processes," J. Opt. B 4, S133-S141 (2002).
[CrossRef]

M. Abdel-Aty, M. S. Abdalla, and A.-S. F. Obada, "Quantum information and entropy squeezing of a two-level atom with a non-linear medium," J. Phys. A 34, 9129-9141 (2001).
[CrossRef]

Bandilla, A.

A. Bandilla, G. Drobny, and I. Jex, "Nondegenerate parametric interactions and nonclassical effects," Phys. Rev. A 53, 507-516 (1996).
[CrossRef] [PubMed]

Barakat, R.

D. F. Walls and R. Barakat, "Quantum-mechanical amplification and frequency conversion with a trilinear Hamiltonian," Phys. Rev. A 1, 446-453 (1970).
[CrossRef]

Berglund, A. J.

D. S. Naik, C. G. Peterson, A. G. White, A. J. Berglund, and P. G. Kwiat, "Entangled state quantum cryptography: eavesdropping on the Ekert protocol," Phys. Rev. Lett. 84, 4733-4736 (2000).
[CrossRef] [PubMed]

Bloembergen, N.

N. Bloembergen, Nonlinear Optics (Benjamin, 1965).

Bouwmeester, D.

D. Bouwmeester, J.-W. Pan, M. Daniell, H. Weinfurter, and A. Zeilinger, "Observation of three-photon Greenberger-Horne-Zeilinger entanglement," Phys. Rev. Lett. 82, 1345-1349 (1999).
[CrossRef]

D. Bouwmeester, Jian-Weipan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, "Experimental quantum teleportation," Nature 390, 575-579 (1997).
[CrossRef]

D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Information (Springer-Verlag, 2000).

Brendel, J.

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, "Quantum cryptography using entangled photons in energy-time Bell states," Phys. Rev. Lett. 84, 4737-4740 (2000).
[CrossRef] [PubMed]

Carusotto, S.

S. Carusotto, "Time evolution in stimulated Compton scattering," Phys. Rev. A 39, 1848-1859 (1989).
[CrossRef] [PubMed]

Caves, C. M.

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

Chrostowski, J.

W. J. Mielniczuk and J. Chrostowski, "Incoherent effects in the optical up-conversion process," Phys. Rev. A 23, 1382-1389 (1981).
[CrossRef]

Chumakov, S. M.

M. Kozierowski and S. M. Chumakov, "Revivals in the off-resonant Jaynes-Cummings model with a Kerr medium," Acta Phys. Slov. 47, 301-310 (1997).

Collett, M. J.

M. J. Collett and R. Loudon, "Output properties of parametric amplifier in cavities," J. Opt. Soc. Am. B 4, 1525-1533 (1987).
[CrossRef]

M. J. Collett and C. W. Gardiner, "Squeezing of intracavity and traveling-wave light fields produced in parametric amplification," Phys. Rev. A 30, 1386-1391 (1984).
[CrossRef]

Cummings, F. W.

M. Tavis and F. W. Cummings, "Approximate solutions for an N-molecule-radiation-field Hamiltonian," Phys. Rev. 188, 692-695 (1969).
[CrossRef]

M. Tavis and F. W. Cummings, "Exact solution for an N-molecule—radiation-field Hamiltonian," Phys. Rev. 170, 379-384 (1968).
[CrossRef]

E. T. Jaynes and F. W. Cummings, "Comparison of quantum and semiclassical radiation theories with application to the beam maser," Proc. IEEE 51, 89-109 (1965).
[CrossRef]

F. W. Cummings, "Stimulated emission of radiation in a single mode," Phys. Rev. 140, A1051-A1056 (1965).
[CrossRef]

Daniell, M.

J. W. Pan, M. Daniell, S. Gasparoni, G. Weihs, and A. Zeilinger, "Experimental demonstration of four-photon entanglement and high-fidelity teleportation," Phys. Rev. Lett. 86, 4435-4438 (2001).
[CrossRef] [PubMed]

D. Bouwmeester, J.-W. Pan, M. Daniell, H. Weinfurter, and A. Zeilinger, "Observation of three-photon Greenberger-Horne-Zeilinger entanglement," Phys. Rev. Lett. 82, 1345-1349 (1999).
[CrossRef]

Drobny, G.

A. Bandilla, G. Drobny, and I. Jex, "Nondegenerate parametric interactions and nonclassical effects," Phys. Rev. A 53, 507-516 (1996).
[CrossRef] [PubMed]

Eibl, M.

D. Bouwmeester, Jian-Weipan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, "Experimental quantum teleportation," Nature 390, 575-579 (1997).
[CrossRef]

Ekert, A.

D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Information (Springer-Verlag, 2000).

El-Orany, F. A. A.

F. A. A. El-Orany, J. Perina, and M. S. Abdalla, "Statistical properties of three quantized interacting oscillators," Phys. Scr. 63, 128-140 (2001).
[CrossRef]

F. A. A. El-Orany, J. Perina, and M. S. Abdalla, "Evolution of Fock states in three mixed harmonic oscillators: quantum statistics," Int. J. Mod. Phys. B 15, 2125-2146 (2001).
[CrossRef]

F. A. A. El-Orany, J. Perina, and M. S. Abdalla, "Generation of squeezed light in a nonlinear asymmetric directional coupler," J. Opt. B 3, 67-75 (2001).
[CrossRef]

M. S. Abdalla, F. A. A. El-Orany, and J. Perina, "Statistical properties of a solvable three-Boson squeeze operator model," Eur. Phys. J. D 13, 423-438 (2001).
[CrossRef]

M. S. Abdalla, F. A. A. El-Orany, and J. Perina, "SU(2) and SU(1,1) squeezing of interacting radiation modes," Acta Phys. Slov. 50, 613-616 (2000).

M. S. Abdalla, F. A. A. El-Orany, and J. Perina, "Quantum statistical properties of nondegenerate optical parametric symmetric coupler," J. Phys. A 32, 3457-3483(1999).
[CrossRef]

Gardiner, C. W.

M. J. Collett and C. W. Gardiner, "Squeezing of intracavity and traveling-wave light fields produced in parametric amplification," Phys. Rev. A 30, 1386-1391 (1984).
[CrossRef]

Gasparoni, S.

J. W. Pan, M. Daniell, S. Gasparoni, G. Weihs, and A. Zeilinger, "Experimental demonstration of four-photon entanglement and high-fidelity teleportation," Phys. Rev. Lett. 86, 4435-4438 (2001).
[CrossRef] [PubMed]

Giordmaine, J. A.

J. A. Giordmaine and R. C. Miller, "Tunable coherent parametric oscillation in LiNbO3 at optical frequencies," Phys. Rev. Lett. 14, 973-976 (1965).
[CrossRef]

Gisin, N.

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, "Quantum cryptography using entangled photons in energy-time Bell states," Phys. Rev. Lett. 84, 4737-4740 (2000).
[CrossRef] [PubMed]

Glauber, R. J.

B. R. Mollow and R. J. Glauber, "Quantum theory of parametric amplification. I," Phys. Rev. 160, 1076-1096 (1967).
[CrossRef]

B. R. Mollow and R. J. Glauber, "Quantum theory of parametric amplification. II," Phys. Rev. 160, 1097-1108 (1967).
[CrossRef]

Graham, R.

R. Graham, "Photon statistics of the optical parametric oscillator including the threshold region," Z. Phys. 211, 469-482 (1968).
[CrossRef]

Hillery, M.

M. Hillery, "Squeezing of the square of the field amplitude in second harmonic generation," Opt. Commun. 62, 135-138 (1987).
[CrossRef]

Holstein, T.

T. Holstein and H. Primakoff, "Field dependence of the intrinsic domain magnetization of a ferromagnet," Phys. Rev. 58, 1098-1113 (1940).
[CrossRef]

Hong, C. K.

C. K. Hong and L. Mandel, "Higher-order squeezing of a quantum field," Phys. Rev. Lett. 54, 323-325 (1985).
[CrossRef] [PubMed]

C. K. Hong and L. Mandel, "Generation of higher-order squeezing of quantum electromagnetic fields," Phys. Rev. A 32, 974-982 (1985).
[CrossRef] [PubMed]

Hradil, Z.

A. Luks, V. Perinová, and Z. Hradil, "Principal squeezing," Acta Phys. Pol. A 74, 713-721 (1988).

Hummer, D. G.

J. Katriel and D. G. Hummer, "Analytic solutions for three- and four-wave mixing via generalised Bose operators," J. Phys. A 14, 1211-1224 (1981).
[CrossRef]

Jaynes, E. T.

E. T. Jaynes and F. W. Cummings, "Comparison of quantum and semiclassical radiation theories with application to the beam maser," Proc. IEEE 51, 89-109 (1965).
[CrossRef]

Jex, I.

A. Bandilla, G. Drobny, and I. Jex, "Nondegenerate parametric interactions and nonclassical effects," Phys. Rev. A 53, 507-516 (1996).
[CrossRef] [PubMed]

Jian-Weipan,

D. Bouwmeester, Jian-Weipan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, "Experimental quantum teleportation," Nature 390, 575-579 (1997).
[CrossRef]

Katriel, J.

J. Katriel and D. G. Hummer, "Analytic solutions for three- and four-wave mixing via generalised Bose operators," J. Phys. A 14, 1211-1224 (1981).
[CrossRef]

Kielich, S.

R. Tanas, A. Miranowicz, and S. Kielich, "Squeezing and its graphical representations in the anharmonic oscillator model," Phys. Rev. A 43, 4014-4021 (1991).
[CrossRef] [PubMed]

Kimble, H. J.

H. J.Kimble and D. P. Walls, "Squeezed states of the electromagnetic field, " J. Opt. Soc. Am. B 4, (1987).

Kozierowski, M.

M. Kozierowski, "Thermal and squeezed vacuum Jaynes-Cummings models with a Kerr medium," J. Mol. Spectrosc. 48, 773-781 (2001).

M. Kozierowski, "Second-order collapses and revivals," Acta Phys. Slov. 48, 281-292 (1998).

M. Kozierowski and S. M. Chumakov, "Revivals in the off-resonant Jaynes-Cummings model with a Kerr medium," Acta Phys. Slov. 47, 301-310 (1997).

Kwiat, P. G.

D. S. Naik, C. G. Peterson, A. G. White, A. J. Berglund, and P. G. Kwiat, "Entangled state quantum cryptography: eavesdropping on the Ekert protocol," Phys. Rev. Lett. 84, 4733-4736 (2000).
[CrossRef] [PubMed]

K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, "Dense Coding in Experimental Quantum Communication," Phys. Rev. Lett. 76, 4656-4659 (1996).
[CrossRef] [PubMed]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, "New high-intensity source of polarization-entangled photon pairs," Phys. Rev. Lett. 75, 4337-4341 (1995).
[CrossRef] [PubMed]

Loudon, R.

Louisell, W. H.

W. H. Louisell, A. Yariv, and A. E. Siegman, "Quantum fluctuations and noise in parametric processes," Phys. Rev. 124, 1646-1654 (1961).
[CrossRef]

Lugiato, L. A.

L. A. Lugiato and G. Strini, "Fluctuations in multiphoton optical bistability," Opt. Commun. 41, 447-449 (1982).
[CrossRef]

L. A. Lugiato and G. Strini, "On the squeezing obtainable in parametric oscillators and bistable absorption," Opt. Commun. 41, 67-70 (1982).
[CrossRef]

Luks, A.

A. Luks, V. Perinová, and Z. Hradil, "Principal squeezing," Acta Phys. Pol. A 74, 713-721 (1988).

A. Luks, V. Perinová, and J. Perina, "Principal squeezing of vacuum fluctuations," Opt. Commun. 67, 149-151 (1988).
[CrossRef]

Mandel, L.

Z. Y. Ou, L. J. Wang, and L. Mandel, "Photon amplification by parametric downconversion," J. Opt. Soc. Am. B 7, 211-214 (1990).
[CrossRef]

C. K. Hong and L. Mandel, "Generation of higher-order squeezing of quantum electromagnetic fields," Phys. Rev. A 32, 974-982 (1985).
[CrossRef] [PubMed]

C. K. Hong and L. Mandel, "Higher-order squeezing of a quantum field," Phys. Rev. Lett. 54, 323-325 (1985).
[CrossRef] [PubMed]

Mattle, K.

D. Bouwmeester, Jian-Weipan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, "Experimental quantum teleportation," Nature 390, 575-579 (1997).
[CrossRef]

K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, "Dense Coding in Experimental Quantum Communication," Phys. Rev. Lett. 76, 4656-4659 (1996).
[CrossRef] [PubMed]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, "New high-intensity source of polarization-entangled photon pairs," Phys. Rev. Lett. 75, 4337-4341 (1995).
[CrossRef] [PubMed]

Mielniczuk, W. J.

W. J. Mielniczuk and J. Chrostowski, "Incoherent effects in the optical up-conversion process," Phys. Rev. A 23, 1382-1389 (1981).
[CrossRef]

Milburn, G. J.

G. J. Milburn and D. F. Walls, "Production of squeezed states in a degenerate parametric amplifier," Opt. Commun. 39, 401-404 (1981).
[CrossRef]

Miller, R. C.

J. A. Giordmaine and R. C. Miller, "Tunable coherent parametric oscillation in LiNbO3 at optical frequencies," Phys. Rev. Lett. 14, 973-976 (1965).
[CrossRef]

Miranowicz, A.

R. Tanas, A. Miranowicz, and S. Kielich, "Squeezing and its graphical representations in the anharmonic oscillator model," Phys. Rev. A 43, 4014-4021 (1991).
[CrossRef] [PubMed]

Mishkin, E. A.

E. A. Mishkin and D. F. Walls, "Quantum statistics of three interacting boson field modes," Phys. Rev. 185, 1618-1628 (1969).
[CrossRef]

Mollow, B. R.

B. R. Mollow and R. J. Glauber, "Quantum theory of parametric amplification. I," Phys. Rev. 160, 1076-1096 (1967).
[CrossRef]

B. R. Mollow and R. J. Glauber, "Quantum theory of parametric amplification. II," Phys. Rev. 160, 1097-1108 (1967).
[CrossRef]

Naik, D. S.

D. S. Naik, C. G. Peterson, A. G. White, A. J. Berglund, and P. G. Kwiat, "Entangled state quantum cryptography: eavesdropping on the Ekert protocol," Phys. Rev. Lett. 84, 4733-4736 (2000).
[CrossRef] [PubMed]

Obada, A.-S. F.

M. Abdel-Aty, M. S. Abdalla, and A.-S. F. Obada, "Entropy squeezing of a two-mode multiphoton Jaynes-Cummings model in the presence of a nonlinear medium," J. Opt. B 4, 134-142 (2002).
[CrossRef]

M. Abdel-Aty, M. S. Abdalla, and A.-S. F. Obada, "Entropy and phase properties of isotropic coupled oscillators interacting with a single atom: one- and two-photon processes," J. Opt. B 4, S133-S141 (2002).
[CrossRef]

M. Abdel-Aty, M. S. Abdalla, and A.-S. F. Obada, "Quantum information and entropy squeezing of a two-level atom with a non-linear medium," J. Phys. A 34, 9129-9141 (2001).
[CrossRef]

Ou, Z. Y.

Pan, J. W.

J. W. Pan, M. Daniell, S. Gasparoni, G. Weihs, and A. Zeilinger, "Experimental demonstration of four-photon entanglement and high-fidelity teleportation," Phys. Rev. Lett. 86, 4435-4438 (2001).
[CrossRef] [PubMed]

Pan, J.-W.

D. Bouwmeester, J.-W. Pan, M. Daniell, H. Weinfurter, and A. Zeilinger, "Observation of three-photon Greenberger-Horne-Zeilinger entanglement," Phys. Rev. Lett. 82, 1345-1349 (1999).
[CrossRef]

Perina, J.

F. A. A. El-Orany, J. Perina, and M. S. Abdalla, "Statistical properties of three quantized interacting oscillators," Phys. Scr. 63, 128-140 (2001).
[CrossRef]

M. S. Abdalla, F. A. A. El-Orany, and J. Perina, "Statistical properties of a solvable three-Boson squeeze operator model," Eur. Phys. J. D 13, 423-438 (2001).
[CrossRef]

F. A. A. El-Orany, J. Perina, and M. S. Abdalla, "Generation of squeezed light in a nonlinear asymmetric directional coupler," J. Opt. B 3, 67-75 (2001).
[CrossRef]

F. A. A. El-Orany, J. Perina, and M. S. Abdalla, "Evolution of Fock states in three mixed harmonic oscillators: quantum statistics," Int. J. Mod. Phys. B 15, 2125-2146 (2001).
[CrossRef]

M. S. Abdalla, F. A. A. El-Orany, and J. Perina, "SU(2) and SU(1,1) squeezing of interacting radiation modes," Acta Phys. Slov. 50, 613-616 (2000).

M. S. Abdalla, F. A. A. El-Orany, and J. Perina, "Quantum statistical properties of nondegenerate optical parametric symmetric coupler," J. Phys. A 32, 3457-3483(1999).
[CrossRef]

A. Luks, V. Perinová, and J. Perina, "Principal squeezing of vacuum fluctuations," Opt. Commun. 67, 149-151 (1988).
[CrossRef]

V. Perinová and J. Perina, "Quantum statistics of quadratic optical parametric processes with intense coherent or stochastic pumping," Opt. Acta 28, 769-793 (1981).
[CrossRef]

J. Perina, Quantum Statistics of Linear and Nonlinear Optical Phenomena, 2nd ed. (Kluwer, 1991).

Perinová, V.

A. Luks, V. Perinová, and J. Perina, "Principal squeezing of vacuum fluctuations," Opt. Commun. 67, 149-151 (1988).
[CrossRef]

A. Luks, V. Perinová, and Z. Hradil, "Principal squeezing," Acta Phys. Pol. A 74, 713-721 (1988).

V. Perinová, "Quantum statistics of quadratic optical parametric processes with intense coherent or stochastic pumping," Opt. Acta 28, 747-768 (1981).
[CrossRef]

V. Perinová and J. Perina, "Quantum statistics of quadratic optical parametric processes with intense coherent or stochastic pumping," Opt. Acta 28, 769-793 (1981).
[CrossRef]

Peterson, C. G.

D. S. Naik, C. G. Peterson, A. G. White, A. J. Berglund, and P. G. Kwiat, "Entangled state quantum cryptography: eavesdropping on the Ekert protocol," Phys. Rev. Lett. 84, 4733-4736 (2000).
[CrossRef] [PubMed]

Primakoff, H.

T. Holstein and H. Primakoff, "Field dependence of the intrinsic domain magnetization of a ferromagnet," Phys. Rev. 58, 1098-1113 (1940).
[CrossRef]

Schwinger, J.

J. Schwinger, Quantum Theory of Angular Momentum, L.C.Bidenharn and H.Van Dam, eds. (Academic, 1965), pp. 229-279.

Sergienko, A. V.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, "New high-intensity source of polarization-entangled photon pairs," Phys. Rev. Lett. 75, 4337-4341 (1995).
[CrossRef] [PubMed]

Shapiro, J. H.

Shih, Y. H.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, "New high-intensity source of polarization-entangled photon pairs," Phys. Rev. Lett. 75, 4337-4341 (1995).
[CrossRef] [PubMed]

Shumovsky, A. S.

A. S. Shumovsky and B. Tanatar, "Quantum-statistical properties of a Raman-type model," Phys. Rev. A 48, 4735-4741 (1993).
[CrossRef] [PubMed]

Siegman, A. E.

W. H. Louisell, A. Yariv, and A. E. Siegman, "Quantum fluctuations and noise in parametric processes," Phys. Rev. 124, 1646-1654 (1961).
[CrossRef]

Strini, G.

L. A. Lugiato and G. Strini, "On the squeezing obtainable in parametric oscillators and bistable absorption," Opt. Commun. 41, 67-70 (1982).
[CrossRef]

L. A. Lugiato and G. Strini, "Fluctuations in multiphoton optical bistability," Opt. Commun. 41, 447-449 (1982).
[CrossRef]

Tanas, R.

R. Tanas, A. Miranowicz, and S. Kielich, "Squeezing and its graphical representations in the anharmonic oscillator model," Phys. Rev. A 43, 4014-4021 (1991).
[CrossRef] [PubMed]

Tanatar, B.

A. S. Shumovsky and B. Tanatar, "Quantum-statistical properties of a Raman-type model," Phys. Rev. A 48, 4735-4741 (1993).
[CrossRef] [PubMed]

Tavis, M.

M. Tavis and F. W. Cummings, "Approximate solutions for an N-molecule-radiation-field Hamiltonian," Phys. Rev. 188, 692-695 (1969).
[CrossRef]

M. Tavis and F. W. Cummings, "Exact solution for an N-molecule—radiation-field Hamiltonian," Phys. Rev. 170, 379-384 (1968).
[CrossRef]

Tittel, W.

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, "Quantum cryptography using entangled photons in energy-time Bell states," Phys. Rev. Lett. 84, 4737-4740 (2000).
[CrossRef] [PubMed]

Tucker, J.

J. Tucker and D. F. Walls, "Quantum theory of parametric frequency conversion," Ann. Phys. (N.Y.) 52, 1-15 (1969).
[CrossRef]

Walls, D. F.

G. J. Milburn and D. F. Walls, "Production of squeezed states in a degenerate parametric amplifier," Opt. Commun. 39, 401-404 (1981).
[CrossRef]

D. F. Walls and P. Zoller, "Reduced quantum fluctuations in resonance fluorescence," Phys. Rev. Lett. 47, 709-711 (1981).
[CrossRef]

D. F. Walls and R. Barakat, "Quantum-mechanical amplification and frequency conversion with a trilinear Hamiltonian," Phys. Rev. A 1, 446-453 (1970).
[CrossRef]

J. Tucker and D. F. Walls, "Quantum theory of parametric frequency conversion," Ann. Phys. (N.Y.) 52, 1-15 (1969).
[CrossRef]

E. A. Mishkin and D. F. Walls, "Quantum statistics of three interacting boson field modes," Phys. Rev. 185, 1618-1628 (1969).
[CrossRef]

Walls, D. P.

H. J.Kimble and D. P. Walls, "Squeezed states of the electromagnetic field, " J. Opt. Soc. Am. B 4, (1987).

Wang, L. J.

Weihs, G.

J. W. Pan, M. Daniell, S. Gasparoni, G. Weihs, and A. Zeilinger, "Experimental demonstration of four-photon entanglement and high-fidelity teleportation," Phys. Rev. Lett. 86, 4435-4438 (2001).
[CrossRef] [PubMed]

Weinfurter, H.

D. Bouwmeester, J.-W. Pan, M. Daniell, H. Weinfurter, and A. Zeilinger, "Observation of three-photon Greenberger-Horne-Zeilinger entanglement," Phys. Rev. Lett. 82, 1345-1349 (1999).
[CrossRef]

D. Bouwmeester, Jian-Weipan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, "Experimental quantum teleportation," Nature 390, 575-579 (1997).
[CrossRef]

K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, "Dense Coding in Experimental Quantum Communication," Phys. Rev. Lett. 76, 4656-4659 (1996).
[CrossRef] [PubMed]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, "New high-intensity source of polarization-entangled photon pairs," Phys. Rev. Lett. 75, 4337-4341 (1995).
[CrossRef] [PubMed]

White, A. G.

D. S. Naik, C. G. Peterson, A. G. White, A. J. Berglund, and P. G. Kwiat, "Entangled state quantum cryptography: eavesdropping on the Ekert protocol," Phys. Rev. Lett. 84, 4733-4736 (2000).
[CrossRef] [PubMed]

Yariv, A.

W. H. Louisell, A. Yariv, and A. E. Siegman, "Quantum fluctuations and noise in parametric processes," Phys. Rev. 124, 1646-1654 (1961).
[CrossRef]

Yuen, H. P.

Zbinden, H.

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, "Quantum cryptography using entangled photons in energy-time Bell states," Phys. Rev. Lett. 84, 4737-4740 (2000).
[CrossRef] [PubMed]

Zeilinger, A.

J. W. Pan, M. Daniell, S. Gasparoni, G. Weihs, and A. Zeilinger, "Experimental demonstration of four-photon entanglement and high-fidelity teleportation," Phys. Rev. Lett. 86, 4435-4438 (2001).
[CrossRef] [PubMed]

D. Bouwmeester, J.-W. Pan, M. Daniell, H. Weinfurter, and A. Zeilinger, "Observation of three-photon Greenberger-Horne-Zeilinger entanglement," Phys. Rev. Lett. 82, 1345-1349 (1999).
[CrossRef]

D. Bouwmeester, Jian-Weipan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, "Experimental quantum teleportation," Nature 390, 575-579 (1997).
[CrossRef]

K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, "Dense Coding in Experimental Quantum Communication," Phys. Rev. Lett. 76, 4656-4659 (1996).
[CrossRef] [PubMed]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, "New high-intensity source of polarization-entangled photon pairs," Phys. Rev. Lett. 75, 4337-4341 (1995).
[CrossRef] [PubMed]

D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Information (Springer-Verlag, 2000).

Zoller, P.

D. F. Walls and P. Zoller, "Reduced quantum fluctuations in resonance fluorescence," Phys. Rev. Lett. 47, 709-711 (1981).
[CrossRef]

Acta Phys. Pol. A (1)

A. Luks, V. Perinová, and Z. Hradil, "Principal squeezing," Acta Phys. Pol. A 74, 713-721 (1988).

Acta Phys. Slov. (3)

M. Kozierowski and S. M. Chumakov, "Revivals in the off-resonant Jaynes-Cummings model with a Kerr medium," Acta Phys. Slov. 47, 301-310 (1997).

M. Kozierowski, "Second-order collapses and revivals," Acta Phys. Slov. 48, 281-292 (1998).

M. S. Abdalla, F. A. A. El-Orany, and J. Perina, "SU(2) and SU(1,1) squeezing of interacting radiation modes," Acta Phys. Slov. 50, 613-616 (2000).

Ann. Phys. (N.Y.) (1)

J. Tucker and D. F. Walls, "Quantum theory of parametric frequency conversion," Ann. Phys. (N.Y.) 52, 1-15 (1969).
[CrossRef]

Eur. Phys. J. D (1)

M. S. Abdalla, F. A. A. El-Orany, and J. Perina, "Statistical properties of a solvable three-Boson squeeze operator model," Eur. Phys. J. D 13, 423-438 (2001).
[CrossRef]

Int. J. Mod. Phys. B (1)

F. A. A. El-Orany, J. Perina, and M. S. Abdalla, "Evolution of Fock states in three mixed harmonic oscillators: quantum statistics," Int. J. Mod. Phys. B 15, 2125-2146 (2001).
[CrossRef]

J. Mol. Spectrosc. (1)

M. Kozierowski, "Thermal and squeezed vacuum Jaynes-Cummings models with a Kerr medium," J. Mol. Spectrosc. 48, 773-781 (2001).

J. Opt. B (3)

F. A. A. El-Orany, J. Perina, and M. S. Abdalla, "Generation of squeezed light in a nonlinear asymmetric directional coupler," J. Opt. B 3, 67-75 (2001).
[CrossRef]

M. Abdel-Aty, M. S. Abdalla, and A.-S. F. Obada, "Entropy squeezing of a two-mode multiphoton Jaynes-Cummings model in the presence of a nonlinear medium," J. Opt. B 4, 134-142 (2002).
[CrossRef]

M. Abdel-Aty, M. S. Abdalla, and A.-S. F. Obada, "Entropy and phase properties of isotropic coupled oscillators interacting with a single atom: one- and two-photon processes," J. Opt. B 4, S133-S141 (2002).
[CrossRef]

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

J. Phys. A (3)

J. Katriel and D. G. Hummer, "Analytic solutions for three- and four-wave mixing via generalised Bose operators," J. Phys. A 14, 1211-1224 (1981).
[CrossRef]

M. Abdel-Aty, M. S. Abdalla, and A.-S. F. Obada, "Quantum information and entropy squeezing of a two-level atom with a non-linear medium," J. Phys. A 34, 9129-9141 (2001).
[CrossRef]

M. S. Abdalla, F. A. A. El-Orany, and J. Perina, "Quantum statistical properties of nondegenerate optical parametric symmetric coupler," J. Phys. A 32, 3457-3483(1999).
[CrossRef]

Nature (1)

D. Bouwmeester, Jian-Weipan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, "Experimental quantum teleportation," Nature 390, 575-579 (1997).
[CrossRef]

Opt. Acta (2)

V. Perinová, "Quantum statistics of quadratic optical parametric processes with intense coherent or stochastic pumping," Opt. Acta 28, 747-768 (1981).
[CrossRef]

V. Perinová and J. Perina, "Quantum statistics of quadratic optical parametric processes with intense coherent or stochastic pumping," Opt. Acta 28, 769-793 (1981).
[CrossRef]

Opt. Commun. (5)

M. Hillery, "Squeezing of the square of the field amplitude in second harmonic generation," Opt. Commun. 62, 135-138 (1987).
[CrossRef]

A. Luks, V. Perinová, and J. Perina, "Principal squeezing of vacuum fluctuations," Opt. Commun. 67, 149-151 (1988).
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Figures (9)

Fig. 1
Fig. 1

Evolution of principle squeeze parameters for compound modes ( 1 , 2 ) , ( 1 , 3 ) , and ( 2 , 3 ) . Left column (a), (b) for n 0 = 0 , n ¯ 0 = 1 ; right column (c), (d) for n 0 = 50 , n ¯ 0 = 10 ; upper row for m = 2 ; and bottom row for m = 3 . g μ = 1 , g 2 g = 0.1 , g 1 g = [ 1 + ( g 2 g ) 2 ] 1 2 ; α 1 ( 0 ) = 5 i , α 2 ( 0 ) = 4 , and α 3 ( 0 ) = 2.5 .

Fig. 2
Fig. 2

Evolution of variances for compound modes ( 1 , 2 ) , ( 1 , 3 ) , and ( 2 , 3 ) . Left (a) for m = 2 and right (b) for m = 3 ; n 0 = 50 , n ¯ 0 = 10 , other parameters as in Fig. 1.

Fig. 3
Fig. 3

Evolution of factorial moments for the compound modes ( 1 , 2 ) , ( 1 , 3 ) , and ( 2 , 3 ) in the succession from upper to lower, m = 3 , n 0 = 50 , n ¯ 0 = 10 ; all for other parameters as in Fig. 1. Left column (a), (b), (c) for m = 2 ; right column (d), (e), (f) for m = 3 .

Fig. 4
Fig. 4

Phase-averaged joint quasi-distribution for the mode ( 2 , 3 ) . Left (a) for m = 2 (the minimal variance is about 3.989 when μ t = 0.106 ) and right (b) for m = 3 (the minimal variance is about 3.988 when μ t = 0.0052 ), n 0 = 50 , n ¯ 0 = 10 ; other parameters as in Fig. 1.

Fig. 5
Fig. 5

Evolution of the photon-number distribution for the compound modes ( 1 , 2 ) , ( 1 , 3 ) , and ( 2 , 3 ) in the succession from upper to lower, n 0 = 50 , n ¯ 0 = 10 ; other parameters as in Fig. 1. Left column for m = 2 , right column for m = 3 .

Fig. 6
Fig. 6

Evolution of the differences of actual photon-number distributions and corresponding Poissonian distributions for the compound modes ( 1 , 2 ) , ( 1 , 3 ) , and ( 2 , 3 ) in the succession from upper to lower, m = 3 , n 0 = 50 , n ¯ 0 = 10 ; for other parameters as in Fig. 1. Left column for m = 2 , right column for m = 3 .

Fig. 7
Fig. 7

Variances for compound modes ( 1 , 2 ) , ( 1 , 3 ) , and ( 2 , 3 ) ; left for m = 2 and right for m = 3 ; n 0 = 50 , n ¯ 0 = 10 , g μ = 1 , g 2 g = 0.1 , g 1 g = [ 1 + ( g 2 g ) 2 ] 1 2 ; α 1 ( 0 ) = α 2 ( 0 ) = α 3 ( 0 ) = 0 .

Fig. 8
Fig. 8

Evolution of factorial moments for the compound modes ( 1 , 2 ) , ( 1 , 3 ) , and ( 2 , 3 ) in the succession from upper to lower, all for the values of parameters as in Fig. 7. Left column for m = 2 , right column for m = 3 .

Fig. 9
Fig. 9

Scaling curves for higher-order transitions, g μ = 1 . Left for n 0 = 0 , n ¯ 0 = 1 , right for n 0 = 50 , n ¯ 0 = 10 .

Equations (75)

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H ̂ = j = 1 3 ω ¯ j a ̂ j a ̂ j + g ¯ 1 { a ̂ 2 a ̂ 1 m exp [ i ( m ω ¯ 1 ω ¯ 2 ) t ] + a ̂ 2 a ̂ 1 m exp [ i ( m ω ¯ 1 ω ¯ 2 ) t ] } + g ¯ 2 { a ̂ 3 a ̂ 1 m exp [ i ( ω ¯ 3 + m ω ¯ 1 ) t ] + a ̂ 3 a ̂ 1 m exp [ i ( ω ¯ 3 + m ω ¯ 1 ) t ] } ,
V I = e r ¯ 1     E ¯ 1 e r ¯ 2         E ¯ 2 ,
V I = g λ [ ( σ ̂ ( 1 ) + σ ̂ + ( 1 ) ) + ( σ ̂ ( 2 ) + σ ̂ + ( 2 ) ) ] ( a ̂ λ + a ̂ λ ) ,
V I = g 1 ( σ ̂ ( 1 ) a ̂ λ + σ ̂ + ( 1 ) a ̂ λ ) + g 2 ( σ ̂ ( 2 ) a ̂ λ + σ ̂ + ( 2 ) a ̂ λ ) ,
V I = g 1 ( J ̂ ( 1 ) a ̂ 1 m + J ̂ + ( 1 ) a ̂ 1 m ) + g 2 ( J ̂ ( 2 ) a ̂ 1 m + J ̂ + ( 2 ) a ̂ 1 m ) ,
H ̂ = ω a ̂ 1 a ̂ 1 + 1 2 [ ω 0 ( 1 ) J ̂ z ( 1 ) + ω 0 ( 2 ) J ̂ z ( 2 ) ] + g 1 ( J ̂ ( 1 ) a ̂ 1 m + J ̂ + ( 1 ) a ̂ 1 m ) + g 2 ( J ̂ ( 2 ) a ̂ 1 m + J ̂ + ( 2 ) a ̂ 1 m ) ,
J ̂ + ( 1 ) = a ̂ 2 a ̂ 3 , J ̂ ( 1 ) = a ̂ 2 a ̂ 3 , J ̂ z ( 1 ) = ( a ̂ 2 a ̂ 2 a ̂ 3 a ̂ 3 ) ,
J ̂ ( 2 ) = a ̂ 4 a ̂ 5 , J ̂ + ( 2 ) = a ̂ 4 a ̂ 5 , J ̂ z ( 2 ) = ( a ̂ 4 a ̂ 4 a ̂ 5 a ̂ 5 ) ,
H ̂ = ω a ̂ 1 a ̂ 1 + 1 2 [ ω 0 ( 1 ) ( a ̂ 2 a ̂ 2 a ̂ 3 a ̂ 3 ) + ω 0 ( 2 ) ( a ̂ 4 a ̂ 4 a ̂ 5 a ̂ 5 ) ] + g 1 ( a ̂ 2 a ̂ 3 a ̂ 1 m + a ̂ 2 a ̂ 3 a ̂ 1 m ) + g 2 ( a ̂ 4 a ̂ 5 a ̂ 1 m + a ̂ 4 a ̂ 5 a ̂ 1 m ) .
a ̂ 1 m = b ̂ 1 f ( n ̂ 1 ) , a ̂ 1 m = f ( n ̂ 1 ) b ̂ 1 ,
H ̂ ( t ) = i = 1 3 ω i b ̂ i b ̂ i + λ 1 ( t ) { b ̂ 1 b ̂ 2 exp [ i Γ 1 ( t ) ] + b ̂ 1 b ̂ 2 exp [ i Γ 1 ( t ) ] } + λ 2 ( t ) { b ̂ 1 b ̂ 3 exp [ i Γ 2 ( t ) ] + b ̂ 1 b ̂ 3 exp [ i Γ 2 ( t ) ] } ,
λ 1 ( t ) = g 1 f ( n ̂ 1 ) , λ 2 ( t ) = g 2 f ( n ̂ 1 ) .
f ( n ̂ 1 ) = { Γ [ m ( n ̂ 1 + 1 ) + 1 ] ( n ̂ 1 + 1 ) Γ ( m n ̂ 1 + 1 ) } 1 2 ,
d b ̂ 1 d t = i ω 1 b ̂ 1 i λ 1 ( t ) b ̂ 2 exp [ i Γ 1 ( t ) ] i λ 2 ( t ) b ̂ 3 exp [ i Γ 2 ( t ) ] ,
d b ̂ 2 d t = i ω 2 b ̂ 2 i λ 1 ( t ) b ̂ 1 exp [ i Γ 1 ( t ) ] ,
d b ̂ 3 d t = i ω 3 b ̂ 3 i λ 2 ( t ) b ̂ 1 exp [ i Γ 2 ( t ) ] .
b ̂ 1 = B ̂ 1 exp [ i ( ω 1 t ϕ 1 ) ] , b ̂ 2 = B ̂ 2 exp [ i ω 2 t ] ,
b ̂ 3 = B ̂ 3 exp [ i ( ω 3 t + ϕ 1 + ϕ 2 ) ] ,
d 2 B ̂ 1 d t 2 λ ̇ 1 λ 1 d B ̂ 1 d t + ( λ 1 2 λ 2 2 ) B ̂ 1 = 0 ,
B ̂ 1 ( t ) = K ̂ 1 cos Ω ( t ) + K ̂ 2 sin Ω ( t ) , Ω ( t ) = g 0 t f ( n ̂ 1 ) d t ,
b ̂ 1 ( t ) = exp ( i ω 1 t ) { b ̂ 1 ( 0 ) cos Ω ( t ) i [ g 1 g b ̂ 2 ( 0 ) exp ( i ϕ 1 ) + g 2 g b ̂ 3 ( 0 ) exp ( i ϕ 2 ) ] sin Ω ( t ) } ,
b ̂ 2 ( t ) = exp ( i ω 2 t ) { b ̂ 2 ( 0 ) [ 1 2 g 1 2 g 2 sin 2 Ω ( t ) 2 ] 2 g 1 g 2 g 2 b ̂ 3 ( 0 ) exp [ i ( ϕ 1 + ϕ 2 ) ] sin 2 Ω ( t ) 2 } i g 1 g b ̂ 1 ( 0 ) exp [ i ( ω 2 t + ϕ 1 ) ] sin Ω ( t ) ,
b ̂ 3 ( t ) = exp ( i ω 3 t ) { b ̂ 3 ( 0 ) [ 1 + 2 g 2 2 g 2 sin 2 Ω ( t ) 2 ] + 2 g 1 g 2 g 2 b ̂ 2 ( 0 ) exp [ i ( ϕ 1 + ϕ 2 ) ] sin 2 Ω ( t ) 2 } i g 2 g b ̂ 1 ( 0 ) exp { i [ ω 3 t + ϕ 2 ] } sin Ω ( t ) .
n ̂ 1 ( t ) = n ¯ 1 cos 2 Ω ( t ) + [ g 1 2 g 2 n ¯ 2 + g 2 2 g 2 ( n ¯ 3 + 1 ) ] sin 2 Ω ( t ) ,
n ̂ 2 ( t ) = n ¯ 2 [ 1 2 g 1 2 g 2 sin 2 Ω ( t ) 2 ] 2 + 4 g 1 2 g 2 2 g 4 ( n ¯ 3 + 1 ) sin 4 Ω ( t ) 2 + g 1 2 g 2 n ¯ 1 sin 2 Ω ( t ) ,
n ̂ 3 ( t ) = n ¯ 3 [ 1 + 2 g 1 2 g 2 sin 2 Ω ( t ) 2 ] 2 + 4 g 1 2 g 2 2 g 4 ( n ¯ 2 + 1 ) sin 4 Ω ( t ) 2 + g 2 2 g 2 ( n ¯ 1 + 1 ) sin 2 Ω ( t ) .
Ω ( t ) = 2 g 0 t [ 1 + 2 n 1 ( t ) ] 1 2 d t ,
Ω ( t ) = 3 g 0 t [ 9 n 1 2 ( t ) + 9 n 1 ( t ) + 2 ] 1 2 d t .
( Δ X ̂ ) 2 ( Δ Y ̂ ) 2 C ̂ 2 4
S ( t ) = 2 [ Δ X ̂ ( t ) ] 2 C ̂ 1 .
η j ( t ) = 1 + 2 [ b ̂ j ( t ) b ̂ j ( t ) b ̂ j b ̂ j b ̂ j 2 ( t ) b ̂ j 2 ] ,
α i = i = 1 3 exp ( 1 2 α i 2 ) n i = 0 α i n i n i ! n i                   i = 1 , 2 , 3
B 1 ( t ) = Δ b ̂ 1 ( t ) Δ b ̂ 1 ( t ) = g 2 2 g 2 sin 2 Ω ( t ) ,
B 2 ( t ) = Δ b ̂ 2 ( t ) Δ b ̂ 2 ( t ) = 4 g 1 2 g 2 2 g 4 sin 4 Ω ( t ) 2 ,
B 3 ( t ) = Δ b ̂ 3 ( t ) Δ b ̂ 3 ( t ) = B 1 ( t ) + B 2 ( t ) ,
η j ( t ) = 1 + 2 B j ( t ) 1 j = 1 , 2 , 3 ,
η 12 ( t ) = 1 + B 3 ( t ) 2 Re D ¯ 12 ( t ) ,
η 13 ( t ) = 1 + B 1 ( t ) + B 3 ( t ) 2 D 13 ( t ) ,
η 23 ( t ) = 1 + B 2 ( t ) + B 3 ( t ) 2 D 23 ( t ) ,
D 13 ( t ) = Δ b ̂ 1 ( t ) Δ b ̂ 3 ( t ) = i g 2 g exp [ i Γ 2 ( t ) ] sin Ω ( t ) [ 1 + 2 g 2 2 g 2 sin 2 Ω ( t ) 2 ] ,
D 23 ( t ) = Δ b ̂ 2 ( t ) Δ b ̂ 3 ( t ) = 2 g 1 g 2 g 2 exp { i [ Γ 1 ( t ) Γ 2 ( t ) ] } sin 2 Ω ( t ) 2 [ 1 + 2 g 2 2 g 2 sin 2 Ω ( t ) 2 ] ,
D ¯ 12 ( t ) = Δ b ̂ 1 ( t ) Δ b ̂ 2 ( t ) = 2 i g 1 g 2 2 g 3 exp [ i Γ 1 ( t ) ] sin Ω ( t ) sin 2 Ω ( t ) 2 ,
( Δ W j ) 2 N = B j 2 ( t ) + 2 B j ( t ) α j ( t ) 2 0 ,
Δ W 1 Δ W 2 N = D ¯ 12 ( t ) 2 [ D ¯ 12 ( t ) α 1 ( t ) α 2 * ( t ) + c.c. ] ,
Δ W 1 Δ W 3 N = D 13 ( t ) 2 + [ D 13 ( t ) α 1 * ( t ) α 3 * ( t ) + c.c. ] ,
Δ W 2 Δ W 3 N = D 23 ( t ) 2 + [ D 23 ( t ) α 2 * ( t ) α 3 * ( t ) + c.c. ] ,
Δ W i Δ W j N = : b ̂ i ( t ) b ̂ i ( t ) b ̂ j ( t ) b ̂ j ( t ) : b ̂ i ( t ) b ̂ i ( t ) b ̂ j ( t ) b ̂ j ( t ) , i j
( Δ W j k ) 2 = ( Δ W j ) 2 + ( Δ W k ) 2 + 2 Δ W j Δ W k < 0                                               j , k = 1 , 2 , 3 , j k .
C ( 3 ) ( β ̱ , t , s ) = T r ( ρ ̂ ( 0 ) exp { [ j = 1 3 [ β j b ̂ j ( t ) β j * b ̂ j ( t ) + s 2 β j 2 ] } ) ,
C N ( β ̱ , t ) = exp { B 1 β 1 2 B 2 β 2 2 B 3 β 3 2 + ( D 13 β 1 * β 3 * + D 23 β 2 * β 3 * + D ¯ 12 β 1 β 2 * + c . c . ) + [ β 1 α 1 * ( t ) + β 2 α 2 * ( t ) + β 3 α 3 * ( t ) c . c . ] } ,
Φ s ( α ̱ , t , s ) = 1 π 6 C ( 3 ) ( β ̱ , t , s ) j = 1 3 exp [ ( α j β j * α j * β j ) ] d 3 β j .
Φ s ( α 1 , α 3 , t ) = 1 D s exp [ α 3 α 3 ( t ) 2 B s 3 ] exp { B s 3 1 2 [ α 1 α 1 ( t ) ] B s 3 1 2 D 13 [ α 3 * α 3 * ( t ) ] 2 D s } ,
Φ s ( α 1 , α 2 , t ) = 1 D s exp [ α 2 α 2 ( t ) 2 B s 2 ] exp { B s 2 1 2 [ α 1 α 1 ( t ) ] + B s 2 1 2 D ¯ 12 * [ α 2 α 2 ( t ) ] 2 D s } ,
Φ A ( α j , α k ) = 0 2 π 0 2 π Φ A ( α j , α k , ϕ j , ϕ k ) d ϕ j d ϕ k ,
Φ N ( α 1 , α 2 , t ) = π B 2 exp [ α 2 α 2 ( t ) 2 B 2 ] δ { α 1 α 1 ( t ) + D ¯ 12 * B 2 [ α 2 α 2 ( t ) ] } ,
p ( n , t ) = n ρ ̂ ( t ) n ,
p ( n , t ) = p ( 0 , t ) ( 1 Λ 1 ) n m = 0 n [ ( 1 Λ 1 1 Λ 2 ) m L m 0 ( B Λ 2 2 Λ 2 1 ) L n m 0 ( A Λ 1 2 Λ 1 1 ) ] ,
p ( 0 , t ) = Λ 1 Λ 2 ( 1 Λ 1 ) ( 1 Λ 2 ) exp [ ( A Λ 1 1 Λ 1 + B Λ 2 1 Λ 2 ) ] ,
W k = ( 1 ) k k ! Λ 2 k l = 0 k ( Λ 2 Λ 1 ) l L l 0 ( A Λ 1 ) L k l 0 ( B Λ 2 ) ,
L s γ ( x ) = r = 0 s Γ ( s + γ + 1 ) ( x ) r r ! ( s r ) ! Γ ( r + γ + 1 ) .
Λ 1 , 2 = { ( B j + B k ) ± [ ( B j + B k ) 2 4 D ] 1 2 2 D } ,
A = 1 D ( Λ 1 Λ 2 ) { α j ( t ) 2 ( B k + 1 Λ 1 ) + α k ( t ) 2 ( B j + 1 Λ 1 ) [ D j k α j * ( t ) α k * ( t ) + c . c . ] } ,
B = 1 D ( Λ 2 Λ 1 ) { α j ( t ) 2 ( B k + 1 Λ 2 ) + α k ( t ) 2 ( B j + 1 Λ 2 ) [ D j k α j * ( t ) α k * ( t ) + c . c . ] } ,
p ( n , t ) = 1 1 + y ( z y ) n exp ( z y u 1 + y ) k = 0 n 1 ( n k ) ! [ y 2 z ( 1 + y ) ] k     L k 0 [ u y ( 1 + y ) ] ,
W k = k ! ( z y ) k l = 0 k 1 ( k l ) ! ( y 2 z ) l L l 0 ( u y ) ,
P N ( W ) = 1 y exp ( W + u z y y ) I 0 { 2 [ u ( W z y ) ] 1 2 y }                   W z y .
y = B 1 + B 2 ,
z = B 1 α 2 ( t ) 2 + B 2 α 1 ( t ) 2 + [ D ¯ 12 α 1 ( t ) α 2 * ( t ) + c . c . ] ,
y u = B 1 α 1 ( t ) 2 + B 2 α 2 ( t ) 2 + [ D ¯ 12 α 1 ( t ) α 2 * ( t ) + c . c . ] .
C s ( λ 1 , 2 , λ 3 ) = [ 1 + λ 1 , 2 B s 1 , 2 + λ 3 B s 3 + λ 1 , 2 λ 3 D s ] 1 ,
C N ( λ 1 , λ 2 ) = ( 1 + λ 1 B 1 + λ 2 B 2 ) 1 ,
P s ( W 1 , 2 , W 3 ) = 1 D s exp ( B s 1 , 2 W 3 + B s 3 W 1 D s ) I 0 [ 2 D 1 , 23 ( W 1 , 2 W 3 ) 1 2 D s ]     D s > 0 ,
P N ( W 1 , W 2 ) = 1 B 1 exp ( W 1 B 1 ) δ ( W 2 B 2 B 1 W 1 ) ,
C N ( λ ) C N ( λ , λ ) = { [ 1 + λ ( B 1 , 2 D 1 , 23 ) ] [ 1 + λ ( B 3 + D 1 , 23 ) ] } 1 ,
Ω ( t ) = g m μ 0 μ t ( i = 1 m 1 { m [ n 0 + n ¯ 0 cos 2 ( x ) ] + m i } ) 1 2 d x .

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