Abstract

We introduce a new quantum treatment for the problem of the interaction between a two-level atom and field, which is expressed by the Jaynes–Cummings model. The treatment is built up on the construction of a quadratic invariant for the free-field Hamiltonian and used to reformulate the interaction term. We concentrate our study on atomic inversion as well as the phenomenon of squeezing. For the atomic inversion there is a delay during the revival period beside a period of partial collapse. The phenomenon of squeezing is also observed in the normal, the variance, and the entropy squeezing. However, the maximum value of the squeezing beside its period depends in general on the variation of the field frequency ω as well as the μ-parameter, both of which play a crucial role to control this phenomenon.

© 2012 Optical Society of America

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  1. 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 (1963).
    [CrossRef]
  2. H. I. Yoo and J. H. Eberly, “Dynamical theory of an atom with two or three levels interacting with quantized cavity fields,” Phys. Rep. 118, 239–337 (1985).
    [CrossRef]
  3. B. W. Shore and P. L. Knight, “The Jaynes–Cummings model,” J. Mod. Opt. 40, 1195–1238 (1993).
    [CrossRef]
  4. H. Walther, B. T. H. Varcoe, B.-G. Englert, and T. Becker, “Cavity quantum electrodynamics,” Rep. Prog. Phys. 69, 1325–1382 (2006).
    [CrossRef]
  5. M. S. Abdalla, M. M. A. Ahmed, and A.-S. F. Obada, “Dynamics of a nonlinear Jaynes–Cummings model,” Physica A 162, 215–240 (1990).
    [CrossRef]
  6. M. S. Abdalla, M. M. A. Ahmed, and A.-S. F. Obada, “Multimode and multiphoton processes in a nonlinear Jaynes–Cummings model,” Physica A 170, 393–414 (1991).
    [CrossRef]
  7. 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 in a nonlinear medium,” J. Opt. B 4, 134–142 (2002).
    [CrossRef]
  8. H. Eleuch and R. Bennaceur, “Nonlinear dissipation and the quantum noise of light in semiconductor microcavities,” J. Opt. B 6, 189–195 (2004).
    [CrossRef]
  9. H. Eleuch, “Autocorrelation function of microcavity-emitting field in the linear regime,” Eur. Phys. J. D 48, 139–143 (2008).
    [CrossRef]
  10. H. Elech, Y. V. Rostovtsev, and M. S. Abdalla, “Autocorrelation function for a two-level atom with counter-rotating terms,” J. Russ. Laser Res. 32, 269–276 (2011).
    [CrossRef]
  11. M. S. Abdalla, E. M. Khalil, and A. S.-F. Obada, “Exact treatment of the Jaynes-Cummings model under the action of external classical field,” Ann. Phys. 326, 2486–2498 (2011).
    [CrossRef]
  12. H. Eleuch, “Noise spectra of microcavity-emitting field in the linear regime,” Eur. Phys. J. D 49, 391–395 (2008).
    [CrossRef]
  13. M. S. Abdalla, J. Křepelka, and J. Peřina, “Effect of Kerr-like medium on a two-level atom in interaction with bimodal oscillators,” J. Phys. B 39, 1563–1577 (2006).
    [CrossRef]
  14. M. Abdel-Aty, M. S. Abdalla, and A.-S. F. Obada, “Quantum information and entropy squeezing of a two-level atom with a nonlinear medium,” J. Phys. A: Math. Gen. 34, 9129–9141 (2001).
    [CrossRef]
  15. H. Eleuch, “Photon statistics of light in semiconductor microcavities,” J. Phys. B 41, 055502 (2008).
    [CrossRef]
  16. M. Abdel-Aty, M. S. Abdalla, and A. S.-F. Obada, “Uncertainty relation and information entropy of a time dependent bimodel two-level system,” J. Phys. B 35, 4773–4786 (2002).
    [CrossRef]
  17. M. S. Abdalla, M. Abdel-Aty, and A. S.-F. Obada, “Entropy and entanglement of time-dependent Jaynes–Cummings model,” Physica A 326, 203–219 (2003).
    [CrossRef]
  18. A. S.-F. Obada, M. Abdel-Aty, and M. S. Abdalla, “Quantum treatment of a time-dependent single trapped ion interacting with a bimodal cavity field,” Int. J. Mod. Phys. B 17, 5925–5941 (2003).
    [CrossRef]
  19. Y. V. Rostovtsev, H. Eleuch, A. Svidzinsky, H. Li, V. Sautenkov, and M. O. Scully, “Excitation of atomic coherence using off-resonant strong laser pulses,” Phys. Rev. A 79, 063833 (2009).
    [CrossRef]
  20. P. K. Jha, H. Eleuch, and Y. V. Rostovtsev, “Coherent control of atomic excitation using off-resonant strong few-cycle pulses,” Phys. Rev. A 82, 045805 (2010).
    [CrossRef]
  21. M. S. Abdalla and A. I. El-kasapy, “Lie algebraic approach of a charged particle in presence of a constant magnetic field via the quadratic invariant,” Ann. Phys. 325, 1667–1678 (2010).
    [CrossRef]
  22. J.-Q. Shen and H.-Y. Zhu, “Exact solutions to the time-dependent supersymmetric Jaynes-Cummings model and the Chiao-Wu model,” Ann. Phys. 12, 131–145 (2003).
    [CrossRef]
  23. Q. Xian-Ming, Y. Zhao-Xian, J. Zhi-Yong, and X. Bing-Hao, “Geometric phase in a time-dependent Λ-type Jaynes–Cummings model,” Commun. Theor. Phys. 51, 407–410 (2009).
    [CrossRef]
  24. J.-Q. Shen, H.-Y. Zhu, and P. Chen, “Exact solutions and geometric phase factor of time-dependent three-generator quantum systems,” Eur. Phys. J. D 23, 305–313 (2003).
    [CrossRef]
  25. H. R. Lewis and W. B. Riesenfeld, “An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field,” J. Math. Phys. 10, 1458–1474 (1969).
    [CrossRef]
  26. H. R. Lewis, “Classical and quantum systems with time-dependent harmonic-oscillator-type Hamiltonians,” Phys. Rev. Lett. 18, 510–512 (1967).
    [CrossRef]
  27. V. Ermakov, “Second order differential equations. Conditions of complete integrability,” Universita Izvestia Kiev Series III 9, 1–25 (1880) (translated from Russian by A. O. Harin).
  28. H. R. Lewis, “Class of exact invariants for classical and quantum time-dependent harmonic oscillators,” J. Math. Phys. 9, 1976–1987 (1968).
    [CrossRef]
  29. M. S. Abdalla and J.-R. Choi, “Propagator for the time-dependent charged oscillator via linear and quadratic invariants,” Ann. Phys. 322, 2795–2810 (2007).
    [CrossRef]
  30. M. S. Abdalla and M. M. Nassar, “Quantum treatment of the time-dependent coupled oscillators under the action of random driving force,” Ann. Phys. 324, 637–669 (2009).
    [CrossRef]
  31. M. S. Abdalla and N. Al-Ismael, “An alternative model of the damped harmonic oscillator under the influence of external force,” Int. J. Theor. Phys. 48, 2757–2776 (2009).
    [CrossRef]
  32. M. S. Abdalla and L. Al-Rageb, “Nonclassical properties of the time-dependent harmonic oscillator under the action of external force,” Appl. Math. Inf. 5, 570–588 (2011).
  33. R. K. Colegrave and M. S. Abdalla, “Invariants for the time-dependent harmonic oscillator. I.,” J. Phys. A: Math. Gen. 16, 3805–3815 (1983).
    [CrossRef]
  34. R. K. Colegrave, M. S. Abdalla, and M. A. Mannan, “Invariants for the time-dependent harmonic oscillator. II. Cubic and quartic invariants,” J. Phys. A: Math. Gen. 17, 1567–1571 (1984).
    [CrossRef]
  35. M. S. Abdalla and P. G. L. Leach, “Lie algebraic treatment of the quadratic invariants for the degenerate parametric amplifier,” Theor. Math. Phys. 159, 535–550 (2009).
    [CrossRef]
  36. M. S. Abdalla and P. G. L. Leach, “Linear and quadratic invariants for the transformed Tavis-Cummings model,” J. Phys. A: Math. Gen. 36, 12205–12221 (2003).
    [CrossRef]
  37. M. S. Abdalla and P. G. L. Leach, “Wigner function for time dependent coupled linear oscillators via linear and quadratic invariant processes,” J. Phys. A: Math. Gen. 38, 881–893 (2005).
    [CrossRef]
  38. M. S. Abdalla and P. G. L. Leach, “Lie algebraic approach and quantum treatment of an anisotropic charged particle via the quadratic invariant,” J. Math. Phys. 52, 083504 (2011).
    [CrossRef]
  39. E. Pinney, “The nonlinear differential equation y′′+p(x)y+cy-3=0,” Proc. Am. Math. Soc. 1, 681 (1950).
  40. M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley1974).
  41. E. Majernikova, V. Majernik, and S. Shpyrko, “Entropic uncertainty measure for fluctuations in two-level electron-phonon models,” Eur. Phys. J. B 38, 25–35 (2004).
    [CrossRef]
  42. X.-P. Liao and M. F. Fang, “Entropy squeezing for a two-level atom in motion interacting with a quantized field,” Physica A 332, 176–184 (2004).
    [CrossRef]

2011 (4)

H. Elech, Y. V. Rostovtsev, and M. S. Abdalla, “Autocorrelation function for a two-level atom with counter-rotating terms,” J. Russ. Laser Res. 32, 269–276 (2011).
[CrossRef]

M. S. Abdalla, E. M. Khalil, and A. S.-F. Obada, “Exact treatment of the Jaynes-Cummings model under the action of external classical field,” Ann. Phys. 326, 2486–2498 (2011).
[CrossRef]

M. S. Abdalla and L. Al-Rageb, “Nonclassical properties of the time-dependent harmonic oscillator under the action of external force,” Appl. Math. Inf. 5, 570–588 (2011).

M. S. Abdalla and P. G. L. Leach, “Lie algebraic approach and quantum treatment of an anisotropic charged particle via the quadratic invariant,” J. Math. Phys. 52, 083504 (2011).
[CrossRef]

2010 (2)

P. K. Jha, H. Eleuch, and Y. V. Rostovtsev, “Coherent control of atomic excitation using off-resonant strong few-cycle pulses,” Phys. Rev. A 82, 045805 (2010).
[CrossRef]

M. S. Abdalla and A. I. El-kasapy, “Lie algebraic approach of a charged particle in presence of a constant magnetic field via the quadratic invariant,” Ann. Phys. 325, 1667–1678 (2010).
[CrossRef]

2009 (5)

Y. V. Rostovtsev, H. Eleuch, A. Svidzinsky, H. Li, V. Sautenkov, and M. O. Scully, “Excitation of atomic coherence using off-resonant strong laser pulses,” Phys. Rev. A 79, 063833 (2009).
[CrossRef]

Q. Xian-Ming, Y. Zhao-Xian, J. Zhi-Yong, and X. Bing-Hao, “Geometric phase in a time-dependent Λ-type Jaynes–Cummings model,” Commun. Theor. Phys. 51, 407–410 (2009).
[CrossRef]

M. S. Abdalla and P. G. L. Leach, “Lie algebraic treatment of the quadratic invariants for the degenerate parametric amplifier,” Theor. Math. Phys. 159, 535–550 (2009).
[CrossRef]

M. S. Abdalla and M. M. Nassar, “Quantum treatment of the time-dependent coupled oscillators under the action of random driving force,” Ann. Phys. 324, 637–669 (2009).
[CrossRef]

M. S. Abdalla and N. Al-Ismael, “An alternative model of the damped harmonic oscillator under the influence of external force,” Int. J. Theor. Phys. 48, 2757–2776 (2009).
[CrossRef]

2008 (3)

H. Eleuch, “Autocorrelation function of microcavity-emitting field in the linear regime,” Eur. Phys. J. D 48, 139–143 (2008).
[CrossRef]

H. Eleuch, “Noise spectra of microcavity-emitting field in the linear regime,” Eur. Phys. J. D 49, 391–395 (2008).
[CrossRef]

H. Eleuch, “Photon statistics of light in semiconductor microcavities,” J. Phys. B 41, 055502 (2008).
[CrossRef]

2007 (1)

M. S. Abdalla and J.-R. Choi, “Propagator for the time-dependent charged oscillator via linear and quadratic invariants,” Ann. Phys. 322, 2795–2810 (2007).
[CrossRef]

2006 (2)

M. S. Abdalla, J. Křepelka, and J. Peřina, “Effect of Kerr-like medium on a two-level atom in interaction with bimodal oscillators,” J. Phys. B 39, 1563–1577 (2006).
[CrossRef]

H. Walther, B. T. H. Varcoe, B.-G. Englert, and T. Becker, “Cavity quantum electrodynamics,” Rep. Prog. Phys. 69, 1325–1382 (2006).
[CrossRef]

2005 (1)

M. S. Abdalla and P. G. L. Leach, “Wigner function for time dependent coupled linear oscillators via linear and quadratic invariant processes,” J. Phys. A: Math. Gen. 38, 881–893 (2005).
[CrossRef]

2004 (3)

E. Majernikova, V. Majernik, and S. Shpyrko, “Entropic uncertainty measure for fluctuations in two-level electron-phonon models,” Eur. Phys. J. B 38, 25–35 (2004).
[CrossRef]

X.-P. Liao and M. F. Fang, “Entropy squeezing for a two-level atom in motion interacting with a quantized field,” Physica A 332, 176–184 (2004).
[CrossRef]

H. Eleuch and R. Bennaceur, “Nonlinear dissipation and the quantum noise of light in semiconductor microcavities,” J. Opt. B 6, 189–195 (2004).
[CrossRef]

2003 (5)

M. S. Abdalla, M. Abdel-Aty, and A. S.-F. Obada, “Entropy and entanglement of time-dependent Jaynes–Cummings model,” Physica A 326, 203–219 (2003).
[CrossRef]

A. S.-F. Obada, M. Abdel-Aty, and M. S. Abdalla, “Quantum treatment of a time-dependent single trapped ion interacting with a bimodal cavity field,” Int. J. Mod. Phys. B 17, 5925–5941 (2003).
[CrossRef]

M. S. Abdalla and P. G. L. Leach, “Linear and quadratic invariants for the transformed Tavis-Cummings model,” J. Phys. A: Math. Gen. 36, 12205–12221 (2003).
[CrossRef]

J.-Q. Shen and H.-Y. Zhu, “Exact solutions to the time-dependent supersymmetric Jaynes-Cummings model and the Chiao-Wu model,” Ann. Phys. 12, 131–145 (2003).
[CrossRef]

J.-Q. Shen, H.-Y. Zhu, and P. Chen, “Exact solutions and geometric phase factor of time-dependent three-generator quantum systems,” Eur. Phys. J. D 23, 305–313 (2003).
[CrossRef]

2002 (2)

M. Abdel-Aty, M. S. Abdalla, and A. S.-F. Obada, “Uncertainty relation and information entropy of a time dependent bimodel two-level system,” J. Phys. B 35, 4773–4786 (2002).
[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 in a nonlinear medium,” J. Opt. B 4, 134–142 (2002).
[CrossRef]

2001 (1)

M. Abdel-Aty, M. S. Abdalla, and A.-S. F. Obada, “Quantum information and entropy squeezing of a two-level atom with a nonlinear medium,” J. Phys. A: Math. Gen. 34, 9129–9141 (2001).
[CrossRef]

1993 (1)

B. W. Shore and P. L. Knight, “The Jaynes–Cummings model,” J. Mod. Opt. 40, 1195–1238 (1993).
[CrossRef]

1991 (1)

M. S. Abdalla, M. M. A. Ahmed, and A.-S. F. Obada, “Multimode and multiphoton processes in a nonlinear Jaynes–Cummings model,” Physica A 170, 393–414 (1991).
[CrossRef]

1990 (1)

M. S. Abdalla, M. M. A. Ahmed, and A.-S. F. Obada, “Dynamics of a nonlinear Jaynes–Cummings model,” Physica A 162, 215–240 (1990).
[CrossRef]

1985 (1)

H. I. Yoo and J. H. Eberly, “Dynamical theory of an atom with two or three levels interacting with quantized cavity fields,” Phys. Rep. 118, 239–337 (1985).
[CrossRef]

1984 (1)

R. K. Colegrave, M. S. Abdalla, and M. A. Mannan, “Invariants for the time-dependent harmonic oscillator. II. Cubic and quartic invariants,” J. Phys. A: Math. Gen. 17, 1567–1571 (1984).
[CrossRef]

1983 (1)

R. K. Colegrave and M. S. Abdalla, “Invariants for the time-dependent harmonic oscillator. I.,” J. Phys. A: Math. Gen. 16, 3805–3815 (1983).
[CrossRef]

1969 (1)

H. R. Lewis and W. B. Riesenfeld, “An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field,” J. Math. Phys. 10, 1458–1474 (1969).
[CrossRef]

1968 (1)

H. R. Lewis, “Class of exact invariants for classical and quantum time-dependent harmonic oscillators,” J. Math. Phys. 9, 1976–1987 (1968).
[CrossRef]

1967 (1)

H. R. Lewis, “Classical and quantum systems with time-dependent harmonic-oscillator-type Hamiltonians,” Phys. Rev. Lett. 18, 510–512 (1967).
[CrossRef]

1963 (1)

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 (1963).
[CrossRef]

1950 (1)

E. Pinney, “The nonlinear differential equation y′′+p(x)y+cy-3=0,” Proc. Am. Math. Soc. 1, 681 (1950).

1880 (1)

V. Ermakov, “Second order differential equations. Conditions of complete integrability,” Universita Izvestia Kiev Series III 9, 1–25 (1880) (translated from Russian by A. O. Harin).

Abdalla, M. S.

H. Elech, Y. V. Rostovtsev, and M. S. Abdalla, “Autocorrelation function for a two-level atom with counter-rotating terms,” J. Russ. Laser Res. 32, 269–276 (2011).
[CrossRef]

M. S. Abdalla, E. M. Khalil, and A. S.-F. Obada, “Exact treatment of the Jaynes-Cummings model under the action of external classical field,” Ann. Phys. 326, 2486–2498 (2011).
[CrossRef]

M. S. Abdalla and L. Al-Rageb, “Nonclassical properties of the time-dependent harmonic oscillator under the action of external force,” Appl. Math. Inf. 5, 570–588 (2011).

M. S. Abdalla and P. G. L. Leach, “Lie algebraic approach and quantum treatment of an anisotropic charged particle via the quadratic invariant,” J. Math. Phys. 52, 083504 (2011).
[CrossRef]

M. S. Abdalla and A. I. El-kasapy, “Lie algebraic approach of a charged particle in presence of a constant magnetic field via the quadratic invariant,” Ann. Phys. 325, 1667–1678 (2010).
[CrossRef]

M. S. Abdalla and M. M. Nassar, “Quantum treatment of the time-dependent coupled oscillators under the action of random driving force,” Ann. Phys. 324, 637–669 (2009).
[CrossRef]

M. S. Abdalla and N. Al-Ismael, “An alternative model of the damped harmonic oscillator under the influence of external force,” Int. J. Theor. Phys. 48, 2757–2776 (2009).
[CrossRef]

M. S. Abdalla and P. G. L. Leach, “Lie algebraic treatment of the quadratic invariants for the degenerate parametric amplifier,” Theor. Math. Phys. 159, 535–550 (2009).
[CrossRef]

M. S. Abdalla and J.-R. Choi, “Propagator for the time-dependent charged oscillator via linear and quadratic invariants,” Ann. Phys. 322, 2795–2810 (2007).
[CrossRef]

M. S. Abdalla, J. Křepelka, and J. Peřina, “Effect of Kerr-like medium on a two-level atom in interaction with bimodal oscillators,” J. Phys. B 39, 1563–1577 (2006).
[CrossRef]

M. S. Abdalla and P. G. L. Leach, “Wigner function for time dependent coupled linear oscillators via linear and quadratic invariant processes,” J. Phys. A: Math. Gen. 38, 881–893 (2005).
[CrossRef]

M. S. Abdalla and P. G. L. Leach, “Linear and quadratic invariants for the transformed Tavis-Cummings model,” J. Phys. A: Math. Gen. 36, 12205–12221 (2003).
[CrossRef]

M. S. Abdalla, M. Abdel-Aty, and A. S.-F. Obada, “Entropy and entanglement of time-dependent Jaynes–Cummings model,” Physica A 326, 203–219 (2003).
[CrossRef]

A. S.-F. Obada, M. Abdel-Aty, and M. S. Abdalla, “Quantum treatment of a time-dependent single trapped ion interacting with a bimodal cavity field,” Int. J. Mod. Phys. B 17, 5925–5941 (2003).
[CrossRef]

M. Abdel-Aty, M. S. Abdalla, and A. S.-F. Obada, “Uncertainty relation and information entropy of a time dependent bimodel two-level system,” J. Phys. B 35, 4773–4786 (2002).
[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 in a nonlinear medium,” J. Opt. B 4, 134–142 (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 nonlinear medium,” J. Phys. A: Math. Gen. 34, 9129–9141 (2001).
[CrossRef]

M. S. Abdalla, M. M. A. Ahmed, and A.-S. F. Obada, “Multimode and multiphoton processes in a nonlinear Jaynes–Cummings model,” Physica A 170, 393–414 (1991).
[CrossRef]

M. S. Abdalla, M. M. A. Ahmed, and A.-S. F. Obada, “Dynamics of a nonlinear Jaynes–Cummings model,” Physica A 162, 215–240 (1990).
[CrossRef]

R. K. Colegrave, M. S. Abdalla, and M. A. Mannan, “Invariants for the time-dependent harmonic oscillator. II. Cubic and quartic invariants,” J. Phys. A: Math. Gen. 17, 1567–1571 (1984).
[CrossRef]

R. K. Colegrave and M. S. Abdalla, “Invariants for the time-dependent harmonic oscillator. I.,” J. Phys. A: Math. Gen. 16, 3805–3815 (1983).
[CrossRef]

Abdel-Aty, M.

M. S. Abdalla, M. Abdel-Aty, and A. S.-F. Obada, “Entropy and entanglement of time-dependent Jaynes–Cummings model,” Physica A 326, 203–219 (2003).
[CrossRef]

A. S.-F. Obada, M. Abdel-Aty, and M. S. Abdalla, “Quantum treatment of a time-dependent single trapped ion interacting with a bimodal cavity field,” Int. J. Mod. Phys. B 17, 5925–5941 (2003).
[CrossRef]

M. Abdel-Aty, M. S. Abdalla, and A. S.-F. Obada, “Uncertainty relation and information entropy of a time dependent bimodel two-level system,” J. Phys. B 35, 4773–4786 (2002).
[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 in a nonlinear medium,” J. Opt. B 4, 134–142 (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 nonlinear medium,” J. Phys. A: Math. Gen. 34, 9129–9141 (2001).
[CrossRef]

Ahmed, M. M. A.

M. S. Abdalla, M. M. A. Ahmed, and A.-S. F. Obada, “Multimode and multiphoton processes in a nonlinear Jaynes–Cummings model,” Physica A 170, 393–414 (1991).
[CrossRef]

M. S. Abdalla, M. M. A. Ahmed, and A.-S. F. Obada, “Dynamics of a nonlinear Jaynes–Cummings model,” Physica A 162, 215–240 (1990).
[CrossRef]

Al-Ismael, N.

M. S. Abdalla and N. Al-Ismael, “An alternative model of the damped harmonic oscillator under the influence of external force,” Int. J. Theor. Phys. 48, 2757–2776 (2009).
[CrossRef]

Al-Rageb, L.

M. S. Abdalla and L. Al-Rageb, “Nonclassical properties of the time-dependent harmonic oscillator under the action of external force,” Appl. Math. Inf. 5, 570–588 (2011).

Becker, T.

H. Walther, B. T. H. Varcoe, B.-G. Englert, and T. Becker, “Cavity quantum electrodynamics,” Rep. Prog. Phys. 69, 1325–1382 (2006).
[CrossRef]

Bennaceur, R.

H. Eleuch and R. Bennaceur, “Nonlinear dissipation and the quantum noise of light in semiconductor microcavities,” J. Opt. B 6, 189–195 (2004).
[CrossRef]

Bing-Hao, X.

Q. Xian-Ming, Y. Zhao-Xian, J. Zhi-Yong, and X. Bing-Hao, “Geometric phase in a time-dependent Λ-type Jaynes–Cummings model,” Commun. Theor. Phys. 51, 407–410 (2009).
[CrossRef]

Chen, P.

J.-Q. Shen, H.-Y. Zhu, and P. Chen, “Exact solutions and geometric phase factor of time-dependent three-generator quantum systems,” Eur. Phys. J. D 23, 305–313 (2003).
[CrossRef]

Choi, J.-R.

M. S. Abdalla and J.-R. Choi, “Propagator for the time-dependent charged oscillator via linear and quadratic invariants,” Ann. Phys. 322, 2795–2810 (2007).
[CrossRef]

Colegrave, R. K.

R. K. Colegrave, M. S. Abdalla, and M. A. Mannan, “Invariants for the time-dependent harmonic oscillator. II. Cubic and quartic invariants,” J. Phys. A: Math. Gen. 17, 1567–1571 (1984).
[CrossRef]

R. K. Colegrave and M. S. Abdalla, “Invariants for the time-dependent harmonic oscillator. I.,” J. Phys. A: Math. Gen. 16, 3805–3815 (1983).
[CrossRef]

Cummings, F. W.

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 (1963).
[CrossRef]

Eberly, J. H.

H. I. Yoo and J. H. Eberly, “Dynamical theory of an atom with two or three levels interacting with quantized cavity fields,” Phys. Rep. 118, 239–337 (1985).
[CrossRef]

Elech, H.

H. Elech, Y. V. Rostovtsev, and M. S. Abdalla, “Autocorrelation function for a two-level atom with counter-rotating terms,” J. Russ. Laser Res. 32, 269–276 (2011).
[CrossRef]

Eleuch, H.

P. K. Jha, H. Eleuch, and Y. V. Rostovtsev, “Coherent control of atomic excitation using off-resonant strong few-cycle pulses,” Phys. Rev. A 82, 045805 (2010).
[CrossRef]

Y. V. Rostovtsev, H. Eleuch, A. Svidzinsky, H. Li, V. Sautenkov, and M. O. Scully, “Excitation of atomic coherence using off-resonant strong laser pulses,” Phys. Rev. A 79, 063833 (2009).
[CrossRef]

H. Eleuch, “Photon statistics of light in semiconductor microcavities,” J. Phys. B 41, 055502 (2008).
[CrossRef]

H. Eleuch, “Noise spectra of microcavity-emitting field in the linear regime,” Eur. Phys. J. D 49, 391–395 (2008).
[CrossRef]

H. Eleuch, “Autocorrelation function of microcavity-emitting field in the linear regime,” Eur. Phys. J. D 48, 139–143 (2008).
[CrossRef]

H. Eleuch and R. Bennaceur, “Nonlinear dissipation and the quantum noise of light in semiconductor microcavities,” J. Opt. B 6, 189–195 (2004).
[CrossRef]

El-kasapy, A. I.

M. S. Abdalla and A. I. El-kasapy, “Lie algebraic approach of a charged particle in presence of a constant magnetic field via the quadratic invariant,” Ann. Phys. 325, 1667–1678 (2010).
[CrossRef]

Englert, B.-G.

H. Walther, B. T. H. Varcoe, B.-G. Englert, and T. Becker, “Cavity quantum electrodynamics,” Rep. Prog. Phys. 69, 1325–1382 (2006).
[CrossRef]

Ermakov, V.

V. Ermakov, “Second order differential equations. Conditions of complete integrability,” Universita Izvestia Kiev Series III 9, 1–25 (1880) (translated from Russian by A. O. Harin).

Fang, M. F.

X.-P. Liao and M. F. Fang, “Entropy squeezing for a two-level atom in motion interacting with a quantized field,” Physica A 332, 176–184 (2004).
[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 (1963).
[CrossRef]

Jha, P. K.

P. K. Jha, H. Eleuch, and Y. V. Rostovtsev, “Coherent control of atomic excitation using off-resonant strong few-cycle pulses,” Phys. Rev. A 82, 045805 (2010).
[CrossRef]

Khalil, E. M.

M. S. Abdalla, E. M. Khalil, and A. S.-F. Obada, “Exact treatment of the Jaynes-Cummings model under the action of external classical field,” Ann. Phys. 326, 2486–2498 (2011).
[CrossRef]

Knight, P. L.

B. W. Shore and P. L. Knight, “The Jaynes–Cummings model,” J. Mod. Opt. 40, 1195–1238 (1993).
[CrossRef]

Krepelka, J.

M. S. Abdalla, J. Křepelka, and J. Peřina, “Effect of Kerr-like medium on a two-level atom in interaction with bimodal oscillators,” J. Phys. B 39, 1563–1577 (2006).
[CrossRef]

Lamb, W. E.

M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley1974).

Leach, P. G. L.

M. S. Abdalla and P. G. L. Leach, “Lie algebraic approach and quantum treatment of an anisotropic charged particle via the quadratic invariant,” J. Math. Phys. 52, 083504 (2011).
[CrossRef]

M. S. Abdalla and P. G. L. Leach, “Lie algebraic treatment of the quadratic invariants for the degenerate parametric amplifier,” Theor. Math. Phys. 159, 535–550 (2009).
[CrossRef]

M. S. Abdalla and P. G. L. Leach, “Wigner function for time dependent coupled linear oscillators via linear and quadratic invariant processes,” J. Phys. A: Math. Gen. 38, 881–893 (2005).
[CrossRef]

M. S. Abdalla and P. G. L. Leach, “Linear and quadratic invariants for the transformed Tavis-Cummings model,” J. Phys. A: Math. Gen. 36, 12205–12221 (2003).
[CrossRef]

Lewis, H. R.

H. R. Lewis and W. B. Riesenfeld, “An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field,” J. Math. Phys. 10, 1458–1474 (1969).
[CrossRef]

H. R. Lewis, “Class of exact invariants for classical and quantum time-dependent harmonic oscillators,” J. Math. Phys. 9, 1976–1987 (1968).
[CrossRef]

H. R. Lewis, “Classical and quantum systems with time-dependent harmonic-oscillator-type Hamiltonians,” Phys. Rev. Lett. 18, 510–512 (1967).
[CrossRef]

Li, H.

Y. V. Rostovtsev, H. Eleuch, A. Svidzinsky, H. Li, V. Sautenkov, and M. O. Scully, “Excitation of atomic coherence using off-resonant strong laser pulses,” Phys. Rev. A 79, 063833 (2009).
[CrossRef]

Liao, X.-P.

X.-P. Liao and M. F. Fang, “Entropy squeezing for a two-level atom in motion interacting with a quantized field,” Physica A 332, 176–184 (2004).
[CrossRef]

Majernik, V.

E. Majernikova, V. Majernik, and S. Shpyrko, “Entropic uncertainty measure for fluctuations in two-level electron-phonon models,” Eur. Phys. J. B 38, 25–35 (2004).
[CrossRef]

Majernikova, E.

E. Majernikova, V. Majernik, and S. Shpyrko, “Entropic uncertainty measure for fluctuations in two-level electron-phonon models,” Eur. Phys. J. B 38, 25–35 (2004).
[CrossRef]

Mannan, M. A.

R. K. Colegrave, M. S. Abdalla, and M. A. Mannan, “Invariants for the time-dependent harmonic oscillator. II. Cubic and quartic invariants,” J. Phys. A: Math. Gen. 17, 1567–1571 (1984).
[CrossRef]

Nassar, M. M.

M. S. Abdalla and M. M. Nassar, “Quantum treatment of the time-dependent coupled oscillators under the action of random driving force,” Ann. Phys. 324, 637–669 (2009).
[CrossRef]

Obada, A. S.-F.

M. S. Abdalla, E. M. Khalil, and A. S.-F. Obada, “Exact treatment of the Jaynes-Cummings model under the action of external classical field,” Ann. Phys. 326, 2486–2498 (2011).
[CrossRef]

M. S. Abdalla, M. Abdel-Aty, and A. S.-F. Obada, “Entropy and entanglement of time-dependent Jaynes–Cummings model,” Physica A 326, 203–219 (2003).
[CrossRef]

A. S.-F. Obada, M. Abdel-Aty, and M. S. Abdalla, “Quantum treatment of a time-dependent single trapped ion interacting with a bimodal cavity field,” Int. J. Mod. Phys. B 17, 5925–5941 (2003).
[CrossRef]

M. Abdel-Aty, M. S. Abdalla, and A. S.-F. Obada, “Uncertainty relation and information entropy of a time dependent bimodel two-level system,” J. Phys. B 35, 4773–4786 (2002).
[CrossRef]

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 in a nonlinear medium,” J. Opt. B 4, 134–142 (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 nonlinear medium,” J. Phys. A: Math. Gen. 34, 9129–9141 (2001).
[CrossRef]

M. S. Abdalla, M. M. A. Ahmed, and A.-S. F. Obada, “Multimode and multiphoton processes in a nonlinear Jaynes–Cummings model,” Physica A 170, 393–414 (1991).
[CrossRef]

M. S. Abdalla, M. M. A. Ahmed, and A.-S. F. Obada, “Dynamics of a nonlinear Jaynes–Cummings model,” Physica A 162, 215–240 (1990).
[CrossRef]

Perina, J.

M. S. Abdalla, J. Křepelka, and J. Peřina, “Effect of Kerr-like medium on a two-level atom in interaction with bimodal oscillators,” J. Phys. B 39, 1563–1577 (2006).
[CrossRef]

Pinney, E.

E. Pinney, “The nonlinear differential equation y′′+p(x)y+cy-3=0,” Proc. Am. Math. Soc. 1, 681 (1950).

Riesenfeld, W. B.

H. R. Lewis and W. B. Riesenfeld, “An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field,” J. Math. Phys. 10, 1458–1474 (1969).
[CrossRef]

Rostovtsev, Y. V.

H. Elech, Y. V. Rostovtsev, and M. S. Abdalla, “Autocorrelation function for a two-level atom with counter-rotating terms,” J. Russ. Laser Res. 32, 269–276 (2011).
[CrossRef]

P. K. Jha, H. Eleuch, and Y. V. Rostovtsev, “Coherent control of atomic excitation using off-resonant strong few-cycle pulses,” Phys. Rev. A 82, 045805 (2010).
[CrossRef]

Y. V. Rostovtsev, H. Eleuch, A. Svidzinsky, H. Li, V. Sautenkov, and M. O. Scully, “Excitation of atomic coherence using off-resonant strong laser pulses,” Phys. Rev. A 79, 063833 (2009).
[CrossRef]

Sargent, M.

M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley1974).

Sautenkov, V.

Y. V. Rostovtsev, H. Eleuch, A. Svidzinsky, H. Li, V. Sautenkov, and M. O. Scully, “Excitation of atomic coherence using off-resonant strong laser pulses,” Phys. Rev. A 79, 063833 (2009).
[CrossRef]

Scully, M. O.

Y. V. Rostovtsev, H. Eleuch, A. Svidzinsky, H. Li, V. Sautenkov, and M. O. Scully, “Excitation of atomic coherence using off-resonant strong laser pulses,” Phys. Rev. A 79, 063833 (2009).
[CrossRef]

M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley1974).

Shen, J.-Q.

J.-Q. Shen and H.-Y. Zhu, “Exact solutions to the time-dependent supersymmetric Jaynes-Cummings model and the Chiao-Wu model,” Ann. Phys. 12, 131–145 (2003).
[CrossRef]

J.-Q. Shen, H.-Y. Zhu, and P. Chen, “Exact solutions and geometric phase factor of time-dependent three-generator quantum systems,” Eur. Phys. J. D 23, 305–313 (2003).
[CrossRef]

Shore, B. W.

B. W. Shore and P. L. Knight, “The Jaynes–Cummings model,” J. Mod. Opt. 40, 1195–1238 (1993).
[CrossRef]

Shpyrko, S.

E. Majernikova, V. Majernik, and S. Shpyrko, “Entropic uncertainty measure for fluctuations in two-level electron-phonon models,” Eur. Phys. J. B 38, 25–35 (2004).
[CrossRef]

Svidzinsky, A.

Y. V. Rostovtsev, H. Eleuch, A. Svidzinsky, H. Li, V. Sautenkov, and M. O. Scully, “Excitation of atomic coherence using off-resonant strong laser pulses,” Phys. Rev. A 79, 063833 (2009).
[CrossRef]

Varcoe, B. T. H.

H. Walther, B. T. H. Varcoe, B.-G. Englert, and T. Becker, “Cavity quantum electrodynamics,” Rep. Prog. Phys. 69, 1325–1382 (2006).
[CrossRef]

Walther, H.

H. Walther, B. T. H. Varcoe, B.-G. Englert, and T. Becker, “Cavity quantum electrodynamics,” Rep. Prog. Phys. 69, 1325–1382 (2006).
[CrossRef]

Xian-Ming, Q.

Q. Xian-Ming, Y. Zhao-Xian, J. Zhi-Yong, and X. Bing-Hao, “Geometric phase in a time-dependent Λ-type Jaynes–Cummings model,” Commun. Theor. Phys. 51, 407–410 (2009).
[CrossRef]

Yoo, H. I.

H. I. Yoo and J. H. Eberly, “Dynamical theory of an atom with two or three levels interacting with quantized cavity fields,” Phys. Rep. 118, 239–337 (1985).
[CrossRef]

Zhao-Xian, Y.

Q. Xian-Ming, Y. Zhao-Xian, J. Zhi-Yong, and X. Bing-Hao, “Geometric phase in a time-dependent Λ-type Jaynes–Cummings model,” Commun. Theor. Phys. 51, 407–410 (2009).
[CrossRef]

Zhi-Yong, J.

Q. Xian-Ming, Y. Zhao-Xian, J. Zhi-Yong, and X. Bing-Hao, “Geometric phase in a time-dependent Λ-type Jaynes–Cummings model,” Commun. Theor. Phys. 51, 407–410 (2009).
[CrossRef]

Zhu, H.-Y.

J.-Q. Shen and H.-Y. Zhu, “Exact solutions to the time-dependent supersymmetric Jaynes-Cummings model and the Chiao-Wu model,” Ann. Phys. 12, 131–145 (2003).
[CrossRef]

J.-Q. Shen, H.-Y. Zhu, and P. Chen, “Exact solutions and geometric phase factor of time-dependent three-generator quantum systems,” Eur. Phys. J. D 23, 305–313 (2003).
[CrossRef]

Ann. Phys. (5)

M. S. Abdalla, E. M. Khalil, and A. S.-F. Obada, “Exact treatment of the Jaynes-Cummings model under the action of external classical field,” Ann. Phys. 326, 2486–2498 (2011).
[CrossRef]

M. S. Abdalla and A. I. El-kasapy, “Lie algebraic approach of a charged particle in presence of a constant magnetic field via the quadratic invariant,” Ann. Phys. 325, 1667–1678 (2010).
[CrossRef]

J.-Q. Shen and H.-Y. Zhu, “Exact solutions to the time-dependent supersymmetric Jaynes-Cummings model and the Chiao-Wu model,” Ann. Phys. 12, 131–145 (2003).
[CrossRef]

M. S. Abdalla and J.-R. Choi, “Propagator for the time-dependent charged oscillator via linear and quadratic invariants,” Ann. Phys. 322, 2795–2810 (2007).
[CrossRef]

M. S. Abdalla and M. M. Nassar, “Quantum treatment of the time-dependent coupled oscillators under the action of random driving force,” Ann. Phys. 324, 637–669 (2009).
[CrossRef]

Appl. Math. Inf. (1)

M. S. Abdalla and L. Al-Rageb, “Nonclassical properties of the time-dependent harmonic oscillator under the action of external force,” Appl. Math. Inf. 5, 570–588 (2011).

Commun. Theor. Phys. (1)

Q. Xian-Ming, Y. Zhao-Xian, J. Zhi-Yong, and X. Bing-Hao, “Geometric phase in a time-dependent Λ-type Jaynes–Cummings model,” Commun. Theor. Phys. 51, 407–410 (2009).
[CrossRef]

Eur. Phys. J. B (1)

E. Majernikova, V. Majernik, and S. Shpyrko, “Entropic uncertainty measure for fluctuations in two-level electron-phonon models,” Eur. Phys. J. B 38, 25–35 (2004).
[CrossRef]

Eur. Phys. J. D (3)

J.-Q. Shen, H.-Y. Zhu, and P. Chen, “Exact solutions and geometric phase factor of time-dependent three-generator quantum systems,” Eur. Phys. J. D 23, 305–313 (2003).
[CrossRef]

H. Eleuch, “Noise spectra of microcavity-emitting field in the linear regime,” Eur. Phys. J. D 49, 391–395 (2008).
[CrossRef]

H. Eleuch, “Autocorrelation function of microcavity-emitting field in the linear regime,” Eur. Phys. J. D 48, 139–143 (2008).
[CrossRef]

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

A. S.-F. Obada, M. Abdel-Aty, and M. S. Abdalla, “Quantum treatment of a time-dependent single trapped ion interacting with a bimodal cavity field,” Int. J. Mod. Phys. B 17, 5925–5941 (2003).
[CrossRef]

Int. J. Theor. Phys. (1)

M. S. Abdalla and N. Al-Ismael, “An alternative model of the damped harmonic oscillator under the influence of external force,” Int. J. Theor. Phys. 48, 2757–2776 (2009).
[CrossRef]

J. Math. Phys. (3)

H. R. Lewis and W. B. Riesenfeld, “An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field,” J. Math. Phys. 10, 1458–1474 (1969).
[CrossRef]

M. S. Abdalla and P. G. L. Leach, “Lie algebraic approach and quantum treatment of an anisotropic charged particle via the quadratic invariant,” J. Math. Phys. 52, 083504 (2011).
[CrossRef]

H. R. Lewis, “Class of exact invariants for classical and quantum time-dependent harmonic oscillators,” J. Math. Phys. 9, 1976–1987 (1968).
[CrossRef]

J. Mod. Opt. (1)

B. W. Shore and P. L. Knight, “The Jaynes–Cummings model,” J. Mod. Opt. 40, 1195–1238 (1993).
[CrossRef]

J. Opt. B (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 in a nonlinear medium,” J. Opt. B 4, 134–142 (2002).
[CrossRef]

H. Eleuch and R. Bennaceur, “Nonlinear dissipation and the quantum noise of light in semiconductor microcavities,” J. Opt. B 6, 189–195 (2004).
[CrossRef]

J. Phys. A: Math. Gen. (5)

M. Abdel-Aty, M. S. Abdalla, and A.-S. F. Obada, “Quantum information and entropy squeezing of a two-level atom with a nonlinear medium,” J. Phys. A: Math. Gen. 34, 9129–9141 (2001).
[CrossRef]

M. S. Abdalla and P. G. L. Leach, “Linear and quadratic invariants for the transformed Tavis-Cummings model,” J. Phys. A: Math. Gen. 36, 12205–12221 (2003).
[CrossRef]

M. S. Abdalla and P. G. L. Leach, “Wigner function for time dependent coupled linear oscillators via linear and quadratic invariant processes,” J. Phys. A: Math. Gen. 38, 881–893 (2005).
[CrossRef]

R. K. Colegrave and M. S. Abdalla, “Invariants for the time-dependent harmonic oscillator. I.,” J. Phys. A: Math. Gen. 16, 3805–3815 (1983).
[CrossRef]

R. K. Colegrave, M. S. Abdalla, and M. A. Mannan, “Invariants for the time-dependent harmonic oscillator. II. Cubic and quartic invariants,” J. Phys. A: Math. Gen. 17, 1567–1571 (1984).
[CrossRef]

J. Phys. B (3)

H. Eleuch, “Photon statistics of light in semiconductor microcavities,” J. Phys. B 41, 055502 (2008).
[CrossRef]

M. Abdel-Aty, M. S. Abdalla, and A. S.-F. Obada, “Uncertainty relation and information entropy of a time dependent bimodel two-level system,” J. Phys. B 35, 4773–4786 (2002).
[CrossRef]

M. S. Abdalla, J. Křepelka, and J. Peřina, “Effect of Kerr-like medium on a two-level atom in interaction with bimodal oscillators,” J. Phys. B 39, 1563–1577 (2006).
[CrossRef]

J. Russ. Laser Res. (1)

H. Elech, Y. V. Rostovtsev, and M. S. Abdalla, “Autocorrelation function for a two-level atom with counter-rotating terms,” J. Russ. Laser Res. 32, 269–276 (2011).
[CrossRef]

Phys. Rep. (1)

H. I. Yoo and J. H. Eberly, “Dynamical theory of an atom with two or three levels interacting with quantized cavity fields,” Phys. Rep. 118, 239–337 (1985).
[CrossRef]

Phys. Rev. A (2)

Y. V. Rostovtsev, H. Eleuch, A. Svidzinsky, H. Li, V. Sautenkov, and M. O. Scully, “Excitation of atomic coherence using off-resonant strong laser pulses,” Phys. Rev. A 79, 063833 (2009).
[CrossRef]

P. K. Jha, H. Eleuch, and Y. V. Rostovtsev, “Coherent control of atomic excitation using off-resonant strong few-cycle pulses,” Phys. Rev. A 82, 045805 (2010).
[CrossRef]

Phys. Rev. Lett. (1)

H. R. Lewis, “Classical and quantum systems with time-dependent harmonic-oscillator-type Hamiltonians,” Phys. Rev. Lett. 18, 510–512 (1967).
[CrossRef]

Physica A (4)

M. S. Abdalla, M. M. A. Ahmed, and A.-S. F. Obada, “Dynamics of a nonlinear Jaynes–Cummings model,” Physica A 162, 215–240 (1990).
[CrossRef]

M. S. Abdalla, M. M. A. Ahmed, and A.-S. F. Obada, “Multimode and multiphoton processes in a nonlinear Jaynes–Cummings model,” Physica A 170, 393–414 (1991).
[CrossRef]

M. S. Abdalla, M. Abdel-Aty, and A. S.-F. Obada, “Entropy and entanglement of time-dependent Jaynes–Cummings model,” Physica A 326, 203–219 (2003).
[CrossRef]

X.-P. Liao and M. F. Fang, “Entropy squeezing for a two-level atom in motion interacting with a quantized field,” Physica A 332, 176–184 (2004).
[CrossRef]

Proc. Am. Math. Soc. (1)

E. Pinney, “The nonlinear differential equation y′′+p(x)y+cy-3=0,” Proc. Am. Math. Soc. 1, 681 (1950).

Proc. IEEE (1)

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 (1963).
[CrossRef]

Rep. Prog. Phys. (1)

H. Walther, B. T. H. Varcoe, B.-G. Englert, and T. Becker, “Cavity quantum electrodynamics,” Rep. Prog. Phys. 69, 1325–1382 (2006).
[CrossRef]

Theor. Math. Phys. (1)

M. S. Abdalla and P. G. L. Leach, “Lie algebraic treatment of the quadratic invariants for the degenerate parametric amplifier,” Theor. Math. Phys. 159, 535–550 (2009).
[CrossRef]

Universita Izvestia Kiev Series III (1)

V. Ermakov, “Second order differential equations. Conditions of complete integrability,” Universita Izvestia Kiev Series III 9, 1–25 (1880) (translated from Russian by A. O. Harin).

Other (1)

M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley1974).

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Figures (4)

Fig. 1.
Fig. 1.

The atomic inversion σz(t) against the scaled time τ=λt for n¯=4, ϑ=π/3, θ=π, and C=0.02. (a) for μ=ω=2, (b) as (a) but for μ=4, (c) as (a) but for n¯=16, (d) as (c) but for ω=1.

Fig. 2.
Fig. 2.

The quadrature variances Nx(t) (dashed line) and Ny(t) (solid line) against the scaled time λt, for n¯=9, ϑ=π/3, θ=π,δ=π/2, and C=0.1. (a) For μ=1, ω=1; (b) as (a) but for μ=4; (c) as (a) but for ω=2; (d) as (b) but for ω=6.

Fig. 3.
Fig. 3.

The variance squeezing Vx (solid line) and Vy (dashed line) against the scaled time τ=λt, for n¯=1, ϑ=π/3, θ=π,δ=π/2, and C=0.02. (a) For μ=ω=1, (b) as (a) but for μ=6, (c) for μ=6,ω=2, (d) as (c) but for ω=4.

Fig. 4.
Fig. 4.

The entropy squeezing Sx(τ) (dashed line) and Sy(τ) (solid line) against the scaled time τ=λt, for n¯=4, ϑ=π/3, θ=π,δ=π/2, and C=0.02. (a) For μ=ω=1, (b) as (a) but for n¯=11, (c) as (b) but for μ=6, (d) as (b) but for ω=6.

Equations (39)

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

H^=ω(a^a^+12)+ω02σ^z+λ(a^+a^)(σ^++σ^),
[σ^±,σ^z]=2σ^±and[σ^+,σ^]=σ^z.
H^=12(p^2+ω2q^2)+ω02σ^z+gq^σ^x,
a^=(2ω)12(ωq^+ip^),a^=(2ω)12(ωq^ip^).
dI^dt=I^t+1i[I^,H^],
I^=ζ(t)p^2+ξ(t)q^2+ς(t)(q^p^+p^q^),
I^=Cρ2(t)q^2+(ρ(t)p^ρ˙(t)q^)2,
d2dt2ρ(t)+ω2ρ(t)=Cρ3(t),
ρ(t)=1ωμ2+νsin(2ωt+ϑ),
A^=(2C)12[(Cρ(t)iρ˙(t))q^+iρ(t)p^],A^=(2C)12[(Cρ(t)+iρ˙(t))q^iρ(t)p^],
I^=Ω(A^A^+12),whereΩ=2C.
q^=Ωρ(t)(A^+A^).
H^(t)=Ω(A^A^+12)+ω02σ^z+λ1(t)(A^+A^)(σ^++σ^),
V(t)=λ1(t)(σ^+e(iω0t)+σ^-e(iω0t))(A^e(iΩt)+A^e(iΩt)).
V(t)=λ1(t)(A^σ^+ei(Ωω0)t+A^σ^-ei(Ωω0)t).
|ψ(0)A=sin(θ2)|e+cos(θ2)eiϕ|g,
|α=n=0Qn|n,Qn=αnn!exp(|α|22),
|ψ(0)s=n=0Qn[sin(θ2)|e+cos(θ2)eiϕ|g]|n.
|ψ(t)=n=0Ce,n(t)|n,e+Cg,n+1(t)|n+1,g;
ddtCe,n(t)=iλ1(t)n+1ei(Ωω0)tCg,n+1(t),ddtCg,n+1(t)=iλ1(t)n+1ei(Ωω0)tCe,n(t).
Ce,n(0)=Qnsin(θ2)andCg,n+1(0)=eiϕQn+1cos(θ2),
Ce,n(t)=Qnsin(θ2)cos(λIn(t))ieiϕQn+1cos(θ2)sin(λIn(t)),Cg,n+1(t)=eiϕQn+1cos(θ2)cos(λIn(t))iQnsin(θ2)sin(λIn(t)),
In(t)=ω(n+1)C0tρ(τ)dτ.
Λ^atom(t)=Trfield|ψ(t)ψ(t)|.
σ^z(t)=n=0[|Ce,n(t)|2|Cg,n+1(t)|2],
σ^z(t)=exp(|α|2)n=0|α|2nn!cos(2λIn(t)).
ΔA^.ΔB^12|C^|,
X^=a^+a^2,Y^=a^a^i2
a^(t)=|α|e(|α|2+iδ)n=0|α|2nn![cos(λIn(t))cos(λIn+1(t))+(n+2)(n+1)sin(λIn(t)sin(λIn+1(t))],
a^2(t)=|α|2e(|α|2+2iδ)n=0|α|2nn![cos(λIn(t))cos(λIn+2(t))+(n+3)(n+1)sin(λIn(t)sin(λIn+2(t))],
a^(t)a^(t)=|α|2+n=0|α|2nn!e|α|2sin2(λIn(t)),
Vγ(t)=(Δσ^γ(t)|σ^z(t)|)<0,γ=xory.
Δσ^x(t)=14K2(t)sin2δ,Δσ^y(t)=14K2(t)cos2δ,
K(t)=n=0|α|2n+1e|α|2n!(n+1)!cos(λIn+1(t))sin(λIn(t)).
r=0N+1H(S^β)[N2ln(N2)+(1+N2)ln(1+N2)],
H(S^β)=j=1NPj(S^β)lnPj(S^β)β=x,y,z,
H(S^β)=[(12+S^β)ln(12+S^β)+(12S^β)ln(12S^β)].
δH(S^x)δH(S^y)δH(S^z)4.
E(S^β)=(δH(S^β)2|δH(S^z)|)<0,β=x,y.

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