Abstract

The analytical solution of the problem of two two-level atoms with degenerate two-photon transitions interacting with a single-mode radiation field in the presence of a parametric amplifier term is presented. The purity of the atomic state has been used to measure the degree of entanglement between the atom and the field. The temporal evolution of variance and entropy squeezing as well as atomic inversion for the single-atom case are studied. It has been shown that maximum squeezing for the variance and entropy squeezing occurs when the ratio between the amplifier coupling λ3 and the field frequency ω equals 0.26. Increasing the value of the ratio λ3ω further leads to the vanishing of squeezing from the system. It is also noted that the existence of the coupling parameter results in the system never reaching the pure state except at the points of revival times. The Q function has been also considered to give more information in the phase space about the system. These aspects are sensitive to changes in the amplifier parameter.

© 2010 Optical Society of America

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2009 (1)

J. S. Zhang, J. B. Xu, and Q. Lin, “Controlling entanglement sudden death in cavity QED by classical driving fields,” Eur. Phys. J. D 51, 283-288 (2009).
[CrossRef]

2008 (2)

M. S. Abdalla, E. Lashin, and G. Sadiek, “Entropy and variance squeezing for time-dependent two-coupled atoms in an external magnetic field,” J. Phys. B 41, 015502 (2008).
[CrossRef]

Faisal A. A. El-Orany, M. R. B. Wahiddin, and A.-S. F. Obada, “Single-atom entropy squeezing for two two-level atoms interacting with a single-mode radiation field,” Opt. Commun. 281, 2854-2863 (2008).
[CrossRef]

2007 (3)

E. M. Khalil, “Generation of a nonlinear two-mode stark shift via nondegenerate Raman transition,” Int. J. Mod. Phys. B 30, 5143-5158 (2007).
[CrossRef]

M. S. Abdalla, E. M. Khalil, and A. S.-F. Obada, “Statistical properties of a two-photon cavity mode in the presence of degenerate parametric amplifier,” Ann. Phys. 11, 2554-2568 (2007).
[CrossRef]

M. S. Abdalla, M. Abdel-Aty, and A.-S. F. Obada, “Sensitive response of the quantum entropies to Jaynes-Cummings model in presence of a second-harmonic generation,” Int. J. Theor. Phys. 46, 637-651 (2007).
[CrossRef]

2006 (4)

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

E. M. Khalil, M. S. Abdalla, and A. S.-F. Obada, “Entropy and variance squeezing of two coupled modes interacting with a two-level atom: frequency converter type,” Ann. Physics 321, 421-434 (2006).
[CrossRef]

A.-S. F. Obada, M. M. A. Ahmed, F. K. Faramawy, and E. M. Khalil, “Influence of Kerr-like medium on a nonlinear two-level atom,” Chaos, Solitons Fractals 28, 983-993 (2006).
[CrossRef]

2005 (2)

M. M. A. Ahmed, E. M. Khalil, and A.-S. F. Obada, “Generation of a nonlinear stark shift through the adiabatic elimination method,” Opt. Commun. 254, 76-87 (2005).
[CrossRef]

M. S. Abdalla, S. S. Hassan, and M. Abdel-Aty, “Entropic uncertainty in the Jaynes-Cummings model in presence of a second-harmonic generation,” Opt. Commun. 244, 431-443 (2005).
[CrossRef]

2004 (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]

2003 (2)

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]

2002 (4)

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: Quantum Semiclassical Opt. 4, 134-142 (2002).
[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]

Y. F. Gao, J. Feng, and S. R. Shi, “Cavity field spectra of the intensity-dependent two-mode Jaynes-Cummings model,” Int. J. Theor. Phys. 41, 867-875 (2002).
[CrossRef]

M. S. Kim, J. Y. Lee, D. Ahn, and P. L. Knight, “Entanglement induced by a single-mode heat environment,” Phys. Rev. A 65, 040101 (2002).
[CrossRef]

2001 (3)

J. M. Raimond, M. Brune, and S. Haroche, “Manipulating quantum entanglement with atoms and photons in a cavity,” Rev. Mod. Phys. 73, 565-582 (2001).
[CrossRef]

S. Bose, I. Fuentes-Guridi, P. L. Knight, and V. Vedral, “Subsystem purity as an enforcer of entanglement,” Phys. Rev. Lett. 87, 050401 (2001).
[CrossRef] [PubMed]

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. 34, 9129-9141 (2001).
[CrossRef]

2000 (3)

J. I. Cirac and P. Zoller, “A scalable quantum computer with ions in an array of microtraps,” Nature 404, 579-581 (2000).
[CrossRef] [PubMed]

V. Vedral and M. B. Plenio, “Entanglement measures and purification procedures,” Phys. Rev. A 57, 1619-1622 (2000).
[CrossRef]

M.-F. Fang, P. Zhou, and S. Swain, “Entropy squeezing for a two-level atom,” J. Mod. Opt. 47, 1043-1053 (2000).
[CrossRef]

1999 (1)

A. Carlson, M. Koashi, and N. Imoto, “Quantum entanglement for secret sharing and secret splitting,” Phys. Rev. A 59, 162-168 (1999).
[CrossRef]

1998 (2)

S. Bose, V. Vedral, and P. L. Knight, “Multiparticle generalization of entanglement swapping,” Phys. Rev. A 57, 822-829 (1998).
[CrossRef]

M. Murao, M. B. Plenio, S. Popescue, V. Vedral, and P. L. Knight, “Multiparticle entanglement purification protocols,” Phys. Rev. A 57, R4075-R4078 (1998).
[CrossRef]

1997 (2)

S. F. Huelga, C. Macchiavello, T. Pellizzari, A. K. Ekert, M. B. Plenio, and J. I. Cirac, “Improvement of frequency standards with quantum entanglement,” Phys. Rev. Lett. 79, 3865-3868 (1997).
[CrossRef]

J. Sanchez-Ruiz, “Asymptotic formula for the quantum entropy of position in energy eigenstates,” Phys. Lett. A 226, 7-13 (1997).
[CrossRef]

1995 (3)

J. Sanchez-Ruiz, “Improved bounds in the entropic uncertainty and certainty relations for complementary observables,” Phys. Lett. A 201, 125-131 (1995).
[CrossRef]

P. W. Shor, “Scheme for reducing decoherence in quantum computer memory,” Phys. Rev. A 52, R2493-R2496 (1995).
[CrossRef] [PubMed]

D. P. Di Vincenzo, “Quantum computation,” Science 270, 255-261 (1995).
[CrossRef]

1993 (4)

Y. Yamamoto and R. E. Slusher, “Optical processes in Microcavities,” Phys. Today 46, 66-73 (1993).
[CrossRef]

T. Nasreen and M. S. K. Razmi, “Atomic emission and cavity field spectra for a two-photon Jaynes-Cummings model in the presence of Stark shift,” J. Opt. Soc. Am. 10, 1292-1300 (1993).
[CrossRef]

C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895-1899 (1993).
[CrossRef] [PubMed]

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

1992 (4)

C. H. Bennet and S. J. Weisner, “Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states,” Phys. Rev. Lett. 69, 2881-2884 (1992).
[CrossRef]

D. J. Gauthier, L. Q. Wu, S. E. Morrin, and T. W. Mossberg, “Realization of a continuous-wave, two-photon optical laser,” Phys. Rev. Lett. 68, 464-467 (1992).
[CrossRef] [PubMed]

V. Buźek, H. Moya-Cessa, and P. L. Knight, “Schrödinger-cat states in the resonant Jaynes-Cummings model: collapse and revival of oscillations of the photon-number distribution,” Phys. Rev. A 45, 8190-8203 (1992).
[CrossRef] [PubMed]

P. Szlachetka, K. Grygiel, J. Bajer, and J. Peřina, “Chaos and order in second-harmonic generation: cumulant approach,” Phys. Rev. A 46, 7311-7314 (1992).
[CrossRef] [PubMed]

1991 (4)

S. J. D. Phoenix and P. L. Knight, “Establishment of an entangled atom-field state in the Jaynes-Cummings model,” Phys. Rev. A 44, 6023-6029 (1991).
[CrossRef] [PubMed]

S. J. D. Phoenix and P. L. Knight, “An example of state preparation by a quantum apparatus,” Phys. Rev. Lett. 66, 2833-2833 (1991).
[CrossRef] [PubMed]

P. A. M. Netto, L. Davidovich, and J. M. Raimond, “Theory of the nondegenerate two-photon micromaser,” Phys. Rev. A 43, 5073-5089 (1991).
[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]

1990 (3)

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]

S. C. Gau, “Time evolution of a two-mode Jaynes-Cummings model in the presence of pair-coherent states,” J. Mod. Opt. 37, 1469-1486 (1990).
[CrossRef]

S. J. D. Phoenix and P. L. Knight, “Periodicity, phase, and entropy in models of two photon resonance,” J. Opt. Soc. Am. B 7, 116-124 (1990).
[CrossRef]

1988 (1)

R. R. Puri and G. S. Agarwal, “Coherent two-photon transitions in Rydberg atoms in a cavity with finite Q,” Phys. Rev. A 37, 3879-3883 (1988).
[CrossRef] [PubMed]

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]

1982 (1)

P. L. Knight and P. M. Radamore, “Quantum origin of dephasing and revivals in the coherent-state Jaynes-Cummings model,” Phys. Rev. A 26, 676-679 (1982).
[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]

Abdalla, M. S.

M. S. Abdalla, E. Lashin, and G. Sadiek, “Entropy and variance squeezing for time-dependent two-coupled atoms in an external magnetic field,” J. Phys. B 41, 015502 (2008).
[CrossRef]

M. S. Abdalla, E. M. Khalil, and A. S.-F. Obada, “Statistical properties of a two-photon cavity mode in the presence of degenerate parametric amplifier,” Ann. Phys. 11, 2554-2568 (2007).
[CrossRef]

M. S. Abdalla, M. Abdel-Aty, and A.-S. F. Obada, “Sensitive response of the quantum entropies to Jaynes-Cummings model in presence of a second-harmonic generation,” Int. J. Theor. Phys. 46, 637-651 (2007).
[CrossRef]

E. M. Khalil, M. S. Abdalla, and A. S.-F. Obada, “Entropy and variance squeezing of two coupled modes interacting with a two-level atom: frequency converter type,” Ann. Physics 321, 421-434 (2006).
[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, S. S. Hassan, and M. Abdel-Aty, “Entropic uncertainty in the Jaynes-Cummings model in presence of a second-harmonic generation,” Opt. Commun. 244, 431-443 (2005).
[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, M. Abdel-Aty, and A. S.-F. Obada, “Entropy and entanglement of time-dependent Jaynes-Cummings model,” Physica A 326, 203-219 (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: Quantum Semiclassical Opt. 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. 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]

Abdel-Aty, M.

M. S. Abdalla, M. Abdel-Aty, and A.-S. F. Obada, “Sensitive response of the quantum entropies to Jaynes-Cummings model in presence of a second-harmonic generation,” Int. J. Theor. Phys. 46, 637-651 (2007).
[CrossRef]

M. S. Abdalla, S. S. Hassan, and M. Abdel-Aty, “Entropic uncertainty in the Jaynes-Cummings model in presence of a second-harmonic generation,” Opt. Commun. 244, 431-443 (2005).
[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, M. Abdel-Aty, and A. S.-F. Obada, “Entropy and entanglement of time-dependent Jaynes-Cummings model,” Physica A 326, 203-219 (2003).
[CrossRef]

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M. S. Kim, J. Y. Lee, D. Ahn, and P. L. Knight, “Entanglement induced by a single-mode heat environment,” Phys. Rev. A 65, 040101 (2002).
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S. Bose, I. Fuentes-Guridi, P. L. Knight, and V. Vedral, “Subsystem purity as an enforcer of entanglement,” Phys. Rev. Lett. 87, 050401 (2001).
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M. Murao, M. B. Plenio, S. Popescue, V. Vedral, and P. L. Knight, “Multiparticle entanglement purification protocols,” Phys. Rev. A 57, R4075-R4078 (1998).
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B. W. Shore and P. L. Knight, “The Jaynes-Cummings model,” J. Mod. Opt. 40, 1195-1238 (1993).
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V. Buźek, H. Moya-Cessa, and P. L. Knight, “Schrödinger-cat states in the resonant Jaynes-Cummings model: collapse and revival of oscillations of the photon-number distribution,” Phys. Rev. A 45, 8190-8203 (1992).
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S. J. D. Phoenix and P. L. Knight, “An example of state preparation by a quantum apparatus,” Phys. Rev. Lett. 66, 2833-2833 (1991).
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A. Carlson, M. Koashi, and N. Imoto, “Quantum entanglement for secret sharing and secret splitting,” Phys. Rev. A 59, 162-168 (1999).
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M. S. Abdalla, E. Lashin, and G. Sadiek, “Entropy and variance squeezing for time-dependent two-coupled atoms in an external magnetic field,” J. Phys. B 41, 015502 (2008).
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M. S. Kim, J. Y. Lee, D. Ahn, and P. L. Knight, “Entanglement induced by a single-mode heat environment,” Phys. Rev. A 65, 040101 (2002).
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D. J. Gauthier, L. Q. Wu, S. E. Morrin, and T. W. Mossberg, “Realization of a continuous-wave, two-photon optical laser,” Phys. Rev. Lett. 68, 464-467 (1992).
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D. J. Gauthier, L. Q. Wu, S. E. Morrin, and T. W. Mossberg, “Realization of a continuous-wave, two-photon optical laser,” Phys. Rev. Lett. 68, 464-467 (1992).
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V. Buźek, H. Moya-Cessa, and P. L. Knight, “Schrödinger-cat states in the resonant Jaynes-Cummings model: collapse and revival of oscillations of the photon-number distribution,” Phys. Rev. A 45, 8190-8203 (1992).
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M. Murao, M. B. Plenio, S. Popescue, V. Vedral, and P. L. Knight, “Multiparticle entanglement purification protocols,” Phys. Rev. A 57, R4075-R4078 (1998).
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T. Nasreen and M. S. K. Razmi, “Atomic emission and cavity field spectra for a two-photon Jaynes-Cummings model in the presence of Stark shift,” J. Opt. Soc. Am. 10, 1292-1300 (1993).
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P. A. M. Netto, L. Davidovich, and J. M. Raimond, “Theory of the nondegenerate two-photon micromaser,” Phys. Rev. A 43, 5073-5089 (1991).
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M. S. Abdalla, E. M. Khalil, and A. S.-F. Obada, “Statistical properties of a two-photon cavity mode in the presence of degenerate parametric amplifier,” Ann. Phys. 11, 2554-2568 (2007).
[CrossRef]

E. M. Khalil, M. S. Abdalla, and A. S.-F. Obada, “Entropy and variance squeezing of two coupled modes interacting with a two-level atom: frequency converter type,” Ann. Physics 321, 421-434 (2006).
[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).
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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]

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Faisal A. A. El-Orany, M. R. B. Wahiddin, and A.-S. F. Obada, “Single-atom entropy squeezing for two two-level atoms interacting with a single-mode radiation field,” Opt. Commun. 281, 2854-2863 (2008).
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M. S. Abdalla, M. Abdel-Aty, and A.-S. F. Obada, “Sensitive response of the quantum entropies to Jaynes-Cummings model in presence of a second-harmonic generation,” Int. J. Theor. Phys. 46, 637-651 (2007).
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A.-S. F. Obada, M. M. A. Ahmed, F. K. Faramawy, and E. M. Khalil, “Influence of Kerr-like medium on a nonlinear two-level atom,” Chaos, Solitons Fractals 28, 983-993 (2006).
[CrossRef]

M. M. A. Ahmed, E. M. Khalil, and A.-S. F. Obada, “Generation of a nonlinear stark shift through the adiabatic elimination method,” Opt. Commun. 254, 76-87 (2005).
[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: Quantum Semiclassical Opt. 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. 34, 9129-9141 (2001).
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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).
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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]

Pellizzari, T.

S. F. Huelga, C. Macchiavello, T. Pellizzari, A. K. Ekert, M. B. Plenio, and J. I. Cirac, “Improvement of frequency standards with quantum entanglement,” Phys. Rev. Lett. 79, 3865-3868 (1997).
[CrossRef]

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C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895-1899 (1993).
[CrossRef] [PubMed]

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).
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Phoenix, S. J. D.

S. J. D. Phoenix and P. L. Knight, “Establishment of an entangled atom-field state in the Jaynes-Cummings model,” Phys. Rev. A 44, 6023-6029 (1991).
[CrossRef] [PubMed]

S. J. D. Phoenix and P. L. Knight, “An example of state preparation by a quantum apparatus,” Phys. Rev. Lett. 66, 2833-2833 (1991).
[CrossRef] [PubMed]

S. J. D. Phoenix and P. L. Knight, “Periodicity, phase, and entropy in models of two photon resonance,” J. Opt. Soc. Am. B 7, 116-124 (1990).
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V. Vedral and M. B. Plenio, “Entanglement measures and purification procedures,” Phys. Rev. A 57, 1619-1622 (2000).
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M. Murao, M. B. Plenio, S. Popescue, V. Vedral, and P. L. Knight, “Multiparticle entanglement purification protocols,” Phys. Rev. A 57, R4075-R4078 (1998).
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S. F. Huelga, C. Macchiavello, T. Pellizzari, A. K. Ekert, M. B. Plenio, and J. I. Cirac, “Improvement of frequency standards with quantum entanglement,” Phys. Rev. Lett. 79, 3865-3868 (1997).
[CrossRef]

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M. Murao, M. B. Plenio, S. Popescue, V. Vedral, and P. L. Knight, “Multiparticle entanglement purification protocols,” Phys. Rev. A 57, R4075-R4078 (1998).
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R. R. Puri and G. S. Agarwal, “Coherent two-photon transitions in Rydberg atoms in a cavity with finite Q,” Phys. Rev. A 37, 3879-3883 (1988).
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P. L. Knight and P. M. Radamore, “Quantum origin of dephasing and revivals in the coherent-state Jaynes-Cummings model,” Phys. Rev. A 26, 676-679 (1982).
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J. M. Raimond, M. Brune, and S. Haroche, “Manipulating quantum entanglement with atoms and photons in a cavity,” Rev. Mod. Phys. 73, 565-582 (2001).
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P. A. M. Netto, L. Davidovich, and J. M. Raimond, “Theory of the nondegenerate two-photon micromaser,” Phys. Rev. A 43, 5073-5089 (1991).
[CrossRef]

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T. Nasreen and M. S. K. Razmi, “Atomic emission and cavity field spectra for a two-photon Jaynes-Cummings model in the presence of Stark shift,” J. Opt. Soc. Am. 10, 1292-1300 (1993).
[CrossRef]

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M. S. Abdalla, E. Lashin, and G. Sadiek, “Entropy and variance squeezing for time-dependent two-coupled atoms in an external magnetic field,” J. Phys. B 41, 015502 (2008).
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M. Sargent, M. O. Scully, and W. E. Lamb Jr., Laser Physics (Addison-Wesley, 1974).

Shi, S. R.

Y. F. Gao, J. Feng, and S. R. Shi, “Cavity field spectra of the intensity-dependent two-mode Jaynes-Cummings model,” Int. J. Theor. Phys. 41, 867-875 (2002).
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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).
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M.-F. Fang, P. Zhou, and S. Swain, “Entropy squeezing for a two-level atom,” J. Mod. Opt. 47, 1043-1053 (2000).
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P. Szlachetka, K. Grygiel, J. Bajer, and J. Peřina, “Chaos and order in second-harmonic generation: cumulant approach,” Phys. Rev. A 46, 7311-7314 (1992).
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Figures (5)

Fig. 1
Fig. 1

Time evolution of linear entropy as a function of scaled time λ t with the atoms initially in excited state, and the field is prepared in a coherent state with fixed parameter α = 5 ; (a) λ 3 0 , (b) λ 3 ω = 0.3 , (c) λ 3 ω = 0.49 , and (d) λ 3 ω = 0.4999 .

Fig. 2
Fig. 2

Time evolution of variance squeezing as a function of scaled time λ t with the atoms initially in excited state, and the field is prepared in a coherent state with fixed parameter α = 5 ; (a) λ 3 0 , (b) λ 3 ω = 0.26 , (c) λ 3 ω = 0.35 , (d) λ 3 ω = 0.45 , (e) λ 3 ω = 0.49 , and (f) λ 3 ω = 0.4999 .

Fig. 3
Fig. 3

Time evolution of entropy squeezing where the parameters are the same as Fig. 2.

Fig. 4
Fig. 4

Time evolution of atomic inversion as a function of scaled time λ t with the atoms initially in excited state, and the field is prepared in a coherent state with fixed parameter α = 5 ; (a) λ 3 0 , (b) λ 3 ω = 0.3 , (c) λ 3 ω = 0.49 , and (d) λ 3 ω 0.5.

Fig. 5
Fig. 5

Q function at t = π 2 λ with fixed parameters α = 5 where (a) λ 3 ω 0 , (b) λ 3 ω = 0.3 , (c) λ 3 ω = 0.45 , and (d) λ 3 ω = 0.4999.

Equations (66)

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H ̂ ( t ) = ω a ̂ a ̂ + ω 0 2 σ ̂ z + k ( a ̂ + a ̂ ) ( σ ̂ + + σ ̂ ) + ( a ̂ 2 ξ ( t ) + a ̂ 2 ξ * ( t ) ) ,
H ̂ = ω a ̂ a ̂ + i = 1 2 Ω i S ̂ i i + i g 0 ( a ̂ 2 a ̂ 2 ) ( S ̂ + + S ̂ ) + g 1 ( a ̂ 2 + a ̂ 2 ) ,
H ̂ = ω a ̂ a ̂ + j = 1 2 Ω j 2 S ̂ z ( j ) + i ( a ̂ 2 a ̂ 2 ) i = 1 2 λ i ( S ̂ + ( i ) + S ̂ ( i ) ) + λ 3 ( a ̂ 2 + a ̂ 2 ) ,
a ̂ = b ̂ cosh ζ b ̂ sinh ζ , a ̂ = b ̂ cosh ζ b ̂ sinh ζ ,
H ̂ = Ω n ̂ + j = 1 2 ( Ω 0 2 S ̂ z ( j ) + i λ j ( b ̂ 2 S ̂ ( j ) b ̂ 2 S ̂ + ( j ) ) ) ,
d n ̂ d t = 2 j = 1 2 λ j ( b ̂ 2 S ̂ ( j ) + b ̂ 2 S ̂ + ( j ) ) ,
d S ̂ z ( 1 ) d t = 2 λ 1 ( b ̂ 2 S ̂ ( 1 ) + b ̂ 2 S ̂ + ( 1 ) ) ,
d S ̂ z ( 2 ) d t = 2 λ 2 ( b ̂ 2 S ̂ ( 2 ) + b ̂ 2 S ̂ + ( 2 ) ) ,
N ̂ = n ̂ + 1 2 j = 1 2 S ̂ z ( j ) ,
H ̂ = ω N ̂ + C ̂ ,
C ̂ = j = 1 2 ( Δ 2 S ̂ z ( j ) + i λ j ( b ̂ 2 S ̂ ( j ) b ̂ 2 S ̂ + ( j ) ) ) .
Δ = Ω 0 2 Ω .
| ψ ( 0 ) = [ c 1 | + , + + c 2 | + , + c 3 | , + + c 4 | , ] | β ,
| c 1 | 2 + | c 2 | 2 + | c 3 | 2 + | c 4 | 2 = 1 .
β = α cosh ζ + α * sinh ζ ,
| β = n = 0 Q n | n , Q n = 1 2 n n ! cosh ( β sinh 2 ζ ) × ( tanh ζ ) n 2 H n ( β sinh 2 ζ ) exp ( 1 2 ( | β | 2 + β 2 tanh ζ ) ) ,
| ψ ( t ) = n = 0 c ¯ 1 ( n , t ) | + , + | n + c ¯ 2 ( n , t ) | + , | n + 2 + c ¯ 3 ( n , t ) | , + | n + 2 + c ¯ 4 ( n , t ) | , | n + 4 .
c ¯ i ( n , t ) = j = 1 4 A i j ( n , t ) ,
A 11 ( n , t ) = c 1 Q n [ 1 ν 1 2 ( n ) sin 2 μ ( n ) t ( μ ( n ) ) 2 ] ,
A 12 ( n , t ) = i c 2 Q n + 2 ν 1 ( n ) sin 2 μ ( n ) t 2 μ ( n ) ,
A 13 ( n , t ) = i c 3 Q n + 2 ν 1 ( n ) sin 2 μ ( n ) t 2 μ ( n ) ,
A 14 ( n , t ) = c 4 Q n + 4 ν 1 ( n ) ν 2 ( n ) sin 2 μ ( n ) t ( μ ( n ) ) 2 ,
A 21 ( n , t ) = i c 1 Q n ν 1 ( n ) sin 2 μ ( n ) t 2 μ ( n ) ,
A 22 ( n , t ) = c 2 Q n + 2 cos 2 μ ( n ) t ,
A 23 ( n , t ) = c 3 Q n + 2 sin 2 μ ( n ) t ,
A 24 ( n , t ) = i c 4 Q n + 4 ν 2 ( n ) sin 2 μ ( n ) t 2 μ ( n ) ,
A 33 ( n , t ) = c 3 Q n + 2 cos 2 μ ( n ) t ,
A 32 ( n , t ) = c 2 Q n + 2 sin 2 μ ( n ) t ,
A 41 ( n , t ) = c 1 Q n ν 1 ( n ) ν 2 ( n ) sin 2 μ ( n ) t ( μ ( n ) ) 2 ,
A 42 ( n , t ) = i c 2 Q n + 2 ν 2 ( n ) sin 2 μ ( n ) t 2 μ ( n ) ,
A 43 ( n , t ) = i c 3 Q n + 2 ν 2 ( n ) sin 2 μ ( n ) t 2 μ ( n ) ,
A 44 ( n , t ) = c 4 Q n + 4 [ 1 ν 2 2 ( n ) sin 2 μ ( n ) t ( μ ( n ) ) 2 ] ,
| ψ ( t ) = n = 0 { A 11 ( n , t ) | + , + , n + A 21 ( n , t ) | + , , n + 2 + A 31 ( n , t ) | , + , n + 2 + A 41 ( n , t ) | , , n + 4 } ,
ρ ̂ f ( t ) = T r a | ψ ( t ) ψ ( t ) | .
P ( t ) = 1 T r [ ρ ̂ f 2 ( t ) ] ,
P ( t ) = 1 ( n = 0 | A 11 ( n , t ) | 2 ) 2 4 ( n = 0 | A 21 ( n , t ) | 2 ) 2 ( n = 0 | A 41 ( n , t ) | 2 ) 2 4 | n = 0 A 11 ( n + 2 , t ) A 21 * ( n , t ) | 2 2 | n = 0 A 11 ( n + 4 , t ) A 41 * ( n , t ) | 2 4 | n = 0 A 21 ( n + 2 , t ) A 41 * ( n , t ) | 2 .
( Δ A ̂ ) 2 ( Δ B ̂ ) 2 1 4 | C ̂ | 2 ,
Δ S ̂ x Δ S ̂ y 1 2 | S ̂ z | .
V ( S ̂ α ) = ( Δ S ̂ α ( t ) | S ̂ z ( t ) 2 | ) < 0 , α = x or y .
ρ ̂ a ( t ) = ρ ̂ 11 ( t ) + ρ ̂ 12 ( t ) + ρ ̂ 21 ( t ) + ρ ̂ 22 ( t ) ,
ρ ̂ 11 ( t ) = n = 0 ( | A 11 ( n , t ) | 2 + | A 12 ( n , t ) | 2 ) | + + | ,
ρ ̂ 12 ( t ) = ( ρ ̂ 21 ( t ) ) = n = 0 A 12 ( n + 2 , t ) A 14 ( n , t ) | + | ,
ρ ̂ 22 ( t ) = n = 0 ( | A 13 ( n , t ) | 2 + | A 14 ( n , t ) | 2 ) | | .
S ̂ z ( t ) = n = 0 ( | A 11 ( n , t ) | 2 + | A 12 ( n , t ) | 2 ) ( | A 13 ( n , t ) | 2 + | A 14 ( n , t ) | 2 )
S ̂ y ( t ) = 2 n = 0 ( A 12 ( n + 2 , t ) A 14 ( n , t ) A 11 ( n + 2 , t ) A 13 ( n , t ) ) ,
k = 1 N + 1 H ( S ̂ k ) N 2 ln ( N 2 ) + ( 1 + N 2 ) ln ( 1 + N 2 ) ,
H ( S ̂ x ) + H ( S ̂ y ) + H ( S ̂ z ) 2 ln 2 .
δ H ( S ̂ x ) δ H ( S ̂ y ) δ H ( S ̂ z ) 4 .
S ̂ α | Ψ α i = λ α i | Ψ α i , α = x , y , z i = 1 , 2 , , N .
H ( S ̂ α ) = i = 1 N P i ( S ̂ α ) ln P i ( S ̂ α ) , α = x , y , z .
H ( S ̂ α ) = 1 2 [ ρ α ( t ) + 1 ] ln [ 1 2 [ ρ α ( t ) + 1 ] ] 1 2 [ 1 ρ α ( t ) ] ln [ 1 2 [ 1 ρ α ( t ) ] ] , α = x , y , z
ρ y ( t ) = 2 Im ( n = 0 [ A 11 ( n + 2 , t ) A 31 * ( n , t ) A 21 ( n + 2 , t ) A 41 * ( n , t ) ] ) ,
ρ z ( t ) = ( n = 0 [ | A 11 ( n , t ) | 2 + | A 21 ( n , t ) | 2 ] ) ( n = 0 [ | A 31 ( n , t ) | 2 + | A 41 ( n , t ) | 2 ] ) ,
ρ x ( t ) = 0.
E ( S ̂ α ) = δ H ( S ̂ α ) 2 δ H ( S ̂ z ) < 0 ,
Q ( γ , t ) = 1 π γ | ρ ̂ f ( t ) | γ .
Q ( γ , t ) = 1 π l , m = 0 ρ l , m ( t ) ( γ * ) l ( γ ) m l ! m ! ,
Q ( γ , t ) = 1 π exp [ ( | γ | 2 + | β | 2 ) ] × ( | n = 0 γ n A 11 ( n , t ) n ! | 2 + | n = 0 γ n + 2 A 21 ( n , t ) ( n + 2 ) ! | 2 + | n = 0 γ n + 2 A 31 ( n , t ) ( n + 2 ) ! | 2 + | n = 0 γ n + 4 A 41 ( n , t ) ( n + 4 ) ! | 2 ) .
M t = i D M ,
M = [ c 1 c 2 c 3 c 4 ] , D = [ 0 ν 1 ( n ) ν 1 ( n ) 0 ν 1 ( n ) 0 0 ν 2 ( n ) ν 1 ( n ) 0 0 ν 2 ( n ) 0 ν 2 ( n ) ν 2 ( n ) 0 ] ,
ν 1 ( n ) = λ ( n + 1 ) ( n + 2 ) , ν 2 ( n ) = λ ( n + 3 ) ( n + 4 ) .
[ c ¯ 1 ( n , t ) c ¯ 2 ( n , t ) c ¯ 3 ( n , t ) c ¯ 4 ( n , t ) ] = j = 0 4 A i j ( n , t ) c j = exp [ i D t ] [ c ¯ 1 ( n , o ) c ¯ 2 ( n , o ) c ¯ 3 ( n , o ) c ¯ 4 ( n , o ) ] ,
D 2 = 2 [ ν 1 2 ( n ) 0 0 ν 2 ( n ) ν 1 ( n ) 0 μ 2 ( n ) μ 2 ( n ) 0 0 μ 2 ( n ) μ 2 ( n ) 0 ν 2 ( n ) ν 1 ( n ) 0 0 ν 2 2 ( n ) ] ,
D 2 n = ( μ ( n ) ) 2 n 2 D 2 and D 2 n + 1 = ( 2 μ ( n ) ) 2 n D ,
exp ( i D t ) = I ̂ i D 1 ! t D 2 2 ! t 2 + i D 3 3 ! t 3 + D 4 4 ! t 4 + . + ( i ) n D n n ! t n + .
exp ( i D t ) = ( I i D 2 μ ( n ) sin 2 μ ( n ) t ) D 2 2 μ 2 ( n ) sin 2 μ ( n ) t .

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