Abstract

We study how intrinsic decoherence leads to growing entropy and a strong degradation of the maximal generated entanglement in the multiquanta Jaynes–Cummings model. We find an exact solution of the Milburn equation in multiquanta precesses and calculate the partial entropy of the particle (atom or trapped ion) and field subsystem as well as total entropy. As the total entropy is not conserved, and it is shown to increase as time develops, one cannot use the partial field or atomic entropy as a direct measure of particle–field entanglement. For a good entropy measure, we also calculate the negativity of the eigenvalues of the partially transposed density matrix. We find that, at least qualitatively, the difference of the total entropy to the sum of field and atom partial entropies can be also used as an entanglement measure. Our results show that the degree of entanglement is very sensitive to any change in the intrinsic decoherence parameter.

© 2004 Optical Society of America

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  45. N. B. Narozhny, J. J. Sanchez-Mondragon, and J. H. Eberly, “Coherence versus incoherence: collapse and revival in a simple quantum model,” Phys. Rev. A 23, 236–247 (1981).
    [CrossRef]
  46. M. Brune, J. M. Raimond, and S. Haroche, “Theory of the Rydberg-atom two-photon micromaser,” Phys. Rev. A 35, 154–163 (1987).
    [CrossRef] [PubMed]
  47. L. Davidovich, J. M. Raimond, M. Brune, and S. Haroche, “Quantum theory of a two-photon micromaser,” Phys. Rev. A 36, 3771–3787 (1987).
    [CrossRef] [PubMed]
  48. M. Brune, J. M. Raimond, P. Goy, L. Davidovich, and S. Haroche, “Realization of a two-photon maser oscillator,” Phys. Rev. Lett. 59, 1899–1902 (1987).
    [CrossRef] [PubMed]
  49. J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, “Periodic spontaneous collapse and revival in a simple quantum model,” Phys. Rev. Lett. 44, 1323–1326 (1980).
    [CrossRef]
  50. G. Rempe, H. Walther, and N. Klein, “Observation of quantum collapse and revival in a one-atom maser,” Phys. Rev. Lett. 58, 353–356 (1987).
    [CrossRef] [PubMed]
  51. D. Meschede, H. Walther, and G. Muller, “One-atom maser,” Phys. Rev. Lett. 54, 551–554 (1985).
    [CrossRef] [PubMed]
  52. P. Filipowicz, J. Javanainen, and P. Meystre, “Theory of a microscopic maser,” Phys. Rev. A 34, 3077–3087 (1986).
    [CrossRef] [PubMed]

2002 (2)

M. Abdel-Aty, S. Furuichi, and A.-S. F. Obada, “Entanglement degree of a nonlinear multiphoton Jaynes-Cummings model,” J. Opt. B: Quantum Semiclass. Opt. 4, 37–43 (2002).
[CrossRef]

G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65, 032314 (2002).
[CrossRef]

2001 (2)

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]

S. Bose, I. Fuentes-Guridi, P. L. Knight, and V. Vedral, “Erratum: Subsystem purity as an enforcer of entanglement [Phys. Rev. Lett. 87, 050401 (2001)],” Phys. Rev. Lett. 87, 279901 (2001).
[CrossRef]

2000 (3)

A.-S. F. Obada and M. Abdel-Aty, “Influence of the Stark shift and Kerr-like medium on the evolution of field entropy and entanglement in two-photon processes,” Acta Phys. Pol. B 31, 589–599 (2000).

L. Henderson and V. Vedral, “Information, relative entropy of entanglement, and irreversibility,” Phys. Rev. Lett. 84, 2263–2266 (2000).
[CrossRef] [PubMed]

M. Horodecki, P. Horodecki, and R. Horodecki, “Limits for entanglement measures,” Phys. Rev. Lett. 84, 2014–2017 (2000).
[CrossRef] [PubMed]

1999 (1)

M. Hillery, V. Buzseek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A 59, 1829–1834 (1999).
[CrossRef]

1998 (1)

A.-S. F. Obada, A. M. Abdel-Hafez, and H. A. Hessian, “Influence of the intrinsic decoherence on nonclassical effects in the nondegenerate bimodal multiquanta Jaynes–Cummings model,” J. Phys. B: At. Mol. Opt. Phys. 31, 5085–5104 (1998).
[CrossRef]

1997 (3)

R. Omnes, “General theory of the decoherence effect in quantum mechanics,” Phys. Rev. A 56, 3383 (1997).
[CrossRef]

I. L. Chuang and Y. Yamamoto, “Creation of a persistent quantum bit using error correction,” Phys. Rev. A 55, 114–127 (1997).
[CrossRef]

V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight, “Quantifying entanglement,” Phys. Rev. Lett. 78, 2275–2279 (1997).
[CrossRef]

1996 (3)

C. H. Bennett, D. P. Di Vincenzo, J. A. Smolin, and W. K. Wootters, “Mixed-state entanglement and quantum error correction,” Phys. Rev. A 54, 3824–3851 (1996).
[CrossRef] [PubMed]

C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, “Concentrating partial entanglement by local operations,” Phys. Rev. A 53, 2046–2052 (1996).
[CrossRef] [PubMed]

M. Brune, E. Hagley, J. Dreyer, X. Matre, A. Maali, C. Wunderlich, J. M. Raimond, and S. Haroche, “Observing the progressive decoherence of the ‘meter’ in a quantum measurement,” Phys. Rev. Lett. 77, 4887 (1996).
[CrossRef] [PubMed]

1995 (1)

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

1994 (1)

M. F. Fang and G. H. Zhou, “Influence of atomic coherence on the evolution of field entropy in multiphoton processes,” Phys. Lett. A 184, 397–402 (1994).
[CrossRef]

1993 (3)

G. J. Milburn, “Reply to comment on intrinsic decoherence in quantum mechanics,” Phys. Rev. A 47, 2415–2416 (1993).
[CrossRef] [PubMed]

H. Moya-Cessa, V. Buzeek, M. S. Kim, and P. L. Knight, “Intrinsic decoherence in the atom–field interaction,” Phys. Rev. A 48, 3900–3905 (1993).
[CrossRef] [PubMed]

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

1992 (3)

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

V. Buzseek, H. Moya-Cessa, P. L. Knight, and S. J. D. Phoenix, “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]

J. Gea-Bancloche, “A new look at the Jaynes–Cummings model for large fields: Bloch sphere evolution and detuning effects,” Opt. Commun. 88, 531 (1992).
[CrossRef]

1991 (4)

J. Gea-Bancloche, “Atom- and field-state evolution in the Jaynes–Cummings model for large initial fields,” Phys. Rev. A 44, 5913–5931 (1991).
[CrossRef]

G. J. Milburn, “Intrinsic decoherence in quantum mechanics,” Phys. Rev. A 44, 5401–5406 (1991).
[CrossRef] [PubMed]

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, “Comment on ‘Collapse and revival of the state vector in the Jaynes–Cummings model: an example of state preparation by a quantum apparatus, ’” Phys. Rev. Lett. 66, 2833 (1991).
[CrossRef]

1990 (5)

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]

J. Gea-Bancloche, “Collapse and revival of the state vector in the Jaynes–Cummings model: an example of state preparation by a quantum apparatus,” Phys. Rev. Lett. 65, 3385–3388 (1990).
[CrossRef]

J. Ellis, S. Mohanty, and D. V. Nanopaulos, “Wormholes violate quantum mechanics in SQUIDs,” Phys. Lett. B 235, 305 (1990).
[CrossRef]

G. C. Ghirardi, P. Pearle, and A. Rimini, “Markov processes in Hilbert space and continuous spontaneous localization of systems of identical particles,” Phys. Rev. A 42, 78–89 (1990).
[CrossRef] [PubMed]

G. C. Ghirardi, R. Grassi, and A. Rimini, “Continuous-spontaneous-reduction model involving gravity,” Phys. Rev. A 42, 1057–1064 (1990).
[CrossRef] [PubMed]

1989 (2)

L. Disi, “Models for universal reduction of macroscopic quantum fluctuations,” Phys. Rev. A 40, 1165–1174 (1989).
[CrossRef]

J. Ellis, S. Mohanty, and D. V. Nanopaulos, “Quantum gravity and the collapse of the wavefunction,” Phys. Lett. B 221, 113 (1989).
[CrossRef]

1988 (1)

S. J. D. Phoenix and P. L. Knight, “Fluctuation and entropy in models of quantum optical resonance,” Ann. Phys. (N.Y.) 186, 381 (1988).
[CrossRef]

1987 (5)

C. M. Caves and G. J. Milburn, “Quantum-mechanical model for continuous position measurements,” Phys. Rev. A 36, 5543 (1987).
[CrossRef] [PubMed]

M. Brune, J. M. Raimond, and S. Haroche, “Theory of the Rydberg-atom two-photon micromaser,” Phys. Rev. A 35, 154–163 (1987).
[CrossRef] [PubMed]

L. Davidovich, J. M. Raimond, M. Brune, and S. Haroche, “Quantum theory of a two-photon micromaser,” Phys. Rev. A 36, 3771–3787 (1987).
[CrossRef] [PubMed]

M. Brune, J. M. Raimond, P. Goy, L. Davidovich, and S. Haroche, “Realization of a two-photon maser oscillator,” Phys. Rev. Lett. 59, 1899–1902 (1987).
[CrossRef] [PubMed]

G. Rempe, H. Walther, and N. Klein, “Observation of quantum collapse and revival in a one-atom maser,” Phys. Rev. Lett. 58, 353–356 (1987).
[CrossRef] [PubMed]

1986 (2)

P. Filipowicz, J. Javanainen, and P. Meystre, “Theory of a microscopic maser,” Phys. Rev. A 34, 3077–3087 (1986).
[CrossRef] [PubMed]

G. C. Ghirardi, A. Rimini, and T. Weber, “Unified dynamics for microscopic and macroscopic systems,” Phys. Rev. D 34, 470–491 (1986).
[CrossRef]

1985 (1)

D. Meschede, H. Walther, and G. Muller, “One-atom maser,” Phys. Rev. Lett. 54, 551–554 (1985).
[CrossRef] [PubMed]

1981 (2)

N. B. Narozhny, J. J. Sanchez-Mondragon, and J. H. Eberly, “Coherence versus incoherence: collapse and revival in a simple quantum model,” Phys. Rev. A 23, 236–247 (1981).
[CrossRef]

C. V. Sukumar and B. Buck, “Multi-phonon generalization of the Jaynes–Cummings model,” Phys. Lett. A 83, 211 (1981).
[CrossRef]

1980 (1)

J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, “Periodic spontaneous collapse and revival in a simple quantum model,” Phys. Rev. Lett. 44, 1323–1326 (1980).
[CrossRef]

1970 (1)

H. Araki and E. Lieb, “Entropy inqualities,” Commun. Math. Phys. 18, 160 (1970).
[CrossRef]

Abdel-Aty, M.

M. Abdel-Aty, S. Furuichi, and A.-S. F. Obada, “Entanglement degree of a nonlinear multiphoton Jaynes-Cummings model,” J. Opt. B: Quantum Semiclass. Opt. 4, 37–43 (2002).
[CrossRef]

A.-S. F. Obada and M. Abdel-Aty, “Influence of the Stark shift and Kerr-like medium on the evolution of field entropy and entanglement in two-photon processes,” Acta Phys. Pol. B 31, 589–599 (2000).

Abdel-Hafez, A. M.

A.-S. F. Obada, A. M. Abdel-Hafez, and H. A. Hessian, “Influence of the intrinsic decoherence on nonclassical effects in the nondegenerate bimodal multiquanta Jaynes–Cummings model,” J. Phys. B: At. Mol. Opt. Phys. 31, 5085–5104 (1998).
[CrossRef]

Araki, H.

H. Araki and E. Lieb, “Entropy inqualities,” Commun. Math. Phys. 18, 160 (1970).
[CrossRef]

Bennett, C. H.

C. H. Bennett, D. P. Di Vincenzo, J. A. Smolin, and W. K. Wootters, “Mixed-state entanglement and quantum error correction,” Phys. Rev. A 54, 3824–3851 (1996).
[CrossRef] [PubMed]

C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, “Concentrating partial entanglement by local operations,” Phys. Rev. A 53, 2046–2052 (1996).
[CrossRef] [PubMed]

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

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

Bernstein, H. J.

C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, “Concentrating partial entanglement by local operations,” Phys. Rev. A 53, 2046–2052 (1996).
[CrossRef] [PubMed]

Berthiaume, A.

M. Hillery, V. Buzseek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A 59, 1829–1834 (1999).
[CrossRef]

Bose, S.

S. Bose, I. Fuentes-Guridi, P. L. Knight, and V. Vedral, “Erratum: Subsystem purity as an enforcer of entanglement [Phys. Rev. Lett. 87, 050401 (2001)],” Phys. Rev. Lett. 87, 279901 (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]

Brassard, G.

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

Brune, M.

M. Brune, E. Hagley, J. Dreyer, X. Matre, A. Maali, C. Wunderlich, J. M. Raimond, and S. Haroche, “Observing the progressive decoherence of the ‘meter’ in a quantum measurement,” Phys. Rev. Lett. 77, 4887 (1996).
[CrossRef] [PubMed]

M. Brune, J. M. Raimond, and S. Haroche, “Theory of the Rydberg-atom two-photon micromaser,” Phys. Rev. A 35, 154–163 (1987).
[CrossRef] [PubMed]

M. Brune, J. M. Raimond, P. Goy, L. Davidovich, and S. Haroche, “Realization of a two-photon maser oscillator,” Phys. Rev. Lett. 59, 1899–1902 (1987).
[CrossRef] [PubMed]

L. Davidovich, J. M. Raimond, M. Brune, and S. Haroche, “Quantum theory of a two-photon micromaser,” Phys. Rev. A 36, 3771–3787 (1987).
[CrossRef] [PubMed]

Buck, B.

C. V. Sukumar and B. Buck, “Multi-phonon generalization of the Jaynes–Cummings model,” Phys. Lett. A 83, 211 (1981).
[CrossRef]

Buzseek, V.

M. Hillery, V. Buzseek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A 59, 1829–1834 (1999).
[CrossRef]

H. Moya-Cessa, V. Buzeek, M. S. Kim, and P. L. Knight, “Intrinsic decoherence in the atom–field interaction,” Phys. Rev. A 48, 3900–3905 (1993).
[CrossRef] [PubMed]

V. Buzseek, H. Moya-Cessa, P. L. Knight, and S. J. D. Phoenix, “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]

Caves, C. M.

C. M. Caves and G. J. Milburn, “Quantum-mechanical model for continuous position measurements,” Phys. Rev. A 36, 5543 (1987).
[CrossRef] [PubMed]

Chuang, I. L.

I. L. Chuang and Y. Yamamoto, “Creation of a persistent quantum bit using error correction,” Phys. Rev. A 55, 114–127 (1997).
[CrossRef]

Crépeau, C.

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

Davidovich, L.

M. Brune, J. M. Raimond, P. Goy, L. Davidovich, and S. Haroche, “Realization of a two-photon maser oscillator,” Phys. Rev. Lett. 59, 1899–1902 (1987).
[CrossRef] [PubMed]

L. Davidovich, J. M. Raimond, M. Brune, and S. Haroche, “Quantum theory of a two-photon micromaser,” Phys. Rev. A 36, 3771–3787 (1987).
[CrossRef] [PubMed]

Di Vincenzo, D. P.

C. H. Bennett, D. P. Di Vincenzo, J. A. Smolin, and W. K. Wootters, “Mixed-state entanglement and quantum error correction,” Phys. Rev. A 54, 3824–3851 (1996).
[CrossRef] [PubMed]

Disi, L.

L. Disi, “Models for universal reduction of macroscopic quantum fluctuations,” Phys. Rev. A 40, 1165–1174 (1989).
[CrossRef]

Dreyer, J.

M. Brune, E. Hagley, J. Dreyer, X. Matre, A. Maali, C. Wunderlich, J. M. Raimond, and S. Haroche, “Observing the progressive decoherence of the ‘meter’ in a quantum measurement,” Phys. Rev. Lett. 77, 4887 (1996).
[CrossRef] [PubMed]

Eberly, J. H.

N. B. Narozhny, J. J. Sanchez-Mondragon, and J. H. Eberly, “Coherence versus incoherence: collapse and revival in a simple quantum model,” Phys. Rev. A 23, 236–247 (1981).
[CrossRef]

J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, “Periodic spontaneous collapse and revival in a simple quantum model,” Phys. Rev. Lett. 44, 1323–1326 (1980).
[CrossRef]

Ellis, J.

J. Ellis, S. Mohanty, and D. V. Nanopaulos, “Wormholes violate quantum mechanics in SQUIDs,” Phys. Lett. B 235, 305 (1990).
[CrossRef]

J. Ellis, S. Mohanty, and D. V. Nanopaulos, “Quantum gravity and the collapse of the wavefunction,” Phys. Lett. B 221, 113 (1989).
[CrossRef]

Fang, M. F.

M. F. Fang and G. H. Zhou, “Influence of atomic coherence on the evolution of field entropy in multiphoton processes,” Phys. Lett. A 184, 397–402 (1994).
[CrossRef]

Filipowicz, P.

P. Filipowicz, J. Javanainen, and P. Meystre, “Theory of a microscopic maser,” Phys. Rev. A 34, 3077–3087 (1986).
[CrossRef] [PubMed]

Fuentes-Guridi, I.

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]

S. Bose, I. Fuentes-Guridi, P. L. Knight, and V. Vedral, “Erratum: Subsystem purity as an enforcer of entanglement [Phys. Rev. Lett. 87, 050401 (2001)],” Phys. Rev. Lett. 87, 279901 (2001).
[CrossRef]

Furuichi, S.

M. Abdel-Aty, S. Furuichi, and A.-S. F. Obada, “Entanglement degree of a nonlinear multiphoton Jaynes-Cummings model,” J. Opt. B: Quantum Semiclass. Opt. 4, 37–43 (2002).
[CrossRef]

Gea-Bancloche, J.

J. Gea-Bancloche, “A new look at the Jaynes–Cummings model for large fields: Bloch sphere evolution and detuning effects,” Opt. Commun. 88, 531 (1992).
[CrossRef]

J. Gea-Bancloche, “Atom- and field-state evolution in the Jaynes–Cummings model for large initial fields,” Phys. Rev. A 44, 5913–5931 (1991).
[CrossRef]

J. Gea-Bancloche, “Collapse and revival of the state vector in the Jaynes–Cummings model: an example of state preparation by a quantum apparatus,” Phys. Rev. Lett. 65, 3385–3388 (1990).
[CrossRef]

Ghirardi, G. C.

G. C. Ghirardi, P. Pearle, and A. Rimini, “Markov processes in Hilbert space and continuous spontaneous localization of systems of identical particles,” Phys. Rev. A 42, 78–89 (1990).
[CrossRef] [PubMed]

G. C. Ghirardi, R. Grassi, and A. Rimini, “Continuous-spontaneous-reduction model involving gravity,” Phys. Rev. A 42, 1057–1064 (1990).
[CrossRef] [PubMed]

G. C. Ghirardi, A. Rimini, and T. Weber, “Unified dynamics for microscopic and macroscopic systems,” Phys. Rev. D 34, 470–491 (1986).
[CrossRef]

Goy, P.

M. Brune, J. M. Raimond, P. Goy, L. Davidovich, and S. Haroche, “Realization of a two-photon maser oscillator,” Phys. Rev. Lett. 59, 1899–1902 (1987).
[CrossRef] [PubMed]

Grassi, R.

G. C. Ghirardi, R. Grassi, and A. Rimini, “Continuous-spontaneous-reduction model involving gravity,” Phys. Rev. A 42, 1057–1064 (1990).
[CrossRef] [PubMed]

Hagley, E.

M. Brune, E. Hagley, J. Dreyer, X. Matre, A. Maali, C. Wunderlich, J. M. Raimond, and S. Haroche, “Observing the progressive decoherence of the ‘meter’ in a quantum measurement,” Phys. Rev. Lett. 77, 4887 (1996).
[CrossRef] [PubMed]

Haroche, S.

M. Brune, E. Hagley, J. Dreyer, X. Matre, A. Maali, C. Wunderlich, J. M. Raimond, and S. Haroche, “Observing the progressive decoherence of the ‘meter’ in a quantum measurement,” Phys. Rev. Lett. 77, 4887 (1996).
[CrossRef] [PubMed]

M. Brune, J. M. Raimond, and S. Haroche, “Theory of the Rydberg-atom two-photon micromaser,” Phys. Rev. A 35, 154–163 (1987).
[CrossRef] [PubMed]

M. Brune, J. M. Raimond, P. Goy, L. Davidovich, and S. Haroche, “Realization of a two-photon maser oscillator,” Phys. Rev. Lett. 59, 1899–1902 (1987).
[CrossRef] [PubMed]

L. Davidovich, J. M. Raimond, M. Brune, and S. Haroche, “Quantum theory of a two-photon micromaser,” Phys. Rev. A 36, 3771–3787 (1987).
[CrossRef] [PubMed]

Henderson, L.

L. Henderson and V. Vedral, “Information, relative entropy of entanglement, and irreversibility,” Phys. Rev. Lett. 84, 2263–2266 (2000).
[CrossRef] [PubMed]

Hessian, H. A.

A.-S. F. Obada, A. M. Abdel-Hafez, and H. A. Hessian, “Influence of the intrinsic decoherence on nonclassical effects in the nondegenerate bimodal multiquanta Jaynes–Cummings model,” J. Phys. B: At. Mol. Opt. Phys. 31, 5085–5104 (1998).
[CrossRef]

Hillery, M.

M. Hillery, V. Buzseek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A 59, 1829–1834 (1999).
[CrossRef]

Horodecki, M.

M. Horodecki, P. Horodecki, and R. Horodecki, “Limits for entanglement measures,” Phys. Rev. Lett. 84, 2014–2017 (2000).
[CrossRef] [PubMed]

Horodecki, P.

M. Horodecki, P. Horodecki, and R. Horodecki, “Limits for entanglement measures,” Phys. Rev. Lett. 84, 2014–2017 (2000).
[CrossRef] [PubMed]

Horodecki, R.

M. Horodecki, P. Horodecki, and R. Horodecki, “Limits for entanglement measures,” Phys. Rev. Lett. 84, 2014–2017 (2000).
[CrossRef] [PubMed]

Javanainen, J.

P. Filipowicz, J. Javanainen, and P. Meystre, “Theory of a microscopic maser,” Phys. Rev. A 34, 3077–3087 (1986).
[CrossRef] [PubMed]

Jozsa, R.

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

Kim, M. S.

H. Moya-Cessa, V. Buzeek, M. S. Kim, and P. L. Knight, “Intrinsic decoherence in the atom–field interaction,” Phys. Rev. A 48, 3900–3905 (1993).
[CrossRef] [PubMed]

Klein, N.

G. Rempe, H. Walther, and N. Klein, “Observation of quantum collapse and revival in a one-atom maser,” Phys. Rev. Lett. 58, 353–356 (1987).
[CrossRef] [PubMed]

Knight, P. L.

S. Bose, I. Fuentes-Guridi, P. L. Knight, and V. Vedral, “Erratum: Subsystem purity as an enforcer of entanglement [Phys. Rev. Lett. 87, 050401 (2001)],” Phys. Rev. Lett. 87, 279901 (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]

V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight, “Quantifying entanglement,” Phys. Rev. Lett. 78, 2275–2279 (1997).
[CrossRef]

H. Moya-Cessa, V. Buzeek, M. S. Kim, and P. L. Knight, “Intrinsic decoherence in the atom–field interaction,” Phys. Rev. A 48, 3900–3905 (1993).
[CrossRef] [PubMed]

V. Buzseek, H. Moya-Cessa, P. L. Knight, and S. J. D. Phoenix, “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]

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, “Comment on ‘Collapse and revival of the state vector in the Jaynes–Cummings model: an example of state preparation by a quantum apparatus, ’” Phys. Rev. Lett. 66, 2833 (1991).
[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]

S. J. D. Phoenix and P. L. Knight, “Fluctuation and entropy in models of quantum optical resonance,” Ann. Phys. (N.Y.) 186, 381 (1988).
[CrossRef]

Lieb, E.

H. Araki and E. Lieb, “Entropy inqualities,” Commun. Math. Phys. 18, 160 (1970).
[CrossRef]

Maali, A.

M. Brune, E. Hagley, J. Dreyer, X. Matre, A. Maali, C. Wunderlich, J. M. Raimond, and S. Haroche, “Observing the progressive decoherence of the ‘meter’ in a quantum measurement,” Phys. Rev. Lett. 77, 4887 (1996).
[CrossRef] [PubMed]

Matre, X.

M. Brune, E. Hagley, J. Dreyer, X. Matre, A. Maali, C. Wunderlich, J. M. Raimond, and S. Haroche, “Observing the progressive decoherence of the ‘meter’ in a quantum measurement,” Phys. Rev. Lett. 77, 4887 (1996).
[CrossRef] [PubMed]

Meschede, D.

D. Meschede, H. Walther, and G. Muller, “One-atom maser,” Phys. Rev. Lett. 54, 551–554 (1985).
[CrossRef] [PubMed]

Meystre, P.

P. Filipowicz, J. Javanainen, and P. Meystre, “Theory of a microscopic maser,” Phys. Rev. A 34, 3077–3087 (1986).
[CrossRef] [PubMed]

Milburn, G. J.

G. J. Milburn, “Reply to comment on intrinsic decoherence in quantum mechanics,” Phys. Rev. A 47, 2415–2416 (1993).
[CrossRef] [PubMed]

G. J. Milburn, “Intrinsic decoherence in quantum mechanics,” Phys. Rev. A 44, 5401–5406 (1991).
[CrossRef] [PubMed]

C. M. Caves and G. J. Milburn, “Quantum-mechanical model for continuous position measurements,” Phys. Rev. A 36, 5543 (1987).
[CrossRef] [PubMed]

Mohanty, S.

J. Ellis, S. Mohanty, and D. V. Nanopaulos, “Wormholes violate quantum mechanics in SQUIDs,” Phys. Lett. B 235, 305 (1990).
[CrossRef]

J. Ellis, S. Mohanty, and D. V. Nanopaulos, “Quantum gravity and the collapse of the wavefunction,” Phys. Lett. B 221, 113 (1989).
[CrossRef]

Moya-Cessa, H.

H. Moya-Cessa, V. Buzeek, M. S. Kim, and P. L. Knight, “Intrinsic decoherence in the atom–field interaction,” Phys. Rev. A 48, 3900–3905 (1993).
[CrossRef] [PubMed]

V. Buzseek, H. Moya-Cessa, P. L. Knight, and S. J. D. Phoenix, “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]

Muller, G.

D. Meschede, H. Walther, and G. Muller, “One-atom maser,” Phys. Rev. Lett. 54, 551–554 (1985).
[CrossRef] [PubMed]

Nanopaulos, D. V.

J. Ellis, S. Mohanty, and D. V. Nanopaulos, “Wormholes violate quantum mechanics in SQUIDs,” Phys. Lett. B 235, 305 (1990).
[CrossRef]

J. Ellis, S. Mohanty, and D. V. Nanopaulos, “Quantum gravity and the collapse of the wavefunction,” Phys. Lett. B 221, 113 (1989).
[CrossRef]

Narozhny, N. B.

N. B. Narozhny, J. J. Sanchez-Mondragon, and J. H. Eberly, “Coherence versus incoherence: collapse and revival in a simple quantum model,” Phys. Rev. A 23, 236–247 (1981).
[CrossRef]

J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, “Periodic spontaneous collapse and revival in a simple quantum model,” Phys. Rev. Lett. 44, 1323–1326 (1980).
[CrossRef]

Obada, A.-S. F.

M. Abdel-Aty, S. Furuichi, and A.-S. F. Obada, “Entanglement degree of a nonlinear multiphoton Jaynes-Cummings model,” J. Opt. B: Quantum Semiclass. Opt. 4, 37–43 (2002).
[CrossRef]

A.-S. F. Obada and M. Abdel-Aty, “Influence of the Stark shift and Kerr-like medium on the evolution of field entropy and entanglement in two-photon processes,” Acta Phys. Pol. B 31, 589–599 (2000).

A.-S. F. Obada, A. M. Abdel-Hafez, and H. A. Hessian, “Influence of the intrinsic decoherence on nonclassical effects in the nondegenerate bimodal multiquanta Jaynes–Cummings model,” J. Phys. B: At. Mol. Opt. Phys. 31, 5085–5104 (1998).
[CrossRef]

Omnes, R.

R. Omnes, “General theory of the decoherence effect in quantum mechanics,” Phys. Rev. A 56, 3383 (1997).
[CrossRef]

Pearle, P.

G. C. Ghirardi, P. Pearle, and A. Rimini, “Markov processes in Hilbert space and continuous spontaneous localization of systems of identical particles,” Phys. Rev. A 42, 78–89 (1990).
[CrossRef] [PubMed]

Peres, A.

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

Phoenix, S. J. D.

V. Buzseek, H. Moya-Cessa, P. L. Knight, and S. J. D. Phoenix, “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]

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, “Comment on ‘Collapse and revival of the state vector in the Jaynes–Cummings model: an example of state preparation by a quantum apparatus, ’” Phys. Rev. Lett. 66, 2833 (1991).
[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]

S. J. D. Phoenix and P. L. Knight, “Fluctuation and entropy in models of quantum optical resonance,” Ann. Phys. (N.Y.) 186, 381 (1988).
[CrossRef]

Plenio, M. B.

V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight, “Quantifying entanglement,” Phys. Rev. Lett. 78, 2275–2279 (1997).
[CrossRef]

Popescu, S.

C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, “Concentrating partial entanglement by local operations,” Phys. Rev. A 53, 2046–2052 (1996).
[CrossRef] [PubMed]

Raimond, J. M.

M. Brune, E. Hagley, J. Dreyer, X. Matre, A. Maali, C. Wunderlich, J. M. Raimond, and S. Haroche, “Observing the progressive decoherence of the ‘meter’ in a quantum measurement,” Phys. Rev. Lett. 77, 4887 (1996).
[CrossRef] [PubMed]

L. Davidovich, J. M. Raimond, M. Brune, and S. Haroche, “Quantum theory of a two-photon micromaser,” Phys. Rev. A 36, 3771–3787 (1987).
[CrossRef] [PubMed]

M. Brune, J. M. Raimond, P. Goy, L. Davidovich, and S. Haroche, “Realization of a two-photon maser oscillator,” Phys. Rev. Lett. 59, 1899–1902 (1987).
[CrossRef] [PubMed]

M. Brune, J. M. Raimond, and S. Haroche, “Theory of the Rydberg-atom two-photon micromaser,” Phys. Rev. A 35, 154–163 (1987).
[CrossRef] [PubMed]

Rempe, G.

G. Rempe, H. Walther, and N. Klein, “Observation of quantum collapse and revival in a one-atom maser,” Phys. Rev. Lett. 58, 353–356 (1987).
[CrossRef] [PubMed]

Rimini, A.

G. C. Ghirardi, R. Grassi, and A. Rimini, “Continuous-spontaneous-reduction model involving gravity,” Phys. Rev. A 42, 1057–1064 (1990).
[CrossRef] [PubMed]

G. C. Ghirardi, P. Pearle, and A. Rimini, “Markov processes in Hilbert space and continuous spontaneous localization of systems of identical particles,” Phys. Rev. A 42, 78–89 (1990).
[CrossRef] [PubMed]

G. C. Ghirardi, A. Rimini, and T. Weber, “Unified dynamics for microscopic and macroscopic systems,” Phys. Rev. D 34, 470–491 (1986).
[CrossRef]

Rippin, M. A.

V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight, “Quantifying entanglement,” Phys. Rev. Lett. 78, 2275–2279 (1997).
[CrossRef]

Sanchez-Mondragon, J. J.

N. B. Narozhny, J. J. Sanchez-Mondragon, and J. H. Eberly, “Coherence versus incoherence: collapse and revival in a simple quantum model,” Phys. Rev. A 23, 236–247 (1981).
[CrossRef]

J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, “Periodic spontaneous collapse and revival in a simple quantum model,” Phys. Rev. Lett. 44, 1323–1326 (1980).
[CrossRef]

Schumacher, B.

C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, “Concentrating partial entanglement by local operations,” Phys. Rev. A 53, 2046–2052 (1996).
[CrossRef] [PubMed]

Shor, P. W.

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

Smolin, J. A.

C. H. Bennett, D. P. Di Vincenzo, J. A. Smolin, and W. K. Wootters, “Mixed-state entanglement and quantum error correction,” Phys. Rev. A 54, 3824–3851 (1996).
[CrossRef] [PubMed]

Sukumar, C. V.

C. V. Sukumar and B. Buck, “Multi-phonon generalization of the Jaynes–Cummings model,” Phys. Lett. A 83, 211 (1981).
[CrossRef]

Vedral, V.

S. Bose, I. Fuentes-Guridi, P. L. Knight, and V. Vedral, “Erratum: Subsystem purity as an enforcer of entanglement [Phys. Rev. Lett. 87, 050401 (2001)],” Phys. Rev. Lett. 87, 279901 (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]

L. Henderson and V. Vedral, “Information, relative entropy of entanglement, and irreversibility,” Phys. Rev. Lett. 84, 2263–2266 (2000).
[CrossRef] [PubMed]

V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight, “Quantifying entanglement,” Phys. Rev. Lett. 78, 2275–2279 (1997).
[CrossRef]

Vidal, G.

G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65, 032314 (2002).
[CrossRef]

Walther, H.

G. Rempe, H. Walther, and N. Klein, “Observation of quantum collapse and revival in a one-atom maser,” Phys. Rev. Lett. 58, 353–356 (1987).
[CrossRef] [PubMed]

D. Meschede, H. Walther, and G. Muller, “One-atom maser,” Phys. Rev. Lett. 54, 551–554 (1985).
[CrossRef] [PubMed]

Weber, T.

G. C. Ghirardi, A. Rimini, and T. Weber, “Unified dynamics for microscopic and macroscopic systems,” Phys. Rev. D 34, 470–491 (1986).
[CrossRef]

Werner, R. F.

G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65, 032314 (2002).
[CrossRef]

Wiesner, S.

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

Wootters, K. W.

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

Wootters, W. K.

C. H. Bennett, D. P. Di Vincenzo, J. A. Smolin, and W. K. Wootters, “Mixed-state entanglement and quantum error correction,” Phys. Rev. A 54, 3824–3851 (1996).
[CrossRef] [PubMed]

Wunderlich, C.

M. Brune, E. Hagley, J. Dreyer, X. Matre, A. Maali, C. Wunderlich, J. M. Raimond, and S. Haroche, “Observing the progressive decoherence of the ‘meter’ in a quantum measurement,” Phys. Rev. Lett. 77, 4887 (1996).
[CrossRef] [PubMed]

Yamamoto, Y.

I. L. Chuang and Y. Yamamoto, “Creation of a persistent quantum bit using error correction,” Phys. Rev. A 55, 114–127 (1997).
[CrossRef]

Zhou, G. H.

M. F. Fang and G. H. Zhou, “Influence of atomic coherence on the evolution of field entropy in multiphoton processes,” Phys. Lett. A 184, 397–402 (1994).
[CrossRef]

Acta Phys. Pol. B (1)

A.-S. F. Obada and M. Abdel-Aty, “Influence of the Stark shift and Kerr-like medium on the evolution of field entropy and entanglement in two-photon processes,” Acta Phys. Pol. B 31, 589–599 (2000).

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

S. J. D. Phoenix and P. L. Knight, “Fluctuation and entropy in models of quantum optical resonance,” Ann. Phys. (N.Y.) 186, 381 (1988).
[CrossRef]

Commun. Math. Phys. (1)

H. Araki and E. Lieb, “Entropy inqualities,” Commun. Math. Phys. 18, 160 (1970).
[CrossRef]

J. Opt. B: Quantum Semiclass. Opt. (1)

M. Abdel-Aty, S. Furuichi, and A.-S. F. Obada, “Entanglement degree of a nonlinear multiphoton Jaynes-Cummings model,” J. Opt. B: Quantum Semiclass. Opt. 4, 37–43 (2002).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Phys. B: At. Mol. Opt. Phys. (1)

A.-S. F. Obada, A. M. Abdel-Hafez, and H. A. Hessian, “Influence of the intrinsic decoherence on nonclassical effects in the nondegenerate bimodal multiquanta Jaynes–Cummings model,” J. Phys. B: At. Mol. Opt. Phys. 31, 5085–5104 (1998).
[CrossRef]

Opt. Commun. (1)

J. Gea-Bancloche, “A new look at the Jaynes–Cummings model for large fields: Bloch sphere evolution and detuning effects,” Opt. Commun. 88, 531 (1992).
[CrossRef]

Phys. Lett. A (2)

M. F. Fang and G. H. Zhou, “Influence of atomic coherence on the evolution of field entropy in multiphoton processes,” Phys. Lett. A 184, 397–402 (1994).
[CrossRef]

C. V. Sukumar and B. Buck, “Multi-phonon generalization of the Jaynes–Cummings model,” Phys. Lett. A 83, 211 (1981).
[CrossRef]

Phys. Lett. B (2)

J. Ellis, S. Mohanty, and D. V. Nanopaulos, “Quantum gravity and the collapse of the wavefunction,” Phys. Lett. B 221, 113 (1989).
[CrossRef]

J. Ellis, S. Mohanty, and D. V. Nanopaulos, “Wormholes violate quantum mechanics in SQUIDs,” Phys. Lett. B 235, 305 (1990).
[CrossRef]

Phys. Rev. A (21)

G. C. Ghirardi, P. Pearle, and A. Rimini, “Markov processes in Hilbert space and continuous spontaneous localization of systems of identical particles,” Phys. Rev. A 42, 78–89 (1990).
[CrossRef] [PubMed]

G. C. Ghirardi, R. Grassi, and A. Rimini, “Continuous-spontaneous-reduction model involving gravity,” Phys. Rev. A 42, 1057–1064 (1990).
[CrossRef] [PubMed]

G. J. Milburn, “Intrinsic decoherence in quantum mechanics,” Phys. Rev. A 44, 5401–5406 (1991).
[CrossRef] [PubMed]

G. J. Milburn, “Reply to comment on intrinsic decoherence in quantum mechanics,” Phys. Rev. A 47, 2415–2416 (1993).
[CrossRef] [PubMed]

H. Moya-Cessa, V. Buzeek, M. S. Kim, and P. L. Knight, “Intrinsic decoherence in the atom–field interaction,” Phys. Rev. A 48, 3900–3905 (1993).
[CrossRef] [PubMed]

R. Omnes, “General theory of the decoherence effect in quantum mechanics,” Phys. Rev. A 56, 3383 (1997).
[CrossRef]

C. M. Caves and G. J. Milburn, “Quantum-mechanical model for continuous position measurements,” Phys. Rev. A 36, 5543 (1987).
[CrossRef] [PubMed]

L. Disi, “Models for universal reduction of macroscopic quantum fluctuations,” Phys. Rev. A 40, 1165–1174 (1989).
[CrossRef]

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

I. L. Chuang and Y. Yamamoto, “Creation of a persistent quantum bit using error correction,” Phys. Rev. A 55, 114–127 (1997).
[CrossRef]

J. Gea-Bancloche, “Atom- and field-state evolution in the Jaynes–Cummings model for large initial fields,” Phys. Rev. A 44, 5913–5931 (1991).
[CrossRef]

C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, “Concentrating partial entanglement by local operations,” Phys. Rev. A 53, 2046–2052 (1996).
[CrossRef] [PubMed]

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Figures (7)

Fig. 1
Fig. 1

Partial entropy SA for the atom (upper solid curve), the field SF (dotted curve), as well as the total entropy S (dashed curve), and SA+SF (upper dot–dashed curve), as a function of the scaled time λt of the particle initially prepared in the excited state and the field initially prepared in a coherent state |α>(n¯=16), for small decoherence γ=106. The lower dot–dashed curve shows the sum of the negative eigenvalues EN of the partially transposed density matrix ρTA in comparison with the entropy difference ED (lower solid curve).

Fig. 2
Fig. 2

Same as Fig. 1 but for the decoherence γ=104.

Fig. 3
Fig. 3

Same as Fig. 1 but for the decoherence γ=103.

Fig. 4
Fig. 4

The same as in Fig. 1, but with the small mean photon number n¯=1.

Fig. 5
Fig. 5

Partial entropy SA for the atom (upper solid curve), the field SF (dotted curve), as well as the total entropy S (dashed curve) and SA+SF (upper dot–dashed curve), as a function of the scaled time λt of the particle initially prepared in the excited state and the field initially prepared in a coherent state |α(n¯=16), for small decoherence γ=106. The lower dot–dashed curve shows the sum of the negative eigenvalues EN of the partially transposed density matrix ρTA in comparison with the entropy difference ED (lower solid curve).

Fig. 6
Fig. 6

Same as Fig. 5 but for the decoherence γ=104.

Fig. 7
Fig. 7

Same as Fig. 5 but for the decoherence γ=103.

Equations (31)

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ddtρˆ(t)=γexp-iγHˆρˆ(t)expiγHˆ-ρˆ(t),
ddtρˆ(t)=-i[Hˆ, ρˆ]-122γ[Hˆ, [Hˆ, ρˆ]],
Hˆ=ωaˆaˆ+ω02σˆz+λ(aˆmσˆ-+aˆmσˆ+)(=1),
exp(Jˆτ)ρˆ(t)=k=0 1k! τγkHˆkρˆ(t)Hˆk,
exp(Sˆτ)ρˆ(t)=exp(-iHˆτ)ρˆ(t)exp(iHˆτ),
exp(Lˆτ)ρˆ(t)=exp-τ2γHˆ2ρˆ(t)exp-τ2γHˆ2,
ρˆ(t)=exp(Jˆt)exp(Sˆt)exp(Lˆt)ρˆ(0),
|α=n=0 exp-12|α|2 αnn!|n=n=0Qn|n,
ρˆ(0)=|αα||ee|.
ρˆ(t)=k=0 1k! tγk[Mˆee(k)(t)|ee|+Mˆeg(k)(t)|eg|+Mˆge(k)(t)|ge|+Mˆgg(k)(t)|gg|],
Mˆee(k)(t)=fˆ(k)(n+m)+Δ2λgˆ(k)(n+m)Ψˆ11(t)×fˆ(k)(n+m)+Δ2λgˆ(k)(n+m)+fˆ(k)(n+m)+Δ2λgˆ(k)(n+m) × Ψˆ12(t)gˆ(k)(n)aˆm+aˆmgˆ(k)(n)Ψˆ21(t) × fˆ(k)(n+m)+Δ2λgˆ(k)(n+m)+aˆmgˆ(k)(n)Ψˆ22(t)gˆ(k)(n)aˆm,
Mˆgg(k)(t)=gˆ(k)(n)aˆmΨˆ11(t)aˆmgˆ(k)(n)+gˆ(k)(n)aˆmΨˆ12(t) × fˆ(k)(n)-Δ2λgˆ(k)(n)+fˆ(k)(n)-Δ2λgˆ(k)(n) × Ψˆ21(t)aˆmgˆ(k)(n)+fˆ(k)(n)-Δ2λgˆ(k)(n)Ψˆ22(t) × fˆ(k)(n)-Δ2λgˆ(k)(n),
Mˆge(k)(t)=[Mˆeg(k)(t)]=gˆ(k)(n)aˆmΨˆ11(t) × fˆ(k)(n+m)+Δ2λgˆ(k)(n+m)+gˆ(k)(n)aˆmΨˆ12(t)gˆ(k)(n)aˆm+fˆ(k)(n)-Δ2λgˆ(k)(n)Ψˆ21(t)×fˆ(k)(n+m)+Δ2λgˆ(k)(n+m)+fˆ(k)(n)-Δ2λgˆ(k)(n) × Ψˆ22(t)gˆ(k)(n)aˆm,
Ψˆij(t)=|Ψˆi(t)Ψˆj(t)|(i, j=1, 2),
|Ψˆ1(t)=Rˆ(n+m, t)-Δ2λ Vˆ(n+m, t)Fˆ(n+m)|Ψˆ(t),
|Ψˆ2(t)=-aˆm Vˆ(n+m, t)Fˆ(n+m)|Ψˆ(t),
|Ψˆ(t)=exp-t2γ[ζˆ2(n+m)+λ2Fˆ2(n+m)]exp[-iζˆ(n+m)t]|α,
Rˆ(n, t)=cos λtFˆ(n)coshλtγζˆ(n)Fˆ(n)+i sin λtFˆ(n)sinhλtγζˆ(n)Fˆ(n),
Vˆ(n, t)=cos λtFˆ(n)sinhλtγζˆ(n)Fˆ(n)+i sin λtFˆ(n)coshλtγζˆ(n)Fˆ(n),
gˆ(k)(n)=gˆ(k)(n)Fˆ(n),
fˆ(k)(n)=12[Φˆk(n,+)+Φˆk(n,-)],
gˆ(k)(n)=12[Φˆk(n,+)-Φˆk(n,-)],
Φˆ(n,±)=ζˆ(n)±λFˆ(n),
Fˆ2(n)=Δ2λ2+νˆ2(n),νˆ2(n)=n!(n-m)!,
ζˆ(n)=ωnˆ-m2.
ρˆA(t)=TrF ρˆ(t)=Cee(t)Ceg(t)Cge(t)Cgg(t),
Cij(t)=k,n=0 1k! tγkn|Mˆij(k)(t)|n(i, j=e,g),
ρˆF(t)=TrA ρˆ(t)=k=0 1k! tγk[Mˆee(k)(t)+Mˆgg(k)(t)].
S=-Tr{ρˆ ln ρˆ},
SA(F)=-TrA(F){ρˆA(F) ln ρA(F)}.
ED=14(S-SA-SF).

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