Abstract

We study the dynamics of the Jaynes–Cummings model for two two-level systems (or qubits) interacting with a quantized single mode electromagnetic cavity (or quantum bus), extending this to the macroscopic case of an array of Nq qubits. For an initial cavity coherent state |α and the qubit system in a specified “basin of attraction” in its Hilbert space, we demonstrate the oscillation of a superposition of two macroscopic quantum states between the qubit system and the field mode. From the perspective of either the qubit or the field system, there is collapse and revival of a “Schrödinger cat” state.

© 2010 Optical Society of America

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    [CrossRef] [PubMed]
  9. J. M. Fink, R. Bianchetti, M. Baur, M. Goeppl, L. Steffen, S. Filipp, P. J. Leek, A. Blais, and A. Wallraff, “Collective qubit states and the Tavis–Cummings model in circuit QED,” arXiv:0812.2651.
  10. P.R.Berman, ed., Cavity Quantum Electrodynamics (Academic, 1994).
  11. J. I. Cirac and P. Zoller, “Quantum computations with cold trapped ions,” Phys. Rev. Lett. 74, 4091–4094 (1995).
    [CrossRef] [PubMed]
  12. 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]
  13. M. Tavis and F. W. Cummings, “Exact solution for an n-molecule-radiation-field Hamiltonian,” Phys. Rev. 170, 379–384 (1968).
    [CrossRef]
  14. J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mongragon, “Periodic spontaneous collapse and revival in a simple quantum model,” Phys. Rev. Lett. 44, 1323–1326 (1980).
    [CrossRef]
  15. J. Gea-Banacloche, “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] [PubMed]
  16. 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]
  17. S. M. Chumakov, A. B. Klimov, and J. J. Sanchez-Mondragon, “General properties of quantum optical systems in a strong-field limit,” Phys. Rev. A 49, 4972–4978 (1994).
    [CrossRef] [PubMed]
  18. J. Gea-Banacloche, “Atom and field state evolution in the Jaynes–Cummings model for large initial fields,” Phys. Rev. A 44, 5913–5931 (1991).
    [CrossRef] [PubMed]
  19. J. M. Radcliffe, “Some properties of coherent spin states,” J. Phys. A 4, 313–323 (1971).
    [CrossRef]
  20. S. M. Chumakov, A. B. Klimov, and J. J. Sanchez-Mondragon, “Collective atomic dynamics in a strong quantum field,” Opt. Commun. 118, 529–536 (1995).
    [CrossRef]
  21. C. E. A. Jarvis, D. A. Rodrigues, B. L. Györffy, T. P. Spiller, A. J. Short, and J. F. Annett, “Dynamics of entanglement and ‘attractor’ states in the Tavis–Cummings model,” New J. Phys. 11, 103047 (2009).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  24. D. A. Rodrigues, B. L. Györffy, and T. P. Spiller, “Arrays of cooper pair boxes coupled to a superconducting reservoir: ‘superradiance’ and ‘revival’,” J. Phys. Condens. Matter 16, 4477–4494 (2004).
    [CrossRef]
  25. C. W. Gardiner and P. Zoller, Quantum Noise (Springer, 1991).
  26. Qs(β,t)∝⟨β,Nq|ρq(t)|β,Nq⟩.
  27. A. Auffeves, P. Maioli, T. Meunier, S. Gleyzes, G. Nogues, M. Brune, J. M. Raimond, and S. Haroche, “Entanglement of a mesoscopic field with an atom induced by photon graininess in a cavity,” Phys. Rev. Lett. 91, 230405 (2003).
    [CrossRef] [PubMed]
  28. T. Meunier, A. Le Diffon, C. Ruef, P. Degiovanni, and J.-M. Raimond, “Entanglement and decoherence of n atoms and a mesoscopic field in a cavity,” Phys. Rev. A 74, 033802 (2006).
    [CrossRef]
  29. V. Bonzom, H. Bouzidi, and P. Degiovanni, “Dissipative dynamics of circuit-QED in the mesoscopic regime,” Eur. Phys. J. D 47, 133–149 (2008).
    [CrossRef]

2009

C. E. A. Jarvis, D. A. Rodrigues, B. L. Györffy, T. P. Spiller, A. J. Short, and J. F. Annett, “Dynamics of entanglement and ‘attractor’ states in the Tavis–Cummings model,” New J. Phys. 11, 103047 (2009).
[CrossRef]

2008

L. P. Pryadko and G. Quiroz, “An introduction to entanglement measures,” Phys. Rev. A 77, 012330 (2008).
[CrossRef]

V. Bonzom, H. Bouzidi, and P. Degiovanni, “Dissipative dynamics of circuit-QED in the mesoscopic regime,” Eur. Phys. J. D 47, 133–149 (2008).
[CrossRef]

2006

T. Meunier, A. Le Diffon, C. Ruef, P. Degiovanni, and J.-M. Raimond, “Entanglement and decoherence of n atoms and a mesoscopic field in a cavity,” Phys. Rev. A 74, 033802 (2006).
[CrossRef]

2004

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004).
[CrossRef] [PubMed]

D. A. Rodrigues, B. L. Györffy, and T. P. Spiller, “Arrays of cooper pair boxes coupled to a superconducting reservoir: ‘superradiance’ and ‘revival’,” J. Phys. Condens. Matter 16, 4477–4494 (2004).
[CrossRef]

2003

A. Auffeves, P. Maioli, T. Meunier, S. Gleyzes, G. Nogues, M. Brune, J. M. Raimond, and S. Haroche, “Entanglement of a mesoscopic field with an atom induced by photon graininess in a cavity,” Phys. Rev. Lett. 91, 230405 (2003).
[CrossRef] [PubMed]

2001

W. J. Munro, D. F. V. James, A. G. White, and P. G. Kwiat, “Maximizing the entanglement of two mixed qubits,” Phys. Rev. A 64, 030302(R) (2001).
[CrossRef]

1998

W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80, 2245–2248 (1998).
[CrossRef]

1995

S. M. Chumakov, A. B. Klimov, and J. J. Sanchez-Mondragon, “Collective atomic dynamics in a strong quantum field,” Opt. Commun. 118, 529–536 (1995).
[CrossRef]

J. I. Cirac and P. Zoller, “Quantum computations with cold trapped ions,” Phys. Rev. Lett. 74, 4091–4094 (1995).
[CrossRef] [PubMed]

1994

S. M. Chumakov, A. B. Klimov, and J. J. Sanchez-Mondragon, “General properties of quantum optical systems in a strong-field limit,” Phys. Rev. A 49, 4972–4978 (1994).
[CrossRef] [PubMed]

1991

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

1990

J. Gea-Banacloche, “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] [PubMed]

1980

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

1971

J. M. Radcliffe, “Some properties of coherent spin states,” J. Phys. A 4, 313–323 (1971).
[CrossRef]

1968

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

1963

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]

Annett, J. F.

C. E. A. Jarvis, D. A. Rodrigues, B. L. Györffy, T. P. Spiller, A. J. Short, and J. F. Annett, “Dynamics of entanglement and ‘attractor’ states in the Tavis–Cummings model,” New J. Phys. 11, 103047 (2009).
[CrossRef]

Auffeves, A.

A. Auffeves, P. Maioli, T. Meunier, S. Gleyzes, G. Nogues, M. Brune, J. M. Raimond, and S. Haroche, “Entanglement of a mesoscopic field with an atom induced by photon graininess in a cavity,” Phys. Rev. Lett. 91, 230405 (2003).
[CrossRef] [PubMed]

Baur, M.

J. M. Fink, R. Bianchetti, M. Baur, M. Goeppl, L. Steffen, S. Filipp, P. J. Leek, A. Blais, and A. Wallraff, “Collective qubit states and the Tavis–Cummings model in circuit QED,” arXiv:0812.2651.

Bianchetti, R.

J. M. Fink, R. Bianchetti, M. Baur, M. Goeppl, L. Steffen, S. Filipp, P. J. Leek, A. Blais, and A. Wallraff, “Collective qubit states and the Tavis–Cummings model in circuit QED,” arXiv:0812.2651.

Blais, A.

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004).
[CrossRef] [PubMed]

J. M. Fink, R. Bianchetti, M. Baur, M. Goeppl, L. Steffen, S. Filipp, P. J. Leek, A. Blais, and A. Wallraff, “Collective qubit states and the Tavis–Cummings model in circuit QED,” arXiv:0812.2651.

Bonzom, V.

V. Bonzom, H. Bouzidi, and P. Degiovanni, “Dissipative dynamics of circuit-QED in the mesoscopic regime,” Eur. Phys. J. D 47, 133–149 (2008).
[CrossRef]

Bouzidi, H.

V. Bonzom, H. Bouzidi, and P. Degiovanni, “Dissipative dynamics of circuit-QED in the mesoscopic regime,” Eur. Phys. J. D 47, 133–149 (2008).
[CrossRef]

Brune, M.

A. Auffeves, P. Maioli, T. Meunier, S. Gleyzes, G. Nogues, M. Brune, J. M. Raimond, and S. Haroche, “Entanglement of a mesoscopic field with an atom induced by photon graininess in a cavity,” Phys. Rev. Lett. 91, 230405 (2003).
[CrossRef] [PubMed]

Chuang, I. L.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge Univ. Press, 2001).

Chumakov, S. M.

S. M. Chumakov, A. B. Klimov, and J. J. Sanchez-Mondragon, “Collective atomic dynamics in a strong quantum field,” Opt. Commun. 118, 529–536 (1995).
[CrossRef]

S. M. Chumakov, A. B. Klimov, and J. J. Sanchez-Mondragon, “General properties of quantum optical systems in a strong-field limit,” Phys. Rev. A 49, 4972–4978 (1994).
[CrossRef] [PubMed]

Cirac, J. I.

J. I. Cirac and P. Zoller, “Quantum computations with cold trapped ions,” Phys. Rev. Lett. 74, 4091–4094 (1995).
[CrossRef] [PubMed]

Cummings, F. W.

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

E. T. Jaynes and F. W. Cummings, “Comparison of quantum and semiclassical radiation theories with application to the beam maser,” Proc. IEEE 51, 89–109 (1963).
[CrossRef]

Degiovanni, P.

V. Bonzom, H. Bouzidi, and P. Degiovanni, “Dissipative dynamics of circuit-QED in the mesoscopic regime,” Eur. Phys. J. D 47, 133–149 (2008).
[CrossRef]

T. Meunier, A. Le Diffon, C. Ruef, P. Degiovanni, and J.-M. Raimond, “Entanglement and decoherence of n atoms and a mesoscopic field in a cavity,” Phys. Rev. A 74, 033802 (2006).
[CrossRef]

Eberly, J. H.

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

Filipp, S.

J. M. Fink, R. Bianchetti, M. Baur, M. Goeppl, L. Steffen, S. Filipp, P. J. Leek, A. Blais, and A. Wallraff, “Collective qubit states and the Tavis–Cummings model in circuit QED,” arXiv:0812.2651.

Fink, J. M.

J. M. Fink, R. Bianchetti, M. Baur, M. Goeppl, L. Steffen, S. Filipp, P. J. Leek, A. Blais, and A. Wallraff, “Collective qubit states and the Tavis–Cummings model in circuit QED,” arXiv:0812.2651.

Frunzio, L.

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004).
[CrossRef] [PubMed]

Gardiner, C. W.

C. W. Gardiner and P. Zoller, Quantum Noise (Springer, 1991).

Gea-Banacloche, J.

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

J. Gea-Banacloche, “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] [PubMed]

Gerry, C. C.

C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge Univ. Press, 2005).

Girvin, S. M.

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004).
[CrossRef] [PubMed]

Gleyzes, S.

A. Auffeves, P. Maioli, T. Meunier, S. Gleyzes, G. Nogues, M. Brune, J. M. Raimond, and S. Haroche, “Entanglement of a mesoscopic field with an atom induced by photon graininess in a cavity,” Phys. Rev. Lett. 91, 230405 (2003).
[CrossRef] [PubMed]

Goeppl, M.

J. M. Fink, R. Bianchetti, M. Baur, M. Goeppl, L. Steffen, S. Filipp, P. J. Leek, A. Blais, and A. Wallraff, “Collective qubit states and the Tavis–Cummings model in circuit QED,” arXiv:0812.2651.

Györffy, B. L.

C. E. A. Jarvis, D. A. Rodrigues, B. L. Györffy, T. P. Spiller, A. J. Short, and J. F. Annett, “Dynamics of entanglement and ‘attractor’ states in the Tavis–Cummings model,” New J. Phys. 11, 103047 (2009).
[CrossRef]

D. A. Rodrigues, B. L. Györffy, and T. P. Spiller, “Arrays of cooper pair boxes coupled to a superconducting reservoir: ‘superradiance’ and ‘revival’,” J. Phys. Condens. Matter 16, 4477–4494 (2004).
[CrossRef]

Haroche, S.

A. Auffeves, P. Maioli, T. Meunier, S. Gleyzes, G. Nogues, M. Brune, J. M. Raimond, and S. Haroche, “Entanglement of a mesoscopic field with an atom induced by photon graininess in a cavity,” Phys. Rev. Lett. 91, 230405 (2003).
[CrossRef] [PubMed]

S. Haroche and J. M. Raimond, Cavity Quantum Electrodynamics (Academic, 1994), p. 123.

Huang, R. -S.

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004).
[CrossRef] [PubMed]

James, D. F. V.

W. J. Munro, D. F. V. James, A. G. White, and P. G. Kwiat, “Maximizing the entanglement of two mixed qubits,” Phys. Rev. A 64, 030302(R) (2001).
[CrossRef]

Jarvis, C. E. A.

C. E. A. Jarvis, D. A. Rodrigues, B. L. Györffy, T. P. Spiller, A. J. Short, and J. F. Annett, “Dynamics of entanglement and ‘attractor’ states in the Tavis–Cummings model,” New J. Phys. 11, 103047 (2009).
[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]

Klimov, A. B.

S. M. Chumakov, A. B. Klimov, and J. J. Sanchez-Mondragon, “Collective atomic dynamics in a strong quantum field,” Opt. Commun. 118, 529–536 (1995).
[CrossRef]

S. M. Chumakov, A. B. Klimov, and J. J. Sanchez-Mondragon, “General properties of quantum optical systems in a strong-field limit,” Phys. Rev. A 49, 4972–4978 (1994).
[CrossRef] [PubMed]

Knight, P. L.

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]

C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge Univ. Press, 2005).

Kumar, S.

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004).
[CrossRef] [PubMed]

Kwiat, P. G.

W. J. Munro, D. F. V. James, A. G. White, and P. G. Kwiat, “Maximizing the entanglement of two mixed qubits,” Phys. Rev. A 64, 030302(R) (2001).
[CrossRef]

Le Diffon, A.

T. Meunier, A. Le Diffon, C. Ruef, P. Degiovanni, and J.-M. Raimond, “Entanglement and decoherence of n atoms and a mesoscopic field in a cavity,” Phys. Rev. A 74, 033802 (2006).
[CrossRef]

Leek, P. J.

J. M. Fink, R. Bianchetti, M. Baur, M. Goeppl, L. Steffen, S. Filipp, P. J. Leek, A. Blais, and A. Wallraff, “Collective qubit states and the Tavis–Cummings model in circuit QED,” arXiv:0812.2651.

Maioli, P.

A. Auffeves, P. Maioli, T. Meunier, S. Gleyzes, G. Nogues, M. Brune, J. M. Raimond, and S. Haroche, “Entanglement of a mesoscopic field with an atom induced by photon graininess in a cavity,” Phys. Rev. Lett. 91, 230405 (2003).
[CrossRef] [PubMed]

Majer, J.

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004).
[CrossRef] [PubMed]

Meunier, T.

T. Meunier, A. Le Diffon, C. Ruef, P. Degiovanni, and J.-M. Raimond, “Entanglement and decoherence of n atoms and a mesoscopic field in a cavity,” Phys. Rev. A 74, 033802 (2006).
[CrossRef]

A. Auffeves, P. Maioli, T. Meunier, S. Gleyzes, G. Nogues, M. Brune, J. M. Raimond, and S. Haroche, “Entanglement of a mesoscopic field with an atom induced by photon graininess in a cavity,” Phys. Rev. Lett. 91, 230405 (2003).
[CrossRef] [PubMed]

Munro, W. J.

W. J. Munro, D. F. V. James, A. G. White, and P. G. Kwiat, “Maximizing the entanglement of two mixed qubits,” Phys. Rev. A 64, 030302(R) (2001).
[CrossRef]

Narozhny, N. B.

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

Nielsen, M. A.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge Univ. Press, 2001).

Nogues, G.

A. Auffeves, P. Maioli, T. Meunier, S. Gleyzes, G. Nogues, M. Brune, J. M. Raimond, and S. Haroche, “Entanglement of a mesoscopic field with an atom induced by photon graininess in a cavity,” Phys. Rev. Lett. 91, 230405 (2003).
[CrossRef] [PubMed]

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]

Pryadko, L. P.

L. P. Pryadko and G. Quiroz, “An introduction to entanglement measures,” Phys. Rev. A 77, 012330 (2008).
[CrossRef]

Quiroz, G.

L. P. Pryadko and G. Quiroz, “An introduction to entanglement measures,” Phys. Rev. A 77, 012330 (2008).
[CrossRef]

Radcliffe, J. M.

J. M. Radcliffe, “Some properties of coherent spin states,” J. Phys. A 4, 313–323 (1971).
[CrossRef]

Raimond, J. M.

A. Auffeves, P. Maioli, T. Meunier, S. Gleyzes, G. Nogues, M. Brune, J. M. Raimond, and S. Haroche, “Entanglement of a mesoscopic field with an atom induced by photon graininess in a cavity,” Phys. Rev. Lett. 91, 230405 (2003).
[CrossRef] [PubMed]

S. Haroche and J. M. Raimond, Cavity Quantum Electrodynamics (Academic, 1994), p. 123.

Raimond, J. -M.

T. Meunier, A. Le Diffon, C. Ruef, P. Degiovanni, and J.-M. Raimond, “Entanglement and decoherence of n atoms and a mesoscopic field in a cavity,” Phys. Rev. A 74, 033802 (2006).
[CrossRef]

Rodrigues, D. A.

C. E. A. Jarvis, D. A. Rodrigues, B. L. Györffy, T. P. Spiller, A. J. Short, and J. F. Annett, “Dynamics of entanglement and ‘attractor’ states in the Tavis–Cummings model,” New J. Phys. 11, 103047 (2009).
[CrossRef]

D. A. Rodrigues, B. L. Györffy, and T. P. Spiller, “Arrays of cooper pair boxes coupled to a superconducting reservoir: ‘superradiance’ and ‘revival’,” J. Phys. Condens. Matter 16, 4477–4494 (2004).
[CrossRef]

Ruef, C.

T. Meunier, A. Le Diffon, C. Ruef, P. Degiovanni, and J.-M. Raimond, “Entanglement and decoherence of n atoms and a mesoscopic field in a cavity,” Phys. Rev. A 74, 033802 (2006).
[CrossRef]

Sanchez-Mondragon, J. J.

S. M. Chumakov, A. B. Klimov, and J. J. Sanchez-Mondragon, “Collective atomic dynamics in a strong quantum field,” Opt. Commun. 118, 529–536 (1995).
[CrossRef]

S. M. Chumakov, A. B. Klimov, and J. J. Sanchez-Mondragon, “General properties of quantum optical systems in a strong-field limit,” Phys. Rev. A 49, 4972–4978 (1994).
[CrossRef] [PubMed]

Sanchez-Mongragon, J. J.

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

Schoelkopf, R. J.

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004).
[CrossRef] [PubMed]

Schuster, D. I.

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Qs(β,t)∝⟨β,Nq|ρq(t)|β,Nq⟩.

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Figures (5)

Fig. 1
Fig. 1

Time evolution for a system with one qubit. (a) The probability of being in the qubit’s initial state | g . (b) The entropy of the qubit. (c) The probability of being in the state | ψ a t t + . At t r / 2 the probability of being in the attractor state approaches unity while the entropy approaches zero. The qubit starts in the initial state | g and the value of n ¯ = 50 .

Fig. 2
Fig. 2

Time evolution for a system with two qubits. (a) The probability of being in the initial state. (b) The entropy of the qubits. (c) The probability of being in the two-qubit attractor state | ψ 2 a t t + when the initial phase of the radiation field is θ = 0 . The two-qubit attractor state is reached at t r / 4 . The initial state of the qubits is ( | e e + | g g ) / 2 and the value of n ¯ = 50 .

Fig. 3
Fig. 3

The qubit system started in the maximally entangled state ( | e e + | g g ) / 2 and n ¯ = 50 . (a) The probability of being in the initial state. (b) The entropy of the qubit system. (c) The mixed state tangle of the qubit system.

Fig. 4
Fig. 4

Diagrams of the Q function [25] (left) and spin Q function [26] (right) at three different times. (a) The time t = 0 , where the cavity is in a coherent state and the qubits are in a spin Schrödinger cat state. (b) The time t = t r / ( 2 N q ) which is the time of the first attractor. (c) The time t = t r / N q , when the field states are again overlapping and in a coherent state. These show the Schrödinger cat state moving from the qubits to the radiation field and back again in the limit n ¯ , N q = 40 , θ = 0 . The Q function for the field has been scaled to a unit circle.  

Fig. 5
Fig. 5

Time evolution for a system with two qubits starting in the product state ( | e e + | g e + | e g + | g g ) / 2 . (a) The probability of being in the initial state. (b) The entropy of the qubits. (c) The probability of being in the two-qubit attractor state | ψ 2 a t t + . For comparison (d) shows the evolution of the entropy of qubits started in the state ( | e e + | g g ) / 2 . The initial phase of the radiation field is θ = 0 , and the size of the coherent state is again n ¯ = 50 .

Equations (10)

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H ̂ = ω a ̂ a ̂ + 2 i = 1 N q Ω i σ ̂ i z + i = 1 N q λ i ( a ̂ σ ̂ i + + a ̂ σ ̂ i ) ,
| ψ a t t ± = 1 2 ( e i θ | e ± i | g ) ,
| ψ N q a t t ± = 1 2 N q ( e i θ | e ± i | g ) N q
| β , N q = 1 N m = N q / 2 N q / 2 C ( N q / 2 ) + m N q β ( N q / 2 ) + m | N q , m ,
| ψ 2 = C e e | e e + C e g | e g + C g e | g e + C g g | g g ,
| ψ 2 = a ( e i θ | e e + e i θ | g g ) + 1 2 | a | 2 ( | e g + | g e ) ,
| ψ N q = m = N q / 2 N q / 2 A ( N q , a , m ) e i [ ( N q / 2 ) + m ] θ N q ! ( N q 2 + m ) ! ( N q 2 m ) ! | N q , m ,
A ( N q , a , m ) = { a if   ( N q 2 m )   is   even 1 2 N q 1 | a | 2 if   ( N q 2 m )   is   odd , }
| ψ N q = 2 N q 2 ( a + 1 2 N q 1 | a | 2 ) | e i θ , N q + 2 N q 2 ( a 1 2 N q 1 | a | 2 ) | e i θ , N q .
| Φ N q ( t r 2 N q ) = [ ( a 1 2 N q 1 | a | 2 ) e i π n ¯ / 2 | i α ( a + 1 2 N q 1 | a | 2 ) e i π n ¯ / 2 | i α ] 2 N q 2 .

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