Abstract

It is known that a macroscopic quantum superposition (MQS), when it is exposed to an environment, decoheres at a rate scaling with the separation of its component states in phase space. This is more or less consistent with the well-known proposition that a more macroscopic quantum state is reduced more quickly to a classical state in general. Effects of initial mixedness, however, on the subsequent decoherence of MQSs have been less known. We study the evolution of a highly mixed MQS interacting with an environment and compare it with that of a pure MQS having the same size as the central distance between its component states. Although the decoherence develops more rapidly for the mixed MQS in short times, its rate can be significantly suppressed after a certain time and becomes smaller than the decoherence rate of its corresponding pure MQS. In an optics experiment to generate a MQS, our result has the practical implication that nonclassicality of a MQS can still be observable in moderate times, even though a large amount of noise is added to the initial state.

© 2008 Optical Society of America

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References

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  1. E. Schrödinger, “Die gegenwartige Situation in der Quantenmechanik,” Naturwiss. 23, 807-812, 823-828, 844-849 (1935).
    [CrossRef]
  2. M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. van der Zouw, and A. Zeilinger, “Wave-particle duality of C60 molecules,” Nature 401, 680-682 (1999).
    [CrossRef]
  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]
  4. A. J. Leggett, “Testing the limits of quantum mechanics: motivation, state of play, prospects,” J. Phys.: Condens. Matter 14, R415-R451 (2002).
    [CrossRef]
  5. A. J. Leggett, “The quantum measurement problem,” Science 307, 871-872 (2005).
    [CrossRef] [PubMed]
  6. J. I. Korsbakken, K. B. Whaley, J. Dubois, and J. I. Cirac, “Measurement-based measure of the size of macroscopic quantum superpositions,” Phys. Rev. A 75, 042106 (2007).
    [CrossRef]
  7. W. H. Zurek, “Decoherence, einselection, and the quantum origins of the classical,” Rev. Mod. Phys. 75, 715-775 (2003), and references therein.
    [CrossRef]
  8. 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, 4887-4890 (1996).
    [CrossRef] [PubMed]
  9. B. Yurke and D. Stoler, “Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion,” Phys. Rev. Lett. 57, 13-16 (1986).
    [CrossRef] [PubMed]
  10. W. Schleich, M. Pernigo, and F. L. Kien, “Nonclassical state from two pseudoclassical states,” Phys. Rev. A 44, 2172-2187 (1991).
    [CrossRef] [PubMed]
  11. G. W. Gardiner and P. Zoller, Quantum Noise, 3rd ed. (Springer-Verlag, 2004).
  12. M. S. Kim and V. Bužek, “Schrodinger-cat states at finite temperature: influence of a finite-temperature heat bath on quantum interferences,” Phys. Rev. A 46, 4239-4251 (1992).
    [CrossRef] [PubMed]
  13. H. J. Carmichael, “Quantum future,” in Proceedings of the Xth Max Born Symposium, Poland, Ph.Blanchard and A.Jadczyk, eds. (Springer, 1999), pp. 15-36.
  14. H. Jeong, J. Lee, and M. S. Kim, “Dynamics of nonlocality for a two-mode squeezed state in a thermal environment,” Phys. Rev. A 61, 052101 (2000).
    [CrossRef]
  15. H. Jeong and T. C. Ralph, “Transfer of nonclassical properties from a microscopic superposition to macroscopic thermal states in the high temperature limit,” Phys. Rev. Lett. 97, 100401 (2006).
    [CrossRef] [PubMed]
  16. E. Schrödinger, “The constant crossover of micro- to macro-mechanics,” Naturwiss. 14, 664-666 (1926).
    [CrossRef]
  17. A more general form of such a superposition state is ∣α⟩+eiφ∣−α⟩, where the normalization factor is omitted and φ is arbitrary, while we choose φ=π for convenience. However, all the conclusions in this paper remain the same regardless of the choice of φ.
  18. A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical 'Schrodinger cats' from photon number states,” Nature 448, 784-786 (2007).
    [CrossRef] [PubMed]
  19. E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749-759 (1932).
    [CrossRef]
  20. D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, 1994).
  21. M. Hillery, R. F. O'Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121-167 (1984).
    [CrossRef]
  22. S. J. D. Phoenix, “Wave-packet evolution in the damped oscillator,” Phys. Rev. A 41, 5132-5138 (1990).
    [CrossRef] [PubMed]
  23. S. M. Barnett and P. M. Radmore, Methods in Theoretical Quantum Optics (Oxford U. Press, 1997).
  24. M. Paternostro, H. Jeong, and M. S. Kim, “Entanglement of mixed macroscopic superpositions: an entangling-power study,” Phys. Rev. A 73, 012338 (2006).
    [CrossRef]

2007 (2)

J. I. Korsbakken, K. B. Whaley, J. Dubois, and J. I. Cirac, “Measurement-based measure of the size of macroscopic quantum superpositions,” Phys. Rev. A 75, 042106 (2007).
[CrossRef]

A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical 'Schrodinger cats' from photon number states,” Nature 448, 784-786 (2007).
[CrossRef] [PubMed]

2006 (2)

H. Jeong and T. C. Ralph, “Transfer of nonclassical properties from a microscopic superposition to macroscopic thermal states in the high temperature limit,” Phys. Rev. Lett. 97, 100401 (2006).
[CrossRef] [PubMed]

M. Paternostro, H. Jeong, and M. S. Kim, “Entanglement of mixed macroscopic superpositions: an entangling-power study,” Phys. Rev. A 73, 012338 (2006).
[CrossRef]

2005 (1)

A. J. Leggett, “The quantum measurement problem,” Science 307, 871-872 (2005).
[CrossRef] [PubMed]

2003 (1)

W. H. Zurek, “Decoherence, einselection, and the quantum origins of the classical,” Rev. Mod. Phys. 75, 715-775 (2003), and references therein.
[CrossRef]

2002 (1)

A. J. Leggett, “Testing the limits of quantum mechanics: motivation, state of play, prospects,” J. Phys.: Condens. Matter 14, R415-R451 (2002).
[CrossRef]

2001 (1)

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]

2000 (1)

H. Jeong, J. Lee, and M. S. Kim, “Dynamics of nonlocality for a two-mode squeezed state in a thermal environment,” Phys. Rev. A 61, 052101 (2000).
[CrossRef]

1999 (1)

M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. van der Zouw, and A. Zeilinger, “Wave-particle duality of C60 molecules,” Nature 401, 680-682 (1999).
[CrossRef]

1996 (1)

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, 4887-4890 (1996).
[CrossRef] [PubMed]

1992 (1)

M. S. Kim and V. Bužek, “Schrodinger-cat states at finite temperature: influence of a finite-temperature heat bath on quantum interferences,” Phys. Rev. A 46, 4239-4251 (1992).
[CrossRef] [PubMed]

1991 (1)

W. Schleich, M. Pernigo, and F. L. Kien, “Nonclassical state from two pseudoclassical states,” Phys. Rev. A 44, 2172-2187 (1991).
[CrossRef] [PubMed]

1990 (1)

S. J. D. Phoenix, “Wave-packet evolution in the damped oscillator,” Phys. Rev. A 41, 5132-5138 (1990).
[CrossRef] [PubMed]

1986 (1)

B. Yurke and D. Stoler, “Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion,” Phys. Rev. Lett. 57, 13-16 (1986).
[CrossRef] [PubMed]

1984 (1)

M. Hillery, R. F. O'Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121-167 (1984).
[CrossRef]

1935 (1)

E. Schrödinger, “Die gegenwartige Situation in der Quantenmechanik,” Naturwiss. 23, 807-812, 823-828, 844-849 (1935).
[CrossRef]

1932 (1)

E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749-759 (1932).
[CrossRef]

1926 (1)

E. Schrödinger, “The constant crossover of micro- to macro-mechanics,” Naturwiss. 14, 664-666 (1926).
[CrossRef]

Arndt, M.

M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. van der Zouw, and A. Zeilinger, “Wave-particle duality of C60 molecules,” Nature 401, 680-682 (1999).
[CrossRef]

Barnett, S. M.

S. M. Barnett and P. M. Radmore, Methods in Theoretical Quantum Optics (Oxford U. Press, 1997).

Brune, M.

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]

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, 4887-4890 (1996).
[CrossRef] [PubMed]

Bužek, V.

M. S. Kim and V. Bužek, “Schrodinger-cat states at finite temperature: influence of a finite-temperature heat bath on quantum interferences,” Phys. Rev. A 46, 4239-4251 (1992).
[CrossRef] [PubMed]

Carmichael, H. J.

H. J. Carmichael, “Quantum future,” in Proceedings of the Xth Max Born Symposium, Poland, Ph.Blanchard and A.Jadczyk, eds. (Springer, 1999), pp. 15-36.

Cirac, J. I.

J. I. Korsbakken, K. B. Whaley, J. Dubois, and J. I. Cirac, “Measurement-based measure of the size of macroscopic quantum superpositions,” Phys. Rev. A 75, 042106 (2007).
[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, 4887-4890 (1996).
[CrossRef] [PubMed]

Dubois, J.

J. I. Korsbakken, K. B. Whaley, J. Dubois, and J. I. Cirac, “Measurement-based measure of the size of macroscopic quantum superpositions,” Phys. Rev. A 75, 042106 (2007).
[CrossRef]

Gardiner, G. W.

G. W. Gardiner and P. Zoller, Quantum Noise, 3rd ed. (Springer-Verlag, 2004).

Grangier, P.

A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical 'Schrodinger cats' from photon number states,” Nature 448, 784-786 (2007).
[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, 4887-4890 (1996).
[CrossRef] [PubMed]

Haroche, S.

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]

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, 4887-4890 (1996).
[CrossRef] [PubMed]

Hillery, M.

M. Hillery, R. F. O'Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121-167 (1984).
[CrossRef]

Jeong, H.

A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical 'Schrodinger cats' from photon number states,” Nature 448, 784-786 (2007).
[CrossRef] [PubMed]

H. Jeong and T. C. Ralph, “Transfer of nonclassical properties from a microscopic superposition to macroscopic thermal states in the high temperature limit,” Phys. Rev. Lett. 97, 100401 (2006).
[CrossRef] [PubMed]

M. Paternostro, H. Jeong, and M. S. Kim, “Entanglement of mixed macroscopic superpositions: an entangling-power study,” Phys. Rev. A 73, 012338 (2006).
[CrossRef]

H. Jeong, J. Lee, and M. S. Kim, “Dynamics of nonlocality for a two-mode squeezed state in a thermal environment,” Phys. Rev. A 61, 052101 (2000).
[CrossRef]

Keller, C.

M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. van der Zouw, and A. Zeilinger, “Wave-particle duality of C60 molecules,” Nature 401, 680-682 (1999).
[CrossRef]

Kien, F. L.

W. Schleich, M. Pernigo, and F. L. Kien, “Nonclassical state from two pseudoclassical states,” Phys. Rev. A 44, 2172-2187 (1991).
[CrossRef] [PubMed]

Kim, M. S.

M. Paternostro, H. Jeong, and M. S. Kim, “Entanglement of mixed macroscopic superpositions: an entangling-power study,” Phys. Rev. A 73, 012338 (2006).
[CrossRef]

H. Jeong, J. Lee, and M. S. Kim, “Dynamics of nonlocality for a two-mode squeezed state in a thermal environment,” Phys. Rev. A 61, 052101 (2000).
[CrossRef]

M. S. Kim and V. Bužek, “Schrodinger-cat states at finite temperature: influence of a finite-temperature heat bath on quantum interferences,” Phys. Rev. A 46, 4239-4251 (1992).
[CrossRef] [PubMed]

Korsbakken, J. I.

J. I. Korsbakken, K. B. Whaley, J. Dubois, and J. I. Cirac, “Measurement-based measure of the size of macroscopic quantum superpositions,” Phys. Rev. A 75, 042106 (2007).
[CrossRef]

Lee, J.

H. Jeong, J. Lee, and M. S. Kim, “Dynamics of nonlocality for a two-mode squeezed state in a thermal environment,” Phys. Rev. A 61, 052101 (2000).
[CrossRef]

Leggett, A. J.

A. J. Leggett, “The quantum measurement problem,” Science 307, 871-872 (2005).
[CrossRef] [PubMed]

A. J. Leggett, “Testing the limits of quantum mechanics: motivation, state of play, prospects,” J. Phys.: Condens. Matter 14, R415-R451 (2002).
[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, 4887-4890 (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, 4887-4890 (1996).
[CrossRef] [PubMed]

Milburn, G. J.

D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, 1994).

Nairz, O.

M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. van der Zouw, and A. Zeilinger, “Wave-particle duality of C60 molecules,” Nature 401, 680-682 (1999).
[CrossRef]

O'Connell, R. F.

M. Hillery, R. F. O'Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121-167 (1984).
[CrossRef]

Ourjoumtsev, A.

A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical 'Schrodinger cats' from photon number states,” Nature 448, 784-786 (2007).
[CrossRef] [PubMed]

Paternostro, M.

M. Paternostro, H. Jeong, and M. S. Kim, “Entanglement of mixed macroscopic superpositions: an entangling-power study,” Phys. Rev. A 73, 012338 (2006).
[CrossRef]

Pernigo, M.

W. Schleich, M. Pernigo, and F. L. Kien, “Nonclassical state from two pseudoclassical states,” Phys. Rev. A 44, 2172-2187 (1991).
[CrossRef] [PubMed]

Phoenix, S. J. D.

S. J. D. Phoenix, “Wave-packet evolution in the damped oscillator,” Phys. Rev. A 41, 5132-5138 (1990).
[CrossRef] [PubMed]

Radmore, P. M.

S. M. Barnett and P. M. Radmore, Methods in Theoretical Quantum Optics (Oxford U. Press, 1997).

Raimond, J. M.

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]

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, 4887-4890 (1996).
[CrossRef] [PubMed]

Ralph, T. C.

H. Jeong and T. C. Ralph, “Transfer of nonclassical properties from a microscopic superposition to macroscopic thermal states in the high temperature limit,” Phys. Rev. Lett. 97, 100401 (2006).
[CrossRef] [PubMed]

Schleich, W.

W. Schleich, M. Pernigo, and F. L. Kien, “Nonclassical state from two pseudoclassical states,” Phys. Rev. A 44, 2172-2187 (1991).
[CrossRef] [PubMed]

Schrödinger, E.

E. Schrödinger, “Die gegenwartige Situation in der Quantenmechanik,” Naturwiss. 23, 807-812, 823-828, 844-849 (1935).
[CrossRef]

E. Schrödinger, “The constant crossover of micro- to macro-mechanics,” Naturwiss. 14, 664-666 (1926).
[CrossRef]

Scully, M. O.

M. Hillery, R. F. O'Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121-167 (1984).
[CrossRef]

Stoler, D.

B. Yurke and D. Stoler, “Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion,” Phys. Rev. Lett. 57, 13-16 (1986).
[CrossRef] [PubMed]

Tualle-Brouri, R.

A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical 'Schrodinger cats' from photon number states,” Nature 448, 784-786 (2007).
[CrossRef] [PubMed]

van der Zouw, G.

M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. van der Zouw, and A. Zeilinger, “Wave-particle duality of C60 molecules,” Nature 401, 680-682 (1999).
[CrossRef]

Vos-Andreae, J.

M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. van der Zouw, and A. Zeilinger, “Wave-particle duality of C60 molecules,” Nature 401, 680-682 (1999).
[CrossRef]

Walls, D. F.

D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, 1994).

Whaley, K. B.

J. I. Korsbakken, K. B. Whaley, J. Dubois, and J. I. Cirac, “Measurement-based measure of the size of macroscopic quantum superpositions,” Phys. Rev. A 75, 042106 (2007).
[CrossRef]

Wigner, E. P.

M. Hillery, R. F. O'Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121-167 (1984).
[CrossRef]

E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749-759 (1932).
[CrossRef]

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, 4887-4890 (1996).
[CrossRef] [PubMed]

Yurke, B.

B. Yurke and D. Stoler, “Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion,” Phys. Rev. Lett. 57, 13-16 (1986).
[CrossRef] [PubMed]

Zeilinger, A.

M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. van der Zouw, and A. Zeilinger, “Wave-particle duality of C60 molecules,” Nature 401, 680-682 (1999).
[CrossRef]

Zoller, P.

G. W. Gardiner and P. Zoller, Quantum Noise, 3rd ed. (Springer-Verlag, 2004).

Zurek, W. H.

W. H. Zurek, “Decoherence, einselection, and the quantum origins of the classical,” Rev. Mod. Phys. 75, 715-775 (2003), and references therein.
[CrossRef]

J. Phys.: Condens. Matter (1)

A. J. Leggett, “Testing the limits of quantum mechanics: motivation, state of play, prospects,” J. Phys.: Condens. Matter 14, R415-R451 (2002).
[CrossRef]

Nature (2)

M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. van der Zouw, and A. Zeilinger, “Wave-particle duality of C60 molecules,” Nature 401, 680-682 (1999).
[CrossRef]

A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical 'Schrodinger cats' from photon number states,” Nature 448, 784-786 (2007).
[CrossRef] [PubMed]

Naturwiss. (2)

E. Schrödinger, “Die gegenwartige Situation in der Quantenmechanik,” Naturwiss. 23, 807-812, 823-828, 844-849 (1935).
[CrossRef]

E. Schrödinger, “The constant crossover of micro- to macro-mechanics,” Naturwiss. 14, 664-666 (1926).
[CrossRef]

Phys. Rep. (1)

M. Hillery, R. F. O'Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121-167 (1984).
[CrossRef]

Phys. Rev. (1)

E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749-759 (1932).
[CrossRef]

Phys. Rev. A (6)

S. J. D. Phoenix, “Wave-packet evolution in the damped oscillator,” Phys. Rev. A 41, 5132-5138 (1990).
[CrossRef] [PubMed]

M. Paternostro, H. Jeong, and M. S. Kim, “Entanglement of mixed macroscopic superpositions: an entangling-power study,” Phys. Rev. A 73, 012338 (2006).
[CrossRef]

H. Jeong, J. Lee, and M. S. Kim, “Dynamics of nonlocality for a two-mode squeezed state in a thermal environment,” Phys. Rev. A 61, 052101 (2000).
[CrossRef]

W. Schleich, M. Pernigo, and F. L. Kien, “Nonclassical state from two pseudoclassical states,” Phys. Rev. A 44, 2172-2187 (1991).
[CrossRef] [PubMed]

M. S. Kim and V. Bužek, “Schrodinger-cat states at finite temperature: influence of a finite-temperature heat bath on quantum interferences,” Phys. Rev. A 46, 4239-4251 (1992).
[CrossRef] [PubMed]

J. I. Korsbakken, K. B. Whaley, J. Dubois, and J. I. Cirac, “Measurement-based measure of the size of macroscopic quantum superpositions,” Phys. Rev. A 75, 042106 (2007).
[CrossRef]

Phys. Rev. Lett. (3)

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, 4887-4890 (1996).
[CrossRef] [PubMed]

B. Yurke and D. Stoler, “Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion,” Phys. Rev. Lett. 57, 13-16 (1986).
[CrossRef] [PubMed]

H. Jeong and T. C. Ralph, “Transfer of nonclassical properties from a microscopic superposition to macroscopic thermal states in the high temperature limit,” Phys. Rev. Lett. 97, 100401 (2006).
[CrossRef] [PubMed]

Rev. Mod. Phys. (2)

W. H. Zurek, “Decoherence, einselection, and the quantum origins of the classical,” Rev. Mod. Phys. 75, 715-775 (2003), and references therein.
[CrossRef]

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]

Science (1)

A. J. Leggett, “The quantum measurement problem,” Science 307, 871-872 (2005).
[CrossRef] [PubMed]

Other (5)

A more general form of such a superposition state is ∣α⟩+eiφ∣−α⟩, where the normalization factor is omitted and φ is arbitrary, while we choose φ=π for convenience. However, all the conclusions in this paper remain the same regardless of the choice of φ.

H. J. Carmichael, “Quantum future,” in Proceedings of the Xth Max Born Symposium, Poland, Ph.Blanchard and A.Jadczyk, eds. (Springer, 1999), pp. 15-36.

G. W. Gardiner and P. Zoller, Quantum Noise, 3rd ed. (Springer-Verlag, 2004).

S. M. Barnett and P. M. Radmore, Methods in Theoretical Quantum Optics (Oxford U. Press, 1997).

D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, 1994).

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Figures (4)

Fig. 1
Fig. 1

Minimum negativity of the Wigner functions, W ( 0 ) , of pure MQSs for α = 20 (solid curve), α = 30 (dashed curve), and α = 50 (dotted-dashed curve) against the scaled time γ t . The minimum negativity of a MQS approaches zero more rapidly when α is larger.

Fig. 2
Fig. 2

Minimum negative values of the Wigner function of pure MQSs (dashed curves) and of highly mixed MQSs (solid curves) for the same separation (i.e., the same α) between the component states. (a) A pure MQS with α = 30 , and a highly mixed MQS with V = 10 3 and α = 30 . The average photon number of the pure MQS is 900 while that of the highly mixed MQS is 1.4 × 10 3 . (b) A pure MQS with α = 100 , and a highly mixed MQS with V = 10 4 and α = 100 . The average photon number of the pure MQS is 10 4 while that of the highly mixed MQS is 1.5 × 10 4 . In both cases, the minimum negative values of pure MQSs approach zero faster than those of highly mixed MQSs after a certain time.

Fig. 3
Fig. 3

Minimum negative values for the Wigner functions of pure MQSs (dashed curves) and highly mixed MQSs (solid curves) for the same average photon numbers: (a) a pure MQS with α = 30 and a highly mixed MQS with V = 10 3 and α = 20 where the average photon number of each state is equally 900 , and (b) a pure MQS with α 100 and a highly mixed MQS with V = 1.5 × 10 4 and α = 50 where the average photon number of each state is equally 1.0 × 10 4 . The minimum negative values of the pure MQSs obviously approach zero faster than those of the highly mixed MQSs.

Fig. 4
Fig. 4

Probabilistic mixture C ( t ) in Eq. (16) (solid curves) of exponentially decaying terms, as compared with the single decaying term e 2 γ α 2 t (dashed curves), for (a) α = 30 , V = 10 3 , and (b) α = 100 , V = 10 4 .

Equations (18)

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Ψ α = N α ( α α ) ,
ρ t h ( V , α ) = d 2 β P β t h ( V , α ) β β ,
P β t h ( V , α ) = 2 π ( V 1 ) exp [ 2 β α 2 V 1 ] .
ρ = N ( ρ t h ( V , α ) + ρ t h ( V , α ) σ ( V , α ) ) ,
S ( ρ ) = 1 4 ( N ) 2 ( 1 + exp [ α 2 V ] V 4 exp [ 4 α 2 V 1 + V 2 ] 1 + V 2 ) .
n ¯ = Tr [ a ̂ a ̂ ρ ] = N exp [ 2 α 2 V 1 ] { V ( V 1 ) Q ( 5 ) + 2 α 2 Q ( 7 ) } V 3 ( V 1 ) ,
Q ( n ) = exp [ 2 α 2 V 1 ] V ( n 2 ) ( V 1 ) V + V 1 2 .
ρ t = J ̂ ρ + L ̂ ρ , J ̂ ρ = γ a ρ a , L ̂ ρ = γ 2 ( a a ρ + ρ a a ) ,
α β e ( 1 κ ) { ( α 2 + β 2 ) 2 α β * } κ α κ β ,
ρ ( t ) = N ( t ) ( ρ t h ( V , α ) + ρ t h ( V , α ) σ C ( V , α ) ) ,
σ C ( V , α ) = d 2 β P t h ( V , α ) e 2 ( 1 κ ) β 2 β β + H.c. ,
N ( t ) = 2 8 exp [ 2 α 2 ( 3 κ + 1 ) 3 κ ( V 1 ) + ( V + 3 ) ] 3 κ ( V 1 ) + ( V + 3 ) .
W ( η ) = 1 π 2 d 2 ξ e η ξ * η * ξ χ ( ξ ) ,
W ( 0 ) = 4 [ e 2 α 2 κ A A 4 e 2 α 2 ( 1 κ ) B B ] π [ 2 8 e 2 α 2 ( 1 + 3 κ ) C C ] ,
ρ = d 2 β P { V , α } ( β ) Ψ β Ψ β ,
P { V , α } ( β ) = 2 π ( V 1 ) 1 e 2 β 2 1 1 V e ( 2 V ) α 2 e ( 2 ( V 1 ) ) β α 2 ,
C ( t ) = d 2 β P { V , α } ( β ) e 2 γ β 2 t , = 1 1 1 V e ( 2 V ) α 2 [ D ( γ t ) D ( γ t + 1 ) ] ,
D ( x ) 1 1 + ( V 1 ) x e 2 α 2 x ( 1 + ( V 1 ) x ) .

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