Abstract

We investigate the long-lived quantum coherence of two-level spontaneous emission models within structured environments. The population of the system under the asymptotic non-Markovian dynamics is linked to the spectral density of the reservoir through a general functional relation between them. We figure out explicitly the preservation of quantum coherence, via notions of entanglement and quantum discord, in connection with the spectral parameters of Ohmic class reservoirs, and then show how to achieve them optimally. We expect these results to contribute to reservoir engineering with the aim of enhancing stationary quantum coherence in noisy environments.

© 2013 Optical Society of America

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  1. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University, 2002).
  2. G. Benenti, G. Casati, and G. Strini, Principles of Quantum Computation and Information (World Scientific, 2007).
  3. T. Yu and J. H. Eberly, Phys. Rev. Lett. 93, 140404 (2004).
    [CrossRef]
  4. P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, Nature 430, 654 (2004).
    [CrossRef]
  5. X. Xu, T. Yamada, R. Ueda, and A. Otomo, Opt. Lett. 33, 1768 (2008).
    [CrossRef]
  6. U. Hoeppe, C. Wolff, J. Küchenmeister, J. Niegemann, M. Drescher, H. Benner, and K. Busch, Phys. Rev. Lett. 108, 043603 (2012).
    [CrossRef]
  7. C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Rev. Mod. Phys. 82, 1225 (2010).
    [CrossRef]
  8. S. John and J. Wang, Phys. Rev. Lett. 64, 2418 (1990).
    [CrossRef]
  9. S. John and J. Wang, Phys. Rev. B 43, 12772 (1991).
    [CrossRef]
  10. S. John and T. Quang, Phys. Rev. A 50, 1764 (1994).
    [CrossRef]
  11. A. G. Kofman, G. Kurizki, and B. Sherman, J. Mod. Opt. 41, 353 (1994).
    [CrossRef]
  12. B. Bellomo, R. LoFranco, S. Maniscalco, and G. Compagno, Phys. Rev. A 78060302(R) (2008).
    [CrossRef]
  13. Q.-J. Tong, J.-H. An, H.-G. Luo, and C. H. Oh, Phys. Rev. A 81, 052330 (2010).
    [CrossRef]
  14. R. Lo Franco, B. Bellomo, S. Maniscalco, and G. Compagno, Int. J. Mod. Phys. B 27, 1345053 (2013).
    [CrossRef]
  15. G. M. Nikolopoulos, S. Bay, and P. Lambropoulos, Phys. Rev. A 60, 5079 (1999).
    [CrossRef]
  16. B. M. Garraway, Phys. Rev. A 55, 4636 (1997).
    [CrossRef]
  17. C. Lazarou, K. Luoma, S. Maniscalco, J. Piilo, and B. M. Garraway, Phys. Rev. A 86, 012331 (2012).
    [CrossRef]
  18. K. O. Friedrichs, Commun. Pure Appl. Math. 1, 361 (1948).
    [CrossRef]
  19. P. Exner, Open Quantum Systems and Feynman Integrals (Reidel, 1985).
  20. J. von Neumann and E. Wigner, Phys. Z. 30, 465 (1929).
  21. L. P. Horwitz and J.-P. Marchand, Rocky Mt. J. Math. 1, 225 (1971).
    [CrossRef]
  22. M. Miyamoto, Phys. Rev. A 72, 063405 (2005).
    [CrossRef]
  23. Q.-J. Tong, J.-H. An, H.-G. Luo, and C. H. Oh, Phys. Rev. B 84, 174301 (2011).
    [CrossRef]
  24. W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).
    [CrossRef]
  25. H. Ollivier and W. H. Zurek, Phys. Rev. Lett. 88, 017901 (2002).
    [CrossRef]
  26. L. Henderson and V. Vedral, J. Phys. A 34, 6899 (2001).
    [CrossRef]
  27. L.-X. Cen, X.-Q. Li, J. S. Shao, and Y. J. Yan, Phys. Rev. A 83, 054101 (2011).
    [CrossRef]

2013 (1)

R. Lo Franco, B. Bellomo, S. Maniscalco, and G. Compagno, Int. J. Mod. Phys. B 27, 1345053 (2013).
[CrossRef]

2012 (2)

C. Lazarou, K. Luoma, S. Maniscalco, J. Piilo, and B. M. Garraway, Phys. Rev. A 86, 012331 (2012).
[CrossRef]

U. Hoeppe, C. Wolff, J. Küchenmeister, J. Niegemann, M. Drescher, H. Benner, and K. Busch, Phys. Rev. Lett. 108, 043603 (2012).
[CrossRef]

2011 (2)

Q.-J. Tong, J.-H. An, H.-G. Luo, and C. H. Oh, Phys. Rev. B 84, 174301 (2011).
[CrossRef]

L.-X. Cen, X.-Q. Li, J. S. Shao, and Y. J. Yan, Phys. Rev. A 83, 054101 (2011).
[CrossRef]

2010 (2)

C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Rev. Mod. Phys. 82, 1225 (2010).
[CrossRef]

Q.-J. Tong, J.-H. An, H.-G. Luo, and C. H. Oh, Phys. Rev. A 81, 052330 (2010).
[CrossRef]

2008 (2)

B. Bellomo, R. LoFranco, S. Maniscalco, and G. Compagno, Phys. Rev. A 78060302(R) (2008).
[CrossRef]

X. Xu, T. Yamada, R. Ueda, and A. Otomo, Opt. Lett. 33, 1768 (2008).
[CrossRef]

2005 (1)

M. Miyamoto, Phys. Rev. A 72, 063405 (2005).
[CrossRef]

2004 (2)

T. Yu and J. H. Eberly, Phys. Rev. Lett. 93, 140404 (2004).
[CrossRef]

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, Nature 430, 654 (2004).
[CrossRef]

2002 (1)

H. Ollivier and W. H. Zurek, Phys. Rev. Lett. 88, 017901 (2002).
[CrossRef]

2001 (1)

L. Henderson and V. Vedral, J. Phys. A 34, 6899 (2001).
[CrossRef]

1999 (1)

G. M. Nikolopoulos, S. Bay, and P. Lambropoulos, Phys. Rev. A 60, 5079 (1999).
[CrossRef]

1998 (1)

W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).
[CrossRef]

1997 (1)

B. M. Garraway, Phys. Rev. A 55, 4636 (1997).
[CrossRef]

1994 (2)

S. John and T. Quang, Phys. Rev. A 50, 1764 (1994).
[CrossRef]

A. G. Kofman, G. Kurizki, and B. Sherman, J. Mod. Opt. 41, 353 (1994).
[CrossRef]

1991 (1)

S. John and J. Wang, Phys. Rev. B 43, 12772 (1991).
[CrossRef]

1990 (1)

S. John and J. Wang, Phys. Rev. Lett. 64, 2418 (1990).
[CrossRef]

1971 (1)

L. P. Horwitz and J.-P. Marchand, Rocky Mt. J. Math. 1, 225 (1971).
[CrossRef]

1948 (1)

K. O. Friedrichs, Commun. Pure Appl. Math. 1, 361 (1948).
[CrossRef]

1929 (1)

J. von Neumann and E. Wigner, Phys. Z. 30, 465 (1929).

An, J.-H.

Q.-J. Tong, J.-H. An, H.-G. Luo, and C. H. Oh, Phys. Rev. B 84, 174301 (2011).
[CrossRef]

Q.-J. Tong, J.-H. An, H.-G. Luo, and C. H. Oh, Phys. Rev. A 81, 052330 (2010).
[CrossRef]

Bay, S.

G. M. Nikolopoulos, S. Bay, and P. Lambropoulos, Phys. Rev. A 60, 5079 (1999).
[CrossRef]

Bellomo, B.

R. Lo Franco, B. Bellomo, S. Maniscalco, and G. Compagno, Int. J. Mod. Phys. B 27, 1345053 (2013).
[CrossRef]

B. Bellomo, R. LoFranco, S. Maniscalco, and G. Compagno, Phys. Rev. A 78060302(R) (2008).
[CrossRef]

Benenti, G.

G. Benenti, G. Casati, and G. Strini, Principles of Quantum Computation and Information (World Scientific, 2007).

Benner, H.

U. Hoeppe, C. Wolff, J. Küchenmeister, J. Niegemann, M. Drescher, H. Benner, and K. Busch, Phys. Rev. Lett. 108, 043603 (2012).
[CrossRef]

Breuer, H.-P.

H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University, 2002).

Busch, K.

U. Hoeppe, C. Wolff, J. Küchenmeister, J. Niegemann, M. Drescher, H. Benner, and K. Busch, Phys. Rev. Lett. 108, 043603 (2012).
[CrossRef]

Casati, G.

G. Benenti, G. Casati, and G. Strini, Principles of Quantum Computation and Information (World Scientific, 2007).

Cen, L.-X.

L.-X. Cen, X.-Q. Li, J. S. Shao, and Y. J. Yan, Phys. Rev. A 83, 054101 (2011).
[CrossRef]

Chin, C.

C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Rev. Mod. Phys. 82, 1225 (2010).
[CrossRef]

Compagno, G.

R. Lo Franco, B. Bellomo, S. Maniscalco, and G. Compagno, Int. J. Mod. Phys. B 27, 1345053 (2013).
[CrossRef]

B. Bellomo, R. LoFranco, S. Maniscalco, and G. Compagno, Phys. Rev. A 78060302(R) (2008).
[CrossRef]

Drescher, M.

U. Hoeppe, C. Wolff, J. Küchenmeister, J. Niegemann, M. Drescher, H. Benner, and K. Busch, Phys. Rev. Lett. 108, 043603 (2012).
[CrossRef]

Eberly, J. H.

T. Yu and J. H. Eberly, Phys. Rev. Lett. 93, 140404 (2004).
[CrossRef]

Exner, P.

P. Exner, Open Quantum Systems and Feynman Integrals (Reidel, 1985).

Friedrichs, K. O.

K. O. Friedrichs, Commun. Pure Appl. Math. 1, 361 (1948).
[CrossRef]

Garraway, B. M.

C. Lazarou, K. Luoma, S. Maniscalco, J. Piilo, and B. M. Garraway, Phys. Rev. A 86, 012331 (2012).
[CrossRef]

B. M. Garraway, Phys. Rev. A 55, 4636 (1997).
[CrossRef]

Grimm, R.

C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Rev. Mod. Phys. 82, 1225 (2010).
[CrossRef]

Henderson, L.

L. Henderson and V. Vedral, J. Phys. A 34, 6899 (2001).
[CrossRef]

Hoeppe, U.

U. Hoeppe, C. Wolff, J. Küchenmeister, J. Niegemann, M. Drescher, H. Benner, and K. Busch, Phys. Rev. Lett. 108, 043603 (2012).
[CrossRef]

Horwitz, L. P.

L. P. Horwitz and J.-P. Marchand, Rocky Mt. J. Math. 1, 225 (1971).
[CrossRef]

Irman, A.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, Nature 430, 654 (2004).
[CrossRef]

John, S.

S. John and T. Quang, Phys. Rev. A 50, 1764 (1994).
[CrossRef]

S. John and J. Wang, Phys. Rev. B 43, 12772 (1991).
[CrossRef]

S. John and J. Wang, Phys. Rev. Lett. 64, 2418 (1990).
[CrossRef]

Julienne, P.

C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Rev. Mod. Phys. 82, 1225 (2010).
[CrossRef]

Kofman, A. G.

A. G. Kofman, G. Kurizki, and B. Sherman, J. Mod. Opt. 41, 353 (1994).
[CrossRef]

Küchenmeister, J.

U. Hoeppe, C. Wolff, J. Küchenmeister, J. Niegemann, M. Drescher, H. Benner, and K. Busch, Phys. Rev. Lett. 108, 043603 (2012).
[CrossRef]

Kurizki, G.

A. G. Kofman, G. Kurizki, and B. Sherman, J. Mod. Opt. 41, 353 (1994).
[CrossRef]

Lambropoulos, P.

G. M. Nikolopoulos, S. Bay, and P. Lambropoulos, Phys. Rev. A 60, 5079 (1999).
[CrossRef]

Lazarou, C.

C. Lazarou, K. Luoma, S. Maniscalco, J. Piilo, and B. M. Garraway, Phys. Rev. A 86, 012331 (2012).
[CrossRef]

Li, X.-Q.

L.-X. Cen, X.-Q. Li, J. S. Shao, and Y. J. Yan, Phys. Rev. A 83, 054101 (2011).
[CrossRef]

Lo Franco, R.

R. Lo Franco, B. Bellomo, S. Maniscalco, and G. Compagno, Int. J. Mod. Phys. B 27, 1345053 (2013).
[CrossRef]

Lodahl, P.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, Nature 430, 654 (2004).
[CrossRef]

LoFranco, R.

B. Bellomo, R. LoFranco, S. Maniscalco, and G. Compagno, Phys. Rev. A 78060302(R) (2008).
[CrossRef]

Luo, H.-G.

Q.-J. Tong, J.-H. An, H.-G. Luo, and C. H. Oh, Phys. Rev. B 84, 174301 (2011).
[CrossRef]

Q.-J. Tong, J.-H. An, H.-G. Luo, and C. H. Oh, Phys. Rev. A 81, 052330 (2010).
[CrossRef]

Luoma, K.

C. Lazarou, K. Luoma, S. Maniscalco, J. Piilo, and B. M. Garraway, Phys. Rev. A 86, 012331 (2012).
[CrossRef]

Maniscalco, S.

R. Lo Franco, B. Bellomo, S. Maniscalco, and G. Compagno, Int. J. Mod. Phys. B 27, 1345053 (2013).
[CrossRef]

C. Lazarou, K. Luoma, S. Maniscalco, J. Piilo, and B. M. Garraway, Phys. Rev. A 86, 012331 (2012).
[CrossRef]

B. Bellomo, R. LoFranco, S. Maniscalco, and G. Compagno, Phys. Rev. A 78060302(R) (2008).
[CrossRef]

Marchand, J.-P.

L. P. Horwitz and J.-P. Marchand, Rocky Mt. J. Math. 1, 225 (1971).
[CrossRef]

Miyamoto, M.

M. Miyamoto, Phys. Rev. A 72, 063405 (2005).
[CrossRef]

Niegemann, J.

U. Hoeppe, C. Wolff, J. Küchenmeister, J. Niegemann, M. Drescher, H. Benner, and K. Busch, Phys. Rev. Lett. 108, 043603 (2012).
[CrossRef]

Nikolaev, I. S.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, Nature 430, 654 (2004).
[CrossRef]

Nikolopoulos, G. M.

G. M. Nikolopoulos, S. Bay, and P. Lambropoulos, Phys. Rev. A 60, 5079 (1999).
[CrossRef]

Oh, C. H.

Q.-J. Tong, J.-H. An, H.-G. Luo, and C. H. Oh, Phys. Rev. B 84, 174301 (2011).
[CrossRef]

Q.-J. Tong, J.-H. An, H.-G. Luo, and C. H. Oh, Phys. Rev. A 81, 052330 (2010).
[CrossRef]

Ollivier, H.

H. Ollivier and W. H. Zurek, Phys. Rev. Lett. 88, 017901 (2002).
[CrossRef]

Otomo, A.

Overgaag, K.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, Nature 430, 654 (2004).
[CrossRef]

Petruccione, F.

H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University, 2002).

Piilo, J.

C. Lazarou, K. Luoma, S. Maniscalco, J. Piilo, and B. M. Garraway, Phys. Rev. A 86, 012331 (2012).
[CrossRef]

Quang, T.

S. John and T. Quang, Phys. Rev. A 50, 1764 (1994).
[CrossRef]

Shao, J. S.

L.-X. Cen, X.-Q. Li, J. S. Shao, and Y. J. Yan, Phys. Rev. A 83, 054101 (2011).
[CrossRef]

Sherman, B.

A. G. Kofman, G. Kurizki, and B. Sherman, J. Mod. Opt. 41, 353 (1994).
[CrossRef]

Strini, G.

G. Benenti, G. Casati, and G. Strini, Principles of Quantum Computation and Information (World Scientific, 2007).

Tiesinga, E.

C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Rev. Mod. Phys. 82, 1225 (2010).
[CrossRef]

Tong, Q.-J.

Q.-J. Tong, J.-H. An, H.-G. Luo, and C. H. Oh, Phys. Rev. B 84, 174301 (2011).
[CrossRef]

Q.-J. Tong, J.-H. An, H.-G. Luo, and C. H. Oh, Phys. Rev. A 81, 052330 (2010).
[CrossRef]

Ueda, R.

van Driel, A. F.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, Nature 430, 654 (2004).
[CrossRef]

Vanmaekelbergh, D.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, Nature 430, 654 (2004).
[CrossRef]

Vedral, V.

L. Henderson and V. Vedral, J. Phys. A 34, 6899 (2001).
[CrossRef]

von Neumann, J.

J. von Neumann and E. Wigner, Phys. Z. 30, 465 (1929).

Vos, W. L.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, Nature 430, 654 (2004).
[CrossRef]

Wang, J.

S. John and J. Wang, Phys. Rev. B 43, 12772 (1991).
[CrossRef]

S. John and J. Wang, Phys. Rev. Lett. 64, 2418 (1990).
[CrossRef]

Wigner, E.

J. von Neumann and E. Wigner, Phys. Z. 30, 465 (1929).

Wolff, C.

U. Hoeppe, C. Wolff, J. Küchenmeister, J. Niegemann, M. Drescher, H. Benner, and K. Busch, Phys. Rev. Lett. 108, 043603 (2012).
[CrossRef]

Wootters, W. K.

W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).
[CrossRef]

Xu, X.

Yamada, T.

Yan, Y. J.

L.-X. Cen, X.-Q. Li, J. S. Shao, and Y. J. Yan, Phys. Rev. A 83, 054101 (2011).
[CrossRef]

Yu, T.

T. Yu and J. H. Eberly, Phys. Rev. Lett. 93, 140404 (2004).
[CrossRef]

Zurek, W. H.

H. Ollivier and W. H. Zurek, Phys. Rev. Lett. 88, 017901 (2002).
[CrossRef]

Commun. Pure Appl. Math. (1)

K. O. Friedrichs, Commun. Pure Appl. Math. 1, 361 (1948).
[CrossRef]

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

R. Lo Franco, B. Bellomo, S. Maniscalco, and G. Compagno, Int. J. Mod. Phys. B 27, 1345053 (2013).
[CrossRef]

J. Mod. Opt. (1)

A. G. Kofman, G. Kurizki, and B. Sherman, J. Mod. Opt. 41, 353 (1994).
[CrossRef]

J. Phys. A (1)

L. Henderson and V. Vedral, J. Phys. A 34, 6899 (2001).
[CrossRef]

Nature (1)

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, Nature 430, 654 (2004).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (8)

B. Bellomo, R. LoFranco, S. Maniscalco, and G. Compagno, Phys. Rev. A 78060302(R) (2008).
[CrossRef]

Q.-J. Tong, J.-H. An, H.-G. Luo, and C. H. Oh, Phys. Rev. A 81, 052330 (2010).
[CrossRef]

S. John and T. Quang, Phys. Rev. A 50, 1764 (1994).
[CrossRef]

G. M. Nikolopoulos, S. Bay, and P. Lambropoulos, Phys. Rev. A 60, 5079 (1999).
[CrossRef]

B. M. Garraway, Phys. Rev. A 55, 4636 (1997).
[CrossRef]

C. Lazarou, K. Luoma, S. Maniscalco, J. Piilo, and B. M. Garraway, Phys. Rev. A 86, 012331 (2012).
[CrossRef]

L.-X. Cen, X.-Q. Li, J. S. Shao, and Y. J. Yan, Phys. Rev. A 83, 054101 (2011).
[CrossRef]

M. Miyamoto, Phys. Rev. A 72, 063405 (2005).
[CrossRef]

Phys. Rev. B (2)

Q.-J. Tong, J.-H. An, H.-G. Luo, and C. H. Oh, Phys. Rev. B 84, 174301 (2011).
[CrossRef]

S. John and J. Wang, Phys. Rev. B 43, 12772 (1991).
[CrossRef]

Phys. Rev. Lett. (5)

U. Hoeppe, C. Wolff, J. Küchenmeister, J. Niegemann, M. Drescher, H. Benner, and K. Busch, Phys. Rev. Lett. 108, 043603 (2012).
[CrossRef]

T. Yu and J. H. Eberly, Phys. Rev. Lett. 93, 140404 (2004).
[CrossRef]

S. John and J. Wang, Phys. Rev. Lett. 64, 2418 (1990).
[CrossRef]

W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).
[CrossRef]

H. Ollivier and W. H. Zurek, Phys. Rev. Lett. 88, 017901 (2002).
[CrossRef]

Phys. Z. (1)

J. von Neumann and E. Wigner, Phys. Z. 30, 465 (1929).

Rev. Mod. Phys. (1)

C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Rev. Mod. Phys. 82, 1225 (2010).
[CrossRef]

Rocky Mt. J. Math. (1)

L. P. Horwitz and J.-P. Marchand, Rocky Mt. J. Math. 1, 225 (1971).
[CrossRef]

Other (3)

P. Exner, Open Quantum Systems and Feynman Integrals (Reidel, 1985).

H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University, 2002).

G. Benenti, G. Casati, and G. Strini, Principles of Quantum Computation and Information (World Scientific, 2007).

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Figures (2)

Fig. 1.
Fig. 1.

Long-lived population P=b4 in relation to the parameters of Ohmic class spectra. (a) P as a function of η with ωc=0.3ω0, s=5, 5.25, and 5.5, respectively. The green-dotted line figures out the boundary determined by the critical condition ηΓ(s)=ω0/ωc. For s=5.5, the maximal P=0.33 is achieved at η=0.08. (b) P as a function of s with η=0.08, ωc/ω0=15, 50, and . In the limit ωc/ω0, the maximal P=0.90 is achieved at s=2.34.

Fig. 2.
Fig. 2.

Long-lived entanglement and quantum discord as a function of s for the Ohmic class reservoirs with η=0.08 and ωcω0. The two-qubit system is initially in a pure state |ψAB(0)=α|++β|+. The maximal values CAB0.95 and QAB0.88 are obtained at s2.34 for the maximally entangled input state.

Equations (9)

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

H=ω0σ+σ+k[ωkakak+(gkakσ++gk*akσ)],
ω00J(ω)ωEdω=E,
b=[1+0J(ω)(ωE)2dω]1/2.
P(t)|c(t)|2tP=b4.
c˙(t)+iω0c(t)+0tc(τ)f(tτ)dτ=0
P(η,s,ω0/ωc)=[1+0ηxsex(xκ)2dx]2,
ω0/ωc0ηxsexxκdx=κ.
Γ1(t)=[00c¯(t)0],Γ2(t)=[c(t)001]
ρAB(t)=i=12Γi(t)IBρAB(0)Γi(t)IB.

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