Abstract

Generating entangled states is a vital ingredient for quantum information engineering. Here, we investigate the entanglement generation between two quantum dots coupled to nanoring surface plasmons with asymmetric coupling strength g1 and g2. The dynamics of concurrence C is obtained by solving the corresponding master equation. High entanglement can be generated at appropriate times through the scatterings of the incident field and its scattered field. Furthermore, we find that maximum entanglement can be created when rg1/g2 is the ratio of odd numbers. Contrary to intuition, relative high entanglement (C1) can remain even if the ratio r is far off the required values, which is useful in real experiments.

© 2012 Optical Society of America

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  1. R. Zia and M. L. Brongersma, Nature Nanotechnol. 2, 426 (2007).
    [CrossRef]
  2. K. Yu. Bliokh, Yu. P. Bliokh, V. Freilikher, S. Savel’ev, and F. Nori, Rev. Mod. Phys. 80, 1201 (2008).
    [CrossRef]
  3. S. Savel’ev, V. A. Yampol’skii, A. L. Rakhmanov, and F. Nori, Rep. Prog. Phys. 73, 026501 (2010).
    [CrossRef]
  4. Y. N. Chen, G. Y. Chen, D. S. Chuu, and T. Brandes, Phys. Rev. A 79, 033815 (2009).
  5. S. D. Liu, Z. Yang, R. P. Liu, and X. Y. Li, J. Phys. Chem. C 115, 24469 (2011).
  6. Z. R. Lin, G. P. Guo, T. Tu, H. O. Li, C. L. Zou, J. X. Chen, Y. H. Lu, X. F. Ren, and G. C. Guo, Phys. Rev. B 82, 241401 (2010).
  7. L. Zhou, Z. R. Gong, Y. X. Liu, C. P. Sun, and F. Nori, Phys. Rev. Lett. 101, 100501 (2008).
  8. G. Y. Chen, N. Lambert, C. H. Chou, Y. N. Chen, and F. Nori, Phys. Rev. B 84, 045310 (2011).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  14. Here we assume that the interdot distance is not long, such that the two quantum dots can simultaneously couple to the k-mode field.
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    [CrossRef]
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    [CrossRef]
  17. C. Ropp, R. Probst, Z. Cummins, R. Kumar, A. J. Berglund, S. R. Raghavan, E. Waks, and B. Shapiro, Nano Lett. 10, 2525 (2010).
  18. J. K. Yang, Jpn. J. Appl. Phys. 50, 060205 (2011).
    [CrossRef]
  19. L. O. Diniz, E. Marega, F. D. Nunes, and B.-H. V. Borges, J. Opt. 13, 115001 (2011).
    [CrossRef]
  20. J. S. Jin, C. S. Yu, and H. S. Song, Phys. Rev. A 81, 042309 (2010).

2011

G. Y. Chen, N. Lambert, C. H. Chou, Y. N. Chen, and F. Nori, Phys. Rev. B 84, 045310 (2011).

A. Gonzalez-Tudela, D. Martin-Cano, E. Moreno, L. Martin-Moreno, C. Tejedor, and F. J. Garcia-Vidal, Phys. Rev. Lett. 106, 020501 (2011).
[CrossRef]

S. D. Liu, Z. Yang, R. P. Liu, and X. Y. Li, J. Phys. Chem. C 115, 24469 (2011).

M. J. Kastoryano, F. Reiter, and A. S. Sørensen, Phys. Rev. Lett. 106, 090502 (2011).
[CrossRef]

J. K. Yang, Jpn. J. Appl. Phys. 50, 060205 (2011).
[CrossRef]

L. O. Diniz, E. Marega, F. D. Nunes, and B.-H. V. Borges, J. Opt. 13, 115001 (2011).
[CrossRef]

2010

J. S. Jin, C. S. Yu, and H. S. Song, Phys. Rev. A 81, 042309 (2010).

Z. R. Lin, G. P. Guo, T. Tu, H. O. Li, C. L. Zou, J. X. Chen, Y. H. Lu, X. F. Ren, and G. C. Guo, Phys. Rev. B 82, 241401 (2010).

S. Savel’ev, V. A. Yampol’skii, A. L. Rakhmanov, and F. Nori, Rep. Prog. Phys. 73, 026501 (2010).
[CrossRef]

C. Ropp, R. Probst, Z. Cummins, R. Kumar, A. J. Berglund, S. R. Raghavan, E. Waks, and B. Shapiro, Nano Lett. 10, 2525 (2010).

2009

Y. N. Chen, G. Y. Chen, D. S. Chuu, and T. Brandes, Phys. Rev. A 79, 033815 (2009).

2008

K. Yu. Bliokh, Yu. P. Bliokh, V. Freilikher, S. Savel’ev, and F. Nori, Rev. Mod. Phys. 80, 1201 (2008).
[CrossRef]

L. Zhou, Z. R. Gong, Y. X. Liu, C. P. Sun, and F. Nori, Phys. Rev. Lett. 101, 100501 (2008).

2007

D. E. Chang, A. S. Sørensen, E. A. Demler, and M. D. Lukin, Nat. Phys. 3, 807 (2007).
[CrossRef]

R. Zia and M. L. Brongersma, Nature Nanotechnol. 2, 426 (2007).
[CrossRef]

K. Srinivasan and O. Painter, Nature 450, 862 (2007).
[CrossRef]

2002

M. S. Kim, Jinhyoung Lee, D. Ahn, and P. L. Knight, Phys. Rev. A 65, 040101 (2002).

2000

S. B. Zheng and G. C. Guo, Phys. Rev. Lett. 85, 2392 (2000).
[CrossRef]

1998

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

Ahn, D.

M. S. Kim, Jinhyoung Lee, D. Ahn, and P. L. Knight, Phys. Rev. A 65, 040101 (2002).

Berglund, A. J.

C. Ropp, R. Probst, Z. Cummins, R. Kumar, A. J. Berglund, S. R. Raghavan, E. Waks, and B. Shapiro, Nano Lett. 10, 2525 (2010).

Bliokh, K. Yu.

K. Yu. Bliokh, Yu. P. Bliokh, V. Freilikher, S. Savel’ev, and F. Nori, Rev. Mod. Phys. 80, 1201 (2008).
[CrossRef]

Bliokh, Yu. P.

K. Yu. Bliokh, Yu. P. Bliokh, V. Freilikher, S. Savel’ev, and F. Nori, Rev. Mod. Phys. 80, 1201 (2008).
[CrossRef]

Borges, B.-H. V.

L. O. Diniz, E. Marega, F. D. Nunes, and B.-H. V. Borges, J. Opt. 13, 115001 (2011).
[CrossRef]

Brandes, T.

Y. N. Chen, G. Y. Chen, D. S. Chuu, and T. Brandes, Phys. Rev. A 79, 033815 (2009).

Brongersma, M. L.

R. Zia and M. L. Brongersma, Nature Nanotechnol. 2, 426 (2007).
[CrossRef]

Chang, D. E.

D. E. Chang, A. S. Sørensen, E. A. Demler, and M. D. Lukin, Nat. Phys. 3, 807 (2007).
[CrossRef]

Chen, G. Y.

G. Y. Chen, N. Lambert, C. H. Chou, Y. N. Chen, and F. Nori, Phys. Rev. B 84, 045310 (2011).

Y. N. Chen, G. Y. Chen, D. S. Chuu, and T. Brandes, Phys. Rev. A 79, 033815 (2009).

Chen, J. X.

Z. R. Lin, G. P. Guo, T. Tu, H. O. Li, C. L. Zou, J. X. Chen, Y. H. Lu, X. F. Ren, and G. C. Guo, Phys. Rev. B 82, 241401 (2010).

Chen, Y. N.

G. Y. Chen, N. Lambert, C. H. Chou, Y. N. Chen, and F. Nori, Phys. Rev. B 84, 045310 (2011).

Y. N. Chen, G. Y. Chen, D. S. Chuu, and T. Brandes, Phys. Rev. A 79, 033815 (2009).

Chou, C. H.

G. Y. Chen, N. Lambert, C. H. Chou, Y. N. Chen, and F. Nori, Phys. Rev. B 84, 045310 (2011).

Chuu, D. S.

Y. N. Chen, G. Y. Chen, D. S. Chuu, and T. Brandes, Phys. Rev. A 79, 033815 (2009).

Cummins, Z.

C. Ropp, R. Probst, Z. Cummins, R. Kumar, A. J. Berglund, S. R. Raghavan, E. Waks, and B. Shapiro, Nano Lett. 10, 2525 (2010).

Demler, E. A.

D. E. Chang, A. S. Sørensen, E. A. Demler, and M. D. Lukin, Nat. Phys. 3, 807 (2007).
[CrossRef]

Diniz, L. O.

L. O. Diniz, E. Marega, F. D. Nunes, and B.-H. V. Borges, J. Opt. 13, 115001 (2011).
[CrossRef]

Freilikher, V.

K. Yu. Bliokh, Yu. P. Bliokh, V. Freilikher, S. Savel’ev, and F. Nori, Rev. Mod. Phys. 80, 1201 (2008).
[CrossRef]

Garcia-Vidal, F. J.

A. Gonzalez-Tudela, D. Martin-Cano, E. Moreno, L. Martin-Moreno, C. Tejedor, and F. J. Garcia-Vidal, Phys. Rev. Lett. 106, 020501 (2011).
[CrossRef]

Gong, Z. R.

L. Zhou, Z. R. Gong, Y. X. Liu, C. P. Sun, and F. Nori, Phys. Rev. Lett. 101, 100501 (2008).

Gonzalez-Tudela, A.

A. Gonzalez-Tudela, D. Martin-Cano, E. Moreno, L. Martin-Moreno, C. Tejedor, and F. J. Garcia-Vidal, Phys. Rev. Lett. 106, 020501 (2011).
[CrossRef]

Guo, G. C.

Z. R. Lin, G. P. Guo, T. Tu, H. O. Li, C. L. Zou, J. X. Chen, Y. H. Lu, X. F. Ren, and G. C. Guo, Phys. Rev. B 82, 241401 (2010).

S. B. Zheng and G. C. Guo, Phys. Rev. Lett. 85, 2392 (2000).
[CrossRef]

Guo, G. P.

Z. R. Lin, G. P. Guo, T. Tu, H. O. Li, C. L. Zou, J. X. Chen, Y. H. Lu, X. F. Ren, and G. C. Guo, Phys. Rev. B 82, 241401 (2010).

Jin, J. S.

J. S. Jin, C. S. Yu, and H. S. Song, Phys. Rev. A 81, 042309 (2010).

Kastoryano, M. J.

M. J. Kastoryano, F. Reiter, and A. S. Sørensen, Phys. Rev. Lett. 106, 090502 (2011).
[CrossRef]

Kim, M. S.

M. S. Kim, Jinhyoung Lee, D. Ahn, and P. L. Knight, Phys. Rev. A 65, 040101 (2002).

Knight, P. L.

M. S. Kim, Jinhyoung Lee, D. Ahn, and P. L. Knight, Phys. Rev. A 65, 040101 (2002).

Kumar, R.

C. Ropp, R. Probst, Z. Cummins, R. Kumar, A. J. Berglund, S. R. Raghavan, E. Waks, and B. Shapiro, Nano Lett. 10, 2525 (2010).

Lambert, N.

G. Y. Chen, N. Lambert, C. H. Chou, Y. N. Chen, and F. Nori, Phys. Rev. B 84, 045310 (2011).

Lee, Jinhyoung

M. S. Kim, Jinhyoung Lee, D. Ahn, and P. L. Knight, Phys. Rev. A 65, 040101 (2002).

Li, H. O.

Z. R. Lin, G. P. Guo, T. Tu, H. O. Li, C. L. Zou, J. X. Chen, Y. H. Lu, X. F. Ren, and G. C. Guo, Phys. Rev. B 82, 241401 (2010).

Li, X. Y.

S. D. Liu, Z. Yang, R. P. Liu, and X. Y. Li, J. Phys. Chem. C 115, 24469 (2011).

Lin, Z. R.

Z. R. Lin, G. P. Guo, T. Tu, H. O. Li, C. L. Zou, J. X. Chen, Y. H. Lu, X. F. Ren, and G. C. Guo, Phys. Rev. B 82, 241401 (2010).

Liu, R. P.

S. D. Liu, Z. Yang, R. P. Liu, and X. Y. Li, J. Phys. Chem. C 115, 24469 (2011).

Liu, S. D.

S. D. Liu, Z. Yang, R. P. Liu, and X. Y. Li, J. Phys. Chem. C 115, 24469 (2011).

Liu, Y. X.

L. Zhou, Z. R. Gong, Y. X. Liu, C. P. Sun, and F. Nori, Phys. Rev. Lett. 101, 100501 (2008).

Lu, Y. H.

Z. R. Lin, G. P. Guo, T. Tu, H. O. Li, C. L. Zou, J. X. Chen, Y. H. Lu, X. F. Ren, and G. C. Guo, Phys. Rev. B 82, 241401 (2010).

Lukin, M. D.

D. E. Chang, A. S. Sørensen, E. A. Demler, and M. D. Lukin, Nat. Phys. 3, 807 (2007).
[CrossRef]

Marega, E.

L. O. Diniz, E. Marega, F. D. Nunes, and B.-H. V. Borges, J. Opt. 13, 115001 (2011).
[CrossRef]

Martin-Cano, D.

A. Gonzalez-Tudela, D. Martin-Cano, E. Moreno, L. Martin-Moreno, C. Tejedor, and F. J. Garcia-Vidal, Phys. Rev. Lett. 106, 020501 (2011).
[CrossRef]

Martin-Moreno, L.

A. Gonzalez-Tudela, D. Martin-Cano, E. Moreno, L. Martin-Moreno, C. Tejedor, and F. J. Garcia-Vidal, Phys. Rev. Lett. 106, 020501 (2011).
[CrossRef]

Moreno, E.

A. Gonzalez-Tudela, D. Martin-Cano, E. Moreno, L. Martin-Moreno, C. Tejedor, and F. J. Garcia-Vidal, Phys. Rev. Lett. 106, 020501 (2011).
[CrossRef]

Nori, F.

G. Y. Chen, N. Lambert, C. H. Chou, Y. N. Chen, and F. Nori, Phys. Rev. B 84, 045310 (2011).

S. Savel’ev, V. A. Yampol’skii, A. L. Rakhmanov, and F. Nori, Rep. Prog. Phys. 73, 026501 (2010).
[CrossRef]

L. Zhou, Z. R. Gong, Y. X. Liu, C. P. Sun, and F. Nori, Phys. Rev. Lett. 101, 100501 (2008).

K. Yu. Bliokh, Yu. P. Bliokh, V. Freilikher, S. Savel’ev, and F. Nori, Rev. Mod. Phys. 80, 1201 (2008).
[CrossRef]

Nunes, F. D.

L. O. Diniz, E. Marega, F. D. Nunes, and B.-H. V. Borges, J. Opt. 13, 115001 (2011).
[CrossRef]

Painter, O.

K. Srinivasan and O. Painter, Nature 450, 862 (2007).
[CrossRef]

Probst, R.

C. Ropp, R. Probst, Z. Cummins, R. Kumar, A. J. Berglund, S. R. Raghavan, E. Waks, and B. Shapiro, Nano Lett. 10, 2525 (2010).

Raghavan, S. R.

C. Ropp, R. Probst, Z. Cummins, R. Kumar, A. J. Berglund, S. R. Raghavan, E. Waks, and B. Shapiro, Nano Lett. 10, 2525 (2010).

Rakhmanov, A. L.

S. Savel’ev, V. A. Yampol’skii, A. L. Rakhmanov, and F. Nori, Rep. Prog. Phys. 73, 026501 (2010).
[CrossRef]

Reiter, F.

M. J. Kastoryano, F. Reiter, and A. S. Sørensen, Phys. Rev. Lett. 106, 090502 (2011).
[CrossRef]

Ren, X. F.

Z. R. Lin, G. P. Guo, T. Tu, H. O. Li, C. L. Zou, J. X. Chen, Y. H. Lu, X. F. Ren, and G. C. Guo, Phys. Rev. B 82, 241401 (2010).

Ropp, C.

C. Ropp, R. Probst, Z. Cummins, R. Kumar, A. J. Berglund, S. R. Raghavan, E. Waks, and B. Shapiro, Nano Lett. 10, 2525 (2010).

Savel’ev, S.

S. Savel’ev, V. A. Yampol’skii, A. L. Rakhmanov, and F. Nori, Rep. Prog. Phys. 73, 026501 (2010).
[CrossRef]

K. Yu. Bliokh, Yu. P. Bliokh, V. Freilikher, S. Savel’ev, and F. Nori, Rev. Mod. Phys. 80, 1201 (2008).
[CrossRef]

Shapiro, B.

C. Ropp, R. Probst, Z. Cummins, R. Kumar, A. J. Berglund, S. R. Raghavan, E. Waks, and B. Shapiro, Nano Lett. 10, 2525 (2010).

Song, H. S.

J. S. Jin, C. S. Yu, and H. S. Song, Phys. Rev. A 81, 042309 (2010).

Sørensen, A. S.

M. J. Kastoryano, F. Reiter, and A. S. Sørensen, Phys. Rev. Lett. 106, 090502 (2011).
[CrossRef]

D. E. Chang, A. S. Sørensen, E. A. Demler, and M. D. Lukin, Nat. Phys. 3, 807 (2007).
[CrossRef]

Srinivasan, K.

K. Srinivasan and O. Painter, Nature 450, 862 (2007).
[CrossRef]

Sun, C. P.

L. Zhou, Z. R. Gong, Y. X. Liu, C. P. Sun, and F. Nori, Phys. Rev. Lett. 101, 100501 (2008).

Tejedor, C.

A. Gonzalez-Tudela, D. Martin-Cano, E. Moreno, L. Martin-Moreno, C. Tejedor, and F. J. Garcia-Vidal, Phys. Rev. Lett. 106, 020501 (2011).
[CrossRef]

Tu, T.

Z. R. Lin, G. P. Guo, T. Tu, H. O. Li, C. L. Zou, J. X. Chen, Y. H. Lu, X. F. Ren, and G. C. Guo, Phys. Rev. B 82, 241401 (2010).

Waks, E.

C. Ropp, R. Probst, Z. Cummins, R. Kumar, A. J. Berglund, S. R. Raghavan, E. Waks, and B. Shapiro, Nano Lett. 10, 2525 (2010).

Wootters, W. K.

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

Yampol’skii, V. A.

S. Savel’ev, V. A. Yampol’skii, A. L. Rakhmanov, and F. Nori, Rep. Prog. Phys. 73, 026501 (2010).
[CrossRef]

Yang, J. K.

J. K. Yang, Jpn. J. Appl. Phys. 50, 060205 (2011).
[CrossRef]

Yang, Z.

S. D. Liu, Z. Yang, R. P. Liu, and X. Y. Li, J. Phys. Chem. C 115, 24469 (2011).

Yu, C. S.

J. S. Jin, C. S. Yu, and H. S. Song, Phys. Rev. A 81, 042309 (2010).

Zheng, S. B.

S. B. Zheng and G. C. Guo, Phys. Rev. Lett. 85, 2392 (2000).
[CrossRef]

Zhou, L.

L. Zhou, Z. R. Gong, Y. X. Liu, C. P. Sun, and F. Nori, Phys. Rev. Lett. 101, 100501 (2008).

Zia, R.

R. Zia and M. L. Brongersma, Nature Nanotechnol. 2, 426 (2007).
[CrossRef]

Zou, C. L.

Z. R. Lin, G. P. Guo, T. Tu, H. O. Li, C. L. Zou, J. X. Chen, Y. H. Lu, X. F. Ren, and G. C. Guo, Phys. Rev. B 82, 241401 (2010).

J. Opt.

L. O. Diniz, E. Marega, F. D. Nunes, and B.-H. V. Borges, J. Opt. 13, 115001 (2011).
[CrossRef]

J. Phys. Chem. C

S. D. Liu, Z. Yang, R. P. Liu, and X. Y. Li, J. Phys. Chem. C 115, 24469 (2011).

Jpn. J. Appl. Phys.

J. K. Yang, Jpn. J. Appl. Phys. 50, 060205 (2011).
[CrossRef]

Nano Lett.

C. Ropp, R. Probst, Z. Cummins, R. Kumar, A. J. Berglund, S. R. Raghavan, E. Waks, and B. Shapiro, Nano Lett. 10, 2525 (2010).

Nat. Phys.

D. E. Chang, A. S. Sørensen, E. A. Demler, and M. D. Lukin, Nat. Phys. 3, 807 (2007).
[CrossRef]

Nature

K. Srinivasan and O. Painter, Nature 450, 862 (2007).
[CrossRef]

Nature Nanotechnol.

R. Zia and M. L. Brongersma, Nature Nanotechnol. 2, 426 (2007).
[CrossRef]

Phys. Rev. A

Y. N. Chen, G. Y. Chen, D. S. Chuu, and T. Brandes, Phys. Rev. A 79, 033815 (2009).

M. S. Kim, Jinhyoung Lee, D. Ahn, and P. L. Knight, Phys. Rev. A 65, 040101 (2002).

J. S. Jin, C. S. Yu, and H. S. Song, Phys. Rev. A 81, 042309 (2010).

Phys. Rev. B

Z. R. Lin, G. P. Guo, T. Tu, H. O. Li, C. L. Zou, J. X. Chen, Y. H. Lu, X. F. Ren, and G. C. Guo, Phys. Rev. B 82, 241401 (2010).

G. Y. Chen, N. Lambert, C. H. Chou, Y. N. Chen, and F. Nori, Phys. Rev. B 84, 045310 (2011).

Phys. Rev. Lett.

A. Gonzalez-Tudela, D. Martin-Cano, E. Moreno, L. Martin-Moreno, C. Tejedor, and F. J. Garcia-Vidal, Phys. Rev. Lett. 106, 020501 (2011).
[CrossRef]

L. Zhou, Z. R. Gong, Y. X. Liu, C. P. Sun, and F. Nori, Phys. Rev. Lett. 101, 100501 (2008).

M. J. Kastoryano, F. Reiter, and A. S. Sørensen, Phys. Rev. Lett. 106, 090502 (2011).
[CrossRef]

S. B. Zheng and G. C. Guo, Phys. Rev. Lett. 85, 2392 (2000).
[CrossRef]

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

Rep. Prog. Phys.

S. Savel’ev, V. A. Yampol’skii, A. L. Rakhmanov, and F. Nori, Rep. Prog. Phys. 73, 026501 (2010).
[CrossRef]

Rev. Mod. Phys.

K. Yu. Bliokh, Yu. P. Bliokh, V. Freilikher, S. Savel’ev, and F. Nori, Rev. Mod. Phys. 80, 1201 (2008).
[CrossRef]

Other

Here we assume that the interdot distance is not long, such that the two quantum dots can simultaneously couple to the k-mode field.

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Figures (3)

Fig. 1.
Fig. 1.

Schematic diagram of a metal nanoring coupled to two quantum dots. The two dots are coherently interconnected via the metal nanoring surface plasmons. The incident clockwise propagating surface plasmon from an evanescently coupled waveguide photon would be scattered by the two dots, leading to the counterclockwise propagating surface plasmon.

Fig. 2.
Fig. 2.

Concurrence dynamics of the two QDs as functions of time for interdot distance kd= (a) (2n+1)π/2, (b) nπ, (c) (4n+1)π/4, (d) (3n+1)π/3. Here we have set the coupling strength g1=g2=g. The field-damping rate and the polarization decay rate of each qubit are chosen to be Γk=Γk=γ1=γ2=1/30g.

Fig. 3.
Fig. 3.

Concurrence dynamics of the two QDs as functions of time for the ratio rg1/g2= (a) 1/2, (b) 1/3, (c) 1/5, (d) 3/5. The continuous black (dashed red) curves represent the results for kd=(2n+1)π/2(nπ), while the dotted blue curves represent the results for kd=(2n+1)π/2 but only the k-mode SP is included. (e) Density plot of the concurrence dynamics of the two QDs as functions of the ratio r and time. The white regions correspond to high entanglement with concurrence around 1. In the white region marked by the arrow, high entanglement is easy to be generated experimentally.

Equations (5)

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

H=j=1,2ω0jσej,ej+k~=k,kωk~ak~ak~+k~=k,k[(g1σ1+ak~+g2σ2+eik~dak~)+H.c.],
{|e1,g2|0,|g1,e2|0,|g1,g2|1k~=k,k,|g1,g2|0},
ρ.=1i[H,ρ]+k~=k,kΓk~(ak~ρak~12ak~ak~ρ12ρak~ak~)+j=1,2γj(σjρσj+12σj+σjρ12ρσj+σj),
sin(2rτ)sin(2τ)=±1,
τ=(2α+1)π22andr=2β+12α+1(α,β=0,1,2,3,).

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