Abstract

We theoretically demonstrate an anisotropic quantum vacuum created by a judiciously designed hyperbolic metamaterial. An electric dipole located nearby shows strong orientation dependence in the decay rate. With a proper arrangement of the ellipsoid-shaped CdSe/ZnSe quantum dot relative to the Ag/TiO2 metamaterial, the anisotropies of quantum vacuum and quantum dot are harnessed to achieve an extraordinary quantum interference between radiative decay channels of orthogonal transitions. The ratio between cross damping term and spontaneous decay rate, Γijii, which never exceeds unity in previously reported works reaches 1.04 in our numerical results. The corresponding evolution of excited state population in quantum dot is also dramatically modified.

© 2016 Optical Society of America

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References

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  1. S. Menon and G. S. Agarwal, “Gain components in the Autler-Townes doublet from quantum interferences in decay channels,” Phys. Rev. A 61(1), 013807 (1999).
    [Crossref]
  2. M. Macovei and C. H. Keitel, “Laser control of collective spontaneous emission,” Phys. Rev. Lett. 91(12), 123601 (2003).
    [Crossref] [PubMed]
  3. J. H. Wu, H. F. Zhang, and J. Y. Gao, “Probe gain with population inversion in a four-level atomic system with vacuum-induced coherence,” Opt. Lett. 28(8), 654–656 (2003).
    [Crossref] [PubMed]
  4. G. S. Agarwal, “Anisotropic vacuum-induced interference in decay channels,” Phys. Rev. Lett. 84(24), 5500–5503 (2000).
    [Crossref] [PubMed]
  5. V. Yannopapas, E. Paspalakis, and N. V. Vitanov, “Plasmon-induced enhancement of quantum interference near metallic nanostructures,” Phys. Rev. Lett. 103(6), 063602 (2009).
    [Crossref] [PubMed]
  6. Y. Gu, L. Wang, P. Ren, J. Zhang, T. Zhang, O. J. Martin, and Q. Gong, “Surface-plasmon-induced modification on the spontaneous emission spectrum via subwavelength-confined anisotropic Purcell factor,” Nano Lett. 12(5), 2488–2493 (2012).
    [Crossref] [PubMed]
  7. G.-X. Li, F.-L. Li, and S.-Y. Zhu, “Quantum interference between decay channels of a three-level atom in a multilayer dielectric medium,” Phys. Rev. A 64(1), 013819 (2001).
    [Crossref]
  8. J.-P. Xu, L.-G. Wang, Y.-P. Yang, Q. Lin, and S.-Y. Zhu, “Quantum interference between two orthogonal transitions of an atom in one-dimensional photonic crystals,” Opt. Lett. 33(17), 2005–2007 (2008).
    [Crossref] [PubMed]
  9. A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B 74(7), 075103 (2006).
    [Crossref]
  10. S. Thongrattanasiri and V. A. Podolskiy, “Hypergratings: nanophotonics in planar anisotropic metamaterials,” Opt. Lett. 34(7), 890–892 (2009).
    [Crossref] [PubMed]
  11. A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
    [Crossref] [PubMed]
  12. M. Y. Shalaginov, S. Ishii, J. Liu, J. Irudayaraj, A. Lagutchev, A. V. Kildishev, and V. M. Shalaev, “Broadband enhancement of spontaneous emission from nitrogen-vacancy centers in nanodiamonds by hyperbolic metamaterials,” Appl. Phys. Lett. 102(17), 173114 (2013).
    [Crossref]
  13. M. Bayer, A. Kuther, A. Forchel, A. Gorbunov, V. B. Timofeev, F. Schäfer, J. P. Reithmaier, T. L. Reinecke, and S. N. Walck, “Electron and hole g factors and exchange interaction from studies of the exciton fine structure in In0.60Ga0.40As quantum dots,” Phys. Rev. Lett. 82(8), 1748–1751 (1999).
    [Crossref]
  14. I. A. Akimov, J. T. Andrews, and F. Henneberger, “Stimulated emission from the biexciton in a single self-assembled II-VI quantum dot,” Phys. Rev. Lett. 96(6), 067401 (2006).
    [Crossref] [PubMed]
  15. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1997).
  16. C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14(6), 063001 (2012).
    [Crossref]
  17. Z. Jacob, I. I. Smolyaninov, and E. E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012).
    [Crossref]
  18. D. P. Craig and T. Thirunamachandran, Molecular Quantum Electrodynamics (Dover Publications, 1998).
  19. A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
    [Crossref]
  20. M. J. Weber, Handbook of Optical Materials (CRC, 2002).
  21. H. N. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
    [Crossref] [PubMed]
  22. R. X. Bian, R. C. Dunn, X. S. Xie, and P. T. Leung, “Single molecule emission characteristics in near-field microscopy,” Phys. Rev. Lett. 75(26), 4772–4775 (1995).
    [Crossref] [PubMed]
  23. L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, New York, 2012).
  24. L. Sun, B. Tang, and C. Jiang, “Enhanced spontaneous emission of mid-infrared dipole emitter in double-layer graphene waveguide,” Opt. Express 22(22), 26487–26497 (2014).
    [Crossref] [PubMed]
  25. P. K. Jha, X. Ni, C. Wu, Y. Wang, and X. Zhang, “Metasurface-enabled remote quantum interference,” Phys. Rev. Lett. 115(2), 025501 (2015).
    [Crossref] [PubMed]
  26. V. D. Kulakovskii, G. Bacher, R. Weigand, T. Kümmell, A. Forchel, E. Borovitskaya, K. Leonardi, and D. Hommel, “Fine structure of biexciton emission in symmetric and asymmetric CdSe/ZnSe single quantum dots,” Phys. Rev. Lett. 82(8), 1780–1783 (1999).
    [Crossref]
  27. Y. Yang, J. Xu, H. Chen, and S. Zhu, “Quantum interference enhancement with left-handed materials,” Phys. Rev. Lett. 100(4), 043601 (2008).
    [Crossref] [PubMed]
  28. G. S. Agarwal, Quantum Optics, Springer Tracts in Modern Physics Vol. 70 (Springer, 1974).
  29. H. T. Dung, S. Y. Buhmann, L. Knöll, D. G. Welsch, S. Scheel, and J. Kästel, “Electromagnetic-field quantization and spontaneous decay in left-handed media,” Phys. Rev. A 68(4), 043816 (2003).
    [Crossref]

2015 (1)

P. K. Jha, X. Ni, C. Wu, Y. Wang, and X. Zhang, “Metasurface-enabled remote quantum interference,” Phys. Rev. Lett. 115(2), 025501 (2015).
[Crossref] [PubMed]

2014 (1)

2013 (2)

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
[Crossref]

M. Y. Shalaginov, S. Ishii, J. Liu, J. Irudayaraj, A. Lagutchev, A. V. Kildishev, and V. M. Shalaev, “Broadband enhancement of spontaneous emission from nitrogen-vacancy centers in nanodiamonds by hyperbolic metamaterials,” Appl. Phys. Lett. 102(17), 173114 (2013).
[Crossref]

2012 (4)

Y. Gu, L. Wang, P. Ren, J. Zhang, T. Zhang, O. J. Martin, and Q. Gong, “Surface-plasmon-induced modification on the spontaneous emission spectrum via subwavelength-confined anisotropic Purcell factor,” Nano Lett. 12(5), 2488–2493 (2012).
[Crossref] [PubMed]

H. N. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[Crossref] [PubMed]

C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14(6), 063001 (2012).
[Crossref]

Z. Jacob, I. I. Smolyaninov, and E. E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012).
[Crossref]

2009 (2)

V. Yannopapas, E. Paspalakis, and N. V. Vitanov, “Plasmon-induced enhancement of quantum interference near metallic nanostructures,” Phys. Rev. Lett. 103(6), 063602 (2009).
[Crossref] [PubMed]

S. Thongrattanasiri and V. A. Podolskiy, “Hypergratings: nanophotonics in planar anisotropic metamaterials,” Opt. Lett. 34(7), 890–892 (2009).
[Crossref] [PubMed]

2008 (2)

2007 (1)

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

2006 (2)

I. A. Akimov, J. T. Andrews, and F. Henneberger, “Stimulated emission from the biexciton in a single self-assembled II-VI quantum dot,” Phys. Rev. Lett. 96(6), 067401 (2006).
[Crossref] [PubMed]

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B 74(7), 075103 (2006).
[Crossref]

2003 (3)

M. Macovei and C. H. Keitel, “Laser control of collective spontaneous emission,” Phys. Rev. Lett. 91(12), 123601 (2003).
[Crossref] [PubMed]

H. T. Dung, S. Y. Buhmann, L. Knöll, D. G. Welsch, S. Scheel, and J. Kästel, “Electromagnetic-field quantization and spontaneous decay in left-handed media,” Phys. Rev. A 68(4), 043816 (2003).
[Crossref]

J. H. Wu, H. F. Zhang, and J. Y. Gao, “Probe gain with population inversion in a four-level atomic system with vacuum-induced coherence,” Opt. Lett. 28(8), 654–656 (2003).
[Crossref] [PubMed]

2001 (1)

G.-X. Li, F.-L. Li, and S.-Y. Zhu, “Quantum interference between decay channels of a three-level atom in a multilayer dielectric medium,” Phys. Rev. A 64(1), 013819 (2001).
[Crossref]

2000 (1)

G. S. Agarwal, “Anisotropic vacuum-induced interference in decay channels,” Phys. Rev. Lett. 84(24), 5500–5503 (2000).
[Crossref] [PubMed]

1999 (3)

S. Menon and G. S. Agarwal, “Gain components in the Autler-Townes doublet from quantum interferences in decay channels,” Phys. Rev. A 61(1), 013807 (1999).
[Crossref]

M. Bayer, A. Kuther, A. Forchel, A. Gorbunov, V. B. Timofeev, F. Schäfer, J. P. Reithmaier, T. L. Reinecke, and S. N. Walck, “Electron and hole g factors and exchange interaction from studies of the exciton fine structure in In0.60Ga0.40As quantum dots,” Phys. Rev. Lett. 82(8), 1748–1751 (1999).
[Crossref]

V. D. Kulakovskii, G. Bacher, R. Weigand, T. Kümmell, A. Forchel, E. Borovitskaya, K. Leonardi, and D. Hommel, “Fine structure of biexciton emission in symmetric and asymmetric CdSe/ZnSe single quantum dots,” Phys. Rev. Lett. 82(8), 1780–1783 (1999).
[Crossref]

1995 (1)

R. X. Bian, R. C. Dunn, X. S. Xie, and P. T. Leung, “Single molecule emission characteristics in near-field microscopy,” Phys. Rev. Lett. 75(26), 4772–4775 (1995).
[Crossref] [PubMed]

Agarwal, G. S.

G. S. Agarwal, “Anisotropic vacuum-induced interference in decay channels,” Phys. Rev. Lett. 84(24), 5500–5503 (2000).
[Crossref] [PubMed]

S. Menon and G. S. Agarwal, “Gain components in the Autler-Townes doublet from quantum interferences in decay channels,” Phys. Rev. A 61(1), 013807 (1999).
[Crossref]

Akimov, I. A.

I. A. Akimov, J. T. Andrews, and F. Henneberger, “Stimulated emission from the biexciton in a single self-assembled II-VI quantum dot,” Phys. Rev. Lett. 96(6), 067401 (2006).
[Crossref] [PubMed]

Alekseyev, L.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Andrews, J. T.

I. A. Akimov, J. T. Andrews, and F. Henneberger, “Stimulated emission from the biexciton in a single self-assembled II-VI quantum dot,” Phys. Rev. Lett. 96(6), 067401 (2006).
[Crossref] [PubMed]

Bacher, G.

V. D. Kulakovskii, G. Bacher, R. Weigand, T. Kümmell, A. Forchel, E. Borovitskaya, K. Leonardi, and D. Hommel, “Fine structure of biexciton emission in symmetric and asymmetric CdSe/ZnSe single quantum dots,” Phys. Rev. Lett. 82(8), 1780–1783 (1999).
[Crossref]

Bayer, M.

M. Bayer, A. Kuther, A. Forchel, A. Gorbunov, V. B. Timofeev, F. Schäfer, J. P. Reithmaier, T. L. Reinecke, and S. N. Walck, “Electron and hole g factors and exchange interaction from studies of the exciton fine structure in In0.60Ga0.40As quantum dots,” Phys. Rev. Lett. 82(8), 1748–1751 (1999).
[Crossref]

Belov, P.

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
[Crossref]

Bian, R. X.

R. X. Bian, R. C. Dunn, X. S. Xie, and P. T. Leung, “Single molecule emission characteristics in near-field microscopy,” Phys. Rev. Lett. 75(26), 4772–4775 (1995).
[Crossref] [PubMed]

Borovitskaya, E.

V. D. Kulakovskii, G. Bacher, R. Weigand, T. Kümmell, A. Forchel, E. Borovitskaya, K. Leonardi, and D. Hommel, “Fine structure of biexciton emission in symmetric and asymmetric CdSe/ZnSe single quantum dots,” Phys. Rev. Lett. 82(8), 1780–1783 (1999).
[Crossref]

Buhmann, S. Y.

H. T. Dung, S. Y. Buhmann, L. Knöll, D. G. Welsch, S. Scheel, and J. Kästel, “Electromagnetic-field quantization and spontaneous decay in left-handed media,” Phys. Rev. A 68(4), 043816 (2003).
[Crossref]

Chen, H.

Y. Yang, J. Xu, H. Chen, and S. Zhu, “Quantum interference enhancement with left-handed materials,” Phys. Rev. Lett. 100(4), 043601 (2008).
[Crossref] [PubMed]

Cortes, C. L.

C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14(6), 063001 (2012).
[Crossref]

Dung, H. T.

H. T. Dung, S. Y. Buhmann, L. Knöll, D. G. Welsch, S. Scheel, and J. Kästel, “Electromagnetic-field quantization and spontaneous decay in left-handed media,” Phys. Rev. A 68(4), 043816 (2003).
[Crossref]

Dunn, R. C.

R. X. Bian, R. C. Dunn, X. S. Xie, and P. T. Leung, “Single molecule emission characteristics in near-field microscopy,” Phys. Rev. Lett. 75(26), 4772–4775 (1995).
[Crossref] [PubMed]

Engheta, N.

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B 74(7), 075103 (2006).
[Crossref]

Forchel, A.

M. Bayer, A. Kuther, A. Forchel, A. Gorbunov, V. B. Timofeev, F. Schäfer, J. P. Reithmaier, T. L. Reinecke, and S. N. Walck, “Electron and hole g factors and exchange interaction from studies of the exciton fine structure in In0.60Ga0.40As quantum dots,” Phys. Rev. Lett. 82(8), 1748–1751 (1999).
[Crossref]

V. D. Kulakovskii, G. Bacher, R. Weigand, T. Kümmell, A. Forchel, E. Borovitskaya, K. Leonardi, and D. Hommel, “Fine structure of biexciton emission in symmetric and asymmetric CdSe/ZnSe single quantum dots,” Phys. Rev. Lett. 82(8), 1780–1783 (1999).
[Crossref]

Franz, K. J.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Gao, J. Y.

Gmachl, C.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Gong, Q.

Y. Gu, L. Wang, P. Ren, J. Zhang, T. Zhang, O. J. Martin, and Q. Gong, “Surface-plasmon-induced modification on the spontaneous emission spectrum via subwavelength-confined anisotropic Purcell factor,” Nano Lett. 12(5), 2488–2493 (2012).
[Crossref] [PubMed]

Gorbunov, A.

M. Bayer, A. Kuther, A. Forchel, A. Gorbunov, V. B. Timofeev, F. Schäfer, J. P. Reithmaier, T. L. Reinecke, and S. N. Walck, “Electron and hole g factors and exchange interaction from studies of the exciton fine structure in In0.60Ga0.40As quantum dots,” Phys. Rev. Lett. 82(8), 1748–1751 (1999).
[Crossref]

Gu, Y.

Y. Gu, L. Wang, P. Ren, J. Zhang, T. Zhang, O. J. Martin, and Q. Gong, “Surface-plasmon-induced modification on the spontaneous emission spectrum via subwavelength-confined anisotropic Purcell factor,” Nano Lett. 12(5), 2488–2493 (2012).
[Crossref] [PubMed]

Henneberger, F.

I. A. Akimov, J. T. Andrews, and F. Henneberger, “Stimulated emission from the biexciton in a single self-assembled II-VI quantum dot,” Phys. Rev. Lett. 96(6), 067401 (2006).
[Crossref] [PubMed]

Hoffman, A. J.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Hommel, D.

V. D. Kulakovskii, G. Bacher, R. Weigand, T. Kümmell, A. Forchel, E. Borovitskaya, K. Leonardi, and D. Hommel, “Fine structure of biexciton emission in symmetric and asymmetric CdSe/ZnSe single quantum dots,” Phys. Rev. Lett. 82(8), 1780–1783 (1999).
[Crossref]

Howard, S. S.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Iorsh, I.

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
[Crossref]

Irudayaraj, J.

M. Y. Shalaginov, S. Ishii, J. Liu, J. Irudayaraj, A. Lagutchev, A. V. Kildishev, and V. M. Shalaev, “Broadband enhancement of spontaneous emission from nitrogen-vacancy centers in nanodiamonds by hyperbolic metamaterials,” Appl. Phys. Lett. 102(17), 173114 (2013).
[Crossref]

Ishii, S.

M. Y. Shalaginov, S. Ishii, J. Liu, J. Irudayaraj, A. Lagutchev, A. V. Kildishev, and V. M. Shalaev, “Broadband enhancement of spontaneous emission from nitrogen-vacancy centers in nanodiamonds by hyperbolic metamaterials,” Appl. Phys. Lett. 102(17), 173114 (2013).
[Crossref]

Jacob, Z.

C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14(6), 063001 (2012).
[Crossref]

Z. Jacob, I. I. Smolyaninov, and E. E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012).
[Crossref]

H. N. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[Crossref] [PubMed]

Jha, P. K.

P. K. Jha, X. Ni, C. Wu, Y. Wang, and X. Zhang, “Metasurface-enabled remote quantum interference,” Phys. Rev. Lett. 115(2), 025501 (2015).
[Crossref] [PubMed]

Jiang, C.

Kästel, J.

H. T. Dung, S. Y. Buhmann, L. Knöll, D. G. Welsch, S. Scheel, and J. Kästel, “Electromagnetic-field quantization and spontaneous decay in left-handed media,” Phys. Rev. A 68(4), 043816 (2003).
[Crossref]

Keitel, C. H.

M. Macovei and C. H. Keitel, “Laser control of collective spontaneous emission,” Phys. Rev. Lett. 91(12), 123601 (2003).
[Crossref] [PubMed]

Kildishev, A. V.

M. Y. Shalaginov, S. Ishii, J. Liu, J. Irudayaraj, A. Lagutchev, A. V. Kildishev, and V. M. Shalaev, “Broadband enhancement of spontaneous emission from nitrogen-vacancy centers in nanodiamonds by hyperbolic metamaterials,” Appl. Phys. Lett. 102(17), 173114 (2013).
[Crossref]

Kivshar, Y.

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
[Crossref]

Knöll, L.

H. T. Dung, S. Y. Buhmann, L. Knöll, D. G. Welsch, S. Scheel, and J. Kästel, “Electromagnetic-field quantization and spontaneous decay in left-handed media,” Phys. Rev. A 68(4), 043816 (2003).
[Crossref]

Kretzschmar, I.

H. N. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[Crossref] [PubMed]

Krishnamoorthy, H. N.

H. N. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[Crossref] [PubMed]

Kulakovskii, V. D.

V. D. Kulakovskii, G. Bacher, R. Weigand, T. Kümmell, A. Forchel, E. Borovitskaya, K. Leonardi, and D. Hommel, “Fine structure of biexciton emission in symmetric and asymmetric CdSe/ZnSe single quantum dots,” Phys. Rev. Lett. 82(8), 1780–1783 (1999).
[Crossref]

Kümmell, T.

V. D. Kulakovskii, G. Bacher, R. Weigand, T. Kümmell, A. Forchel, E. Borovitskaya, K. Leonardi, and D. Hommel, “Fine structure of biexciton emission in symmetric and asymmetric CdSe/ZnSe single quantum dots,” Phys. Rev. Lett. 82(8), 1780–1783 (1999).
[Crossref]

Kuther, A.

M. Bayer, A. Kuther, A. Forchel, A. Gorbunov, V. B. Timofeev, F. Schäfer, J. P. Reithmaier, T. L. Reinecke, and S. N. Walck, “Electron and hole g factors and exchange interaction from studies of the exciton fine structure in In0.60Ga0.40As quantum dots,” Phys. Rev. Lett. 82(8), 1748–1751 (1999).
[Crossref]

Lagutchev, A.

M. Y. Shalaginov, S. Ishii, J. Liu, J. Irudayaraj, A. Lagutchev, A. V. Kildishev, and V. M. Shalaev, “Broadband enhancement of spontaneous emission from nitrogen-vacancy centers in nanodiamonds by hyperbolic metamaterials,” Appl. Phys. Lett. 102(17), 173114 (2013).
[Crossref]

Leonardi, K.

V. D. Kulakovskii, G. Bacher, R. Weigand, T. Kümmell, A. Forchel, E. Borovitskaya, K. Leonardi, and D. Hommel, “Fine structure of biexciton emission in symmetric and asymmetric CdSe/ZnSe single quantum dots,” Phys. Rev. Lett. 82(8), 1780–1783 (1999).
[Crossref]

Leung, P. T.

R. X. Bian, R. C. Dunn, X. S. Xie, and P. T. Leung, “Single molecule emission characteristics in near-field microscopy,” Phys. Rev. Lett. 75(26), 4772–4775 (1995).
[Crossref] [PubMed]

Li, F.-L.

G.-X. Li, F.-L. Li, and S.-Y. Zhu, “Quantum interference between decay channels of a three-level atom in a multilayer dielectric medium,” Phys. Rev. A 64(1), 013819 (2001).
[Crossref]

Li, G.-X.

G.-X. Li, F.-L. Li, and S.-Y. Zhu, “Quantum interference between decay channels of a three-level atom in a multilayer dielectric medium,” Phys. Rev. A 64(1), 013819 (2001).
[Crossref]

Lin, Q.

Liu, J.

M. Y. Shalaginov, S. Ishii, J. Liu, J. Irudayaraj, A. Lagutchev, A. V. Kildishev, and V. M. Shalaev, “Broadband enhancement of spontaneous emission from nitrogen-vacancy centers in nanodiamonds by hyperbolic metamaterials,” Appl. Phys. Lett. 102(17), 173114 (2013).
[Crossref]

Macovei, M.

M. Macovei and C. H. Keitel, “Laser control of collective spontaneous emission,” Phys. Rev. Lett. 91(12), 123601 (2003).
[Crossref] [PubMed]

Martin, O. J.

Y. Gu, L. Wang, P. Ren, J. Zhang, T. Zhang, O. J. Martin, and Q. Gong, “Surface-plasmon-induced modification on the spontaneous emission spectrum via subwavelength-confined anisotropic Purcell factor,” Nano Lett. 12(5), 2488–2493 (2012).
[Crossref] [PubMed]

Menon, S.

S. Menon and G. S. Agarwal, “Gain components in the Autler-Townes doublet from quantum interferences in decay channels,” Phys. Rev. A 61(1), 013807 (1999).
[Crossref]

Menon, V. M.

H. N. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[Crossref] [PubMed]

Molesky, S.

C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14(6), 063001 (2012).
[Crossref]

Narimanov, E.

H. N. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[Crossref] [PubMed]

Narimanov, E. E.

Z. Jacob, I. I. Smolyaninov, and E. E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012).
[Crossref]

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Newman, W.

C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14(6), 063001 (2012).
[Crossref]

Ni, X.

P. K. Jha, X. Ni, C. Wu, Y. Wang, and X. Zhang, “Metasurface-enabled remote quantum interference,” Phys. Rev. Lett. 115(2), 025501 (2015).
[Crossref] [PubMed]

Paspalakis, E.

V. Yannopapas, E. Paspalakis, and N. V. Vitanov, “Plasmon-induced enhancement of quantum interference near metallic nanostructures,” Phys. Rev. Lett. 103(6), 063602 (2009).
[Crossref] [PubMed]

Poddubny, A.

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
[Crossref]

Podolskiy, V. A.

S. Thongrattanasiri and V. A. Podolskiy, “Hypergratings: nanophotonics in planar anisotropic metamaterials,” Opt. Lett. 34(7), 890–892 (2009).
[Crossref] [PubMed]

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Reinecke, T. L.

M. Bayer, A. Kuther, A. Forchel, A. Gorbunov, V. B. Timofeev, F. Schäfer, J. P. Reithmaier, T. L. Reinecke, and S. N. Walck, “Electron and hole g factors and exchange interaction from studies of the exciton fine structure in In0.60Ga0.40As quantum dots,” Phys. Rev. Lett. 82(8), 1748–1751 (1999).
[Crossref]

Reithmaier, J. P.

M. Bayer, A. Kuther, A. Forchel, A. Gorbunov, V. B. Timofeev, F. Schäfer, J. P. Reithmaier, T. L. Reinecke, and S. N. Walck, “Electron and hole g factors and exchange interaction from studies of the exciton fine structure in In0.60Ga0.40As quantum dots,” Phys. Rev. Lett. 82(8), 1748–1751 (1999).
[Crossref]

Ren, P.

Y. Gu, L. Wang, P. Ren, J. Zhang, T. Zhang, O. J. Martin, and Q. Gong, “Surface-plasmon-induced modification on the spontaneous emission spectrum via subwavelength-confined anisotropic Purcell factor,” Nano Lett. 12(5), 2488–2493 (2012).
[Crossref] [PubMed]

Salandrino, A.

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B 74(7), 075103 (2006).
[Crossref]

Schäfer, F.

M. Bayer, A. Kuther, A. Forchel, A. Gorbunov, V. B. Timofeev, F. Schäfer, J. P. Reithmaier, T. L. Reinecke, and S. N. Walck, “Electron and hole g factors and exchange interaction from studies of the exciton fine structure in In0.60Ga0.40As quantum dots,” Phys. Rev. Lett. 82(8), 1748–1751 (1999).
[Crossref]

Scheel, S.

H. T. Dung, S. Y. Buhmann, L. Knöll, D. G. Welsch, S. Scheel, and J. Kästel, “Electromagnetic-field quantization and spontaneous decay in left-handed media,” Phys. Rev. A 68(4), 043816 (2003).
[Crossref]

Shalaev, V. M.

M. Y. Shalaginov, S. Ishii, J. Liu, J. Irudayaraj, A. Lagutchev, A. V. Kildishev, and V. M. Shalaev, “Broadband enhancement of spontaneous emission from nitrogen-vacancy centers in nanodiamonds by hyperbolic metamaterials,” Appl. Phys. Lett. 102(17), 173114 (2013).
[Crossref]

Shalaginov, M. Y.

M. Y. Shalaginov, S. Ishii, J. Liu, J. Irudayaraj, A. Lagutchev, A. V. Kildishev, and V. M. Shalaev, “Broadband enhancement of spontaneous emission from nitrogen-vacancy centers in nanodiamonds by hyperbolic metamaterials,” Appl. Phys. Lett. 102(17), 173114 (2013).
[Crossref]

Sivco, D. L.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Smolyaninov, I. I.

Z. Jacob, I. I. Smolyaninov, and E. E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012).
[Crossref]

Sun, L.

Tang, B.

Thongrattanasiri, S.

Timofeev, V. B.

M. Bayer, A. Kuther, A. Forchel, A. Gorbunov, V. B. Timofeev, F. Schäfer, J. P. Reithmaier, T. L. Reinecke, and S. N. Walck, “Electron and hole g factors and exchange interaction from studies of the exciton fine structure in In0.60Ga0.40As quantum dots,” Phys. Rev. Lett. 82(8), 1748–1751 (1999).
[Crossref]

Vitanov, N. V.

V. Yannopapas, E. Paspalakis, and N. V. Vitanov, “Plasmon-induced enhancement of quantum interference near metallic nanostructures,” Phys. Rev. Lett. 103(6), 063602 (2009).
[Crossref] [PubMed]

Walck, S. N.

M. Bayer, A. Kuther, A. Forchel, A. Gorbunov, V. B. Timofeev, F. Schäfer, J. P. Reithmaier, T. L. Reinecke, and S. N. Walck, “Electron and hole g factors and exchange interaction from studies of the exciton fine structure in In0.60Ga0.40As quantum dots,” Phys. Rev. Lett. 82(8), 1748–1751 (1999).
[Crossref]

Wang, L.

Y. Gu, L. Wang, P. Ren, J. Zhang, T. Zhang, O. J. Martin, and Q. Gong, “Surface-plasmon-induced modification on the spontaneous emission spectrum via subwavelength-confined anisotropic Purcell factor,” Nano Lett. 12(5), 2488–2493 (2012).
[Crossref] [PubMed]

Wang, L.-G.

Wang, Y.

P. K. Jha, X. Ni, C. Wu, Y. Wang, and X. Zhang, “Metasurface-enabled remote quantum interference,” Phys. Rev. Lett. 115(2), 025501 (2015).
[Crossref] [PubMed]

Wasserman, D.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Weigand, R.

V. D. Kulakovskii, G. Bacher, R. Weigand, T. Kümmell, A. Forchel, E. Borovitskaya, K. Leonardi, and D. Hommel, “Fine structure of biexciton emission in symmetric and asymmetric CdSe/ZnSe single quantum dots,” Phys. Rev. Lett. 82(8), 1780–1783 (1999).
[Crossref]

Welsch, D. G.

H. T. Dung, S. Y. Buhmann, L. Knöll, D. G. Welsch, S. Scheel, and J. Kästel, “Electromagnetic-field quantization and spontaneous decay in left-handed media,” Phys. Rev. A 68(4), 043816 (2003).
[Crossref]

Wu, C.

P. K. Jha, X. Ni, C. Wu, Y. Wang, and X. Zhang, “Metasurface-enabled remote quantum interference,” Phys. Rev. Lett. 115(2), 025501 (2015).
[Crossref] [PubMed]

Wu, J. H.

Xie, X. S.

R. X. Bian, R. C. Dunn, X. S. Xie, and P. T. Leung, “Single molecule emission characteristics in near-field microscopy,” Phys. Rev. Lett. 75(26), 4772–4775 (1995).
[Crossref] [PubMed]

Xu, J.

Y. Yang, J. Xu, H. Chen, and S. Zhu, “Quantum interference enhancement with left-handed materials,” Phys. Rev. Lett. 100(4), 043601 (2008).
[Crossref] [PubMed]

Xu, J.-P.

Yang, Y.

Y. Yang, J. Xu, H. Chen, and S. Zhu, “Quantum interference enhancement with left-handed materials,” Phys. Rev. Lett. 100(4), 043601 (2008).
[Crossref] [PubMed]

Yang, Y.-P.

Yannopapas, V.

V. Yannopapas, E. Paspalakis, and N. V. Vitanov, “Plasmon-induced enhancement of quantum interference near metallic nanostructures,” Phys. Rev. Lett. 103(6), 063602 (2009).
[Crossref] [PubMed]

Zhang, H. F.

Zhang, J.

Y. Gu, L. Wang, P. Ren, J. Zhang, T. Zhang, O. J. Martin, and Q. Gong, “Surface-plasmon-induced modification on the spontaneous emission spectrum via subwavelength-confined anisotropic Purcell factor,” Nano Lett. 12(5), 2488–2493 (2012).
[Crossref] [PubMed]

Zhang, T.

Y. Gu, L. Wang, P. Ren, J. Zhang, T. Zhang, O. J. Martin, and Q. Gong, “Surface-plasmon-induced modification on the spontaneous emission spectrum via subwavelength-confined anisotropic Purcell factor,” Nano Lett. 12(5), 2488–2493 (2012).
[Crossref] [PubMed]

Zhang, X.

P. K. Jha, X. Ni, C. Wu, Y. Wang, and X. Zhang, “Metasurface-enabled remote quantum interference,” Phys. Rev. Lett. 115(2), 025501 (2015).
[Crossref] [PubMed]

Zhu, S.

Y. Yang, J. Xu, H. Chen, and S. Zhu, “Quantum interference enhancement with left-handed materials,” Phys. Rev. Lett. 100(4), 043601 (2008).
[Crossref] [PubMed]

Zhu, S.-Y.

J.-P. Xu, L.-G. Wang, Y.-P. Yang, Q. Lin, and S.-Y. Zhu, “Quantum interference between two orthogonal transitions of an atom in one-dimensional photonic crystals,” Opt. Lett. 33(17), 2005–2007 (2008).
[Crossref] [PubMed]

G.-X. Li, F.-L. Li, and S.-Y. Zhu, “Quantum interference between decay channels of a three-level atom in a multilayer dielectric medium,” Phys. Rev. A 64(1), 013819 (2001).
[Crossref]

Appl. Phys. Lett. (2)

M. Y. Shalaginov, S. Ishii, J. Liu, J. Irudayaraj, A. Lagutchev, A. V. Kildishev, and V. M. Shalaev, “Broadband enhancement of spontaneous emission from nitrogen-vacancy centers in nanodiamonds by hyperbolic metamaterials,” Appl. Phys. Lett. 102(17), 173114 (2013).
[Crossref]

Z. Jacob, I. I. Smolyaninov, and E. E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012).
[Crossref]

J. Opt. (1)

C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14(6), 063001 (2012).
[Crossref]

Nano Lett. (1)

Y. Gu, L. Wang, P. Ren, J. Zhang, T. Zhang, O. J. Martin, and Q. Gong, “Surface-plasmon-induced modification on the spontaneous emission spectrum via subwavelength-confined anisotropic Purcell factor,” Nano Lett. 12(5), 2488–2493 (2012).
[Crossref] [PubMed]

Nat. Mater. (1)

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Nat. Photonics (1)

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
[Crossref]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. A (3)

H. T. Dung, S. Y. Buhmann, L. Knöll, D. G. Welsch, S. Scheel, and J. Kästel, “Electromagnetic-field quantization and spontaneous decay in left-handed media,” Phys. Rev. A 68(4), 043816 (2003).
[Crossref]

G.-X. Li, F.-L. Li, and S.-Y. Zhu, “Quantum interference between decay channels of a three-level atom in a multilayer dielectric medium,” Phys. Rev. A 64(1), 013819 (2001).
[Crossref]

S. Menon and G. S. Agarwal, “Gain components in the Autler-Townes doublet from quantum interferences in decay channels,” Phys. Rev. A 61(1), 013807 (1999).
[Crossref]

Phys. Rev. B (1)

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B 74(7), 075103 (2006).
[Crossref]

Phys. Rev. Lett. (9)

G. S. Agarwal, “Anisotropic vacuum-induced interference in decay channels,” Phys. Rev. Lett. 84(24), 5500–5503 (2000).
[Crossref] [PubMed]

V. Yannopapas, E. Paspalakis, and N. V. Vitanov, “Plasmon-induced enhancement of quantum interference near metallic nanostructures,” Phys. Rev. Lett. 103(6), 063602 (2009).
[Crossref] [PubMed]

M. Macovei and C. H. Keitel, “Laser control of collective spontaneous emission,” Phys. Rev. Lett. 91(12), 123601 (2003).
[Crossref] [PubMed]

M. Bayer, A. Kuther, A. Forchel, A. Gorbunov, V. B. Timofeev, F. Schäfer, J. P. Reithmaier, T. L. Reinecke, and S. N. Walck, “Electron and hole g factors and exchange interaction from studies of the exciton fine structure in In0.60Ga0.40As quantum dots,” Phys. Rev. Lett. 82(8), 1748–1751 (1999).
[Crossref]

I. A. Akimov, J. T. Andrews, and F. Henneberger, “Stimulated emission from the biexciton in a single self-assembled II-VI quantum dot,” Phys. Rev. Lett. 96(6), 067401 (2006).
[Crossref] [PubMed]

R. X. Bian, R. C. Dunn, X. S. Xie, and P. T. Leung, “Single molecule emission characteristics in near-field microscopy,” Phys. Rev. Lett. 75(26), 4772–4775 (1995).
[Crossref] [PubMed]

P. K. Jha, X. Ni, C. Wu, Y. Wang, and X. Zhang, “Metasurface-enabled remote quantum interference,” Phys. Rev. Lett. 115(2), 025501 (2015).
[Crossref] [PubMed]

V. D. Kulakovskii, G. Bacher, R. Weigand, T. Kümmell, A. Forchel, E. Borovitskaya, K. Leonardi, and D. Hommel, “Fine structure of biexciton emission in symmetric and asymmetric CdSe/ZnSe single quantum dots,” Phys. Rev. Lett. 82(8), 1780–1783 (1999).
[Crossref]

Y. Yang, J. Xu, H. Chen, and S. Zhu, “Quantum interference enhancement with left-handed materials,” Phys. Rev. Lett. 100(4), 043601 (2008).
[Crossref] [PubMed]

Science (1)

H. N. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[Crossref] [PubMed]

Other (5)

L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, New York, 2012).

G. S. Agarwal, Quantum Optics, Springer Tracts in Modern Physics Vol. 70 (Springer, 1974).

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1997).

M. J. Weber, Handbook of Optical Materials (CRC, 2002).

D. P. Craig and T. Thirunamachandran, Molecular Quantum Electrodynamics (Dover Publications, 1998).

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Figures (4)

Fig. 1
Fig. 1

Schematic of (a) metamaterial structure and (b) anisotropic QD. (c) Three lowest energy levels and allowed optical transitions in QD.

Fig. 2
Fig. 2

(a) Electric dipole and effective medium model for the system. (b) Hyperbolic dispersion relation of p-polarized waves in HMM. (c) Anisotropic decay rate of a quantum emitter versus the distance d.

Fig. 3
Fig. 3

Cross section of the structure in the YOZ plane. The angle between y' and y axes is denoted by θ.

Fig. 4
Fig. 4

Evolution of (a) excited state populations and (b) transient coherence between the excited states in a QD placed adjacent to the HMM, initially prepared in | y' . The populations are derived at d = 50 nm while the coherence is plotted as a function of distance d. Counterparts of (a) and (b) for a QD initially prepared in | z' are depicted in (c) and (d), respectively. Decay of the excited state population ρy'y' with (e) θ = 0°, 22° and (f) θ = 22°, 45°. The QD is placed at a distance of 50 nm from the HMM and initially prepared in (| y' | z' )/ 2 .

Equations (10)

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

ε xx = ε yy = ε || =f ε m +(1f) ε d , ε zz = ε = ε m ε d f ε d +(1f) ε m ,
k x 2 + k y 2 | ε | k z 2 | ε || | = k 0 2 ,
k x 2 + k y 2 + k z 2 =| ε || | k 0 2 ,
μ= μ 0 (|yv| e y +|zv| e z +H.c.)
ρ ˙ y'y' = Γ y'y' ρ y'y' 1 2 ( Γ z'y' ρ y'z' + Γ y'z' ρ z'y' ), ρ ˙ z'z' = Γ z'z' ρ z'z' 1 2 ( Γ y'z' ρ z'y' + Γ z'y' ρ y'z' ), ρ ˙ y'z' = Γ y'y' + Γ z'z' 2 ρ y'z' Γ y'z' 2 ρ z'z' Γ y'z' 2 ρ y'y' , ρ y'y' + ρ z'z' + ρ vv =1,
Γ= 2 ω 0 2 ε 0 c 2 μIm[ G ( r 0 , r 0 ,ω)]μ,
( Γ y'y' Γ y'z' Γ z'y' Γ z'z' )=( cosθ sinθ sinθ cosθ )( Γ yy 0 0 Γ zz )( cosθ sinθ sinθ cosθ ),
Γ y'z' Γ y'y' = ( Γ zz Γ yy )sinθcosθ Γ yy cos 2 θ+ Γ zz sin 2 θ ,
ρ y'y' =0.73 e 0.52 Γ 0 t 0.25 e 1.86 Γ 0 t +0.02 e 3.21 Γ 0 t
ρ y'y' =0.5 e 0.52 Γ 0 t

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