Abstract

We propose an efficient scheme in helium or alkaline earth atomic vapor to achieve a parity-time symmetric Bragg structure using coherent lights. Unidirectional invisibility can be realized in this scheme, i.e., the atomic vapor shows total transparency for probe light incident from one particular direction, but exhibits enhanced Bragg reflection for probe from the opposite side. By changing the relative phase between the coherent lights, this direction can easily be manipulated, providing a convenient way for investigating special properties of 𝒫𝒯 -symmetric Bragg structures.

© 2014 Optical Society of America

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References

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  1. K. G. Makris, R. El-Ganainy, and D.N. Christodoulides, “Beam Dynamics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008).
    [Crossref] [PubMed]
  2. F. Nazari, M. Nazari, and M. K. Morawej-Farshi, “A 2 × 2 spatial optical switch based on PT-symmetry,” Opt. Lett. 36, 4368–4370 (2011).
    [Crossref] [PubMed]
  3. A. A. Sukhorukov, Z. Y. Xu, and Y. S. Kivshar, “Nonlinear suppression of time reversals in PT-symmetric optical couplers,” Phys. Rev. A 82, 043818 (2010).
    [Crossref]
  4. M. Miri, P. LiKamWa, and D. N. Christodoulides, “Large area single-mode parity-time-symmetric laser amplifiers,” Opt. Lett. 37, 764–766 (2012).
    [Crossref] [PubMed]
  5. C. M. Bender, D. C. Brody, and H. F. Jones, “Extension of PT-symmetric quantum mechanics to quantum field theory with cubic interaction,” Phys. Rev. D 70, 025001 (2004).
    [Crossref]
  6. Y. Goldsheid and B. A. Khoruzhenko, “Distribution of Eigenvalues in Non-Hermitian Anderson Models,” Phys. Rev. Lett. 80, 2897–2900 (1998).
    [Crossref]
  7. I. Rotter, “A non-Hermitian Hamilton operator and the physics of open quantum systems,” J. Phys. A, 42, 153001 (2009).
    [Crossref]
  8. S. Klaiman, N. Moiseyev, and Gunther, “Visualization of Branch Points in PT-Symmetric Waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
    [Crossref] [PubMed]
  9. A. Mostafazadeh, “Pseudo-Hermiticity versus PT-Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian,” J. Math. Phys. 43, 3944 (2002).
    [Crossref]
  10. C. E. Rüter, K.G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
    [Crossref]
  11. V. Yannopapas, “Spontaneous PT -symmetry breaking in complex frequency band structures,” Phys. Rev. A 89, 013808 (2014).
    [Crossref]
  12. A. Guo, G. J. Salamo, D. Duchesne, R. Morandomdotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of -Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett, 103, 093902 (2009).
    [Crossref]
  13. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional Invisibility Induced by PT-Symmetric Periodic Structures,” Phys. Rev. Lett. 106, 213901 (2011).
    [Crossref] [PubMed]
  14. L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mat. 12(2), 108–113 (2013).
    [Crossref]
  15. A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices”, Nature 488, 167–171 (2012).
    [Crossref] [PubMed]
  16. V. Yannopapas, “One-way photonic band gaps and optical isolation with three-dimensional photonic crystals of low symmetry,” Phys. Rev. A 88, 043837 (2013).
    [Crossref]
  17. C. Wang, C. Zhou, and Z. Li, “On-chip optical diode based on silicon photonic crystal heterojunctions,” Opt. Express,  19, 26948–26955 (2011).
    [Crossref]
  18. M. A. Miri, A. B. Aceves, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Bragg solitons in nonlinear PT-symmetric periodic potentials,” Phys. Rev. A 86, 033801 (2012).
    [Crossref]
  19. M. O. Scully, “Enhancement of the index of refraction via quantum coherence,” Phys. Rev. Lett. 67, 1855–1858 (1991).
    [Crossref] [PubMed]
  20. J. P. Dowling and C. M. Bowden, “Near dipole-dipole effects in lasing without inversion: An enhancement of gain and absorptionless index of refraction,” Phys. Rev. Lett. 70, 1421–1424 (1993).
    [Crossref] [PubMed]
  21. D. D. Yavuz, “Refractive Index Enhancement in a Far-Off Resonant Atomic System,” Phys. Rev. Lett. 95, 223601 (2005).
    [Crossref] [PubMed]
  22. C. O’Brian, P. M. Anisimov, Y. Rostovtsev, and O. Kocharovskaya, “Coherent control of refractive index in far-detuned Λ systems,” Phys. Rev. A 84, 063835 (2011).
    [Crossref]
  23. C. O’Brian and O. Kocharovskaya, “Optically Controllable Photonic Structures with Zero Absorption,” Phys. Rev. Lett. 107, 137401 (2011).
    [Crossref]
  24. Z. Chen, B. Luo, and H. Guo, “Absorption-free Bragg reflector using Zeeman sublevels in atomic vapor,” Opt. Express,  22, 15564–15570 (2014).
    [Crossref] [PubMed]
  25. C. Hang and G. Huang, “𝒫𝒯 -Symmetry with a System of Three-Level Atoms,” and V. V. Konotop, Phys. Rev. Lett. 110, 083604 (2013).
    [Crossref]
  26. H. Li, J. Dou, and G. Huang, “PT symmetry via electromagnetically induced transparency, Opt. Express, 21, 32053–32062 (2013).
    [Crossref]
  27. J. Sheng, M-A. Miri, D. N. Christodoulides, and M. Xiao, “PT-symmetric optical potentials in a coherent atomic medium,” Phys. Rev. A 88, 041803 (R) (2013).
    [Crossref]
  28. W. L. Wiese and J. R. Fuhr, “Accurate Atomic Transition Probabilities for Hydrogen, Helium, and Lithium”, Journal of physical and chemical reference data,  38, 565–719 (2009).
    [Crossref]

2014 (2)

V. Yannopapas, “Spontaneous PT -symmetry breaking in complex frequency band structures,” Phys. Rev. A 89, 013808 (2014).
[Crossref]

Z. Chen, B. Luo, and H. Guo, “Absorption-free Bragg reflector using Zeeman sublevels in atomic vapor,” Opt. Express,  22, 15564–15570 (2014).
[Crossref] [PubMed]

2013 (5)

H. Li, J. Dou, and G. Huang, “PT symmetry via electromagnetically induced transparency, Opt. Express, 21, 32053–32062 (2013).
[Crossref]

C. Hang and G. Huang, “𝒫𝒯 -Symmetry with a System of Three-Level Atoms,” and V. V. Konotop, Phys. Rev. Lett. 110, 083604 (2013).
[Crossref]

J. Sheng, M-A. Miri, D. N. Christodoulides, and M. Xiao, “PT-symmetric optical potentials in a coherent atomic medium,” Phys. Rev. A 88, 041803 (R) (2013).
[Crossref]

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mat. 12(2), 108–113 (2013).
[Crossref]

V. Yannopapas, “One-way photonic band gaps and optical isolation with three-dimensional photonic crystals of low symmetry,” Phys. Rev. A 88, 043837 (2013).
[Crossref]

2012 (3)

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices”, Nature 488, 167–171 (2012).
[Crossref] [PubMed]

M. A. Miri, A. B. Aceves, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Bragg solitons in nonlinear PT-symmetric periodic potentials,” Phys. Rev. A 86, 033801 (2012).
[Crossref]

M. Miri, P. LiKamWa, and D. N. Christodoulides, “Large area single-mode parity-time-symmetric laser amplifiers,” Opt. Lett. 37, 764–766 (2012).
[Crossref] [PubMed]

2011 (5)

F. Nazari, M. Nazari, and M. K. Morawej-Farshi, “A 2 × 2 spatial optical switch based on PT-symmetry,” Opt. Lett. 36, 4368–4370 (2011).
[Crossref] [PubMed]

C. Wang, C. Zhou, and Z. Li, “On-chip optical diode based on silicon photonic crystal heterojunctions,” Opt. Express,  19, 26948–26955 (2011).
[Crossref]

C. O’Brian, P. M. Anisimov, Y. Rostovtsev, and O. Kocharovskaya, “Coherent control of refractive index in far-detuned Λ systems,” Phys. Rev. A 84, 063835 (2011).
[Crossref]

C. O’Brian and O. Kocharovskaya, “Optically Controllable Photonic Structures with Zero Absorption,” Phys. Rev. Lett. 107, 137401 (2011).
[Crossref]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional Invisibility Induced by PT-Symmetric Periodic Structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref] [PubMed]

2010 (2)

C. E. Rüter, K.G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

A. A. Sukhorukov, Z. Y. Xu, and Y. S. Kivshar, “Nonlinear suppression of time reversals in PT-symmetric optical couplers,” Phys. Rev. A 82, 043818 (2010).
[Crossref]

2009 (3)

A. Guo, G. J. Salamo, D. Duchesne, R. Morandomdotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of -Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett, 103, 093902 (2009).
[Crossref]

I. Rotter, “A non-Hermitian Hamilton operator and the physics of open quantum systems,” J. Phys. A, 42, 153001 (2009).
[Crossref]

W. L. Wiese and J. R. Fuhr, “Accurate Atomic Transition Probabilities for Hydrogen, Helium, and Lithium”, Journal of physical and chemical reference data,  38, 565–719 (2009).
[Crossref]

2008 (2)

S. Klaiman, N. Moiseyev, and Gunther, “Visualization of Branch Points in PT-Symmetric Waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
[Crossref] [PubMed]

K. G. Makris, R. El-Ganainy, and D.N. Christodoulides, “Beam Dynamics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

2005 (1)

D. D. Yavuz, “Refractive Index Enhancement in a Far-Off Resonant Atomic System,” Phys. Rev. Lett. 95, 223601 (2005).
[Crossref] [PubMed]

2004 (1)

C. M. Bender, D. C. Brody, and H. F. Jones, “Extension of PT-symmetric quantum mechanics to quantum field theory with cubic interaction,” Phys. Rev. D 70, 025001 (2004).
[Crossref]

2002 (1)

A. Mostafazadeh, “Pseudo-Hermiticity versus PT-Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian,” J. Math. Phys. 43, 3944 (2002).
[Crossref]

1998 (1)

Y. Goldsheid and B. A. Khoruzhenko, “Distribution of Eigenvalues in Non-Hermitian Anderson Models,” Phys. Rev. Lett. 80, 2897–2900 (1998).
[Crossref]

1993 (1)

J. P. Dowling and C. M. Bowden, “Near dipole-dipole effects in lasing without inversion: An enhancement of gain and absorptionless index of refraction,” Phys. Rev. Lett. 70, 1421–1424 (1993).
[Crossref] [PubMed]

1991 (1)

M. O. Scully, “Enhancement of the index of refraction via quantum coherence,” Phys. Rev. Lett. 67, 1855–1858 (1991).
[Crossref] [PubMed]

Aceves, A. B.

M. A. Miri, A. B. Aceves, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Bragg solitons in nonlinear PT-symmetric periodic potentials,” Phys. Rev. A 86, 033801 (2012).
[Crossref]

Aimez, V.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandomdotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of -Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett, 103, 093902 (2009).
[Crossref]

Almeida, V. R.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mat. 12(2), 108–113 (2013).
[Crossref]

Anisimov, P. M.

C. O’Brian, P. M. Anisimov, Y. Rostovtsev, and O. Kocharovskaya, “Coherent control of refractive index in far-detuned Λ systems,” Phys. Rev. A 84, 063835 (2011).
[Crossref]

Bender, C. M.

C. M. Bender, D. C. Brody, and H. F. Jones, “Extension of PT-symmetric quantum mechanics to quantum field theory with cubic interaction,” Phys. Rev. D 70, 025001 (2004).
[Crossref]

Bersch, C.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices”, Nature 488, 167–171 (2012).
[Crossref] [PubMed]

Bowden, C. M.

J. P. Dowling and C. M. Bowden, “Near dipole-dipole effects in lasing without inversion: An enhancement of gain and absorptionless index of refraction,” Phys. Rev. Lett. 70, 1421–1424 (1993).
[Crossref] [PubMed]

Brody, D. C.

C. M. Bender, D. C. Brody, and H. F. Jones, “Extension of PT-symmetric quantum mechanics to quantum field theory with cubic interaction,” Phys. Rev. D 70, 025001 (2004).
[Crossref]

Cao, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional Invisibility Induced by PT-Symmetric Periodic Structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref] [PubMed]

Chen, Y. F.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mat. 12(2), 108–113 (2013).
[Crossref]

Chen, Z.

Christodoulides, D. N.

J. Sheng, M-A. Miri, D. N. Christodoulides, and M. Xiao, “PT-symmetric optical potentials in a coherent atomic medium,” Phys. Rev. A 88, 041803 (R) (2013).
[Crossref]

M. A. Miri, A. B. Aceves, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Bragg solitons in nonlinear PT-symmetric periodic potentials,” Phys. Rev. A 86, 033801 (2012).
[Crossref]

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices”, Nature 488, 167–171 (2012).
[Crossref] [PubMed]

M. Miri, P. LiKamWa, and D. N. Christodoulides, “Large area single-mode parity-time-symmetric laser amplifiers,” Opt. Lett. 37, 764–766 (2012).
[Crossref] [PubMed]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional Invisibility Induced by PT-Symmetric Periodic Structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref] [PubMed]

C. E. Rüter, K.G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandomdotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of -Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett, 103, 093902 (2009).
[Crossref]

Christodoulides, D.N.

K. G. Makris, R. El-Ganainy, and D.N. Christodoulides, “Beam Dynamics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

Dou, J.

Dowling, J. P.

J. P. Dowling and C. M. Bowden, “Near dipole-dipole effects in lasing without inversion: An enhancement of gain and absorptionless index of refraction,” Phys. Rev. Lett. 70, 1421–1424 (1993).
[Crossref] [PubMed]

Duchesne, D.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandomdotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of -Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett, 103, 093902 (2009).
[Crossref]

Eichelkraut, T.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional Invisibility Induced by PT-Symmetric Periodic Structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref] [PubMed]

El-Ganainy, R.

C. E. Rüter, K.G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

K. G. Makris, R. El-Ganainy, and D.N. Christodoulides, “Beam Dynamics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

Fegadolli, W. S.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mat. 12(2), 108–113 (2013).
[Crossref]

Feng, L.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mat. 12(2), 108–113 (2013).
[Crossref]

Fuhr, J. R.

W. L. Wiese and J. R. Fuhr, “Accurate Atomic Transition Probabilities for Hydrogen, Helium, and Lithium”, Journal of physical and chemical reference data,  38, 565–719 (2009).
[Crossref]

Goldsheid, Y.

Y. Goldsheid and B. A. Khoruzhenko, “Distribution of Eigenvalues in Non-Hermitian Anderson Models,” Phys. Rev. Lett. 80, 2897–2900 (1998).
[Crossref]

Gunther,

S. Klaiman, N. Moiseyev, and Gunther, “Visualization of Branch Points in PT-Symmetric Waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
[Crossref] [PubMed]

Guo, A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandomdotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of -Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett, 103, 093902 (2009).
[Crossref]

Guo, H.

Hang, C.

C. Hang and G. Huang, “𝒫𝒯 -Symmetry with a System of Three-Level Atoms,” and V. V. Konotop, Phys. Rev. Lett. 110, 083604 (2013).
[Crossref]

Huang, G.

C. Hang and G. Huang, “𝒫𝒯 -Symmetry with a System of Three-Level Atoms,” and V. V. Konotop, Phys. Rev. Lett. 110, 083604 (2013).
[Crossref]

H. Li, J. Dou, and G. Huang, “PT symmetry via electromagnetically induced transparency, Opt. Express, 21, 32053–32062 (2013).
[Crossref]

Jones, H. F.

C. M. Bender, D. C. Brody, and H. F. Jones, “Extension of PT-symmetric quantum mechanics to quantum field theory with cubic interaction,” Phys. Rev. D 70, 025001 (2004).
[Crossref]

Khoruzhenko, B. A.

Y. Goldsheid and B. A. Khoruzhenko, “Distribution of Eigenvalues in Non-Hermitian Anderson Models,” Phys. Rev. Lett. 80, 2897–2900 (1998).
[Crossref]

Kip, D.

C. E. Rüter, K.G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

Kivshar, Y. S.

A. A. Sukhorukov, Z. Y. Xu, and Y. S. Kivshar, “Nonlinear suppression of time reversals in PT-symmetric optical couplers,” Phys. Rev. A 82, 043818 (2010).
[Crossref]

Klaiman, S.

S. Klaiman, N. Moiseyev, and Gunther, “Visualization of Branch Points in PT-Symmetric Waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
[Crossref] [PubMed]

Kocharovskaya, O.

C. O’Brian, P. M. Anisimov, Y. Rostovtsev, and O. Kocharovskaya, “Coherent control of refractive index in far-detuned Λ systems,” Phys. Rev. A 84, 063835 (2011).
[Crossref]

C. O’Brian and O. Kocharovskaya, “Optically Controllable Photonic Structures with Zero Absorption,” Phys. Rev. Lett. 107, 137401 (2011).
[Crossref]

Kottos, T.

M. A. Miri, A. B. Aceves, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Bragg solitons in nonlinear PT-symmetric periodic potentials,” Phys. Rev. A 86, 033801 (2012).
[Crossref]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional Invisibility Induced by PT-Symmetric Periodic Structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref] [PubMed]

Kovanis, V.

M. A. Miri, A. B. Aceves, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Bragg solitons in nonlinear PT-symmetric periodic potentials,” Phys. Rev. A 86, 033801 (2012).
[Crossref]

Li, H.

Li, Z.

LiKamWa, P.

Lin, Z.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional Invisibility Induced by PT-Symmetric Periodic Structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref] [PubMed]

Lu, M. H.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mat. 12(2), 108–113 (2013).
[Crossref]

Luo, B.

Makris, K. G.

K. G. Makris, R. El-Ganainy, and D.N. Christodoulides, “Beam Dynamics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

Makris, K.G.

C. E. Rüter, K.G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

Miri, M.

Miri, M. A.

M. A. Miri, A. B. Aceves, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Bragg solitons in nonlinear PT-symmetric periodic potentials,” Phys. Rev. A 86, 033801 (2012).
[Crossref]

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices”, Nature 488, 167–171 (2012).
[Crossref] [PubMed]

Miri, M-A.

J. Sheng, M-A. Miri, D. N. Christodoulides, and M. Xiao, “PT-symmetric optical potentials in a coherent atomic medium,” Phys. Rev. A 88, 041803 (R) (2013).
[Crossref]

Moiseyev, N.

S. Klaiman, N. Moiseyev, and Gunther, “Visualization of Branch Points in PT-Symmetric Waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
[Crossref] [PubMed]

Morandomdotti, R.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandomdotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of -Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett, 103, 093902 (2009).
[Crossref]

Morawej-Farshi, M. K.

Mostafazadeh, A.

A. Mostafazadeh, “Pseudo-Hermiticity versus PT-Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian,” J. Math. Phys. 43, 3944 (2002).
[Crossref]

Nazari, F.

Nazari, M.

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C. O’Brian and O. Kocharovskaya, “Optically Controllable Photonic Structures with Zero Absorption,” Phys. Rev. Lett. 107, 137401 (2011).
[Crossref]

C. O’Brian, P. M. Anisimov, Y. Rostovtsev, and O. Kocharovskaya, “Coherent control of refractive index in far-detuned Λ systems,” Phys. Rev. A 84, 063835 (2011).
[Crossref]

Oliveira, J. E. B.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mat. 12(2), 108–113 (2013).
[Crossref]

Onishchukov, G.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices”, Nature 488, 167–171 (2012).
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Peschel, U.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices”, Nature 488, 167–171 (2012).
[Crossref] [PubMed]

Ramezani, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional Invisibility Induced by PT-Symmetric Periodic Structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref] [PubMed]

Regensburger, A.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices”, Nature 488, 167–171 (2012).
[Crossref] [PubMed]

Rostovtsev, Y.

C. O’Brian, P. M. Anisimov, Y. Rostovtsev, and O. Kocharovskaya, “Coherent control of refractive index in far-detuned Λ systems,” Phys. Rev. A 84, 063835 (2011).
[Crossref]

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I. Rotter, “A non-Hermitian Hamilton operator and the physics of open quantum systems,” J. Phys. A, 42, 153001 (2009).
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C. E. Rüter, K.G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

Salamo, G. J.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandomdotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of -Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett, 103, 093902 (2009).
[Crossref]

Scherer, A.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mat. 12(2), 108–113 (2013).
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C. E. Rüter, K.G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
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J. Sheng, M-A. Miri, D. N. Christodoulides, and M. Xiao, “PT-symmetric optical potentials in a coherent atomic medium,” Phys. Rev. A 88, 041803 (R) (2013).
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Siviloglou, G. A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandomdotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of -Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett, 103, 093902 (2009).
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A. A. Sukhorukov, Z. Y. Xu, and Y. S. Kivshar, “Nonlinear suppression of time reversals in PT-symmetric optical couplers,” Phys. Rev. A 82, 043818 (2010).
[Crossref]

Volatier-Ravat, M.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandomdotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of -Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett, 103, 093902 (2009).
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Wang, C.

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W. L. Wiese and J. R. Fuhr, “Accurate Atomic Transition Probabilities for Hydrogen, Helium, and Lithium”, Journal of physical and chemical reference data,  38, 565–719 (2009).
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Xiao, M.

J. Sheng, M-A. Miri, D. N. Christodoulides, and M. Xiao, “PT-symmetric optical potentials in a coherent atomic medium,” Phys. Rev. A 88, 041803 (R) (2013).
[Crossref]

Xu, Y. L.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mat. 12(2), 108–113 (2013).
[Crossref]

Xu, Z. Y.

A. A. Sukhorukov, Z. Y. Xu, and Y. S. Kivshar, “Nonlinear suppression of time reversals in PT-symmetric optical couplers,” Phys. Rev. A 82, 043818 (2010).
[Crossref]

Yannopapas, V.

V. Yannopapas, “Spontaneous PT -symmetry breaking in complex frequency band structures,” Phys. Rev. A 89, 013808 (2014).
[Crossref]

V. Yannopapas, “One-way photonic band gaps and optical isolation with three-dimensional photonic crystals of low symmetry,” Phys. Rev. A 88, 043837 (2013).
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Yavuz, D. D.

D. D. Yavuz, “Refractive Index Enhancement in a Far-Off Resonant Atomic System,” Phys. Rev. Lett. 95, 223601 (2005).
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Zhou, C.

J. Math. Phys. (1)

A. Mostafazadeh, “Pseudo-Hermiticity versus PT-Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian,” J. Math. Phys. 43, 3944 (2002).
[Crossref]

J. Phys. A, (1)

I. Rotter, “A non-Hermitian Hamilton operator and the physics of open quantum systems,” J. Phys. A, 42, 153001 (2009).
[Crossref]

Journal of physical and chemical reference data (1)

W. L. Wiese and J. R. Fuhr, “Accurate Atomic Transition Probabilities for Hydrogen, Helium, and Lithium”, Journal of physical and chemical reference data,  38, 565–719 (2009).
[Crossref]

Nat. Phys. (1)

C. E. Rüter, K.G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

Nature (1)

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices”, Nature 488, 167–171 (2012).
[Crossref] [PubMed]

Nature Mat. (1)

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mat. 12(2), 108–113 (2013).
[Crossref]

Opt. Express (3)

Opt. Lett. (2)

Phys. Rev. A (6)

A. A. Sukhorukov, Z. Y. Xu, and Y. S. Kivshar, “Nonlinear suppression of time reversals in PT-symmetric optical couplers,” Phys. Rev. A 82, 043818 (2010).
[Crossref]

V. Yannopapas, “Spontaneous PT -symmetry breaking in complex frequency band structures,” Phys. Rev. A 89, 013808 (2014).
[Crossref]

M. A. Miri, A. B. Aceves, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Bragg solitons in nonlinear PT-symmetric periodic potentials,” Phys. Rev. A 86, 033801 (2012).
[Crossref]

J. Sheng, M-A. Miri, D. N. Christodoulides, and M. Xiao, “PT-symmetric optical potentials in a coherent atomic medium,” Phys. Rev. A 88, 041803 (R) (2013).
[Crossref]

V. Yannopapas, “One-way photonic band gaps and optical isolation with three-dimensional photonic crystals of low symmetry,” Phys. Rev. A 88, 043837 (2013).
[Crossref]

C. O’Brian, P. M. Anisimov, Y. Rostovtsev, and O. Kocharovskaya, “Coherent control of refractive index in far-detuned Λ systems,” Phys. Rev. A 84, 063835 (2011).
[Crossref]

Phys. Rev. D (1)

C. M. Bender, D. C. Brody, and H. F. Jones, “Extension of PT-symmetric quantum mechanics to quantum field theory with cubic interaction,” Phys. Rev. D 70, 025001 (2004).
[Crossref]

Phys. Rev. Lett, (1)

A. Guo, G. J. Salamo, D. Duchesne, R. Morandomdotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of -Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett, 103, 093902 (2009).
[Crossref]

Phys. Rev. Lett. (9)

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional Invisibility Induced by PT-Symmetric Periodic Structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref] [PubMed]

Y. Goldsheid and B. A. Khoruzhenko, “Distribution of Eigenvalues in Non-Hermitian Anderson Models,” Phys. Rev. Lett. 80, 2897–2900 (1998).
[Crossref]

M. O. Scully, “Enhancement of the index of refraction via quantum coherence,” Phys. Rev. Lett. 67, 1855–1858 (1991).
[Crossref] [PubMed]

J. P. Dowling and C. M. Bowden, “Near dipole-dipole effects in lasing without inversion: An enhancement of gain and absorptionless index of refraction,” Phys. Rev. Lett. 70, 1421–1424 (1993).
[Crossref] [PubMed]

D. D. Yavuz, “Refractive Index Enhancement in a Far-Off Resonant Atomic System,” Phys. Rev. Lett. 95, 223601 (2005).
[Crossref] [PubMed]

K. G. Makris, R. El-Ganainy, and D.N. Christodoulides, “Beam Dynamics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

S. Klaiman, N. Moiseyev, and Gunther, “Visualization of Branch Points in PT-Symmetric Waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
[Crossref] [PubMed]

C. O’Brian and O. Kocharovskaya, “Optically Controllable Photonic Structures with Zero Absorption,” Phys. Rev. Lett. 107, 137401 (2011).
[Crossref]

C. Hang and G. Huang, “𝒫𝒯 -Symmetry with a System of Three-Level Atoms,” and V. V. Konotop, Phys. Rev. Lett. 110, 083604 (2013).
[Crossref]

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Figures (6)

Fig. 1
Fig. 1 (a) Basic experimental setup of the study of ���� -symmetric media based on the similarity between scalar wave equation with paraxial approximation and Schrödinger equation. (b) Basic experimental setup of the study of ���� -symmetric Bragg structure.
Fig. 2
Fig. 2 (a) Level configuration in helium or alkaline atoms to realize the ���� -symmetric Bragg structure. (b) A possible experimental setup. Both control lights are set to be standing wave with phase difference of π/4.
Fig. 3
Fig. 3 (a) Equivalent Ξ-system of the first Λ-system. (b) Equivalent Ξ-system of the second Λ-system.
Fig. 4
Fig. 4 Numerical simulation results of χ. When we turn off control light 1 and set ωp = ωJ2 − 99.9975γ1, the real (red, dotted) and imaginary (red, dot dashed) part of χ are shown. In this case, we obtain a conventional Bragg reflector. When we set Ω1,82 = Ω1,93 = 0.7653γ1 cos (kx + π/4) and ωp = ωJ2 − 99.9916γ1, we can get the real (blue, solid) and imaginary(blue, dashed) part of χ. In this case, unidirectional invisibility can be observed when the probe light is parallel to +x-direction. The case when Ω1,82 = Ω1,93 = 0.7653γ1 cos (kxπ/4) and ωp = ωJ2 − 99.9916γ1 is not shown here. In this case χ is the complex conjugate of that when Ω1 = 0.7653γ1 cos (kx + π/4) and unidirectional invisibility occurs when the probe light is antiparallel to the +x-direction.
Fig. 5
Fig. 5 The band structure of the conventional Bragg reflector and the ���� -symmetric Bragg structure working at breaking point. kB is the Brillouin wavenumber (normalized by k). (a) Real part of kB of both Bragg reflector and ���� -symmetric Bragg structure working at breaking point. (b) Imaginary part of kB of conventional Bragg reflector (blue, dashed) and ���� -symmetric Bragg structure working at the breaking point.
Fig. 6
Fig. 6 The phenomenon of unidirectional invisibility. 1000 periods are used to simulate the transmittance and reflectivity, kp is the wavevector of the probe light. (a) Unidirectional invisibility occurs when the probe light is parallel to the +x-direction. In this case, Ω1,82 = Ω1,93 = 0.7653γ1 cos (kx + π/4) and ωp = ωJ2 − 99.9916γ1. rA, rP and t are shown. (b) Unidirectional invisibility occurs when the probe light is antiparallel to the +x-direction. In this case, Ω1,82 = Ω1,93 = 0.7653γ1 cos (kxπ/4) and ωp = ωJ2 − 99.9916γ1. rA, rP and t are shown. (c) The vapor is a conventional Bragg reflector when Ω1,82 = Ω1,93 = 0 and ωp = ωJ2 − 99.9975γ1. r and t are shown. (d) The argument of transmittance t when unidirectional invisibility occurs (blue, solid) and when the vapor acs as a conventional Bragg reflector (red, dashed).

Tables (2)

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Table 1 Definition of parameters

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Table 2 Values of parameters

Equations (5)

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χ = N | μ 41 | 2 ε 0 h ¯ [ ξ 1 ( ρ 1 ρ 2 ) ω 2 1 ω p + i γ 2 1 + ( ρ 1 ρ 4 ) ω 4 1 ω p + i γ 4 1 ] + N | μ 63 | 2 ε 0 h ¯ [ ξ 2 ( ρ 3 ρ 1 ) ω 1 3 ω p + i γ 1 3 + ( ρ 3 ρ 6 ) ω 6 3 ω p + i γ 6 3 ] + N | μ 52 | 2 ε 0 h ¯ [ ( ρ 5 ρ 2 ) ω 52 + Ω 1 , 82 2 / Δ 1 , 82 + Ω c , 42 2 / Δ 42 ω p + i γ 1 ] ,
ξ 1 = Ω c , 42 2 / Δ c , 42 2 , ξ 2 = Ω c , 61 2 / Δ c , 61 2 ,
γ 2 1 = ( 1 ξ 1 ) γ 2 + ξ 1 γ 1 , γ 4 1 = ( 1 ξ 1 ) γ 1 + ξ 1 γ 2 γ 3 1 = ( 1 ξ 2 ) γ 2 + ξ 2 γ 1 , γ 6 3 = ( 1 ξ 2 ) γ 1 + ξ 2 γ 2 .
χ c = N | μ 41 | 2 ( ρ 4 ρ 1 ) ε 0 h ¯ Δ p , 41 + N | μ 63 | 2 ( ρ 6 ρ 3 ) ε 0 h ¯ Δ p , 63 + N | μ 52 | 2 ( ρ 5 ρ 2 ) ε 0 h ¯ Δ p , 52 ,
χ N ε 0 h ¯ [ | μ 41 | 2 ξ 1 ( ρ 1 ρ 2 ) ω 2 1 ω p + i γ 2 1 + | μ 63 | 2 ξ 2 ( ρ 3 ρ 1 ) ω 1 3 ω p + i γ 1 3 ] + χ c = χ N ( 1 δ + δ ω i γ 2 1 δ δ ω i γ 2 ) + χ c ,

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