Abstract

We propose a scheme to realize parity-time (PT) symmetry via electromagnetically induced transparency (EIT). The system we consider is an ensemble of cold four-level atoms with an EIT core. We show that the cross-phase modulation contributed by an assisted field, the optical lattice potential provided by a far-detuned laser field, and the optical gain resulted from an incoherent pumping can be used to construct a PT-symmetric complex optical potential for probe field propagation in a controllable way. Comparing with previous study, the present scheme uses only a single atomic species and hence is easy for the physical realization of PT-symmetric Hamiltonian via atomic coherence.

© 2013 Optical Society of America

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  1. C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys.70, 947–1018 (2007).
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  2. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT symmetric periodic optical potentials,” Int. J. Theor. Phys.50, 1019–1041 (2011).
    [CrossRef]
  3. 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. D70, 025001 (2004).
    [CrossRef]
  4. I. Y. Goldsheid and B. A. Khoruzhenko, “Distribution of Eigenvalues in Non-Hermitian Anderson Models,” Phys. Rev. Lett.80, 2897–2900 (1998).
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    [CrossRef]
  7. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett.32, 2632–2634 (2007).
    [CrossRef] [PubMed]
  8. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam Dynamics in PT Symmetric Optical Lattices,” Phys. Rev. Lett.100, 103904 (2008).
    [CrossRef] [PubMed]
  9. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical Solitons in PT Periodic Potentials,” Phys. Rev. Lett.100, 030402 (2008).
    [CrossRef] [PubMed]
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    [CrossRef]
  11. M. Kulishov, J. M. Laniel, N. Be langer, J. Azaña, and D. V. Plant, “Nonreciprocal waveguide Bragg gratings,” Opt. Express13, 3068–3078 (2005).
    [CrossRef] [PubMed]
  12. 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]
  13. 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, 108–113 (2013).
    [CrossRef]
  14. S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A82, 031801 (2010).
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  15. Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent Perfect Absorbers: Time-Reversed Lasers,” Phys. Rev. Lett.105, 053901 (2010).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  23. W. Demtröder, Laser Spectroscopy: Basic Concepts and Instrumentation (3rd ed.) (Springer, Berlin, 2003), Chap. 10.
    [CrossRef]
  24. G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E72, 016617 (2005).
    [CrossRef]
  25. H.-j. Li, Y.-p. Wu, and G. Huang, “Stable weak-light ultraslow spatiotemporal solitons via atomic coherence,” Phys. Rev. A84, 033816 (2011).
    [CrossRef]
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2013

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, 108–113 (2013).
[CrossRef]

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

2012

V. V. Konotop, V. S. Shchesnovich, and D. A. Zezyulin, “Giant amplification of modes in parity-time symmetric waveguides,” Phys. Lett. A376, 2750–2753 (2012).
[CrossRef]

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

2011

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with PT symmetries,” Phys. Rev. A84, 040101 (2011).
[CrossRef]

H. Benisty, A. Degiron, A. Lupu, A. DeLustrac, S. Cheńais, S. Forget, M. Besbes, G. Barbillon, A. Bruyant, S. Blaize, and G. Lérondel, “Implementation of PT symmetric devices using plasmonics: principle and applications,” Opt. Express19, 18004–18019 (2011).
[CrossRef] [PubMed]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT symmetric periodic optical potentials,” Int. J. Theor. Phys.50, 1019–1041 (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]

H.-j. Li, Y.-p. Wu, and G. Huang, “Stable weak-light ultraslow spatiotemporal solitons via atomic coherence,” Phys. Rev. A84, 033816 (2011).
[CrossRef]

2010

S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A82, 031801 (2010).
[CrossRef]

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent Perfect Absorbers: Time-Reversed Lasers,” Phys. Rev. Lett.105, 053901 (2010).
[CrossRef] [PubMed]

C. E. Rüter, K. R. 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]

2009

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

2008

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

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical Solitons in PT Periodic Potentials,” Phys. Rev. Lett.100, 030402 (2008).
[CrossRef] [PubMed]

2007

2005

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A: Math. Gen.38, L171–L176 (2005).
[CrossRef]

M. Kulishov, J. M. Laniel, N. Be langer, J. Azaña, and D. V. Plant, “Nonreciprocal waveguide Bragg gratings,” Opt. Express13, 3068–3078 (2005).
[CrossRef] [PubMed]

G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E72, 016617 (2005).
[CrossRef]

M. Fleischhauer, A. Imamoǧlu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys.77, 633–673 (2005).
[CrossRef]

2004

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. D70, 025001 (2004).
[CrossRef]

1998

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

1996

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, 108–113 (2013).
[CrossRef]

Azaña, J.

Barbillon, G.

Be langer, N.

Bender, C. M.

C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys.70, 947–1018 (2007).
[CrossRef]

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. D70, 025001 (2004).
[CrossRef]

Benisty, H.

Bersch, C.

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

Besbes, M.

Blaize, S.

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. D70, 025001 (2004).
[CrossRef]

Bruyant, A.

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]

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent Perfect Absorbers: Time-Reversed Lasers,” Phys. Rev. Lett.105, 053901 (2010).
[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, 108–113 (2013).
[CrossRef]

Chenais, S.

Chong, Y. D.

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent Perfect Absorbers: Time-Reversed Lasers,” Phys. Rev. Lett.105, 053901 (2010).
[CrossRef] [PubMed]

Christodoulides, D. N.

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

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT symmetric periodic optical potentials,” Int. J. Theor. Phys.50, 1019–1041 (2011).
[CrossRef]

C. E. Rüter, K. R. 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, D. N. Christodoulides, and Z. H. Musslimani, “Beam Dynamics in PT Symmetric Optical Lattices,” Phys. Rev. Lett.100, 103904 (2008).
[CrossRef] [PubMed]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical Solitons in PT Periodic Potentials,” Phys. Rev. Lett.100, 030402 (2008).
[CrossRef] [PubMed]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett.32, 2632–2634 (2007).
[CrossRef] [PubMed]

Christodoulides, D.N.

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]

Degiron, A.

Delgado, F.

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A: Math. Gen.38, L171–L176 (2005).
[CrossRef]

DeLustrac, A.

Demtröder, W.

W. Demtröder, Laser Spectroscopy: Basic Concepts and Instrumentation (3rd ed.) (Springer, Berlin, 2003), Chap. 10.
[CrossRef]

Deng, L.

G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E72, 016617 (2005).
[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.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT symmetric periodic optical potentials,” Int. J. Theor. Phys.50, 1019–1041 (2011).
[CrossRef]

C. E. Rüter, K. R. 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]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical Solitons in PT Periodic Potentials,” Phys. Rev. Lett.100, 030402 (2008).
[CrossRef] [PubMed]

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

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett.32, 2632–2634 (2007).
[CrossRef] [PubMed]

Ellis, F. M.

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with PT symmetries,” Phys. Rev. A84, 040101 (2011).
[CrossRef]

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, 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, 108–113 (2013).
[CrossRef]

Fleischhauer, M.

M. Fleischhauer, A. Imamoǧlu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys.77, 633–673 (2005).
[CrossRef]

Forget, S.

Ge, L.

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent Perfect Absorbers: Time-Reversed Lasers,” Phys. Rev. Lett.105, 053901 (2010).
[CrossRef] [PubMed]

Goldsheid, I. Y.

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

Hang, C.

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

Huang, G.

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

H.-j. Li, Y.-p. Wu, and G. Huang, “Stable weak-light ultraslow spatiotemporal solitons via atomic coherence,” Phys. Rev. A84, 033816 (2011).
[CrossRef]

G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E72, 016617 (2005).
[CrossRef]

Imamog?lu, A.

M. Fleischhauer, A. Imamoǧlu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys.77, 633–673 (2005).
[CrossRef]

H. Schmidt and A. Imamoǧlu, “Giant Kerr nonlinearities obtained by electromagnetically induced transparency,” Opt. Lett.21, 1936–1938 (1996).
[CrossRef] [PubMed]

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. D70, 025001 (2004).
[CrossRef]

Khoruzhenko, B. A.

I. 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. R. 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]

Konotop, V. V.

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

V. V. Konotop, V. S. Shchesnovich, and D. A. Zezyulin, “Giant amplification of modes in parity-time symmetric waveguides,” Phys. Lett. A376, 2750–2753 (2012).
[CrossRef]

Kottos, 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]

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with PT symmetries,” Phys. Rev. A84, 040101 (2011).
[CrossRef]

Kulishov, M.

Laniel, J. M.

Lérondel, G.

Li, A.

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with PT symmetries,” Phys. Rev. A84, 040101 (2011).
[CrossRef]

Li, H.-j.

H.-j. Li, Y.-p. Wu, and G. Huang, “Stable weak-light ultraslow spatiotemporal solitons via atomic coherence,” Phys. Rev. A84, 033816 (2011).
[CrossRef]

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]

Longhi, S.

S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A82, 031801 (2010).
[CrossRef]

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, 108–113 (2013).
[CrossRef]

Lupu, A.

Makris, K. G.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT symmetric periodic optical potentials,” Int. J. Theor. Phys.50, 1019–1041 (2011).
[CrossRef]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical Solitons in PT Periodic Potentials,” Phys. Rev. Lett.100, 030402 (2008).
[CrossRef] [PubMed]

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

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett.32, 2632–2634 (2007).
[CrossRef] [PubMed]

Makris, K. R.

C. E. Rüter, K. R. 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]

Marangos, J. P.

M. Fleischhauer, A. Imamoǧlu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys.77, 633–673 (2005).
[CrossRef]

Miri, M.-A.

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

Muga, J. G.

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A: Math. Gen.38, L171–L176 (2005).
[CrossRef]

Musslimani, Z. H.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT symmetric periodic optical potentials,” Int. J. Theor. Phys.50, 1019–1041 (2011).
[CrossRef]

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

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical Solitons in PT Periodic Potentials,” Phys. Rev. Lett.100, 030402 (2008).
[CrossRef] [PubMed]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett.32, 2632–2634 (2007).
[CrossRef] [PubMed]

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, 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 (London)488, 167–171 (2012).
[CrossRef]

Payne, M. G.

G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E72, 016617 (2005).
[CrossRef]

Peschel, U.

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

Plant, D. V.

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 (London)488, 167–171 (2012).
[CrossRef]

Rotter, I.

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

Ruschhaupt, A.

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A: Math. Gen.38, L171–L176 (2005).
[CrossRef]

Rüter, C. E.

C. E. Rüter, K. R. 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]

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, 108–113 (2013).
[CrossRef]

Schindler, J.

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with PT symmetries,” Phys. Rev. A84, 040101 (2011).
[CrossRef]

Schmidt, H.

Segev, M.

C. E. Rüter, K. R. 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]

Shchesnovich, V. S.

V. V. Konotop, V. S. Shchesnovich, and D. A. Zezyulin, “Giant amplification of modes in parity-time symmetric waveguides,” Phys. Lett. A376, 2750–2753 (2012).
[CrossRef]

Stone, A. D.

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent Perfect Absorbers: Time-Reversed Lasers,” Phys. Rev. Lett.105, 053901 (2010).
[CrossRef] [PubMed]

Wu, Y.-p.

H.-j. Li, Y.-p. Wu, and G. Huang, “Stable weak-light ultraslow spatiotemporal solitons via atomic coherence,” Phys. Rev. A84, 033816 (2011).
[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, 108–113 (2013).
[CrossRef]

Zezyulin, D. A.

V. V. Konotop, V. S. Shchesnovich, and D. A. Zezyulin, “Giant amplification of modes in parity-time symmetric waveguides,” Phys. Lett. A376, 2750–2753 (2012).
[CrossRef]

Zheng, M. C.

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with PT symmetries,” Phys. Rev. A84, 040101 (2011).
[CrossRef]

Int. J. Theor. Phys.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT symmetric periodic optical potentials,” Int. J. Theor. Phys.50, 1019–1041 (2011).
[CrossRef]

J. Phys. A: Math. Gen.

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A: Math. Gen.38, L171–L176 (2005).
[CrossRef]

J. Phys. A: Math. Theor.

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

Nat. Phys.

C. E. Rüter, K. R. 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 (London)

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

Nature Mat.

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, 108–113 (2013).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Lett. A

V. V. Konotop, V. S. Shchesnovich, and D. A. Zezyulin, “Giant amplification of modes in parity-time symmetric waveguides,” Phys. Lett. A376, 2750–2753 (2012).
[CrossRef]

Phys. Rev. A

H.-j. Li, Y.-p. Wu, and G. Huang, “Stable weak-light ultraslow spatiotemporal solitons via atomic coherence,” Phys. Rev. A84, 033816 (2011).
[CrossRef]

S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A82, 031801 (2010).
[CrossRef]

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with PT symmetries,” Phys. Rev. A84, 040101 (2011).
[CrossRef]

Phys. Rev. D

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. D70, 025001 (2004).
[CrossRef]

Phys. Rev. E

G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E72, 016617 (2005).
[CrossRef]

Phys. Rev. Lett.

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

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

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical Solitons in PT Periodic Potentials,” Phys. Rev. Lett.100, 030402 (2008).
[CrossRef] [PubMed]

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

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent Perfect Absorbers: Time-Reversed Lasers,” Phys. Rev. Lett.105, 053901 (2010).
[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]

Rep. Prog. Phys.

C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys.70, 947–1018 (2007).
[CrossRef]

Rev. Mod. Phys.

M. Fleischhauer, A. Imamoǧlu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys.77, 633–673 (2005).
[CrossRef]

Other

D. A. Steck, Rubidium 87 D Line Data, http://steck.us/alkalidata/ .

W. Demtröder, Laser Spectroscopy: Basic Concepts and Instrumentation (3rd ed.) (Springer, Berlin, 2003), Chap. 10.
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

(a) Energy-level diagram and excitation scheme used for obtaining a PT symmetric model. (b) Possible experimental arrangement. All the notation are defined in the text.

Fig. 2
Fig. 2

The imaginary part ImK of K as a function of Δ3/γ3 for Δ2 = Δ3. Solid (red), dashed (green), and dashed-dotted (blue) lines correspond to (Ωc, Γ31) = (0, 0), (5 × 107 Hz, 0), and (5 × 107 Hz, 0.7γ3), respectively. For illustration, the value of dashed-dotted (green) line has been amplified 7.8 times.

Equations (33)

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E a = e y a ( x ) exp [ i ( k a z ω a t ) ] + c . c .
E Stark = e y 2 E s ( x ) cos ( ω L t )
Δ j = Δ j + α j 2 h ¯ | E s ( x ) | 2 .
σ t = i h ¯ [ H ^ int , σ ] Γ σ ,
i ( z + 1 c t ) Ω p + c 2 ω p 2 Ω p x 2 + κ 13 σ 31 = 0 ,
Ω p ( 1 ) = F e i K z 0 ,
σ 21 ( 1 ) = Ω c * ( σ 33 ( 0 ) σ 11 ( 0 ) ) d 31 ( 0 ) σ 23 ( 0 ) D 1 F e i K z 0 α 21 ( 1 ) F e i K z 0 ,
σ 31 ( 1 ) = K κ 13 F e i K z 0 α 31 ( 1 ) F e i K z 0 ,
σ 42 ( 1 ) = d 43 ( 0 ) σ 22 ( 0 ) + Ω c σ 23 ( 0 ) D 2 Ω a ( 1 ) α 42 ( 1 ) Ω a ( 1 ) ,
σ 43 ( 1 ) = Ω c * σ 22 ( 0 ) + d 42 ( 0 ) σ 23 ( 0 ) D 2 Ω a ( 1 ) α 43 ( 1 ) Ω a ( 1 ) ,
K = κ 13 d 21 ( 0 ) ( σ 11 ( 0 ) σ 33 ( 0 ) ) + Ω c σ 23 ( 0 ) D 1 .
i Ω p z + c 2 ω p 2 Ω p x 2 + V ˜ ( x ) Ω p = 0
V ˜ ( x ) = α 12 | e y p 24 | 2 h ¯ 2 | a ( x ) | 2 + α 13 | E s ( x ) | 2 + K ,
a ( x ) = E a 0 [ cos ( x / R ) + sin ( x / R ) ] ,
E s ( x ) = E s 0 cos ( x / R ) ,
i u s + 2 u ξ 2 + V ( ξ ) u = 0 ,
V ( ξ ) = ( g 12 + g 12 sin 2 ξ ) + g 13 cos 2 ξ + K 0 ,
a ( x ) = 0.1 ( cos ξ + sin ξ ) V / cm ,
E s ( x ) = 4.51 × 10 5 cos ξ V / cm ,
Γ 31 = 7.0 × 10 5 Hz .
V ( ξ ) = 11.7 + cos 2 ξ + 0.4 i sin 2 ξ + 𝒪 ( 10 2 ) .
i t σ 11 + i Γ 31 σ 11 i Γ 13 σ 33 + Ω p * σ 31 Ω p σ 31 * = 0 ,
i t σ 22 i Γ 23 σ 33 i Γ 24 σ 44 + Ω c * σ 32 Ω c σ 32 * + Ω a * σ 42 Ω a σ 42 * = 0 ,
i ( t + Γ 3 ) σ 33 i Γ 31 σ 11 Ω p * σ 31 + Ω p σ 31 * Ω c * σ 32 + Ω c σ 32 * = 0 ,
i ( t + Γ 4 ) σ 44 Ω a * σ 42 + Ω a σ 42 * = 0 ,
( i t + d 21 ) σ 21 + Ω c * σ 31 + Ω a * σ 41 Ω p σ 32 * = 0 ,
( i t + d 31 ) σ 31 + Ω p ( σ 11 σ 33 ) + Ω c σ 21 = 0 ,
( i t + d 41 ) σ 41 + Ω a σ 21 Ω p σ 43 = 0 ,
( i t + d 32 ) σ 32 + Ω c ( σ 22 σ 33 ) + Ω p σ 21 * Ω a σ 43 * = 0 ,
( i t + d 42 ) σ 42 + Ω a ( σ 22 σ 44 ) Ω c σ 43 = 0 ,
( i t + d 43 ) σ 43 + Ω a σ 32 * Ω p * σ 41 Ω c * σ 42 = 0 ,
α 12 = κ 13 Ω c D 1 α 41 ( 2 ) + κ 13 Ω c D 1 α 23 G ( 2 ) + κ 13 d 21 ( 0 ) D 1 ( α 11 G ( 2 ) α 33 G ( 2 ) ) ,
α 13 = κ 13 ( α 3 α 1 ) 2 h ¯ D 1 d 21 ( 0 ) α 31 ( 1 ) ,

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