Abstract

A collection of cold rubidium atoms in three-level configuration trapped in two dimensional (2D) optical lattices is revisited. The trapped atoms are considered in the Gaussian density distribution and we study the realization of $\mathcal {PT}$-, non-$\mathcal {PT}$-, and $\mathcal {PT}$-antisymmetry in 2D optical lattices. Such a fascinating modulation is achieved by spatially modulating the intensity of the driving field. Interestingly, control over $\mathcal {PT}$- to non-$\mathcal {PT}$-symmetry and vice versa in 2D optical lattices is achieved via a single knob such as microwave field, probe field and relative phase of optical and microwave fields. In addition, control over $\mathcal {PT}$-antisymmetry to non-$\mathcal {PT}$-symmetry and vice versa is also achieved via relative phase. The coherent control of $\mathcal {PT}$- non-$\mathcal {PT}$- and $\mathcal {PT}$-antisymmetry in optical susceptibility of 2D atomic lattices can be extended to 2D optical devices including modulators, detectors, and the 2D atomic lattices can also be extended to photonic transistors and diodes.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref]
  2. C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40(5), 2201–2229 (1999).
    [Crossref]
  3. 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(9), L171–L176 (2005).
    [Crossref]
  4. S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of Branch Points in PT-Symmetric Waveguides,” Phys. Rev. Lett. 101(8), 080402 (2008).
    [Crossref]
  5. S. Longhi, “Bloch Oscillations in Complex Crystals with PT Symmetry,” Phys. Rev. Lett. 103(12), 123601 (2009).
    [Crossref]
  6. C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
    [Crossref]
  7. A. Regensburger, C. Bersch, M. -A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012).
    [Crossref]
  8. 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,” Nat. Mater. 12(2), 108–113 (2013).
    [Crossref]
  9. N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect Metamaterial Absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
    [Crossref]
  10. Y. -D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent Perfect Absorbers: Time-Reversed Lasers,” Phys. Rev. Lett. 105(5), 053901 (2010).
    [Crossref]
  11. L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal Light Propagation in a Silicon Photonic Circuit,” Science 333(6043), 729–733 (2011).
    [Crossref]
  12. A. Szameit, M. C. Rechtsman, O. Bahat-Treidel, and M. Segev, “PT-symmetry in honeycomb photonic lattices,” Phys. Rev. A 84(2), 021806 (2011).
    [Crossref]
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    [Crossref]
  14. B. He, S. -B Yan, J. Wang, and M. Xiao, “Quantum noise effects with Kerr-nonlinearity enhancement in coupled gain-loss waveguides,” Phys. Rev. A 91(5), 053832 (2015).
    [Crossref]
  15. H. Benisty, A. Lupu, and A. Degiron, “Transverse periodic PT symmetry for modal demultiplexing in optical waveguides,” Phys. Rev. A 91(5), 053825 (2015).
    [Crossref]
  16. B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
    [Crossref]
  17. H. Jing, S. K. Özdemir, X.-Y. Lü, J. Zhang, L. Yang, and F. Nori, “PT-Symmetric Phonon Laser,” Phys. Rev. Lett. 113(5), 053604 (2014).
    [Crossref]
  18. X.-Y. Lü, H. Jing, J.-Y. Ma, and Y. Wu, “PT-Symmetry-Breaking Chaos in Optomechanics,” Phys. Rev. Lett. 114(25), 253601 (2015).
    [Crossref]
  19. X. -W. Xu, Y. -X. Liu, C. -P. Sun, and Y. Li, “Mechanical PT symmetry in coupled optomechanical systems,” Phys. Rev. A 92(1), 013852 (2015).
    [Crossref]
  20. J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537(7618), 76–79 (2016).
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  21. D. Chatzidimitrious and E. E. Kriezis, “Optical switching through graphene-induced exceptional points,” J. Opt. Soc. Am. B 35(7), 1525 (2018).
    [Crossref]
  22. Y. Sun, W. Tan, H. Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with PT phase transition,” Phys. Rev. Lett. 112(14), 143903 (2014).
    [Crossref]
  23. C. Hang, G. Huang, and V. V. Konotop, “$\mathcal {PT}$PT symmetry with a system of three-level atoms,” Phys. Rev. Lett. 110(8), 083604 (2013).
    [Crossref]
  24. C. Hang, G. Gabadadze, and G. Huang, “Realization of non-$\mathcal {PT}$PT-symmetric optical potentials with all-real spectra in a coherent atomic system,” Phys. Rev. A 95(2), 023833 (2017).
    [Crossref]
  25. J. Sheng, M. A. Miri, D. N. Christodoulides, and M. Xiao, “$\mathcal {PT}$PT-symmetric optical potentials in a coherent atomic medium,” Phys. Rev. A 88(4), 041803 (2013).
    [Crossref]
  26. Ziauddin, Y.-L. Chaung, and R.-K. Lee, “PT -symmetry in Rydberg atoms,” Europhys. Lett. 115(1), 14005 (2016).
    [Crossref]
  27. H. -J. Li, J. -P. Dou, and G. -X. Huang, “PT symmetry via electromagnetically induced transparency,” Opt. Express 21(26), 32053 (2013).
    [Crossref]
  28. J. -H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113(12), 123004 (2014).
    [Crossref]
  29. X. Wang and J. -H. Wu, “Optical PT -symmetry and PT -antisymmetry in coherently driven atomic lattices,” Opt. Express 24(4), 4289 (2016).
    [Crossref]
  30. P. Peng, W. Cao, C. Shen, W. Qu, J. Wen, L. Jiang, and Y. Xiao, “Anti-parity time symmetry with flying atoms,” Nat. Phys. 12(12), 1139–1145 (2016).
    [Crossref]
  31. K. Staliunas, R. Herrero, and R. Vilaseca, “Subdiffraction and spatial filtering due to periodic spatial modulation of the gain-loss profile,” Phys. Rev. A 80(1), 013821 (2009).
    [Crossref]
  32. M. Botey, R. Herrero, and K. Staliunas, “Light in materials with periodic gain-loss modulation on a wavelength scale,” Phys. Rev. A 82(1), 013828 (2010).
    [Crossref]
  33. R. Herrero, M. Botey, M. Radziunas, and K. Staliunas, “Beam shaping in spatially modulated broad-area semiconductor amplifiers,” Opt. Lett. 37(24), 5253–5255 (2012).
    [Crossref]
  34. M. Radziunas, M. Botey R. Herrero, and K. Staliunas, “Intrinsic beam shaping mechanism in spatially modulated broad area semiconductor amplifiers,” Appl. Phys. Lett. 103(13), 132101 (2013).
    [Crossref]
  35. H. Li, V. A. Sautenkov, Y. V. Rostovtsev, G. R. Welch, P. R. Hemmer, and M. O. Scully, “Electromagnetically induced transparency controlled by a microwave field,” Phys. Rev. A 80(2), 023820 (2009).
    [Crossref]
  36. J. Joo, J. Bourassa, A. Blais, and B. C. Sanders, “Electromagnetically Induced Transparency with Amplification in Superconducting Circuits,” Phys. Rev. Lett. 105(7), 073601 (2010).
    [Crossref]

2018 (1)

2017 (1)

C. Hang, G. Gabadadze, and G. Huang, “Realization of non-$\mathcal {PT}$PT-symmetric optical potentials with all-real spectra in a coherent atomic system,” Phys. Rev. A 95(2), 023833 (2017).
[Crossref]

2016 (4)

Ziauddin, Y.-L. Chaung, and R.-K. Lee, “PT -symmetry in Rydberg atoms,” Europhys. Lett. 115(1), 14005 (2016).
[Crossref]

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537(7618), 76–79 (2016).
[Crossref]

X. Wang and J. -H. Wu, “Optical PT -symmetry and PT -antisymmetry in coherently driven atomic lattices,” Opt. Express 24(4), 4289 (2016).
[Crossref]

P. Peng, W. Cao, C. Shen, W. Qu, J. Wen, L. Jiang, and Y. Xiao, “Anti-parity time symmetry with flying atoms,” Nat. Phys. 12(12), 1139–1145 (2016).
[Crossref]

2015 (5)

X.-Y. Lü, H. Jing, J.-Y. Ma, and Y. Wu, “PT-Symmetry-Breaking Chaos in Optomechanics,” Phys. Rev. Lett. 114(25), 253601 (2015).
[Crossref]

X. -W. Xu, Y. -X. Liu, C. -P. Sun, and Y. Li, “Mechanical PT symmetry in coupled optomechanical systems,” Phys. Rev. A 92(1), 013852 (2015).
[Crossref]

M. Turduev, M. Botey, I. Giden, R. Herrero, H. Kurt, E. Ozbay, and K. Staliunas, “Two-dimensional complex parity-time-symmetric photonic structures,” Phys. Rev. A 91(2), 023825 (2015).
[Crossref]

B. He, S. -B Yan, J. Wang, and M. Xiao, “Quantum noise effects with Kerr-nonlinearity enhancement in coupled gain-loss waveguides,” Phys. Rev. A 91(5), 053832 (2015).
[Crossref]

H. Benisty, A. Lupu, and A. Degiron, “Transverse periodic PT symmetry for modal demultiplexing in optical waveguides,” Phys. Rev. A 91(5), 053825 (2015).
[Crossref]

2014 (4)

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

H. Jing, S. K. Özdemir, X.-Y. Lü, J. Zhang, L. Yang, and F. Nori, “PT-Symmetric Phonon Laser,” Phys. Rev. Lett. 113(5), 053604 (2014).
[Crossref]

Y. Sun, W. Tan, H. Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with PT phase transition,” Phys. Rev. Lett. 112(14), 143903 (2014).
[Crossref]

J. -H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113(12), 123004 (2014).
[Crossref]

2013 (5)

M. Radziunas, M. Botey R. Herrero, and K. Staliunas, “Intrinsic beam shaping mechanism in spatially modulated broad area semiconductor amplifiers,” Appl. Phys. Lett. 103(13), 132101 (2013).
[Crossref]

C. Hang, G. Huang, and V. V. Konotop, “$\mathcal {PT}$PT symmetry with a system of three-level atoms,” Phys. Rev. Lett. 110(8), 083604 (2013).
[Crossref]

H. -J. Li, J. -P. Dou, and G. -X. Huang, “PT symmetry via electromagnetically induced transparency,” Opt. Express 21(26), 32053 (2013).
[Crossref]

J. Sheng, M. A. Miri, D. N. Christodoulides, and M. Xiao, “$\mathcal {PT}$PT-symmetric optical potentials in a coherent atomic medium,” Phys. Rev. A 88(4), 041803 (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,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref]

2012 (2)

A. Regensburger, C. Bersch, M. -A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012).
[Crossref]

R. Herrero, M. Botey, M. Radziunas, and K. Staliunas, “Beam shaping in spatially modulated broad-area semiconductor amplifiers,” Opt. Lett. 37(24), 5253–5255 (2012).
[Crossref]

2011 (2)

L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal Light Propagation in a Silicon Photonic Circuit,” Science 333(6043), 729–733 (2011).
[Crossref]

A. Szameit, M. C. Rechtsman, O. Bahat-Treidel, and M. Segev, “PT-symmetry in honeycomb photonic lattices,” Phys. Rev. A 84(2), 021806 (2011).
[Crossref]

2010 (4)

Y. -D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent Perfect Absorbers: Time-Reversed Lasers,” Phys. Rev. Lett. 105(5), 053901 (2010).
[Crossref]

C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

M. Botey, R. Herrero, and K. Staliunas, “Light in materials with periodic gain-loss modulation on a wavelength scale,” Phys. Rev. A 82(1), 013828 (2010).
[Crossref]

J. Joo, J. Bourassa, A. Blais, and B. C. Sanders, “Electromagnetically Induced Transparency with Amplification in Superconducting Circuits,” Phys. Rev. Lett. 105(7), 073601 (2010).
[Crossref]

2009 (3)

H. Li, V. A. Sautenkov, Y. V. Rostovtsev, G. R. Welch, P. R. Hemmer, and M. O. Scully, “Electromagnetically induced transparency controlled by a microwave field,” Phys. Rev. A 80(2), 023820 (2009).
[Crossref]

K. Staliunas, R. Herrero, and R. Vilaseca, “Subdiffraction and spatial filtering due to periodic spatial modulation of the gain-loss profile,” Phys. Rev. A 80(1), 013821 (2009).
[Crossref]

S. Longhi, “Bloch Oscillations in Complex Crystals with PT Symmetry,” Phys. Rev. Lett. 103(12), 123601 (2009).
[Crossref]

2008 (2)

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect Metamaterial Absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[Crossref]

S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of Branch Points in PT-Symmetric Waveguides,” Phys. Rev. Lett. 101(8), 080402 (2008).
[Crossref]

2005 (1)

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(9), L171–L176 (2005).
[Crossref]

1999 (1)

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40(5), 2201–2229 (1999).
[Crossref]

1998 (1)

C. M. Bender and S. Boettcher, “Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
[Crossref]

-A. Miri, M.

A. Regensburger, C. Bersch, M. -A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012).
[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,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref]

Artoni, M.

J. -H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113(12), 123004 (2014).
[Crossref]

Ayache, M.

L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal Light Propagation in a Silicon Photonic Circuit,” Science 333(6043), 729–733 (2011).
[Crossref]

-B Yan, S.

B. He, S. -B Yan, J. Wang, and M. Xiao, “Quantum noise effects with Kerr-nonlinearity enhancement in coupled gain-loss waveguides,” Phys. Rev. A 91(5), 053832 (2015).
[Crossref]

Bahat-Treidel, O.

A. Szameit, M. C. Rechtsman, O. Bahat-Treidel, and M. Segev, “PT-symmetry in honeycomb photonic lattices,” Phys. Rev. A 84(2), 021806 (2011).
[Crossref]

Bender, C. M.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40(5), 2201–2229 (1999).
[Crossref]

C. M. Bender and S. Boettcher, “Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
[Crossref]

Benisty, H.

H. Benisty, A. Lupu, and A. Degiron, “Transverse periodic PT symmetry for modal demultiplexing in optical waveguides,” Phys. Rev. A 91(5), 053825 (2015).
[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(7410), 167–171 (2012).
[Crossref]

Blais, A.

J. Joo, J. Bourassa, A. Blais, and B. C. Sanders, “Electromagnetically Induced Transparency with Amplification in Superconducting Circuits,” Phys. Rev. Lett. 105(7), 073601 (2010).
[Crossref]

Boettcher, S.

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40(5), 2201–2229 (1999).
[Crossref]

C. M. Bender and S. Boettcher, “Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
[Crossref]

Böhm, J.

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537(7618), 76–79 (2016).
[Crossref]

Botey, M.

M. Turduev, M. Botey, I. Giden, R. Herrero, H. Kurt, E. Ozbay, and K. Staliunas, “Two-dimensional complex parity-time-symmetric photonic structures,” Phys. Rev. A 91(2), 023825 (2015).
[Crossref]

R. Herrero, M. Botey, M. Radziunas, and K. Staliunas, “Beam shaping in spatially modulated broad-area semiconductor amplifiers,” Opt. Lett. 37(24), 5253–5255 (2012).
[Crossref]

M. Botey, R. Herrero, and K. Staliunas, “Light in materials with periodic gain-loss modulation on a wavelength scale,” Phys. Rev. A 82(1), 013828 (2010).
[Crossref]

Botey R. Herrero, M.

M. Radziunas, M. Botey R. Herrero, and K. Staliunas, “Intrinsic beam shaping mechanism in spatially modulated broad area semiconductor amplifiers,” Appl. Phys. Lett. 103(13), 132101 (2013).
[Crossref]

Bourassa, J.

J. Joo, J. Bourassa, A. Blais, and B. C. Sanders, “Electromagnetically Induced Transparency with Amplification in Superconducting Circuits,” Phys. Rev. Lett. 105(7), 073601 (2010).
[Crossref]

Cao, H.

Y. -D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent Perfect Absorbers: Time-Reversed Lasers,” Phys. Rev. Lett. 105(5), 053901 (2010).
[Crossref]

Cao, W.

P. Peng, W. Cao, C. Shen, W. Qu, J. Wen, L. Jiang, and Y. Xiao, “Anti-parity time symmetry with flying atoms,” Nat. Phys. 12(12), 1139–1145 (2016).
[Crossref]

Chatzidimitrious, D.

Chaung, Y.-L.

Ziauddin, Y.-L. Chaung, and R.-K. Lee, “PT -symmetry in Rydberg atoms,” Europhys. Lett. 115(1), 14005 (2016).
[Crossref]

Chen, H.

Y. Sun, W. Tan, H. Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with PT phase transition,” Phys. Rev. Lett. 112(14), 143903 (2014).
[Crossref]

Chen, Y. F.

L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal Light Propagation in a Silicon Photonic Circuit,” Science 333(6043), 729–733 (2011).
[Crossref]

Christodoulides, D. N.

J. Sheng, M. A. Miri, D. N. Christodoulides, and M. Xiao, “$\mathcal {PT}$PT-symmetric optical potentials in a coherent atomic medium,” Phys. Rev. A 88(4), 041803 (2013).
[Crossref]

A. Regensburger, C. Bersch, M. -A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012).
[Crossref]

C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

-D. Chong, Y.

Y. -D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent Perfect Absorbers: Time-Reversed Lasers,” Phys. Rev. Lett. 105(5), 053901 (2010).
[Crossref]

Degiron, A.

H. Benisty, A. Lupu, and A. Degiron, “Transverse periodic PT symmetry for modal demultiplexing in optical waveguides,” Phys. Rev. A 91(5), 053825 (2015).
[Crossref]

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(9), L171–L176 (2005).
[Crossref]

Doppler, J.

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537(7618), 76–79 (2016).
[Crossref]

El-Ganainy, R.

C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

-F. Chen, Y.

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,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref]

Fainman, Y.

L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal Light Propagation in a Silicon Photonic Circuit,” Science 333(6043), 729–733 (2011).
[Crossref]

Fan, S.

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M. Radziunas, M. Botey R. Herrero, and K. Staliunas, “Intrinsic beam shaping mechanism in spatially modulated broad area semiconductor amplifiers,” Appl. Phys. Lett. 103(13), 132101 (2013).
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N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect Metamaterial Absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
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J. Joo, J. Bourassa, A. Blais, and B. C. Sanders, “Electromagnetically Induced Transparency with Amplification in Superconducting Circuits,” Phys. Rev. Lett. 105(7), 073601 (2010).
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[Crossref]

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P. Peng, W. Cao, C. Shen, W. Qu, J. Wen, L. Jiang, and Y. Xiao, “Anti-parity time symmetry with flying atoms,” Nat. Phys. 12(12), 1139–1145 (2016).
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J. Sheng, M. A. Miri, D. N. Christodoulides, and M. Xiao, “$\mathcal {PT}$PT-symmetric optical potentials in a coherent atomic medium,” Phys. Rev. A 88(4), 041803 (2013).
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N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect Metamaterial Absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
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M. Turduev, M. Botey, I. Giden, R. Herrero, H. Kurt, E. Ozbay, and K. Staliunas, “Two-dimensional complex parity-time-symmetric photonic structures,” Phys. Rev. A 91(2), 023825 (2015).
[Crossref]

M. Radziunas, M. Botey R. Herrero, and K. Staliunas, “Intrinsic beam shaping mechanism in spatially modulated broad area semiconductor amplifiers,” Appl. Phys. Lett. 103(13), 132101 (2013).
[Crossref]

R. Herrero, M. Botey, M. Radziunas, and K. Staliunas, “Beam shaping in spatially modulated broad-area semiconductor amplifiers,” Opt. Lett. 37(24), 5253–5255 (2012).
[Crossref]

M. Botey, R. Herrero, and K. Staliunas, “Light in materials with periodic gain-loss modulation on a wavelength scale,” Phys. Rev. A 82(1), 013828 (2010).
[Crossref]

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Y. -D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent Perfect Absorbers: Time-Reversed Lasers,” Phys. Rev. Lett. 105(5), 053901 (2010).
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Y. Sun, W. Tan, H. Q. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with PT phase transition,” Phys. Rev. Lett. 112(14), 143903 (2014).
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Science (1)

L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal Light Propagation in a Silicon Photonic Circuit,” Science 333(6043), 729–733 (2011).
[Crossref]

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Figures (12)

Fig. 1.
Fig. 1. (a) Energy-level configuration for three-level atomic configuration. (b) In the $x$ - and $y$ -directions, three-level atomic medium is tapped in 2D optical lattices in a Gaussian distribution.
Fig. 2.
Fig. 2. (a) Real and (b) imaginary parts of the optical susceptibility vs probe field detuning $\Delta$ . The parameters are presented at the text.
Fig. 3.
Fig. 3. Density plot of imaginary part of optical susceptibility vs probe field detuning $\Delta$ and lattice position $x/a_1$ or ( $y/a_1$ ). The parameters are $\Omega _{11}=1\gamma$ , $\delta \Omega _{2x}=\delta \Omega _{2y}=0.3\gamma$ , $\sigma =0.2a_1$ , $a_1=0.5\lambda$ and $\Omega _\mu =0.05\gamma$ , the other parameters remains the same as presented in the text.
Fig. 4.
Fig. 4. 3D plot of (a) real and (b) imaginary parts of the optical susceptibility vs lattice position $x/a_1$ and $y/a_1$ with probe field detuning $\Delta =0.48\gamma$ and $\Omega _{\mu }=0.05\gamma$ , the other parameters remains the same as that in Fig. 3.
Fig. 5.
Fig. 5. Real part of optical susceptibility vs lattice position $x/a_1$ and $y/a_1$ with probe field detuning $\Delta =0.48\gamma$ , $\delta \Omega _{2x}=\pm 0.3\gamma , \delta \Omega _{2y}=\pm 0.3\gamma$ , and $\Omega _{\mu }=0.05\gamma$ , the other parameters remains the same as that in Fig. 4 .
Fig. 6.
Fig. 6. Imaginary part of the optical susceptibility vs lattice position $x/a_1$ and $y/a_1$ with probe field detuning $\Delta =0.48\gamma$ , (a) $\delta \Omega _{2x}=0.3\gamma , \delta \Omega _{2y}=0.3\gamma$ , (b) $\delta \Omega _{2x}=-0.3\gamma , \delta \Omega _{2y}=0.3\gamma$ , (c) $\delta \Omega _{2x}=0.3\gamma , \delta \Omega _{2y}=-0.3\gamma$ , (d) $\delta \Omega _{2x}=-0.3\gamma , \delta \Omega _{2y}=-0.3\gamma$ and $\Omega _{\mu }=0.05\gamma$ , the other parameters remains the same as that in Fig. 4.
Fig. 7.
Fig. 7. (a, b) Real and Imaginary parts of the optical susceptibility vs lattice position $x/a_1$ and $y/a_1$ with probe field $\Omega _{p}=0.01\gamma$ , (c, d) Real and Imaginary parts of the optical susceptibility vs lattice position $x/a_1$ and $y/a_1$ with microwave field $\Omega _{\mu }=0.01\gamma$ , all the other parameters remain the same as that of Fig. 4.
Fig. 8.
Fig. 8. Density plot of real part of optical susceptibility vs probe field detuning $\Delta$ and lattice position $x/a_1$ or ( $y/a_1$ ) by considering $\phi =\pi$ . The other parameters remains the same as presented in the text.
Fig. 9.
Fig. 9. Real and imaginary parts of the optical susceptibility vs lattice position $x/a_1$ and $y/a_1$ with probe field detuning $\Delta =0.57\gamma$ and $\phi =\pi$ , all the other parameters remain the same as that in Fig. 4.
Fig. 10.
Fig. 10. Real part of the optical susceptibility vs lattice position $x/a_1$ and $y/a_1$ with probe field detuning $\Delta =0.57\gamma$ , (a) $\delta \Omega _{2x}=0.3\gamma , \delta \Omega _{2y}=0.3\gamma$ , (b) $\delta \Omega _{2x}=-0.3\gamma , \delta \Omega _{2y}=0.3\gamma$ , (c) $\delta \Omega _{2x}=0.3\gamma , \delta \Omega _{2y}=-0.3\gamma$ , (d) $\delta \Omega _{2x}=-0.3\gamma , \delta \Omega _{2y}=-0.3\gamma$ and $\Omega _{\mu }=0.05\gamma$ , the other parameters remains the same as that in Fig. 4.
Fig. 11.
Fig. 11. Imaginary part of optical susceptibility vs lattice position $x/a_1$ and $y/a_1$ with probe field detuning $\Delta =0.57\gamma$ , $\delta \Omega _{2x}=\pm 0.3\gamma , \delta \Omega _{2y}=\pm 0.3\gamma$ , and $\Omega _{\mu }=0.05\gamma$ , the other parameters remains the same as that in Fig. 4
Fig. 12.
Fig. 12. Real and imaginary parts of optical susceptibility vs lattice position $x/a_1$ and $y/a_1$ with relative phase $\phi =\pi /2$ , the other parameters remain the same as that in Figs. 10 and 11.

Equations (6)

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H = 2 ( Ω 2 | a   c | e i ϕ 2 + Ω μ | c   b | e i ϕ μ + Ω p e i Δ t | a   b | e i ϕ p + H c )
ρ ˙ a b = ( i Δ γ a b ) ρ a b + i 2 Ω 2 ρ c b e i ϕ 2 + i 2 Ω p e i ϕ p , ρ ˙ c b = ( i Δ γ c b ) ρ c b + i 2 Ω 2 ρ a b e i ϕ 2 + i 2 Ω μ e i ϕ μ ,
ρ a b = 2 i γ c b Ω p + 2 Δ Ω p Ω 2 Ω μ e i ϕ 4 ( γ a b i Δ ) ( γ c b i Δ ) + Ω 2 2
χ = N 0 | a b | 2 2 ϵ 0 ρ a b .
N j ( x , y ) = N 0 e ( ( x x j ) 2 / σ 2 ( y y j ) 2 / σ 2 ) , x ( x j a 1 / 2 , x j + a 1 / 2 ) , y ( y j a 1 / 2 , y j + a 1 / 2 ) ,
Ω 2 = Ω 11 + δ Ω 2 x sin [ 2 π ( x x j ) / a 1 + δ Ω 2 y sin [ 2 π ( y y j ) / a 1 ] ,

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