Abstract

A collection of cold rubidium atoms in three-level configuration trapped in one dimensional (1D) optical lattice is revisited. The trapped atoms are considered in the Gaussian density distribution and study the realization of $\mathcal {PT}$-, non-$\mathcal {PT}$- and $\mathcal {PT}$ anti-symmetry in optical susceptibility in 1D atomic lattices in a periodic structure. Such a fascinating modulation is achieved by spatially modulating the intensity of the driving field. Interestingly, a nonreciprocal optical propagation phenomenon is investigated. In this system, we have introduced a microwave that couples to the two ground states, spatial modulation of the coupling field, and the atomic density with Gaussian distribution in practice. With a proper detuning and coupling field Rabi frequencies, we can find the condition of $\mathcal {PT}$-symmetry along with field propagation direction, and the novel properties of transmission and reflections have been discussed. The large difference of field reflections from the two ends of the atomic lattice medium shows strong evidence that the nonreciprocal behavior can be greatly enhanced by increasing the spatial modulation amplitude.

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

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]

2018 (1)

2016 (3)

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

X. Wang and J.-H. Wu, “Optical PT -symmetry and PT -antisymmetry in coherently driven atomic lattices,” Opt. Express 24(4), 4289–4298 (2016).
[Crossref]

Z. Zhang, Y. Zhang, J. Sheng, L. Yang, M.-A. Miri, D. N. Christodoulides, B. He, Y. Zhang, and M. Xiao, “Observation of Parity-Time Symmetry in Optically Induced Atomic Lattices,” Phys. Rev. Lett. 117(12), 123601 (2016).
[Crossref]

2015 (5)

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]

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]

2014 (4)

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]

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]

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

2013 (5)

H.-J. Li, J.-P. Dou, and G.-X. Huang, “PT symmetry via electromagnetically induced transparency,” Opt. Express 21(26), 32053–32062 (2013).
[Crossref]

C. Hang, G. Huang, and V. V. Konotop, “Localization of light in a parity-time-symmetric quasiperiodic lattice,” Phys. Rev. Lett. 110(8), 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(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]

S. A. R. Horsley, J.-H. Wu, M. Artoni, and G. C. La Rocca, “Optical Nonreciprocity of Cold Atom Bragg Mirrors in Motion,” Phys. Rev. Lett. 110(22), 223602 (2013).
[Crossref]

2012 (1)

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]

2011 (3)

A. Szameit, M. C. Rechtsman, O. B. Treidel, and M. Segev, “PT-symmetry in honeycomb photonic lattices,” Phys. Rev. A 84(2), 021806 (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(21), 213901 (2011).
[Crossref]

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]

2010 (2)

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

2009 (3)

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. V.-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref]

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

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]

2008 (2)

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

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]

2005 (1)

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]

Aimez, V.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. V.-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103(9), 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,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref]

Artoni, M.

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

S. A. R. Horsley, J.-H. Wu, M. Artoni, and G. C. La Rocca, “Optical Nonreciprocity of Cold Atom Bragg Mirrors in Motion,” Phys. Rev. Lett. 110(22), 223602 (2013).
[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]

Azaña, J.

Bélanger, N.

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]

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]

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]

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(21), 213901 (2011).
[Crossref]

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]

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]

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

Chong, Y.-D.

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]

Christodoulides, D. N.

Z. Zhang, Y. Zhang, J. Sheng, L. Yang, M.-A. Miri, D. N. Christodoulides, B. He, Y. Zhang, and M. Xiao, “Observation of Parity-Time Symmetry in Optically Induced Atomic Lattices,” Phys. Rev. Lett. 117(12), 123601 (2016).
[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(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]

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(21), 213901 (2011).
[Crossref]

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

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. V.-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref]

Chuang, Y.-L.

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]

Dou, J.-P.

Duchesne, D.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. V.-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref]

E.-Ganainy, R.

C. E. Ruter, K. G. Makris, R. E.-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[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(21), 213901 (2011).
[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.

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]

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

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]

Ge, L.

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]

Gianfreda, 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]

Giden, I.

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]

Gunther, U.

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

Guo, A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. V.-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref]

Hang, C.

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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).
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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).
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Nat. Phys. (2)

C. E. Ruter, K. G. Makris, R. E.-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
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Nature (1)

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).
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Opt. Express (4)

Phys. Rev. A (7)

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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).
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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).
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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).
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Science (1)

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[Crossref]

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Figures (13)

Fig. 1.
Fig. 1. (a) Energy-level configuration for three-level atomic configuration. (b) In the $x$ -direction, three-level atomic medium is tapped in 1D 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 $\gamma =1$ MHz, $\gamma _{eb}=1\gamma$ , $\gamma _{cb}=0.01\gamma$ , $\Omega _p=0.01\gamma$ , $\Omega _2=5\gamma$ and $\phi =\pi /2$ .
Fig. 3.
Fig. 3. Density plot of imaginary part of the optical susceptibility vs probe field detuning $\Delta$ and lattice position $x/a$ . The parameters are $\gamma =1$ MHz, $\gamma _{eb}=1\gamma$ , $\gamma _{cb}=0.01\gamma$ , $\Omega _p=0.01\gamma$ , $\Omega _2=5\gamma$ , $\phi =\pi /2$ , $\Omega _{11}=5\gamma$ , $\delta \Omega _2=0.3\gamma$ , $\sigma =0.35 a$ , $a= 1 \lambda$ and $\Omega _\mu =0.05\gamma$ .
Fig. 4.
Fig. 4. (a) real and (b) imaginary parts of the optical susceptibility for a single cycle vs lattice position $x/a$ with probe field detuning $\Delta =2.404\gamma$ , the other parameters remains the same as that in Fig. 3.
Fig. 5.
Fig. 5. (a) real and (b) imaginary parts of the optical susceptibility for many cycles vs lattice position $x/a$ with probe field detuning $\Delta =2.404\gamma$ , the other parameters remains the same as that in Fig. 3.
Fig. 6.
Fig. 6. (a) real and (b) imaginary parts of the optical susceptibility for many cycles vs lattice position $x/a$ with probe field detuning $\Delta =2.404\gamma$ and $\Omega _{\mu }=0$ , the other parameters remains the same as that in Fig. 3.
Fig. 7.
Fig. 7. (a) real and (b) imaginary parts of the optical susceptibility for one cycle vs lattice position $x/a$ with probe field detuning $\Delta =2.502\gamma$ and $\phi =\pi$ , the other parameters remains the same as that in Fig. 3.
Fig. 8.
Fig. 8. Field scattering in one-dimension under the condition of $\mathcal {PT}$ -symmetry. The forward fields at the two ends of lattice are $E_f(-L/2)$ (green) and $E_f(+L/2)$ (red), and backward fields are $E_b(-L/2)$ (yellow) and $E_b(+L/2)$ (blue).
Fig. 9.
Fig. 9. Spatial refractive index. The blue and red lines represent $\eta _1(x)$ and $\eta _2(x)$ , respectively. The parameters are given as $\Omega _{11} = 5\gamma$ , $\delta \Omega _2 = 0.3\gamma$ , $\Omega _{\mu } = 0.05\gamma$ , $\Omega _p = 0.01\gamma$ , $\sigma = 0.35a$ , and $\phi = \pi /2$ .
Fig. 10.
Fig. 10. Transmission and reflections versus spatial detuning under $\delta \Omega _2 = 0.3\gamma$ (figure a) and $\delta \Omega _2 = 1.0\gamma$ (figure b). The red curve represents the field transmission $T$ , and the reflections $R_R$ and $R_L$ are plotted by blue and green curves. The other parameters we’ve used are given by $\Omega _{11} = 5\gamma$ , $\Omega _{\mu } = 0.05\gamma$ , $\Omega _p = 0.01\gamma$ , $\sigma = 0.35a$ , $k = 500$ and $\phi = \pi /2$ .
Fig. 11.
Fig. 11. Transmission and reflections v.s. modulation amplitude $\delta \Omega _2$ . The transmission is plotted by the red curve, and the two reflections, $R_R$ and $R_L$ , are represented by blue and green curves, respectively. Other parameters are the same as that of in Fig. 10.
Fig. 12.
Fig. 12. The values of $B^2$ and $D^2$ v.s. $\delta \Omega _2$ . Blue curve represents square of $B$ , and blue curve represents square of $D$ . We’ve set the same parameters used in Fig. 11.
Fig. 13.
Fig. 13. The modulation refraction index with two different $\delta \Omega _2$ ’s (a and c), and the corresponding reflections and transmissions (b and d) under $\mathcal {PT}$ -antisymmetry condition. Blue lines and red lines in (a) and (c) represent $\eta _1$ and $\eta _2$ , respectively. In (a) and (b), $\delta \Omega _2 = 0.3\gamma$ , and $\delta \Omega _2 = 0.7\gamma$ for (c) and (d). Other parameters are given by $\Omega _{11} = 5\gamma$ , $\Omega _{\mu } = 0.05\gamma$ , $\Omega _p = 0.01\gamma$ , $\sigma = 0.35a$ , and $\phi = \pi$ .

Equations (18)

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H = 2 ( Ω 2 | e   c | e i φ 2 + Ω μ | c   b | e i φ μ + Ω p | e   b | e i φ p + H c )
ρ ˙ e b = ( i Δ γ e b ) ρ e 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 e b e i φ 2 + i 2 Ω μ e i φ μ .
ρ e b = 2 i γ c b Ω p + 2 Δ Ω p Ω 2 Ω μ e i φ 4 ( γ e b i Δ ) ( γ c b i Δ ) + Ω 2 2 ,
χ ( x ) = N j ( x ) | e b | 2 2 ϵ 0 ρ e b ,
N j ( x ) = N 0 e ( x x j ) 2 / σ 2 , x ( x j a / 2 , x j + a / 2 ) .
Ω 2 ( x ) = Ω 11 + δ Ω 2 sin [ 2 π ( x x j ) / a ] ,
[ d 2 d x 2 + ω 2 c 2 n 2 ( x ) ] E ( x ) = 0 ,
η 1 ( x ) = a 0 2 + n = 1 a n cos ( 2 n β x ) ,
η 2 ( x ) = n = 1 b n sin ( 2 n β x ) . ,
E ( x ) = E f ( z ) e + i k x + E b ( z ) e i k x .
d d x E f ( x ) = i δ E f ( x ) + i k 2 B E b ( x ) ,
d d x E b ( x ) = + i δ E b ( x ) i k 2 D E f ( x ) ,
B = ( a 1 + b 1 ) + ( I 1 + I 2 ) / 2 ,
D = ( a 1 b 1 ) + ( I 1 I 2 ) / 2 ,
( E f ( + L / 2 ) E b ( + L / 2 ) ) = ( M 11 M 12 M 21 M 22 ) ( E f ( L / 2 ) E b ( L / 2 ) ) .
T = | λ | 2 | λ | 2 cos 2 ( λ L ) + δ 2 sin 2 ( λ L )
R L = k 2 | D | 2 / 4 δ 2 + | λ cot ( λ L ) | 2
R R = k 2 | B | 2 / 4 δ 2 + | λ cot ( λ L ) | 2

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