Abstract

Topological phases of matter are of fundamental interest and have promising applications. Fascinating topological properties of light have been unveiled in classical optical materials. However, the manifestation of topological physics in quantum optics has not been discovered. Here we study the topological phases in a two-dimensional momentum-space superradiance lattice composed of timed Dicke states (TDSs) in electromagnetically induced transparency (EIT). By periodically modulating the three EIT coupling fields, we can create a Haldane model with in situ tunable topological properties, which manifest themselves in the contrast between diffraction signals emitted by superradiant TDSs. The topological superradiance lattices provide a controllable platform for simulating exotic phenomena in condensed matter physics and offer a basis of topological quantum optics and novel photonic devices.

© 2015 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]

2015 (2)

A. V. Nalitov, G. Malpuech, H. Terças, and D. D. Solnyshkov, Phys. Rev. Lett. 114, 026803 (2015).
[Crossref]

D.-W. Wang, R.-B. Liu, S.-Y. Zhu, and M. O. Scully, Phys. Rev. Lett. 114, 043602 (2015).
[Crossref]

2014 (1)

G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, Nature 515, 237 (2014).
[Crossref]

2013 (5)

C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang, J.-Q. Ji, Y. Feng, S. Ji, X. Chen, J. Jia, X. Dai, Z. Fang, S.-C. Zhang, K. He, Y. Wang, L. Lu, X.-C. Ma, and Q.-K. Xue, Science 340, 167 (2013).
[Crossref]

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, Nat. Photonics 7, 1001 (2013).
[Crossref]

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, Nature 496, 196 (2013).
[Crossref]

Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, Phys. Rev. Lett. 111, 243905 (2013).
[Crossref]

D.-W. Wang, H.-T. Zhou, M.-J. Guo, J.-X. Zhang, J. Evers, and S.-Y. Zhu, Phys. Rev. Lett. 110, 093901 (2013).
[Crossref]

2012 (1)

A. B. Khanikaev, M. S. Hossein, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, Nat. Mater. 12, 233 (2012).
[Crossref]

2011 (2)

N. H. Lindner, G. Refael, and V. Galitski, Nat. Phys. 7, 490 (2011).
[Crossref]

C. W. Thiel, T. Böttger, and R. L. Cone, J. Lumin. 131, 353 (2011).
[Crossref]

2010 (3)

V. B. Tiwari, S. Singh, H. S. Rawat, M. P. Singh, and S. C. Mehendale, J. Phys. B 43, 095503 (2010).
[Crossref]

J.-I. Inoue and A. Tanaka, Phys. Rev. Lett. 105, 017401 (2010).
[Crossref]

T. Kitagawa, E. Berg, M. Rudner, and E. Demler, Phys. Rev. B 82, 235114 (2010).
[Crossref]

2009 (2)

T. Oka and H. Aoki, Phys. Rev. B 79, 081406 (2009).
[Crossref]

X.-L. Qi, T. L. Hughes, S. Raghu, and S.-C. Zhang, Phys. Rev. Lett. 102, 187001 (2009).
[Crossref]

2008 (1)

F. D. M. Haldane and S. Raghu, Phys. Rev. Lett. 100, 013904 (2008).
[Crossref]

2007 (1)

M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L. W. Molenkamp, X.-L. Qi, and S.-C. Zhang, Science 318, 766 (2007).
[Crossref]

2006 (2)

M. O. Scully, E. S. Fry, C. H. R. Ooi, and K. Wódkiewicz, Phys. Rev. Lett. 96, 010501 (2006).
[Crossref]

B. A. Bernevig and S. C. Zhang, Phys. Rev. Lett. 96, 106802 (2006).
[Crossref]

2005 (1)

C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005).
[Crossref]

2001 (1)

S.-C. Zhang and J. Hu, Science 294, 823 (2001).
[Crossref]

1991 (1)

K.-J. Boller, A. Imamoğlu, and S. E. Harris, Phys. Rev. Lett. 66, 2593 (1991).
[Crossref]

1988 (1)

F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988).
[Crossref]

1980 (1)

K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, 494 (1980).
[Crossref]

1965 (1)

J. H. Shirley, Phys. Rev. 138, B979 (1965).
[Crossref]

1954 (1)

R. H. Dicke, Phys. Rev. 93, 99 (1954).
[Crossref]

Aoki, H.

T. Oka and H. Aoki, Phys. Rev. B 79, 081406 (2009).
[Crossref]

Berg, E.

T. Kitagawa, E. Berg, M. Rudner, and E. Demler, Phys. Rev. B 82, 235114 (2010).
[Crossref]

Bernevig, B. A.

B. A. Bernevig and S. C. Zhang, Phys. Rev. Lett. 96, 106802 (2006).
[Crossref]

Boller, K.-J.

K.-J. Boller, A. Imamoğlu, and S. E. Harris, Phys. Rev. Lett. 66, 2593 (1991).
[Crossref]

Böttger, T.

C. W. Thiel, T. Böttger, and R. L. Cone, J. Lumin. 131, 353 (2011).
[Crossref]

Brüne, C.

M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L. W. Molenkamp, X.-L. Qi, and S.-C. Zhang, Science 318, 766 (2007).
[Crossref]

Buhmann, H.

M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L. W. Molenkamp, X.-L. Qi, and S.-C. Zhang, Science 318, 766 (2007).
[Crossref]

Chang, C.-Z.

C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang, J.-Q. Ji, Y. Feng, S. Ji, X. Chen, J. Jia, X. Dai, Z. Fang, S.-C. Zhang, K. He, Y. Wang, L. Lu, X.-C. Ma, and Q.-K. Xue, Science 340, 167 (2013).
[Crossref]

Chen, X.

C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang, J.-Q. Ji, Y. Feng, S. Ji, X. Chen, J. Jia, X. Dai, Z. Fang, S.-C. Zhang, K. He, Y. Wang, L. Lu, X.-C. Ma, and Q.-K. Xue, Science 340, 167 (2013).
[Crossref]

Cone, R. L.

C. W. Thiel, T. Böttger, and R. L. Cone, J. Lumin. 131, 353 (2011).
[Crossref]

Dai, X.

C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang, J.-Q. Ji, Y. Feng, S. Ji, X. Chen, J. Jia, X. Dai, Z. Fang, S.-C. Zhang, K. He, Y. Wang, L. Lu, X.-C. Ma, and Q.-K. Xue, Science 340, 167 (2013).
[Crossref]

Demler, E.

T. Kitagawa, E. Berg, M. Rudner, and E. Demler, Phys. Rev. B 82, 235114 (2010).
[Crossref]

Desbuquois, R.

G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, Nature 515, 237 (2014).
[Crossref]

Dicke, R. H.

R. H. Dicke, Phys. Rev. 93, 99 (1954).
[Crossref]

Dorda, G.

K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, 494 (1980).
[Crossref]

Dreisow, F.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, Nature 496, 196 (2013).
[Crossref]

Esslinger, T.

G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, Nature 515, 237 (2014).
[Crossref]

Evers, J.

D.-W. Wang, H.-T. Zhou, M.-J. Guo, J.-X. Zhang, J. Evers, and S.-Y. Zhu, Phys. Rev. Lett. 110, 093901 (2013).
[Crossref]

Fan, J.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, Nat. Photonics 7, 1001 (2013).
[Crossref]

Fang, Z.

C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang, J.-Q. Ji, Y. Feng, S. Ji, X. Chen, J. Jia, X. Dai, Z. Fang, S.-C. Zhang, K. He, Y. Wang, L. Lu, X.-C. Ma, and Q.-K. Xue, Science 340, 167 (2013).
[Crossref]

Feng, X.

C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang, J.-Q. Ji, Y. Feng, S. Ji, X. Chen, J. Jia, X. Dai, Z. Fang, S.-C. Zhang, K. He, Y. Wang, L. Lu, X.-C. Ma, and Q.-K. Xue, Science 340, 167 (2013).
[Crossref]

Feng, Y.

C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang, J.-Q. Ji, Y. Feng, S. Ji, X. Chen, J. Jia, X. Dai, Z. Fang, S.-C. Zhang, K. He, Y. Wang, L. Lu, X.-C. Ma, and Q.-K. Xue, Science 340, 167 (2013).
[Crossref]

Fry, E. S.

M. O. Scully, E. S. Fry, C. H. R. Ooi, and K. Wódkiewicz, Phys. Rev. Lett. 96, 010501 (2006).
[Crossref]

Galitski, V.

N. H. Lindner, G. Refael, and V. Galitski, Nat. Phys. 7, 490 (2011).
[Crossref]

Greif, D.

G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, Nature 515, 237 (2014).
[Crossref]

Guo, M.

C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang, J.-Q. Ji, Y. Feng, S. Ji, X. Chen, J. Jia, X. Dai, Z. Fang, S.-C. Zhang, K. He, Y. Wang, L. Lu, X.-C. Ma, and Q.-K. Xue, Science 340, 167 (2013).
[Crossref]

Guo, M.-J.

D.-W. Wang, H.-T. Zhou, M.-J. Guo, J.-X. Zhang, J. Evers, and S.-Y. Zhu, Phys. Rev. Lett. 110, 093901 (2013).
[Crossref]

Hafezi, M.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, Nat. Photonics 7, 1001 (2013).
[Crossref]

Haldane, F. D. M.

F. D. M. Haldane and S. Raghu, Phys. Rev. Lett. 100, 013904 (2008).
[Crossref]

F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988).
[Crossref]

Harris, S. E.

K.-J. Boller, A. Imamoğlu, and S. E. Harris, Phys. Rev. Lett. 66, 2593 (1991).
[Crossref]

He, K.

C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang, J.-Q. Ji, Y. Feng, S. Ji, X. Chen, J. Jia, X. Dai, Z. Fang, S.-C. Zhang, K. He, Y. Wang, L. Lu, X.-C. Ma, and Q.-K. Xue, Science 340, 167 (2013).
[Crossref]

Hossein, M. S.

A. B. Khanikaev, M. S. Hossein, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, Nat. Mater. 12, 233 (2012).
[Crossref]

Hu, J.

S.-C. Zhang and J. Hu, Science 294, 823 (2001).
[Crossref]

Hughes, T. L.

X.-L. Qi, T. L. Hughes, S. Raghu, and S.-C. Zhang, Phys. Rev. Lett. 102, 187001 (2009).
[Crossref]

Imamoglu, A.

K.-J. Boller, A. Imamoğlu, and S. E. Harris, Phys. Rev. Lett. 66, 2593 (1991).
[Crossref]

Inoue, J.-I.

J.-I. Inoue and A. Tanaka, Phys. Rev. Lett. 105, 017401 (2010).
[Crossref]

Ji, J.-Q.

C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang, J.-Q. Ji, Y. Feng, S. Ji, X. Chen, J. Jia, X. Dai, Z. Fang, S.-C. Zhang, K. He, Y. Wang, L. Lu, X.-C. Ma, and Q.-K. Xue, Science 340, 167 (2013).
[Crossref]

Ji, S.

C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang, J.-Q. Ji, Y. Feng, S. Ji, X. Chen, J. Jia, X. Dai, Z. Fang, S.-C. Zhang, K. He, Y. Wang, L. Lu, X.-C. Ma, and Q.-K. Xue, Science 340, 167 (2013).
[Crossref]

Jia, J.

C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang, J.-Q. Ji, Y. Feng, S. Ji, X. Chen, J. Jia, X. Dai, Z. Fang, S.-C. Zhang, K. He, Y. Wang, L. Lu, X.-C. Ma, and Q.-K. Xue, Science 340, 167 (2013).
[Crossref]

Jotzu, G.

G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, Nature 515, 237 (2014).
[Crossref]

Kane, C. L.

C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005).
[Crossref]

Kargarian, M.

A. B. Khanikaev, M. S. Hossein, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, Nat. Mater. 12, 233 (2012).
[Crossref]

Khanikaev, A. B.

A. B. Khanikaev, M. S. Hossein, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, Nat. Mater. 12, 233 (2012).
[Crossref]

Kitagawa, T.

T. Kitagawa, E. Berg, M. Rudner, and E. Demler, Phys. Rev. B 82, 235114 (2010).
[Crossref]

König, M.

M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L. W. Molenkamp, X.-L. Qi, and S.-C. Zhang, Science 318, 766 (2007).
[Crossref]

Lebrat, M.

G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, Nature 515, 237 (2014).
[Crossref]

Li, K.

C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang, J.-Q. Ji, Y. Feng, S. Ji, X. Chen, J. Jia, X. Dai, Z. Fang, S.-C. Zhang, K. He, Y. Wang, L. Lu, X.-C. Ma, and Q.-K. Xue, Science 340, 167 (2013).
[Crossref]

Lindner, N. H.

N. H. Lindner, G. Refael, and V. Galitski, Nat. Phys. 7, 490 (2011).
[Crossref]

Liu, R.-B.

D.-W. Wang, R.-B. Liu, S.-Y. Zhu, and M. O. Scully, Phys. Rev. Lett. 114, 043602 (2015).
[Crossref]

Lu, L.

C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang, J.-Q. Ji, Y. Feng, S. Ji, X. Chen, J. Jia, X. Dai, Z. Fang, S.-C. Zhang, K. He, Y. Wang, L. Lu, X.-C. Ma, and Q.-K. Xue, Science 340, 167 (2013).
[Crossref]

Lumer, Y.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, Nature 496, 196 (2013).
[Crossref]

Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, Phys. Rev. Lett. 111, 243905 (2013).
[Crossref]

Ma, X.-C.

C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang, J.-Q. Ji, Y. Feng, S. Ji, X. Chen, J. Jia, X. Dai, Z. Fang, S.-C. Zhang, K. He, Y. Wang, L. Lu, X.-C. Ma, and Q.-K. Xue, Science 340, 167 (2013).
[Crossref]

MacDonald, A. H.

A. B. Khanikaev, M. S. Hossein, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, Nat. Mater. 12, 233 (2012).
[Crossref]

Malpuech, G.

A. V. Nalitov, G. Malpuech, H. Terças, and D. D. Solnyshkov, Phys. Rev. Lett. 114, 026803 (2015).
[Crossref]

Mehendale, S. C.

V. B. Tiwari, S. Singh, H. S. Rawat, M. P. Singh, and S. C. Mehendale, J. Phys. B 43, 095503 (2010).
[Crossref]

Mele, E. J.

C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005).
[Crossref]

Messer, M.

G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, Nature 515, 237 (2014).
[Crossref]

Migdall, A.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, Nat. Photonics 7, 1001 (2013).
[Crossref]

Mittal, S.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, Nat. Photonics 7, 1001 (2013).
[Crossref]

Molenkamp, L. W.

M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L. W. Molenkamp, X.-L. Qi, and S.-C. Zhang, Science 318, 766 (2007).
[Crossref]

Nalitov, A. V.

A. V. Nalitov, G. Malpuech, H. Terças, and D. D. Solnyshkov, Phys. Rev. Lett. 114, 026803 (2015).
[Crossref]

Nolte, S.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, Nature 496, 196 (2013).
[Crossref]

Oka, T.

T. Oka and H. Aoki, Phys. Rev. B 79, 081406 (2009).
[Crossref]

Ooi, C. H. R.

M. O. Scully, E. S. Fry, C. H. R. Ooi, and K. Wódkiewicz, Phys. Rev. Lett. 96, 010501 (2006).
[Crossref]

Ou, Y.

C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang, J.-Q. Ji, Y. Feng, S. Ji, X. Chen, J. Jia, X. Dai, Z. Fang, S.-C. Zhang, K. He, Y. Wang, L. Lu, X.-C. Ma, and Q.-K. Xue, Science 340, 167 (2013).
[Crossref]

Pepper, M.

K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, 494 (1980).
[Crossref]

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Supplementary Material (1)

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Figures (4)

Fig. 1.
Fig. 1.

Realization of the Haldane model in a superradiance lattice of timed Dicke states in electromagnetically induced transparency. (a) Schematic configuration of the probe field kp, EIT coupling fields along k1/2/3, and diffraction fields along k+kp+k1k3 and kkp+k1k2, (b) the energy level diagram of the EIT coupling, probe, and scattering fields, (c) honeycomb structure of the SL. |ek and |mk correspond to the two sublattices. The NNN hopping Ω31 (red solid arrows), Ω23 (blue dotted arrows), and Ω12 (green dashed–dotted arrows) enclose a nonzero effective magnetic flux in momentum space. The arrow on the circle denotes the direction of the excitation current when the Chern number C=1.

Fig. 2.
Fig. 2.

Topological phases of the superradiance lattice. (a) The value of α [defined in Eq. (5)] as a function of the modulation phases of the coupling fields ϕ2 and ϕ3. ϕ1=0. The “+” and “−” signs are for the Chern numbers C=±1. The dashed line separates the two topological phases. (b) The two Chern numbers corresponding to two topologically distinctive configurations of the three phase angles ϕ1, ϕ2, and ϕ3 on a unit circle.

Fig. 3.
Fig. 3.

Topological phase transitions signatured by superradiance contrast. (a) The contrast η [defined in Eq. (6)] as a function of the modulation phases ϕ2 and ϕ3 with ϕ1=0. The red dotted lines are zero points. Δp=Δc=0. Ω=0.01. (b) The topological phase transition between C=1 and 0. Δp=Δc/2. Ω=0.01 (red), 0.1 (green), 0.5 (blue), and 1.0 (black). (c) The SL band structures in the topological phase transition. Ω=0.1. K and K are the two valley points. For (b) and (c), ϕ1=0, ϕ2=4π/3, and ϕ3=2π/3. For all three figures, Ωs=3, γe=1, and γm=0.1.

Fig. 4.
Fig. 4.

Real-space propagation of a light pulse through an atom ensemble driven into trivial or topological superradiance lattices. Diffraction of a weak Gaussian probe pulse in the xy plane for (a) a trivial SL with Ωd=0 and (b) a TSL with C=1. ϕ1=0, ϕ2=4π/3, ϕ3=2π/3, and Ω=0.01. The derived linear and nonlinear susceptibilities are χ(1)=i0.1410, χ+=χ=i0.0522 in (a) and χ(1)=i0.1446, χ+=i0.1057, χ=i0.0021 in (b). The parameters are such that Ωs=3, Δp=Δc=0, γe=1, and γm=0.1. The square denotes the area occupied by atoms. The three figures in each group are for scaled time t=1,80,100, successively. We assume the group velocity to be uniform everywhere.

Equations (6)

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|ekp=1Nj=1Neikp·rj|g1,g2,,ej,,gN,
H=Δc2k(|ekek||mkk1mkk1|)kl=13Ωl[|ekmkkl|+h.c.],
Ωl=Ωs+2Ωdcos(νdt+ϕl),
H=klj=13Ωlj(|ek+klkjek|+|mkk1mkk1+klkj|),
α=l=13sin(ϕl+1ϕl)=4l=13sinϕl+1ϕl2.
η=|ck+|2|ck|2|ck+|2+|ck|2.

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