Abstract

We propose a scheme of creating a tunable highly nonlinear defect in a one-dimensional photonic crystal. The defect consists of an atomic cell filled in with two isotopes of three-level atoms. The probe-field refractive index of the defect can be made parity–time (PT) symmetric, which is achieved by proper combination of a control field and of Stark shifts induced by a far-off-resonance field. In the PT-symmetric system, families of stable nonlinear defect modes can be formed by the probe field.

© 2013 Optical Society of America

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  1. W. C. K. Mak, B. A. Malomed, and P. L. Chu, Phys. Rev. E 67, 026608 (2003).
    [CrossRef]
  2. D. A. Zezyulin, Y. V. Kartashov, and V. V. Konotop, Opt. Lett. 36, 1200 (2011).
    [CrossRef]
  3. Y. V. Kartashov, V. V. Konotop, V. A. Vysloukh, and D. A. Zezyulin, in Spontaneous Symmetry Breaking, Self-Trapping, and Josephson Oscillations, Progress in Optical Science and Photonics (Springer, 2013), pp. 167–200.
  4. Y. K. Ha, Y. C. Yang, J. E. Kim, and H. Y. Park, Appl. Phys. Lett. 79, 15 (2001).
    [CrossRef]
  5. R. Ozaki, T. Matsui, M. Ozaki, and K. Yoshino, Appl. Phys. Lett. 82, 3593 (2003).
    [CrossRef]
  6. R. Ozaki, H. Miyoshi, M. Ozaki, and K. Yosjino, Mol. Cryst. Liq. Cryst. 433, 247 (2006).
  7. V. Ya. Zyryanov, S. A. Myslivets, V. A. Gunyakov, A. M. Parshin, V. G. Arkhipkin, V. F. Shabanov, and W. Lee, Opt. Express 18, 1283 (2010).
    [CrossRef]
  8. V. V. Konotop and V. Kuzmiak, Phys. Rev. B 64, 125120 (2001).
  9. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
    [CrossRef]
  10. C. M. Bender, Rep. Prog. Phys. 70, 947 (2007).
    [CrossRef]
  11. A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A 38, L171 (2005).
    [CrossRef]
  12. S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, Phys. Rev. A 85, 043826 (2012).
    [CrossRef]
  13. H. Wang and J. Wang, Opt. Express 19, 4030 (2011).
    [CrossRef]
  14. C. Hang, G. Huang, and V. V. Konotop, Phys. Rev. Lett. 110, 083604 (2013).
    [CrossRef]
  15. C. O’Brien, P. M. Anisimov, Y. Rostovtsev, and O. Kocharovskaya, Phys. Rev. A 84, 063835 (2011).
    [CrossRef]
  16. Z. J. Simmons, N. A. Proite, J. Miles, D. E. Sikes, and D. D. Yavuz, Phys. Rev. A 85, 053810 (2012).
    [CrossRef]
  17. V. S. Letokhov, Laser Control of Atoms and Molecules (Oxford University, 2007).
  18. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, Rev. Mod. Phys. 77, 633 (2005).
    [CrossRef]
  19. D. Steck, “85Rb D line data,” http://steck.us/alkalidata .
  20. J. P. Dowling and C. M. Bowden, J. Mod. Opt. 41, 345 (1994).
    [CrossRef]
  21. J. M. Bendickson, J. P. Dowling, and M. Scalora, Phys. Rev. E 53, 4107 (1996).
    [CrossRef]
  22. D. A. Zezyulin and V. V. Konotop, Phys. Rev. A 85, 043840 (2012).
    [CrossRef]
  23. N. Moiseyev, L. D. Carr, B. A. Malomed, and Y. B. Band, J. Phys. B 37, L193 (2004).

2013 (1)

C. Hang, G. Huang, and V. V. Konotop, Phys. Rev. Lett. 110, 083604 (2013).
[CrossRef]

2012 (3)

Z. J. Simmons, N. A. Proite, J. Miles, D. E. Sikes, and D. D. Yavuz, Phys. Rev. A 85, 053810 (2012).
[CrossRef]

S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, Phys. Rev. A 85, 043826 (2012).
[CrossRef]

D. A. Zezyulin and V. V. Konotop, Phys. Rev. A 85, 043840 (2012).
[CrossRef]

2011 (3)

2010 (2)

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

V. Ya. Zyryanov, S. A. Myslivets, V. A. Gunyakov, A. M. Parshin, V. G. Arkhipkin, V. F. Shabanov, and W. Lee, Opt. Express 18, 1283 (2010).
[CrossRef]

2007 (1)

C. M. Bender, Rep. Prog. Phys. 70, 947 (2007).
[CrossRef]

2006 (1)

R. Ozaki, H. Miyoshi, M. Ozaki, and K. Yosjino, Mol. Cryst. Liq. Cryst. 433, 247 (2006).

2005 (2)

A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A 38, L171 (2005).
[CrossRef]

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, Rev. Mod. Phys. 77, 633 (2005).
[CrossRef]

2004 (1)

N. Moiseyev, L. D. Carr, B. A. Malomed, and Y. B. Band, J. Phys. B 37, L193 (2004).

2003 (2)

W. C. K. Mak, B. A. Malomed, and P. L. Chu, Phys. Rev. E 67, 026608 (2003).
[CrossRef]

R. Ozaki, T. Matsui, M. Ozaki, and K. Yoshino, Appl. Phys. Lett. 82, 3593 (2003).
[CrossRef]

2001 (2)

Y. K. Ha, Y. C. Yang, J. E. Kim, and H. Y. Park, Appl. Phys. Lett. 79, 15 (2001).
[CrossRef]

V. V. Konotop and V. Kuzmiak, Phys. Rev. B 64, 125120 (2001).

1996 (1)

J. M. Bendickson, J. P. Dowling, and M. Scalora, Phys. Rev. E 53, 4107 (1996).
[CrossRef]

1994 (1)

J. P. Dowling and C. M. Bowden, J. Mod. Opt. 41, 345 (1994).
[CrossRef]

Anisimov, P. M.

C. O’Brien, P. M. Anisimov, Y. Rostovtsev, and O. Kocharovskaya, Phys. Rev. A 84, 063835 (2011).
[CrossRef]

Arkhipkin, V. G.

Band, Y. B.

N. Moiseyev, L. D. Carr, B. A. Malomed, and Y. B. Band, J. Phys. B 37, L193 (2004).

Bender, C. M.

C. M. Bender, Rep. Prog. Phys. 70, 947 (2007).
[CrossRef]

Bendickson, J. M.

J. M. Bendickson, J. P. Dowling, and M. Scalora, Phys. Rev. E 53, 4107 (1996).
[CrossRef]

Bowden, C. M.

J. P. Dowling and C. M. Bowden, J. Mod. Opt. 41, 345 (1994).
[CrossRef]

Carr, L. D.

N. Moiseyev, L. D. Carr, B. A. Malomed, and Y. B. Band, J. Phys. B 37, L193 (2004).

Christodoulides, D. N.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

Chu, P. L.

W. C. K. Mak, B. A. Malomed, and P. L. Chu, Phys. Rev. E 67, 026608 (2003).
[CrossRef]

Delgado, F.

A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A 38, L171 (2005).
[CrossRef]

Dowling, J. P.

J. M. Bendickson, J. P. Dowling, and M. Scalora, Phys. Rev. E 53, 4107 (1996).
[CrossRef]

J. P. Dowling and C. M. Bowden, J. Mod. Opt. 41, 345 (1994).
[CrossRef]

El-Ganainy, R.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

Fleischhauer, M.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, Rev. Mod. Phys. 77, 633 (2005).
[CrossRef]

Gunyakov, V. A.

Ha, Y. K.

Y. K. Ha, Y. C. Yang, J. E. Kim, and H. Y. Park, Appl. Phys. Lett. 79, 15 (2001).
[CrossRef]

Hang, C.

C. Hang, G. Huang, and V. V. Konotop, Phys. Rev. Lett. 110, 083604 (2013).
[CrossRef]

Hu, S.

S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, Phys. Rev. A 85, 043826 (2012).
[CrossRef]

Hu, W.

S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, Phys. Rev. A 85, 043826 (2012).
[CrossRef]

Huang, G.

C. Hang, G. Huang, and V. V. Konotop, Phys. Rev. Lett. 110, 083604 (2013).
[CrossRef]

Imamoglu, A.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, Rev. Mod. Phys. 77, 633 (2005).
[CrossRef]

Kartashov, Y. V.

D. A. Zezyulin, Y. V. Kartashov, and V. V. Konotop, Opt. Lett. 36, 1200 (2011).
[CrossRef]

Y. V. Kartashov, V. V. Konotop, V. A. Vysloukh, and D. A. Zezyulin, in Spontaneous Symmetry Breaking, Self-Trapping, and Josephson Oscillations, Progress in Optical Science and Photonics (Springer, 2013), pp. 167–200.

Kim, J. E.

Y. K. Ha, Y. C. Yang, J. E. Kim, and H. Y. Park, Appl. Phys. Lett. 79, 15 (2001).
[CrossRef]

Kip, D.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

Kocharovskaya, O.

C. O’Brien, P. M. Anisimov, Y. Rostovtsev, and O. Kocharovskaya, Phys. Rev. A 84, 063835 (2011).
[CrossRef]

Konotop, V. V.

C. Hang, G. Huang, and V. V. Konotop, Phys. Rev. Lett. 110, 083604 (2013).
[CrossRef]

D. A. Zezyulin and V. V. Konotop, Phys. Rev. A 85, 043840 (2012).
[CrossRef]

D. A. Zezyulin, Y. V. Kartashov, and V. V. Konotop, Opt. Lett. 36, 1200 (2011).
[CrossRef]

V. V. Konotop and V. Kuzmiak, Phys. Rev. B 64, 125120 (2001).

Y. V. Kartashov, V. V. Konotop, V. A. Vysloukh, and D. A. Zezyulin, in Spontaneous Symmetry Breaking, Self-Trapping, and Josephson Oscillations, Progress in Optical Science and Photonics (Springer, 2013), pp. 167–200.

Kuzmiak, V.

V. V. Konotop and V. Kuzmiak, Phys. Rev. B 64, 125120 (2001).

Lee, W.

Letokhov, V. S.

V. S. Letokhov, Laser Control of Atoms and Molecules (Oxford University, 2007).

Lu, D.

S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, Phys. Rev. A 85, 043826 (2012).
[CrossRef]

Ma, X.

S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, Phys. Rev. A 85, 043826 (2012).
[CrossRef]

Mak, W. C. K.

W. C. K. Mak, B. A. Malomed, and P. L. Chu, Phys. Rev. E 67, 026608 (2003).
[CrossRef]

Makris, K. G.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

Malomed, B. A.

N. Moiseyev, L. D. Carr, B. A. Malomed, and Y. B. Band, J. Phys. B 37, L193 (2004).

W. C. K. Mak, B. A. Malomed, and P. L. Chu, Phys. Rev. E 67, 026608 (2003).
[CrossRef]

Marangos, J. P.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, Rev. Mod. Phys. 77, 633 (2005).
[CrossRef]

Matsui, T.

R. Ozaki, T. Matsui, M. Ozaki, and K. Yoshino, Appl. Phys. Lett. 82, 3593 (2003).
[CrossRef]

Miles, J.

Z. J. Simmons, N. A. Proite, J. Miles, D. E. Sikes, and D. D. Yavuz, Phys. Rev. A 85, 053810 (2012).
[CrossRef]

Miyoshi, H.

R. Ozaki, H. Miyoshi, M. Ozaki, and K. Yosjino, Mol. Cryst. Liq. Cryst. 433, 247 (2006).

Moiseyev, N.

N. Moiseyev, L. D. Carr, B. A. Malomed, and Y. B. Band, J. Phys. B 37, L193 (2004).

Muga, J. G.

A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A 38, L171 (2005).
[CrossRef]

Myslivets, S. A.

O’Brien, C.

C. O’Brien, P. M. Anisimov, Y. Rostovtsev, and O. Kocharovskaya, Phys. Rev. A 84, 063835 (2011).
[CrossRef]

Ozaki, M.

R. Ozaki, H. Miyoshi, M. Ozaki, and K. Yosjino, Mol. Cryst. Liq. Cryst. 433, 247 (2006).

R. Ozaki, T. Matsui, M. Ozaki, and K. Yoshino, Appl. Phys. Lett. 82, 3593 (2003).
[CrossRef]

Ozaki, R.

R. Ozaki, H. Miyoshi, M. Ozaki, and K. Yosjino, Mol. Cryst. Liq. Cryst. 433, 247 (2006).

R. Ozaki, T. Matsui, M. Ozaki, and K. Yoshino, Appl. Phys. Lett. 82, 3593 (2003).
[CrossRef]

Park, H. Y.

Y. K. Ha, Y. C. Yang, J. E. Kim, and H. Y. Park, Appl. Phys. Lett. 79, 15 (2001).
[CrossRef]

Parshin, A. M.

Proite, N. A.

Z. J. Simmons, N. A. Proite, J. Miles, D. E. Sikes, and D. D. Yavuz, Phys. Rev. A 85, 053810 (2012).
[CrossRef]

Rostovtsev, Y.

C. O’Brien, P. M. Anisimov, Y. Rostovtsev, and O. Kocharovskaya, Phys. Rev. A 84, 063835 (2011).
[CrossRef]

Ruschhaupt, A.

A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A 38, L171 (2005).
[CrossRef]

Rüter, C. E.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

Scalora, M.

J. M. Bendickson, J. P. Dowling, and M. Scalora, Phys. Rev. E 53, 4107 (1996).
[CrossRef]

Segev, M.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

Shabanov, V. F.

Sikes, D. E.

Z. J. Simmons, N. A. Proite, J. Miles, D. E. Sikes, and D. D. Yavuz, Phys. Rev. A 85, 053810 (2012).
[CrossRef]

Simmons, Z. J.

Z. J. Simmons, N. A. Proite, J. Miles, D. E. Sikes, and D. D. Yavuz, Phys. Rev. A 85, 053810 (2012).
[CrossRef]

Vysloukh, V. A.

Y. V. Kartashov, V. V. Konotop, V. A. Vysloukh, and D. A. Zezyulin, in Spontaneous Symmetry Breaking, Self-Trapping, and Josephson Oscillations, Progress in Optical Science and Photonics (Springer, 2013), pp. 167–200.

Wang, H.

Wang, J.

Ya. Zyryanov, V.

Yang, Y. C.

Y. K. Ha, Y. C. Yang, J. E. Kim, and H. Y. Park, Appl. Phys. Lett. 79, 15 (2001).
[CrossRef]

Yavuz, D. D.

Z. J. Simmons, N. A. Proite, J. Miles, D. E. Sikes, and D. D. Yavuz, Phys. Rev. A 85, 053810 (2012).
[CrossRef]

Yoshino, K.

R. Ozaki, T. Matsui, M. Ozaki, and K. Yoshino, Appl. Phys. Lett. 82, 3593 (2003).
[CrossRef]

Yosjino, K.

R. Ozaki, H. Miyoshi, M. Ozaki, and K. Yosjino, Mol. Cryst. Liq. Cryst. 433, 247 (2006).

Zezyulin, D. A.

D. A. Zezyulin and V. V. Konotop, Phys. Rev. A 85, 043840 (2012).
[CrossRef]

D. A. Zezyulin, Y. V. Kartashov, and V. V. Konotop, Opt. Lett. 36, 1200 (2011).
[CrossRef]

Y. V. Kartashov, V. V. Konotop, V. A. Vysloukh, and D. A. Zezyulin, in Spontaneous Symmetry Breaking, Self-Trapping, and Josephson Oscillations, Progress in Optical Science and Photonics (Springer, 2013), pp. 167–200.

Zheng, Y.

S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, Phys. Rev. A 85, 043826 (2012).
[CrossRef]

Appl. Phys. Lett. (2)

Y. K. Ha, Y. C. Yang, J. E. Kim, and H. Y. Park, Appl. Phys. Lett. 79, 15 (2001).
[CrossRef]

R. Ozaki, T. Matsui, M. Ozaki, and K. Yoshino, Appl. Phys. Lett. 82, 3593 (2003).
[CrossRef]

J. Mod. Opt. (1)

J. P. Dowling and C. M. Bowden, J. Mod. Opt. 41, 345 (1994).
[CrossRef]

J. Phys. A (1)

A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A 38, L171 (2005).
[CrossRef]

J. Phys. B (1)

N. Moiseyev, L. D. Carr, B. A. Malomed, and Y. B. Band, J. Phys. B 37, L193 (2004).

Mol. Cryst. Liq. Cryst. (1)

R. Ozaki, H. Miyoshi, M. Ozaki, and K. Yosjino, Mol. Cryst. Liq. Cryst. 433, 247 (2006).

Nat. Phys. (1)

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. A (4)

S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, Phys. Rev. A 85, 043826 (2012).
[CrossRef]

D. A. Zezyulin and V. V. Konotop, Phys. Rev. A 85, 043840 (2012).
[CrossRef]

C. O’Brien, P. M. Anisimov, Y. Rostovtsev, and O. Kocharovskaya, Phys. Rev. A 84, 063835 (2011).
[CrossRef]

Z. J. Simmons, N. A. Proite, J. Miles, D. E. Sikes, and D. D. Yavuz, Phys. Rev. A 85, 053810 (2012).
[CrossRef]

Phys. Rev. B (1)

V. V. Konotop and V. Kuzmiak, Phys. Rev. B 64, 125120 (2001).

Phys. Rev. E (2)

J. M. Bendickson, J. P. Dowling, and M. Scalora, Phys. Rev. E 53, 4107 (1996).
[CrossRef]

W. C. K. Mak, B. A. Malomed, and P. L. Chu, Phys. Rev. E 67, 026608 (2003).
[CrossRef]

Phys. Rev. Lett. (1)

C. Hang, G. Huang, and V. V. Konotop, Phys. Rev. Lett. 110, 083604 (2013).
[CrossRef]

Rep. Prog. Phys. (1)

C. M. Bender, Rep. Prog. Phys. 70, 947 (2007).
[CrossRef]

Rev. Mod. Phys. (1)

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, Rev. Mod. Phys. 77, 633 (2005).
[CrossRef]

Other (3)

D. Steck, “85Rb D line data,” http://steck.us/alkalidata .

V. S. Letokhov, Laser Control of Atoms and Molecules (Oxford University, 2007).

Y. V. Kartashov, V. V. Konotop, V. A. Vysloukh, and D. A. Zezyulin, in Spontaneous Symmetry Breaking, Self-Trapping, and Josephson Oscillations, Progress in Optical Science and Photonics (Springer, 2013), pp. 167–200.

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Figures (2)

Fig. 1.
Fig. 1.

(a) Geometry of the photonic crystal, the energy-level diagram, and Raman resonance scheme of a binary mixture. The initially populated levels are indicated by the black dots. (b) Distribution of real n0r (solid line) and imaginary n0i (dashed line) parts of the linear refractive index induced by the fields (4a) and (4b).

Fig. 2.
Fig. 2.

(a) Two families of nonlinear modes bifurcating from the eigenvalues b0,1 (see the text). The shaded domain corresponds to propagation constants belonging to the continuous spectrum. (b) The amplitude |u| (solid line) and the current S (dashed line) for the nonlinear mode with b=3. The shaded domain corresponds to the cladding. Evolution of the nonlinear mode intensity |u|2 for the (c) PT-symmetric and (d) non PT-symmetric refractive indexes.

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

iρ˙ggs=iΓegρeesΩ¯pρegs+Ωpρ¯egs,
iρ˙aas=iΓeaρeesΩ¯cρeas+Ωcρ¯eas,
iρ˙ees=i(Γeg+Γea)ρees+Ω¯pρegsΩpρ¯egs+Ω¯cρeasΩcρ¯eas,
iρ˙ags=dagsρags+Ωpρ¯easΩ¯cρegs,
iρ˙egs=degsρegs+Ωp(ρeesρggs)Ωcρags,
iρ˙eas=deasρeas+Ωc(ρeesρaas)Ωpρ¯ags.
(dagsΩ¯cΩcdegs)(ρag,msρeg,ms)=Ωp(ρ¯ea,m1sρee,m1sρgg,m1s),
(iΓegiΓeg00iΓeaiΓeaΩ¯cΩcΩc2Ωcdeas0Ω¯c2Ω¯c0d¯eas)(ρgg,msρaa,msρea,msρae,ms)=Mm1,
Mm=(Ωpρ¯eg,msΩ¯pρeg,ms0Ωpρ¯ag,msΩ¯pρag,ms).
χp,1=peg2ε0(N1A1+N2A2),
χp,3=peg2ϵ0Γeas=12NsdagsCsBs|Ωc|2dagsdegs
Bs=ΓeadeasΓegIm(As)|Ωc|2Im(Ds),
Cs=Im(deasDs6ΓeaΓegAs+|deas|2|Ωc|2ΓeaΓegAs),
D1=1dag1deg1|Ωc|2,D2=deg2d¯ea2(|Ωc|2dag2deg2).
Ωc(ξ)=(2.5+0.025ξ+2.4741×104ξ2)Γeg,
ES(ξ)=(1.93940.2542ξ)E0,
χp,1(ξ){χ˜0+iχ˜1ξ+χ˜2ξ2,|ξ|1,χclad,|ξ|>1,
iuζ+2uξ2(iV1ξ+V2ξ2)u|u|2u=0,|ξ|<1,
iuζ+2uξ2V0u=0|ξ|1.

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