Abstract

We consider the coupling effect between interdot tunneling coupling and external optical control field to study the linear optical property and the formation of temporal optical solitons in a quantum dot molecules system, analytically. The results show that the double tunneling induced transparency (TIT) windows are appeared in the absorption curve of probe field because of the formation of dynamic Stark splitting and quantum destructive interference effect from the two upper levels. Interestingly, the width of the TIT window becomes wider with the increasing intensity of the optical control field. We also find that the Kerr nonlinear effect of the probe field can be modulated effectively through coherent control both the control field and the interdot tunneling coupling in this system. Meanwhile, we demonstrate that the formation of dark or bright solitons can be practical regulated by varying the intensity of the optical control field.

© 2013 OSA

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    [CrossRef]

2012

K. Mller, A. Bechtold, C. Ruppert, M. Zecherle, G. Reithmaier, M. Bichler, H. J. Krenner, G. Abstreiter, A. W. Holleitner, J. M. Villas-Bôas, M. Betz, and J. J. Finley, “Electrical control of interdot electron tunneling in a double InGaAs quantum-dot nanostructure,” Phys. Rev. Lett.108(19), 197402(1–4) (2012).

H. S. Borges, L. Sanz, J. M. Villas-Boas, O. O. Diniz Neto, and A. M. Alcalde, “Tunneling induced transparency and slow light in quantum dot molecules,” Phys. Rev. B85(11), 115425 (2012).
[CrossRef]

X. Hao, J. Wu, and Y Wang, “Steady-state absorptionCdispersion properties and four-wave mixing process in a quantum dot nanostructure,” J. Opt. Soc. Am. B29(3), 420–428 (2012).
[CrossRef]

H. Sun, S. Fan, X. Feng, C. Wu, S. Gong, G. Huang, and C. H. Oh, “Strongly interacting photons in asymmetric quantum well via resonant tunneling,” Opt. Express20(8), 8485–8495 (2012).
[CrossRef] [PubMed]

2011

2010

2009

H. Zhang, D. Tang, L. Zhao, and X. Wu, “Observation of polarization domain wall solitons in weakly birefringent cavity fiber lasers,” Phys. Rev. B80(5), 052302(1–4) (2009).

H. Zhang, D. Tang, L. Zhao, and X. Wu, “Dark pulse emission of a fiber laser,” Phys. Rev. A80(4), 045803(1–4) (2009).
[CrossRef]

V. V. Nikolaev, N. S. Averkiev, M. M. Sobolev, I. M. Gadzhiyev, I. O. Bakshaev, M. S. Buyalo, and E. L. Portnoi, “Tunnel coupling in an ensemble of vertically aligned quantum dot at room temperature,” Phys. Rev. B80(20), 205304 (2009).
[CrossRef]

2008

W. Yang and R. Lee, “Slow optical solitons via intersubband transitions in a semiconductor quantum well,” Europhys. Lett.83(1), 14002(1–6) (2008).
[CrossRef]

M. Ohtsu, T. Kawazoe, T. Yatsui, and M. Naruse, “Nanophotonics: application of dressed photons to novel photonic devices and systems,” IEEE J. Sel. Top. Quantum Electron.14(6), 1404–1417 (2008).
[CrossRef]

S. Marcinkevicius, A. Gushterov, and J. P. Reithmaier, “Transient electromagnetically induced transparency in self-assembled quantum dots,” Appl. Phys. Lett.92(4), 041113(1–3) (2008).
[CrossRef]

W. Yang and R. Lee, “Controllable entanglement and polarization phase gate in coupled double quantum-well structures,” Opt. Express16(22), 17161–17170 (2008).
[CrossRef] [PubMed]

2007

2006

H. Su and S. L. Chuang, “Room-temperature slow light with semiconductor quantum-dot devices,” Opt. Lett.31(2), 271–273 (2006).
[CrossRef] [PubMed]

P. C. Peng, C. T. Lin, H. C. Kuo, W. K. Tsai, J. N. Liu, S. Chi, S. C. Wang, G. Lin, H. P. Yang, K. F. Lin, and J. Y. Chi, “Tunable slow light device using quantum dot semiconductor laser,” Opt. Express14(26), 12880–12886 (2006).
[CrossRef] [PubMed]

R. Espiau de Lamaëstre and H. Bernas1, “Significance of lognormal nanocrystal size distributions,” Phys. Rev. B73(12), 125317(1–18) (2006).
[CrossRef]

C. Yuan and K. Zhu, “Voltage-controlled slow light in asymmetry double quantum dots,” Appl. Phys. Lett.89(5), 052115 (1–3) (2006).
[CrossRef]

G. J. Beirne, C. Hermannstädter, L. Wang, A. Rastelli, O. G. Schmidt, and P. Michler, “Quantum light emission of two lateral tunnel-coupled (In,Ga)As/GaAs quantum dots controlled by a tunable static electric field,” Phys. Rev. Lett.96(13), 137401(1–4) (2006).
[CrossRef]

2005

B. Krause, T. H. Metzger, A Rastelli, R. Songmuang, S. Kiravittaya, and O. G. Schmidt, “Shape, strain, and ordering of lateral InAs quantum dot molecules,” Phys. Rev. B72(8), 085339(1–12) (2005).
[CrossRef]

K. Kobayashi, S. Sangu, T. Kawazoe, and M. Ohtsu, “Exciton dynamics and logic operations in a near-field optically coupled quantum-dot system,” J. Lumin.112, 117–121 (2005).
[CrossRef]

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys.77(2), 633–673 (2005).
[CrossRef]

2004

2003

X. Li, Y. Wu, D. Steel, D. Gammon, T. H. Stievater, D. S. Katzer, D. Park, C. Piermarocchi, and L. J. Sham, “An all-optical quantum gate in a semiconductor quantum dot,” Science301(5634), 809–811 (2003).
[CrossRef] [PubMed]

C. J. Chang-Hasnain, P. C Ku, J. Kim, and S. L. Chuang, “Variable optical buffer using slow light in semiconductor nanostructure,” Proc. of IEEE91(11), 1884–1897 (2003).
[CrossRef]

P. Borri, W. Langbein, U. Woggon, M. Schwab, M. Bayer, S. Fafard, Z. Wasilewski, and P. Hawrylak, “Exciton dephasing in quantum dot molecules,” Phys. Rev. Lett.91(26), 267401(1–4) (2003).
[CrossRef]

R. Songmuang, S. Kiravittaya, and O. G. Schmidt, “Formation of lateral quantum dot molecules around self-assembled nanoholes,” Appl. Phys. Lett.82(17), 2892(1–3) (2003).
[CrossRef]

2001

M. Bayer, P. Hawrylak, K. Hinzer, S. Fafard, M. Korkusinski, Z. R. Wasilewski, O. Stern1, and A. Forchel, “Coupling and entangling of quantum states in quantum dot molecules,” Science291(5503), 451–453 (2001).
[CrossRef] [PubMed]

Abstreiter, G.

K. Mller, A. Bechtold, C. Ruppert, M. Zecherle, G. Reithmaier, M. Bichler, H. J. Krenner, G. Abstreiter, A. W. Holleitner, J. M. Villas-Bôas, M. Betz, and J. J. Finley, “Electrical control of interdot electron tunneling in a double InGaAs quantum-dot nanostructure,” Phys. Rev. Lett.108(19), 197402(1–4) (2012).

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics4th ed. (Academic, 2009).

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

Alcalde, A. M.

H. S. Borges, L. Sanz, J. M. Villas-Boas, O. O. Diniz Neto, and A. M. Alcalde, “Tunneling induced transparency and slow light in quantum dot molecules,” Phys. Rev. B85(11), 115425 (2012).
[CrossRef]

Averkiev, N. S.

V. V. Nikolaev, N. S. Averkiev, M. M. Sobolev, I. M. Gadzhiyev, I. O. Bakshaev, M. S. Buyalo, and E. L. Portnoi, “Tunnel coupling in an ensemble of vertically aligned quantum dot at room temperature,” Phys. Rev. B80(20), 205304 (2009).
[CrossRef]

Bakshaev, I. O.

V. V. Nikolaev, N. S. Averkiev, M. M. Sobolev, I. M. Gadzhiyev, I. O. Bakshaev, M. S. Buyalo, and E. L. Portnoi, “Tunnel coupling in an ensemble of vertically aligned quantum dot at room temperature,” Phys. Rev. B80(20), 205304 (2009).
[CrossRef]

Bayer, M.

P. Borri, W. Langbein, U. Woggon, M. Schwab, M. Bayer, S. Fafard, Z. Wasilewski, and P. Hawrylak, “Exciton dephasing in quantum dot molecules,” Phys. Rev. Lett.91(26), 267401(1–4) (2003).
[CrossRef]

M. Bayer, P. Hawrylak, K. Hinzer, S. Fafard, M. Korkusinski, Z. R. Wasilewski, O. Stern1, and A. Forchel, “Coupling and entangling of quantum states in quantum dot molecules,” Science291(5503), 451–453 (2001).
[CrossRef] [PubMed]

Bechtold, A.

K. Mller, A. Bechtold, C. Ruppert, M. Zecherle, G. Reithmaier, M. Bichler, H. J. Krenner, G. Abstreiter, A. W. Holleitner, J. M. Villas-Bôas, M. Betz, and J. J. Finley, “Electrical control of interdot electron tunneling in a double InGaAs quantum-dot nanostructure,” Phys. Rev. Lett.108(19), 197402(1–4) (2012).

Beirne, G. J.

G. J. Beirne, C. Hermannstädter, L. Wang, A. Rastelli, O. G. Schmidt, and P. Michler, “Quantum light emission of two lateral tunnel-coupled (In,Ga)As/GaAs quantum dots controlled by a tunable static electric field,” Phys. Rev. Lett.96(13), 137401(1–4) (2006).
[CrossRef]

Bernas1, H.

R. Espiau de Lamaëstre and H. Bernas1, “Significance of lognormal nanocrystal size distributions,” Phys. Rev. B73(12), 125317(1–18) (2006).
[CrossRef]

Betz, M.

K. Mller, A. Bechtold, C. Ruppert, M. Zecherle, G. Reithmaier, M. Bichler, H. J. Krenner, G. Abstreiter, A. W. Holleitner, J. M. Villas-Bôas, M. Betz, and J. J. Finley, “Electrical control of interdot electron tunneling in a double InGaAs quantum-dot nanostructure,” Phys. Rev. Lett.108(19), 197402(1–4) (2012).

Bichler, M.

K. Mller, A. Bechtold, C. Ruppert, M. Zecherle, G. Reithmaier, M. Bichler, H. J. Krenner, G. Abstreiter, A. W. Holleitner, J. M. Villas-Bôas, M. Betz, and J. J. Finley, “Electrical control of interdot electron tunneling in a double InGaAs quantum-dot nanostructure,” Phys. Rev. Lett.108(19), 197402(1–4) (2012).

Borges, H. S.

H. S. Borges, L. Sanz, J. M. Villas-Boas, O. O. Diniz Neto, and A. M. Alcalde, “Tunneling induced transparency and slow light in quantum dot molecules,” Phys. Rev. B85(11), 115425 (2012).
[CrossRef]

Borri, P.

P. Borri, W. Langbein, U. Woggon, M. Schwab, M. Bayer, S. Fafard, Z. Wasilewski, and P. Hawrylak, “Exciton dephasing in quantum dot molecules,” Phys. Rev. Lett.91(26), 267401(1–4) (2003).
[CrossRef]

Buyalo, M. S.

V. V. Nikolaev, N. S. Averkiev, M. M. Sobolev, I. M. Gadzhiyev, I. O. Bakshaev, M. S. Buyalo, and E. L. Portnoi, “Tunnel coupling in an ensemble of vertically aligned quantum dot at room temperature,” Phys. Rev. B80(20), 205304 (2009).
[CrossRef]

Chang-Hasnain, C. J.

C. J. Chang-Hasnain, P. C Ku, J. Kim, and S. L. Chuang, “Variable optical buffer using slow light in semiconductor nanostructure,” Proc. of IEEE91(11), 1884–1897 (2003).
[CrossRef]

Chen, A.

W. Yang, A. Chen, R. Lee, and Y Wu, “Matched slow optical soliton pairs via biexciton coherence in quantum dots,” Phys. Rev. A84(1), 013835(1–11) (2011).
[CrossRef]

Chi, J. Y.

Chi, S.

Chuang, S. L.

H. Su and S. L. Chuang, “Room-temperature slow light with semiconductor quantum-dot devices,” Opt. Lett.31(2), 271–273 (2006).
[CrossRef] [PubMed]

C. J. Chang-Hasnain, P. C Ku, J. Kim, and S. L. Chuang, “Variable optical buffer using slow light in semiconductor nanostructure,” Proc. of IEEE91(11), 1884–1897 (2003).
[CrossRef]

Deng, L.

Ding, J.

Diniz Neto, O. O.

H. S. Borges, L. Sanz, J. M. Villas-Boas, O. O. Diniz Neto, and A. M. Alcalde, “Tunneling induced transparency and slow light in quantum dot molecules,” Phys. Rev. B85(11), 115425 (2012).
[CrossRef]

Espiau de Lamaëstre, R.

R. Espiau de Lamaëstre and H. Bernas1, “Significance of lognormal nanocrystal size distributions,” Phys. Rev. B73(12), 125317(1–18) (2006).
[CrossRef]

Fafard, S.

P. Borri, W. Langbein, U. Woggon, M. Schwab, M. Bayer, S. Fafard, Z. Wasilewski, and P. Hawrylak, “Exciton dephasing in quantum dot molecules,” Phys. Rev. Lett.91(26), 267401(1–4) (2003).
[CrossRef]

M. Bayer, P. Hawrylak, K. Hinzer, S. Fafard, M. Korkusinski, Z. R. Wasilewski, O. Stern1, and A. Forchel, “Coupling and entangling of quantum states in quantum dot molecules,” Science291(5503), 451–453 (2001).
[CrossRef] [PubMed]

Fan, S.

Feng, X.

Finley, J. J.

K. Mller, A. Bechtold, C. Ruppert, M. Zecherle, G. Reithmaier, M. Bichler, H. J. Krenner, G. Abstreiter, A. W. Holleitner, J. M. Villas-Bôas, M. Betz, and J. J. Finley, “Electrical control of interdot electron tunneling in a double InGaAs quantum-dot nanostructure,” Phys. Rev. Lett.108(19), 197402(1–4) (2012).

Fleischhauer, M.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys.77(2), 633–673 (2005).
[CrossRef]

Forchel, A.

M. Bayer, P. Hawrylak, K. Hinzer, S. Fafard, M. Korkusinski, Z. R. Wasilewski, O. Stern1, and A. Forchel, “Coupling and entangling of quantum states in quantum dot molecules,” Science291(5503), 451–453 (2001).
[CrossRef] [PubMed]

Fujita, H.

Gadzhiyev, I. M.

V. V. Nikolaev, N. S. Averkiev, M. M. Sobolev, I. M. Gadzhiyev, I. O. Bakshaev, M. S. Buyalo, and E. L. Portnoi, “Tunnel coupling in an ensemble of vertically aligned quantum dot at room temperature,” Phys. Rev. B80(20), 205304 (2009).
[CrossRef]

Gammon, D.

X. Li, Y. Wu, D. Steel, D. Gammon, T. H. Stievater, D. S. Katzer, D. Park, C. Piermarocchi, and L. J. Sham, “An all-optical quantum gate in a semiconductor quantum dot,” Science301(5634), 809–811 (2003).
[CrossRef] [PubMed]

Gong, S.

H. Sun, S. Fan, X. Feng, C. Wu, S. Gong, G. Huang, and C. H. Oh, “Strongly interacting photons in asymmetric quantum well via resonant tunneling,” Opt. Express20(8), 8485–8495 (2012).
[CrossRef] [PubMed]

Y. Qi, F. Zhou, T. Huang, Y. Niu, and S. Gong, “Spatial vector solitons in a four-level tripod-type atomic system,” Phys. Rev. A84(2), 023814(1–6) (2011)
[CrossRef]

Gushterov, A.

S. Marcinkevicius, A. Gushterov, and J. P. Reithmaier, “Transient electromagnetically induced transparency in self-assembled quantum dots,” Appl. Phys. Lett.92(4), 041113(1–3) (2008).
[CrossRef]

Hang, C.

Hao, X.

Hasegawa, A.

A. Hasegawa and M. Matsumoto, Optical Solitons in Fibers (Springer, 2003).
[CrossRef]

Hawrylak, P.

P. Borri, W. Langbein, U. Woggon, M. Schwab, M. Bayer, S. Fafard, Z. Wasilewski, and P. Hawrylak, “Exciton dephasing in quantum dot molecules,” Phys. Rev. Lett.91(26), 267401(1–4) (2003).
[CrossRef]

M. Bayer, P. Hawrylak, K. Hinzer, S. Fafard, M. Korkusinski, Z. R. Wasilewski, O. Stern1, and A. Forchel, “Coupling and entangling of quantum states in quantum dot molecules,” Science291(5503), 451–453 (2001).
[CrossRef] [PubMed]

He, Z.

Hermannstädter, C.

G. J. Beirne, C. Hermannstädter, L. Wang, A. Rastelli, O. G. Schmidt, and P. Michler, “Quantum light emission of two lateral tunnel-coupled (In,Ga)As/GaAs quantum dots controlled by a tunable static electric field,” Phys. Rev. Lett.96(13), 137401(1–4) (2006).
[CrossRef]

Hinzer, K.

M. Bayer, P. Hawrylak, K. Hinzer, S. Fafard, M. Korkusinski, Z. R. Wasilewski, O. Stern1, and A. Forchel, “Coupling and entangling of quantum states in quantum dot molecules,” Science291(5503), 451–453 (2001).
[CrossRef] [PubMed]

Hoga, M.

Holleitner, A. W.

K. Mller, A. Bechtold, C. Ruppert, M. Zecherle, G. Reithmaier, M. Bichler, H. J. Krenner, G. Abstreiter, A. W. Holleitner, J. M. Villas-Bôas, M. Betz, and J. J. Finley, “Electrical control of interdot electron tunneling in a double InGaAs quantum-dot nanostructure,” Phys. Rev. Lett.108(19), 197402(1–4) (2012).

Huang, G.

Huang, T.

Y. Qi, F. Zhou, T. Huang, Y. Niu, and S. Gong, “Spatial vector solitons in a four-level tripod-type atomic system,” Phys. Rev. A84(2), 023814(1–6) (2011)
[CrossRef]

Imamoglu, A.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys.77(2), 633–673 (2005).
[CrossRef]

Kaer Nielsen, P.

Katzer, D. S.

X. Li, Y. Wu, D. Steel, D. Gammon, T. H. Stievater, D. S. Katzer, D. Park, C. Piermarocchi, and L. J. Sham, “An all-optical quantum gate in a semiconductor quantum dot,” Science301(5634), 809–811 (2003).
[CrossRef] [PubMed]

Kawazoe, T.

N. Tate, M. Naruse, W. Nomura, T. Kawazoe, T. Yatsui, M. Hoga, Y. Ohyagi, Y. Sekine, H. Fujita, and M. Ohtsu, “Demonstration of modulatable optical near-field interactions between dispersed resonant quantum dots,” Opt. Express19(19), 18260–18271 (2011).
[CrossRef] [PubMed]

M. Ohtsu, T. Kawazoe, T. Yatsui, and M. Naruse, “Nanophotonics: application of dressed photons to novel photonic devices and systems,” IEEE J. Sel. Top. Quantum Electron.14(6), 1404–1417 (2008).
[CrossRef]

K. Kobayashi, S. Sangu, T. Kawazoe, and M. Ohtsu, “Exciton dynamics and logic operations in a near-field optically coupled quantum-dot system,” J. Lumin.112, 117–121 (2005).
[CrossRef]

M. Ohtsu, K. Kobayashi, T. Kawazoe, T. Yatsui, and M. Naruse, Principles of Nanophotonics (Taylor and Francis, 2008).
[CrossRef]

Kim, J.

C. J. Chang-Hasnain, P. C Ku, J. Kim, and S. L. Chuang, “Variable optical buffer using slow light in semiconductor nanostructure,” Proc. of IEEE91(11), 1884–1897 (2003).
[CrossRef]

Kiravittaya, S.

B. Krause, T. H. Metzger, A Rastelli, R. Songmuang, S. Kiravittaya, and O. G. Schmidt, “Shape, strain, and ordering of lateral InAs quantum dot molecules,” Phys. Rev. B72(8), 085339(1–12) (2005).
[CrossRef]

R. Songmuang, S. Kiravittaya, and O. G. Schmidt, “Formation of lateral quantum dot molecules around self-assembled nanoholes,” Appl. Phys. Lett.82(17), 2892(1–3) (2003).
[CrossRef]

Kivshar, Y. S.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

Knize, R.

Kobayashi, K.

K. Kobayashi, S. Sangu, T. Kawazoe, and M. Ohtsu, “Exciton dynamics and logic operations in a near-field optically coupled quantum-dot system,” J. Lumin.112, 117–121 (2005).
[CrossRef]

M. Ohtsu, K. Kobayashi, T. Kawazoe, T. Yatsui, and M. Naruse, Principles of Nanophotonics (Taylor and Francis, 2008).
[CrossRef]

Korkusinski, M.

M. Bayer, P. Hawrylak, K. Hinzer, S. Fafard, M. Korkusinski, Z. R. Wasilewski, O. Stern1, and A. Forchel, “Coupling and entangling of quantum states in quantum dot molecules,” Science291(5503), 451–453 (2001).
[CrossRef] [PubMed]

Krause, B.

B. Krause, T. H. Metzger, A Rastelli, R. Songmuang, S. Kiravittaya, and O. G. Schmidt, “Shape, strain, and ordering of lateral InAs quantum dot molecules,” Phys. Rev. B72(8), 085339(1–12) (2005).
[CrossRef]

Krenner, H. J.

K. Mller, A. Bechtold, C. Ruppert, M. Zecherle, G. Reithmaier, M. Bichler, H. J. Krenner, G. Abstreiter, A. W. Holleitner, J. M. Villas-Bôas, M. Betz, and J. J. Finley, “Electrical control of interdot electron tunneling in a double InGaAs quantum-dot nanostructure,” Phys. Rev. Lett.108(19), 197402(1–4) (2012).

Ku, P. C

C. J. Chang-Hasnain, P. C Ku, J. Kim, and S. L. Chuang, “Variable optical buffer using slow light in semiconductor nanostructure,” Proc. of IEEE91(11), 1884–1897 (2003).
[CrossRef]

Kuo, H. C.

Langbein, W.

P. Borri, W. Langbein, U. Woggon, M. Schwab, M. Bayer, S. Fafard, Z. Wasilewski, and P. Hawrylak, “Exciton dephasing in quantum dot molecules,” Phys. Rev. Lett.91(26), 267401(1–4) (2003).
[CrossRef]

Lee, R.

W. Yang, A. Chen, R. Lee, and Y Wu, “Matched slow optical soliton pairs via biexciton coherence in quantum dots,” Phys. Rev. A84(1), 013835(1–11) (2011).
[CrossRef]

W. Yang and R. Lee, “Controllable entanglement and polarization phase gate in coupled double quantum-well structures,” Opt. Express16(22), 17161–17170 (2008).
[CrossRef] [PubMed]

W. Yang and R. Lee, “Slow optical solitons via intersubband transitions in a semiconductor quantum well,” Europhys. Lett.83(1), 14002(1–6) (2008).
[CrossRef]

Li, X.

X. Li, Y. Wu, D. Steel, D. Gammon, T. H. Stievater, D. S. Katzer, D. Park, C. Piermarocchi, and L. J. Sham, “An all-optical quantum gate in a semiconductor quantum dot,” Science301(5634), 809–811 (2003).
[CrossRef] [PubMed]

Lin, C. T.

Lin, G.

Lin, K. F.

Liu, J. N.

Marangos, J. P.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys.77(2), 633–673 (2005).
[CrossRef]

Marcinkevicius, S.

S. Marcinkevicius, A. Gushterov, and J. P. Reithmaier, “Transient electromagnetically induced transparency in self-assembled quantum dots,” Appl. Phys. Lett.92(4), 041113(1–3) (2008).
[CrossRef]

Matsumoto, M.

A. Hasegawa and M. Matsumoto, Optical Solitons in Fibers (Springer, 2003).
[CrossRef]

Metzger, T. H.

B. Krause, T. H. Metzger, A Rastelli, R. Songmuang, S. Kiravittaya, and O. G. Schmidt, “Shape, strain, and ordering of lateral InAs quantum dot molecules,” Phys. Rev. B72(8), 085339(1–12) (2005).
[CrossRef]

Michler, P.

G. J. Beirne, C. Hermannstädter, L. Wang, A. Rastelli, O. G. Schmidt, and P. Michler, “Quantum light emission of two lateral tunnel-coupled (In,Ga)As/GaAs quantum dots controlled by a tunable static electric field,” Phys. Rev. Lett.96(13), 137401(1–4) (2006).
[CrossRef]

Mller, K.

K. Mller, A. Bechtold, C. Ruppert, M. Zecherle, G. Reithmaier, M. Bichler, H. J. Krenner, G. Abstreiter, A. W. Holleitner, J. M. Villas-Bôas, M. Betz, and J. J. Finley, “Electrical control of interdot electron tunneling in a double InGaAs quantum-dot nanostructure,” Phys. Rev. Lett.108(19), 197402(1–4) (2012).

Mørk, J.

Naruse, M.

N. Tate, M. Naruse, W. Nomura, T. Kawazoe, T. Yatsui, M. Hoga, Y. Ohyagi, Y. Sekine, H. Fujita, and M. Ohtsu, “Demonstration of modulatable optical near-field interactions between dispersed resonant quantum dots,” Opt. Express19(19), 18260–18271 (2011).
[CrossRef] [PubMed]

M. Ohtsu, T. Kawazoe, T. Yatsui, and M. Naruse, “Nanophotonics: application of dressed photons to novel photonic devices and systems,” IEEE J. Sel. Top. Quantum Electron.14(6), 1404–1417 (2008).
[CrossRef]

M. Ohtsu, K. Kobayashi, T. Kawazoe, T. Yatsui, and M. Naruse, Principles of Nanophotonics (Taylor and Francis, 2008).
[CrossRef]

Nikolaev, V. V.

V. V. Nikolaev, N. S. Averkiev, M. M. Sobolev, I. M. Gadzhiyev, I. O. Bakshaev, M. S. Buyalo, and E. L. Portnoi, “Tunnel coupling in an ensemble of vertically aligned quantum dot at room temperature,” Phys. Rev. B80(20), 205304 (2009).
[CrossRef]

Niu, Y.

Y. Qi, F. Zhou, T. Huang, Y. Niu, and S. Gong, “Spatial vector solitons in a four-level tripod-type atomic system,” Phys. Rev. A84(2), 023814(1–6) (2011)
[CrossRef]

Nomura, W.

Oh, C. H.

Ohtsu, M.

N. Tate, M. Naruse, W. Nomura, T. Kawazoe, T. Yatsui, M. Hoga, Y. Ohyagi, Y. Sekine, H. Fujita, and M. Ohtsu, “Demonstration of modulatable optical near-field interactions between dispersed resonant quantum dots,” Opt. Express19(19), 18260–18271 (2011).
[CrossRef] [PubMed]

M. Ohtsu, T. Kawazoe, T. Yatsui, and M. Naruse, “Nanophotonics: application of dressed photons to novel photonic devices and systems,” IEEE J. Sel. Top. Quantum Electron.14(6), 1404–1417 (2008).
[CrossRef]

K. Kobayashi, S. Sangu, T. Kawazoe, and M. Ohtsu, “Exciton dynamics and logic operations in a near-field optically coupled quantum-dot system,” J. Lumin.112, 117–121 (2005).
[CrossRef]

M. Ohtsu, K. Kobayashi, T. Kawazoe, T. Yatsui, and M. Naruse, Principles of Nanophotonics (Taylor and Francis, 2008).
[CrossRef]

Ohyagi, Y.

Park, D.

X. Li, Y. Wu, D. Steel, D. Gammon, T. H. Stievater, D. S. Katzer, D. Park, C. Piermarocchi, and L. J. Sham, “An all-optical quantum gate in a semiconductor quantum dot,” Science301(5634), 809–811 (2003).
[CrossRef] [PubMed]

Peng, P. C.

Piermarocchi, C.

X. Li, Y. Wu, D. Steel, D. Gammon, T. H. Stievater, D. S. Katzer, D. Park, C. Piermarocchi, and L. J. Sham, “An all-optical quantum gate in a semiconductor quantum dot,” Science301(5634), 809–811 (2003).
[CrossRef] [PubMed]

Portnoi, E. L.

V. V. Nikolaev, N. S. Averkiev, M. M. Sobolev, I. M. Gadzhiyev, I. O. Bakshaev, M. S. Buyalo, and E. L. Portnoi, “Tunnel coupling in an ensemble of vertically aligned quantum dot at room temperature,” Phys. Rev. B80(20), 205304 (2009).
[CrossRef]

Qi, Y.

Y. Qi, F. Zhou, T. Huang, Y. Niu, and S. Gong, “Spatial vector solitons in a four-level tripod-type atomic system,” Phys. Rev. A84(2), 023814(1–6) (2011)
[CrossRef]

Rastelli, A

B. Krause, T. H. Metzger, A Rastelli, R. Songmuang, S. Kiravittaya, and O. G. Schmidt, “Shape, strain, and ordering of lateral InAs quantum dot molecules,” Phys. Rev. B72(8), 085339(1–12) (2005).
[CrossRef]

Rastelli, A.

G. J. Beirne, C. Hermannstädter, L. Wang, A. Rastelli, O. G. Schmidt, and P. Michler, “Quantum light emission of two lateral tunnel-coupled (In,Ga)As/GaAs quantum dots controlled by a tunable static electric field,” Phys. Rev. Lett.96(13), 137401(1–4) (2006).
[CrossRef]

Reithmaier, G.

K. Mller, A. Bechtold, C. Ruppert, M. Zecherle, G. Reithmaier, M. Bichler, H. J. Krenner, G. Abstreiter, A. W. Holleitner, J. M. Villas-Bôas, M. Betz, and J. J. Finley, “Electrical control of interdot electron tunneling in a double InGaAs quantum-dot nanostructure,” Phys. Rev. Lett.108(19), 197402(1–4) (2012).

Reithmaier, J. P.

S. Marcinkevicius, A. Gushterov, and J. P. Reithmaier, “Transient electromagnetically induced transparency in self-assembled quantum dots,” Appl. Phys. Lett.92(4), 041113(1–3) (2008).
[CrossRef]

Ruppert, C.

K. Mller, A. Bechtold, C. Ruppert, M. Zecherle, G. Reithmaier, M. Bichler, H. J. Krenner, G. Abstreiter, A. W. Holleitner, J. M. Villas-Bôas, M. Betz, and J. J. Finley, “Electrical control of interdot electron tunneling in a double InGaAs quantum-dot nanostructure,” Phys. Rev. Lett.108(19), 197402(1–4) (2012).

Sangu, S.

K. Kobayashi, S. Sangu, T. Kawazoe, and M. Ohtsu, “Exciton dynamics and logic operations in a near-field optically coupled quantum-dot system,” J. Lumin.112, 117–121 (2005).
[CrossRef]

Sanz, L.

H. S. Borges, L. Sanz, J. M. Villas-Boas, O. O. Diniz Neto, and A. M. Alcalde, “Tunneling induced transparency and slow light in quantum dot molecules,” Phys. Rev. B85(11), 115425 (2012).
[CrossRef]

Schmidt, O. G.

G. J. Beirne, C. Hermannstädter, L. Wang, A. Rastelli, O. G. Schmidt, and P. Michler, “Quantum light emission of two lateral tunnel-coupled (In,Ga)As/GaAs quantum dots controlled by a tunable static electric field,” Phys. Rev. Lett.96(13), 137401(1–4) (2006).
[CrossRef]

B. Krause, T. H. Metzger, A Rastelli, R. Songmuang, S. Kiravittaya, and O. G. Schmidt, “Shape, strain, and ordering of lateral InAs quantum dot molecules,” Phys. Rev. B72(8), 085339(1–12) (2005).
[CrossRef]

R. Songmuang, S. Kiravittaya, and O. G. Schmidt, “Formation of lateral quantum dot molecules around self-assembled nanoholes,” Appl. Phys. Lett.82(17), 2892(1–3) (2003).
[CrossRef]

Schwab, M.

P. Borri, W. Langbein, U. Woggon, M. Schwab, M. Bayer, S. Fafard, Z. Wasilewski, and P. Hawrylak, “Exciton dephasing in quantum dot molecules,” Phys. Rev. Lett.91(26), 267401(1–4) (2003).
[CrossRef]

Sekine, Y.

Sham, L. J.

X. Li, Y. Wu, D. Steel, D. Gammon, T. H. Stievater, D. S. Katzer, D. Park, C. Piermarocchi, and L. J. Sham, “An all-optical quantum gate in a semiconductor quantum dot,” Science301(5634), 809–811 (2003).
[CrossRef] [PubMed]

She, Y.

Sobolev, M. M.

V. V. Nikolaev, N. S. Averkiev, M. M. Sobolev, I. M. Gadzhiyev, I. O. Bakshaev, M. S. Buyalo, and E. L. Portnoi, “Tunnel coupling in an ensemble of vertically aligned quantum dot at room temperature,” Phys. Rev. B80(20), 205304 (2009).
[CrossRef]

Songmuang, R.

B. Krause, T. H. Metzger, A Rastelli, R. Songmuang, S. Kiravittaya, and O. G. Schmidt, “Shape, strain, and ordering of lateral InAs quantum dot molecules,” Phys. Rev. B72(8), 085339(1–12) (2005).
[CrossRef]

R. Songmuang, S. Kiravittaya, and O. G. Schmidt, “Formation of lateral quantum dot molecules around self-assembled nanoholes,” Appl. Phys. Lett.82(17), 2892(1–3) (2003).
[CrossRef]

Steel, D.

X. Li, Y. Wu, D. Steel, D. Gammon, T. H. Stievater, D. S. Katzer, D. Park, C. Piermarocchi, and L. J. Sham, “An all-optical quantum gate in a semiconductor quantum dot,” Science301(5634), 809–811 (2003).
[CrossRef] [PubMed]

Stern1, O.

M. Bayer, P. Hawrylak, K. Hinzer, S. Fafard, M. Korkusinski, Z. R. Wasilewski, O. Stern1, and A. Forchel, “Coupling and entangling of quantum states in quantum dot molecules,” Science291(5503), 451–453 (2001).
[CrossRef] [PubMed]

Stievater, T. H.

X. Li, Y. Wu, D. Steel, D. Gammon, T. H. Stievater, D. S. Katzer, D. Park, C. Piermarocchi, and L. J. Sham, “An all-optical quantum gate in a semiconductor quantum dot,” Science301(5634), 809–811 (2003).
[CrossRef] [PubMed]

Su, H.

Sun, H.

Tang, D.

H. Zhang, D. Tang, L. Zhao, and X. Wu, “Dual-wavelength domain wall solitons in a fiber ring laser,” Opt. Express19(4), 3525–3530 (2011).
[CrossRef] [PubMed]

H. Zhang, D. Tang, L. Zhao, and R. Knize, “Vector dark domain wall solitons in a fiber ring laser,” Opt. Express18(5), 4428–4433 (2010).
[CrossRef] [PubMed]

H. Zhang, D. Tang, L. Zhao, and X. Wu, “Observation of polarization domain wall solitons in weakly birefringent cavity fiber lasers,” Phys. Rev. B80(5), 052302(1–4) (2009).

H. Zhang, D. Tang, L. Zhao, and X. Wu, “Dark pulse emission of a fiber laser,” Phys. Rev. A80(4), 045803(1–4) (2009).
[CrossRef]

Tate, N.

Thyrrestrup, H.

Tromborg, B.

Tsai, W. K.

Villas-Boas, J. M.

H. S. Borges, L. Sanz, J. M. Villas-Boas, O. O. Diniz Neto, and A. M. Alcalde, “Tunneling induced transparency and slow light in quantum dot molecules,” Phys. Rev. B85(11), 115425 (2012).
[CrossRef]

Villas-Bôas, J. M.

K. Mller, A. Bechtold, C. Ruppert, M. Zecherle, G. Reithmaier, M. Bichler, H. J. Krenner, G. Abstreiter, A. W. Holleitner, J. M. Villas-Bôas, M. Betz, and J. J. Finley, “Electrical control of interdot electron tunneling in a double InGaAs quantum-dot nanostructure,” Phys. Rev. Lett.108(19), 197402(1–4) (2012).

Wang, D.

Wang, L.

G. J. Beirne, C. Hermannstädter, L. Wang, A. Rastelli, O. G. Schmidt, and P. Michler, “Quantum light emission of two lateral tunnel-coupled (In,Ga)As/GaAs quantum dots controlled by a tunable static electric field,” Phys. Rev. Lett.96(13), 137401(1–4) (2006).
[CrossRef]

Wang, S. C.

Wang, Y

Wasilewski, Z.

P. Borri, W. Langbein, U. Woggon, M. Schwab, M. Bayer, S. Fafard, Z. Wasilewski, and P. Hawrylak, “Exciton dephasing in quantum dot molecules,” Phys. Rev. Lett.91(26), 267401(1–4) (2003).
[CrossRef]

Wasilewski, Z. R.

M. Bayer, P. Hawrylak, K. Hinzer, S. Fafard, M. Korkusinski, Z. R. Wasilewski, O. Stern1, and A. Forchel, “Coupling and entangling of quantum states in quantum dot molecules,” Science291(5503), 451–453 (2001).
[CrossRef] [PubMed]

Woggon, U.

P. Borri, W. Langbein, U. Woggon, M. Schwab, M. Bayer, S. Fafard, Z. Wasilewski, and P. Hawrylak, “Exciton dephasing in quantum dot molecules,” Phys. Rev. Lett.91(26), 267401(1–4) (2003).
[CrossRef]

Wu, C.

Wu, J.

Wu, X.

H. Zhang, D. Tang, L. Zhao, and X. Wu, “Dual-wavelength domain wall solitons in a fiber ring laser,” Opt. Express19(4), 3525–3530 (2011).
[CrossRef] [PubMed]

H. Zhang, D. Tang, L. Zhao, and X. Wu, “Dark pulse emission of a fiber laser,” Phys. Rev. A80(4), 045803(1–4) (2009).
[CrossRef]

H. Zhang, D. Tang, L. Zhao, and X. Wu, “Observation of polarization domain wall solitons in weakly birefringent cavity fiber lasers,” Phys. Rev. B80(5), 052302(1–4) (2009).

Wu, Y

W. Yang, A. Chen, R. Lee, and Y Wu, “Matched slow optical soliton pairs via biexciton coherence in quantum dots,” Phys. Rev. A84(1), 013835(1–11) (2011).
[CrossRef]

Wu, Y.

Y. Wu and L. Deng, “Ultraslow bright and dark optical solitons in a cold three-state medium,” Opt. Lett.29(17), 2064–2066 (2004).
[CrossRef] [PubMed]

X. Li, Y. Wu, D. Steel, D. Gammon, T. H. Stievater, D. S. Katzer, D. Park, C. Piermarocchi, and L. J. Sham, “An all-optical quantum gate in a semiconductor quantum dot,” Science301(5634), 809–811 (2003).
[CrossRef] [PubMed]

Yang, H. P.

Yang, W.

W. Yang, A. Chen, R. Lee, and Y Wu, “Matched slow optical soliton pairs via biexciton coherence in quantum dots,” Phys. Rev. A84(1), 013835(1–11) (2011).
[CrossRef]

W. Yang and R. Lee, “Controllable entanglement and polarization phase gate in coupled double quantum-well structures,” Opt. Express16(22), 17161–17170 (2008).
[CrossRef] [PubMed]

W. Yang and R. Lee, “Slow optical solitons via intersubband transitions in a semiconductor quantum well,” Europhys. Lett.83(1), 14002(1–6) (2008).
[CrossRef]

Yatsui, T.

N. Tate, M. Naruse, W. Nomura, T. Kawazoe, T. Yatsui, M. Hoga, Y. Ohyagi, Y. Sekine, H. Fujita, and M. Ohtsu, “Demonstration of modulatable optical near-field interactions between dispersed resonant quantum dots,” Opt. Express19(19), 18260–18271 (2011).
[CrossRef] [PubMed]

M. Ohtsu, T. Kawazoe, T. Yatsui, and M. Naruse, “Nanophotonics: application of dressed photons to novel photonic devices and systems,” IEEE J. Sel. Top. Quantum Electron.14(6), 1404–1417 (2008).
[CrossRef]

M. Ohtsu, K. Kobayashi, T. Kawazoe, T. Yatsui, and M. Naruse, Principles of Nanophotonics (Taylor and Francis, 2008).
[CrossRef]

Yuan, C.

C. Yuan and K. Zhu, “Voltage-controlled slow light in asymmetry double quantum dots,” Appl. Phys. Lett.89(5), 052115 (1–3) (2006).
[CrossRef]

Zecherle, M.

K. Mller, A. Bechtold, C. Ruppert, M. Zecherle, G. Reithmaier, M. Bichler, H. J. Krenner, G. Abstreiter, A. W. Holleitner, J. M. Villas-Bôas, M. Betz, and J. J. Finley, “Electrical control of interdot electron tunneling in a double InGaAs quantum-dot nanostructure,” Phys. Rev. Lett.108(19), 197402(1–4) (2012).

Zhang, H.

H. Zhang, D. Tang, L. Zhao, and X. Wu, “Dual-wavelength domain wall solitons in a fiber ring laser,” Opt. Express19(4), 3525–3530 (2011).
[CrossRef] [PubMed]

H. Zhang, D. Tang, L. Zhao, and R. Knize, “Vector dark domain wall solitons in a fiber ring laser,” Opt. Express18(5), 4428–4433 (2010).
[CrossRef] [PubMed]

H. Zhang, D. Tang, L. Zhao, and X. Wu, “Observation of polarization domain wall solitons in weakly birefringent cavity fiber lasers,” Phys. Rev. B80(5), 052302(1–4) (2009).

H. Zhang, D. Tang, L. Zhao, and X. Wu, “Dark pulse emission of a fiber laser,” Phys. Rev. A80(4), 045803(1–4) (2009).
[CrossRef]

Zhang, W.

Zhao, L.

H. Zhang, D. Tang, L. Zhao, and X. Wu, “Dual-wavelength domain wall solitons in a fiber ring laser,” Opt. Express19(4), 3525–3530 (2011).
[CrossRef] [PubMed]

H. Zhang, D. Tang, L. Zhao, and R. Knize, “Vector dark domain wall solitons in a fiber ring laser,” Opt. Express18(5), 4428–4433 (2010).
[CrossRef] [PubMed]

H. Zhang, D. Tang, L. Zhao, and X. Wu, “Observation of polarization domain wall solitons in weakly birefringent cavity fiber lasers,” Phys. Rev. B80(5), 052302(1–4) (2009).

H. Zhang, D. Tang, L. Zhao, and X. Wu, “Dark pulse emission of a fiber laser,” Phys. Rev. A80(4), 045803(1–4) (2009).
[CrossRef]

Zhou, F.

Y. Qi, F. Zhou, T. Huang, Y. Niu, and S. Gong, “Spatial vector solitons in a four-level tripod-type atomic system,” Phys. Rev. A84(2), 023814(1–6) (2011)
[CrossRef]

Zhu, C.

Zhu, K.

C. Yuan and K. Zhu, “Voltage-controlled slow light in asymmetry double quantum dots,” Appl. Phys. Lett.89(5), 052115 (1–3) (2006).
[CrossRef]

Appl. Phys. Lett.

S. Marcinkevicius, A. Gushterov, and J. P. Reithmaier, “Transient electromagnetically induced transparency in self-assembled quantum dots,” Appl. Phys. Lett.92(4), 041113(1–3) (2008).
[CrossRef]

R. Songmuang, S. Kiravittaya, and O. G. Schmidt, “Formation of lateral quantum dot molecules around self-assembled nanoholes,” Appl. Phys. Lett.82(17), 2892(1–3) (2003).
[CrossRef]

C. Yuan and K. Zhu, “Voltage-controlled slow light in asymmetry double quantum dots,” Appl. Phys. Lett.89(5), 052115 (1–3) (2006).
[CrossRef]

Europhys. Lett.

W. Yang and R. Lee, “Slow optical solitons via intersubband transitions in a semiconductor quantum well,” Europhys. Lett.83(1), 14002(1–6) (2008).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

M. Ohtsu, T. Kawazoe, T. Yatsui, and M. Naruse, “Nanophotonics: application of dressed photons to novel photonic devices and systems,” IEEE J. Sel. Top. Quantum Electron.14(6), 1404–1417 (2008).
[CrossRef]

J. Lumin.

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Figures (5)

Fig. 1
Fig. 1

Energy-level diagram and excitation scheme of QD molecules system interacting with a strong control field (with half Rabi frequency Ωc) and a weak probe optical field (with half Rabi frequency Ωp). A bias voltage, which is added between two quantum dots, induces electron appearing tunneling effect, and the corresponding tunneling strength is Te. The detail description on the diagram sees the contexts of the paper.

Fig. 2
Fig. 2

The linear absorption ImK(ω) (red dash curve) and linear dispersion Re(K) (black solid curve) of the probe field Ωp as a function of the frequencies ω under the consideration of (a)Te = Ωc = 0, (b)Te = 1.5meV, Ωc = 0, (c)Te = 2.5meV, Ωc = 0, (d)Te = Ωc = 1.5meV, and (e)Te = 1.5meV, Ωc = 2.5meV. The other parameters used are given in the text.

Fig. 3
Fig. 3

The linear absorption ImK(ω) (Fig.3 (a)) and linear dispersion Re(K) (Fig.3(b)) of the probe field Ωp as a function of the frequencies ω under the different level detuning Δ. The black solid, red dash, and blue dotted curves correspond to the detuning Δ = −1.0meV, 0.0meV, 1.0meV, respectively. Here, Te = Ωc = 2.0meV, Δ2 = 0.1meV, Δ4 = Δ + Δ2. The other parameters used are the same as those in Fig. 2.

Fig. 4
Fig. 4

The Kerr nonlinear coefficient |W| as a function of the control field Ωc with different tunneling strength (Fig.4(a)) and one-photon detuning Δ4 (Fig.4(b)). In the Fig.4(a), the black solid, red dash, and blue dotted curves correspond to the tunneling strength Te = 1.5meV, 2.0meV and 2.5meV, respectively. In the Fig.4(b), the black solid, red dash, and blue dotted curves correspond to the detuning Δ4 = 3.0meV, 3.5meV, 4.0meV. Here Δ2 = −3.0meV, Δ3 = 1.0meV. The other parameters used are the same as those in Fig. 2.

Fig. 5
Fig. 5

The ratios of the imaginary parts and the corresponding real parts of the Kerr nonlinearity coefficient W (red solid curve) and group-velocity dispersion coefficient K2 (blue dashed curve) versus the tunneling coupling strength Te with different control fields Ωc. The Figs. 4(a) and 4(b) correspond respectively to the case of bright solitons and dark solitons due to K2rWr > 0 and K2rWr < 0, in the case of Ωc = 2.5meV and 3.5meV. The other parameters used are the same as those in Fig. 4.

Equations (22)

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H I / h ¯ = j = 2 4 Δ j | j j | ( Ω p | 2 1 | + Ω c | 4 3 | T e | 4 2 | + H . c . ) ,
( i t + d 2 ) A 2 + Ω p A 1 T e A 4 = 0 ,
( i t + d 3 ) A 3 + Ω c * A 4 = 0 ,
( i t + d 4 ) A 4 + Ω c * A 3 T e A 2 = 0 ,
| A 1 | 2 + | A 2 | 2 + | A 3 | 2 + | A 4 | 2 = 1.
2 E 1 c 2 2 t 2 E = 1 ε 0 c 2 2 t 2 P ,
i ( z + 1 c t ) Ω p + c 2 ω p ( 2 x 2 + 2 y 2 ) Ω p + κ 12 A 1 * A 2 = 0 ,
K ( ω ) = ω c κ 12 D c D ,
i t 0 A 2 ( j ) + d 2 A 2 ( j ) T e A 4 ( j ) + Ω p ( j ) = α ( j ) ,
i t 0 A 3 ( j ) + d 3 A 3 ( j ) + Ω c A 4 ( j ) = β ( j ) ,
i t 0 A 4 ( j ) + d 4 A 4 ( j ) + Ω c A 3 ( j ) T e A 2 ( j ) = γ ( j ) ,
i z 0 Ω p ( j ) + i 1 c t 0 Ω p ( j ) + κ 12 A 2 ( j ) = ρ ( j ) .
Ω p ( 1 ) = F e i θ = F e i ( K z 0 w t 0 ) ,
A 2 ( 1 ) = 1 κ 12 ( K ω c ) F e i θ ,
A 3 ( 1 ) = 1 Ω c [ T e 1 κ 12 ( K ω c ) 1 κ 12 T e ( ω + d 4 ) ( ω + d 2 ) ( K ω c ) 1 T e ( ω + d 4 ) ] F e i θ ,
A 4 ( 1 ) = 1 T e [ 1 κ 12 ( ω + d 2 ) ( K ω c ) + 1 ] F e i θ ,
i F z 2 + c 2 ω p ( 2 x 1 2 + 2 y 1 2 ) F K 2 2 2 F t 1 2 W | F | 2 F e α ˜ z 2 = 0 ,
i ( z + α ˜ 2 ) U + c 2 ω p ( 2 x 2 + 2 y 2 ) U K 2 2 2 U τ 2 W | U | 2 U = 0 ,
i u s + 2 u σ 2 + 2 | u | 2 u = 0 ,
i u s 2 u σ 2 + 2 | u | 2 u = 0 ,
Ω p = 1 τ 0 K 2 r W r sec h [ 1 τ 0 ( t z V g ) ] exp [ i K z + i z 2 L D ] .
Ω p = 1 τ 0 K 2 r W r tan h [ 1 τ 0 ( t z V g ) ] exp [ i K z + i z 2 L D ] .

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