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Negative refraction in the double quantum dot system

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Abstract

This work proposes a double quantum dot (DQD) system, with a wetting layer (WL) is included, to study the negative refractive index (NRI) under the application of the electric fields: pump, probe, and fields between WL-QD state, in addition to the magnetic field. The density matrix theory is used to write the equation of motion and an orthogonalized plane wave is used between WL-QD states. The results show that the DQD system exhibit NRI ordinarily until with pump and probe signals, only, due to the manipulation between states. A high NRI corresponding to neglected absorption is obtained under applied electric fields between QD-QD, the conduction (CB) and valence bands (VB) WL-QD fields. It is shown that the main requirement in increasing NRI is the high electric gain connected with a low magnetic one. This can be obtained under five applied electric fields in addition to a high VB WL-QD electric field. Neglecting WL reduces NRI by ~ 16 times. In single QD, the NRI is very small compared with DQD.

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Authors and Affiliations

  1. Nasiriya Nanotechnology Research Laboratory (NNRL), Science College, Thi-Qar University, Nasiriya, Iraq

    Hussein G. Al-Toki & Amin Habbeb Al-Khursan

Authors
  1. Hussein G. Al-Toki
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  2. Amin Habbeb Al-Khursan
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Correspondence to Amin Habbeb Al-Khursan.

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Appendix

Appendix

The dynamical equations of the DQD system shown in Fig. 1 are written as,

$$ \begin{aligned} \rho_{00}^{ \cdot } = & - \gamma_{0} \rho_{00} + i[A_{10} (\rho_{10} - \rho_{01} ) + \Omega_{20} (\rho_{20} - \rho_{02} ) \\ & + \Omega_{03} (\rho_{30} - \rho_{03} ) + \Omega_{04} (\rho_{40} - \rho_{04} )] \\ \end{aligned} $$
(36-1)
$$ \rho_{11}^{ \cdot } = - \gamma_{1} \rho_{11} + i[A_{10} (\rho_{01} - \rho_{10} ) + T_{21} (\rho_{21} - \rho_{11} ) + \Omega_{13} (\rho_{31} - \rho_{13} ) + \Omega_{14} (\rho_{41} - \rho_{14} )] $$
(36-2)
$$ \rho_{22}^{ \cdot } = - \gamma_{1} \rho_{22} + i[\Omega_{20} (\rho_{02} - \rho_{20} ) + T_{21} (\rho_{21} - \rho_{11} ) + \Omega_{25} (\rho_{52} - \rho_{25} )] $$
(36-3)
$$ \rho_{33}^{ \cdot } = - \gamma_{33} \rho_{33} + i[\Omega_{30} (\rho_{03} - \rho_{30} ) + \Omega_{31} (\rho_{13} - \rho_{31} ) + \Omega_{03} (\rho_{23} - \rho_{32} ) + \Omega_{35} (\rho_{53} - \rho_{35} )] $$
(36-4)
$$ \rho_{44}^{ \cdot } = - \gamma_{4} \rho_{44} + i[\Omega_{40} (\rho_{04} - \rho_{40} ) + \Omega_{41} (\rho_{14} - \rho_{41} )] $$
(36-5)
$$ \rho_{55}^{ \cdot } = - \gamma_{5} \rho_{55} + i[\Omega_{52} (\rho_{25} - \rho_{52} ) + \Omega_{53} (\rho_{35} - \rho_{53} )] $$
(36-6)
$$ \begin{aligned} \rho_{10}^{ \cdot } & = - [(\gamma_{1} + \gamma_{0} )\rho_{10} ] + i[A_{10} (\rho_{00} - \rho_{11} ) + T_{21} \rho_{20} + \Omega_{31} \rho_{30} + \Omega_{14} \rho_{40} - \,\,\Omega_{20} \rho_{12} \hfill \\ &\quad - \Omega_{30} \rho_{13} - \Omega_{40} \rho_{14} ] \hfill \\ \end{aligned} $$
(36-7)
$$ \rho_{20}^{ \cdot } = - i[(\gamma_{0} + \gamma_{2} )\rho_{20} ]\, + i[\Omega_{20} (\rho_{00} - \rho_{22} )\, + \,(T_{21} \rho_{10} + \Omega_{23}^{m} \rho_{30} + \Omega_{25} \rho_{25} ) - (A_{10} \rho_{21} \, + \Omega_{30} \rho_{23} + \Omega_{25} \rho_{25} )] $$
(36-8)
$$ \begin{aligned} \rho_{30}^{ \cdot } & = - [(\gamma_{0} + \gamma_{3} )\rho_{30} ] + i[\Omega_{30} (\rho_{00} - \rho_{33} ) + \Omega_{31} \rho_{10} + \Omega_{32}^{m} \rho_{20} \hfill \\ &\quad - A_{10} \rho_{31} + \Omega_{20} \rho_{30} ] \hfill \\ \end{aligned} $$
(36-9)
$$ \rho_{40}^{ \cdot } = - [(\gamma_{4} + \gamma_{0} )\rho_{40} ] + i[\Omega_{40} (\rho_{00} - \rho_{44} )\, + \Omega_{41} \rho_{10} + \Omega_{10}^{{}} \rho_{41} ] $$
(36-10)
$$ \rho_{21}^{ \cdot } = - [(\gamma_{2} + \gamma_{1} )\rho_{21} ] + i[T_{21} (\rho_{11} - \rho_{22} ) + \Omega_{20} \rho_{01} + \Omega_{20}^{{}} \rho_{31} + \Omega_{25} \rho_{51} ] $$
(36-11)
$$ \rho_{23}^{ \cdot } { = } - {[}(\gamma_{2} + \gamma_{3} )\rho_{23} ] + i[\Omega_{23}^{m} (\rho_{33} - \rho_{22} )\, + \Omega_{20} \rho_{03} + T_{21} \rho_{31} + \Omega_{25} \rho_{53} ] $$
(36-12)
$$ \rho_{31}^{ \cdot } { = } - {[}(\gamma_{3} + \gamma_{1} )\rho_{31} ] + i[\Omega_{31} (\rho_{11} - \rho_{33} ) + \,\Omega_{30} \rho_{10} + \Omega_{23}^{m} \rho_{21} - \Omega_{10}^{{}} \rho_{30} + T_{21} \rho_{23} ] $$
(36-13)
$$ \begin{aligned} \rho_{41}^{ \cdot } &= - {[}(\gamma_{4} + \gamma_{1} )\rho_{41} ] + i[\Omega_{41} (\rho_{11} - \rho_{44} )\, + \Omega_{40} \rho_{01} \hfill \\ &\quad +\,Omega_{42} \rho_{21} - A_{10} \rho_{40} + T_{21} \rho_{42} ] \hfill \\ \end{aligned} $$
(36-14)
$$ \rho_{24}^{ \cdot } = - [(\gamma_{4} + \gamma_{2} )\rho_{24} ] + i[\Omega_{20} \rho_{04} + T_{21} \rho_{14} - i\Omega_{04} \rho_{20} + \Omega_{14} \rho_{21} + \Omega_{52} \rho_{25} ] $$
(36-15)
$$ \rho_{25}^{ \cdot } = - [(\gamma_{2} + \gamma_{5} )\rho_{25} ] + i[\Omega_{25} (\rho_{55} + \rho_{22} )\, + \Omega_{23}^{{}} \rho - \Omega_{35} \rho_{23} ] $$
(36-16)
$$ \rho_{35}^{ \cdot } = - [(\gamma_{3} + \gamma_{5} )\rho_{35} ] + i[\Omega_{23}^{m} \rho_{25} \, + \Omega_{35} \rho_{55} - \Omega_{25} \rho_{32} ] $$
(36-17)
$$ \rho_{43}^{ \cdot } = - [(\gamma_{4} + \gamma_{3} )\rho_{43} ] + i[\Omega_{40} \rho_{03} + \Omega_{41} \rho_{13} - \Omega_{03} \rho_{40} \Omega_{13} \rho_{41} - \Omega_{30} \rho_{53} ] $$
(36-18)
$$ \rho_{50}^{ \cdot } = - [(\gamma_{5} + \gamma_{0} )\rho_{50} ] + i[\Omega_{52} \rho_{20} + i\Omega_{53} \rho_{30} - \Omega_{10}^{{}} \rho_{50} - \Omega_{20} \rho_{52} - \Omega_{30} \rho_{53} ] $$
(36-19)
$$ \rho_{51}^{ \cdot } = - [(\gamma_{5} + \gamma_{1} )\rho_{51} ] + i[\Omega_{52} \rho_{20} + \,\Omega_{53} \rho_{30} - \Omega_{10}^{{}} \rho_{50} - T_{21} \rho_{52} - \Omega_{30} \rho_{53} ] $$
(36-20)
$$ \rho_{55}^{ \cdot } = - \gamma_{5} \rho_{55} + i[\Omega_{52} (\rho_{25} - \rho_{52} ) + \Omega_{53} (\rho_{35} - \rho_{53} )] $$
(36-21)

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Al-Toki, H.G., Al-Khursan, A.H. Negative refraction in the double quantum dot system. Opt Quant Electron 52, 467 (2020). https://doi.org/10.1007/s11082-020-02580-6

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