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Single-Photon Transport in One-Dimensional Coupled-Resonator Waveguide with Second-Order Nonlinearity oupling to a Nanocavity Containing a Two-Level Atom and Kerr-Nonlinearity

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Abstract

In the recent publication [Opt. Expr 28, 1249 (2020)] the single photon scattering is studied in a second-order nonlinear system. In this paper, we study controllable single photon scattering in a one-dimensional waveguide coupling with an additional cavity by second order nonlinear materials in a non-cascading configuration, where the additional cavity is embedded with a two-level atom and filled with Kerr-nonlinear materials. Considering the second order nonlinear coupling, we analyze the transmission properties of the three different coupling forms as follows: (i) The two-level atom is excited without the Kerr-nonlinearity. (ii)The Kerr-nonlinearity is excited without the two-level atom. (iii) Both of the two-level atom and Kerr-nonlinearity are excited. The transmission and reflection amplitudes are obtained by the discrete coordinates approach for the three cases. The results show that the transmission properties can be adjusted by the above three different coupling forms, which indicate our scheme can be used as a single photon switch to control the transmission and reflection of the single photon in the one-dimensional coupled resonant waveguide, and which is compared to the results with a two level system [Phys. Rev. A 85, 053840 (2012)] and find the advantages.

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Acknowledgment

This work is supported by the National Natural Science Foundation of China with Grants No. 11647054, the Science and Technology Development Program of Jilin province, China with Grant No. 2018-0520165JH, the Jiangxi Education Department Fund under Grant No. GJJ180873.

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Authors and Affiliations

  1. Physics, Changchun University of Science and Technology, Changchun, 130022, China

    Hongyu Lin, Xiaoqian Wang & Zhihai Yao

  2. College of Physics and Electronic Information, Baicheng Normal University, Baicheng, 137000, China

    Hongyu Lin

  3. School of Physics and Electronic Information, Shangrao Normal University, Shangrao, 334001, China

    Dandan Zou

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  1. Hongyu Lin
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Correspondence to Zhihai Yao.

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Appendix

Appendix

Here we give the derivation of (8):

$$ \begin{array}{@{}rcl@{}} a_{k}&=&\frac{1}{\sqrt{N}}{\sum}_{j}a_{j}e^{-ikj},\\ a_{j}&=&\frac{1}{\sqrt{N}}{\sum}_{k}a_{k}e^{ikj}. \end{array} $$
(18)

By substituting (18) into the Hamiltonian Ha, we can obtain

$$ \begin{array}{@{}rcl@{}} &&\omega_{a}{\sum}_{j}{a}_{j}^{\dag}a_{j}-\xi{\sum}_{j}(a_{j+1}^{\dag}a_{j}+{a}_{j}^{\dag}a_{j+1}),\\ &=&{\sum}_{j}\bigg\{\omega_{a}\frac{1}{\sqrt{N}}{\sum}_{k}{a}_{k}^{\dag} e^{-ikj}\frac{1}{\sqrt{N}}{\sum}_{k^{\prime}}a_{k^{\prime}}e^{ik^{\prime}j} -\xi\big[\frac{1}{\sqrt{N}}{\sum}_{k}{a}_{k}^{\dag} e^{-ik(j+1)}\frac{1}{\sqrt{N}}{\sum}_{k^{\prime}}a_{k^{\prime}}e^{ik^{\prime}j}\\ &&+\frac{1}{\sqrt{N}}{\sum}_{k}{a}_{k}^{\dag} e^{-ikj}\frac{1}{\sqrt{N}}{\sum}_{k^{\prime}}a_{k^{\prime}}e^{ik^{\prime}(j+1)}\big] \bigg\}\\ &=&{\sum}_{j}\omega_{a}\frac{1}{\sqrt{N}}{\sum}_{kk^{\prime}}{a}_{k}^{\dag} a_{k^{\prime}} e^{-i(k-k^{\prime})j} -\xi{\sum}_{j}\bigg[\frac{1}{N}{\sum}_{kk^{\prime}}{a}_{k}^{\dag} a_{k^{\prime}} e^{-i(k-k^{\prime})j}e^{-ik}\\ &&+\frac{1}{N}{\sum}_{kk^{\prime}}{a}_{k}^{\dag} a_{k^{\prime}} e^{-i(k-k^{\prime})j}e^{ik^{\prime}}\bigg]. \end{array} $$
(19)

Considering the relation

$$ \frac{1}{N}{\sum}_{j}e^{-i(k-k^{\prime})j}=\delta_{k,k^{\prime}}, $$
(20)

Equation 19 reduces to

$$ \begin{array}{@{}rcl@{}} &&\omega_{a}{\sum}_{kk^{\prime}}{a}_{k}^{\dag} a_{k^{\prime}}\delta_{k,k^{\prime}} -\xi({\sum}_{kk^{\prime}}{a}_{k}^{\dag} a_{k^{\prime}}\delta_{k,k^{\prime}}e^{-ik}+{\sum}_{kk^{\prime}}{a}_{k}^{\dag} a_{k^{\prime}}\delta_{k,k^{\prime}}e^{ik^{\prime}})\\ &=&\omega_{a}{\sum}_{k}{a}_{k}^{\dag} a_{k}-\xi({\sum}_{k}{a}_{k}^{\dag} a_{k}e^{-ik}+{\sum}_{k}{a}_{k}^{\dag} a_{k}e^{ik})\\ &=&(\omega_{a}-2\xi \cos k){\sum}_{k}{a}_{k}^{\dag} a_{k}. \end{array} $$
(21)

From (21) we can easily obtain the energy eigenvalues equation in the form of (8).

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Lin, H., Wang, X., Yao, Z. et al. Single-Photon Transport in One-Dimensional Coupled-Resonator Waveguide with Second-Order Nonlinearity oupling to a Nanocavity Containing a Two-Level Atom and Kerr-Nonlinearity. Int J Theor Phys 59, 2294–2307 (2020). https://doi.org/10.1007/s10773-020-04498-x

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