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Breather-impurity interactions and modulational instability in a quantum 2D Klein–Gordon chain

  • Regular Article - A: Solid State and Materials
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Abstract

We study the breather-impurity interactions and modulational instability in a quantum 2D Klein–Gordon chain. By using the second quantification operators, we transform classical Hamiltonian into its quantum version, through Glauber’s coherent state representation in addition to the multiple-scale method, the 2D nonlinear Schrödinger equation (NLSE) is obtained. This NLSE is analytically solved by adopting the Rayleigh–Ritz variational method. Around impurity’s critical mass, we prove the existence of resonant structure. This critical mass is observed when plotting the frequency spectrum under the effect of the impurity mass and harmonic force constants. The effects of impurity mass and the harmonic force constants are found in the amplitude frequency spectra. When the breather interacts with the impurity, the system exhibit different scenario that are: barrier, well, excitation and chaotic all related to the trapping phenomenon. The modulational instability (MI) which helps to confirm the existence of breather is investigated. We have shown that impurity strength increases the instability regions, and the MI growth rate can be dramatically affected by the impurity mass around the critical value.

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This manuscript has no associated data or the data will not be deposited. [Authors’ comment: The authors declare that all data supporting the findings of this study are available within the article.]

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Acknowledgements

Zacharie Isidore Djoufack acknowledges the African Institute for Mathematical Sciences (AIMS) South Africa for research facilities.

Author information

Authors and Affiliations

  1. Pure Physics Laboratory: Group of Nonlinear physics and Complex systems, Department of Physics, University of Douala, P.O. Box 24157, Douala, Cameroon

    R. Abouem A. Ribama & J. P. Nguenang

  2. Unité de Recherche d’Automatique et Informatique Apliquée (UR-AIA), Fotso Victor University Institute of Technology, University of Dschang, P.O. Box 134, Bandjoun, Cameroon

    Z. I. Djoufack

  3. The African Institute for Mathematical Sciences (AIMS), 6 Melrose Road, Muizenberg, Cape Town, 7945, South Africa

    Z. I. Djoufack

Authors
  1. R. Abouem A. Ribama
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  2. Z. I. Djoufack
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Corresponding author

Correspondence to Z. I. Djoufack.

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Author contribution statement

RAAR: design, calculations, simulations, writing; ZID: design, conception, calculations, simulations, writing, reading, revision; JPN: design, conception, reading, revision and supervision.

Appendix

Appendix

$$\begin{aligned}&u^2_{l,n}=\frac{\hbar }{2m\omega _0}\left( b^{\dagger }_{l,n}b^{\dagger }_{l,n} +b^{\dagger }_{l,n}b_{l,n}+b_{l,n}b^{\dagger }_{l,n}+b_{l,n}b_{l,n}\right) \nonumber \\&P^2_{l,n}=-\frac{\hbar {m\omega _0}}{2}\left( b^{\dagger }_{l,n}b^{\dagger }_{l,n} -b^{\dagger }_{l,n}b_{l,n}-b_{l,n}b^{\dagger }_{l,n}+b_{l,n}b_{l,n}\right) \nonumber \\&u^2_{l-1,n}=\frac{\hbar }{2m\omega _0}\left( b^{\dagger }_{l-1,n}b^{\dagger }_{l-1,n}\right. \nonumber \\&\qquad \left. +b^{\dagger }_{l-1,n}b_{l-1,n}+b_{l-1,n}b^{\dagger }_{l-1,n}+b_{l-1,n}b_{l-1,n}\right) \nonumber \\&u^2_{l,n-1}=\frac{\hbar }{2m\omega _0}\left( b^{\dagger }_{l,n-1}b^{\dagger }_{l,n-1}\right. \nonumber \\&\qquad \left. +b^{\dagger }_{l,n-1}b_{l,n-1}+b_{l,n-1}b^{\dagger }_{l,n-1}+b_{l,n-1}b_{l,n-1}\right) \end{aligned}$$
(54)
$$\begin{aligned}&u^4_{l,n}=\left( \frac{\hbar }{2m\omega _0}\right) ^2\left( b^{\dagger }_{l,n}b^{\dagger }_{l,n} +b^{\dagger }_{l,n}b_{l,n}+b_{l,n}b^{\dagger }_{l,n}+b_{l,n}b_{l,n}\right) \nonumber \\&\qquad \left( b^{\dagger }_{l,n}b^{\dagger }_{l,n}+b^{\dagger }_{l,n}b_{l,n} +b_{l,n}b^{\dagger }_{l,n}+b_{l,n}b_{l,n}\right) \nonumber \\&\quad =\left( \frac{\hbar }{2m\omega _0}\right) ^2(b^{\dagger }_{l,n} b^{\dagger }_{l,n}b^{\dagger }_{l,n}b^{\dagger }_{l,n} +b^{\dagger }_{l,n}b^{\dagger }_{l,n}b^{\dagger }_{l,n}b_{l,n}\nonumber \\&\qquad +b^{\dagger }_{l,n}b^{\dagger }_{l,n}b_{l,n}b^{\dagger }_{l,n} +b^{\dagger }_{l,n}b^{\dagger }_{l,n}b_{l,n}b_{l,n} +b^{\dagger }_{l,n}b_{l,n}b^{\dagger }_{l,n}b^{\dagger }_{l,n}\nonumber \\&\qquad +b^{\dagger }_{l,n}b_{l,n}b^{\dagger }_{l,n}b_{l,n}+b^{\dagger }_{l,n} b_{l,n}b_{l,n}b^{\dagger }_{l,n}+b^{\dagger }_{l,n}b_{l,n}b_{l,n}b_{l,n}\nonumber \\&\qquad +b_{l,n}b^{\dagger }_{l,n}b^{\dagger }_{l,n}b^{\dagger }_{l,n} +b_{l,n}b^{\dagger }_{l,n}b^{\dagger }_{l,n}b_{l,n}\nonumber \\&\qquad +b_{l,n}b^{\dagger }_{l,n}b_{l,n}b^{\dagger }_{l,n}\nonumber \\&\qquad +b_{l,n}b^{\dagger }_{l,n}b_{l,n}b_{l,n} +b_{l,n}b_{l,n}b^{\dagger }_{l,n}b^{\dagger }_{l,n}+b_{l,n}b_{l,n}b^{\dagger }_{l,n}b_{l,n}\nonumber \\&\qquad +b_{l,n}b_{l,n}b_{l,n}b^{\dagger }_{l,n}+b_{l,n}b_{l,n}b_{l,n}b_{l,n}) \end{aligned}$$
(55)
$$\begin{aligned}&u_{l,n}u_{l-1,n}=\left( \frac{\hbar }{2m\omega _0}\right) (b^{\dagger }_{l,n}+b_{l,n}) (b^{\dagger }_{l-1,n}+b_{l-1,n})\nonumber \\&\quad =\left( \frac{\hbar }{2m\omega _0}\right) (b^{\dagger }_{l,n}b^{\dagger }_{l-1,n}\nonumber \\&\qquad +b^{\dagger }_{l,n}b_{l-1,n}+b_{l,n}b^{\dagger }_{l-1,n}+b_{l,n}b_{l-1,n}) \end{aligned}$$
(56)
$$\begin{aligned}&u_{l,n}u_{l,n-1}=\left( \frac{\hbar }{2m\omega _0}\right) (b^{\dagger }_{l,n}+b_{l,n}) (b^{\dagger }_{l,n-1}+b_{l,n-1})\nonumber \\&\quad =\left( \frac{\hbar }{2m\omega _0}\right) (b^{\dagger }_{l,n}b^{\dagger }_{l,n-1}\nonumber \\&\qquad +b^{\dagger }_{l,n}b_{l,n-1}+b_{l,n}b^{\dagger }_{l,n-1}+b_{l,n}b_{l,n-1}) \end{aligned}$$
(57)
$$\begin{aligned}&u^2_{l-1,n}-2u_{l,n}u_{l-1,n}=\left( \frac{\hbar }{2\omega _0}\right) (b^{\dagger }_{l-1,n}b^{\dagger }_{l-1,n}\nonumber \\&\qquad +b^{\dagger }_{l-1,n}b_{l-1,n}+b_{l-1,n}b_{l-1,n}-2b^{\dagger }_{l,n}b^{\dagger }_{l-1,n}\nonumber \\&\qquad -2b^{\dagger }_{l,n}b_{l-1,n}-2b_{l,n}b^{\dagger }_{l-1,n}\nonumber \\&\qquad -2b_{l,n}b_{l-1,n}+b_{l-1,n}b^{\dagger }_{l-1,n}) \end{aligned}$$
(58)
$$\begin{aligned}&u^2_{l,n-1}-2u_{l,n}u_{l,n-1}=\left( \frac{\hbar }{2m\omega _0}\right) (b^{\dagger }_{l,n-1}b^{\dagger }_{l,n-1}\nonumber \\&\qquad +b^{\dagger }_{l,n-1}b_{l,n-1} +b_{l,n-1}b_{l,n-1}-2b^{\dagger }_{l,n}b^{\dagger }_{l,n-1}\nonumber \\&\qquad -2b^{\dagger }_{l,n}b_{l,n-1}-2b_{l,n}b^{\dagger }_{l,n-1}\nonumber \\&\qquad -2b_{l,n}b_{l,n-1}+b_{l,n-1}b^{\dagger }_{l,n-1}). \end{aligned}$$
(59)

Inserting (54), (55), (56), (57), (58) and (59) into (1), we have:

$$\begin{aligned}&{H_{eff}}= \sum _{l}\sum _{n}\frac{\hbar \omega _{0}}{4}\Big (b_{l,n}^{\dagger }b_{l,n}+ b_{l,n}b_{l,n}^{\dagger }\Big )\nonumber \\&\quad + \sum _{l}\sum _{n}\frac{\hbar c_{x}}{4m\omega _{0}}\Big [b_{l,n}^{\dagger }b_{l,n}+b_{l,n}b_{l,n}^{\dagger }+ b_{l-1,n}^{\dagger }b_{l-1,n}\nonumber \\&\quad + b_{l-1,n}b_{l-1,n}^{\dagger }-2 b_{l,n}^{\dagger }b_{l-1,n}-2b_{l,n}b_{l-1,n}^{\dagger }\Big ]\nonumber \\&\quad + \sum _{l}\sum _{n}\frac{\hbar c_{y}}{4m\omega _{0}}\Big [b_{l,n}^{\dagger }b_{l,n}+ b_{l,n}b_{l,n}^{\dagger }+ b_{l,n-1}^{\dagger }b_{l,n-1} \nonumber \\&\quad +b_{l,n-1}b_{l,n-1}^{\dagger }-2 b_{l,n}^{\dagger }b_{l,n-1}-2b_{l,n}b_{l,n-1}^{\dagger }\Big ]\nonumber \\&\quad + \sum _{l}\sum _{n}\frac{\hbar k_{2}}{4m\omega _{0}}\Big (b_{l,n}^{\dagger }b_{l,n}+ b_{l,n}b_{l,n}^{\dagger }\Big )\nonumber \\&\quad +\sum _{l}\sum _{n}\frac{k_{4}}{4} \Big (\frac{\hbar }{2m\omega _{0}}\Big )^{2}\Big [6b_{l,n}^{\dagger }b_{l,n}^{\dagger }b_{l,n}b_{l,n}+ 12b_{l,n}^{\dagger }b_{l,n}+3\Big ]\nonumber \\&H^{'}=-\sum _{l}\sum _{n}\frac{\hbar \omega _{0}}{4}\Big (b_{l,n}^{\dagger }b_{l,n}^{\dagger }+ b_{l,n}b_{l,n}\Big )\nonumber \\&\quad + \sum _{l}\sum _{n}\frac{\hbar k_{2}}{4m\omega _{0}}\Big (b_{l,n}^{\dagger }b_{l,n}^{\dagger }+ b_{l,n}b_{l,n}\Big )\nonumber \\&\quad +\sum _{l}\sum _{n}\frac{k_{4}}{4} \Big (\frac{\hbar }{2m\omega _{0}}\Big )^{2}\Big [b_{l,n}^{\dagger } b_{l,n}^{\dagger }b_{l,n}^{\dagger }b_{l,n}^{\dagger }\nonumber \\&\quad +b_{l,n}^{\dagger }b_{l,n}^{\dagger }b_{l,n}^{\dagger }b_{l,n} +b_{l,n}^{\dagger }b_{l,n}^{\dagger }b_{l,n}b_{l,n}^{\dagger } +b_{l,n}b_{l,n}b_{l,n}b_{l,n}\nonumber \\&\quad +b_{l,n}b_{l,n}b_{l,n}^{\dagger }b_{l,n}+b_{l,n}b_{l,n}b_{l,n}b_{l,n}^{\dagger } +b_{l,n}^{\dagger }b_{l,n}b_{l,n}^{\dagger }b_{l,n}^{\dagger }\nonumber \\&\quad +b_{l,n}^{\dagger }b_{l,n}b_{l,n}b_{l,n} +b_{l,n}b_{l,n}^{\dagger }b_{l,n}^{\dagger }b_{l,n}^{\dagger } +b_{l,n}b_{l,n}^{\dagger }b_{l,n}b_{l,n}\Big ]\nonumber \\&\quad +\sum _{l}\sum _{n}\frac{\hbar c_{x}}{4m\omega _{0}}\Big [b_{l,n}^{\dagger }b_{l,n}^{\dagger }+ b_{l,n}b_{l,n}+ b_{l-1,n}^{\dagger }b_{l-1,n}^{\dagger } \nonumber \\&\quad + b_{l-1,n}b_{l-1,n}-b_{l-1,n}^{\dagger }b_{l,n}^{\dagger }-b_{l-1,n}b_{l,n}\nonumber \\&\quad -b_{l,n}^{\dagger }b_{l-1,n}^{\dagger }-b_{l,n}b_{l-1,n}\Big ]\nonumber \\&\quad +\sum _{l}\sum _{n}\frac{\hbar c_{y}}{4m\omega _{0}}\Big [b_{l,n}^{\dagger }b_{l,n}^{\dagger }\nonumber \\&\quad + b_{l,n}b_{l,n}+ b_{l,n-1}^{\dagger }b_{l,n-1}^{\dagger }\nonumber \\&\quad +b_{l,n-1}b_{l,n-1}-b_{l,n-1}^{\dagger }b_{l,n}^{\dagger }-b_{l,n-1}b_{l,n}\nonumber \\&\quad -b_{l,n}^{\dagger }b_{l,n-1}^{\dagger }-b_{l,n}b_{l,n-1}\Big ]\nonumber \\&\quad i\frac{d\langle b_{l, n}\rangle }{dt}=\langle [b_{l, n}, H_{eff}]\rangle \end{aligned}$$
(60)
$$\begin{aligned}&{[}b_{l,n},H_{eff}]=\sum _{l}\sum _{n}\frac{\hbar {\omega _0}}{4}[b_{l,n}, b^{\dagger }_{l,n}b_{l,n}+b_{l,n}b^{\dagger }_{l,n}]\nonumber \\&\quad +\sum _{l}\sum _{n} \frac{\hbar {c_x}}{4m\omega _0}[b_{l,n},b^{\dagger }_{l,n}b_{l,n}+b_{l,n}b^{\dagger }_{l,n}]\nonumber \\&\quad +\sum _{l}\sum _{n}\frac{\hbar {c_x}}{4m\omega _0}[b_{l,n},b^{\dagger }_{l-1,n} b_{l-1,n}+b_{l-1,n}b^{\dagger }_{l-1,n}]\nonumber \\&\quad -\sum _{l}\sum _{n} \frac{\hbar {c_x}}{4m\omega _0}[b_{l,n},2(b^{\dagger }_{l,n}b_{l-1,n}+b_{l,n}b^{\dagger }_{l-1,n})]\nonumber \\&\quad +\sum _{l}\sum _{n}\frac{\hbar {c_y}}{4m\omega _0}[b_{l,n}, b^{\dagger }_{l,n}b_{l,n}+b_{l,n}b^{\dagger }_{l,n}]\nonumber \\&\quad +\sum _{l}\sum _{n} \frac{\hbar {c_y}}{4m\omega _0}[b_{l,n},b^{\dagger }_{l,n-1}b_{l,n-1} +b_{l,n-1}b^{\dagger }_{l,n-1}]\nonumber \\&\quad -\sum _{l}\sum _{n}\frac{\hbar {c_y}}{4m\omega _0}[b_{l,n},2(b^{\dagger }_{l,n}b_{l,n-1} +b_{l,n}b^{\dagger }_{l,n-1})]\nonumber \\&\quad +\sum _{l}\sum _{n}\frac{k_2\hbar }{4m\omega _0} [b_{l,n},b^{\dagger }_{l,n}b_{l,n}+b_{l,n}b^{\dagger }_{l,n}]\nonumber \\&\quad +\sum _{l}\sum _{n}\frac{k_4}{4}\left( \frac{\hbar }{2m\omega _0}\right) ^2[b_{l,n}, 6b^{\dagger }_{l,n}b^{\dagger }_{l,n}b_{l,n}b_{l,n}]\nonumber \\&\quad +\sum _{l}\sum _{n}\frac{k_4}{4}\left( \frac{\hbar }{2m\omega _0}\right) ^2 \left( [b_{l,n},12b^{\dagger }_{l,n}b_{l,n}]+[b_{l,n},3]\right) \end{aligned}$$
(61)
$$\begin{aligned}&i\hbar \frac{\partial {b_{l,n}}}{\partial {t}} =\hbar \bigg [\frac{\omega _0}{2}b_{l,n}+\frac{c_x}{m\omega _0}b_{l,n}\nonumber \\&\quad +\frac{c_y}{m\omega _0}b_{l,n}+3k_4\hbar \left( \frac{1}{2m\omega _0}\right) ^2b_{l,n}\nonumber \\&\quad +\frac{k_2}{2m\omega _0}b_{l,n}\nonumber \\&\quad -\frac{c_x}{2m\omega _0}(b_{l,n+1}-b_{l,n-1}) -\frac{c_y}{2m\omega _0}(b_{l+1,n}-b_{l-1,n})\nonumber \\&\quad +3k_4\hbar \left( \frac{1}{2m\omega _0}\right) ^2b^{\dagger }_{l,n}b_{l,n}b_{l,n}\bigg ] \end{aligned}$$
(62)

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Ribama, R.A.A., Djoufack, Z.I. & Nguenang, J.P. Breather-impurity interactions and modulational instability in a quantum 2D Klein–Gordon chain. Eur. Phys. J. B 95, 86 (2022). https://doi.org/10.1140/epjb/s10051-022-00337-6

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