中国物理B ›› 2019, Vol. 28 ›› Issue (1): 10501-010501.doi: 10.1088/1674-1056/28/1/010501

• SPECIAL TOPIC—Recent advances in thermoelectric materials and devices • 上一篇    下一篇

Solitons in nonlinear systems and eigen-states in quantum wells

Li-Chen Zhao(赵立臣), Zhan-Ying Yang(杨战营), Wen-Li Yang(杨文力)   

  1. 1 School of Physics, Northwest University, Xi'an 710069, China;
    2 Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi'an 710069, China;
    3 Institute of Modern Physics, Northwest University, Xi'an 710069, China
  • 收稿日期:2018-10-09 修回日期:2018-10-24 出版日期:2019-01-05 发布日期:2019-01-05
  • 通讯作者: Li-Chen Zhao E-mail:zhaolichen3@nwu.edu.cn
  • 基金资助:

    Project supported by the National Natural Science Foundation of China (Grant No. 11775176), the Basic Research Program of Natural Science of Shaanxi Province, China (Grant No. 2018KJXX-094), the Key Innovative Research Team of Quantum Many-Body Theory and Quantum Control in Shaanxi Province, China (Grant No. 2017KCT-12), and the Major Basic Research Program of Natural Science of Shaanxi Province, China (Grant No. 2017ZDJC-32).

Solitons in nonlinear systems and eigen-states in quantum wells

Li-Chen Zhao(赵立臣)1,2, Zhan-Ying Yang(杨战营)1,2, Wen-Li Yang(杨文力)1,2,3   

  1. 1 School of Physics, Northwest University, Xi'an 710069, China;
    2 Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi'an 710069, China;
    3 Institute of Modern Physics, Northwest University, Xi'an 710069, China
  • Received:2018-10-09 Revised:2018-10-24 Online:2019-01-05 Published:2019-01-05
  • Contact: Li-Chen Zhao E-mail:zhaolichen3@nwu.edu.cn
  • Supported by:

    Project supported by the National Natural Science Foundation of China (Grant No. 11775176), the Basic Research Program of Natural Science of Shaanxi Province, China (Grant No. 2018KJXX-094), the Key Innovative Research Team of Quantum Many-Body Theory and Quantum Control in Shaanxi Province, China (Grant No. 2017KCT-12), and the Major Basic Research Program of Natural Science of Shaanxi Province, China (Grant No. 2017ZDJC-32).

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摘要：

We study the relations between solitons of nonlinear Schrödinger equation and eigen-states of linear Schrödinger equation with some quantum wells. Many different non-degenerated solitons are re-derived from the eigen-states in the quantum wells. We show that the vector solitons for the coupled system with attractive interactions correspond to the identical eigen-states with the ones of the coupled systems with repulsive interactions. Although their energy eigenvalues seem to be different, they can be reduced to identical ones in the same quantum wells. The non-degenerated solitons for multi-component systems can be used to construct much abundant degenerated solitons in more components coupled cases. Meanwhile, we demonstrate that soliton solutions in nonlinear systems can also be used to solve the eigen-problems of quantum wells. As an example, we present the eigenvalue and eigen-state in a complicated quantum well for which the Hamiltonian belongs to the non-Hermitian Hamiltonian having parity-time symmetry. We further present the ground state and the first exited state in an asymmetric quantum double-well from asymmetric solitons. Based on these results, we expect that many nonlinear physical systems can be used to observe the quantum states evolution of quantum wells, such as a water wave tank, nonlinear fiber, Bose-Einstein condensate, and even plasma, although some of them are classical physical systems. These relations provide another way to understand the stability of solitons in nonlinear Schrödinger equation described systems, in contrast to the balance between dispersion and nonlinearity.

关键词: non-degenerated soliton, eigen-states, quantum well, nonlinear Schrö, dinger equation

Abstract:

We study the relations between solitons of nonlinear Schrödinger equation and eigen-states of linear Schrödinger equation with some quantum wells. Many different non-degenerated solitons are re-derived from the eigen-states in the quantum wells. We show that the vector solitons for the coupled system with attractive interactions correspond to the identical eigen-states with the ones of the coupled systems with repulsive interactions. Although their energy eigenvalues seem to be different, they can be reduced to identical ones in the same quantum wells. The non-degenerated solitons for multi-component systems can be used to construct much abundant degenerated solitons in more components coupled cases. Meanwhile, we demonstrate that soliton solutions in nonlinear systems can also be used to solve the eigen-problems of quantum wells. As an example, we present the eigenvalue and eigen-state in a complicated quantum well for which the Hamiltonian belongs to the non-Hermitian Hamiltonian having parity-time symmetry. We further present the ground state and the first exited state in an asymmetric quantum double-well from asymmetric solitons. Based on these results, we expect that many nonlinear physical systems can be used to observe the quantum states evolution of quantum wells, such as a water wave tank, nonlinear fiber, Bose-Einstein condensate, and even plasma, although some of them are classical physical systems. These relations provide another way to understand the stability of solitons in nonlinear Schrödinger equation described systems, in contrast to the balance between dispersion and nonlinearity.

Key words: non-degenerated soliton, eigen-states, quantum well, nonlinear Schrö, dinger equation

中图分类号:  (Solitons)

  • 05.45.Yv
02.30.Ik (Integrable systems) 42.65.Tg (Optical solitons; nonlinear guided waves)