Departamento de Física Teórica, GIyA, Comisión Nacional de Energía Atómica, Avenida del Libertador 8250, 1429 Buenos Aires, Argentina.
Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Buenos Aires, Godoy Cruz 2290, C1425FQB, CABA, Argentina.
Phys Rev E. 2019 Jan;99(1-1):012215. doi: 10.1103/PhysRevE.99.012215.
We introduce and study the extension of the Chirikov standard map when the kick potential has two and three incommensurate spatial harmonics. This system is called the incommensurate standard map. At small kick amplitudes, the dynamics is bounded by the isolating Kolmogorov-Arnold-Moser surfaces, whereas above a certain kick strength, it becomes unbounded and diffusive. The quantum evolution at small quantum kick amplitudes is somewhat similar to the case of the Aubru-André model studied in mathematics and experiments with cold atoms in a static incommensurate potential. We show that for the quantum map there is also a metal-insulator transition in space whereas in momentum we have localization similar to the case of two-dimensional Anderson localization. In the case of three incommensurate frequencies of the space potential, the quantum evolution is characterized by the Anderson transition similar to the three-dimensional case of the disordered potential. We discuss possible physical systems with such a map description including dynamics of comets and dark matter in planetary systems.
我们介绍并研究了 Chirikov 标准映射的扩展，当踢力势具有两个和三个非简谐空间谐波时。这个系统被称为非简谐标准映射。在小的踢力幅度下，动力学被隔离的 Kolmogorov-Arnold-Moser 表面所限制，而在一定的踢力强度以上，它变得无界和扩散。在小的量子踢力幅度下的量子演化在某种程度上类似于数学中的 Aubru-André 模型的情况，以及在静态非简谐势中冷原子的实验。我们表明，对于量子映射，在空间中也存在金属-绝缘体转变，而在动量中，我们具有类似于二维 Anderson 局域化的局域化。在空间势的三个非简谐频率的情况下，量子演化的特点是类似于无序势的三维情况下的 Anderson 转变。我们讨论了具有这种映射描述的可能的物理系统，包括彗星和行星系统中暗物质的动力学。
