Permuting quantum eigenmodes by a quasi-adiabatic motion of a potential wall
Permuting quantum eigenmodes by a quasi-adiabatic motion of a potential wall
复制标题
DOI：
10.1063/5.0005399
复制
发表时间：
2019-12
期刊：
影响因子：
--
通讯作者：
Alessandro Duca;R. Joly;D. Turaev
中科院分区：
文献类型：
--
作者：
Alessandro Duca;R. Joly;D. Turaev
研究了L^2（（0，1），\mathbb{C}）上的Schr\“odinger方程i\partial_t\psi=-\Delta\psi+V\psi$，其中V$是一个很高的定域势壁.我们的目标是执行置换的本征模式和控制方程的解决方案。我们考虑势壁的位置和高度变化的过程如下。首先，势从零增加到一个非常大的值，因此形成了一个狭窄的势墙，几乎将间隔分成两部分;然后墙移动到不同的位置，之后墙的高度再次衰减到零。我们表明，即使潜在的参数的变化率可以任意慢，这个过程交替绝热和非绝热动力学，导致一个非平凡的本征态的排列。此外，我们考虑了具有多个窄壁的势，并证明了壁的任意慢运动是如何将系统从任意给定的状态引导到任意小的邻域的，从而证明了上述Schr“odinger方程通过势的软准绝热变化的近似可控性.
We study the Schr\"odinger equation $i\partial_t\psi=-\Delta\psi+V\psi$ on $L^2((0,1),\mathbb{C})$ where $V$ is a very high and localized potential wall. We aim to perform permutations of the eigenmodes and to control the solution of the equation. We consider the process where the position and the height of the potential wall change as follows. First, the potential increases from zero to a very large value, so a narrow potential wall is formed that almost splits the interval into two parts; then the wall moves to a different position, after which the height of the wall decays to zero again. We show that even though the rate of the variation of the potential's parameters can be arbitrarily slow, this process alternates adiabatic and non-adiabatic dynamics, leading to a non-trivial permutation of the eigenstates. Furthermore, we consider potentials with several narrow walls and we show how an arbitrarily slow motion of the walls can lead the system from any given state to an arbitrarily small neighborhood of any other state, thus proving the approximate controllability of the above Schr\"odinger equation by means of a soft, quasi-adiabatic variation of the potential.