Quantum dynamics and localization in fluctuating potentials

凝聚态理论将有限温度介质中的原子和电子视为具有有限空间范围的对象，这些对象在时间演化过程中保持空间集中。这与薛定谔方程对许多相互作用的粒子系统的预测相反，在薛定谔方程中，粒子之间的相互作用导致不同多粒子状态的线性叠加。如何解决这个难题的问题与解释量子测量的问题密切相关，在量子测量中，时间演化导致被测量粒子的随机局部化，从而观察到可能的测量结果之一。在这个提议中，我们追求的假设是，热涨落使量子粒子在有限温度系统中局域化，热涨落被建模为经典的涨落势。这同样是由凝聚态物理推动的，在凝聚态物理中，超过一定距离的环境的影响通常被视为外势，即经典势。因此，我们考虑时空起伏势中电子的一维晶格模型。电子密度矩阵随时间演化的Lindblad方程可以用波函数的随机方程以多种不同的方式解开，其中一部分导致空间局域波函数，一部分导致空间扩展波函数。在这个项目中，我们将(I)比较这些不同的分解，并评估各自波函数的局域化和相干性的程度(Ii)包括来自波函数的势的反馈，使得时间演化在波函数中变为非线性。我们预计，这将解除不同解体之间的简并，并导致波函数保持有限空间扩展的动力学(III)讨论这些研究产生的有趣的哲学问题。这些问题涉及波函数和密度矩阵的解释，以及量子世界和经典世界之间的关系。

Condensed matter theory treats atoms and electrons in finite-temperature media as objects with a limited spatial extension that remain spatially concentrated under time evolution. This is in contrast to what the Schrödinger equation for a system of many interacting particles predicts, where the interaction between the particles leads to the linear superposition of different many-particle states. The question of how to resolve this puzzle is closely related to the problem of interpreting quantum measurement, where time evolution leads to a stochastic localization of the measured particle such that one of the possible measurement outcomes is observed. In this proposal, we pursue the hypothesis that thermal fluctuations localize quantum particles in a finite-temperature system, with thermal fluctuations being modeled as a classically fluctuating potential. This is again motivated by condensed matter physics, where the influence of the environment beyond some short distance is usually treated as an external potential, i.e., as a classical potential.We therefore consider a one-dimensional lattice model for an electron in a spatiotemporally fluctuating potential. The Lindblad equation for the time evolution of the density matrix of the electron can be unravelled in terms of a stochastic equation for the wave function in many different ways, part of which lead to spatially localized wave functions, and part of which lead to spatially extended wave functions. In this project, we will (i) compare these different unravellings and evaluate the extent of localization and phase coherence of the respective wave functions (ii) include a feedback from the wave function on the potential, such that the time evolution becomes nonlinear in the wave function. We expect that this will lift the degeneracy between the different unravellings and lead to a dynamics where the wave function retains a limited spatial extension(iii) discuss the interesting philosophical issues arising from these investigations. These issues related to the interpretation of wave functions and density matrices, and to the relation between the quantum and classical world.