Low-energy effective theory of far-from-equilibrium time-evolving Bose systems

在非常低的能量和高的相空间密度下，物质的波动性质是理解相互作用的多粒子系统的集体量子动力学的关键。在这种情况下，它可以在冷原子气体或（假设的）暗物质晕等不同的系统中实现，需要量子场论描述。根据基本场自由度制定的理论，例如，冷原子气体玻色场，通常需要在高度非微扰近似下进行评估，以解释伴随的强相关性。然而，在低能量状态下，集体物质波动力学占主导地位，这主要是在量子场相位的相干性中编码的。相位“生活”在一个圆上，而不是在真实的和虚部的笛卡尔坐标系中。考虑到这一点的低能有效模型代表了一种有用的替代方法来描述基本场，因为它们可以描述已经在微扰近似中的相关性，这更容易评估。在这里，我们考虑强相关性所造成的非线性激发，如合奏的拓扑缺陷和其他，较不有序的量子相位和密度自由度的激发。该项目的中心目标是开发低能有效场模型，用于描述远离平衡的动力学。重点将放在单组分和多组分超冷玻色气体上。基于U（N）对称系统的强相位激励的初步工作，我们将把我们的方法应用到不同的模型。这将包括通过对偶变换获得的含缺陷玻色气体的描述。后者导致的统计制定的动态，允许计算场的相关性，而不跟踪系统中的单个缺陷的运动学。该项目的另一个目标是开发一种变分统计方法，用于在空间和时间上普遍缩放的远离平衡系统，通过近似最大化信息的雷尼熵来产生它们的相关性。我们的主要目标是确定这种方法之间的关系，紧密相连的Tsallis的非广泛的热统计学，并确定从动力学方程的缩放解决方案。我们的目标是高度相关的，因为在远离平衡态的动力学和量子系统的热化或平衡态领域仍然有许多悬而未决的问题。

At very low energies and high phase-space densities, the wave nature of matter is a key to understanding the collective quantum dynamics of interacting many-particle systems. In this regime, which can be realized in systems as different as cold atomic gases or (hypothetical) dark-matter halos, a quantum field theoretical description is required. Theories formulated in terms of the fundamental field degrees of freedom, e.g., the Bose field of a cold-atom gas, typically need to be evaluated in highly non-perturbative approximations to account for the concomitant strong correlations. In that low-energy regime, however, collective matter-wave dynamics prevails and this is encoded mainly in the coherence of the phase of the quantum field. The phase “lives” on a circle rather than in the cartesian coordinate system of the real and imaginary parts. Low-energy effective models taking this into account represent a useful alternative approach to descriptions in terms of fundamental fields, as they can describe the correlations of interest already in perturbative approximations, which are easier to evaluate. Here we consider strong correlations which are caused by non-linear excitations such as ensembles of topological defects and other, less ordered excitations of the quantum phase and density degrees of freedom. The central goal of this project is to develop low-energy effective field models to be used for describing dynamics far from equilibrium. The emphasis will be set on single- and multicomponent ultra cold Bose gases. Building on preliminary work for U(N) symmetric systems with strong phase excitations we will apply our methods to different models. This will include descriptions of defect-bearing Bose gases obtained through duality transformations. The latter lead to a statistical formulation of the dynamics which allows the computation of field correlations without tracing the kinematics of single defects in the system. A further goal of this project is to develop a variational statistical approach to far-from-equilibrium systems universally scaling in space and time, yielding their correlations by approximative maximization of a Rényi entropy of information. The main goal is to identify the relation between this approach, closely connected to Tsallis' non-extensive thermostatistics, and the determination of scaling solutions from kinetic equations. Our goals are highly relevant in view of the still many open questions in the realm of far-from-equilibrium dynamics and the thermalization or equilibration of quantum systems.