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

在非常低的能量和很高的相空间密度下，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。在这个可以在冷原子气体或(假设的)暗物质晕这样不同的系统中实现的区域中，需要一个量子场理论描述。用基本场自由度表示的理论，例如冷原子气体的玻色场，通常需要在高度非微扰近似下进行评估，以解释伴随而来的强关联。然而，在这种低能量状态下，集体物质-波动力学占上风，这主要编码在量子场相位的相干性中。相“存在”在一个圆上，而不是在实部和虚部的笛卡尔坐标系中。考虑到这一点的低能量有效模型代表了用基本场描述的一种有用的替代方法，因为它们可以描述已经在微扰近似中的感兴趣的相关性，这更容易评估。在这里，我们考虑了由非线性激发引起的强关联，如拓扑缺陷系综以及其他无序的量子相和密度自由度的激发。这个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的超冷玻色气体上。在对具有强位相激发的U(N)对称系统的前期工作的基础上，我们将把我们的方法应用于不同的模型。这将包括通过对偶变换获得的含缺陷玻色气体的描述。后者导致了动力学的统计公式，该公式允许计算场相关性，而不跟踪系统中单个缺陷的运动学。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的系统，通过近似最大化信息的Rényi熵来产生它们的相关性。主要目的是确定这种与Tsallis的非广泛热统计密切相关的方法与从动力学方程确定标度解之间的关系。考虑到在远离平衡的动力学和量子系统的热化或平衡领域仍有许多悬而未决的问题，我们的目标是高度相关的。

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.