原子-场-介质相互作用 I：介质改变量子场中 $N$ 个谐波原子的分级影响作用
Atom-Field-Medium Interactions I: Graded Influence Actions for $N$ Harmonic Atoms in a Dielectric-Altered Quantum Field
AI总结：
本文发展分级影响作用形式体系，通过三层粗粒化推导原子-场-介质系统中原子内部自由度的随机运动方程，揭示介质非马尔可夫关联及频率相关介电常数的影响。
摘要：
本系列论文有两个更广泛的目标：1) 构建一个包含多个结构层次、具有自洽反作用的多部分开放量子系统理论。发展分级影响作用形式体系 \cite{BehHu10,BH11}，以考虑连续子层对感兴趣变量动力学的影响。2) 将这些方法应用于原子-场-介质相互作用的研究，并突出其相对于传统方法的优势。我们考虑一个由 $N$ 个谐波振子组成的系统，模拟 $N$ 个中性原子（A）的内部自由度（idf），与一个量子场（F）相互作用，此处为简单起见采用标量场，该场因介电介质（M）的存在而改变。在本文中，我们使用影响泛函形式体系中的粗粒化和随机有效作用，推导连续结构层中动力学变量约化密度矩阵的随机方程。“分级”一词指的是粗粒化过程的具体排序。执行了三层粗粒化：首先，对介电振子的公共浴进行积分，不仅导致介电必要耗散性质的出现，还产生了关键细微特征，如介电中的非马尔可夫空间关联。其次，对介质变量整体进行积分，得到一个介质修正的量子场，介质对量子场的影响通过频率相关的介电常数函数体现。最后，对这个与原子idf相互作用的介质改变量子场进行积分，得到一个影响作用。由此我们得到随机运动方程，描述了与介质修正量子场相互作用的原子idf的非平衡随机动力学。
英文摘要：
This series of papers has two broader aims: 1) Construct a theory for multi-partite open quantum systems comprising several layers of structure with self-consistent back-actions. Develop the graded influence action formalism \cite{BehHu10,BH11} to account for the influences of successive sub-layers on the dynamics of the variables of interest. 2) Apply these methods to the study of atom-field-medium interactions and highlight their merits over conventional methods. We consider a system of $N$ harmonic oscillators, modeling the internal degrees of freedom (idf) of $N$ neutral atoms (A), interacting with a quantum field (F), scalar here, for simplicity, altered by the presence of a dielectric medium (M). In this paper we use the coarse-grained and stochastic effective actions in the influence functional formalism to derive the stochastic equations for the reduced density matrices of the dynamical variables in the successive layers of structure. The word `graded' refers to the specific ordering of the coarse-graining procedures. Three layers of coarse-graining are performed, firstly, integrating over the common bath of the dielectric oscillators results not only in the appearance of necessary dissipative properties of the dielectric but also essential nuanced features such as nonMarkovian spatial correlations in the dielectric. Secondly, integrating over the medium variables as a whole results in a dielectric-modified quantum field, the influence of the medium on the quantum field manifesting through a frequency-dependent permittivity function. Finally, integrating over this dielectric-altered quantum field which interacts with the idfs of the atoms yields an influence action. From it we obtain the stochastic equation of motion which describes the nonequilibrium stochastic dynamics of the idf of the atoms interacting with a dielectric medium-modified quantum field.
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