开放量子系统例外点的分析技术与不可逆性出现的QPT类比
Analysis technique for exceptional points in open quantum systems and QPT analogy for the appearance of irreversibility
摘要：
提出了一种分析开放量子系统（OQS）离散谱中异常点（EP）的方法.我们概述了我们的方法来定位OQS中的EP，进一步获得的EP，从而产生的特征指数附近的本征值展开。我们还报告了发生在OQS中的EP的精确数量，该OQS具有由二次色散曲线描述的连续体。特别地，在维度为$n_\textrm{D}$的裸离散哈密顿量中出现的EP的数量由$n_\textrm{D} (n_\textrm{D} - 1)$给出;如果这个离散的哈密顿量然后耦合到连续统（或连续体）以形成OQS，与连续体的相互作用通常产生扩大的离散解空间，其包括更多数量的EP，具体地说，$2^{n_\textrm{C}} (n_\textrm{C} + n_\textrm{D}) [2^{n_\textrm{C}} (n_\textrm{C} + n_\textrm{D}) - 1] $，其中$n_\textrm{C}$是离散扇区所连接的（非退化）连续统的数量。最后，我们提供了一个启发式的量子相变类比的共振（通过指数衰减引起不可逆性）的出现，其中的衰减宽度起着序参量的作用，相关的临界指数，然后由上述本征值展开。
英文摘要：
We propose an analysis technique for the exceptional points (EPs) occurring in the discrete spectrum of open quantum systems (OQS), using a semi-infinite chain coupled to an endpoint impurity as a prototype. We outline our method to locate the EPs in OQS, further obtaining an eigenvalue expansion in the vicinity of the EPs that gives rise to characteristic exponents. We also report the precise number of EPs occurring in an OQS with a continuum described by a quadratic dispersion curve. In particular, the number of EPs occurring in a bare discrete Hamiltonian of dimension $n_\textrm{D}$ is given by $n_\textrm{D} (n_\textrm{D} - 1)$; if this discrete Hamiltonian is then coupled to continuum (or continua) to form an OQS, the interaction with the continuum generally produces an enlarged discrete solution space that includes a greater number of EPs, specifically $2^{n_\textrm{C}} (n_\textrm{C} + n_\textrm{D}) [2^{n_\textrm{C}} (n_\textrm{C} + n_\textrm{D}) - 1] $, in which $n_\textrm{C}$ is the number of (non-degenerate) continua to which the discrete sector is attached. Finally, we offer a heuristic quantum phase transition analogy for the emergence of the resonance (giving rise to irreversibility via exponential decay) in which the decay width plays the role of the order parameter; the associated critical exponent is then determined by the above eigenvalue expansion.
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