论文标题
阿贝尔和非阿贝尔量子几何张量的相空间公式
Phase space formulation of the Abelian and non-Abelian quantum geometric tensor
论文作者
论文摘要
参数空间的几何形状由量子几何张量编码，该量子张量捕获有关量子状态的基本信息，并包含量子度量张量和浆果连接的曲率。我们介绍了浆果连接和相位空间或Wigner函数形式框架中的量子几何张量的表述。该公式是通过将Weyl对应关系直接应用于所考虑的几何结构来获得的。特别是，我们表明只能使用Wigner函数来计算量子度量张量，这为实验测量该张量的组件打开了另一种方法。我们还谈到了非阿布尔的概括，并获得了Wilczek-Zee连接和非亚伯利亚量子几何张量的相空间公式。在这种情况下，非亚伯量子公制张量仅涉及非对角线的函数。然后，我们通过示例验证我们的方法，并将其应用于$ n $耦合的谐波振荡器的系统，表明相关的浆果连接消失并获得量子公制张量的分析表达式。我们的结果表明，开发的方法非常适合研究与量子多体系统相关的参数空间。
The geometry of the parameter space is encoded by the quantum geometric tensor, which captures fundamental information about quantum states and contains both the quantum metric tensor and the curvature of the Berry connection. We present a formulation of the Berry connection and the quantum geometric tensor in the framework of the phase space or Wigner function formalism. This formulation is obtained through the direct application of the Weyl correspondence to the geometric structure under consideration. In particular, we show that the quantum metric tensor can be computed using only the Wigner functions, which opens an alternative way to experimentally measure the components of this tensor. We also address the non-Abelian generalization and obtain the phase space formulation of the Wilczek-Zee connection and the non-Abelian quantum geometric tensor. In this case, the non-Abelian quantum metric tensor involves only the non-diagonal Wigner functions. Then, we verify our approach with examples and apply it to a system of $N$ coupled harmonic oscillators, showing that the associated Berry connection vanishes and obtaining the analytic expression for the quantum metric tensor. Our results indicate that the developed approach is well adapted to study the parameter space associated with quantum many-body systems.
