Bosonic and fermionic Gaussian states from Kähler structures
Bosonic and fermionic Gaussian states from Kähler structures
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DOI：
10.21468/scipostphyscore.4.3.025
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发表时间：
2020-10
影响因子：
3.6
通讯作者：
L. Hackl;Eugenio Bianchi
中科院分区：
文献类型：
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作者：
L. Hackl;Eugenio Bianchi
我们表明，玻色子和费米子高斯态（也被称为“压缩相干态”）可以唯一的特点是他们的线性复杂的结构JJ这是一个线性映射的经典相空间。这扩展了基于协方差矩阵的传统高斯方法，并提供了一个统一的框架来同时处理玻色子和费米子。纯高斯态可以用三元组（G，\Omega，J）（G，Ω，J）的相容Kähler结构来识别，包括正定度量GG、辛形式\OmegaΩ和线性复结构JJ，其中J^2=-\mathbb{1}J2=−1。混合高斯态也可以用这样的三元组来识别，但用J^2\neq -\mathbb{1}J2 <$−1。我们应用这些方法来显示如何计算涉及高斯状态可以减少到这些对象的代数运算，导致许多已知的和一些未知的身份。我们将这些方法应用于研究（A）纠缠和复杂性，（B）稳定系统的动力学，（C）驱动系统的动力学。由此，我们编制了一个全面的数学结构和公式列表，以并排比较玻色子和费米子高斯态。
We show that bosonic and fermionic Gaussian states (also known as ``squeezed coherent states’’) can be uniquely characterized by their linear complex structure JJ which is a linear map on the classical phase space. This extends conventional Gaussian methods based on covariance matrices and provides a unified framework to treat bosons and fermions simultaneously. Pure Gaussian states can be identified with the triple (G,\Omega,J)(G,Ω,J) of compatible Kähler structures, consisting of a positive definite metric GG, a symplectic form \OmegaΩ and a linear complex structure JJ with J^2=-\mathbb{1}J2=−1. Mixed Gaussian states can also be identified with such a triple, but with J^2\neq -\mathbb{1}J2≠−1. We apply these methods to show how computations involving Gaussian states can be reduced to algebraic operations of these objects, leading to many known and some unknown identities. We apply these methods to the study of (A) entanglement and complexity, (B) dynamics of stable systems, (C) dynamics of driven systems. From this, we compile a comprehensive list of mathematical structures and formulas to compare bosonic and fermionic Gaussian states side-by-side.