量子拉比模型的热核

IF 1 4区 物理与天体物理 Q3 PHYSICS, MATHEMATICAL Advances in Theoretical and Mathematical Physics Pub Date : 2019-06-23 DOI:10.4310/ATMP.2022.v26.n5.a8
Cid Reyes-Bustos, M. Wakayama
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引用次数: 6

摘要

量子拉比模型(QRM)被广泛认为是量子光学中一个特别重要的模型。它被认为是描述量子光-物质相互作用的最简单和最基本的系统。本文的目的是利用无穷级数的迭代积分,给出哈密顿函数的热核的解析表达式。该公式的推导是基于对Trotter-Kato积公式的直接评估,而不使用费曼-卡茨路径积分。更准确地说,热核表达式中的无限和源于将Trotter-Kato积公式简化为无限对称群$\mathfrak{S}_\infty$对群$\mathbb{Z}_2^{\infty}$的作用的轨道和,然后将迭代积分视为每个轨道的轨道积分。这里,组$ \mathbb{Z}_2^{\infty} $和$\mathfrak{S}_\infty$分别是族$\{\mathbb{Z}_2^n\}_{n\geq0}$和族$\{\mathfrak{S}_n\}_{n\geq0}$的归纳极限。为了完成约化,对阿贝尔群$\mathbb{Z}_2^n\, (n \geq0)$归纳族的调和(傅立叶)分析的广泛研究与图论研究是至关重要的。据作者所知,这是获得非平凡现实相互作用量子系统的热核封闭公式的第一个显式计算。该模型的热核进一步由一个2乘2的矩阵值函数给出,并表示为两个各自的热核的直接和,表示哈密顿量通过宇称分解的宇称($\mathbb{Z}_2$ -对称)。
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Heat kernel for the quantum Rabi model
The quantum Rabi model (QRM) is widely recognized as a particularly important model in quantum optics. It is considered to be the simplest and most fundamental system describing quantum light-matter interaction. The objective of the paper is to give an analytical formula of the heat kernel of the Hamiltonian explicitly by infinite series of iterated integrals. The derivation of the formula is based on the direct evaluation of the Trotter-Kato product formula without the use of Feynman-Kac path integrals. More precisely, the infinite sum in the expression of the heat kernel arises from the reduction of the Trotter-Kato product formula into sums over the orbits of the action of the infinite symmetric group $\mathfrak{S}_\infty$ on the group $\mathbb{Z}_2^{\infty}$, and the iterated integrals are then considered as the orbital integral for each orbit. Here, the groups $ \mathbb{Z}_2^{\infty} $ and $\mathfrak{S}_\infty$ are the inductive limit of the families $\{\mathbb{Z}_2^n\}_{n\geq0}$ and $\{\mathfrak{S}_n\}_{n\geq0}$, respectively. In order to complete the reduction, an extensive study of harmonic (Fourier) analysis on the inductive family of abelian groups $\mathbb{Z}_2^n\, (n \geq0)$ together with a graph theoretical investigation is crucial. To the best knowledge of the authors, this is the first explicit computation for obtaining a closed formula of the heat kernel for a non-trivial realistic interacting quantum system. The heat kernel of this model is further given by a two-by-two matrix valued function and is expressed as a direct sum of two respective heat kernels representing the parity ($\mathbb{Z}_2$-symmetry) decomposition of the Hamiltonian by parity.
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来源期刊
Advances in Theoretical and Mathematical Physics
Advances in Theoretical and Mathematical Physics 物理-物理:粒子与场物理
CiteScore
2.20
自引率
6.70%
发文量
0
审稿时长
>12 weeks
期刊介绍: Advances in Theoretical and Mathematical Physics is a bimonthly publication of the International Press, publishing papers on all areas in which theoretical physics and mathematics interact with each other.
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