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P T $\mathcal {P}\mathcal {T}$ -symmetry in Compact Phase Space for a Linear Hamiltonian
International Journal of Theoretical Physics ( IF 1.4 ) Pub Date : 2021-07-28 , DOI: 10.1007/s10773-021-04905-x
Ivan F. Valtierra 1 , Mario B. Gaeta 1 , Adrian Ortega 1 , Thomas Gorin 1
Affiliation  

We study the time evolution of a \(\mathcal {P}\mathcal {T}\)-symmetric, non-Hermitian quantum system for which the associated phase space is compact. We focus on the simplest non-trivial example of such a Hamiltonian, which is linear in the angular momentum operators. In order to describe the evolution of the system, we use a particular disentangling decomposition of the evolution operator, which remains numerically accurate even in the vicinity of the Exceptional Point. We then analyze how the non-Hermitian part of the Hamiltonian affects the time evolution of two archetypical quantum states, coherent and Dicke states. For that purpose we calculate the Husimi distribution or Q function and study its evolution in phase space. For coherent states, the characteristics of the evolution equation of the Husimi function agree with the trajectories of the corresponding angular momentum expectation values. This allows to consider these curves as the trajectories of a classical system. For other types of quantum states, e.g. Dicke states, the equivalence of characteristics and trajectories of expectation values is lost.



中文翻译:

PT $\mathcal {P}\mathcal {T}$ - 线性哈密顿量在紧凑相空间中的对称性

我们研究了一个\(\mathcal {P}\mathcal {T}\)对称的非厄米量子系统的时间演化,其中相关的相空间是紧凑的。我们专注于这种哈密顿量的最简单的非平凡示例,它在角动量算子中是线性的。为了描述系统的演化,我们使用演化算子的​​特定解缠分解分解,即使在异常点附近也保持数值准确。然后我们分析哈密顿量的非厄米部分如何影响两个原型量子态,相干态和迪克态的时间演化。为此,我们计算 Husimi 分布或Q函数并研究其在相空间中的演化。对于相干态,胡西米函数的演化方程特征与相应角动量期望值的轨迹一致。这允许将这些曲线视为经典系统的轨迹。对于其他类型的量子态,例如 Dicke 态,特性的等价性和期望值的轨迹会丢失。

更新日期:2021-07-29
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