统一转子类量子系统的不确定性度量
Unifying uncertainties for rotor-like quantum systems
AI总结：
本文统一了量子转子中平移算符的多种不确定性度量，证明它们均可视为单一通用角不确定性度量的特例，并利用该框架推导了角动量与平移算符的最优联合测量，为量子信息与计量学应用奠定基础。
摘要：
量子转子是继谐振子之后，研究互补变量对——角动量与角动量空间中的幺正平移算符——的下一个自然量子系统。因此，对这些算符的不确定性及其不相容性进行恰当量化，对于转子类量子系统的应用至关重要。角动量的不确定性通常用方差来刻画，而对于平移算符，人们已提出多种不确定性度量，其中色散是最简单的例子。我们建立了这些度量之间的层级关系以及相应的不确定性关系，并且这些关系均可被一组层析完备的冯·米塞斯态完美或近乎完美地饱和。基于将色散解释为单位圆环的转动惯量这一观点，我们进一步证明其他度量也具有相同的力学解释。这一统一视角使我们能够将所有度量表达为单一通用角不确定性度量的特例。随后，通过应用其中最简单的两种度量来推导角动量与平移算符的最优联合测量，凸显了这些度量的重要性。最后，我们论证量子转子模型的意义超越了其力学含义，在奇异光学、含约瑟夫森结的超导电路以及时频域中的最优脉冲整形等领域具有广阔的应用前景。我们的发现为量子转子的量子信息与计量学应用奠定了基础，并揭示了其跨学科性质。
英文摘要：
The quantum rotor represents, after the harmonic oscillator, the next obvious quantum system to study the complementary pair of variables: the angular momentum and the unitary shift operator in angular momentum. Proper quantification of uncertainties and the incompatibility of these two operators are thus essential for applications of rotor-like quantum systems. While angular momentum uncertainty is characterized by variance, several uncertainty measures have been proposed for the shift operator, with dispersion the simplest example. We establish a hierarchy of those measures and corresponding uncertainty relations which are all perfectly or almost perfectly saturated by a tomographically complete set of von Mises states. Building on the interpretation of dispersion as the moment of inertia of the unit ring we then show that the other measures also possess the same mechanical interpretation. This unifying perspective allows us to express all measures as a particular instance of a single generic angular uncertainty measure. The importance of these measures is then highlighted by applying the simplest two of them to derive optimal simultaneous measurements of the angular momentum and the shift operator. Finally, we argue that the model of quantum rotor extends beyond its mechanical meaning with promising applications in the fields of singular optics, super-conductive circuits with a Josephson junction or optimal pulse shaping in the time-frequency domain. Our findings lay the groundwork for quantum-information and metrological applications of the quantum rotor and point to its interdisciplinary nature.
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