arXiv 2302.054772023-02-10quant-ph

量子光学中的傍轴近似I：标量场的单色模

The paraxial approximation in quantum optics I: Henochromatic modes of a scalar field

Christopher Beetle, M. Fernanda Jongewaard de Boer

备注：: 8 pages

摘要：

本文研究了如何最好地将单个粒子的量子态与近轴近似下的窄准直经典辐射束的不同模式联系起来。我们的分析强调了问题中自然产生的两个内积之间关系的重要性。这两个参数是：一方面，用于在傍轴近似下将一般光束展开为正交模叠加的内积;另一方面，量子（场）理论的统计解释所依据的正则内积。虽然在文献中已经提出了几个候选人之间的关联光束模式和单粒子量子态的排序，在这里，我们认为，其中之一是唯一适合的任务。具体地说，束模式到“单色”量子态的映射在以前由Sudarshan、Simon和Mukunda引入的一大类类似映射中是独特的，因为它在数学精确意义上是酉的。

英文摘要：

This paper examines how best to associate quantum states of a single particle to different modes of a narrowly collimated beam of classical radiation modeled in the paraxial approximation. Our analysis stresses the importance of the relationship between two inner products naturally arising in the problem. These are the inner product used to expand a general beam as a superposition of orthogonal modes in the paraxial approximation, on the one hand, and the canonical inner product on which the statistical interpretation of quantum (field) theory is founded, on the other. While several candidates for the sort of association between beam modes and single-particle quantum states have been proposed in the literature, here we argue that one of them is uniquely well suited to the task. Specifically, the mapping of beam modes to the ``henochromatic'' quantum states previously introduced by Sudarshan, Simon and Mukunda is unique within a large class of similar mappings in that it is unitary in a mathematically precise sense.

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