Title: Mapping continuous-variable quantum states onto optical scalar beams Show Chinese title

Title: 将连续变量量子态映射到光学标量光束上

Abstract: Optical fields provide an accessible platform to explore connections between classical and quantum mechanics. We introduce a group-theoretic framework based on the $\mathrm{su}(1,1)$ Lie algebra to construct classical analogs of continuous-variable quantum states using the spatial degree of freedom of paraxial scalar beams. Our framework maps squeezed number states onto scalar beams expanded in orthonormal Gaussian modal bases, encompassing both Gaussian and non-Gaussian classical analogs, including one- and two-mode squeezed beams. To characterize the structural changes induced by squeezing, we examine phase-space redistribution through Fourier analysis and optical Wigner distribution functions. We derive analytical expressions for the waist, curvature, and Gouy phase of two-mode squeezed Laguerre-Gaussian beams, and establish a relation between the number of accessible modes and the achievable squeezing under finite numerical aperture. While squeezing introduces spatial and spectral correlations that reshape the beam structure, these beams remain constrained by the diffraction limit, as confirmed by the numerical propagation of apodized beams. These correlations give rise to classical entanglement. We establish a classical analog of the Duan--Simon inseparability criterion for continuous-variable two-mode Gaussian states. For non-Gaussian squeezed states, we analyze the marginal optical Wigner distribution functions and identify phase-space features, such as negativity, that act as witnesses of classical continuous-variable entanglement. Our framework unifies classical analogs of continuous-variable quantum states through beam engineering, enabling quantum-inspired applications in optical imaging, metrology, and communication. Show Chinese abstract

Abstract: 光学场为探索经典力学和量子力学之间的联系提供了一个可访问的平台。 我们引入一个基于$\mathrm{su}(1,1)$李代数的群论框架，利用傍轴标量光束的空间自由度来构建连续变量量子态的经典类比。 我们的框架将压缩数态映射到在正交高斯模态基下展开的标量光束，包括高斯和非高斯的经典类比，包括单模和双模压缩光束。 为了表征压缩引起的结构变化，我们通过傅里叶分析和光学魏格纳分布函数来研究相空间的重新分布。 我们推导出双模压缩拉盖尔-高斯光束的束腰、曲率和戈伊相位的解析表达式，并建立了在有限数值孔径下可访问模式数与可实现压缩之间的关系。 虽然压缩引入了空间和光谱相关性，重塑了光束结构，但这些光束仍受衍射极限的约束，这通过加权光束的数值传播得到了证实。 这些相关性产生了经典纠缠。 我们建立了一个连续变量双模高斯态的杜安-西蒙不可分性准则的经典类比。 对于非高斯压缩态，我们分析了边缘光学魏格纳分布函数，并识别了相空间特征，如负性，这些特征作为经典连续变量纠缠的见证者。 我们的框架通过光束工程统一了连续变量量子态的经典类比，使得在光学成像、计量学和通信中的量子启发应用成为可能。