二维平面中单光子的运动：限制与边界逃逸
Single-Photon Motion in a Two-Dimensional Plane: Confinement and Boundary Escape
AI总结：
本文研究了二维平面中单光子在封闭与开放边界条件下的运动，通过两种方法比较，发现方法B在描述光子逃逸时更准确，且在后期演化中表现出更稳定的概率分布。
摘要：
本文研究了在封闭和开放边界条件下，二维平面中单光子的运动。我们采用两种方法来构建希尔伯特空间：方法A基于标准的二次量子化形式，方法B基于非标准方法。通过消除冗余的量子态，我们得到一个维度显著降低的简化希尔伯特空间，从而提高了数值模拟的效率。在封闭系统中，两种方法等价，其幺正演化结果相同。概率分布向外扩散，从中心出发并在达到边界时表现出显著的反弹。在开放系统中，方法B通过引入更多的耗散通道，提供了更准确的描述，以描述光子在边界处的逃逸过程。两种方法在达到边界前获得的概率曲线完全重合。在边界被达到后，出现轻微差异，但这种差异不会随演化放大，后期趋于收敛。方法B产生稍高的耗散态概率，表明光子逃逸更快。二维概率分布的可视化显示，三种情景（封闭系统、使用方法A的开放系统、使用方法B的开放系统）在达到边界前具有相同的概率分布。在边界被达到后，开放系统表现出显著的概率损失，随着演化迅速增加。两种开放系统的概率分布模式高度相似，表现出同步的演化。
英文摘要：
This paper investigates the motion of a single photon in a two-dimensional plane under closed and open boundary conditions. We employ two methods to construct the Hilbert space: Method A, based on the standard second-quantization formalism, and Method B, based on a non-standard approach. By eliminating redundant quantum states, we obtain a reduced Hilbert space with significantly lower dimensionality, thereby improving the efficiency of numerical simulations. In a closed system, the two methods are equivalent, and their unitary evolution results are identical. The probability distribution diffuses outward from the center and exhibits a significant rebound after reaching the boundary. In an open system, Method B, by incorporating more dissipation channels, provides a more accurate description of the photon escape process at the boundary. The probability curves obtained from the two methods completely overlap before reaching the boundary. After the boundary is reached, a slight difference appears, but this difference does not amplify with evolution and tends to converge in the later stage. Method B yields a slightly higher dissipative-state probability, indicating that the photon escapes faster. Visualization of the two-dimensional probability distribution shows that the three scenarios (closed system, open system with Method A, and open system with Method B) exhibit identical probability distributions before reaching the boundary. After the boundary is reached, the open systems exhibit significant probability loss, which increases rapidly with evolution. The probability distribution patterns of the two open systems are highly similar, exhibiting synchronized evolution.
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