Single-Photon Motion in a Two-Dimensional Plane: Confinement and Boundary Escape

TL;DR

This study explores the behavior of a single photon in a two-dimensional plane with different boundary conditions, comparing two methods for constructing the Hilbert space to improve simulation efficiency and accuracy.

Contribution

It introduces a non-standard Hilbert space construction method that reduces dimensionality and enhances the simulation of photon dynamics in open and closed boundary conditions.

Findings

Both methods produce identical results in closed systems.

Method B captures faster photon escape due to more dissipation channels.

Probability distributions are highly similar in open systems before and after boundary reach.

Abstract

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.