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

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.

Figures (5)

• Figure 1: (online color) Schematic diagram of single-photon motion in a two-dimensional plane. Panel (a) shows the coupling scheme between optical cavities. All optical cavities are arranged in a rectangular (or square) grid on the plane. Each cavity is connected to its four nearest neighbors via optical fibers along the horizontal and vertical directions. The photon tunneling strength between different cavities is $\zeta$. We use $l$ and $h$ to denote the horizontal and vertical coordinates of the cavity, respectively. The total numbers of cavities along the horizontal and vertical directions are denoted as $l_{\max}=L$ and $h_{\max}=H$, respectively. Under open boundary conditions, dissipation channels exist on all four boundaries of this two-dimensional rectangular plane, with a dissipation strength $\gamma$. Panel (b) illustrates this dissipation channel, using the upper-right corner as an example. Panel (c) shows the initial state of the system: the single photon is initially located at the exact center of the two-dimensional plane.
• Figure 2: (online color) Comparison of evolution: unitary in the closed system vs non-unitary in the open system. Since the photon is initially located at the center of the square plane, the probability distribution remains centrally symmetric throughout the subsequent evolution. To facilitate the analysis, we sum the probabilities over all cavity sites with the same horizontal coordinate $l$, i.e., we integrate along the vertical direction. Black and red curves represent the results of unitary and non-unitary evolution, respectively. Panels (a)--(h) show the probability distribution curves at iteration steps 300, 400, 500, 750, 1000, 2000, 5000, and 10000, respectively.
• Figure 3: (online color) Probability distributions for non-unitary evolution in an open system: Method A vs Method B. Similarly, we sum the probabilities over all cavity sites with the same horizontal coordinate $l$. Red and blue curves represent the results of non-unitary evolution obtained from Method A and Method B, respectively. Panels (a)--(h) show the probability distribution curves at iteration steps 300, 400, 500, 750, 1000, 2000, 5000, and 10000, respectively. Unlike in Fig. \ref{['fig:ComparisonClosedOpen']}, the vertical axis range for each row of panels has been adjusted appropriately to facilitate comparison between the red and blue curves.
• Figure 4: (online color) Probability curves for the dissipative state: Method A vs Method B. Red and blue curves represent the results of non-unitary evolution obtained from Method A and Method B, respectively.
• Figure 5: (online color) Probability distributions on the two-dimensional plane. Comparison of three cases: closed system (first row), open system with Method A (second row), and open system with Method B (third row). Each row contains four panels corresponding to iteration steps 300, 500, 700, and 1000 (from left to right). The upper bound of the color bar is set to the global maximum probability of $0.024963$.