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

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.

💡 Research Summary

This paper investigates the dynamics of a single photon propagating on a two‑dimensional lattice of optical cavities, focusing on the differences between closed (reflective) and open (dissipative) boundary conditions. The authors introduce two distinct approaches for constructing the Hilbert space of the system. Method A follows the conventional second‑quantization formalism, assigning a binary occupation variable to each cavity. Although the full Fock space would have dimension 2^{L·H}, the single‑photon restriction allows the authors to discard all multi‑occupancy states, reducing the effective dimension to L·H for a closed system and L·H + 1 for an open system. Method B adopts a non‑standard representation that records the last cavity coordinates (l, h) together with a set of four directional flags (left, right, up, down) that become active only when the photon escapes through a boundary. This representation initially yields a space of size 5·L·H, but after eliminating redundant configurations the reduced dimensions become L·H for the closed case and L·H + 2(L + H) for the open case. The reduction dramatically lowers memory requirements and computational cost, enabling simulations on a 31 × 31 lattice (961 cavities) with modest hardware.

The dynamics are governed by a Jaynes‑Cummings‑Hubbard Hamiltonian under the rotating‑wave approximation, with a tunneling strength ζ between nearest‑neighbor cavities. For open boundaries, Lindblad operators with a uniform dissipation rate γ are added on all four edges, allowing photons to leave the lattice. Time evolution is performed using a Markovian quantum master equation; the authors employ a split‑step scheme where the unitary part is applied first, followed by a first‑order update of the dissipative term.

Simulation results show that in the closed system both construction methods produce identical unitary evolution, as expected because the reduced Hilbert spaces coincide. Starting from a photon localized at the lattice centre, the probability distribution spreads symmetrically outward. Upon reaching the edge, a pronounced rebound appears, reflecting the loss‑less nature of the closed system. In the open system, the probability curves from Method A and Method B overlap perfectly before the photon reaches the boundary. After the boundary is reached, a small divergence emerges: Method B, which includes four independent escape channels (one per direction), yields a slightly lower probability of finding the photon inside the lattice and a correspondingly higher probability of the dissipative (escaped) state. This indicates a faster escape rate in Method B. Importantly, the divergence does not grow with time; instead the two curves tend to reconverge as the photon population is depleted, and the oscillations of the two curves remain tightly synchronized throughout the evolution.

Two‑dimensional visualizations of the probability density confirm these observations. Prior to boundary contact, all three scenarios (closed, open‑A, open‑B) display identical circular diffusion patterns. After contact, the open systems exhibit rapid loss of probability, while the spatial patterns of the two open methods remain virtually indistinguishable, evolving in lockstep.

The authors conclude that (i) systematic elimination of redundant states can reduce the Hilbert space dimension by orders of magnitude without sacrificing physical accuracy; (ii) Method B provides a more realistic description of boundary dissipation by distinguishing directional escape channels, leading to a modest but consistent increase in escape speed; and (iii) the overall dynamics are robust against the specific choice of Hilbert‑space construction once the appropriate dissipation channels are included. The work offers a practical framework for efficient simulation of photon transport in large‑scale cavity arrays and suggests extensions to multi‑photon, non‑Markovian, or irregular lattice configurations for future research.