Fig. 1: Two-boson correlation dynamics and associated spatial density distributions in a boundary-interacting system.

The dramatic enhancement of bosonic clustering at site x = 1 is captured by P₁₁ [Eq. (5)], which measures the probability of finding both bosons at this site. This correlation is significantly stronger (red curves in (e) when both the non-Hermitian skin effect (NHSE, r > 0) and boundary interaction (U > 0) are present, demonstrating their synergistic effect on boson trapping. This enhanced clustering is studied for different initial states (1, 1), (4, 4), and (1, 4) evolving under H [Eq. (1)]. Supporting this observation, (a−d) show the spatial density evolution ρ_x(t) [Eq. (3)] over time t ∈ [0, 10], with corresponding time-averaged profiles \({\bar{\rho }}_{i}\) [Eq. (4)] plotted above. The density distributions contrast four scenarios: Hermitian (r = 1) without (a) and with (b) interaction, and non-Hermitian (r = 4) without (c) and with (d) interaction. The interaction term U, despite being typically repulsive, acts as an effective trap at x₀ = 1 during non-equilibrium evolution when combined with leftward NHSE, as evidenced by the bright regions in the heatmaps. Parameters: t₀ = 3, and for r = 4, t_L = 1.6 and t_R = 0.4. While we only show up to time  <10 evolution, we have verified that this enhanced clustering behavior persists at longer times (up to t = 100). See Supplementary Note 2.