Fig. 1. Bandgap structures for 2D PT symmetric moiré optical lattices at θ=arctan(3/4) with increasing imaginary potential strength V0 (a) and strength contrast p (b). (c) First Brillouin zone in 2D reciprocal space. Contour plots of the real (d), (g) and imaginary (e), (h) parts of the lattice (blue, lattice potential minima; red, lattice potential maxima) at θ=arctan(3/4) (d), (e) and θ=arctan(5/12) (g), (h), and their corresponding bandgap diagrams (f), (i) at V0=0.02 and p=1 in reduced zone representation. I and II in (f), (i) represent the first and second bandgaps.

Fig. 2. Typical profile of a fundamental GS supported by the 2D PT symmetric moiré optical lattice at θ=arctan(3/4) (a)–(c). Corresponding real (a) and imaginary (b) parts, and contour plot of the module (c). Condensate population, N, as a function of chemical potential μ (d), strength contrast p (e), and imaginary potential strength V0 (f) at θ=arctan(3/4). Other parameters: μ=5, N=47.4 in (a)–(c). μ=6.7 in (f).

Fig. 3. Typical profiles of higher-order GSs grouped as two out-of-phase (a), (b) and in-phase (c), (d) fundamental GSs at θ=arctan(3/4); corresponding contour plots are displayed in the second line (e)–(h). (i) Condensate population, N, as a function of chemical potential μ at θ=arctan(3/4) (black, out-of-phase mode; red dashed, in-phase mode). Other parameters: μ=4.4, N=43.9 for B and μ=4.6, N=59.3 for C.

Fig. 4. Profiles of gap vortices consisting of four fundamental GSs with vortex charge S=1 prepared within (a) and near the upper edge (b) of the first finite gap; corresponding condensate population, N, as a function of chemical potential μ at θ=arctan(3/4) (c). Panels for the top and center lines denote, respectively, contour plot of the module, real and imaginary parts, as well as the associated phase structure. Other parameters for gap vortices marked by points D and E: (a) μ=4.8, N=153.5; (b) μ=6, N=430.

Fig. 5. Profiles of fundamental GSs (a), (b), higher-order GSs (c), (d), and gap vortices with S=1 (e), (f) prepared within the first (a), (c), (e), (f) and second (b), (d) finite gaps. Corresponding perturbed evolutions and linear eigenvalue spectra obtained from linear-stability analysis are displayed in the second and third lines, respectively. Other parameters: (a) μ=5, N=47.4; (b) μ=6.29, N=128.2; (c) μ=5.2, N=114.7; (d) μ=6.28, N=252.6; (e) μ=4.8, N=153.5; (f) μ=6, N=430. White noise with 10% of its soliton’s amplitude is applied for all.

Table 1. Stability Regions (Characterized by μ) of Nonlinear Localized Modes within the First and Second Finite Gaps