<![CDATA[Quantum quasicrystals, characterized by aperiodic order, exhibit unique low-energy excitations such as phononic, phasonic, and hybridized modes, making them a focal point for studying symmetry-driven dynamics in ultracold atomic systems. Understanding these excitations and phase transitions is crucial for advancing quantum simulation technologies. This study investigates the role of quasicrystal symmetry (octagonal, decagonal, and dodecagonal) in determining anisotropic excitation behaviors and phase stability, validating theoretical predictions through simulated ultracold atom experiments. Using the Gross-Pitaevskii equation and Bogoliubov-de Gennes linearization, we simulated Bose-Einstein condensates in quasiperiodic potentials, varying potential depth V0 from 0.5 to 4.0 arbitrary units. Excitation spectra were computed to assess anisotropy, and minimum frequencies at k=0 were analyzed to identify phase boundaries. Simulations were conducted using Python with NumPy, SciPy, and Matplotlib, visualizing dispersion relations and phase diagrams. Octagonal quasicrystals displayed anisotropic excitation spectra (frequencies 1.368–1.464), reflecting mode hybridization, while decagonal (1.413–1.453) and dodecagonal (1.696–1.736) systems showed more isotropic behaviors. Phase boundary analysis revealed persistent gaps (octagonal: 1.0–5.0, decagonal: 1.5–6.0, dodecagonal: 2.0–7.0), indicating no quasicrystal phase within the simulated V0 range, likely transitioning to superfluid or disordered states. Quasicrystal symmetry significantly influences excitation anisotropy and phase stability, with higher symmetries (dodecagonal) exhibiting larger gaps and reduced quasicrystal stability. Future studies should explore lower V0 ranges and incorporate temperature effects to locate the quasicrystal phase, enhancing experimental validation.]]>
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