Generation of patterns and higher harmonics in 1D quantum droplets in tilted and driven quasi-periodic confinements

The generation of patterns by breaking the spatial symmetry in external confinement is a captivating area of physics. The emergence of patterns is a fundamental inquiry spanning various disciplines such as nonlinear optics, condensed matter physics, and fluid dynamics. The article investigates the generation of a variety of patterns in a one-dimensional binary mixture of Bose-Einstein condensate forming quantum droplets. By solving the extended Gross-Pitaevskii equation in the presence of tilted and driven engineered bi-chromatic optical lattices (BOL), the out-of-equilibrium dynamics of droplets under strong dc and ac fields are illustrated. Under the influence of a dc field, a stripe-like pattern emerges in the temporal domain, while the scenario with ac fields demonstrates temporal periodic and bi-periodic oscillations of density waves. The width and period of formed patterns are directly correlated with the strength of ac and dc fields. Moreover, temporal modulation of the BOL potential depth yields various harmonics in the oscillations of the condensate density pattern. Through Fast Fourier Transform (FFT) analysis, it is confirmed that these harmonics encompass multiple and combinational frequencies, suggesting potential applications in generating desired frequency combs within quantum droplets. We have also carried out a thorough numerical stability check of the obtained solutions and found them sufficiently stable.