Sieve of Eratosthenes for Bose-Einstein Condensates in Optical Moir\'e Lattices

We catalog known optical moir\'e lattices and uncover exotic lattice configurations following a geometric analog of the ancient sieve of Eratosthenes algorithm for finding prime numbers. Rich dynamics of Bose-Einstein condensates loaded into these optical lattices is revealed from numerical simulations of time-of-flight interference patterns. What sets this method apart is the ability to tune the periodicity of the optical lattices without changing the wavelength of the laser, yet maintaining the local potential at the individual lattices sites. In addition, we discuss the ability to spatially translate the optical lattice through applying a structured phase only.