  The exploration of dimensional crossover carries profound fundamental
significance, serving as a crucial bridge in comprehending the remarkable
disparities observed in transitional phenomena across the two distinct
dimensions of a physical system. The prevalent strategy for manipulating the
dimensionality involves meticulously controlling the external trapping
geometry, thereby restricting the degrees of freedom of the kinetic energy from
three-dimensional (3D) to lower-dimensional spaces, while maintaining the 3D
nature of the interaction energy degrees of freedom. The aim of this work is to
introduce an innovative scenario to achieve dimensional crossover,
characterized by lower-D nature of both the kinetic and the interaction energy
degrees of freedom. To accomplish this objective, we delve deeply into the
realm of a 2D optically trapped Bose gas, focusing specifically on its
finite-range interaction. Our emphasis lies in exploring the lattice-induced
dimensional crossover from full 3D to 1D in both kinetic and interaction terms.
Utilizing the functional path integral method, we derive the equation of states
of the model system, encompassing crucial quantities such as the ground state
energy and quantum depletion. These equations enable us to analyze the combined
effects of finite range interaction and an optical lattice on quantum
fluctuations of the BEC system. Notably, our analytical findings reconcile the
Lee-Huang-Yang (LHY) correction to the ground state energy in 3D and
Lieb-Liniger (LL) ones in 1D limit, thereby providing fresh insights into the
intriguing disparities between LHY and LL corrections.
