  The essential role of synthetic spin-orbit coupling in discovering new
topological matter phases with cold atoms is widely acknowledged. However, the
engineering of spin-orbit coupling remains unclear for arbitrary-spin models
due to the complexity of spin matrices. In this work, we develop a more general
but relatively straightforward method to achieve spin-orbit coupling for
multi-spin models. Our approach hinges on controlling the coupling between
distinct pseudo-spins through two intermediary states, resulting in tunneling
with spin flips that have direction-dependent strength. The engineered
spin-orbit coupling can facilitate topological phase transitions with Chern
numbers over 1, a unique characteristic of multi-spin models compared to
spin-1/2 models. By utilizing existing cold atom techniques, our proposed
method provides an ideal platform for investigating topological properties
related to large Chern numbers.
