Spin exchange of two spin-1/2 atoms

The quantum Cheshire cat effect is an important phenomenon in quantum mechanics that reveals the separability of physical properties from their carriers. This effect transcends the classical framework whose attributes must be inherently attached to objects, providing new perspectives for quantum information and precision measurement. According to the quantum Cheshire cat effect, we prepare a pre-selected state of a spin1/2 atomic system composed of two particles through a pre-selection process. We conduct quantum weak measurements on the spins and positions of these two atoms and extract weak values by using the method of imaginary time evolution(ITE). Subsequently, we perform post-selection on these two atoms and design two distinct post-selected states. Initially, we calculate analytical solutions when both atoms encounter these two different post-selected states separately. We also compare the analytical and numerical solutions. Our research theoretically confirms the feasibility of fermionic systems within bipartite quantum Cheshire cat effects and illustrates how delayed-choice influences quantum Cheshire cat effects in spin-1/2 atomic systems.