  We propose an experiment to directly prove the commutation relation between
bosonic annihilation and creation operators, based on the recent experimental
success in single-photon subtraction and addition. We devise a single-photon
interferometer to realize coherent superpositions of two sequences of photon
addition and subtraction. Depending on the interference outcome, the
commutation relation is directly proven or a highly nonclassical state is
produced. Experimental imperfections are assessed to show that the realization
of the scheme is highly feasible.
