Experimental realization of creation and annihilation perators and direct proof of the bosonic commutation relation

The recent development of experimental techniques to implement the creation and the annihilation operators in the optical domain has found great interest for quantum information manipulation and processing. Quantum states of light obtained by the application of alternated sequences of the creation and annihilation operators $\hat{a}^\dagger$ and $\hat{a}$ to a thermal state of light depend on the order in which the two operations are performed. The non-commutativity of the bosonic operators, $\hat{a}\hat{a}^\dagger\ne \hat{a}^\dagger\hat{a}\ne 1$, has been experimentally verified in terms of density matrixes and Wigner functions, which are reconstructed by quantum homodyne tomography. Recently, a setup has been proposed to directly and completely prove the bosonic commutation relation where a single-photon interferometer is used to create coherent superpositions of two alternated sequences of operators, such as $[\hat{a}\hat{a}^\dagger \pm \hat{a}^\dagger\hat{a}]$. Some of the latest experimental results are discussed.