Free probability is a kind of noncommutative probability, which was introduced by D. Voiculescu in 1990`s. In this theory, we study freely independent operators with respect to a state instead of independent random variables. In free probability, free semicircular elements are of central inportance. They are limit objects not only for free analoge of classical central limit theorem, but also for empirical eigenvalue distributions of independent Gaussian unitary ensembles which are typical random matrix models. In this talk, I will explain an equivalence between rationality of an operator obtained from free semicircular elements and finiteness of the rank of its commutator with right annihilation (creation) operators via representation on full Fock space. Our results are analogues of the results for free group which were conjectured by A. Connes and solved by G. Duchamp and C. Reutenauer, and extended by P. A. Linnel. For the proof, I will explain how it involves a combination of the tools for noncommutative rational power series, Haagerup type inequality and matrix tricks. This talk is based on my preprint, arXiv:2109.08841.