First passage time for an underdamped harmonic oscillator and application to the power of an information engine

The distribution of the first passage time $t_{fp}$ for the position $x$ to overcome a threshold $x_B$ is calculated in an underdamped harmonic oscillator. The proof combines several approaches based on the determination of the eigenvalues of the Kramers differential operator for the intermediate and long time regimes and on a Hamiltonian approximation for the short times. The theoretical predictions are in excellent agreement with the results of an experiment on an underdamped micro-cantilever. The knowledge of the $t_{fp}$ distribution opens the way to several applications, among them the precise estimation of the power of information engines, which we have also experimentally checked.