A quantum parametric oscillator in a radiofrequency trap

An ion confined within a Paul trap can be assimilated, in good approximation, with a harmonic oscillator. A method which enables the construction of an invariant operator based on the Lewis and Riesenfeld approach is reported. The method is then applied to the case of an ion confined in a Paul trap, treated as a quantum harmonic oscillator. An invariant operator is associated with the system. The spectrum of the quasienergy operator finally results. The eigenvectors of the system are found using the Fock state basis.