  The transitionless tracking (TT) algorithm enables the exact tracking of
quantum adiabatic dynamics in an arbitrary short time by adding a
counterdiabatic Hamiltonian to the original adiabatic Hamiltonian. By applying
Husimi's method originally developed for a quantum parametric oscillator (QPO)
to the transitionless QPO achieved using the TT algorithm, we obtain the
transition probability generating function with a time-dependent parameter
constituted with solutions of the corresponding classical parametric oscillator
(CPO). By obtaining the explicit solutions of this CPO using the
phase-amplitude method, we find that the time-dependent parameter can be
reduced to the frequency ratio between the Hamiltonians without and with the
counterdiabatic Hamiltonian, from which we can easily characterize the result
achieved by the TT algorithm. We illustrate our theory by showing the
trajectories of the CPO on the classical phase space, which elucidate the
effect of the counterdiabatic Hamiltonian of the QPO.
