Eigenstates of the higher powers of an annihilation operator for a time-dependent harmonic oscillator and their properties

In the framework of dynamical invariant theory, we introduce the annihilation operator and creation operator for a time-dependent harmonic oscillator. The orthonormalized eigenstates of the operator are constructed, and their mathematical and quantum statistical properties are discussed in detail. Specifically, a particle moving in a Paul trap with periodically varying frequency is treated as an example to demonstrate the dynamical squeezing effects for the eigenstates of the operator .