Classical and quantum action-phase variables for time-dependent oscillators

For a time-dependent classical quadratic oscillator we introduce pairs of real and complex invariants that are linear in position and momentum. Each pair of invariants realize explicitly a canonical transformation from the phase space to the invariant space, in which the action-phase variables are defined. We find the action operator for the time-dependent quantum oscillator via the classical-quantum correspondence. Candidate phase operators conjugate to the action operator are discussed, but no satisfactory ones are found.