The notion of f-oscillators generalizing q-oscillators is introduced. For classical and quantum cases, an interpretation of the f-oscillator is provided as corresponding to a special nonlinearity of vibration for which the frequency of oscillation depends on the energy. The f-coherent states (nonlinear coherent states) generalizing q-coherent states are constructed. Applied to quantum optics, photon distribution function, photon number means, and dispersions are calculated for the f-coherent states as well as the Wigner function and Q-function. As an example, it is shown how this nonlinearity may affect the Planck distribution formula.

It is shown that because of the radiation pressure a Schrödinger cat state can be generated in a resonator with an oscillating wall. The optomechanical control of quantum macroscopic coherence and its detection is taken into account by introducing new cat states. The effects due to the environmental couplings with this nonlinear system are considered by developing an operator perturbation procedure to solve the master equation for the field mode density operator.

A scheme for measuring the quantum state for an arbitrary spin is proposed that is analogous to the symplectic tomography scheme used to measure quantum states associated with continuous observables s

Invertible maps from operators of quantum obvservables onto functions of c-number arguments and their associative products are first assessed. Different types of maps like Weyl-Wigner-Stratonovich map and s-ordered quasidistribution are discussed. The recently introduced symplectic tomography map of observables (tomograms) related to the Heisenberg-Weyl group is shown to belong to the standard framework of the maps from quantum observables onto the c-number functions. The star-product for symbols of the quantum-observable for each one of the maps (including the tomographic map) and explicit relations among different star-products are obtained. Deformations of the Moyal star-product and alternative commutation relations are also considered.