Probability Representation of Nonclassical States of the Inverted Oscillator

We determine the evolving probability representations of several important nonclassical states of the inverted oscillator by applying the method of integrals of motion for this system. The considered nonclassical states initially prepared in the potential of the harmonic oscillator are even and odd Schrödinger cat states, squeezed coherent states, and lattice superpositions of coherent states. The latter superpositions can approximate several nonclassical states with high precision, hence their probability representation can describe various nonclassical states of the inverted oscillators. Explicit results are shown for the approximation of number states, photon number superpositions, and amplitude squeezed states by determining the parameters of the superposition appearing in the probability