  Optical Schr\"odinger "cat" states created by superpositions of coherent
light states, correspond to an optical analog of the Schr\"odinger's cat in his
$\textit{Gedankenexperiment}$. These non-classical light states are generated
by means of quantum state engineering methods, and they are considered as one
of the main resources for fundamental tests of quantum theory and the
development of new quantum technologies. However, the power of existing optical
"cat" state sources is limited by their low average photon number, which
prevents their use in nonlinear optics. Here, we demonstrate the generation of
a femtosecond duration optical "cat" state in the infrared spectral range, with
mean photon number orders of magnitude higher than those delivered by current
available sources. These states exhibit intensities sufficient to induce
nonlinear processes in matter. This is shown using the process of second
harmonic generation in an optical crystal, in which the infrared-frequency
photons of an optical "cat" state are up-converted into blue-frequency photons.
We create the light states driving the second harmonic generation process, by
means of conditioning operations applied on the quantum state of an intense
infrared femtosecond laser field after its nonlinear interaction with atoms.
Due to the presence of quantum interference between the coherent states
composing the optical "cat" state, the quantum properties of the state are
imprinted in the measured second-order interferometric autocorrelation traces.
The findings introduce the optical "cat" states into the realm of nonlinear
quantum optics, opening up exciting new paths in quantum information science.
