  Optical downconversion is a key resource for generating nonclassical states.
Very recently, direct nondegenerate triple-photon spontaneous downconversion
(NTPSD) with bright photon triplets and strong third-order correlations has
been demonstrated in a superconducting device (2020 Phys. Rev. X 10 011011).
Besides, linear and nonlinear tripartite entanglement in this process have also
been predicted (2018 Phys. Rev. Lett. 120 043601; 2020 Phys. Rev. Lett. 125
020502). In this paper, we consider the generation of nonclassical optical
quantum superpositions and investigate nonlinear quantum steering effects in
NTPSD.We find that large-size Schr\"odinger cat states of one downconverted
mode can be achieved when the other two modes are subjected to homodyne
detection. Also, a two-photon Bell entangled state can be generated when only
one mode is homodyned.We further reveal that such ability of remote state
steering originates from nonlinear quantum steerable correlations among the
triplets. This is specifically embodied by the seeming violation of the
Heisenberg uncertainty relation for the inferred variances of two
noncommutating higher-order quadratures of downconverted modes, based on the
outcomes of homodyne detection on the other mode, i.e., nonlinear quantum
steering, compared to original EPR steering. Our results demonstrate
non-Gaussian nonclassical features in NTPSD and would be useful for the
fundamental tests of quantum physics and implementations of optical quantum
technologies.
