  Quantum non-Gaussian gate is a missing piece to the realization of
continuous-variable universal quantum operations in the optical system. In a
measurement-based implementation of the cubic phase gate, a lowest-order
non-Gaussian gate, non-Gaussian ancillary states that has a property we call
nonlinear squeezing are required. This property, however, has never been
experimentally verified. In this paper, we generate a superposition between a
vacuum state and a single photon state whose nonlinear squeezing are maximized
by the optimization of the superposition coefficients. The nonlinear squeezing
is observed via real-time quadrature measurements, meaning that the generated
states are compatible with the real-time feedforward and are suitable as the
ancillary states for the cubic phase gate in time domain. Moreover, by
increasing the number of the photons, it is expected that nonlinear squeezing
can be further improved. The idea presented here can be readily extended to the
higher-order phase gates [P. Marek et al., Phys. Rev. A 97, 022329 (2018)]. As
such, this work presents an important step to extend the CV quantum information
processing from Gaussian regime to non-Gaussian regime.
