  Non-classical state generation is an important component throughout
experimental quantum science for quantum information applications and probing
the fundamentals of physics. Here, we investigate permutations of quantum
non-demolition quadrature measurements and single quanta addition/subtraction
to prepare quantum superposition states in bosonic systems. The performance of
each permutation is quantified and compared using several different
non-classicality criteria including Wigner negativity, non-classical depth, and
optimal fidelity with a coherent state superposition. We also compare the
performance of our protocol using squeezing instead of a quadrature measurement
and find that the purification provided by the quadrature measurement can
significantly increase the non-classicality generated. Our approach is ideally
suited for implementation in light-matter systems such as quantum optomechanics
and atomic spin ensembles, and offers considerable robustness to initial
thermal occupation.
