  We investigate a generic tripartite quantum system featuring a single qubit
interacting concurrently with two quantized harmonic oscillators via nonlinear
multiphoton Jaynes-Cummings (MPJC) interactions. Assuming the qubit is
initially prepared in a superposition state and the two oscillators are in
arbitrary Fock states, we analytically trace the temporal evolution of this
tripartite pure initial state. We identify four broad cases, each further
divided into two subcases, and derive exact analytical solutions for most
cases. Notably, we obtain perfect swapping of arbitrary Fock states between the
oscillators by carefully selecting system parameters. In addition, we
extensively examine the manner in which the nonclassicalities of various
initial oscillator Fock states, quantified by the volume of negative regions in
the associated Wigner functions, evolve under the MPJC Hamiltonian, considering
diverse system parameters including environmentally induced effects. Besides
producing substantial enhancements in the initial value for higher photon
number states, our analysis reveals that driven solely by the initial qubit
energy, with both oscillators initialized in the vacuum state, the nonlinear
MPJC interaction yields nontrivial Wigner negativities in the oscillators. The
additional nonlinearity introduced by the multiphoton process plays a pivotal
role in surpassing the initial nonclassicalities of the photon number states.
