  By using both the Lewis-Riesenfeld invariant theory and the invariant-related
unitary transformation formulation, the present paper obtains the exact
solutions to the time-dependent supersymmetric two-level multiphoton
Jaynes-Cummings model and the Chiao-Wu model that describes the propagation of
a photon inside the optical fiber. On the basis of the fact that the two-level
multiphoton Jaynes-Cummings model possesses the supersymmetric structure, an
invariant is constructed in terms of the supersymmetric generators by working
in the sub-Hilbert-space corresponding to a particular eigenvalue of the
conserved supersymmetric generators (i.e., the time-independent invariant). By
constructing the effective Hamiltonian that describes the interaction of the
photon with the medium of the optical fiber, it is further verified that the
particular solution to the Schr\"{o}dinger equation is the eigenfunction of the
second-quantized momentum operator of photons field. This, therefore, means
that the explicit expression (rather than the hidden form that involves the
chronological product) for the time-evolution operator of wave function is
obtained by means of the invariant theories.
