  For the two-mode exciton system formed by the quasi-spin wave collective
excitation of many $\Lambda$ atoms fixed at the lattice sites of a crystal, we
discover a dynamic symmetry depicted by the semi-direct product algebra
$SU(2)\bar{\otimes} h_2 $ in the large $N$ limit with low excitations. With the
help of the spectral generating algebra method, we obtain a larger class of
exact zero-eigenvalue states adiabatically interpolating between the initial
state of photon-type and the final state of quasi-spin wave exciton-type. The
conditions for the adiabatic passage of dark states are shown to be valid, even
with the presence of the level degeneracy. These theoretical results can lead
to propose new protocol of implementing quantum memory robust against quantum
decoherence.
