  In the present paper, we shall study effect of a random quenched external
field for spin order and also multiple Bose-Einstein condensation (BEC). This
system is realized by the cold atomic gases in an optical lattice. In
particular, we are interested in the strong-repulsion region of two-component
gases for which the bosonic t-J model is a good effective model. In the bosonic
t-J model, a long-range order of the pseudo-spin and also BEC of atoms appear
quite naturally as in the fermion t-J model for the high-temperature
superconducting materials. Random Raman scattering between two internal states
of a single atom plays a role of the random external field, and we study its
effects on the pseudo-spin order and the BEC by means of quantum Monte-Carlo
simulations. The random external field breaks a continuous U(1) symmetry
existing in the original bosonic t-J model and it induces new orders named
random-field-induced order (RFIO). We show a phase diagram of the bosonic t-J
model with the random external magnetic field and study the robustness of the
RFIO states. We also study topological excitations like vortices and domain
wall in the RFIO state. Finally we point out the possibility of a quantum bit
by the RFIO.
