= Solution
For identical <fermions>, $(b_j^\dagger)^2=0$ and $b_2^\dagger b_1^\dagger=-b_1^\dagger b_2^\dagger$. Therefore
$$
a_1^\dagger a_2^\dagger|0\rangle
=\frac12(b_1^\dagger+b_2^\dagger)(b_1^\dagger-b_2^\dagger)|0\rangle
=-b_1^\dagger b_2^\dagger|0\rangle.
$$
The minus sign is a <global phase> for this output. \b[The only occupancy is $(n_1,n_2)=(1,1)$, with <probability> one.] Double occupancy of either specified single-particle mode is excluded; no extra internal states are present in this model.
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