I refer to this pdf.
On Page 41 (Quantum-state transformation of number states) the output state on a beam splitter is derived, based on
$$b_1^{+} = T^{*} a_1^{+} + R^{*} a_2^{+}$$
$$b_2^{+} = -R a_1^{+} + T a_2^{+}$$
The matrix, making up this transformation ist unitary:
$$U_{T ^*} =\begin{pmatrix} T^* & R^*\\ -R & T \end{pmatrix}$$
Let us assume that the radiation field was initially prepared in a product state of two single-photon Fock states,
$$| \Psi_{in}\rangle = | 1\rangle |1 \rangle$$
For my opinion this must be
$$| 1\rangle |1 \rangle = a_1^{+}a_2^{+} | 0\rangle |0 \rangle$$
since
$a_{1,2}^{+}$ are the creation operators for the input fields, and $b_{1,2}^{+}$ are for the output fields.
Why do they write $| \Psi_{out}\rangle = a_1^{+} a_2^{+}| 0\rangle |0 \rangle$? Shouldn't it be $| \Psi_{out}\rangle = b_1^{+} b_2^{+}| 0\rangle |0 \rangle$ instead?
EDIT:
$b_1^{+} b_2^{+}| 0\rangle |0 \rangle = | 1\rangle | 1\rangle$, but this cannot be the output state...
So it seems, the first variant is ok. But I still do not understand the formalism...it seems, that in order to get the output state, $a$ has to replaced by $b$, according to the inverse of $U_{T^*}$.