What do these subscripts mean? I am reading an introductory article about quantum optics, I am confused what $\hat{a_j}$ means, I tried to interpret it as applying $\hat{a}$ then taking the component along $|j>$ but it's not consistent with the relation.
Another interpretation is that $\hat{a_j}|j>=\sqrt{j}|j-1>$ and null for the other Fock states but it doesn't work out too.

 A: The index $j$ in $\hat{a}_j$ refers to a mode from which a particle is annihilated.
If you say there is one photon, that doesn't completely specify the quantum state of this photon. The photon might be a standing wave in a box-shaped cavity, for example. In that case, you have to specify the wavelength in each direction of the 3-dimensional space. (The wavelength may be any integer multiple of half the edge length.) There are different 1-photon states, and you can use the index $j$ to specify which state is meant.
A: You can think of the Fock space vacuum state as being a tensor product like
\begin{equation}
|{\rm vac}\rangle = |0\rangle_1 |0\rangle_2 |0\rangle_3 \cdots |0\rangle_n \cdots = \prod_{k=1}^\infty |0\rangle_k
\end{equation}
(no one actually ever writes this explicitly but I am just trying to make it clear here).
Then, the creation and annihilation operators $a_j$ and $a_j^\dagger$ operate on the state $|0\rangle_j$ and ignore the others. For example:
\begin{eqnarray}
a_j^\dagger |{\rm vac}\rangle &=& a_j^\dagger\left[ |0\rangle_1 |0\rangle_2 |0\rangle_3 \cdots |0\rangle_n \cdots\right] \\
&=& |0\rangle_1 |0\rangle_2 |0\rangle_3 \cdots a_j^\dagger |0 \rangle_j \cdots |0\rangle_n \cdots \\
&=& |0\rangle_1 |0\rangle_2 |0\rangle_3 \cdots  |1 \rangle_j \cdots |0\rangle_n \cdots \\
&=& \left( \prod_{k=1}^{j-1} |0 \rangle_k \right) |1\rangle_j \left(\prod_{k=j+1}^\infty |0\rangle_k \right)
\end{eqnarray}
Your text is lazy and drops the $j$ index from $a$ and $a^\dagger$ from all equations except Eqs A4a and A4b, although it is easy enough to put the indices in the other equations (and, essentially the main point of Eqs A4a and A4b is that you can treat each part of the whole Fock space independently, unless you want to consider interactions that cause transitions between subspaces of the Fock space, the indices are rarely important for practical calculations, so long as you know in the back of your mind they are there). People are never as explicit as I am in this answer in practice, because it would lead to ridiculous equations.
