11. In a square $A B C D$ with side length 4, if it is folded along $B D$ to form a $60^{\circ}$ dihedral angle, then the distance between the midpoint of side $B C$ and point $A$ is $\qquad$
11. $2 \sqrt{2}$.

Take the midpoints $O$ and $E$ of $B D$ and $B C$, respectively, and connect $A O$ and $C O$. Then, $\angle A O C=60^{\circ}$. Therefore,
$$
A C=A O=C O=2 \sqrt{2} \text {. }
$$

In $\triangle A B C$, by the median length formula, we have
$$
\begin{array}{l}
A E^{2}=\frac{1}{2}\left(A B^{2}+A C^{2}\right)-\frac{1}{4} B C^{2}=8 \\
\Rightarrow A E=2 \sqrt{2} .
\end{array}
$$