11. Let the complex numbers $z_{1}, z_{2}$ satisfy $\left|z_{1}\right|=\left|z_{1}+z_{2}\right|=3,\left|z_{1}-z_{2}\right|=3 \sqrt{3}$, then $\log _{3}\left|\left(z_{1} \overline{z_{2}}\right)^{2000}+\left(\overline{z_{1}} z_{2}\right)^{2000}\right|=$ $\qquad$ .
According to the problem, we get
\[
\left\{
\begin{array}{l}
z_{1} \overline{z_{1}}=9, \\
(z_{1}+z_{2})(\overline{z_{1}}+\overline{z_{2}})=9, \\
(z_{1}-z_{2})(\overline{z_{1}}-\overline{z_{2}})=27
\end{array}
\Rightarrow
\left\{
\begin{array}{l}
z_{1} \overline{z_{1}}=9, \\
z_{2} \overline{z_{2}}=9, \\
z_{1} \overline{z_{2}}+\overline{z_{1}} z_{2}=-9 .
\end{array}
\right.
\right.
\]
Let \( x = z_{1} \overline{z_{2}}, y = \overline{z_{1}} z_{2} \), then \( x + y = -9, xy = 81 \Rightarrow x^2 + y^2 = (x + y)^2 - 2xy = -81 \).
In general, \( x^{n+1} + y^{n+1} = (x + y)(x^n + y^n) - xy(x^{n-1} + y^{n-1}) \).
Let \( a_n = x^n + y^n \Rightarrow a_{n+1} = -9a_n - 81a_{n-1} \xrightarrow{b_n = \frac{a_n}{9^n}} b_{n+1} = -b_n - b_{n-1} \),
and \( b_1 = b_2 = -1 \). The characteristic equation \( x^2 + x + 1 = 0 \) has roots \( x_1 = \omega, x_2 = \omega^2 \),
so \( b_n = A \omega^n + B \omega^{2n} \). Substituting \( b_1 = b_2 = -1 \Rightarrow A = B = 1 \).
Thus, \( b_{2000} = \omega^{2000} + \omega^{4000} = \omega^2 + \omega = -1 \Rightarrow a_{2000} = -9^{2000} \).
Therefore, \( \log_3 \left| (z_{1} \overline{z_{2}})^{2000} + (\overline{z_{1}} z_{2})^{2000} \right| = \log_3 9^{2000} = 4000 \).