Let point $P$ be inside the ellipse $\Gamma$ and $P(t, t)$, then $K\left(\sqrt{4\left(1-\frac{t^{2}}{3}\right)}, t\right), L\left(t, \sqrt{3\left(1-\frac{t^{2}}{4}\right)}\right)$.
Let $Q(x, y)$, according to the problem, we have $\left\{\begin{array}{l}x=\sqrt{4\left(1-\frac{t^{2}}{3}\right)}, \\ y=\sqrt{3\left(1-\frac{t^{2}}{4}\right)}\end{array} \Rightarrow\left\{\begin{array}{l}x^{2}=4-\frac{4 t^{2}}{3}, \\ y^{2}=3-\frac{3 t^{2}}{4}\end{array}\right.\right.$
$$
\Rightarrow\left\{\begin{array}{l}
\frac{3}{4} x^{2}=3-t^{2}, \\
\frac{4}{3} y^{2}=4-t^{2}
\end{array} \Rightarrow \frac{4}{3} y^{2}-\frac{3}{4} x^{2}=1 \Rightarrow \frac{y^{2}}{\frac{3}{4}}-\frac{x^{2}}{\frac{4}{3}}=1\right. \text {. }
$$

Therefore, the trajectory of point $Q$ is a segment of a hyperbola $\Omega: \frac{y^{2}}{\frac{3}{4}}-\frac{x^{2}}{\frac{4}{3}}=1$, with $a^{2}=\frac{3}{4}, b^{2}=\frac{4}{3}$, then $c^{2}=a^{2}+b^{2}=\frac{25}{12}$.
Take points $A, B$ as $\left(0, \pm \frac{5 \sqrt{3}}{6}\right)$, at this time $|Q A|-|Q B|= \pm 2 a= \pm \sqrt{3}$.
If there are other points $A^{\prime}, B^{\prime}$ such that $\left|Q A^{\prime}\right|-\left|Q B^{\prime}\right|$ is a non-zero constant, by the definition of a hyperbola, $Q$ lies on a certain hyperbola $\Omega^{\prime}$ with foci $A^{\prime}, B^{\prime}$. Since the position of moving point $Q$ has infinitely many possibilities, hyperbola $\Omega^{\prime}$ and $\Omega$ have infinitely many common points. Thus, $\Omega^{\prime}=\Omega$, and $\left\{A^{\prime}, B^{\prime}\right\}=\{A, B\}$. That is, the line segment $A B$ is unique, and its length is $\frac{5 \sqrt{3}}{3}$.
11. In the Cartesian coordinate system, the ellipse $\Gamma: \frac{x^{2}}{4}+\frac{y^{2}}{3}=1$, point $P$ is inside the ellipse $\Gamma$ and moves along the line $y=x$. Points $K, L$ are on $\Gamma$, such that $\overrightarrow{P K}, \overrightarrow{P L}$ are in the positive directions of the $x$-axis and $y$-axis, respectively, and point $Q$ makes $P K Q L$ a rectangle. Does there exist two points $A, B$ in the plane such that when the rectangle $P K Q L$ changes, $|Q A|-|Q B|$ is a non-zero constant? If so, find all possible lengths of segment $A B$; if not, explain the reason.