  We present detailed numerical calculations of the mechanical torque induced
by quantum fluctuations on two parallel birefringent plates with in plane
optical anisotropy, separated by either vacuum or a liquid (ethanol). The
torque is found to vary as $\sin(2\theta)$, where $\theta$ represents the angle
between the two optical axes, and its magnitude rapidly increases with
decreasing plate separation $d$. For a 40 $\mu$m diameter disk, made out of
either quartz or calcite, kept parallel to a Barium Titanate plate at $d\simeq
100$ nm, the maximum torque (at $\theta={\pi\over 4}$) is of the order of
$\simeq 10^{-19}$ N$\cdot$m. We propose an experiment to observe this torque
when the Barium Titanate plate is immersed in ethanol and the other
birefringent disk is placed on top of it. In this case the retarded van der
Waals (or Casimir-Lifshitz) force between the two birefringent slabs is
repulsive. The disk would float parallel to the plate at a distance where its
net weight is counterbalanced by the retarded van der Waals repulsion, free to
rotate in response to very small driving torques.
