Torque on birefringent plates induced by quantum fluctuations

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 $\sin \left(2\theta \right)$, where $\theta$ represents the angle between the two optical axes, and its magnitude rapidly increases with decreasing plate separation $d$. For a $40\mu \mathrm{m}$ diameter disk, made out of either quartz or calcite, kept parallel to a barium titanate plate at $d\simeq 100\mathrm{nm}$, the maximum torque (at $\theta =\pi ∕4$) is of the order of $\simeq {10}^{-19}\mathrm{N}\mathrm{m}$. 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.

Figure 1

Sketch of the system under investigation. Two birefringent parallel plates with in-plane optical axis are placed in close proximity. The $z$ axis is chosen to be orthogonal to the plates. $\theta$ is the angle between the two optical axes.

Figure 2

Dielectric functions of the materials used in our calculations as a function of imaginary angular frequency. Solid lines and dotted lines represent, respectively, ${ϵ}_{\Vert }$ and ${ϵ}_{\perp }$ of the birefringent slabs: (a) quartz, (b) calcite, and (c) barium titanate.

Figure 3

Calculated torque as a function of $\theta$ between a $40\mu \mathrm{m}$ diameter disk made of quartz (a) or calcite (b) and a barium titanate plate separated in vacuum by a distance $d=100\mathrm{nm}$. The lines represent a fit of the numerical results with $M=a\sin \left(2\theta \right)$.

Figure 4

Calculated torque as a function of plate separation between a $40\mu \mathrm{m}$ diameter disk made of quartz $(•)$ or calcite $(\times )$ and a barium titanate plate in vacuum at $\theta =\pi ∕4$.

Figure 5

Calculated retarded van der Waals force as a function of plate separation for a $40\mu \mathrm{m}$ diameter disk made of quartz (a) or calcite (b) and a plate of barium titanate immersed in ethanol, calculated for $\theta =\pi ∕4$. The arrow represents the distance at which the retarded van der Waals repulsion is in equilibrium with the weight of the disk. Insets: Dielectric functions of the interacting materials. For the three birefringent samples, the curve represents the average of ${ϵ}_{\perp }$ and ${ϵ}_{\Vert }$.

Figure 6

Calculated torque as a function of plate separation between a $40\mu \mathrm{m}$ diameter disk made of quartz (a) or calcite (b) and a barium titanate plate immersed in ethanol $(•)$ or vacuum $(\circ )$ for $\theta =\pi ∕4$.

Figure 7

A sketch of the experimental setup proposed in this paper. Inset: calculated value of the angle between the optical axes of the two birefringent crystals as a function of time.