Mechanical response of the quantum vacuum to dynamic deformations of a cavity

We consider two dynamically deforming plates immersed in the vacuum of a scalar field. We study both the (generalized) Neumann and the Dirichlet boundary conditions on the surfaces. For rough plates undergoing small displacements parallel or perpendicular to the normal axis, the mass correction, the vacuum viscosity, and the dissipation rate arising from photon emission inside the cavity, are calculated. In the case of motion along the normal axis, these quantities have no dependence on the shape of plates, and the Neumann ones are greater. We examine the specific example of two corrugated surfaces of a wave number $k_{1p}$ and mean separation $H$, performing lateral oscillation with a frequency ${\omega }_{1p}$. In contrast to Dirichlet photons, Neumann photons can be excited for all values of $k_{1p}$, $H$, and ${\omega }_{1p}$. For short cavity lengths, the number of Neumann photons increases as $H^{-2}$. The phase difference between the harmonic motion of plates can be used to tune the dissipation rate. Neumann mass correction and dissipation rate are greater than Dirichlet ones. We calculated the force exerted on two corrugated plates that are moving uniformly in the lateral directions. This lateral force decreases as $\exp (-k_{1p}H)$ if $k_{1p}H>1$, increases as $H^{-5}$ if $k_{1p}H<0.1$, and is stronger in the case of the Dirichlet boundary condition.

Figure 1

Various components of $\delta M$ in units of $(\hslash A{h}_p^2{k}_{1p}^5)∕\left(288{\pi }^{2}c\right)$ vs dimensionless variable $k_{1p}H$. Note that for $k_{1p}H\to \infty$, $\delta M_{11,N}^{\left(11\right)}=11$, $\delta M_{11,N}^{\left(12\right)}=0$, $\delta M_{11,D}^{\left(11\right)}=1$ and $\delta M_{11,D}^{\left(12\right)}=0$. $\delta M_{11,D}^{\left(11\right)}$ is positive for $k_{1p}H>2.13$.

Figure 2

Mass corrections $\mathrm{\Delta }{M}_{11}=\delta M_{11}^{\left(11\right)}-\delta M_{11}^{\left(12\right)}$ and ${\Delta }^{\prime }{M}_{11}=2\delta M_{11}^{\left(11\right)}+2\delta M_{11}^{\left(12\right)}$ in units of $(\hslash A{h}_p^2{k}_{1p}^5)∕\left(288{\pi }^{2}c\right)$ vs dimensionless variable $k_{1p}H$.

Figure 3

The dissipation rate $\mathcal{P}$ in units of $r_{1p}^2{\omega }_{1p}^2{\eta }_{\infty }^{\left(1\right)}\left({\omega }_{1p}\right)∕4$ vsdimensionless variable $H{\omega }_{1p}∕\left(\pi c\right)\sqrt{1-c^2{k}_{1p}^2∕{\omega }_{1p}^2}$. One of the plates is motionless and $k_{1p}=k_{2p}$.

Figure 4

The Neumann dissipation rate ${\mathcal{P}}_N$ in units of $r_{1p}^2{\omega }_{1p}^2{\eta }_{\infty }^{\left(1\right)}\left({\omega }_{1p}\right)∕4$ vs dimensionless variable$H{\omega }_{1p}∕\left(\pi c\right)\sqrt{1-c^2{k}_{1p}^2∕{\omega }_{1p}^2}$ for various ${\theta }_p$. $k_{1p}=k_{2p}$, ${\omega }_{1p}={\omega }_{2p}$, and $r_{1p}=r_{2p}$. Independent of ${\theta }_p$, ${\mathcal{P}}_N=22$ at large values of $H$.

Figure 5

The Dirichlet dissipation rate ${\mathcal{P}}_D$ in units of $r_{1p}^2{\omega }_{1p}^2{\eta }_{\infty }^{\left(1\right)}\left({\omega }_{1p}\right)∕4$ vs dimensionless variable$H{\omega }_{1p}∕\left(\pi c\right)\sqrt{1-c^2{k}_{1p}^2∕{\omega }_{1p}^2}$ for various ${\theta }_p$. $k_{1p}=k_{2p}$, ${\omega }_{1p}={\omega }_{2p}$, and $r_{1p}=r_{2p}$. Independent of ${\theta }_p$, ${\mathcal{P}}_D=2$ at large values of $H$.

Figure 6

The maximum force $F_J$ in units of $\hslash cA{h}_{1p}{h}_{2p}{k}_{1p}^6∕2$ vs dimensionless variable $k_{1p}H$.

Figure 7

$F_{J,D}∕F_{J,N}$ vs dimensionless variable $k_{1p}H$. At $k_{1p}H=5.32$, $F_{J,D}∕F_{J,N}$ reaches its maximum value $3.88$. Note that both $F_{J,D}$ and $F_{J,N}$ are quite small when $k_{1p}H>1$.