Vacuum energy densities of a field in a cavity with a mobile boundary

We consider the zero-point field fluctuations, and the related field energy densities, inside a one-dimensional and a three-dimensional cavity with a mobile wall. The mechanical degrees of freedom of the mobile wall are described quantum mechanically, and they are fully included in the overall system dynamics. In this optomechanical system, the field and the wall can interact with each other through the radiation pressure on the wall, given by the photons inside the cavity or even by vacuum fluctuations. We consider two cases—the one-dimensional electromagnetic field and the three-dimensional scalar field—and use the Green’s functions formalism, which allows extension of the results obtained for the scalar field to the electromagnetic field. We show that the quantum fluctuations of the position of the cavity’s mobile wall significantly affect the field energy density inside the cavity, in particular at the very proximity of the mobile wall. The dependence of this effect from the ultraviolet cutoff frequency, related to the plasma frequency of the cavity walls, is discussed. We also compare our new results for the one-dimensional electromagnetic field and the three-dimensional massless scalar field to results recently obtained for the one-dimensional massless scalar field. We show that the presence of a mobile wall also changes the Casimir-Polder force on a polarizable body placed inside the cavity, giving the possibility to detect experimentally the new effects we have considered.

Figure 1

Plots (a) and (b) show the corrections to the renormalized electric and magnetic energy density, respectively, as a function of the position inside the cavity. The total field energy density is shown in plot (c). The numerical values used for the relevant parameters are $L_0=10\mu \mathrm{m}$, $M={10}^{-11}\mathrm{kg}$, ${\omega }_{\text{osc}}={10}^5{s}^{-1}$, ${\omega }_{\text{cut}}={10}^{15}{\mathrm{s}}^{-1}$, which are typical values of a MEMS (microelectromechanical system).

Figure 2

Change of the renormalized electromagnetic energy density, compared to the static walls case, in the very proximity of the moving mirror, for different values of the cutoff frequency: ${\omega }_{\text{cut}}=6\times {10}^{15}{\mathrm{s}}^{-1}$ (blue dot-dashed line), ${\omega }_{\text{cut}}=8\times {10}^{15}{\mathrm{s}}^{-1}$ (red dotted line), ${\omega }_{\text{cut}}=9\times {10}^{15}{\mathrm{s}}^{-1}$ (green dashed line), ${\omega }_{\text{cut}}={10}^{16}{\mathrm{s}}^{-1}$ (black continuous line). The change of the energy density becomes more and more localized at the wall’s equilibrium position with increasing cutoff frequency. The numerical values of the other parameters are $L_0=10\mu \mathrm{m}$, $M={10}^{-11}\mathrm{k}\mathrm{g}$ and ${\omega }_{\text{osc}}={10}^5{\mathrm{s}}^{-1}$.

Figure 3

Correction to the scalar energy density of the field inside the three-dimensional cavity with respect to the fixed-wall configuration. The numerical values of the parameters used are $L_0=10\mu \mathrm{m}$, $L_y=L_z=0.5\times {10}^{-4}\mathrm{m}$, $M={10}^{-11}\mathrm{kg}$, ${\omega }_{\text{osc}}={10}^5{\mathrm{s}}^{-1}$ e ${\omega }_{\text{cut}}={10}^{15}{\mathrm{s}}^{-1}$.