Figure 1

Absolute electrostatic forces for parallel plane (dotted curve), cylinder-plane (solid), and sphere-plane configurations (dotted-dashed). The force in the plane-plane case is given by $F_{\mathrm{El}}^{\left(0\right)}={ϵ}_{0}A{V}^2∕2d^2$, and in the sphere-plane case it is given by $F_{\mathrm{El}}^{\left(0\right)}=2\pi {ϵ}_0{V}^2{\sum }_{n=1}^{\infty }[\coth \left(u\right)-n\coth \left(nu\right)]∕\sinh \left(nu\right)$, with $\cosh \left(u\right)=1+d∕R$. We use $V=50\mathrm{mV}$ for the bias voltage, $A=1{\mathrm{mm}}^2$ for the surface area of the parallel plane configuration, $L=1\mathrm{cm}$ for the length of the cylinder, $a=100\mu \mathrm{m}$ for the radius of the cylinder, and $R=100\mu \mathrm{m}$ for the radius of curvature of the sphere. For the parallel plane and sphere-plane configurations these values are taken from actual experiments described in Refs. 18, 20, respectively.Reuse & Permissions

Figure 2

Geometrical construction of the nonparallel configuration. For the proximity approximation, the local distance between the cylinder and the plane is $d\left(x\right)=d-(L∕2-x)\tan \theta$.Reuse & Permissions

Figure 3

Absolute Casimir force for plane-plane (dotted curve), cylinder-plane (solid), and sphere-plane configurations (dotted-dashed). For the plane-plane case it is given by $F_{\mathrm{Cas}}^{\left(0\right)}={\pi }^2\hslash cA∕240d^4$. The latter two are evaluated within the proximity force approximation. For the sphere-plane case it is given by $F_{\mathrm{Cas}}^{\left(0\right)}={\pi }^3\hslash cR∕360d^3$. Parameters are the same as in Fig. 1.Reuse & Permissions

Figure 4

Permittivity for Au as a function of frequency calculated from optical data (courtesy of Astrid Lambrecht and Serge Reynaud). Permittivity data for Matsubara frequencies with $m⩾1$ in the range of temperatures of interest $-20°\mathrm{C}⩽T⩽+60°\mathrm{C}$ can be obtained from this plot. The vertical lines show the frequency region used in the $m$ summation over the Matsubara frequencies. The dotted line corresponds to $m=1$ at $T=-20°\mathrm{C}$, the dashed line to $m=1$ at $T=+60°\mathrm{C}$, and the dotted-dashed line to the maximum value $m_{\max }$ used at all temperatures, corresponding to a maximum Matsubara frequency of ${\xi }_{\max }={10}^{17}\mathrm{rad}∕\mathrm{s}$.Reuse & Permissions

Figure 5

Combined thermal and conductivity corrections to the Casimir force in the cylinder-plane geometry. We show the Casimir force, normalized to its bare value for infinite conductivity and zero temperature, versus the distance $d$ in the case of gold metallic surfaces at a temperature $T=300\mathrm{K}$. Optical data are used to calculate the frequency-dependent permittivity of Au. The $m=0$ contribution to the Lifshitz formula is computed by extrapolating the optical data to zero frequency using two different theoretical approaches: the plasma model (for which the TE $m=0$ mode contributes to the force), and the Drude model (for which the TE $m=0$ mode does not contribute). Parameters are the same as in Fig. 1.Reuse & Permissions

Figure 6

Dependence upon temperature of the Casimir force in the cylinder-plane geometry. We show the Casimir force, normalized to its bare value for infinite conductivity and zero temperature, versus the temperature $T$ in the case of gold metallic surfaces separated by a distance $d=3\mu \mathrm{m}$. Extrapolated data for zero frequency $(m=0)$ is obtained from two different theoretical approaches: the plasma model (for which the TE $m=0$ reflection coefficient is nonzero), and the Drude model (for which the TE $m=0$ reflection coefficient vanishes). Parameters are the same as in Fig. 1.Reuse & Permissions

Figure 7

Cylinder-plane geometry (lateral view). In the proximity force approximation, the force is computed as a superposition of forces between infinitesimal parallel plates, separated by a distance $d\left(\phi \right)=d+a(1-\cos \phi )$.Reuse & Permissions

Figure 8

(Color online) Electrostatic force in the cylinder-plane geometry, normalized to the electrostatic force evaluated in the leading proximity approximation of Eq. (2). We show the exact force (continuous line), and the force evaluated through different proximity approximation schemes obtained using the area of the cylinder (squares), the area of the plane (dots), and a geometric mean of the areas (triangles).Reuse & Permissions

Figure 9

(Color online) Casimir force in the cylinder-plane geometry, normalized to the Casimir force evaluated in the leading proximity approximation. We show the different proximity approximation schemes obtained using the area of the cylinder (squares), the area of the plane (dots), and a geometric mean of the areas (triangles).Reuse & Permissions

Figure 10

(Color online) Images of the experimental set-up, with an overall view (left), a close-up on the cylinder-resonator region (right), and an image from the optical microscope (inset). Inside the vacuum chamber are visible the two piezoelectric actuators for the fine approach between the cylinder and the resonator on the bottom side, and the piezoelectric actuator for the fine approach of the optical fiber and its mount on the top side, with the fiber end facing the middle point of the resonator. In between are the resonator and the cylinder, with coarse motion controlled by vertical feedthrough micrometers. The fiber is sent through a feedthrough inside the vacuum chamber (left side in the overall view), while a goniometer stage is used for coarse parallelization (right side in the overall view). The large viewport allows for use of a digital optical microscope with up to $100\times$ magnification for a rough parallelization and a coarse assessment of the fiber-resonator and resonator-cylinder size of the resonator is $2\mathrm{cm}\times 1\mathrm{cm}\times 178\mu \mathrm{m}$ for width, length, and thickness, respectively. The cylinder has a diameter $2a=6.35\mathrm{mm}$ $(1∕4\mathrm{in.})$ and a length of $2\mathrm{cm}$.Reuse & Permissions

Figure 11

Assessment of the parallelism. Resonator frequency versus the difference between the voltages applied to the two piezoelectric actuators driving the cylinder base $\Delta V_{\mathrm{PZT}}=V_L-V_R$, starting from a common mode of $V_L=V_R=50\mathrm{V}$, for $V_0=100\mathrm{V}$. The resonator frequency without bias voltage is ${\nu }_0=(527.30\pm 0.02)\mathrm{Hz}$. The maximum frequency (corresponding to the minimum frequency shift with respect to ${\nu }_0$) realizes the condition for parallelism, and this occurs at $\Delta V_{\mathrm{PZT}}=(54.8\pm 7.1)\mathrm{V}$. As the two piezoelectric actuators are spaced by $2\mathrm{cm}$, this corresponds to achieving parallelism within $\delta \theta =5.3\times {10}^{-5}\mathrm{rad}$. The determination of the frequency is based upon a fit of the mechanical transfer function of the resonator with a Lorentzian function, obtained through 100 averages of the FFT with a frequency span of $12.5\mathrm{Hz}$.Reuse & Permissions

Figure 12

(Color online) Electrostatic calibrations. Squared frequency shift $\Delta {\nu }^2$ versus bias voltage between the cylinder and the resonator for different values of the cylinder-plane distance, corresponding to the application of $V_{\mathrm{PZT}}=0\mathrm{V}$ (continuous line), $30\mathrm{V}$ (dashed line), and $70\mathrm{V}$ (dotted line). The curves are the best fit with a generic parabola. The typical resonance frequency without bias potential is around $527\mathrm{Hz}$, with a bandwidth of $4\mathrm{Hz}$. The peak frequency is determined through the Lorenzian fit with a typical error of $12\mathrm{mHz}$.Reuse & Permissions

Figure 13

Electrostatic calibrations. Plot of the curvature coefficient $k_c$ versus the piezoelectric common voltage $V_{\mathrm{PZT}}$ and best fit with a law $\Delta {\nu }^2=a+b∕{(V_{\mathrm{MAX}}-V_{\mathrm{PZT}})}^{2.5}$ corresponding to the expected frequency-shift dependence from Eq. (16). The best fit gives a parameter $V_{\mathrm{MAX}}=(177.4\pm 24.1)\mathrm{V}$, corresponding to the voltage which should close the gap between the cylinder and the resonator.Reuse & Permissions