Control of the Casimir force by the modification of dielectric properties with light

The experimental demonstration of the modification of the Casimir force [Proc. K. Ned. Akad. Wet. 51, 793 (1948)] between a gold coated sphere and a single-crystal Si membrane by light pulses is performed. The specially designed and fabricated Si membrane was irradiated with $514\mathrm{nm}$ laser pulses of $5\mathrm{ms}$ width in high vacuum, leading to a change of the charge-carrier density. The difference in the Casimir force in the presence and in the absence of laser radiation was measured by means of an atomic force microscope as a function of separation at different powers of the absorbed light. The total experimental error of the measured force differences at a separation of $100\mathrm{nm}$ varies from 10% to 20% in different measurements. The experimental results are compared with theoretical computations using the Lifshitz theory [Zh. Eksp. Teor. Fiz. 29, 94 (1956) [Sov. Phys. JETP 2, 73 (1956)]; Statistical Physics (Pergamon, Oxford, 1981), Pt. II] at both zero and laboratory temperatures. The total theoretical error determined mostly by the uncertainty in the concentration of charge carriers when the light is incident is found to be about 14% at separations less than $140\mathrm{nm}$. The experimental data are consistent with the Lifshitz theory at laboratory temperature, if the static dielectric permittivity of high-resistivity Si in the absence of light is assumed to be finite. If the dc conductivity of high-resistivity Si in the absence of light is included into the model of dielectric response, the Lifshitz theory at nonzero temperature is shown to be experimentally inconsistent at 95% confidence. The demonstrated phenomenon of the modification of the Casimir force through a change of the charge-carrier density is topical for applications of the Lifshitz theory to real materials in fields ranging from nanotechnology and condensed matter physics to the theory of fundamental interactions.

Figure 1

Schematic of the experimental setup, showing its main components (see text).

Figure 2

Fabrication process of Si membrane. (a) The Si substrate (colored black) with a buried $\mathrm{Si}{\mathrm{O}}_2$ layer (white). (b) The substrate is mechanically polished and oxidized, and (c) a window in $\mathrm{Si}{\mathrm{O}}_2$ is etched with HF. (d) Next, TMAH is used to etch the Si. (e) Finally, $\mathrm{Si}{\mathrm{O}}_2$ layer is etched away in HF solution to form a clean Si surface.

Figure 3

The deflection signal of the cantilever in response to the dc voltage and square voltage pulse applied to the Si membrane as a function of separation.

Figure 4

The change of the reflectivity after the termination of the laser pulse.

Figure 5

The differences of the Casimir forces in the presence and in the absence of light versus separation for different absorbed powers: (a) $9.3\mathrm{mW}$, (b) $8.5\mathrm{mW}$, and (c) $4.7\mathrm{mW}$. The measured differences $⟨\Delta F_C^{\mathrm{expt}}⟩$ are shown as dots, differences calculated using the Lifshitz formula at $T=300\mathrm{K}$, $\Delta F_C^{\mathrm{theor}}$, and at $T=0$, $\Delta F_C^{\mathrm{theor}}(T=0)$, as the solid and short-dashed lines, respectively, and those calculated including the dc conductivity of high-resistivity Si, $\Delta {\tilde{F}}_C^{\mathrm{theor}}$, as the long-dashed lines.

Figure 6

The random errors (which are equal to the total) versus separation for the measurements with the different absorbed powers: $a$, $9.3\mathrm{mW}$; $b$, $8.5\mathrm{mW}$; and $c$, $4.7\mathrm{mW}$.

Figure 7

(a) The dielectric permittivity of the Si membrane along the imaginary frequency axis in the absence of light (the long-dashed line is for the model of Si with a finite static permittivity and the short-dashed line includes dc conductivity of high-resistivity Si) and in the presence of light for different absorbed powers [solid lines: $a$, $9.3\mathrm{mW}$; $b$, $8.5\mathrm{mW}$; and $c$, $4.7\mathrm{mW}$]. (b) The same is shown on an enlarged scale in the region of the first Matsubara frequency ${\xi }_1$.

Figure 8

The total theoretical errors versus separation for measurements with different absorbed powers: $a$, $9.3\mathrm{mW}$; $b$, $8.5\mathrm{mW}$; and $c$, $4.7\mathrm{mW}$.

Figure 9

Theoretical minus experimental differences in the Casimir force versus separation for the measurements with different absorbed powers, (a) $9.3\mathrm{mW}$, (b) $8.5\mathrm{mW}$, and (c) $4.7\mathrm{mW}$, are shown as dots. The results computed at $T=300\mathrm{K}$ using the model with a finite static permittivity of high-resistivity Si are labeled 1 and those including the dc conductivity are labeled 2. Solid lines show the 95% confidence intervals.

Figure 10

The experimental differences in the Casimir force with their experimental errors are shown as crosses. Solid and dashed lines represent the theoretical differences computed at $T=300\mathrm{K}$ using the model with a finite static permittivity of high-resistivity Si and that including the dc conductivity, respectively.