Universal experimental test for the role of free charge carriers in the thermal Casimir effect within a micrometer separation range

We propose a universal experiment to measure the differential Casimir force between a Au-coated sphere and two halves of a structured plate covered with a P-doped Si overlayer. The concentration of free charge carriers in the overlayer is chosen slightly below the critical one, for which the phase transition from dielectric to metal occurs. One half of the structured plate is insulating, while the second half is made of gold. For the former we consider two structures, one consisting of bulk high-resistivity Si and the other of a layer of $\mathrm{SiO}{}_2$ followed by bulk high-resistivity Si. The differential Casimir force is computed within the Lifshitz theory using four approaches that have been proposed in the literature to account for the role of free charge carriers in metallic and dielectric materials interacting with quantum fluctuations. According to these approaches, Au at low frequencies is described by either the Drude or the plasma model, whereas the free charge carriers in dielectric materials at room temperature are either taken into account or disregarded. It is shown that the values of differential Casimir forces, computed in the micrometer separation range using these four approaches, are widely distinct from each other and can be easily discriminated experimentally. It is shown that for all approaches the thermal component of the differential Casimir force is sufficiently large for direct observation. The possible errors and uncertainties in the proposed experiment are estimated and its importance for the theory of quantum fluctuations is discussed.

Figure 1

Experimental configuration of a Au sphere moving back and forth above a structured plate covered with a P-doped Si overlayer in the dielectric state. The measured quantity is the differential Casimir force $F_{\mathrm{diff}}$ between the Au sphere and the two halves of the plate when the sphere bottom is far away from their boundaries. The figure displays the two extreme positions of the sphere during its motion. The size of the sphere is not shown to scale.

Figure 2

Differential Casimir forces computed at $T=300$ K for the configuration in Fig. 1 using the plasma model for Au with free charge carriers in dielectric materials disregarded and taken into account [top (red) pair of solid and dashed lines, respectively] and using the Drude model for Au with free charge carriers in dielectrics disregarded and included [bottom (blue) pair of solid and dashed lines, respectively] are shown as functions of the separation. Inset: Region of larger distances.

Figure 3

Differential Casimir forces computed at $T=300$ K in the configuration in Fig. 1 using the plasma model for Au with free charge carriers in dielectric materials disregarded and taken into account [top (red) pair of solid and dashed lines, respectively] and using the Drude model for Au with free charge carriers in dielectrics disregarded and included [bottom (blue) pair of solid and dashed lines, respectively] are shown as functions of the overlayer thickness at $a=1\mu \mathrm{m}$.

Figure 4

Logarithm of the differential Casimir forces computed at $T=300$ K using the plasma and the Drude models for Au with free charge carriers in dielectric materials disregarded is shown as functions of the separation by the top (red) and bottom (blue) lines, respectively. The middle (green) line shows common computational results for the differential Casimir force at zero temperature.

Figure 5

Thermal corrections to the differential Casimir forces at $T=300$ K computed using the plasma model for Au with free charge carriers in dielectric materials disregarded and taken into account [top (red) pair of solid and dashed lines, respectively] and using the Drude model for Au with free charge carriers in dielectrics disregarded and included [bottom (blue) pair of solid and dashed lines, respectively] are shown as functions of the separation. Inset: Region of larger distances.

Figure 6

The experimental configuration of a Au sphere moving back and forth above a structured plate with an additional ${\mathrm{SiO}}_2$ layer. The P-doped Si overlayer is in the dielectric state. The measured quantity is the differential Casimir force $F_{\mathrm{diff}}$ between the Au sphere and the two halves of the plate when the sphere bottom is far away from their boundaries. The figure displays the two extreme positions of the sphere during its motion. The size of the sphere is not shown to scale.

Figure 7

Differential Casimir forces computed at $T=300$ K in the configuration with an additional ${\mathrm{SiO}}_2$ layer (Fig. 6) using the plasma model for Au with free charge carriers in dielectric materials disregarded and taken into account [top (red) pair of solid and dashed lines, respectively] and using the Drude model for Au with free charge carriers in dielectrics disregarded and included [bottom (blue) pair of solid and dashed lines, respectively] are shown as functions of the separation.

Figure 8

Thermal corrections to the differential Casimir forces computed at $T=300$ K in the configuration with an additional ${\mathrm{SiO}}_2$ layer using the plasma model for Au with free charge carriers in dielectric materials disregarded and taken into account [top (red) pair of solid and dashed lines, respectively] and using the Drude model for Au with free charge carriers in dielectrics disregarded and included [bottom (blue) pair of solid and dashed lines, respectively] are shown as functions of the separation.