Impact of dispersion forces on matter-wave scattering near a dielectric disk

The influence of the Casimir-Polder potential upon matter-wave scattering near a dielectric disk is considered. We employ a rescaled Dyson series in order to take into account the disk geometry, and compare this with a previously used approach based on the proximity-force approximation. In the latter approach one uses a simplified potential that is sharply switched on in the region of the disk. This intrinsically neglects edge and saturation effects. We show that under appropriate conditions the previous simpler model works very well for the calculation of the phase accumulated along the full path of the particle due to approximate cancellations of the errors. We present specific results for experimentally relevant scenarios, namely indium atoms or deuterium molecules interacting with a silicon nitride or silicon dioxide disk. Finally we compare our calculations to the analytically solvable problem of an infinitely thin perfectly reflecting disk, finding the expected qualitative agreement.

Figure 1

Schematic of the considered setup and illustration of the improvement our model makes over a previously used one.

Figure 2

Comparison of both polarizabilities as a function of (complex) frequency $\xi$. It can be seen that the polarizability for indium decreases much faster with frequency compared to that for deuterium.

Figure 3

The coordinate system used in our calculations of Casimir-Polder potentials.

Figure 4

Casimir-Polder potential (solid line) for a deuterium molecule at a lateral distance ${\rho }_A-R=100\text{nm}$ from a ${\mathrm{SiN}}_{\text{x}}$ disk. The dashed lines show the potential for the Dyson half-space (14) and half-space approximated (15) for comparison. The shaded area represents the extent of the disk.

Figure 5

Detail of the Casimir-Polder potential for a deuterium molecule near the edge of an ${\mathrm{SiN}}_{\text{x}}$ disk of thickness 1 $\mu \mathrm{m}$ and radius of 30 $\mu \mathrm{m}$.

Figure 6

(Upper curve) Collected phase [found from Eq. (18)] of a deuterium molecule moving at 1060 m/s at various distances ${\rho }_{\text{A}}-r$ from the edge of a disk with the same parameters as that shown in Figs. 4 and 5. (Lower curve) The same quantity but multiplied by $C_{3\text{HS}}/C_{3\text{DHS}}=0.86$ (see Table 1) in order to compensate for the Dyson series overestimation.

Figure 7

The difference between the phase collected in the PFA and the Dyson-compensated phase collection from the disk modeled by Eq. (18). The difference is very small (tens of milliradians), showing that our approximation works well.

Figure 8

Effect of the CP-induced phase shift on matter-wave diffraction. The plots show the relative on-axis intensity of Poisson's spot calculated for the experimental parameters used in Ref. [8] assuming as the diffraction obstacle (a) a silicon-nitride disk, as was used in the experiment, and (b) a silicon-dioxide disk. $I_{\mathrm{rel}}$ is given in intensity units of the undisturbed wave front (without the sphere the plot would show $I_{\mathrm{rel}}=1$). The on-axis intensities were calculated using the two different dispersion constants given in Table 1 as indicated. The graph in (a) labeled as $C_{3,\mathrm{GRI}}$ was calculated using the half-space dispersion constant determined in the experiment of Ref. [27]. The thick graphs assume a disk thickness of $1\mu \mathrm{m}$ and the thin graphs $0.5\mu \mathrm{m}$. The graph indicated with “no CP” shows the relative intensity of Poisson's spot without any phase shift due to the CP potential.

Figure 9

Simulated lateral relative intensity for the deuterium Poisson spot experiment [8] for detector distances (a) and (c) $b=30$ mm, and (b) and (d) $b=321$ mm. The line styles indicate the type of CP potential used (colors analogous to Fig. 8). Dark blue solid lines, no CP potential; light blue dashed lines, $C_{3\text{HS}}$; orange dashdotted lines, $C_{3\text{DHS}}$; green dotted lines, $C_{3,\text{GRI}}$. In (b) and (d) the difference in the simulated intensity for the different dispersion constants is not visible; a small change, mostly on axis, can be seen for the smaller detector distance.

Figure 10

Simulated relative on-axis intensity of Poisson's spot for the deuterium experiment compared to the experimental values deduced from Ref. [8]. Here we only show the result for the normal half-space dispersion constant $C_{3\text{HS}}$ and a disk thickness of $1\mu \mathrm{m}$, but include an estimate for the influence of the support bars that kept the disk in place (intensity reduced by area shaded in gray) and an increasing amount of disk edge corrugation (orange contour lines correspond to increases of peak-to-peak corrugation amplitude ${\sigma }_{\mathrm{corr}}$ in steps of 10 nm). The data points agree with a corrugation amplitude of about 200 nm.

Figure 11

Simulated relative on-axis intensity of Poisson's spot in the shadow of a silicon-dioxide disk 200 nm in diameter and indium atoms as beam species. The thickness of the disk we assume to be 10 nm, 50 nm, and 100 nm as indicated. Besides the CP interaction potential the experimental parameters (wavelength $\lambda =6.67$ pm, source width $w_s=20\mu \mathrm{m}$, $g=600$ mm) are the same as for the experiment discussed in Ref. [25], where the focus is on diffraction of indium atoms at a silicon-dioxide sphere. For comparison we show the relative on-axis intensity behind a 200-nm diameter sphere, which we computed using the sphere-radius dependent dispersion constant $C_{52}=9.365\times {10}^{-22}{\mathrm{m}}^{5/2}$ for the CP phase shift $\mathrm{\Delta }{\phi }_{\text{CP,}\text{BNR}}=C_{52}/z_{\text{A}}^{5/2}$ in the vicinity of a sphere introduced there.