Analysis of the surface potential

5.4. Analysis of the surface potential

Here we describe how we use static and dynamic surface loss measurements to obtain information about the surface potential Us = UCP+Uad and a calibration of the

atom-cantilever distance d=zt,0−zc on both sides of the cantilever.

5.4.1. Additional potential

U

If we assume for the moment that only the CP-potentials are present on both sides of the cantilever (i.e. Uad = 0 on both sides), we would expect from a simulation of

U =Um+UCP that the “effective cantilever thickness”heff, defined by the width of

the window where χ = 0 in Figure 5.4, is heff = 1.4 µm for ωz/2π = 10 kHz and th = 1 ms. However, we observe heff = 2.2 µm. This shows that Us is significantly stronger than the expected contribution from UCP on at least one side of the can-

tilever. We explain this by the presence of an additional potential Uad due to surface

inhomogeneities or contamination [268, 129, 25, 26, 27, 269]. Without taking into account further information about Uad, this leaves an uncertainty in d of ±400 nm,

corresponding to the difference between the observed and expected heff.

The atoms could be used as a three-dimensional scanning probe that allows one to map out the spatial dependence of Uad in detail and to determine whether it

is due to magnetic, electrostatic, or other interactions, see e.g. the measurements in [25, 26, 27, 269]. As the characterization of Uad is not the main focus of our

work, we simply determine its strength relative to UCP in the relevant range of d

by combining the measurements from chapter 5.2 (see Figure 5.4) with information from measurements of resonant atom-cantilever coupling as described in chapter 5.5 (see e.g. Figures 5.8, 5.11).

Such dynamical coupling measurements are performed on both sides of the can- tilever in traps with similar U0. We can determine U0 to 10% from the measured

curves in Figure 5.4 without detailed knowledge of Us or d. Comparing measure- ments on both sides of the cantilever, we find that the dynamical coupling signal is a factor β = 3.2±0.6 larger on the metallized side. Furthermore we observe a linear dependence of the coupling signal with the cantilever amplitude and thus also a linear dependence of the signal on δzt = a. From this we conclude that has to be larger by the same factor, and the ratio β is thus given by the ratio of the coupling strength parameters on the metallized and dielectric side, β = met/diel.

Because ∝ ∂2

zUs, this implies a stronger surface potential on the metallized side. Stronger Us also implies larger d to maintain the same U0. Due to the fast decay

of Us with d, a substantially larger Us is required on the metallized side (not just larger by a factor of order β).

5.4.2. Iterative determination of the strength of

U

To obtain an absolute value for the strength of the surface potential on both sides, we use an iterative procedure to match both the observed effective cantilever thickness and the measured coupling strength ratioβ. We first choose a certain Cad and then

evaluate and d for the given U0 on the dielectric side. This fixes the cantilever

position zc. Then we adjust Cad on the metallized side to be consistent with the

surface loss curves in Figure 5.4 and extract for the given U0 on this side. We

compare the values ofon both sides and start a new iteration with weaker (stronger) Cad on the dielectric side if their ratio is smaller (larger) than the observed β, or

finish if it equals the observed β.

The observed β = 3.2 as well as the surface loss curves can be best explained by potentials of the formUad =Cad/(z−zc)4 with the following coefficients:

Cad = (200±100)C4 metallized side

Cad = (10±10)C4,d dielectric side. (5.18) For these potentials, zc= 64.36µm results.

For the metallized side, we use theC4 coefficient for a perfect conductor according

to Eq.2.57, which bears a small error (on the percent level) due to the small thickness and the finite conductivity of the Au/Cr film. On the dielectric side, the thin SiN layer together with the Au/Cr film acts as a cavity or waveguide for the vacuum modes [127], which results in a correction to the CP-potential (see chapter 2.3.1). With Eq.2.63 we calculate that at d = 1.0 µm this leads to a 25% larger potential than that of a bulk dielectric described by C4,d (see Eq. 2.58). On this side, the inferred potential is thus consistent with a pure Casimir-Polder potential.

To check the robustness of our analysis against changes in the assumed distance dependence of Uad, we perform similar analyses with other distance-dependences,

such asUad ∝(z−zc)−3 on both sides or Uad ∝(z−zc)−4 on the dielectric side and Uad ∝(z−zc)−3 on the metallized side. These analyses result in similar calibrations

of d. The overall error in d is±160 nm, which contains the uncertainty in Uad,zt,0,

U0,β, as well as the uncertainty in the cantilever thickness, and a contribution due

to residual oscillations of the atoms in the trap due to the ramping to the cantilever. We observe that Uad slowly changes over time by up to a factor of four on a

time scale of weeks. The measurements used to determine Uad described above

were all performed on the same day. The change in Uad during the course of these

measurements is negligible. However, for the analysis of measurements from other days, the strength of the surface potential is known with less precision. But as the relative change in the coupling strength is much less than the relative change in the surface potential strength (by a factor β C4/Cad = 1.6×10−2), this affects only