Constraints on Spatially Oscillating Sub-mm Forces from the Stanford Levitated Microsphere Experiment Data

A recent analysis by one of the authors\cite{Perivolaropoulos:2016ucs} has indicated the presence of a $2\sigma$ signal of spatially oscillating new force residuals in the torsion balance data of the Washington experiment. We extend that study and analyse the data of the Stanford Optically Levitated Microsphere Experiment (SOLME) \cite{Rider:2016xaq} (kindly provided by the authors of \cite{Rider:2016xaq}) searching for sub-mm spatially oscillating new force signals. We find a statistically significant oscillating signal for a force residual of the form $F(z)=\alpha \; cos(\frac{2\pi}{\lambda}\; z +c)$ where $z$ is the distance between the macroscopic interacting masses (levitated microsphere and cantilever). The best fit parameter values are $\alpha=(1.1 \pm 0.4)\times 10^{-17}N$, $\lambda=(35.2\pm 0.6)\mu m$. Monte Carlo simulation of the SOLME data under the assumption of zero force residuals has indicated that the statistical significance of this signal is at about $2\sigma$ level. Private communication with members of the SOLME group has indicated that this previously unnoticed signal is most probably due to a systematic effect (diffraction of non-Gaussian tails of the laser beam). Thus it can only be useful as an upper bound of the amplitude of new spatially oscillating forces on sub-mm scales. Assuming gravitational origin emerging from a fundamental modification of the Newtonian potential of the form $V_{eff}(r)=-G\frac{M}{r}(1+\alpha_O \cos(\frac{2\pi}{\lambda}\; r+\theta))\equiv V_{N}(r)+V_{osc}(r)$, we evaluate the source integral and obtain a bound $\alpha_O < 10^7$ for $\lambda \simeq 35 \mu m$. Thus, we get no useful constraints on the modified gravity parameter $\alpha_O$. However, the constraints on the general phenomenological parameter $\alpha$ can be useful in constraining other fifth force models related to dark energy (chameleon oscillating potentials etc).