Improved limits for violations of local position invariance from atomic clock comparisons

We compare two optical clocks based on the $^2$S$_{1/2}(F=0)to {}^2$D$_{3/2}(F=2)$ electric quadrupole (E2) and the $^2$S$_{1/2}(F=0)to {}^2$F$_{7/2}(F=3)$ electric octupole (E3) transition of $^{171}$Yb$^{+}$ and measure the frequency ratio $ u_{mathrm{E3}}/ u_{mathrm{E2}}=0.932,829,404,530,965,376(32)$. We determine the transition frequency $ u_{E3}=642,121,496,772,645.10(8)$ Hz using two caesium fountain clocks. Repeated measurements of both quantities over several years are analyzed for potential violations of local position invariance. We improve by factors of about 20 and 2 the limits for fractional temporal variations of the fine structure constant $alpha$ to $1.0(1.1)times10^{-18}/mathrm{yr}$ and of the proton-to-electron mass ratio $mu$ to $-8(36)times10^{-18}/mathrm{yr}$. Using the annual variation of the Suns gravitational potential at Earth $Phi$, we improve limits for a potential coupling of both constants to gravity, $(c^2/alpha) (dalpha/dPhi)=14(11)times 10^{-9}$ and $(c^2/mu) (dmu/dPhi)=7(45)times 10^{-8}$.