We describe experiments with an optical frequency standard based on a lasercooled $^{171}$Yb$^+$ ion confined in a radiofrequency Paul trap. Theelectric-quadrupole transition from the $^2S_{1/2}(F=0)$ ground state to the$^2D_{3/2}(F=2)$ state at the wavelength of 436 nm is used as the referencetransition. In order to compare two $^{171}$Yb$^+$ standards, separatefrequency servo systems are employed to stabilize two probe laser frequenciesto the reference transition line centers of two independently stored ions. Theexperimental results indicate a relative instability (Allan standard deviation)of the optical frequency difference between the two systems of $\sigma_y(1000{\rm s})=5\cdot 10^{-16}$ only, so that shifts in the sub-hertz range can beresolved. Shifts of several hertz are observed if a stationary electric fieldgradient is superimposed on the radiofrequency trap field. The absolute opticaltransition frequency of Yb$^+$ at 688 THz was measured with a cesium atomicclock at two times separated by 2.8 years. A temporal variation of thisfrequency can be excluded within a $1\sigma$ relative uncertainty of $4.4\cdot10^{-15}$ yr$^{-1}$. Combined with recently published values for the constancyof other transition frequencies this measurement provides a limit on thepresent variability of the fine structure constant $\alpha$ at the level of$2.0\cdot 10^{-15}$ yr$^{-1}$.
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