Testing relativistic time dilation with a two-velocity ion clock in a storage ring

We report on a new, improved test of time dilation in special relativity via the relativistic Doppler-effect. We use an optical transition with rest-frequency $\nu$ in $^7{\rm Li}^+$ ions, stored at two different speeds of $\beta_1=0.064$ and $\beta_2=0.03 $, respectively, in the TSR heavy-ion storage ring in Heidelberg. The Doppler-shifted excitation frequencies $\nu_{\rm p}$ and $\nu_{\rm a}$ for laser beams travelling parallel and antiparallel with respect to the ions are measured simultaneously using saturation spectroscopy in separate measurements at $ \beta_1$ and $\beta_2$. The resonance conditions $\nu=\gamma (1-\beta)\nu_{\rm p}$ and $\nu=\gamma (1+\beta)\nu_{\rm a}$ yield the relation $\nu_{1 \rm p} \nu_{1 \rm a} = \nu_{2 \rm p} \nu_{2 \rm a}$, if $\gamma = 1/\sqrt{1-\beta ^2}$, as predicted by SR. Deviations, e.g. caused by the existence of preferred frames, are parametrized by $\gamma = \gamma_{\rm SR} (1+\alpha \beta^2 + ...)$. We have established a preliminary new limit of $|\alpha| < 9 \times 10^{-8}$, a $25\times$ improvement over non-storage-ring techniques. We will also review the progress towards a measurement at $\beta=0.34$ at the ESR storage ring at GSI in Darmstadt, which will be key to reducing the limit on $\alpha$ even further.