Testing Time Dilation in Special Relativity with Spectroscopy of Fast Ions in Storage Rings: A New, Improved Limit

The relativistic Doppler effect was already proposed as an experimental test of relativity by Einstein in 1907 [1]. Time dilation leads to the ether-independent relativistic Doppler formula {$\nu_0=\nu_{\rm l}\gamma(1-\beta\cos\theta)$}, where {$\nu_{\rm l}$} and {$\nu_0$} denote the frequencies in the laboratory reference frame of the observer and the particles' rest frame moving at velocity
{$v=\beta c$} with respect to the observer, respectively; {$\theta$} is the angle of observation with respect to the particles' movement as measured in the lab frame, and {$\gamma=1/\sqrt{1-v^2/c^2}$}. We are using a modern version of the experiment by Ives and Stilwell [2], where two laser beams, parallel and anti-parallel with the atomic motion, excite the same transition of rest-frame frequency {$\nu_0$}. Within Special Relativity (SR), the lasers will have laboratory frequencies of
{$\nu_p = \nu_0/\gamma (1-\beta)$} and {$\nu_a = \nu_0/\gamma (1+\beta)$}, respectively, and the product of the two expressions yields {$\nu_0^2 = \nu_a \nu_p$}. It is common to parameterize possible deviations from SR using the Robertson-Mansouri-Sexl (RMS) test theory, where deviations in the time dilation sector are quantified by the test parameter {$\hat{\alpha}$}, and we get

{$$\frac{\nu_{\rm p}\nu_{\rm a}}{\nu_{\rm 0}^2} = 1 + 2\,\hat\alpha\, (\beta^2 + 2\,\vec \beta_{\rm lab}\cdot\vec\beta)+{\mathcal O}(c^{-4}), $$}

where {$c \vec{\beta}_{\rm lab}$} is the velocity of the lab against a preferred cosmic frame, which is generally taken to be the cosmic microwave background rest frame. The {$\beta^2 $} term used in our measurement allows to determine {$\hat\alpha$} absolutely without having to rely on the precise knowledge of {$\beta_{\rm lab}$} (for {$\beta \gg \beta_{\rm lab}$}).

Fig. 1 Schematic layout of the TSR. Li{$^+$} ions circulate in the 55 m circumference ring. In the electron cooler, cold electrons are overlapped with the ions and provide cooling. The measurements at the two different velocities are carried out sequentially. In the experiment, the two lasers are coupled into the ring from the same side and are retro-reflected.

In our experiment at the Max Planck Institute for Nuclear Physics, {$^7$}Li{$^+$} ions are accelerated by a tandem Van-de-Graaff accelerator and injected into the Test Storage Ring (TSR) shown in Fig.~1. The helium-like {$^7$Li$^+$} exhibits the strong {$2s~^3S_{1}\rightarrow 2p~^3P_{2}$} transition at 548~nm in its metastable triplet spectrum. To extract time dilation from a measurement of the Doppler shifts at one ion velocity, the rest frame transition frequency needs to be known accurately. Since the best available measurement has an uncertainty of 400~kHz, which was the limiting factor in our previous time dilation measurement [3], The Doppler-shifted frequencies {$\nu^{(1,2)}_{\rm a}$}, {$\nu^{(1,2)}_{\rm p}$} measured at {$\beta_1$} and {$\beta_2$} can be combined(neglecting the sidereal term) to

{$$\frac{\nu_{\rm a}^{(2)}\nu^{(2)}_{\rm p}}{\nu^{(1)}_{\rm a}\nu^{(1)}_{\rm p}}=\frac{1+2\hat{\alpha}\beta_2^2}{1+2\hat{\alpha}\beta_1^2}\approx 1+2\hat{\alpha}(\beta_2^2-\beta_1^2),$$}

independent of the rest frame frequency. As {$\beta_2^2 - \beta_1^2 \approx 0.8 \beta_2^2$}, the sensitivity is not significantly diminished.

The moving clocks are read using laser saturation spectroscopy. The laboratory frequencies {$\nu_{\rm p}$} and {$\nu_{\rm a}$} of the parallel and anti-parallel laser beams (with respect to the ion beam) must obey the above relation for resonance, which is indicated by a dip in the fluorescence spectrum. Through permanent cooling of the ions by a cold electron beam, the ion beam's width shrinks to {$\approx 250~ \mu$} m, the divergence to {$\approx 50~\mu$} rad, and the longitudinal momentum spread to {$\delta p/p=3.5\times 10^{-5}$}, leading to a Doppler width of the transition of about 2.5 GHz full-width half maximum. This broadening is overcome in saturation spectroscopy by selecting a narrow velocity class of the order of the natural linewidth; two lasers are overlapped parallel and anti-parallel with the ion beam, respectively, and excite the clock transition. The co-propagating laser (a Nd:YAG laser at 532 nm for {$\beta_1$} and an argon-ion laser at 514 nm for {$\beta_2$}) is fixed in frequency by locking it to a well-known iodine (I{$_2$}) line, whereas the counter-propagating light is generated by a tuneable dye laser (at 565 nm and 585 nm for {$\beta_1$} and {$\beta_2$}, respectively). The dye laser frequency is referenced to a second, I{$_2$}-stabilised dye laser by a tuneable frequency-offset lock. The iodine lines for the dye laser are calibrated using an optical frequency comb [4,5]. All laser frequencies are known absolutely to 70~kHz during the whole experiment.

The mean velocity of the ion beam is adjusted for the fixed-frequency laser at {$\nu^{(1,2)}_{\rm p}$} to select ions in the centre of the velocity distribution. The dye laser is scanned around {$\nu^{{(1,2)}}_{\rm a}$} and the Lamb dip in the fluorescence is recorded with photomultipliers (PMT) from the side; its frequency is measured with respect to the I{$_2$} clock in the laboratory frame. The observed resonance widths are in accordance with the natural linewidth of the {$2\,^3S \rightarrow 2\,^3P$} transition of 3.7~MHz, once the broadening mechanisms present in our experiment are accounted for.

Taking all systematic errors into account, the transition frequencies {$\nu_{\rm a}$} and {$\nu_{\rm p}$} measured at {$\beta_1=0.030$} and {$\beta_2=0.064$} yield SR values for the rest frame frequency of

{$$ \sqrt{\nu_{\rm a}{^{{(}1{)}}}\nu_{\rm p}^{(1)}} = (546\ 466\ 918\ 577 \pm108)~{\rm kHz} $$} {$$ \sqrt{\nu_{\rm a}^{(2)}\nu_{\rm p}^{(2)}} = (546\ 466\ 918\ 493 \pm98)~{\rm kHz}, $$}

respectively. From above follows a test parameter

{$$\hat{\alpha} = (-4.8\pm 8.4)\times 10^{-8}, $$}

which is consistent with SR. In late 2007 this experiment came to an end with the final analysis and publication [6]. This measurement is the most stringent test of relativistic time dilation, and puts a more than one order of magnitude tighter limit on deviations from special relativity than any non-storage ring bases method.

To make further progress at the TSR, the Lamb-dip frequency would have to be measured to significantly better than 100 kHz absolutely. At this level, distortions of the resonance due to laser forces on the ions and other effects start to become intractable. The obvious alternative is to use faster ion beams. A new experimental initiative has been started at the ESR storage ring at GSI in Darmstadt, Germany, where the Li{$^+$} beam can be stored at much higher speeds. First data was taken at {$\beta = 0.338$}, where the co-propagating laser beam at 386 nm has essentially twice the frequency of the counter-propagating one at 780 nm. A first-order Doppler-free signal has been observed (Lambda-spectrosocopy rather than saturation), and the prospects for a further-improved test of time dilation appear to be very good.

References:

[1] A. Einstein, Ann. Phys. 328, 197(1907).
[2] H. E. Ives and G. R. Stilwell. J. Opt. Soc. Am. 28, 215 (1938).
[3] G. Saathoff, S. Karpuk, U. Eisenbarth, G. Huber, S. Krohn, R. M. Horta, S. Reinhardt, D. Schwalm, A. Wolf, and G. Gwinner. Phys. Rev. Lett. 91, 190403 (2003).
[4] T. Udem, R. Holzwarth, and T. W. H\"ansch. Nature 416, 233 (2002).
[5] S. Reinhardt, G. Saathoff, S. Karpuk, C. Novotny, G. Huber, M. Zimmermann, R. Holzwarth, T. Udem, T. H\"ansch, and G. Gwinner. Optics Communications 261, 282 (2006).
[6] S. Reinhardt, G. Saathoff, H. Buhr, L. A. Carlson, A. Wolf, D. Schwalm, S. Karpuk, C. Novotny, G. Huber, M. Zimmermann, R. Holzwarth, Th. Udem, T. W. H\"ansch, and G. Gwinner. Nature Physics  3, 861 (2007).