An new test of relativistic time dilation with fast atomic clocks at different speeds

SASCHA REINHARDT¹, GUIDO SAATHOFF¹, HENRIK BUHR¹, LARS A. CARLSON¹, ANDREAS WOLF¹, DIRK SCHWALM¹, SERGEI KARPUK², CHRISTIAN NOVOTNY², GERHARD HUBER², MARCUS ZIMMERMANN³, RONALD HOLZWARTH³, THOMAS UDEM³, THEODOR W. HÄNSCH³ AND GERALD GWINNER^4*

¹ Max-Planck-Institut für Kernphysik, 69029 Heidelberg, Germany
² Institut für Physik, Universität Mainz, 55099 Mainz, Germany
³ Max-Planck-Institut für Quantenoptik, 85748 Garching, Germany
⁴ Dept. of Physics & Astronomy, University of Manitoba, Winnipeg R3T 2N2, Canada
^* e-mail: gwinner@physics.umanitoba.ca

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$} absolutely {\em without} having to rely on the precise knowledge of {$\beta_{\rm lab}$} (for {$\beta \gg \beta_{\rm lab}$}).

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.

Experiment

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], we set up a new experiment with ion beams at two different velocities, {$\beta_1 = 0.030$} and {$\beta_{2} = 0.064$}. 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

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

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 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

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

respectively. From Eq. ? follows a test parameter

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

which is consistent with SR [6].

Standard Model extension

In the leading order, the limit on {$\hat{\alpha}$} in RMS theory is identical to the constraint on {$\tilde{\kappa}_{\rm tr}$} in the photon sector of the Standard Model extension (SME), hence we obtain {$\tilde{\kappa}_{\rm tr} \leq 8.4\times 10^{-8}$}. In the particle sector, limits on the quadrupole parameters {$\tilde{c}_Q, \tilde{c}_{TJ}$} for the proton and the electron had been derived by Lane [7] from our previous measurement. Since then, much stronger limits have been measured and deduced from astronomical observations. In the present experiment, we have taken additional data with circularly polarized light, as shown in Fig. 2. At the position of detector PMT2, a residual magnetic field causes a polarization dependence of the Lamb dip frequency via the Zeeman effect. The absence of magnetic fields at PMT3 leads to a polarization independence of the resonance as observed in that detector. Non-vanishing SME dipole parameters would cause a polarization dependence of the Lamb dip, hence the PMT 3 data can constrain them. Preliminary investigations show that a Doppler experiment with circularly polarized light is sensitive to the so far untested parameters {$\tilde{b}_Z, \tilde{d}_{ZX},\tilde{d}_Z,\tilde{g}_{DZ},\tilde{g}_{IJ},\tilde{g}_Q,\tilde{g}_{TJ}$}, even if individual {$m_F$} levels cannot be resolved in the measurement.

Frequency of the Lamb dip as a function of the polarization of the laser light for two fluorescence detectors.

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] Th. Udem, R. Holzwarth, and T. W. Hänsch. Nature 416, 233 (2002).

[5] S. Reinhardt, G. Saathoff, S. Karpuk, C. Novotny, G. Huber, M. Zimmermann, R. Holzwarth, T. Udem, T. Hänsch, 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. Hänsch, and G. Gwinner. Nature Physics, in print (2007).

[7] C.D. Lane, Phys. Rev. D72, 016005(2005).