Experiment

Two nominally identical cylinders (51 mm diameter and 30 mm height) of HEMEX grade ultra-pure sapphire (GT Advanced Technologies) were machined from the same boule. The crystals were cleaned in a solution of 70 % nitric acid containing several drops of hydrofluoric acid and then mounted in copper cavities and sealed in a stainless steel vacuum can, evacuated to sub-10−6 mbar. Whispering gallery resonant mode WGE 16,0,0 was used for both cavities. This mode features a dominant radial electric field with 32 variations around the circumference of the sapphire crystal. The majority of the electromagnetic fields are contained within the sapphire dielectric, with minimal evanescent field leaking out to the copper walls of the cavity structure. The first cavity had a resonant frequency of 12.9688 GHz and a loaded quality factor of 109 at 4.4 K, while the second cavity had a resonant frequency of 12.9685 GHz and a loaded quality factor of 1.5 × 109 at 4.4 K.

Small concentrations of impurities within the sapphire give rise to a temperature/frequency turning point21; for the 347 kHz beat frequency between the two resonators, this turning point occurred at 5.5 K. A temperature controller (model 340, Lake Shore Cryotronics, Inc.) was used in conjunction with a resistive heater and a carbon-glass temperature sensor to operate the resonators at the turning point. Custom microwave circuits and control electronics are used to create a loop oscillator out of each resonator. Pound locking is employed, whereby modulation sidebands reflected back from the resonator are demodulated with a lock-in amplifier (model SR830, Standford Research Systems) and from this a correction signal is applied to a voltage-controlled phase shifter to align the frequency of the oscillator with that of the resonator. Power incident on the cavity is monitored with a detector (model DT8016, Herotek, Inc.), the signal is compared against a user-defined set-point and a correction voltage is applied to a voltage-controlled attenuator placed in situ with the loop oscillator. The oscillator comparison beat frequency was logged on a frequency counter (model 53142A, Agilent Technologies, Inc.) referenced to a 10 MHz rubidium standard.

The oscillators were rotated on a high-precision air-bearing rotation table (Kugler GmbH); an 18,000 point optical encoder was used to track the angular position of the table and maintain a constant rotation velocity. The table sat on three aluminium legs, with each one in turn placed on a force sensor; these were used to align the centre of mass with the axis of rotation. A biaxial high-gain tilt sensor (model 755, Applied Geomechanics) sits at the centre of the experiment. Variations in tilt were compensated for by heating or cooling two of the three aluminium legs. All three legs were heated above ambient temperature to improve performance of the tilt-control system.

Data analysis

Original time tags are converted into time in seconds since the Vernal Equinox before the start of data collection (20 March 2012, 05:14 UTC+0). This format assists with calculating the relevant phase offsets required to analyse the data in the context of the SME22. The rotation turntable features an optical encoder with 18,000 points and a trigger mark to indicate that a full rotation has occurred; the data are scanned and incomplete rotations are discarded. The rotation points are converted into a modular angle value in radians. The data are broken up into subsets containing ten full turntable rotations (corresponding to ∼1,000 s) and an ordinary least-squares regression is used to fit the subset to the following model:

The value of ω R t comes straight from the modular angular position of the turntable recorded in the data. The phase offset, φ R , is the angular difference between the start of data collection and the alignment of the crystal axis of the top cavity with geographical East. The start of data collection is triggered at the same point for each experimental run. The phase offset is once again only needed to aid with the reporting of coefficients in the SME framework. The amplitudes of equation (2) and the mean time of each subset are stored to file, producing 19,597 entries. A histogram of the magnitude of errors for the fits to and is produced (Supplementary Fig. 9); all subsets with an error further than 3σ from the mean are discarded, resulting in 299 entries being removed (1.5% of the full data set). These points correspond to data with excessive additional noise present that does not fit our model or expected signals and would otherwise corrupt the quality of the subsequent analysis.

The demodulated data set is then broken up into subsets containing 100 entries each (∼1.2 days) and fit to the following model via a weighted least-squares regression, using the square of the s.e. of the fits from equation (2) as the weights.

The phase offset φ ⊕ is the difference between the 2012 Vernal Equinox and the alignment of geographical East with the Y axis of the Sun Centred frame used for determination of the SME coefficients22. The value of ω ⊕ used was 7.3 × 10−5 rad s−1 and t is the relevant mean time calculated during the previous demodulation (equation (2)). Once again the amplitudes, s.e. and mean time for each subset is stored to file. The fitted amplitudes and s.e. from equations (3) and (4) are shown in Supplementary Figs 1–8. The eight amplitudes from equations (3) and (4) representing the first two harmonics of daily variations are used to bound the overall sensitivity of the experiment to LIVs. We take the s.e.-weighted average of all the amplitudes, which is equivalent to weighting by the variance. Noting the distribution of the histogram of all the amplitudes (Supplementary Fig. 10), whereby 95% of the points lie within 2 s.d. of the mean, we multiply the associated s.e. by 2, to determine the 95% confidence interval for our bound, Δν/ν≤9.2±10.7 × 10−19.

The final stage of the data analysis is used to set bounds on coefficients of the SME. Each amplitude and s.e. from equations (3) and (4) is used to perform a weighted least-squares regression fit to an offset and variations at harmonics of Earth’s orbital frequency Ω ⊕ and 2Ω ⊕ . Supplementary Table 2 summarizes the relevant amplitudes, their sensitivity to different coefficients of the SME and the corresponding numerical weights.