A thorium-229 optical nuclear clock with feedback loop – Nature

Laser set-up

In the ²²⁹Th interrogation system, a commercial laser (FHG-TA Pro, TOPTICA) provides an output power of approximately 500 mW at a wavelength of 296.8 nm, frequency-quadrupled in two doubling stages and one amplification stage from a diode seed laser at 1,187 nm. The final second-harmonic-generation process is based on nonlinear frequency conversion in an SBO crystal kept under a high-purity N₂ atmosphere (purity 5.0). Fundamental radiation still present in the beam after the final second-harmonic-generation step is separated using three dielectric-coated mirrors. For the absorption measurements, we used a head-on type PMT (R6835, Hamamatsu) mounted inside the vacuum chamber and operated at 2.5 kV. We used the same segment of the X2 sample⁶⁰ as in ref. ⁷. The segment in use has a cylindrical geometry with a diameter of 3.1(1) mm and a length of 4.2(1) mm and was oriented such that the beam traversed along the centre line of the piece. The measured average concentration of ²²⁹Th in the segment was 6.6(5) × 10¹⁵ mm⁻³.

The seed laser of the FHG-TA was locked to the clock laser (CLS, TOPTICA) through an offset frequency phase lock. A fast photodiode with a bandwidth of 10 GHz measured the beat frequency between the two lasers, which was then mixed with a reference frequency. The fast loop of this lock actuated on the seed laser diode current, and the slow loop was used to steer the grating of the laser diode. The laser was scanned by modulating the reference frequency.

Like the actuators of the offset lock, the Pound–Drever–Hall lock of the clock laser to the cavity was also configured such that the fast feedback loop actuated on the laser diode current, whereas the slow feedback signal of the Pound–Drever–Hall scheme controlled the position of the grating for optical feedback to the laser diode.

For clock comparison measurements and drift rate measurements of the cavity, the beat frequency f_b of the clock laser with the frequency comb (FC1500-250-ULN, Menlo) was recorded using a dead-time-free frequency counter (K+K FXE) operated in Π-type counting mode^61,62 at a gate time of 1 s. Our method of comparing the thorium clock at TU Wien with the Yb⁺ clock (TOPTICLOCK, TOPTICA)⁶³ at BEV follows the scheme described in ref. ⁷. The Yb⁺ clock was based on the 435.5-nm E2 transition of a single ¹⁷¹Yb ion.

All frequency synthesizers and counters in the ²²⁹Th clock system and the frequency comb were referenced to the 10-MHz signal of a commercial Rb clock (FS725, SRS).

To perform an absorption measurement at a specific frequency, we modulated (square wave) the offset lock frequency between the seed laser diode and the clock laser with 10 Hz between the target frequency and an off-resonance frequency. Each modulation cycle was also reflected in a synchronization signal connected to a time-resolved pulse counter (TimeTagger Ultra, Swabian Instruments). The modulation ensured that power fluctuations in the laser output did not affect the measurement. The detection set-up consists of a PMT, a radiofrequency amplifier and the pulse counter, which binned the arriving pulses from the PMT relative to the last synchronization edge. The absorption was then calculated by subtracting the on-resonance counts from the off-resonance counts and dividing by the off-resonance counts.

Clock operation

To operate the set-up as a clock, we first acquired a single absorption spectrum, as shown in Extended Data Fig. 2a. We observed that a Lorentzian line shape fits our absorption measurement data well. We then measured the expected error signal (Extended Data Fig. 2b). To measure an error signal, we modulated the offset lock frequency with 10 Hz between two frequencies f_low and f_high. The difference Δf = f_high − f_low was kept constant during the measurement, and only the centre was swept across the resonance. In our measurements, we used a frequency deviation Δf of \({f}_{{\rm{FWHM}}}/\sqrt{3}\), with f_FWHM describing the FWHM of the measured absorption peak (Extended Data Fig. 2a). We fitted this signal with the function

where A_L is the amplitude and γ is the half width at half maximum of the Lorentzian. For the clock operation, the laser frequency was again switched between two frequencies while we adjusted the centre position with the feedback. For both frequency positions, the counts c_low,i and c_high,i were recorded, where i is the measurement index for averaging the signal. The applied error signal E can be calculated as

with the number of cycles L and the inverse fit function \({y}_{{\rm{fit}}}^{-1}(c)\). Extended Data Fig. 2c illustrates an adjustment step inferred by the calculation of E. With the integration times set in the clock operation, the beat was shifted after every interrogation cycle by the calculated value E, as shown in Extended Data Fig. 1.

Variation of the fundamental constants

The variation of α(t) relates to the variation of the ratio of the ²²⁹Th and ytterbium clock frequencies through: \(({k}_{\mathrm{Th}}^{\alpha }-{k}_{\mathrm{Yb}}^{\alpha })\times {\partial }_{t}\,\log \,\alpha



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