GoKawiilTrap magnetic field profile, stabilization and characterization
The design of the flattened magnetic field profile along the axis at the centre of the trap in the trapping region, shown in Fig. 1, represents an important component of the gains in frequency resolution reported here. In particular, the curvature of the magnetic field governs the timescale over which anti-atoms interact with the microwave field as they are brought into resonance. In round numbers, the second axial derivative of the axial magnetic field profile used in the current work is less than 2 T m−2, or a factor of 20 smaller than that of the (unflattened) trapping field used during the measurement of the hyperfine splitting reported in ref. 15. This, in turn, enhances positron spin-flip efficiencies and our ability to probe anti-atoms much closer to the minimum frequencies of the two transitions. However, unlike the maximally flattened field configurations used in previous optical spectroscopy experiments13,39, here we intentionally depress the field below the central mirror coil by about 5 × 10−5 T to ensure the existence of a shallow, centrally located absolute minimum (Extended Data Fig. 1).
We use the electron cyclotron resonance (ECR) technique to map the axial magnetic field profile along the axis of the trap in situ with a frequency resolution of the order of 1 ppm (ref. 32) and a spatial resolution of about 1 mm, and are thus able to directly confirm the formation, location and depth of the central magnetic field minimum along the axis. Moreover, we use ECR to independently resolve and characterize both the initial field drift (attributed to fine-scale flux relaxation or redistribution after the first ramp up of the superconducting magnets at the beginning of the experiment) and the subsequent long-term downward linear field drift (attributed to the decay of persistent currents in the external solenoid used to generate the 1 T background field). These studies show that the drifts depend on the time history of magnet operations. Consequently, magnets are energized in the same sequence in all experiments.
For each experiment, we perform ECR measurements after the magnetic trap is energized as well as after each replicate as a complementary monitor of the magnetic field drift. From a linear fit to the ECR magnetic field measurements following each replicate, we extract a linear magnetic field drift of −0.025 ± 0.001 G h−1 for the 1.03 T experiment and −0.026 ± 0.001 G h−1 for the 1.07 T experiment (Extended Data Fig. 2). Assuming the same magnetic field dependence as in hydrogen, this corresponds to expected positron spin resonance onset frequency drifts of −71 ± 3 kHz h−1 for the 1.03 T experiment and −73 ± 3 kHz h−1 for the 1.07 T experiment. This is consistent with the measured linear drift of the positron spin resonance frequencies of −72.82 ± 0.04 kHz h−1 for the 1.03 T experiment and −75.64 ± 0.05 kHz h−1 for the 1.07 T experiment we find in our analysis (Fig. 4). Note that ECR measurements of the on-axis magnetic field were used solely to characterize magnetic fields in preparation for the experiments described here and as a complementary magnetic field measurement. ECR measurements were not used in our extraction of the ground-state hyperfine splitting of antihydrogen.
During our experiments, the mirror coils are supplied with currents ranging from a few amperes to nearly 500 A. The octupole is operated with a current of 900 A. These currents are individually monitored using high-precision, ultrastable direct current–current transformers based on closed-loop fluxgate magnetometer sensors (ITZ 2000-SB FLEX ULTRASTAB from LEM International SA). The DCCT provides a ±10 V output with a 500 kHz small-signal bandwidth. This signal was digitized by 24-bit National Instruments NI-9239 cRIO (Compact Real-time Input Output) analog-to-digital converter modules at 50 kS s−1 and averaged by the cRIO FPGA (field programmable gate array) firmware to 10 kS s−1. The resulting averaged signal was used for proportional–integral–derivative (PID)-based closed-loop control of the magnet power supplies, stabilizing output currents to within several mA.
Extracting the zero-field ground-state hyperfine splitting
The Breit–Rabi formula40 gives the following energy levels of hydrogen (Fig. 2a) in a magnetic field:
$${E}_{d}=\frac{{a}_{1{\rm{S}}}}{4}+\frac{1}{2}{g}_{{\rm{e}}}{\mu }_{{\rm{B}}}B\left(1-\frac{{g}_{{\rm{p}}}{m}_{{\rm{e}}}}{{g}_{{\rm{e}}}{m}_{{\rm{p}}}}\right),$$
$${E}_{c}=-\frac{{a}_{1{\rm{S}}}}{4}+\frac{1}{2}{\left[{a}_{1{\rm{S}}}^{2}+{\left({g}_{{\rm{e}}}{\mu }_{{\rm{B}}}B\left(1+\frac{{g}_{{\rm{p}}}{m}_{{\rm{e}}}}{{g}_{{\rm{e}}}{m}_{{\rm{p}}}}\right)\right)}^{2}\right]}^{1/2},$$
$${E}_{b}=\frac{{a}_{1{\rm{S}}}}{4}-\frac{1}{2}{g}_{{\rm{e}}}{\mu }_{{\rm{B}}}B\left(1-\frac{{g}_{{\rm{p}}}{m}_{{\rm{e}}}}{{g}_{{\rm{e}}}{m}_{{\rm{p}}}}\right),$$
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