# The g-factor proton experiment / BASE-Mainz

The goal of this experiment is the measurement of the g-factor or the magnetic moment of the proton with highest precision. To this end, a Penning trap located in the bore of a superconducting magnet confines a single proton at the temperature of liquid helium (4.2 K). The proton is non-destructively detected by highly-sensitive superconducting detectors, which are capable of measuring the tiny induced image-current of the proton in the trap electrodes. The ultra-high vacuum in the Penning trap allows storing and measuring with the same single proton over a period of several months.

## Motivation

The motivation for the high-precision measurement of the g-factor of the proton is to compare it to the antiproton g-factor. Comparing both values in high-precision measurements constitutes a stringent test of CPT invariance, which is a fundamenal symmetry in the Standard Model of particle physics.

This symmetry predicted by the Standard Model - which is consistent with our experimental observations up to date - is in contrast to the matter-antimatter asymmetry which we observe in our universe. Any observation of the breaking of this symmetry would be an important contribution to solving this puzzle in modern physics. A difference of the proton and antiproton g-factors could for example be due to a so far unobserved interaction, which is not included in the Standard Model.

## The proton g-factor

The magnetic moment of an elementary particle is one of its fundamental properties. According to Dirac's equation, any point-like charged elementary particle with charge *q*, mass *m* and spin *S* a magnetic moment of:

The g-factor linking the spin and the magnetic moment is predicted by Dirac's theory to be exactly 2. The discovery the the electron g-factor is slightly larger than 2 lead to the development of quantum field theory, which considers self-energy and vacuum-polarization corrections to the g-factor, and is accurate to a tremendous amount of precision.

The first measurements of the proton g-factor, conducted by Otto Stern and Isaac Rabi in the 1930s, showed already a significant difference from Dirac's prediction and were an indication of the substructure of the proton.

## Measurement principle

We reach the high precision in measuring the g-factor by converting it into frequency measurements, since frequencies are the quantities which we can measure most precisely. The g-factor of the proton is determined by measuring the ratio of the Larmor frequency to the cyclotron frequency:

Consequently, our measurement determines the magnetic moment of the particle in units of the nuclear magneton.

The Larmor frequency is the precession frequency of the proton's spin in the magnetic field of the Penning trap:

To determine the Larmor frequency, we flip the spin of the proton by applying an oscillating magnetic field in orthogonal direction to the main magnetic field. We measure the spin-flip probability as function of the frequency of the oscillation magnetic field and extract the Larmor frequency from the spin-flip resonance.

The cyclotron frequency is the revolution frequency of the proton in a magnetic field. The electric field of the Penning trap which is needed for the confinement along the magnetic field lines modifies the motion of the trapped particle resulting in a system with three orthogonal oscillation modes. We determine the cyclotron frequency by measuring the three motional frequencies by non-destructive detection of the image-current signals and using the *invariance theorem* relation:

## Experimental setup

The unique part of our experimental setup is the analysis trap to determine the spin state of the proton. To this end, we superimpose a magnetic bottle to a Penning trap. The magnetic potential of the inhomogeneous magnetic bottle changes the effective trapping potential of the proton depending on its spin state. This results in a tiny difference in the oscillation frequency of the axial mode (along the magnetic field lines) for the two spin states. Since the magnetic moment of the proton is very small - about 660 times smaller compared to the electron - we require a very strong magnetic gradient of 300 000 Tm^{-2}, which corresponds to a magnetic field change of 1 T over a distance of 1.5 mm. This strong magnetic field gradient is necessary to make spin flips detectable. The result is a tiny change of only 170 mHz in the axial frequency of about 630 kHz.

Electrode stack of the Penning trap system of the proton experiment

The strong magnetic gradient limits however the line width of the cyclotron and Larmor resonance to about 100 ppm due to the amplitude dependence of the frequencies in the magnetic bottle. Since this is clearly an obstacle for high-precision measurements, we implemented the double-trap method for measuring magnetic moments for the proton. We make use of a second trap - the *precision trap* - which has a 100 000-fold more homogeneous magnetic field, so that the line width of the spin-flip resonance is on the parts-per-billion level. We shuttle the proton adiabaticly to the analysis trap to identify the initial and final spin-state of the spin-flip drives irradiated in the precision trap. In this way, we can measure both, the Larmor frequency and the cyclotron freqeuncy, in a homogeneous magnetic field. This is the key method to reach a precision on the ppb level in measuring the proton magnetic moment.

## Recent results

The proton g-factor experiment in the Institute of Physics in Mainz has developed to fundamental techniques to make high-precision measurements of the proton and antiproton magnetic moments. The first crucial step was the non-destructive detection of spin flips of a single proton [1]. Further, we reached a high-fidelity detection of single spin transistions [2], and we implemented the double-trap method for the first time for the measurement of a nuclear spin [3]. These developments allowed us to establish the BASE experiment at the antiproton decelerator of CERN to apply these methods to the antiproton to make stringent CPT invariance tests.

In Mainz, we have conducted the first high-precision measurement of the proton magnetic moment in 2014 [4]. We reached a 3.3 ppb uncertainty, which improved the uncertainty of this fundamental property by a factor of 3 compared to a hydrogen maser measurement, which set the record for more than 40 years.

In 2017, we made a further improvement by a factor of 11 and reached an uncertainty of 300 ppt [5], which represents the most precise measurement of a nuclear magnetic moment. Key methods for this result were an improved cyclotron cooling method and an improvement of the homogeneity in the precision trap.

Antiproton g-factor measurements are conducted at CERN by the BASE collaboration, the __B__aryon __A__ntibaryon __S__ymmetrie __E__xperiment, which operates an advanced multi-Penning trap system at the antiproton decelerator/ELENA facility. We capture low energy antiprotons in this Penning trap system and measure the antiproton g-factor with similar methods. We conducted a 1.5 ppb measurement of the antiproton magnetic moment in 2016/2017. More details about the BASE experiment at CERN are found on the website of the BASE collaboration. Combining the antiproton measurement with the results obtained here in Mainz resulted in a factor 3000 improvement of the CPT invariance test over competing efforts.

## Current research

We are presently aiming for a proton g-factor measurement with 30 ppt uncertainty - improving our record measurement by another factor of 10. A key method for this further improvement are the sympathetic cooling methods for protons with laser-cooled beryllium ions, which allow reducing the proton temperature into the Millikelvin range. This increases the spin-state detection fidelity to almost 100%, reduces systematic uncertainties due to finite particle amplitudes, and boosts our measurement statstics to a more than 10-fold reduced cyclotron cooling time. Further essential improvements are the implementation of phase-sensitive methods for the cyclotron frequency measurement and the stablilization of the superconducting magnet to reduce magnetic field fluctuations.

## References

[1] S. Ulmer *et al.*, *Phys. Rev. Lett. 106*, 253001 (2011)

[2] A. Mooser *et al.*, *Phys. **Lett. **B*** ***723*, 78–81 (2013)

*et al., Phys. Rev. Lett. 110*, 140405 (2013) [4] A. Mooser

*et al.*,

**Nature 509**, 596-599 (2014), link to the article

[5] G. Schneider *et al.*, **Science 358**, 1081-1084 (2017), link to the article

## Contact

If you are interested in joining this project for your thesis or as post-doc researcher please contact:

Dr. Christian Smorra (Christian.Smorra@cern.ch)

Univ.-Prof. Dr. Jochen Walz (Jochen.Walz@uni-mainz.de)