## Nailing down the proton magnetic moment

Fundamental physical laws are believed to remain the same if subjected to three simultaneous transformations: flipping the sign of electric charge, taking a mirror image, and running time backward. To test this charge, parity, and time-reversal (CPT) symmetry, it is desirable to know the fundamental properties of particles such as the proton to high precision. Schneider *et al.* used a double ion trap to determine the magnetic moment of a single trapped proton to a precision of 0.3 parts per billion. Comparatively precise measurements of the same quantity in the antiproton are now needed for a rigorous test of CPT symmetry.

*Science*, this issue p. 1081

## Abstract

Precise knowledge of the fundamental properties of the proton is essential for our understanding of atomic structure as well as for precise tests of fundamental symmetries. We report on a direct high-precision measurement of the magnetic moment μ_{p} of the proton in units of the nuclear magneton μ_{N}. The result, μ_{p} = 2.79284734462 (±0.00000000082) μ_{N}, has a fractional precision of 0.3 parts per billion, improves the previous best measurement by a factor of 11, and is consistent with the currently accepted value. This was achieved with the use of an optimized double–Penning trap technique. Provided a similar measurement of the antiproton magnetic moment can be performed, this result will enable a test of the fundamental symmetry between matter and antimatter in the baryonic sector at the 10^{−10} level.

Precise knowledge of the properties of the proton, such as its mass (*1*), lifetime (*2*), charge radius (*3*), and magnetic moment (*4*), is of great fundamental importance. The mass, for example, is an important input parameter for precise calculations in quantum electrodynamics, whereas the lower bound of the proton lifetime sets constraints on baryon number violation (*5*). Measurements of the proton charge radius have recently drawn attention, owing to the discrepancy between proton charge radius measurements done with muonic hydrogen and electronic hydrogen. The origins of this difference triggered the proton-radius puzzle. However, a recent publication (*6*) indicates that a possible reason has been identified. The proton magnetic moment, which is the focus of this work, was directly measured in 2014 with a fractional precision of 10^{−9} (*4*). This direct measurement reached a higher precision than the indirect MASER (microwave amplification by stimulated emission of radiation) measurement, which had stood as the most precise for more than four decades (*7*).

High-precision measurements of these properties provide sensitive probes to investigate fundamental symmetries such as CPT (charge, parity, and time-reversal) symmetry. A violation of CPT invariance could provide essential input in understanding the observed baryon asymmetry in our universe, which has yet to be understood within the Standard Model of particle physics and cosmology. One method to set stringent limits on possible effects caused by CPT violation involves comparison between matter and antimatter systems, such as hydrogen and antihydrogen (*8*) or neutral K^{0} and mesons (*9*), or direct high-precision comparison of the properties of protons and antiprotons. Each of these systems sets constraints on a specific set of CPT- and Lorentz-violating parameters (*10*) and allows tests of CPT symmetry in different sectors.

We have developed techniques that enabled us to improve the measurement of the proton magnetic moment in (*4*) by more than one order of magnitude (Fig. 1). A dedicated Penning trap system was used to store a single isolated proton for spin-transition spectroscopy and simultaneous high-precision frequency measurements. By applying our methods to the antiproton (*11*), an improved test of CPT invariance at the 10^{−10} level in the baryon sector may be possible.

A Penning trap (*12*) is formed by a superposition of an electrostatic quadrupole potential and a homogeneous magnetic field *B*_{0}, in our case at 1.9 T. The coupling between the spin magnetic moment μ_{p} of a proton and *B*_{0} results in an energy splitting Δ*E* = *ħ*ω_{L} between the two spin eigenstates (↑,↓) characterized by the Larmor frequency ω_{L}, where *ħ* is Planck’s constant divided by 2π. Probing this energy splitting in units of the free cyclotron frequency ω_{c} = *eB*_{0}/*m*_{p}, where *e*/*m*_{p} is the charge-to-mass ratio, enables determination of the magnetic moment independent from the magnetic field in units of the nuclear magneton ω_{L}/ω_{c} = μ_{p}/μ_{N} = *g*/2, where μ_{p} and μ_{N} are the proton magnetic moment and the nuclear magneton, respectively, and* g* is the dimensionless *g* factor.

In the field configuration of a Penning trap, the motion of a particle has three harmonic components. In our apparatus, the axial component oscillates at a frequency ω_{z} ≈ 2π × 633,665 Hz along the magnetic field lines. In the radial direction, the motion is composed of the modified cyclotron mode at frequency ω_{+} ≈ 2π × 28.96 MHz and the magnetron mode at ω_{–} ≈ 2π × 6933 Hz. The axial eigenmotion at ω_{z} is detected by a nondestructive measurement of the induced image currents (on the order of femtoamperes) in the trap electrodes that are converted to a measurable voltage by a superconducting resonance circuit and a cryogenic low-noise amplifier (*13*). The proton oscillation frequencies ω_{+} and ω_{–} are determined by sideband coupling (*4*, *12*, *14*). The measurement of these individual mode frequencies enables the determination of ω_{c}, and thus the magnetic field strength, by the invariance theorem (*12*) ω_{c}^{2} = ω_{+}^{2} + ω_{z}^{2} + ω_{–}^{2}.

In contrast, the Larmor frequency ω_{L} is not accompanied by an oscillating charge and thus is not accessible by direct image current detection. However, ω_{L} can be determined by probing the transition probability between the two spin eigenstates as a function of the frequency of an applied radio-frequency drive. To this end, the continuous Stern-Gerlach effect (*15*) is deployed, in which an inhomogeneous magnetic field called a “magnetic bottle,” *B*_{z,AT}(*z*) = *B*_{0,AT} + (*B*_{2,AT} × *z*^{2}) (where *B*_{0,AT} = 1.180 ± 0.001 T and *B*_{2,AT} = 298,000 ± 5000 T/m^{2}) with the axial coordinate *z*, causes a nondestructive coupling of the magnetic moment to the axial oscillation of the proton. As a result, a spin transition from spin-down to spin-up causes the axial frequency to change by Δω_{z,SF} = ω_{z,↓} – ω_{z,↑} = 2μ_{p}*B*_{2,AT}/(*m*_{p}ω_{z}) = 2π × (172 ± 3) mHz.

Measurements of ω_{L} and ω_{c} in a magnetic bottle are limited to a relative precision on the order of 10^{−6} (*16*) by line broadening. To circumvent this limitation, we use the double–Penning trap technique (*17*), which uses two spatially separated Penning traps (*4*). These traps are interconnected by a transport section, which allows adiabatic shuttling of the particle between the traps. The “precision trap” (PT) has a nearly homogeneous magnetic field for high-precision measurements of ω_{c} and ω_{L}: *B*_{z,PT}(*z*) = *B*_{0,PT} + (*B*_{1,PT} × *z*) + (*B*_{2,PT} × *z*^{2}), where *B*_{0,PT} = 1.9 T, *B*_{1,PT} = 66.8 ± 0.1 mT/m, and *B*_{2,PT} = 0.1 ± 0.1 T/m^{2}. The “analysis trap” (AT) contains the magnetic bottle for spin state detection.

Our double–Penning trap system (Fig. 2, A and B) is mounted in the horizontal bore of a superconducting magnet at *B*_{0} = 1.9 T located at the University of Mainz, Germany, with magnetic north pointing north-northwest (342° ± 5°). Both traps are in a five-electrode cylindrical compensated orthogonal configuration (*18*, *19*). All trap electrodes are manufactured from oxygen-free copper, gold-plated, and electrically isolated from each other with sapphire spacers. The center electrode in the AT is made with ferromagnetic Co/Fe, which produces the strong magnetic bottle. A self-shielding solenoid (*20*) around the trap suppresses external magnetic field fluctuations by more than one order of magnitude. Both Penning traps are equipped with superconducting toroidal coils (*21*) for image current detection of the axial frequency (*13*). In addition, a superconducting cyclotron detector in the PT, connected to a segmented electrode, enables a factor of 4 increase in the cooling rate of the cyclotron mode relative to earlier experiments (*4*). This development allows faster particle preparation, which ultimately improves the measurement speed by a factor of 2. Particle manipulation is realized with disc-shaped coils mounted close to the trap electrodes to drive magnetic dipole transitions between the spin eigenstates. In addition, electric-quadrupole excitation lines are connected directly to the trap electrodes to drive sidebands (*14*). The electrode stack is enclosed in a sealed cryogenic “trap can” with a volume of about 300 ml. When cooled to liquid helium temperatures, cryopumping produces pressures below 10^{−14} Pa (*22*), resulting in single-particle storage times of more than 1 year.

To conduct the actual high-precision *g*-factor measurement, we first prepare a single proton and clean the trap from simultaneously stored contaminant particles such as H_{2}^{+}, N^{+}, O^{+}, or any other positively charged ion. Any particle in the Penning trap is only stable if the stability criterion 2ω_{z}^{2} < ω_{c}^{2} (*12*) is met. By manipulating ω_{z}, all ions with larger charge-to-mass ratio than the proton violate the criterion and can be efficiently removed from the trap (see supplementary materials). Afterward, we start the experiment sequence shown in Fig. 2C. A single measurement cycle *i* commences with the preparation of the proton’s modified cyclotron energy to *E*_{+} ≤ *E*_{th} = *k*_{B} × 0.6 K, where *k*_{B} is the Boltzmann constant and *E*_{th} is the threshold energy for the preparation. This is of utmost importance in order to obtain low–axial frequency fluctuations σ(*E*_{+}) < 103 mHz, and thus high spin state detection fidelities, for a particle with an energy below the threshold energy *E*_{+} ≤ *E*_{th} (*23*). To this end, the proton’s modified cyclotron mode is resistively cooled by coupling it to the ω_{+} resonator in the PT. The energy *E*_{+} is determined by transporting the proton to the AT, where the axial frequency strongly depends on *E*_{+} owing to the large inhomogeneous magnetic field (*24*). If *E*_{+} > *E*_{th}, the proton is thermalized again until *E*_{+} ≤ *E*_{th} is obtained. Then the spin state detection is performed in the AT. For this purpose, an initial series of axial frequencies {ω_{z,AT}^{(1)}, …, ω_{z,AT}^{(}^{n}^{0)}}^{(}^{i}^{)} is measured while applying a spin-flip drive in the intervals between the frequency measurements until |Δω_{z}| = |ω_{z,AT}^{(}^{n}^{0)} – ω_{z,AT}^{(}^{n}^{0}^{–}^{1)}| > Δω_{z,th} = Δω_{z,SF} + [2π × 0.5σ(*E*_{+})] is satisfied. This allows the determination of the spin state at the end of the series with a fidelity of 97% on average (*23*, *25*).

Subsequently, the proton is transported to the PT to determine ω_{c}^{(}^{i}^{)} and ω_{L}. For this purpose, a Larmor drive close to ω_{L} with a uniformly distributed randomized frequency offset ω_{L,exc}^{(}^{i}^{)} = ω_{L} + Δω_{L}^{(}^{i}^{)}, where Δω_{L}^{(}^{i}^{)}/2π ∈ [–400, 400] mHz, is applied during a simultaneous measurement of ω_{+}^{(}^{i}^{)}. To calculate ω_{c}^{(}^{i}^{)} using the invariance theorem, the axial frequency ω_{z}^{(}^{i}^{)} is obtained from an interpolation of four ω_{z} measurements (compare fig. S1), two in advance of and two following the ω_{+}^{(}^{i}^{)} measurement. The magnetron frequency is obtained using the relation ω_{–}^{(}^{i}^{)} = (ω_{z}^{(}^{i}^{)})^{2}/(2ω_{+}^{(}^{i}^{)}) (*12*). This allows us to assign the ratio Γ^{(}^{i}^{)} = 2ω_{L,exc}^{(}^{i}^{)}/ω_{c}^{(}^{i}^{)} to each experiment cycle.

To determine whether a spin flip occurred in the PT during the Larmor excitation ω_{L,exc}^{(}^{i}^{)}, the spin state of the particle must be identified a second time in the AT. However, mode coupling in the measurement of the modified cyclotron frequency heats the proton to *E*_{+} ~ *k*_{B} × 200 K owing to energy exchange of both modes. Thus, in a next step the particle is thermalized to *E*_{+} ≤ *E*_{th} again and a second axial frequency measurement {ω_{z,AT}^{(1)}, …, ω_{z,AT}^{(}^{n}^{1)}}^{(}^{i}^{+1)} is recorded. Together with the initial axial frequency series, these two frequency sets yield the probabilities *p*_{ini,↑}^{(}^{i}^{)} and *p*_{fin,↑}^{(}^{i}^{+1)} (*23*, *25*), which are the probabilities of being in the spin-up state at the end of the first series (initial) and the start of the second series (final). Thus, we can calculate the probability that a spin flip occurred in the PT from *p*_{SF,PT}^{(}^{i}^{)} = *p*_{ini,↑}^{(}^{i}^{)}(1 – *p*_{fin,↑}^{(}^{i}^{+1)}) + (1 – *p*_{ini,↑}^{(}^{i}^{)})*p*_{fin,↑}^{(}^{i}^{+1)}.

Typically, one of these sequence cycles takes around 90 min. By repeating this sequence for different drive frequencies several hundred times, we obtain tuples (Γ^{(}^{i}^{)}, *p*_{SF,PT}^{(}^{i}^{)}) that make up the *g*-factor resonance.

Three individual protons were used to conduct a total of 1264 experiment cycles, accumulated over a period of 4 months from the end of August to December 2016. We analyzed the data with a maximum likelihood analysis using a Gaussian line shape (*26*). Monte Carlo simulations of the complete measurement sequence were used to check the consistency, bias, and robustness of the analysis and to ensure that the evaluation leads to the correct statistical error (see supplementary materials). The three individual resonances, one for each proton, are in 1σ agreement with each other, which justifies the combined analysis of all cycles. The result from the maximum likelihood analysis over the combined data set (Fig. 3) is
(1)where the standard error of the mean is 0.00000000075, which corresponds to a fractional precision of 268 parts per trillion (ppt). The resonance has a width of 1.34 ± 0.21 parts per billion (ppb), which is in good agreement with the expected linewidth of 1.41 ± 0.02 ppb from independent experimental measurements of the free cyclotron frequency (see supplementary materials).

Systematic corrections arising from energy- and time-dependent shifts of the Larmor and cyclotron frequency (*27*) shift the *g* factor by(2)These corrections are caused by small deviations of the trap potential from an ideal quadrupole configuration and by inhomogeneities in the magnetic field. An important feature and key difference relative to earlier experiments that measured the proton *g* factor (*4*) is the simultaneous measurement of ω_{L} and ω_{+} instead of a sequential determination. In particular, this ensures that ω_{L} and ω_{+} (which dominates the determination of ω_{c}) are measured at the same energies and times. This eliminates many systematic contributions, such as corrections stemming from the linear term *B*_{1,PT} of the magnetic field. The higher-order *B*_{2,PT} contribution leads to a minor correction of 8 ± 4 ppt (see supplementary materials). Electrostatic imperfections affect only ω_{c} and were thus carefully optimized (fig. S2). The residual uncertainty of Δω_{c} is 9 ppt. The dominant systematic contribution arises from the interaction of the particle with the image charge it induces on the trap surface (*28*). The resulting correction of –98 ± 3 ppt can be calculated very accurately, where the error accounts for deviations from the ideal geometry (*12*). The relativistic correction (*12*) contributes with –44 ± 26 ppt. Furthermore, we add the uncertainty of our cyclotron frequency determination (compare fig. S3), which is at 80 ppt. Correcting for the systematics shifts, as summarized in Table 1, the final result of the proton magnetic moment is
(3)where the statistical uncertainty is 0.00000000075 and the systematic uncertainty is 0.00000000034.

The value reported here is in agreement with the currently accepted CODATA value (*29*) but is an order of magnitude more precise, and improves the result of our previous μ_{p} measurement (*4*) by a factor of 11. This improvement with respect to the apparatus developed in (*24*) and used in (*4*) arises from linewidth narrowing achieved by improvements of the magnetic field homogeneity in the precision trap and optimization of the parameters of the Larmor drive. A considerably improved detector for the modified cyclotron frequency reduced the particle preparation time and allowed a doubling of the data acquisition rate. The improved detection was achieved by replacing a normal conducting copper coil with a superconducting coil together with an improved low-noise, cryogenic amplifier that together allowed for lower detector temperatures and shorter coupling times. Finally, the simultaneous irradiation of the Larmor drive and the measurement of the modified cyclotron frequency in the precision trap eliminated the dominant systematic shift seen in (*4*). The improvements presented here can be directly applied to the planned high-precision measurement of the magnetic moment of the antiproton (*30*), potentially enabling a CPT test at the sub-ppb level. This allows tests of the Standard Model at an energy scale of <10^{−25} GeV in the framework of the Standard Model extension (*10*); these tests may reach a higher absolute sensitivity than lepton magnetic moment comparisons with respect to CPT violation (*31*, *32*).

The data collection rate can be increased by further reducing the experiment cycle times, which are limited by subthermal cyclotron state preparation using the detector for the modified cyclotron frequency. One way to overcome this limitation is to sympathetically cool the proton by resonant coupling to laser-cooled ions (*33*). This will provide quasi-deterministic cooling and will reduce particle preparation times by more than two orders of magnitude. With this method, we expect that proton and antiproton magnetic moment measurements on the parts per trillion level will become possible.

## Supplementary Materials

This is an article distributed under the terms of the Science Journals Default License.

## References and Notes

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**Acknowledgments:**Supported by the Helmholtz-Gemeinschaft, the RIKEN Initiative Research Unit Program, the RIKEN Pioneering Project Funding, the RIKEN FPR Funding, the RIKEN JRA Program, the Max Planck Society, and European Research Council advanced grant 290870-MEFUCO. Data are available upon request to the corresponding author.