A high-precision Penning-trap experiment was operated with a programmable 20 V Josephson voltage source, providing a significantly higher stability of the electrostatic trapping potential. This increased the motional frequency stability of a trapped ion by about a factor of two with respect to state-of-the-art voltage sources. An absolute axial frequency stability of 7.8(6) mHz corresponding to a relative precision of 9.7 ( 8 ) × 10 − 9 at 795 kHz was demonstrated using a trapped 9Be3+ ion as a measurement probe. The enhanced stability in the trapping potential opens up the possibility for improved determinations of the proton and antiproton magnetic moments and direct high-precision measurements of the nuclear magnetic moments of 2H, 3He, and 7Li. In the field of mass spectrometry, the developments will enable measurements of cyclotron frequency ratios and, thus, mass ratio measurements with unprecedented relative statistical uncertainties below the 10 − 12 level.

Topics
Josephson effect, Electrostatics, Magnetic dipole moment, Electronic noise, Precision measurements, Ion-trap, Mass spectrometry, Cyclotron resonance

Penning traps are versatile tools in the determination and direct comparison of particle properties. The measurement of motional eigenfrequencies of single, trapped ions allows to determine masses,1–3 magnetic moments,4,5 charge-to-mass ratios,6 electron binding energies7 as well as the excitation energy of metastable electronic states.8,9 Such high-precision measurements of particle properties play a crucial role in validating fundamental theories in physics, for example, quantum electrodynamics,7,10–13 charge-parity-time reversal symmetry,14–16 special relativity,17 or particle interactions and symmetries.18 They are thus sensitive tests for physics beyond the Standard Model of particle physics, such as fifth force carriers19–21 or the variation of fundamental constants.22 

In a Penning trap, single ions are confined in a well-controlled environment using a homogeneous magnetic field B and a superimposed electrostatic quadrupole potential Φ⁠. The stability of both trapping fields is essential for the precise measurement of the motional eigenfrequencies of the trapped ion, and thus for the to-be-determined particle properties. In state-of-the-art Penning-trap setups, the magnetic field is stabilized to a relative precision of a few 10 − 11 over time periods of hours by combining techniques presented in Refs. 23–25. The electrostatic trapping potential is typically created by ultra-stable voltage supplies based on Zener diodes. They provide relative voltage stabilities of 4 × 10 − 8 for small voltages up to 10 V (e.g., with the UM1-14 voltage source from Stahl electronics26) and 2 × 10 − 8 for high voltages up to 100 V (e.g., with the StaReP voltage source built at the Max Planck Institute for Nuclear Physics27) on time scales of a few minutes using a single trapped ion as a measurement probe.

In our groups, mainly two physical properties are measured, namely masses and g-factors. To exceed the currently achievable relative precision in mass measurements at Pentatrap in the low 10 − 12 range,2,9 the stability of the electrostatic potential must be improved. Likewise, in the case of nuclear magnetic moment measurements at μTEx, voltage stabilities of at least 1.5 × 10 − 8 are required over the measurement time of a few minutes to resolve tiny frequency differences indicating a change in the spin state28 of, e.g., 3He and 7Li. For heavier ions, however, improved stabilities of the electrostatic potential of up to 6 × 10 − 9 are mandatory. Due to its small mass, the first g-factor measurement in a Penning trap was achieved for an electron.29 After that, it took more than 30 years to detect a spin-flip of the next heavier particle, the proton30,31 and measure its magnetic moment32 which is now determined with a precision of 0.3 × 10 − 9 (equivalent to 0.3 ppb).15 Challenges persist in directly measuring the magnetic moments of bare nuclei heavier than a proton. These issues are addressed by implementing a programmable Josephson voltage standard (PJVS) to create an ultra-stable trapping potential.

The electrostatic potential along the z-axis Φ ( z ) = V 0 ( C 0 + C 2 z 2 + C 4 z 4 + ⋯ ) is created by applying voltages to a set of electrodes33 (see Fig. 1), where V0 is the voltage applied to the central ring electrode. In a symmetric trap, odd-order contributions are negligible. Typically, the trap geometry is chosen such that the axial frequency becomes independent of the voltage applied to the correction electrodes VCE (orthogonal trap)34 and the higher-order contributions of the potential (C4, C6) are minimized (compensated trap)35 by adapting the so-called tuning ratio T R = V C E / V 0⁠. In this potential, an ion with charge-to-mass ratio q/m oscillates with the axial eigenfrequency as follows:
(1)
FIG. 1.
Schematic cross section through a Penning trap with five electrodes creating the harmonic potential (red), cooled in liquid helium. Top: UM1-14 voltage source. Left: the PJVS is cooled in a separate liquid helium reservoir. Low-pass filters are used to suppress high-frequency noise, which could be picked up from surrounding electronics, but do not influence the voltage stability. The homogeneous magnetic field along the z-axis (solid petrol line), used for the precise determination of the ions eigenfrequencies, is created by a superconducting magnet. For magnetic moment measurements, an inhomogeneous magnetic field is superimposed (dashed petrol line). For details see text.
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Schematic cross section through a Penning trap with five electrodes creating the harmonic potential (red), cooled in liquid helium. Top: UM1-14 voltage source. Left: the PJVS is cooled in a separate liquid helium reservoir. Low-pass filters are used to suppress high-frequency noise, which could be picked up from surrounding electronics, but do not influence the voltage stability. The homogeneous magnetic field along the z-axis (solid petrol line), used for the precise determination of the ions eigenfrequencies, is created by a superconducting magnet. For magnetic moment measurements, an inhomogeneous magnetic field is superimposed (dashed petrol line). For details see text.

FIG. 1.
Schematic cross section through a Penning trap with five electrodes creating the harmonic potential (red), cooled in liquid helium. Top: UM1-14 voltage source. Left: the PJVS is cooled in a separate liquid helium reservoir. Low-pass filters are used to suppress high-frequency noise, which could be picked up from surrounding electronics, but do not influence the voltage stability. The homogeneous magnetic field along the z-axis (solid petrol line), used for the precise determination of the ions eigenfrequencies, is created by a superconducting magnet. For magnetic moment measurements, an inhomogeneous magnetic field is superimposed (dashed petrol line). For details see text.
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Schematic cross section through a Penning trap with five electrodes creating the harmonic potential (red), cooled in liquid helium. Top: UM1-14 voltage source. Left: the PJVS is cooled in a separate liquid helium reservoir. Low-pass filters are used to suppress high-frequency noise, which could be picked up from surrounding electronics, but do not influence the voltage stability. The homogeneous magnetic field along the z-axis (solid petrol line), used for the precise determination of the ions eigenfrequencies, is created by a superconducting magnet. For magnetic moment measurements, an inhomogeneous magnetic field is superimposed (dashed petrol line). For details see text.

Close modal
A superimposed magnetic field B confines the ion in the radial plane, leading to the modified cyclotron motion (+) and the slow magnetron motion (−) with eigenfrequencies ν ± = 1 2 ( ν c ± ν c 2 − 2 ν z 2 )⁠. The free-space cyclotron frequency νc is retrieved by the Brown–Gabrielse invariance theorem as follows:36 
(2)

For a sufficiently stable magnetic field, the mass can be extracted with the same relative precision as the free cyclotron frequency. Magnetic field drifts are canceled to a large extent when measuring mass ratios37 between one ion with a well-known mass m1 and another ion with a to-be-determined mass m2: ν c , 1 / ν c , 2 = q 1 m 2 / ( q 2 m 1 )⁠. Additionally, for q/m doublets, systematic uncertainties, such as frequency shifts due to, e.g., relativistic effects, are largely reduced,36,38,39 allowing mass determinations with the lowest statistical uncertainty of a few 10 − 12⁠.

At this level, the achievable precision becomes limited by fluctuations of the electrostatic trapping potential Δ V 0 / V 0 which affects the stability of the eigenfrequencies as follows:
(3)
Typical relative voltage fluctuations of 10 − 8 limit the shot-to-shot cyclotron frequency determination to a relative precision of a few 10 − 11⁠, depending on the measurement scheme, resolution, and trapping parameters. This is at least an order of magnitude larger than systematic uncertainties of ideal q/m doublets (more details in the supplementary material). A better voltage stability could in principle be achieved by using lowpass filters with a lower cutoff frequency. However, this increases the settling time of the electrostatic potential whenever the voltage on an electrode is changed. For both mass measurements and magnetic moment measurements, it is essential to shuttle the trapped ions frequently from one trap to another trap, which requires changing the applied voltages. Thus, a compromise between a low cutoff frequency and a fast settling time is needed, which in our case was chosen to be f c = 1 / ( 2 π R C ) = 8.8   Hz⁠, equivalent to a settling time of 0.1 s.
Nuclear g-factors are extracted from the following Larmor precession frequency νL:
(4)
with spin quantum number I, elementary charge e, and proton mass mp. The Larmor frequency is not directly accessible in a Penning trap. Instead, the spin-flip probability is probed for a range of driving frequencies, and νL is subsequently determined from the resonance curve. Spin-flips are observed in a dedicated trap, utilizing the continuous Stern–Gerlach effect.28 To this end, an inhomogeneous magnetic field B z = B 0 + B 2 z 2 is superimposed. This results in a coupling of the ion's magnetic moment, originating from the cyclotron motion and spin μ z cycl + μ z spin⁠, to the axial motion, such that the axial frequency becomes dependent on the spin state,28,31
(5)
where ν z , 0 is the axial frequency without magnetic inhomogeneity, E + is the cyclotron energy, and μN is the nuclear magneton. A nuclear spin-flip shifts the axial frequency by tens of mHz out of several hundred kHz, which is a few orders of magnitude smaller than for electron spin-flips. Instabilities in the electrostatic trapping potential also shift the axial frequency [Eq. (3)] by tens of mHz and, thus, mimic axial frequency shifts caused by nuclear spin-flips [Eq. (5)], which is a limiting factor in the determination of bare nuclear g-factors. Thus, by implementing a Josephson voltage standard as the main potential source, high-precision nuclear g-factor measurements of trapped, light ions become accessible.
Josephson voltage standards (JVS) are used as primary standards for the voltage and to calibrate all other voltage sources. A review of their development until their adoption as representation of the unit volt can be found, e.g., in Ref. 40. The output voltage U of a JVS is given by the microwave frequency f, the number of biased junctions in series M, and the Shapiro step order n expressed as
(6)
where K J = 2 e / h ≃ 483   597.8484   GHz V − 1 is the Josephson constant and h the Planck constant. For use as a variable voltage source, a series array of Josephson junctions is divided into subarrays41 which can be switched on and off remotely. It is, therefore, called programmable Josephson voltage standard (PJVS). The PJVS used in this work consists of three main parts:

  1. a cryoprobe with a double-stacked array of Josephson junctions in a binary arrangement, immersed in liquid helium,

  2. a computer-controlled 18-channel bias current source used to quickly set the number of Josephson junctions M,42 with a resolution of two junctions or 290 μV,43 

  3. and a 68–72 GHz microwave synthesizer with a minimum step size of 4 kHz.

All M = 139   264 Josephson junctions are connected in series for the maximum output voltage of 20 V. At the used output voltages around 3 V, a change in the microwave frequency by 4 kHz corresponds to a voltage adjustability below the nV level. The stability of the PJVS voltage is determined by the microwave frequency stability, which is better than 10 − 11 in the short-term regime and locked to a rubidium standard and GPS reference for long-term stability.

The superior performance of the PJVS compared to the conventional ultra-stable voltage supply UM1-1426 used at μTEx has been verified with a state-of-the-art nanovoltmeter from Agilent, model 3458 A. Two channels of the same voltage supply were measured differentially against each other at an absolute voltage of 3.4 V for the UM1-14 and 10 V for the PJVS. The relative voltage stability of the PJVS is at least a factor of 9 better than the UM1-14 stability [see Fig. 3(a)], the experimental uncertainty of this comparison is limited by the intrinsic noise of the nanovoltmeter.

In a Penning trap, the oscillating ion induces image currents on the trapping electrodes. These are picked up using a superconducting tank circuit and amplified by a cryogenic low-noise amplifier (Fig. 1). An ion in resonance with the tank circuit is resistively cooled until both systems are in thermal equilibrium.44 In the fast Fourier transform (FFT) of the tank circuit's noise signal, such an ion shows a characteristic dip signal at its axial frequency. A peak signal indicates an ion excited to a larger radius of motion than a thermalized ion [see Fig. 2(a)].

FIG. 2.
(a) Detection signal of an ion in thermal equilibrium with the tank circuit, showing a dip (blue) and an ion excited to a larger amplitude while the resonator is shifted by 2.5 kHz, showing a peak (green). The two signals were obtained for slightly different ring voltages V0 and FFT times. (b) Phase-sensitive axial measurement scheme. With the resonator shifted by 2.5 kHz, a dipolar excitation is applied to the axial mode (green). During the short (reference) and long evolution time, a weak coupling to the tank circuit leads to a loss of amplitude (orange). Subsequently, the phase and amplitude are readout (gray) and the resonator is shifted back for cooling the ion (blue).
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(a) Detection signal of an ion in thermal equilibrium with the tank circuit, showing a dip (blue) and an ion excited to a larger amplitude while the resonator is shifted by 2.5 kHz, showing a peak (green). The two signals were obtained for slightly different ring voltages V0 and FFT times. (b) Phase-sensitive axial measurement scheme. With the resonator shifted by 2.5 kHz, a dipolar excitation is applied to the axial mode (green). During the short (reference) and long evolution time, a weak coupling to the tank circuit leads to a loss of amplitude (orange). Subsequently, the phase and amplitude are readout (gray) and the resonator is shifted back for cooling the ion (blue).

FIG. 2.
(a) Detection signal of an ion in thermal equilibrium with the tank circuit, showing a dip (blue) and an ion excited to a larger amplitude while the resonator is shifted by 2.5 kHz, showing a peak (green). The two signals were obtained for slightly different ring voltages V0 and FFT times. (b) Phase-sensitive axial measurement scheme. With the resonator shifted by 2.5 kHz, a dipolar excitation is applied to the axial mode (green). During the short (reference) and long evolution time, a weak coupling to the tank circuit leads to a loss of amplitude (orange). Subsequently, the phase and amplitude are readout (gray) and the resonator is shifted back for cooling the ion (blue).
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(a) Detection signal of an ion in thermal equilibrium with the tank circuit, showing a dip (blue) and an ion excited to a larger amplitude while the resonator is shifted by 2.5 kHz, showing a peak (green). The two signals were obtained for slightly different ring voltages V0 and FFT times. (b) Phase-sensitive axial measurement scheme. With the resonator shifted by 2.5 kHz, a dipolar excitation is applied to the axial mode (green). During the short (reference) and long evolution time, a weak coupling to the tank circuit leads to a loss of amplitude (orange). Subsequently, the phase and amplitude are readout (gray) and the resonator is shifted back for cooling the ion (blue).

Close modal
Highest resolution of the absolute axial frequency and of axial frequency differences can be achieved with a phase-sensitive measurement method.45 This method exploits the coherent evolution of the phase ϕ of a non-thermal, excited ion, which can be converted to a frequency,
(7)
with, in our case, evolution time t = 4.1   s and the subscript R for the reference measurement with short evolution time t R = 0.1   s⁠. The evolution time cannot be chosen arbitrarily high, since a weak coupling to the resonator leads to a loss of coherence. As illustrated in Fig. 2(b), the measurement sequence starts by applying a dipolar excitation pulse at the ion's axial frequency with a well-defined phase, which is imprinted on the ion motion. Then, the ion evolves freely for a time t after which its phase is measured.45 Subsequently, the ion's axial motion is resistively cooled to set the initial conditions for the next measurement. This is achieved by shifting the detection circuit's resonance frequency by 2.5 kHz onto the ion's axial frequency using a switch that changes the parallel capacitance of the tank circuit.

This measurement scheme is carried out for two voltage sources: the high-precision voltage source UM1-1426 and the 20 V PJVS.43 The voltage of the PJVS is calibrated with the UM1-14 ring voltage such that the axial frequency of the trapped ion matches for both voltage supplies.

Both voltage sources were tested on a single 9Be3+ ion stored in a Penning trap with ν z = 795   kHz ,   V 0 = − 3.4   V ,   C 2 = 108 × 10 3   m −2⁠, and a sizable B 2 = 0.28   T mm − 2 necessary for nuclear spin-flip detection. Here, the UM1-14 generates a frequency stability of 17.6(1.5) ppb over a time span of 8 min, which is improved by the PJVS to 9.7(8) ppb, as illustrated in Fig. 3(b). The PJVS improves the stability of the axial frequency by a factor of four compared to the StaReP,27 a voltage source designed for high-precision Penning-trap measurements. At a voltage of −7 V, as needed for trapped 40Ca17+ ions, the StaReP reaches 45 ppb. The ideal averaging time of 8 min can easily be reduced to 4 min by omitting the measurement of the reference phase. At longer averaging times, the frequency uncertainty increases due to a random walk in the cyclotron energy σ E +31 [Eq. (5)]. This effect is well known and is not related to the voltage sources (see the supplementary material and Ref. 31).

FIG. 3.
(a) Allan deviation of two UM1-14 channels at −3.4 V (black) and two 10 V PJVS (red) measured differentially with a nanovoltmeter. The recorded stability of the PJVS is limited by the small instability of the voltmeter used. (b) Measured frequency stability of a trapped 9Be3+ ion at t evol = 4 s. The white noise component associated with voltage fluctuations and readout noise decreases with increasing averaging time as τ − 1 / 2, whereas random-walk noise due to cyclotron frequency jumps increases with increasing averaging time as τ 1 / 2, see Refs. 31 and 46. The minima occur at roughly 8 min with a relative frequency uncertainty of 17.6(1.5) ppb for the UM1-14 and 9.7(8) ppb for the PJVS.
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(a) Allan deviation of two UM1-14 channels at −3.4 V (black) and two 10 V PJVS (red) measured differentially with a nanovoltmeter. The recorded stability of the PJVS is limited by the small instability of the voltmeter used. (b) Measured frequency stability of a trapped 9Be3+ ion at t evol = 4 s. The white noise component associated with voltage fluctuations and readout noise decreases with increasing averaging time as τ − 1 / 2⁠, whereas random-walk noise due to cyclotron frequency jumps increases with increasing averaging time as τ 1 / 2⁠, see Refs. 31 and 46. The minima occur at roughly 8 min with a relative frequency uncertainty of 17.6(1.5) ppb for the UM1-14 and 9.7(8) ppb for the PJVS.

FIG. 3.
(a) Allan deviation of two UM1-14 channels at −3.4 V (black) and two 10 V PJVS (red) measured differentially with a nanovoltmeter. The recorded stability of the PJVS is limited by the small instability of the voltmeter used. (b) Measured frequency stability of a trapped 9Be3+ ion at t evol = 4 s. The white noise component associated with voltage fluctuations and readout noise decreases with increasing averaging time as τ − 1 / 2, whereas random-walk noise due to cyclotron frequency jumps increases with increasing averaging time as τ 1 / 2, see Refs. 31 and 46. The minima occur at roughly 8 min with a relative frequency uncertainty of 17.6(1.5) ppb for the UM1-14 and 9.7(8) ppb for the PJVS.
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(a) Allan deviation of two UM1-14 channels at −3.4 V (black) and two 10 V PJVS (red) measured differentially with a nanovoltmeter. The recorded stability of the PJVS is limited by the small instability of the voltmeter used. (b) Measured frequency stability of a trapped 9Be3+ ion at t evol = 4 s. The white noise component associated with voltage fluctuations and readout noise decreases with increasing averaging time as τ − 1 / 2⁠, whereas random-walk noise due to cyclotron frequency jumps increases with increasing averaging time as τ 1 / 2⁠, see Refs. 31 and 46. The minima occur at roughly 8 min with a relative frequency uncertainty of 17.6(1.5) ppb for the UM1-14 and 9.7(8) ppb for the PJVS.

Close modal

The initial shot-to-shot noise of 37.9(7) ppb and 31.8(6) ppb for the UM1-14 and the PJVS, respectively, mainly originates from the uncertainty in the TR optimization σ T R⁠. Experimentally, the TR is optimized until electrostatic anharmonicities such as C4 become smallest. A non-zero C4 leads to a dependency of the axial frequency on the oscillation amplitude38  Δ ν z / ν z ( C 4 ≠ 0 ) ∝ r z 2⁠. After thermal coupling of the ion to the detection circuit, the ion's amplitude in the axial mode is Boltzmann distributed. This leads to a distribution of hot radii after the excitation pulse45 such that with a non-zero C4, the axial frequency changes slightly for every cooling cycle, generating noise. Here, the TR was optimized with the UM1-14 by adapting the voltage on the correction electrode in 10 − 5 V steps. From the TR measurement in Fig. 4 it can be deduced that a TR deviation of 5 × 10 − 6 from the ideal value leads to shot-to-shot noise in the axial frequency measurements amounting to 28(8) ppb, which ultimately limits the achievable axial frequency stability. Since the TR offset has a major impact on the axial frequency stability, further optimization will be crucial in future measurements where, e.g., a higher sensitivity to frequency differences is required.

FIG. 4.
Tuning ratio (TR) optimization with the UM1-14 with t evol = 0.5 s. The measured data (black dots) are fitted (gray) to σ ν z = α T R 2 ( T R − T R opt ) 2 + σ rest 2, from which the frequency jitter dependency on the TR (green) is extracted. Here, σrest denotes all noise contributions to the shot-to-shot noise, that are not due to the trap anharmonicity, e.g., voltage fluctuations or readout noise. For a TR offset amounting to half the optimization step width ( 5 × 10 − 6) (blue), a frequency uncertainty of 28(8) ppb is expected in the shot-to-shot noise.
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Tuning ratio (TR) optimization with the UM1-14 with t evol = 0.5 s⁠. The measured data (black dots) are fitted (gray) to σ ν z = α T R 2 ( T R − T R opt ) 2 + σ rest 2⁠, from which the frequency jitter dependency on the TR (green) is extracted. Here, σrest denotes all noise contributions to the shot-to-shot noise, that are not due to the trap anharmonicity, e.g., voltage fluctuations or readout noise. For a TR offset amounting to half the optimization step width (⁠ 5 × 10 − 6⁠) (blue), a frequency uncertainty of 28(8) ppb is expected in the shot-to-shot noise.

FIG. 4.
Tuning ratio (TR) optimization with the UM1-14 with t evol = 0.5 s. The measured data (black dots) are fitted (gray) to σ ν z = α T R 2 ( T R − T R opt ) 2 + σ rest 2, from which the frequency jitter dependency on the TR (green) is extracted. Here, σrest denotes all noise contributions to the shot-to-shot noise, that are not due to the trap anharmonicity, e.g., voltage fluctuations or readout noise. For a TR offset amounting to half the optimization step width ( 5 × 10 − 6) (blue), a frequency uncertainty of 28(8) ppb is expected in the shot-to-shot noise.
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Tuning ratio (TR) optimization with the UM1-14 with t evol = 0.5 s⁠. The measured data (black dots) are fitted (gray) to σ ν z = α T R 2 ( T R − T R opt ) 2 + σ rest 2⁠, from which the frequency jitter dependency on the TR (green) is extracted. Here, σrest denotes all noise contributions to the shot-to-shot noise, that are not due to the trap anharmonicity, e.g., voltage fluctuations or readout noise. For a TR offset amounting to half the optimization step width (⁠ 5 × 10 − 6⁠) (blue), a frequency uncertainty of 28(8) ppb is expected in the shot-to-shot noise.

Close modal
The next-largest noise contributions arise from the fluctuations of the voltage sources σ V 0 and the readout jitter, along with the thermal radius distribution of a cooled ion σFFT. A simulation of the ion's detection signal yields a result of 10.5 ( 1.7 ) ppb. This is in agreement with the value obtained from the reference phase with the short evolution time t R = 0.1   s⁠, giving an upper limit of the phase jitter due to readout and thermal ion radius distribution σ FFT < 13.4(1.0)°. This is equivalent to 11.7(9) ppb in relative frequency uncertainty. Details can be found in the supplementary material. All major contributions are summarized in Table I along with their dependency on averaging time τ and add up as
(8)
TABLE I.

Main noise contributions estimated for this measurement, adding quadratically to the measured initial relative frequency shot-to-shot noise [Eq. (8)]. While the contributions to the relative frequency uncertainties of white noise (⁠ σ T R , σ V 0 , …⁠) scale with the square root of the number of measurements n − 1 / 2 = ( τ / τ 0 ) − 1 / 2⁠, for random walk (⁠ σ E +⁠) and flicker noise of the voltage source, they directly depend on the total measurement time via τ 1 / 2 and τ 0⁠, respectively.46 

Noise source UM1-14 (ppb) PJVS (ppb) τ dependency
Trap anharmonicity σTR  28(8)  28(8)  ( τ / τ 0 ) − 1 / 2 
Voltage fluctuations on ring electrode σ V 0  16.6(2.2)  2.2(2.2)  ( τ / τ 0 ) − 1 / 2 
Readout jitter and thermal radius distribution σFFT  10.5(1.7)  10.5(1.7)  ( τ / τ 0 ) − 1 / 2 
Voltage fluctuations on correction electrode σ V C E  3.1(4)  3.1(4)  ( τ / τ 0 ) − 1 / 2 
Cyclotron energy σ E +  4.2(3)  2.4(3)  τ 1 / 2 
Flicker noise  8.0(5)  ⋯  τ 0 
Measured shot-to-shot noise  37.9(7)  31.8(6)   
Noise source UM1-14 (ppb) PJVS (ppb) τ dependency
Trap anharmonicity σTR  28(8)  28(8)  ( τ / τ 0 ) − 1 / 2 
Voltage fluctuations on ring electrode σ V 0  16.6(2.2)  2.2(2.2)  ( τ / τ 0 ) − 1 / 2 
Readout jitter and thermal radius distribution σFFT  10.5(1.7)  10.5(1.7)  ( τ / τ 0 ) − 1 / 2 
Voltage fluctuations on correction electrode σ V C E  3.1(4)  3.1(4)  ( τ / τ 0 ) − 1 / 2 
Cyclotron energy σ E +  4.2(3)  2.4(3)  τ 1 / 2 
Flicker noise  8.0(5)  ⋯  τ 0 
Measured shot-to-shot noise  37.9(7)  31.8(6)   
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Noise limitations are assessed in detail in the supplementary material.

The achieved axial frequency stability of 9.7(8) ppb with the PJVS at an averaging time of roughly 8 min potentially enables a shot-to-shot measurement uncertainty of the free cyclotron frequency in the sub-10 ppt regime. This is a factor of 2–5 (depending on the ion species) smaller than in previous Penning-trap precision mass measurements.

Furthermore, with background fluctuations reduced to 9.7(8) ppb, an axial frequency difference around 45 ppb equivalent to 35 mHz at 795 kHz can be distinguished unambiguously. This is three times smaller than in previous measurements45 and makes the detection of single nuclear spin-flips in light ions, such as 2H, 3He, or 7Li, feasible; the comparison is presented in Table II. Figure 5(a) illustrates the spin-flip fidelity of two light ions, 3He and 7Li, as a function of the axial frequency fluctuations. With the UM1-14 voltage source, the 3He spin-flip fidelity is 84(3)%, which means that about four out of five axial frequency shifts are correctly identified as a spin-flip. The nuclear magnetic moment measurements of heavier ions with a minimum spin-flip fidelity of 80% are not possible with the UM1-14. In contrast, the PJVS increases the 3He spin-flip fidelity to 98.6(8)% and makes direct measurements of the 7Li and 2H g-factors, with spin-flip fidelities of 91(2)% and 88(3)%, respectively, accessible. The observation of a nuclear spin-flip of 9Be, which was trapped in this measurement, is not possible with the current setup.

TABLE II.

Theoretical spin-flip-induced axial frequency shifts for bare, light ions Δ ν z , S F [Eq. (5)]. For an axial potential stability of 17.6 ppb with the UM1-14 and 9.7 ppb with the PJVS, the expected spin-flip fidelity will be limited to FUM and FPJV S, respectively. Additional limitations are expected to arise from other noise contributions, see the supplementary material for more details.

ion g-factor47  I Δ ν z , S F / ν z (ppb) FUM (%) FPJVS (%)
1H+  +5.585 694 68  1/2  187  100  100 
2H+  +0.857 438 2  1  29  66  88 
3H+  +5.957 993 69  1/2  68  95  100 
3He2+  −4.254 995 44  1/2  48  84  99 
7Li3+  +2.170 951  3/2  31  69  91 
11B5+  +1.792 432 6  3/2  16  <50  67 
ion g-factor47  I Δ ν z , S F / ν z (ppb) FUM (%) FPJVS (%)
1H+  +5.585 694 68  1/2  187  100  100 
2H+  +0.857 438 2  1  29  66  88 
3H+  +5.957 993 69  1/2  68  95  100 
3He2+  −4.254 995 44  1/2  48  84  99 
7Li3+  +2.170 951  3/2  31  69  91 
11B5+  +1.792 432 6  3/2  16  <50  67 
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FIG. 5.
(a) Theoretical calculations for the spin-flip fidelity of 3He2+ (green) and 7Li3+ (blue). (b) Spin-flip-induced relative axial frequency shift for stable/long-lived, light ions in the μTEx Penning trap. The UM1-14 resolves a 1H and 3H spin-flip with a fidelity of at least 85% (gray shaded area). The PJVS additionally resolves nuclear spin-flips of 2H, 3He, and 7Li with at least 85% spin-flip fidelity (red shaded area).
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(a) Theoretical calculations for the spin-flip fidelity of 3He2+ (green) and 7Li3+ (blue). (b) Spin-flip-induced relative axial frequency shift for stable/long-lived, light ions in the μTEx Penning trap. The UM1-14 resolves a 1H and 3H spin-flip with a fidelity of at least 85% (gray shaded area). The PJVS additionally resolves nuclear spin-flips of 2H, 3He, and 7Li with at least 85% spin-flip fidelity (red shaded area).

FIG. 5.
(a) Theoretical calculations for the spin-flip fidelity of 3He2+ (green) and 7Li3+ (blue). (b) Spin-flip-induced relative axial frequency shift for stable/long-lived, light ions in the μTEx Penning trap. The UM1-14 resolves a 1H and 3H spin-flip with a fidelity of at least 85% (gray shaded area). The PJVS additionally resolves nuclear spin-flips of 2H, 3He, and 7Li with at least 85% spin-flip fidelity (red shaded area).
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(a) Theoretical calculations for the spin-flip fidelity of 3He2+ (green) and 7Li3+ (blue). (b) Spin-flip-induced relative axial frequency shift for stable/long-lived, light ions in the μTEx Penning trap. The UM1-14 resolves a 1H and 3H spin-flip with a fidelity of at least 85% (gray shaded area). The PJVS additionally resolves nuclear spin-flips of 2H, 3He, and 7Li with at least 85% spin-flip fidelity (red shaded area).

Close modal

An ultra-stable voltage source based on the inverse AC Josephson effect48 was implemented in a Penning-trap setup dedicated to high-precision measurements of electron and nuclear magnetic moments.49 Compared to other ultra-stable voltage sources used in Penning-trap experiments, such as the UM1-14,26 which creates potentials up to 14 V, or the StaReP27 used in the high-voltage range of up to 100 V, the 20 V PJVS43 is significantly more stable on account of its quantized voltage steps [Fig. 3(a)]. With the PJVS, the axial frequency stability of a trapped 9Be3+ ion improved to 9.7(8) ppb, from previously 17.6(1.5) ppb with the UM1-14 [Fig. 3(b)]. The stability of the electrostatic trapping potential directly affects the stability of all three eigenfrequencies, which ultimately limits the precision on the free cyclotron frequency [Eqs. (2) and (3)]. In the context of mass measurements, the PJVS implementation reduces measurement times for reaching a similar precision compared to the UM1-14 or StaReP and paves the way for shot-to-shot measurements of the free cyclotron frequency in the sub-10 ppt regime.

Regarding g-factor measurements, the increased stability will enable direct observation of nuclear spin-flips of bare 2H, 3He, and 7Li, with a detection fidelity of 88(3)%, 98.6(8)%, and 91(2)%, respectively. By adapting the detection circuit to a lower resonance frequency, even the spin-flip of nuclei with higher mass and smaller magnetic moment, such as 10 , 11B, 6Li, and 9Be, can be resolved when using the PJVS [Eq (5)]. In addition, the high stability in the trapping potential can be used for improved determinations of the proton and antiproton g-factors. To further improve the axial frequency stability of a trapped ion, smaller anharmonicities in the axial trapping potential are required which can be achieved by further optimizing the TR.

A detailed noise contribution analysis for the performed measurement is given in the supplementary material. Additionally, the preparation of the ion with low cyclotron energy is described and the impact of voltage fluctuations in mass measurements is elaborated.

This work is part of and funded by the Max Planck Society. Furthermore, this project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program under Grant Agreement No. 832848-FunI, and we acknowledge funding and support by the Physikalisch-Technische Bundesanstalt (PTB), the International Max Planck Research School for Precision Tests of Fundamental Symmetries (IMPRS-PTFS), the Max Planck-RIKEN-PTB Center for Time, Constants and Fundamental Symmetries (TCFS), and the German Research Foundation (DFG) Project-ID 273811115—SFB 1225 ISOQUANT. This work comprises parts of the Ph.D. thesis work of A. Kaiser to be submitted to Heidelberg University, Germany.

The authors have no conflicts to disclose.

A. Kaiser: Data curation (equal); Formal analysis (equal); Investigation (lead); Software (supporting); Validation (equal); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). S. Dickopf: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (lead); Supervision (supporting); Validation (equal); Visualization (supporting); Writing – review & editing (supporting). M. Door: Conceptualization (supporting); Formal analysis (supporting); Investigation (equal); Methodology (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). R. Behr: Conceptualization (equal); Formal analysis (supporting); Funding acquisition (supporting); Investigation (supporting); Resources (equal); Supervision (supporting); Validation (supporting); Writing – original draft (equal); Writing – review & editing (supporting). U. Beutel: Writing – review & editing (supporting). S. Eliseev: Conceptualization (supporting); Investigation (supporting); Writing – review & editing (supporting). A. Kaushik: Data curation (equal); Writing – review & editing (supporting). K. Kromer: Investigation (supporting); Writing – review & editing (supporting). M. Müller: Methodology (equal); Software (supporting); Writing – review & editing (supporting). L. Palafox: Conceptualization (lead); Formal analysis (supporting); Funding acquisition (supporting); Investigation (supporting); Resources (equal); Software (supporting); Supervision (supporting); Validation (supporting); Writing – original draft (equal); Writing – review & editing (supporting). S. Ulmer: Funding acquisition (supporting); Resources (supporting); Supervision (supporting); Writing – review & editing (supporting). A. Mooser: Conceptualization (supporting); Data curation (supporting); Formal analysis (supporting); Investigation (equal); Methodology (equal); Project administration (lead); Software (supporting); Supervision (lead); Validation (equal); Visualization (supporting); Writing – original draft (supporting); Writing – review & editing (equal). K. Blaum: Conceptualization (equal); Funding acquisition (lead); Project administration (supporting); Resources (lead); Supervision (supporting); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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