Introduction

The standard model (SM) of particle physics describes non-gravitational interactions between all particles and fields, while gravitation is described by general relativity in a classical manner. Together, they have explained many physical phenomena observed in the universe remarkably well, but an accurate description of gravity at the quantum level is lacking. A number of theories that attempt to unify the SM and gravitation at the Planck scale suggest that breaking of Lorentz symmetry might occur either spontaneously1 or explicitly2,3,4. Lorentz symmetry states that the outcome of a local experiment does not depend on the orientation or the velocity of the apparatus5. A suppressed effect emerging from Lorentz violation (LV) at the Planck scale could be observed at experimentally accessible energies in the laboratory6. Accurate spectroscopic measurements in trapped particles have reached fractional uncertainties beyond the natural suppression factor7, which makes a hypothetical LV measurable in such systems. Furthermore, at high energies, LV could be suppressed by super-symmetry8. Therefore, accurate low-energy measurements in atoms are suitable to search for LV and complement existing bounds set at high energies with, e.g., particle colliders and astrophysical observations9,10,11,12.

Laboratory tests of Lorentz symmetry are based on a similar principle as introduced by Michelson and Morley, who used a rotating interferometer to measure the isotropy of the speed of light13. Improved bounds on LV for photons have been realized by a variety of experiments involving high-finesse optical and microwave cavities, see e.g. refs. 5, 14,15,16,17. Spectroscopic bounds for protons and neutrons are set using atomic fountain clocks18,19 and co-magnetometers20,21. More recently, the bounds on LV in the combined electron-photon sector have been explored using precision spectroscopy in trapped ions22,23,24. These experiments compare energy levels with differently oriented, relativistic, non-spherical electron orbitals as the Earth rotates. Strong bounds on LV were set with trapped 40Ca+ ions, where a decoherence-free entangled state of two ions in the electronic 2D5/2 manifold was created to suppress ambient noise22,23. The relatively short 1.2 s radiative lifetime of the 2D5/2 state in Ca+ and the requirement for high-fidelity quantum gates limit the scalability and, ultimately, the sensitivity of LV tests with this scheme23. The highly relativistic 2F7/2 state in the Yb+ ion is an order of magnitude more sensitive to LV than the 2D5/2 state in Ca+ 25,26 and its radiative lifetime was measured to be about 1.6 years27. However, gate operation on the electronic octupole (E3) transition, required to efficiently populate the 2F7/2 manifold, suffers from a low fidelity, making entanglement-based techniques unfeasible. The beneficial properties of Yb+ were recently partly exploited in a 45-day comparison of two separate state-of-the-art single-ion optical 171Yb+ clocks, both with a fractional uncertainty at the 10−18 level, reaching a more than ten-fold improvement24. However, operating the Yb+ ion as an optical clock limits the experiment to only probe the Zeeman levels that are least sensitive to LV (mj = ± 1/2).

In this work we present improved bounds on LV in the electron-photon sector using a method that fully exploits the high susceptibility of the stretched mj = ± 7/2 states in the 2F7/2 manifold of Yb+ and takes advantage of its long radiative lifetime. With a robust radio frequency (rf) spin-echoed Ramsey sequence26,28 we populate all Zeeman sublevels in the 2F7/2 manifold and make a direct energy comparison between the orthogonally oriented atomic orbitals within a single trapped ion. We decouple the energy levels from noise in the ambient magnetic field to reach a 5000-fold longer coherence time during the Ramsey measurement and extend the dark time to TD > 1 s. With an unprecedented sensitivity to LV, scaling as \({\sigma }_{{{{{{{{\rm{LV}}}}}}}}}\propto \sqrt{1/{T}_{{{{{{{{\rm{D}}}}}}}}}{N}_{{{{{{{{\rm{ion}}}}}}}}}}\), we reduce the required total averaging time by nearly an order of magnitude already with a single ion (Nion = 1). The measurement scheme is simple, robust, scalable and eliminates the requirement of optical clock operation or high-fidelity quantum gates. The rf sequence is insensitive to both temporal and spatial field inhomogeneities and can be applied to a string of Nion trapped ions for an increased sensitivity to LV in the future.

Results

Theoretical framework

The constraints on LV extracted in this work are quantified in the theoretical framework of the standard model extension (SME)29. The SME is an effective field theory in which the SM Lagrangian is extended with all possible terms that are not Lorentz invariant. It is a platform in which LV of all SM particles are described, enabling comparisons between experimental results from many different fields30. In spectroscopic experiments, a violation of Lorentz symmetry can be interpreted as LV of electrons or photons, because there is no preferred reference system. In this work, we interpret the results as a difference in isotropy between photons and electrons, similar as in refs. 22, 24.

LV in the combined electron-photon sector is quantified by adding a symmetry-breaking tensor \({c}_{\mu \nu }^{{\prime} }={c}_{\mu \nu }+{k}_{\mu \nu }/2\) to the SM Lagrangian25,29, where cμν and kμν describe LV for electrons and photons, respectively. For simplicity, the prime is omitted throughout the rest of this work and the extracted coefficients are those of the combined \({c}_{\mu \nu }^{{\prime} }\) tensor, which is taken as traceless and symmetric. The components of the cμν tensor are frame dependent. A unique definition of the symmetry breaking tensor cMN exists in the Sun-centered, celestial, equatorial frame (SCCEF), illustrated in Fig. 1a. In order to make comparisons with other experiments, the cμν tensor defined in our local laboratory frame is transformed to the SCCEF to constrain the components of the cMN tensor. The full derivation of the transformation can be found in the Supplementary Information.

Fig. 1: A schematic overview of the experimental principle.
figure 1

a In the laboratory frame (LF) coordinate system z points vertically upwards, y points North and x points east. The fixed quantization axis B (pink arrow) in the LF coordinate system lies in the horizontal plane and points 20º south of x and probes different directions in the SCCEF as the Earth rotates around its axis (ω⊕) and orbits the Sun (Ω⊙). b In the LF a single 172Yb+ ion is trapped in a segmented rf trap. Two coils generate a quantization field of 221 μT. The electron orbitals of the mj = ± 1/2 and mj = ± 7/2 Zeeman sublevels in the 2F7/2 state orient themselves orthogonally to each other. The 2F7/2 state is population via the E3 transition near 467 nm. The mj substates are coupled via the rf magnetic field at ωrf created with a coil that is placed 5.5 cm above the ion. c A simplified energy level diagram of 172Yb+, showing the optical E3 transition and the transitions near 370 nm and 935 nm used for Doppler cooling and state detection.

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In a bound electronic systems, LV leads to a small energy shift25,26

$$\delta {{{{{{{\mathcal{H}}}}}}}}=-\frac{1}{6{m}_{e}}{C}_{0}^{(2)}{T}_{0}^{(2)}\,,$$
(1)

where me is the electron mass, the \({T}_{0}^{(2)}={{{{{{{{\bf{p}}}}}}}}}^{2}-3{p}_{z}^{2}\) operator depends on the direction of the electron’s momentum and \({C}_{0}^{(2)}\) contains elements of the cMN tensor. For a state with total angular momentum J and projection mj onto the quantization axis \(\hat{z}\), the matrix element of the \({T}_{0}^{(2)}\) operator is given by26

$$\left\langle J,{m}_{j}\right|{T}_{0}^{(2)}\left|\,J,{m}_{j}\right\rangle=\frac{-J(\,J+1)+3{m}_{j}^{2}}{\sqrt{(2J+3)(\,J+1)(2J+1)\,J(2J-1)}}\times \langle \,J\parallel {T}^{(2)}\parallel J \rangle \,.$$
(2)

Equations (1) and (2) show that in the SCCEF, LV manifests itself as an energy shift that modulates with Earth’s rotation frequency. The magnitude of this shift is dependent on both \({m}_{j}^{2}\) and the reduced matrix element 〈 J∥T(2)∥ J 〉. The value of the latter is particularly high for the 2F7/2 manifold in the Yb+ ion25,26. The goal of this experiment is to test LV in a single trapped 172Yb+ ion by measuring the energy difference between mj substates in the 2F7/2 manifold as the Earth rotates.

Measurement principle

The experiment is performed with a single ion, stored in a linear rf Paul-trap, see Fig. 1b. It is cooled to the Doppler limit of about 0.5 mK on the dipole allowed transition near a wavelength of 370 nm, assisted by a repumper near 935 nm. A set of coils is used to define the quantization axis of B = 221 μT, which lies in the horizontal plane with respect to Earth’s surface and points 20º south of east, see Fig. 1a. Active feedback is applied on auxiliary coils in three directions to stabilize the magnetic field. The 2F7/2 state can be efficiently populated via coherent excitation of the highly-forbidden electric octupole (E3) transition (Fig. 1c) using an ultra-stable frequency-doubled laser at 934 nm31, which is either stabilized only to a cryogenic silicon cavity via a frequency comb32 or, optionally, to the single-ion optical 171Yb+ clock33. A Rabi frequency of ΩE3/2π = 10 Hz is achieved on the E3 transition. More details on the experimental apparatus can be found in the Methods section.

The free evolution of a substate ∣2F7/2, \({m}_{j} \rangle\) interacting with a magnetic field \({{{{{{{\bf{B}}}}}}}}\equiv {B}_{z}\hat{z}\), is given by the Hamiltonian \({{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{free}}}}}}}}}={{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{lin}}}}}}}}}+{{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{quad}}}}}}}}}=\mu {B}_{z}\,{J}_{z}+\kappa {J}_{z}^{2}\), where μ is the magnetic moment. The quadratic term in the Hamiltonian gives rise to an energy shift according to \({E}_{{m}_{j}}/h\equiv \kappa /2\pi \times {m}_{j}^{2}\). The value of κ = κq + κLV is dependent on the quadrupole shift for the trapped ion in the 2F7/2 state and a possible shift due to LV, respectively. The contribution from LV to κ is given by26

$$\frac{{\kappa }_{{{{{{{{\rm{LV}}}}}}}}}}{2\pi }=5.1\times 1{0}^{15}\,{{{{{{{\rm{Hz}}}}}}}}\times {C}_{0}^{(2)}\,,$$
(3)

where \({C}_{0}^{(2)}\) contains components of the cMN tensor in the SCCEF, see the Supplementary Information.

A modulation of the quadratic contribution to the Zeeman splitting in the 2F7/2 manifold is measured with rf spectroscopy. The mj levels are coupled via a rf magnetic field supplied to the ion by a resonant LC circuit. The coupling term in the Hamiltonian is given by \({{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{coupling}}}}}}}}}={{{\Omega }}}_{F}\cos ({\omega }_{{{{{{{{\rm{rf}}}}}}}}}t+\phi ){J}_{x}\), where ΩF/2π = 33 kHz is the multilevel Rabi frequency and ωrf and ϕ are the frequency and the phase of the rf field, respectively. The rf frequency is close to resonance with the, to first order, equidistant mj levels given by ωrf/2π = μBz/h + δ(t)/2π ≈ 3.5 MHz, where δ(t) is a small detuning from temporal drifts in the ambient magnetic field. The full Hamiltonian of the system in the interaction picture after applying the rotating wave approximation is given by

$${{{{{{{\mathcal{H}}}}}}}}=\delta (t){J}_{z}+\kappa {J}_{z}^{2}+{{{\Omega }}}_{F}[{J}_{x}\cos (\phi )-{J}_{y}\sin (\phi )]\,,$$
(4)

where the changes in δ(t) and κ should be much slower than the π-pulse time of tπ = π/ΩF = 15 μs.

A composite rf pulse sequence, based on a spin-echoed Ramsey scheme, is implemented to mitigate the influence of δ(t), while retaining a high sensitivity to variations of κ. A schematic overview of the rf sequence is shown in Fig. 2 a. Starting from either one of the mj = ± 1/2 states, a π/2-pulse, i.e. tπ/2 = tπ/2, with phase ϕ = 0 spreads the population over all the Zeeman sublevels. A modulation sequence, consisting of ten repetitions of the form [tw] − [tπ(ϕi)] − [tw], is applied to cancel dephasing from δ(t). Here tw is the time in which the state freely evolves and tπ(ϕi) indicates a π-pulse with phase ϕi. In our experimental environment, dephasing is successfully canceled for tw = 100 μs. The phases ϕi of the consecutive π-pulses are set to \(\left(0,4,2,4,0,0,4,2,4,0\right)\pi /5\), for which the modulation sequence is shown to be highly robust against pulse errors from, e.g., detuning and intensity variations28. At the end of the sequence, a π/2-pulse with phase ϕ = π retrieves a fraction Pf of the population back into the initial mj = ± 1/2 state. The Ramsey dark-time, i.e. the time in which the state freely evolves, can be extended to TD = N × 20 tw by repeating the modulation sequence N times.

Fig. 2: Pulse sequences.
figure 2

a The applied composite rf pulse sequence is a more robust version of a spin-echoed Ramsey scheme. A Ramsey sequence of Trf = 1236.27 ms starts and ends with a π/2-pulse. A 2.15 ms-long modulation sequence of 10 π-pulses with phases ϕi (i = 1. . . 10), each spaced by 2tw = 200 μs, is repeated N = 575 times to cancel dephasing due to ambient magnetic field noise. The phases ϕi are chosen such (see text) that the sequence is robust against pulse errors. b Every n = 50 measurements, a 4-point servo sequence (E3 servo) of 40 s is applied to follow the E3 center frequency, after which the rf sequence is compiled for 2.5 s. The measurement sequence consists of two 50 ms-long E3 pulses (E3), the rf pulse sequence (rf) of 1236.27 ms, four detection pulses (D) of 2.5 ms and a sequence at the end of 20 ms for state preparation (R) for the next measurement run. During post-selection, the data points are considered valid if the ion was in the desired state during the first three detection pulses.

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The retrieved fraction Pf is dependent on the phase κTD acquired during the free-evolution time, which stems from the quadratic term in the Hamiltonian of Eq. (4). Therefore, κLV can be extracted from a measurement of Pf at a fixed TD. Since a hypothetical LV would manifest itself as a modulation of κ at frequencies related to a sidereal day (ω⊕/2π = 1/23.934 h), a signal of LV is characterized by oscillations of Pf at the same modulation frequencies. A measurement of Pf is realized by de-exciting the ion via the E3 transition to the 2S1/2 state, where it can be detected by collecting fluorescence on the dipole allowed 2S1/2 → 2P1/2 transition.

The highest measurement accuracy is achieved when the measured quantity Pf is most sensitive to variations of κ, i.e. ∣dPf/dκ∣, is maximized, which is the case at κTD = 0.15 rad26. Using the axial secular frequency of 266(5) kHz set during the measurement campaign and the quadrupole moment of the 2F7/2 state34, κq was calculated to be 0.13(3) rad s−1, see the Supplementary Information. The corresponding optimal Ramsey dark time is TD = 1.15 s.

Experimental sequence

A schematic overview of the full experimental sequence is shown in Fig. 2b. Stable long-term operation of the experiment is required to resolve oscillation periods related to a potential LV, i.e. at 11.967 and 23.934 hours. Especially the center frequency of the E3 transition is sensitive to external perturbations from, e.g., magnetic field drifts and intensity fluctuations. Therefore, a 4-point servo-sequence of two opposite Zeeman transitions ∣2S1/2, mj = \(\pm 1/2 \rangle \to {|}^{2}{F}_{7/2}\), \({m}_{j}=\pm 1/2 \rangle\) at half the linewidth is applied every n = 50 measurement runs to follow the E3 center frequency. For details on this technique, see e.g. ref. 35. On average a population transfer of 80 % is realized to the 2F7/2 state via the E3 transition using the servo-sequence.

After the E3 servo and 2.5 s of compilation time for the rf pulse sequence, a measurement starts with a detection pulse to determine if the ion is cooled and prepared in the correct initial ∣2S1/2, mj = \(\pm 1/2 \rangle\) state. If this is not the case, consecutive repumping and recooling sequences are applied. To save overhead time, either the ∣2F7/2,\(\,{m}_{j}=+ 1/2 \rangle\) or the ∣2F7/2,\(\,{m}_{j}=-1/2 \rangle\) state is populated via E3 excitation of the same Zeeman transition as was addressed last by the servo sequence. Another detection pulse is applied to determine if the E3 excitation was successful. A single rf modulation sequence of Tm = 10 × tπ + 20 × tw = 2.15 ms is repeated N = 575 times to reach TD = 1.15 s. The full rf sequence runs for Trf = N × Tm + 2 × tπ/2 = 1236.27 ms. A third detection pulse is applied to determine if the ion was quenched to the ground state during the rf sequence due to, e.g., a collision with background gas. The ion is de-excited on the E3 transition and, with a final detection pulse, Pf is measured. At the end of the measurement sequence, the ion is re-cooled and prepared in the required ∣2S1/2, mj = \(\pm 1/2 \rangle\) electronic ground state for the next measurement run. Data points are only considered valid when the ion was in the correct state at both the second and the third detection stage. After post-selection of valid data points an average contrast of 0.77(6) is reached. Including additional overhead from compilation time, data points are obtained at a rate of 1/191 s−1.

Bounds on Lorentz violation

The data acquired over a period of 912 h, with an up-time of 591 h, is shown in Fig. 3a. Data points are averaged in equidistant bins of 15 min. The measured population is decomposed in two parts Pf = P(κq) + P(κLV), where P(κq) gives rise to a constant offset of P0 = 0.39 and P(κLV) = dPf/dκ × κLV contains a potential LV signal. To extract κLV from Pf, a high-pass filter is applied with a cutoff frequency of νc = 5 μHz, removing P0 and slow variations (τdrift < 2.5 days) caused by drifts in the E3 excitation probability, and the slope dPf/dκ∣κ=0.13(3) = − 4.4(4) is calculated. More details on the data handling and the measurement sensitivity can be found in the Supplementary Information.

Fig. 3: Data acquired over a period of 912 h with an up-time of 591 h.
figure 3

a The measured value of κLV (blue data points) extracted from the retrieved population Pf. Data points are binned at 15 min intervals. The error bars reflect the standard error (1σ) from quantum projection noise. The data is fitted to Eq. (5) (red line) from which bounds are set on components of the cMN tensor. b The residuals of the fit with a reduced χ2 = 0.92. c A Lomb–Scargle periodogram for spectral analysis of irregularly spaced data. The normalized power spectral density is plotted for frequencies between 0 and 100 μHz. The vertical lines indicate the sidereal day frequency, ω⊕, (red) and its second harmonic, 2ω⊕, (blue). The dashed lines indicate the statistical significance level, i.e. p-value, of the peaks36. d The Allan deviation, σκ, of the data points shown in a as a function of averaging time, τ. The error bars correspond to the standard error of the stability. From a fit (gray line) to the data, the stability is extracted to be σκ = 372(9) mrad s−1 τ−1/2. The expected stability from quantum projection noise (dashed-dotted green line) is 29(3)% lower.

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In search of LV at Fourier frequencies of ω⊕ and 2ω⊕, the data is fitted globally to the fit function

$${\kappa }_{{{{{{{{\rm{LV}}}}}}}}}= 2\pi \times 5.1\times 1{0}^{15} \times \left[-3\sin (2\chi ){c}_{XZ}\cos ({\omega }_{\oplus }T)-3\sin (2\chi ){c}_{YZ}\sin ({\omega }_{\oplus }T) \vphantom{-\frac{3}{2}}\right.\\ \left.-\frac{3}{2}{c}_{X-Y}{\sin }^{2}(\chi )\cos (2{\omega }_{\oplus }T)-3{c}_{XY}{\sin }^{2}(\chi )\sin (2{\omega }_{\oplus }T)\right]\,,$$
(5)

from which the individual components of the cMN tensor in the SCCEF are extracted, where cX−Y = cXX − cYY. The fit overlays the data in Fig. 3a and the residuals from the fit are shown in Fig. 3b, from which the reduced chi-square of χ2 = 0.92 is extracted. The fitted values of the components of the cMN tensor are given in Table 1. For comparison, the values obtained in refs. 23, 24 are also presented. The uncorrelated linear combination of the fit parameters, calculated by diagonalizing the covariance matrix, are given in Table 2. The spectral content of the data, shown in Fig. 3c, is analyzed using a Lomb–Scargle periodogram36, which is specifically suited for spectral analysis of irregularly spaced data. The statistical significance level, i.e. p-values36, of 50% and 95% are indicated by horizontal lines.

Table 1 The symmetry breaking tensor components
Full size table
Table 2 The uncorrelated combination of symmetry breaking tensor components
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The fit results show that the extracted values for cX−Y, cXZ and cYZ are consistent with zero within a 1σ uncertainty. Only cXY shows a 1.1σ deviation from zero, but spectral analysis does not show a significant Fourier component at 2ω⊕. Therefore, we conclude that we do not find evidence of LV, in agreement with previous work23,24. The stability of the data points is analyzed by calculating the Allan deviation, as shown in Fig. 3d. The data averages down as σκ = 372(9) mrad s−1 τ−1/2, which is 29(3)% higher than what is expected from quantum projection noise.

Discussion

The presented results set the most stringent bounds on this type of LV in the combined electron-photon sector. With the presented method, the resolution of the previous most sensitive measurement24 is reached 9 times faster. We improve on the state-of-the-art by a factor of 2.2 and constrain all the coefficients of the cMN tensor now at the 10−21 level. Due to the experimental geometry, a higher sensitivity is reached for signals that oscillate at 2ω⊕ than those that oscillate at ω⊕, see the Supplementary Information. Therefore, the tightest constraint of 3.9 × 10−21 is achieved on cXY.

In this work, coefficients of the first and second harmonic order of the sidereal day modulation frequency were considered. However, due to the high total angular momentum of the 2F7/2 state, the applied method is sensitive to LV at harmonics of up to sixth order37,38. Therefore, in combination with improved many-body calculations, experimental constraints can be translated into bounds on a larger number of coefficients in the future38.

The method demonstrated in this work is technically less demanding and more robust than alternative methods requiring simultaneous operation of independent optical clocks or high-fidelity entanglement gates23,24. It is applicable in a wide variety of systems, e.g., highly-charged ions or ultra-cold atoms26 and it is capable of scaling to multiple Nion ions in linear ion Coulomb crystals for a further enhancement of the sensitivity. Up-scaling with an entanglement-based techniques23 is technically demanding when considering the 2F7/2 state in Yb+ due to the low-fidelity (p) single ion gate on the E3 transition, which further deteriorates, according to \({p}^{{N}_{{{{{{{{\rm{ion}}}}}}}}}}\), for larger ion numbers. In contrast, the implemented composite rf pulse sequence is highly robust against errors originating from spatial and temporal fluctuations of both the ambient field and the rf field. Therefore, the coherence time is not expected to significantly decrease for larger ion numbers and a \(\sqrt{{N}_{{{{{{{{\rm{ion}}}}}}}}}}\) higher sensitivity is expected in such a system. Moreover, the long radiative lifetime of the 2F7/2 state27 does not significantly limit the coherence time and, with several technical improvements, longer interrogation times could be reached. Note that for efficient population transfer via the E3 transition, advanced cooling techniques, such as EIT39 or Sisyphus cooling40,41, might be advantageous in larger ion crystals. With the presented method, the boundaries of Lorentz symmetry tests can be pushed to the 10−22 level with a string of 10 Yb+ ions in the future.

Methods

Experimental details

A single 172Yb+ ion is trapped in a segmented rf Paul-trap42,43. The radial confinement is set with an rf electric field supplied by a resonant circuit at a frequency of Ωrf/2π = 24.38 MHz, while the axial confinement is set by dc voltages supplied to the trapping segment and the neighboring segments. With the applied confinement, the secular frequencies are ω(rad1, rad2, ax)/2π = (775, 510, 266) kHz. Micromotion is measured on a daily basis with the photon correlation technique44 and compensated in three directions to typically Erf < 100 V/m. The quantization field of B = 221 μT lies in the horizontal plane under an angle of 25º to the trap axis. The B-field is measured with a sensor near the vacuum chamber and active feedback is applied in three orthogonal directions via current modulation of six auxiliary coils.

The ion is cooled to approximately 0.5 mK, close to the Doppler limit, on the dipole allowed 2S1/2 → 2P1/2 transition assisted by a repumper laser near 935 nm. Fluorescence from the decay of the 2P1/2 state is collected by a lens of N/A = 0.27 and imaged onto the electron-multiplying charge-coupled device (EMCCD) camera45. This enables state detection via the electron shelving technique. The ion can be prepared in either the ∣2S1/2,\(\,{m}_{j}=-1/2 \rangle\) or the ∣2S1/2,\(\,{m}_{j}=+ 1/2 \rangle\) state using circular polarized beam near 370 nm pointing along the direction of the quantization axis. The 467 nm laser for excitation on the E3 transition lies in the radial plane and its beam waist is 26(3) × 38(3) μm at the ion. The power is stabilized to 6.0(2) mW, at which a Rabi frequency of ΩE3/2π = 10 Hz is reached. The light is frequency-shifted and pulsed using acousto-optic modulators.

The resonant rf coil, consisting of 27 turns wound at a diameter of 4.5 cm, is placed 5.5 cm above the ion. The resonance frequency of the coil is \({\omega }_{{{{{{{{\rm{res}}}}}}}}}/2\pi=3.5147(7)\) MHz and it is driven by a signal derived from a direct digital synthesizer (DDS), referenced to a stable 100 MHz signal from a hydrogen maser. At 221 μT, the resonance frequency between the mj levels in the 2F7/2 state is ωrf/2π = 3.52 MHz, close to the resonance frequency of the coil. The achieved multi-level Rabi frequency is ΩF/2π = 33 kHz. The ambient magnetic field is monitored throughout the measurement campaign via data acquired in the E3 servo sequence. Drifts are observed at the level of 100 nT, corresponding to a detuning Δωrf/2π = 1.6 kHz. The frequency supplied to the coil is actively adjusted to remain in resonance with the frequency given by the linear Zeeman splitting. For this purpose, the resonance frequency as a function of magnetic field was calibrated to be \({\omega }_{{{{{{{{\rm{rf}}}}}}}}}(B)/2\pi=\left[1.581(1.6)B+0.0162(3)\right]\) in MHz. For further details on the experimental set-up, see refs. 31, 42, 45, 46.