Improved bounds on Lorentz violation from composite pulse Ramsey spectroscopy in a trapped ion

In attempts to unify the four known fundamental forces in a single quantum-consistent theory, it is suggested that Lorentz symmetry may be broken at the Planck scale. Here we search for Lorentz violation at the low-energy limit by comparing orthogonally oriented atomic orbitals in a Michelson-Morley-type experiment. We apply a robust radiofrequency composite pulse sequence in the 2F7/2 manifold of an Yb+ ion, extending the coherence time from 200 μs to more than 1 s. In this manner, we fully exploit the high intrinsic susceptibility of the 2F7/2 state and take advantage of its exceptionally long lifetime. We match the stability of the previous best Lorentz symmetry test nearly an order of magnitude faster and improve the constraints on the symmetry breaking coefficients to the 10−21 level. These results represent the most stringent test of this type of Lorentz violation. The demonstrated method can be further extended to ion Coulomb crystals.

The standard model (SM) of particle physics describes nongravitational 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 spontaneously 1 or explicitly [2][3][4] . Lorentz symmetry states that the outcome of a local experiment does not depend on the orientation or the velocity of the apparatus 5 . A suppressed effect emerging from Lorentz violation (LV) at the Planck scale could be observed at experimentally accessible energies in the laboratory 6 . Accurate spectroscopic measurements in trapped particles have reached fractional uncertainties beyond the natural suppression factor 7 , which makes a hypothetical LV measurable in such systems. Furthermore, at high energies, LV could be suppressed by super-symmetry 8 . 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 observations [9][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 light 13 . 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, 14-17. Spectroscopic bounds for protons and neutrons are set using atomic fountain clocks 18,19 and co-magnetometers 20,21 . More recently, the bounds on LV in the combined electron-photon sector have been explored using precision spectroscopy in trapped ions [22][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 40 Ca + ions, where a decoherencefree entangled state of two ions in the electronic 2 D 5/2 manifold was created to suppress ambient noise 22,23 . The relatively short 1.2 s radiative lifetime of the 2 D 5/2 state in Ca + and the requirement for highfidelity quantum gates limit the scalability and, ultimately, the sensitivity of LV tests with this scheme 23 . The highly relativistic 2 F 7/2 state in the Yb + ion is an order of magnitude more sensitive to LV than the 2 D 5/2 state in Ca + 25,26 and its radiative lifetime was measured to be about 1.6 years 27 . However, gate operation on the electronic octupole (E3) transition, required to efficiently populate the 2 F 7/2 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 171 Yb + clocks, both with a fractional uncertainty at the 10 −18 level, reaching a more than ten-fold improvement 24 . However, operating the Yb + ion as an optical clock limits the experiment to only probe the Zeeman levels that are least sensitive to LV (m j = ± 1/2).
In this work we present improved bounds on LV in the electronphoton sector using a method that fully exploits the high susceptibility of the stretched m j = ± 7/2 states in the 2 F 7/2 manifold of Yb + and takes advantage of its long radiative lifetime. With a robust radio frequency (rf) spin-echoed Ramsey sequence 26,28 we populate all Zeeman sublevels in the 2 F 7/2 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 T D > 1 s. With an unprecedented sensitivity to LV, scaling as σ LV / ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=T D N ion p , we reduce the required total averaging time by nearly an order of magnitude already with a single ion (N ion = 1). The measurement scheme is simple, robust, scalable and eliminates the requirement of optical clock operation or highfidelity quantum gates. The rf sequence is insensitive to both temporal and spatial field inhomogeneities and can be applied to a string of N ion trapped ions for an increased sensitivity to LV in the future.

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 fields 30 . 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 0 μν = c μν + k μν =2 to the SM Lagrangian 25,29 , where c μν and k μν 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 0 μν tensor, which is taken as traceless and symmetric. The components of the c μν tensor are frame dependent. A unique definition of the symmetry breaking tensor c MN exists in the Sun-centered, celestial, equatorial frame (SCCEF), illustrated in Fig. 1a. In order to make comparisons with other experiments, the c μν tensor defined in our local laboratory frame is transformed to the SCCEF to constrain the components of the c MN tensor. The full derivation of the transformation can be found in the Supplementary Information.
In a bound electronic systems, LV leads to a small energy shift 25,26 where m e is the electron mass, the T ð2Þ 0 = p 2 À 3p 2 z operator depends on the direction of the electron's momentum and C ð2Þ 0 contains elements of the c MN tensor. For a state with total angular momentum J and projection m j onto the quantization axisẑ, the matrix element of the T ð2Þ 0 operator is given by 26 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 2 j and the reduced matrix element 〈 J∥T (2) ∥ J 〉. The value of the latter is particularly high for the 2 F 7/2 manifold in the Yb + ion 25,26 . The goal of this experiment is to test LV in a single trapped 172 Yb + ion by measuring the energy difference between m j substates in the 2 F 7/2 manifold as the Earth rotates.

Measurement principle
The experiment is performed with a single ion, stored in a linear rf Paultrap, 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 2 F 7/2 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 nm 31 , which is either stabilized only to a cryogenic silicon cavity via a frequency comb 32 or, optionally, to the single-ion optical 171 Yb + clock 33 . A Rabi frequency of Ω E3 /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 | 2 F 7/2 , m j i interacting with a magnetic field B B zẑ , is given by the Hamiltonian H free = H lin + H quad = μB z J z + κJ 2 z , where μ is the magnetic moment. The quadratic term in the Hamiltonian gives rise to an energy shift according to E m j =h κ=2π × m 2 j . The value of κ = κ q + κ LV is dependent on the quadrupole shift for the trapped ion in the 2 F 7/2 state and a possible shift due to LV, respectively. The contribution from LV to κ is 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 172 Yb + ion is trapped in a segmented rf trap. Two coils generate a quantization field of 221 μT. The electron orbitals of the m j = ± 1/2 and m j = ± 7/2 Zeeman sublevels in the 2 F 7/2 state orient themselves orthogonally to each other. The 2 F 7/2 state is population via the E3 transition near 467 nm. The m j substates are coupled via the rf magnetic field at ω rf created with a coil that is placed 5.5 cm above the ion. c A simplified energy level diagram of 172 Yb + , showing the optical E3 transition and the transitions near 370 nm and 935 nm used for Doppler cooling and state detection.
given by 26 where C ð2Þ 0 contains components of the c MN tensor in the SCCEF, see the Supplementary Information.
A modulation of the quadratic contribution to the Zeeman splitting in the 2 F 7/2 manifold is measured with rf spectroscopy. The m j 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 H coupling = Ω F cosðω rf t + ϕÞJ x , where Ω F /2π = 33 kHz is the multilevel Rabi frequency and ω rf 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 m j levels given by ω rf /2π = μB z /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 where the changes in δ(t) and κ should be much slower than the πpulse time of t π = π/Ω F = 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 m j = ± 1/2 states, a π/2-pulse, i.e. t π/2 = t π /2, with phase ϕ = 0 spreads the population over all the Zeeman sublevels. A modulation sequence, consisting of ten repetitions of the form , is applied to cancel dephasing from δ(t). Here t w is the time in which the state freely evolves and t π (ϕ i ) indicates a π-pulse with phase ϕ i . In our experimental environment, dephasing is successfully canceled for t w = 100 μs. The phases ϕ i of the consecutive π-pulses are set to 0,4,2,4,0,0,4,2,4,0 ð Þ π=5, for which the modulation sequence is shown to be highly robust against pulse errors from, e.g., detuning and intensity variations 28 . At the end of the sequence, a π/2-pulse with phase ϕ = π retrieves a fraction P f of the population back into the initial m j = ± 1/2 state. The Ramsey dark-time, i.e. the time in which the state freely evolves, can be extended to T D = N × 20 t w by repeating the modulation sequence N times.
The retrieved fraction P f is dependent on the phase κT D acquired during the free-evolution time, which stems from the quadratic term in the Hamiltonian of Eq. (4). Therefore, κ LV can be extracted from a measurement of P f at a fixed T D . 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 P f at the same modulation frequencies. A measurement of P f is realized by de-exciting the ion via the E3 transition to the 2 S 1/2 state, where it can be detected by collecting fluorescence on the dipole allowed 2 S 1/2 → 2 P 1/2 transition.
The highest measurement accuracy is achieved when the measured quantity P f is most sensitive to variations of κ, i.e. |dP f /dκ|, is maximized, which is the case at κT D = 0.15 rad 26 . Using the axial secular frequency of 266(5) kHz set during the measurement campaign and the quadrupole moment of the 2 F 7/2 state 34 , κ q was calculated to be 0.13(3) rad s −1 , see the Supplementary Information. The corresponding optimal Ramsey dark time is T D = 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 | 2 S 1/2 , m j = ± 1=2i ! | 2 F 7=2 , m j = ± 1=2i 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 2 F 7/2 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 | 2 S 1/2 , m j = ± 1=2i state. If this is not the case, consecutive repumping and recooling sequences are applied. To save overhead time, either the | 2 F 7/2 , m j = + 1=2i or the | 2 F 7/2 , m j = À 1=2i 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 T m = 10 × t π + 20 × t w = 2.15 ms is repeated N = 575 times to reach T D = 1.15 s. The full rf sequence runs for T rf = N × T m + 2 × t π/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, P f is measured. At the end of the measurement sequence, the ion is re-cooled and prepared in the required | 2 S 1/2 , m j = ± 1=2i 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 P f = P(κ q ) + P(κ LV ), where P(κ q ) gives rise to a constant offset of P 0 = 0.39 and P(κ LV ) = dP f /dκ × κ LV contains a potential LV signal. To extract κ LV from Fig. 2 | Pulse sequences. a The applied composite rf pulse sequence is a more robust version of a spin-echoed Ramsey scheme. A Ramsey sequence of T rf = 1236.27 ms starts and ends with a π/2-pulse. A 2.15 ms-long modulation sequence of 10 π-pulses with phases ϕ i (i = 1. . . 10), each spaced by 2t w = 200 μs, is repeated N = 575 times to cancel dephasing due to ambient magnetic field noise. The phases ϕ i 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. P f , a high-pass filter is applied with a cutoff frequency of ν c = 5 μHz, removing P 0 and slow variations (τ drift < 2.5 days) caused by drifts in the E3 excitation probability, and the slope dP f /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.
In search of LV at Fourier frequencies of ω ⊕ and 2ω ⊕ , the data is fitted globally to the fit function κ LV = 2π × 5:1 × 10 15 × À3 sinð2χÞc XZ cosðω È TÞ À 3 sinð2χÞc Y Z sinðω È TÞ À 3 2 c X ÀY sin 2 ðχÞ cosð2ω È TÞ À 3c XY sin 2 ðχÞ sinð2ω È TÞ ,  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 periodogram 36 , which is specifically suited for spectral analysis of irregularly spaced data. The statistical significance level, i.e. p-values 36 , of 50% and 95% are indicated by horizontal lines.
The fit results show that the extracted values for c X−Y , c XZ and c YZ are consistent with zero within a 1σ uncertainty. Only c XY 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 work 23,24 . The stability of the data points is analyzed by calculating the Allan deviation, as shown in Fig. 3d. The data averages down as σ κ = 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 measurement 24 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 c MN 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 c XY .
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 2 F 7/2 state, the applied method is sensitive to LV at harmonics of up to sixth order 37,38 . Therefore, in combination with improved many-body calculations, experimental constraints can be translated into bounds on a larger number of coefficients in the future 38 . The correlated components of the cMN tensor extracted from the fit shown in Fig. 3a, compared to existing limits from refs. 23, 24. All uncertainties represent a 1σ interval.  The covariance matrix from the fit shown in Fig. 3a is diagonalized to obtain the linear combination of parameters of the c MN tensor. All uncertainties represent a 1σ interval. 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 gates 23,24 . It is applicable in a wide variety of systems, e.g., highly-charged ions or ultra-cold atoms 26 and it is capable of scaling to multiple N ion ions in linear ion Coulomb crystals for a further enhancement of the sensitivity. Up-scaling with an entanglementbased techniques 23 is technically demanding when considering the 2 F 7/ 2 state in Yb + due to the low-fidelity (p) single ion gate on the E3 transition, which further deteriorates, according to p N 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 ffiffiffiffiffiffiffiffiffi N ion p higher sensitivity is expected in such a system. Moreover, the long radiative lifetime of the 2 F 7/2 state 27 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 EIT 39 or Sisyphus cooling 40,41 , 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.

Experimental details
A single 172 Yb + ion is trapped in a segmented rf Paul-trap 42,43 . The radial confinement is set with an rf electric field supplied by a resonant circuit at a frequency of Ω rf /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 technique 44 and compensated in three directions to typically E rf < 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 2 S 1/2 → 2 P 1/2 transition assisted by a repumper laser near 935 nm. Fluorescence from the decay of the 2 P 1/2 state is collected by a lens of N/A = 0.27 and imaged onto the electronmultiplying charge-coupled device (EMCCD) camera 45 . This enables state detection via the electron shelving technique. The ion can be prepared in either the | 2 S 1/2 , m j = À 1=2i or the | 2 S 1/2 , m j = + 1=2i 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 Ω E3 /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 ω res =2π = 3:5147ð7Þ 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 m j levels in the 2 F 7/2 state is ω rf /2π = 3.52 MHz, close to the resonance frequency of the coil. The achieved multi-level Rabi frequency is Ω F /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 Δω rf /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 ω rf ðBÞ=2π = 1:581ð1:6ÞB + 0:0162ð3Þ ½ in MHz. For further details on the experimental set-up, see refs. 31, 42, 45, 46.

Data availability
The source data generated in this study are provided in the Source Data file Source data are provided with this paper.