Lorentz invariance (LI) stands as a fundamental pillar in both special relativity and the standard model of particle physics, asserting that the local time and space axes hold no inherent significance. While extensively verified and embraced as an accurate depiction of nature at its core, the ongoing examination of LI remains compelling. Testing the foundational assumptions regarding the structure of space and time has profoundly enriched our comprehension of the natural world. Recent insights emerging from string theory and the quantum theory of gravity suggest the potential violation of LI at the Planck scale [1,2]. In response, the standard model extension furnishes an effective framework for delineating Lorentz violation [3–7]. Furthermore, the imperative of CPT invariance in quantum field theories underscores the necessity of LI, as any breach of CPT invariance entails a departure from LI. Therefore, investigating Lorentz invariance serves as a powerful avenue for uncovering new physics that extends beyond the limitations of the standard model of particle physics.

Drawing from these motivations, many theories and experiments have been devised and executed to scrutinize Lorentz invariance [8–14]. Among these, the Ives-Stilwell-type experiments [15] emerge as particularly noteworthy for their ability to subject LI to rigorous examination under substantial Lorentz boosts. After this landmark 1938 experiment of Ives and Stilwell [15] that directly measured time dilation, this form of special relativity testing was subsequently refined based on the theoretical frameworks proposed by Robertson [16] and by Mansouri and Sexl [17–19] (RMS). In the RMS theory, adjustments to the Lorentz transformations are introduced to accommodate possible anisotropic effects in space or time. These adjustments are delineated by additional parameters beyond the typical Lorentz-invariant ones. The theory suggests the existence of preferred directions or frames of reference, and the laws of physics may exhibit disparate behavior in any frame boosted with respect to the preferred one, in contrast to the tenets of conventional special relativity.

To illustrate, let us examine the relativistic Doppler effect between a stationary detector and a moving atomic system traveling at a velocity v relative to the detector. When an atom in an excited state undergoes a transition to a lower stationary state by emitting a photon, the frequency of this photon experiences a Doppler shift as it approaches the detector (νp) or moves away from it (νa). If the intrinsic transition frequency is ν0, according to the relativistic Doppler effect, we have νp=ν0/γ(1−β) and νa=ν0/γ(1+β), where β=v/c, c represents the speed of light, and γ=1/1−β2. Consequently, νaνp/ν02=1, regardless of the time-dilation factor β, in line with conventional special relativity. However, the RMS theory suggests the following modification: νaνpν02≈1+2α^β2.(1)Here, the so-called Robertson-Mansouri-Sexl parameter α^ denotes a departure from conventional special relativity. Furthermore, we have assumed that β≫βlab, where βlab denotes the velocity of the Earth-bound laboratory system relative to the hypothetical preferred reference frame in the RMS theory, commonly acknowledged as the cosmic microwave background rest frame.

The most precise time dilation experiments typically involve high-velocity Li+ ion beams conducted in the storage rings (SRs) [14,20], and optical clock comparison experiments [21]. The corresponding values of α^ for these experiments are (1.3±2.0)×10−8, (−4.8±8.4)×10−8, and (−0.38±1.06)×10−8, respectively.

According to Eq. (1), the tests of time dilation depend on the low-velocity intrinsic frequency ν0 in the denominator being accurately known, as well as νp and νa in the numerator from high-velocity (high β) measurements. Although the precision of the RMS parameter is no longer limited by ν0 at present, a clear issue is the lack of experimental research on ν0. Results of the two existing, typical ν0 experiments [22,23] show a discrepancy of up to 2 MHz, which will inevitably affect the higher accuracy of RMS tests. Here we improve the accuracy of ν0 in the denominator of Eq. (1). We employ a four-traveling-waves optical field [24,25] to extend the Ramsey spectroscopy method to our Li+ ion beam experiments, enhancing the measurement precision of the 23S1−23P2 transition to the order of 10−10, giving a reference to resolve the disagreement between collinear [22] and perpendicular laser spectroscopy [23]. In addition, by integrating our findings of low-energy beam data and complementing them with high-energy results obtained from the SR experiments, we aim to improve the precision of lithium ion spectroscopy measurements and furnish an updated determination of the Lorentz violation parameter α^.

In our experiment, we are measuring the absolute frequency of the transition from the 23S1(F=5/2) hyperfine state to the 23P2(F=7/2). This measurement is conducted utilizing an ion beam with ion speed approximately β∼0.00035. The experimental setup is depicted in Fig. 1. We conduct our measurements by probing the optical Ramsey resonance using four spatially separated traveling waves.

The Li+ ion beam is produced from a low-energy metastable Li+ ion source, the details of which are described in Ref. [26]. The ion source produces an ion beam, with approximately 1% of ions occupying the metastable state 23S1. The interaction between the ion beam and the four separated parallel laser beams is illustrated in Fig. 1. These four laser traveling waves are generated by two cat’s eye reflectors, each comprising a lens with a 500 mm focal length and a mirror. The initial parallelism of the laser beams is optimized by considering the frequency of Doppler and saturation spectroscopy during setup.

Furthermore, to mitigate errors associated with laser direction, such as the residual first-order Doppler effect [27], the probe laser is split into two beams to reverse the direction of the laser interacting with the Li+ ion beam. This method ensures precision in the alignment of the laser beams. The alignment is further corroborated by comparing the frequency difference in Ramsey spectroscopy, which is probed by two counter-propagating beams.

To maintain precise frequency control, the probe laser frequency is locked to a high-precision wavelength meter and an optical frequency comb (FC8004, Menlosystems) using digital feedback [28]. The frequency comb is referenced to an H-maser with a stability of 10−11 at 1 s average time, ensuring excellent long-term stability of the laser frequency. Additionally, a double-pass acousto-optic modulator (AOM1) system with a driver frequency of 205 MHz is employed to scan the laser frequency. Simultaneously, the laser power stabilization is accomplished through amplitude modulation of the AOM2, with the stabilized zeroth-order light selected for use as the probing laser beam.

The fluorescence emitted by the ion beam is captured by the photomultiplier tube (PMT). By scanning the frequency of the probe laser, we observed Ramsey fringes overlaid on the Lamb dip. Interestingly, the four spatially separated traveling-wave light fields yield similar Ramsey interference fringes to those obtained under three consecutive equally spaced standing-wave light fields, as described in Ref. [29]. As depicted in Fig. 1, these four traveling-wave light fields can be generated from two different directions of propagation by toggling the normal and reverse direction laser beams using shutters.

We determine the absolute frequency of the spectroscopy by averaging the central frequency of Ramsey interference probed by these two counter-propagating beam geometries. The kinetic energy of the ion beam is assessed using a retarding field energy analyzer (RFEA, FC-73A, Kimball), and the RFEA signal is recorded by a picoammeter (Keithley 6485). To mitigate potential measurement drift, we periodically reconfigure the beam geometry and adjust the kinetic energy of the ion beam every few days throughout the three-month measurement period, ensuring the repetition of the absolute frequency measurements.

The distribution of the measurement data collected over a single day, comprising 150 data samples, is illustrated in Fig. 2. We calculate the weighted average value to obtain the expectation value and the statistical uncertainty of the measurement results. Notably, these results represent only the centroid of the spectral line fit and do not incorporate various frequency shifts within the system. The resulting statistical uncertainty is about 3 kHz, and a Gaussian function is applied to fit the histogram of the distribution in order to assess its normality.

Given that the experimental equipment is restarted daily, the state of the ion beam varies each day, potentially resulting in unequal ion velocities. To address this, we monitor the ion velocities daily and provide a statistical average of the data for the entire day, along with an additional correction for Doppler frequency shift.

Several systematic effects have been meticulously examined and accounted for in order to refine our final outcomes. Among these, the Doppler effect holds significant importance, encompassing both the residual first-order Doppler effect and the second-order Doppler effect. In our experiment, we employed the laser-beam reversal technique to mitigate the residual first-order Doppler effect. Nonetheless, due to imperfect alignment among the four traveling waves, a residual first-order Doppler effect persists. We actively change the parallelism of the laser beam by adjusting the position of the lens in the “cat’s eye” configuration. Meanwhile, the average and difference between the frequency obtained from two probe schemes, where their laser direction is reversed, are recorded and fitted with a linear function. The uncertainty result from the residual first-order Doppler effect is assessed to ±102kHz according to the slope and the range of the real difference frequency. Additionally, we re-calibrated the probe geometry and conducted repeated measurements of the absolute frequency to ensure the reliability of our uncertainty assessment.

For the second-order Doppler shift, we employ RFEA to extract the velocity distribution of a group of charged particles [30,31] by measuring the ion current. As depicted by the black line in Fig. 3, when the retarding field voltage V varies from low to high, the ion current I as a function of V registered by the picoammeter decreases. Differentiating I with respect to V yields the ion current density distribution at various kinetic energies, represented by the red-dotted line in Fig. 3. A Gaussian fit for a single measurement yields the most probable kinetic energy of 415.36(5) eV, with a half-width at half-maximum (HWHM) of 2 eV. The statistical uncertainty for a one-day monitoring period is about 0.4 eV.

Due to the non-Gaussian nature of the kinetic energy distribution and the resolution of the RFEA being less than 1 V, we consider the HWHM as an intrinsic uncertainty of the ion kinetic energy measurement method. The transfer relation between the kinetic energy and the second-order Doppler shift is νsd=EeVq2mc2ν0,(2)where νsd is the second-order Doppler shift due to the kinetic energy of lithium ion EeV with the unit of eV, q is the element charge, c is the speed of light, m is the mass of lithium ion, and ν0 is the absolute frequency of the 23S1→23P2 transition. The 2 eV value of HWHM serves as an upper limit for the measurement uncertainty, corresponding to a second-order Doppler shift uncertainty of 160 kHz. Additionally, we must account for the electric potential gradient within the laser-ion interaction zone, which may be influenced by the 180 mm distant electrode, causing the measured ion kinetic energy by RFEA to deviate from the actual kinetic energy of the ions within the laser-ion interaction zone. Simulating the potential distribution across the entire area indicates a potential shift of approximately 1.3 V, with an uncertainty of ±0.5V, corresponding to a frequency uncertainty of about 40 kHz.

Moreover, the RFEA measures the kinetic energy of ions normal to the RFEA detection plane, resulting in an angle-induced kinetic offset. The inner radius of the control electrode and the probe area of the RFEA restrict this angle to less than 0.001 mrad, which translates to an uncertainty in the second-order Doppler shift of 78 kHz. The incident angle offset consistently leads to a lower kinetic energy. To account for a symmetric uncertainty, we assign a kinetic energy shift of 39±39kHz, considering this shift as the angle-induced uncertainty. Thus, the total uncertainty of the second-order Doppler effect amounts to 170 kHz.

Other systematic error sources, such as power dependence, the Zeeman effect, and quantum interference, underwent thorough consideration and analysis; see Appendix A. The impact of probe laser power was evaluated by probing spectroscopy frequency across various laser power settings, with an assessed uncertainty of 15 kHz. In our measurements, a linearly polarized probe laser was utilized, ideally leading to symmetrical spectrum broadening. The residual Zeeman effect was quantified as 3.9 kHz through measurements of the background magnetic field and the proportion of circular polarization in the probe laser. Additionally, an asymmetric line shape arising from quantum interference introduced an uncertainty of roughly 9 kHz. For a detailed description, see Appendix A.2. Furthermore, the measurement uncertainty of the probe laser frequency was determined to be 5.5 kHz, calculated by evaluating the uncertainty of the H-maser. The presence of the photon-recoil doublet does not alter the spectroscopy’s central value and is thus disregarded. Moreover, the DC Stark shift resulting from the stray electric fields, calculated based on the static polarizability of Li+ [32], is less than 1 Hz and therefore negligible.

The uncertainty budget for measuring the transition frequency from 23S1(F=5/2) to 23P2(F=7/2) in Li7+ is detailed in Table 1. Currently, the Doppler shift stands out as the primary source of error in determining the absolute frequency. Over a period of three months, repeated measurements were conducted, as illustrated in Fig. 4. Each data point in Fig. 4 represents an average value derived from several hundred measurements taken within a single day. The error bars in the figure encompass both statistical and systematic uncertainties.Table 1.

Uncertainty Budget for the Determination of the Transition Frequency from 23S1(F=5/2) to 23P2(F=7/2) in ⁷Li⁺ (in kHz)

Upon accounting for the various contributions to the overall uncertainty and applying corrections for the second-order Doppler shift, the transition frequency of 23S1(F=5/2)−23P2(F=7/2) is determined to be 546466918.716(199)MHz at the level of four parts in 1010. Notably, our achieved uncertainty is a factor of two smaller than the best previous measurements in the rest frame [22,23]. In the work of Ref. [22], the Lamb dip was obtained through collinear laser spectroscopy, yielding a result of 546466918.790(400)MHz. In the work of Ref. [23], a perpendicular laser spectroscopy method was employed, resulting in 546466916.490(870)MHz, which deviated by 2 MHz from the measurement in Ref. [22]. Our approach utilizes perpendicular laser spectroscopy and measures Ramsey fringes instead of Lamb dip, yielding a more precise outcome, and aligns closely with the findings of Ref. [22].

In addition, Gwinner and colleagues [20,33] conducted experiments to measure the Doppler shifted frequencies of the transition from 23S1(F=5/2) to 23P2(F=7/2) at velocities of 0.03c and 0.064c and determined the values of νa and νp. By combining their values with our value for ν0, we can calculate the α^ parameter in the Robertson-Mansouri-Sexl framework [16–19]. The results for α^ are shown in Table 2 and are found to be (−3±15)×10−8 and (−10.0±9.9)×10−8, respectively. These values are comparable to the results obtained by Saathoff et al. [33] and Reinhardt et al. [20], which are (−0.7±2.2)×10−7 and (−4.8±8.4)×10−8, respectively. These consistent results obtained for α^ suggest that, under the assumption of time dilation being correct, our determination of the Doppler-corrected ν0 with β=0.00035 aligns with the measurements obtained from higher-energy ion beams at β=0.03 and 0.064.Table 2.

α^ Parameter from Different Sources

Moreover, we have conducted measurements on the hyperfine splittings of 23S1(F=3/2−5/2) and 23P2(F=5/2−7/2), yielding the respective results of 19817.677(12) MHz and 11773.066(11) MHz. By combining these findings with the transition frequency of 23S1(F=5/2)−23P2(F=7/2), we have derived the absolute frequencies of a Λ-type three-level system. This system encompasses the ν1 transition, 23S1(F=3/2)−23P2(F=5/2), and the ν2 transition, 23S1(F=5/2)−23P2(F=5/2), with frequencies calculated as ν1=546474963.327(200) MHz and ν2=546455145.650(199) MHz, respectively. This Λ system was also examined at a velocity of 0.338c in Ref. [14]. In their study, the Doppler shift of these Λ energy level transitions was measured with an accuracy of about 1–2 MHz. They subsequently combined the measurements of Riis et al. [22] and Kowalski et al. [34] in the rest frame (note that the numerical values for ν1 and ν2 cited in Ref. [14] should be swapped), achieving one of the highest accuracies for the α^ parameter (1.3±2.0)×10−8. Riis et al. and Kowalski et al. provided two frequencies for the Λ energy levels, ν1 and ν2, with values of ν1=546474960.66(64)=546466918.79(40)−11775.8(5)+19817.673(40) (in MHz) and ν2=546455142.99(64)=546466918.79(40)−11775.8(5) (in MHz). The uncertainty in ν1 and ν2 in Ref. [14] was initially reported as 0.43 MHz, and we have corrected it to 0.63 MHz based on the results from Refs. [22,34], although this adjustment does not impact their final result. A clear discrepancy of more than 2 MHz is observed when compared to our results. Now, we incorporate our findings alongside those from Ref. [14] and substitute them into the following equation: α^≈νa(1)νp(2)ν1ν2−1β2,(3)where νa(1) and νp(2) represent the frequencies of the 23S1(F=3/2)−23P2(F=5/2) and 23S1(F=5/2)−23P2(F=5/2) transitions, respectively, as observed in the relativistic frame in Ref. [14]. This results in α^=(−2.9±2.0)×10−8, with the uncertainty primarily constrained by the measurement precision of the Botermann et al. experiment [14]. Consequently, we assign uncertainties that mirror theirs. Upon undergoing Lorentz transformation, our findings display deviations from theirs at the level of MHz, significantly surpassing our current measurement uncertainty. In this investigation, we substituted our experimental data for the absolute frequency of the 23S1(F=5/2)−23P2(F=7/2) transition outlined in Ref. [22], as well as the hyperfine splitting values for the S and P states provided in Refs. [22,34]. Our analysis revealed that the central value shift primarily arises from a discrepancy of approximately 2 MHz with one of the hyperfine splittings of the P state, namely, 23P2(F=5/2−7/2), measuring 11773.066(11) MHz in our study [35] compared to 11775.8(5) MHz in Ref. [34]. Notably, our measurement aligns well with recent experimental and theoretical values for this specific hyperfine splitting [35].

We have performed precise measurements of the absolute transition frequencies in metastable Li7+. By incorporating the optical Ramsey resonance technique into ion beam spectroscopy, we have significantly narrowed the linewidth of the measured spectra, approaching the natural linewidth. Alongside a comprehensive assessment of systematic errors, we have updated the Robertson-Mansouri-Sexl parameter α^. The accuracy of these results is primarily delimited by relativistic experiment spectroscopy. It is worth noting that our results for α^ based on Ref. [14] require a 1.45σ margin to encompass zero. This might suggest the potential presence of overlooked factors by either party, prompting the need for further exploration. Therefore, we anticipate that conducting further Li7+ spectroscopy experiments in storage rings, in both stationary and relativistic frames, will yield higher precision and consequently offer additional tests of Lorentz invariance.

Furthermore, our determination of the 23S1(F=5/2) to 23P2(F=7/2) transition frequency serves as a reference point for resolving the 2 MHz discrepancy observed between collinear and perpendicular laser spectroscopy. Our findings align with collinear laser spectroscopy by Riis et al. [22] and relativistic experiments by Reinhardt et al. [20] and Saathoff et al. [33]. Also, our determination of the hyperfine splitting of P23(F=5/2to7/2) once again corroborates the measurements from our previous work [36] and Clarke [37], while differing by 5.5σ from the results of Kowalskik [34]. This high-precision spectroscopy of Li+ also holds promise as an important platform for testing quantum electrodynamic theory and determining nuclear structure parameters [38].

Acknowledgment. We gratefully acknowledge Zongchao Yan for his thorough review and invaluable feedback, which greatly enhanced the quality of this work. We also acknowledge Tingyun Shi and Liyan Tang for helpful discussions. Additionally, we appreciate the technical support provided by Qunfeng Chen, Yanqi Xu, and Huanyao Sun in optical frequency comb and electronic circuits.