Abstract
Experimental searches for exotic spin-dependent forces are attracting a lot of attention because they allow to test theoretical extensions to the standard model. Here, we report an experimental search for possible exotic spin-dependent force, specifically spin-and-velocity-dependent forces, by using a K-Rb-21Ne co-magnetometer and a tungsten ring featuring a high nucleon density. Taking advantage of the high sensitivity of the co-magnetometer, the pseudomagnetic field from this exotic force is measured to be ≤7 aT. This sets limits on coupling constants for the neutron-nucleon and proton-nucleon interactions in the range of ≥0.1 m (mediator boson mass ≤2 μeV). The coupling constant limits are established to be \(|{g}_{V}^{n}|\,\le \,8.2\times 1{0}^{-11}\) and \(|{g}_{V}^{p}|\,\le \,3.7\times 1{0}^{-10}\), which are more than one order of magnitude tighter than astronomical and cosmological limits on the coupling between the new gauge boson such as Z’ and standard model particles.
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Introduction
Precision measurements are powerful tools to find new physics beyond the Standard Model. For example, the discrepancies revealed by the precision measurements of the muon anomalous magnetic moment1,2 and the proton radius3,4 have been analyzed as possible indications of “new” physics. Light particles, including the spin-1 boson Z’ and spin-0 Axion Like Particles (ALPs) were proposed to resolve these discrepancies5,6, and are also promising candidates for dark matter7,8,9. If these particles exist, they mediate new long-range spin-dependent forces10,11, and could be discovered in precision measurements.
Many experiments are conducted to search for long-range forces. Typical approaches include torsion pendulums12, torsional oscillator13, atomic magnetometers14,15,16,17, nuclear magnetic resonance18,19,20,21, nitrogen-vacancy (NV) centers in diamond22, magnetic microscopes23, polarized neutron experiments24,25,26, and measurements of atomic and molecular electric dipole moments27.
Mathematically, forces between two particles, which depend on velocity, spin, and distance, can be broken down into 15 terms of spin-dependent forces10,11,28, which provides a guide on how to search them experimentally. Among the 15 terms, the spin-and-velocity-dependent (SVD) terms have received extensive attention in recent years16,19,22,23,29. Considering that these forces are mediated by Z’, the corresponding Lagrangian can be expressed as11
where ψ is the fermion field, γ5 and γμ are Dirac matrices, and gA and gV are axial and vector coupling constants. One of the SVD potentials, V4+5, as being noted in Ref. [10], can be derived from this Lagrangian10,19,
where f4+5 = − (gA gA + 3gV gV)/2 is a dimensionless coupling factor, ℏ is the reduced Planck constant, c is the speed of light, λ is the force range, r and v are the relative distance and velocity between the two particles, \(\hat{\sigma }\) is the spin of one fermion and m is its mass.
One can write V4+5 as V4+5 = − μ ⋅ Bpsd, where Bpsd is the pseudomagnetic field, and μ is the magnetic moment of the particle. Using bulk test material, the total pseudomagnetic field can be computed by integration over the volume of the material,
where ρN(r) is the mass-source nucleon density at location r with the sensor chosen as the origin, where the “N” notes the average nucleon contribution to Bpsd from the mass-source nucleons which is a linear combination of the protons and neutrons in the material (for example, 74 protons and 110 neutrons for tungsten gives 74/184 proton and 110/184 neutron interaction constants). Accordingly, the exotic force decays exponentially with the relative distance and it is beneficial to use high-density test materials. In fact, non-magnetic materials, such as silica (nucleon density 1.33 × 1024 cm−3) and bismuth germanium oxide (BGO; nucleon density 4.3 × 1024 cm−3) are used as the test material in recent works19,22. Because the possible signals are expected to be weak, a high-sensitivity magnetometer like a Spin-Exchange Relaxation Free (SERF) device30,31,32,33 is desired.
In this paper, we report a new measurement of V4+5 by using a K–Rb–21Ne comagnetometer34, and a tungsten test ring. Tungsten has a nucleon density of 1.15 × 1025 cm−3, which is among the highest-density practically-available non-magnetic materials. The sensitivity of SERF comagnetometers to pseudomagnetic fields acting on the sensor nuclei has been demonstrated to be better than 1 fT/Hz1/233,35, which is one order of magnitude more sensitive than the “spin-based amplifier” demonstrated recently19. With these advantages, limits on the V4+5 neutron-nucleon and proton-nucleon interactions have been achieved.
Results
Experimental setup
The experimental setup is shown in Fig. 1. A tungsten–aluminum (W–Al) ring, composed of tungsten wires and its ring-shape duralumin support, serves as the nucleon source and is located behind a K–Rb–21Ne comagnetometer. Driven with a servo motor, the W–Al ring can rotate clockwise or counterclockwise. The rotation axis of the ring is coaxial with the vapor cell along the \(\hat{y}\) direction, along which the comagnetometer has the highest sensitivity. The angular position of the ring is monitored with a photoelectric encoder. If it exists, the pseudomagnetic field induced by the W–Al ring can be measured with the comagnetometer.
The K–Rb–21Ne comagnetometer is similar to that of Refs. [35,36]. Hybrid optical pumping is utilized to improve the polarization homogeneity of alkali spins and hyperpolarization efficiency of noble-gas spins, where the optically thin K atoms are optically polarized with a circularly polarized K D1-line laser along the \(\hat{z}\)-axis and are used to polarize the optically thick Rb atoms via spin-exchange (SE) collisions between K and Rb atoms. With the help of Rb and K, the 21Ne nucleus can be polarized through the spin-exchange-optical-pumping mechanism37. The precession of the Rb polarization is measured via optical rotation of a linearly polarized laser beam propagating along the \(\hat{x}\)-axis. The frequency of the laser is detuned from the Rb D1-line towards lower frequencies by about 240 GHz. Here the optically thick Rb ensemble is used for probing instead of the optically thin K ensemble, in order to improve the signal-to-noise ratio. The precession of the 21Ne nuclei is probed with Rb atoms, and detected via optical rotation of the probe laser beam. The K–Rb–21Ne comagnetometer is operated in the self-compensation regime to suppress magnetic noise. It is also operated in the SERF regime, which results in its high sensitivity to pseudomagnetic signals32.
Principle
After zeroing the normal magnetic field, the leading terms in the comagnetometer signal in the self-compensation regime is given by15
where γe and γNe are the gyromagnetic ratios of electrons and 21Ne nuclei; \({P}_{z}^{e}\) and \({R}_{{{{{{{{\rm{tot}}}}}}}}}^{e}\) are the equilibrium spin polarization and transverse spin relaxation rate of alkali atoms, respectively; K is a factor to transform the \({P}_{x}^{e}\) to the output electric signal S; \({b}_{y}^{{{{{{{{\rm{Ne}}}}}}}}}\) and \({b}_{y}^{e}\) are the exotic fields along the \(\hat{y}\) axis that couple to 21Ne and alkali atoms, Ωy is the inertial rotation rate in \(\hat{y}\) axis. Deploying in Eq.(4), we utilize the inertial rotation to calibrate the comagnetometer response to exotic fields, which is summarized by a s factor \({\kappa }_{n}\equiv K{\gamma }_{e}{P}_{z}^{e}/{R}_{{{{{{{{\rm{tot}}}}}}}}}^{e}\).
If we consider the specific coupling to fermions, the pseudomagnetic field on the 21Ne nuclei can be written as
where \({\zeta }_{n}^{{{{{{{{\rm{Ne}}}}}}}}}=0.58\) and \({\zeta }_{p}^{{{{{{{{\rm{Ne}}}}}}}}}=0.04\) are the fraction factors for neutron and proton polarization in the 21Ne nucleus, respectively15,38, and \({B}_{p}^{p}\) and \({B}_{p}^{n}\) are the exotic fields acting on the proton and neutron, respectively. To detect \({b}_{y}^{{{{{{{{\rm{Ne}}}}}}}}}\), a quantum nondemolition approach is used. First, the precession of 21Ne nuclear spin under the \({b}_{y}^{{{{{{{{\rm{Ne}}}}}}}}}\) is transferred to the Rb atoms through the Fermi-contact interaction. By measuring the precession of Rb atoms based on optical rotation, one can measure the \({b}_{y}^{{{{{{{{\rm{Ne}}}}}}}}}\).
Data taking and simulation
The signal from the comagnetometer, as well as the angular position of the ring, are recorded with a data-acquisition (DAQ) device. Data for an individual set are taken continuously for four hours, and 26 datasets are collected in total. Between the datasets the co-magnetometer performance is optimized. The parameters of the comagnetometer, including the cell temperature, laser power, laser frequency etc., are monitored and feedback controlled throughout the experiments to ensure its stability.
To optimize the sensitivity to the hypothetical force, it is modulated by modulating the rotation frequency of the tungsten ring. This way, the exotic signal is shifted into a frequency region with optimum noise performance of the co-magnetometer. The angular velocity of the ring \({\omega }^{\exp }\) is measured with a high-precision photoelectric encoder and shown in Fig. 2a. The measured angular velocity is fitted with sinusoidal harmonics to obtain the amplitude (\({\omega }_{\max }=3.774\) rad/s) and frequency (0.8369 Hz) of the fundamental harmonic.
The \({b}_{y}^{{{{{{{{\rm{Ne}}}}}}}}}\) induced by the test material can be simulated using Eq. (3) and Eq. (5). The major input parameters for the simulation are listed in Table 1, and the simulated \({b}_{y}^{{{{{{{{\rm{Ne}}}}}}}}}\) is shown in Fig. 2b. When calculating \({b}_{y}^{{{{{{{{\rm{Ne}}}}}}}}}\), a coupling constant \({f}_{4+5}^{(0)}\) is assumed. In Fig. 2b, it is set to be \({f}_{4+5}^{(0)}=1\).
In Fig. 2c, the response of the comagnetometer \({S}^{{{{{{{{\rm{sim}}}}}}}}}\) is presented. This response signal is simulated based on Eq. (7) with parameters measured in the experiment. Because the main component of the \({b}_{y}^{{{{{{{{\rm{Ne}}}}}}}}}\) is at the fundamental harmonic and the bandwidth of the comagnetometer is narrow, the comagnetometer is mainly sensitive to the first harmonic component of \({b}_{y}^{{{{{{{{\rm{Ne}}}}}}}}}\), which results in the approximately sinusoidal shape of \({S}^{{{{{{{{\rm{sim}}}}}}}}}\). Compared with \({b}_{y}^{{{{{{{{\rm{Ne}}}}}}}}}\), the \({S}^{{{{{{{{\rm{sim}}}}}}}}}\) has a phase shift Δϕ = −67∘ ± 2∘ due to the phase response of the comagnetometer (corresponding to a time delay of Δt = 0.222 ± 0.007 s). In Fig. 2d, the corresponding experimental data \({S}^{\exp }\) are shown. There are slight beating patterns in the experimental data, which are due to the resonant vibrations of the isolation station and the table supporting the comagnetometer.
Limits on exotic forces
The coupling constant can be found by
where the \({\bar{b}}^{\exp }\) and \({\bar{b}}^{{{{{{{{\rm{sim}}}}}}}}}\) are exotic fields from the experiment and simulation respectively. The details of the data analysis can be found in the Supplementary Note 3.
A typical \({\bar{S}}^{\exp }\) distribution, deduced from 4-hour data, is shown in the inset of Fig. 3. Note that the proton fraction of polarization in the Rb atom is \({\zeta }_{p}^{Rb}=0.31\)39, but the Rb atoms’ energy sensitivity is three orders of magnitude smaller than that of 21Ne15,40 (see the Supplementary Note 2B). Thus we only consider the proton spin in 21Ne and ignore that in Rb atoms. Using all the 104-hour data, the pseudomagnetic field \({b}_{y}^{{{{{{{{\rm{Ne}}}}}}}}}\) is measured to be (2.4 ± 7.1)aT. This results in new limits on the dimensionless coupling factor of the new force ∣f4+5∣ ≤ 3.4 × 10−21 with 95% confidence level at the force range of 10 m. The limits on the coupling between the source nucleon with the 21Ne neutron and proton are set to be \(|{f}_{4+5}^{n} |=|{f}_{4+5}|/{\zeta }_{n}^{{{{{{{{\rm{Ne}}}}}}}}}\,\le \,5.9\times 1{0}^{-21}\) and \(|{f}_{4+5}^{p} |=|{f}_{4+5}|/{\zeta }_{p}^{{{{{{{{\rm{Ne}}}}}}}}}\,\le \,8.5\times 1{0}^{-20}\) respectively. Our experiment results and a comparison with the results in the literature are shown in Fig. 4. Our result on the coupling with neutron is more than an order of magnitude better than that of previous works. The limits on exotic coupling with proton is set by this work for the first time to the best of our knowledge. For the coupling to electron spins, our experimental result is comparable to that of a recent work in Ref. [29].
It may be useful to consider specific cases for coupling, using the fact that \({f}_{4+5}=\frac{1}{2}{g}_{A}{g}_{A}-\frac{3}{2}{g}_{V}{g}_{V}\). Assuming gAgA = 0, we have \(|({\zeta }_{p}^{{{{{{{{\rm{Ne}}}}}}}}}{g}_{V}^{p}+{\zeta }_{n}^{{{{{{{{\rm{Ne}}}}}}}}}{g}_{V}^{n})({\zeta }_{p}^{W}{g}_{V}^{p}+{\zeta }_{n}^{W}{g}_{V}^{n})|\,\le \,2.3\times 1{0}^{-21}\), where \({\zeta }_{n}^{W}=0.588\) and \({\zeta }_{p}^{W}=0.412\) are the neutron and proton mass contribution in the W–Al mass source. If we assume \({g}_{V}^{p}=0\), we get a limit on \(|{g}_{V}^{n}|\,\le \,8.2\times 1{0}^{-11}\). Conversely, if we assume \({g}_{V}^{n}=0\), we get \(|{g}_{V}^{p}|\,\le \,3.7\times 1{0}^{-10}\). The coupling limit \(|{g}_{A}^{p}|\,\le \,6.4\times 1{0}^{-10}\) and \(|{g}_{A}^{n}|\,\le \,1.4\times 1{0}^{-10}\) is also set using the same method.
A comparison of the limits on gV, the vector coupling constant between Z’ and standard model particles, between this SVD force result and other results including the cosmology and astronomy is shown in Fig. 5. The ‘Z’-Lepton Universe Expansion’ line is excluded by the effective number of neutrino species ΔNnef ≈ 0.2 in the early universe41. The authors of this work assume that Z’ can decay to neutrinos and affect the expansion of the universe. This model can relax the 3σ tension of Hubble constants, i.e., the discrepancy between local measurements and temperature anisotropies of the cosmic microwave background41,42,43. The ‘Z’-μ, g − 2’ is excluded by the muon g−2 experiment41,44,45. Note that the anomalous magnetic moment of muon can also be used to search for the spin-0 boson, such as axion like particles6. The ‘Z’-μ, SN1987A Cooling’ is excluded by the supernova SN1987A, assuming the new gauge boson Z’ decreases the cooling time44. Our results represent more than one order of magnitude tighter constraints than previous ones.
Discussion
The main advantage of this experiment compared to that in Ref. [19] are attributed to the high nucleon density of the nucleon source and the ultrahigh sensitivity of the comagnetometer. Tungsten has nucleon density approximately four times that of BGO, and our comagnetometer sensitivity is approximately one order of magnitude better than the 129Xe based magnetometer used in Ref. [19].
The comagnetometer can also be used to search for many terms of spin-spin-velocity-dependent forces if one can use a spin-polarized source, such as a pure-iron shielded SmCo5 electron-spin source14. For four terms of the spin-spin-velocity-dependent forces, new limits on electron-neutron and electron-proton interaction can be set in the force range ≥1 cm46, which would complement the results of17 for interaction ranges exceeding 1 km.
The techniques used in this work can be used to search for a broad class of beyond-standard-model particles10,11,28. The search for such particles is motivated, among other things, by the attempt to understand the composition of the dark sector (dark matter and dark energy). However, similar to other spin-dependent force searches, it does not, in any way, rely on specific local dark-sector properties, e.g., the local dark-matter density. If the exotic-force mediator is a Z’-boson, the interactions could violate parity, however, the current experiment does not exploit parity-violating effects. Indeed, the interaction in Eq. (2) is P- and T-even.
In conclusion, in this work, we have searched for exotic spin- and velocity-dependent interactions and, for the force range larger than several centimeters, improved the limits on the mass interactions with neutrons by more than an order of magnitude, while also setting a stringent limit on mass interactions with protons. This result demonstrates that the spin-dependent force approach is competitive in terms of sensitivity to new bosons with the cosmological and astronomical searches. Z’ can also be searched through pseudovector parity-violation couplings 47,48.
Methods
Calibration
The Earth rotation speed ΩE = 7.292 × 10−5 rad/s is used to calibrate the system. Mounted on a precision rotary platform, the apparatus can rotate in the horizontal plane as shown in Fig. 6 [a]. The Ωy in Eq. (4) can be written as \({{{\Omega }}}_{y}={{{\Omega }}}_{E}^{h}\sin (\alpha )\), where \({{{\Omega }}}_{E}^{h}={{{\Omega }}}_{E}\cos (\beta )\) is the projection of ΩE in the horizontal plane, β = 39.983∘ is the latitude of the laboratory, and α is the relative azimuth angle of the sensitive \(\hat{y}\) axis of the comagnetometer. Therefore, by fitting the measured signals with \(S(\alpha )={\kappa }_{n}{{{\Omega }}}_{E}^{h}\sin (\alpha )/{\gamma }_{n}\), the calibration factor κn can be obtained. As shown in Fig. 6 [b], the calibration factor at this near-DC frequency is measured to be κn(DC)=(4.18 ± 0.07) × 10−6 V/fT with the \({\bar{\chi }}^{2}=0.687\).
In the self-compensation comagnetometer, the frequency responses to normal magnetic fields, inertial rotations (Bx/y, and Ωx/y) and exotic fields (\({b}_{y}^{{{{{{{{\rm{Ne}}}}}}}}}\) and \({b}_{y}^{e}\)) are determined by four parameters, which are the Fermi-contact-interaction fields between noble-gas atoms and alkali atoms (\(\lambda {M}_{0}^{e}{P}_{z}^{e}\) and \(\lambda {M}_{0}^{n}{P}_{z}^{n}\))49, and the transverse relaxation rates (1/T2e and 1/T2n). These four parameters are independently measured to be (110.6 ± 2.2) nT, (579.4 ± 2.1) nT, (3.7 ± 0.4) × 103 /s, and (0.063 ± 0.007) /s, respectively36. Particularly, the noble-gas nuclear spins and alkali electron spins are strongly coupled around the self-compensation point, such that the damping rate of nuclear spins is sped up by electron spins, while electron spins are slowed down36. The damping rate of nuclear spins is measured to be 2π × (0.907 ± 0.006)/s by fitting the damping oscillation response to step magnetic field excitation, which is significantly larger than the 1/T2n owing to the coupling between nuclear spins and electron spins. More details about the analysis of the comagnetometer can be found in Supplementary Note 2 and 3.
Simulation
The spin dynamics of the comagnetometer could be described by the coupled Bloch equations32,36:
where Q is the slowing-down factor arising from spin-exchange collisions and hyperfine interaction, Pe and Pn are the spin polarizations of alkali electron and 21Ne nuclear, respectively. B and Ω are external magnetic field and inertial rotation. The Fermi-contact interaction between alkali atoms and 21Ne atoms can be described by an effective magnetic field \(\lambda {M}_{0}^{{{{{{{{\rm{e}}}}}}}},{{{{{{{\rm{n}}}}}}}}}{{{{{{{{\bf{P}}}}}}}}}^{{{{{{{{\rm{e}}}}}}}},{{{{{{{\rm{n}}}}}}}}}\), where \({M}_{0}^{n}\) (\({M}_{0}^{e}\)) is the maximum magnetization of 21Ne nucleon (alkali electron)49. For a uniformly spin-polarized spherical cell, λ = 8πκ0/3, where κ0 is the enhancement factor. T1e and T2e are the longitudinal and transverse relaxation rates for alkali electron spin, respectively, and T1n and T2n are the longitudinal and transverse relaxation times for the 21Ne nucleon spin.
As shown in Fig. 7 [a], to further verify the validity of the parameters \(\lambda {M}_{0}^{e}{P}_{z}^{e}\), \(\lambda {M}_{0}^{n}{P}_{z}^{n}\), T2e, and T2n) and the frequency response model, the frequency responses to By and Bx are measured and compared with the simulated results with Eq. (7) based on these four parameters with only one free parameter to describe the scale. The measured signals are consistent with the simulated results. Therefore, the response of the comagnetometer \({S}^{{{{{{{{\rm{sim}}}}}}}}}\) to the exotic field \({b}_{y}^{{{{{{{{\rm{Ne}}}}}}}}}\) can be simulated by solving the Bloch Eqs. (7)40 with these verified parameters. The simulation result of the amplitude and phase response to \({b}_{y}^{{{{{{{{\rm{Ne}}}}}}}}}\) is shown in Fig. 7 [b]. The results are further used to correct the calibration factor κn and the phase retardation Δϕ. For the modulation frequency at 0.83681(1) Hz, the amplitude response is 0.40 ± 0.01 of that in DC. Meanwhile, the phase shift Δϕ is − 67∘ ± 2∘. The corrected calibration factor is κn(0.84 Hz) = (1.67 ± 0.05) × 10−6 V/fT. A detailed analysis of the uncertainty of calibration factor can be found in Supplementary Note 3.
Data availability
The datasets generated during the current study are available from the corresponding author on request.
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Acknowledgements
The authors thank Xing Heng and Zitong Xu for debugging and operating the servo motor. This work is supported by the National Natural Science Foundation of China (NSFC) under Grant Nos. 61925301 for Distinguished Young Scholars, 11875191(CF), and 12235003(CF), the China postdoctoral Science Foundation (Grant No. 2021M700345), the DFG Project ID 390831469: EXC 2118 (PRISMA+ Cluster of Excellence), by the German Federal Ministry of Education and Research (BMBF) within the Quantumtechnologien program (Grant No. 13N15064). and by the QuantERA project LEMAQUME (DFG Project Number 500314265).
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K.W., W.J., C.B.F, J.C.F., and D.B. proposed this study. K.W. and W.J. performed the experiment and analyzed the data. K.W., W.J., C.B.F., A.W., V.F., and D.B. wrote the manuscript.
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Wei, K., Ji, W., Fu, C. et al. Constraints on exotic spin-velocity-dependent interactions. Nat Commun 13, 7387 (2022). https://doi.org/10.1038/s41467-022-34924-z
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DOI: https://doi.org/10.1038/s41467-022-34924-z
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