Nuclear-spin comagnetometer based on a liquid of identical molecules
Teng Wu, John W. Blanchard, Derek F. Jackson Kimball, Min Jiang, Dmitry Budker
NNuclear-spin comagnetometer based on a liquid of identical molecules
Teng Wu, , ∗ John W. Blanchard, Derek F. Jackson Kimball, Min Jiang, and Dmitry Budker , Helmholtz-Institut Mainz, Johannes Gutenberg University, 55128 Mainz, Germany Department of Physics, California State University-East Bay, Hayward, California 94542-3084, USA CAS Key Laboratory of Microscale Magnetic Resonance and Department of Modern Physics,University of Science and Technology of China, Hefei, Anhui 230026, China Department of Physics, University of California at Berkeley, California 94720-7300, USA (Dated: April 9, 2018)Atomic comagnetometers are used in searches for anomalous spin-dependent interactions. Mag-netic field gradients are one of the major sources of systematic errors in such experiments. Here wedescribe a comagnetometer based on the nuclear spins within an ensemble of identical molecules.The dependence of the measured spin-precession frequency ratio on the first-order magnetic fieldgradient is suppressed by over an order of magnitude compared to a comagnetometer based onoverlapping ensembles of different molecules. Our single-species comagnetometer is shown to becapable of measuring the hypothetical spin-dependent gravitational energy of nuclei at the 10 − eV level, comparable to the most stringent existing constraints. Combined with techniques forenhancing the signal such as parahydrogen-induced polarization, this method of comagnetometryoffers the potential to improve constraints on spin-gravity coupling of nucleons by several orders ofmagnitude. Atomic comagnetometers typically consist of overlap-ping ensembles of at least two different species of atomicspins [1, 2]. The basic idea of comagnetometry is thatthe precession frequency of one spin can be used to mon-itor and compensate magnetic field fluctuations, whilethe other spin is used to search for nonmagnetic torques.Practically, it is the ratio of the spin-precession frequen-cies of the different species under the influence of a biasmagnetic field that is measured. The ratio is relativelyinsensitive to changes in the magnetic field, but retainssensitivity to Zeeman-like nonmagnetic spin interactions.Comagnetometers have been widely used for fundamen-tal physics experiments [3], such as measurements of per-manent electric dipole moments (EDMs) [4–8], tests of
CPT and Lorentz invariance [9–14], and searches for ex-otic spin-dependent interactions mediated by hypothet-ical bosonic fields [15–22]. Comagnetometers also findpractical applications as sensitive gyroscopes [23, 24].In fundamental-physics experiments using comagne-tometers based on overlapping ensembles of differentspecies, one of the major systematic effects reducing ac-curacy is due to uncontrolled magnetic field gradients[6, 22, 25]. Previous work demonstrates that there existssome spatial separation between the ensemble-averagedposition of different spin species due to nonuniform po-larization [25], gravity [6], and/or thermodiffusion effects[26]. As a consequence, in the presence of a magneticfield gradient, the average magnetic field sensed by dif-ferent spin species is different. Thus the ratio of spin-precession frequencies acquires a magnetic-field-gradientdependence that can add noise and is difficult to dis-tinguish from other sources of nonmagnetic torques onspins. For this reason, complex arrangements are neededto monitor and reduce the magnetic field gradient foreach cycle of measurement [2, 22, 27].In contrast to comagnetometers which utilize overlap-ping ensembles of different atomic or molecular species, here we introduce and demonstrate a new comagne-tometer configuration based on an ensemble of identicalmolecules. In this single-species comagnetometer, differ-ent nuclear spins are probed within the same molecule.In this way, the spatial sampling of the field by the differ-ent nuclear spins is made nearly identical and systematicerrors related to field gradients are highly suppressed. Bytaking advantage of the techniques of ultralow-field nu-clear magnetic resonance (NMR) and sensitive atomicmagnetometry, the J -coupling (indirect spin-spin cou-pling) spectrum of a liquid-state ensemble of acetonitrile-2- C molecules can be measured with sub-mHz precisionin an ultralow magnetic field with a single scan (10 s mea-surement time). Under the influence of a bias magneticfield, the J -coupling resonance lines at different frequen-cies split into separate peaks. The frequency separationbetween the split peaks for each J -coupling resonance hasdistinct linear coefficients with respect to the magneticfield. Measurements of these splittings can be employedas a comagnetometer. We experimentally demonstratethat in the presence of a temperature gradient, such a co-magnetometer is insensitive to first-order magnetic fieldgradients within experimental uncertainty. We analyze apossible application of this new kind of comagnetometerfor measurement of a coupling between nuclear spins andgravitational fields.The device is based on a zero- to ultralow- field (ZULF)NMR configuration and the experimental setup is de-scribed in detail in Refs. [26, 28]. The spin ensem-ble we use to realize the comagnetometer is liquid-stateacetonitrile-2- C ( CH CN, from Sigma-Aldrich). Theacetonitrile-2- C sample ( ∼ µ L) is flame-sealed un-der vacuum in a standard 5 mm NMR tube. The dis-solved oxygen inside the sample is removed through afew freeze-pump-thaw cycles assisted with liquid nitro-gen. The sample is initially polarized in a 1.8 T Halbachmagnet for about 30 s, and then pneumatically shuttled a r X i v : . [ phy s i c s . a t o m - ph ] A p r
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FIG. 1. (color online). Experimentally measured J -couplingspectrum of acetonitrile-2- C ( CH CN) in a 80 nT biasfield along z . The top and bottom traces show the split spec-trum at J CH and 2 J CH , respectively. The related transitionsused for comagnetometry are shown with solid red arrows. down into a four-layer magnetic shield (Twinleaf MS-1F).During the transit of the sample, a ∼ µ T magneticfield is applied to the sample with a solenoid, which isused to guide the initial spin magnetization along thevertical direction ( y ). After the sample drops into thedetection region ( ∼ µ s. The initialspin magnetization then evolves under the J -coupling in-teraction between C and the three H protons, which inturn generates an oscillating magnetization signal along y and is detected with a rubidium atomic magnetometer(sensitivity ≈
10 fT/Hz / ). A π pulse ( ∼ µ s)for C along x is triggered after switching off the guid-ing field and prior to data acquisition, which is tuned toprovide the maximum signal amplitude. Besides, a smallbias field along z is applied by coils within the innermostshield layer, and can be regarded as a small perturbationto the dominant J -coupling interaction. The J -couplingspectrum is thus split into different peaks under the in-fluence of the bias magnetic field.Acetonitrile-2- C is a CH system with three equiv-alent protons. Thus, the resulting zero-field J -couplingspectrum consists of two resonance lines, with one at J CH and the other at 2 J CH [29, 30]. The measured J -coupling frequency for acetonitrile-2- C in our experi-ment is 140.55002(3) Hz, which is a function of the sam-ple temperature (the frequency shift as a function of tem-perature is ∼ -125 µ Hz/K). With a small bias magneticfield ( ∼
80 nT), the two lines split into different patternsof peaks, see Fig. 1. The spectrum around J CH splitsinto two peaks, while the spectrum around 2 J CH splitsinto six. Within the 2 J CH multiplet, we focus on the cen- D A (a)(b) / ( H z ) ( H z ) Time (hours) B A C A Time (hours)
FIG. 2. (color online). Experimental demonstration of co-magnetometry with an ensemble of identical molecules. (a)The measurements were taken while an oscillating magneticfield was applied in the z direction (see text). The red solidlines are the fitted curves. (b) The calculated ∆ ν / ∆ ν basedon (a). The average value of ∆ ν / ∆ ν (red dashed line) is0.70088(4). tral two peaks (the corresponding transitions are shownwith solid red arrows in Fig. 1), as they have the highestsignal-to-noise ratio compared to the others. Neglect-ing all other nonmagnetic spin interactions, the frequen-cies for the two splittings ∆ ν , are ∆ ν = ( γ h + γ c ) B z ,∆ ν = ( γ h + 3 γ c ) B z , where γ h,c are the gyromagneticratios for H and C, respectively, and B z is the biasmagnetic field [29, 31]. There are no contributions fromthe second-order Zeeman effect on ∆ ν , , and the third-order Zeeman effect is an order of magnitude smaller thanthe current experimental uncertainty (see Appendix B).Since ∆ ν and ∆ ν are both proportional to B z but withdifferent linear coefficients, we can employ them to real-ize a comagnetometer based on an ensemble of identicalmolecules.Comagnetometers should be able to suppress the vari-ations in the bias magnetic field. As a demonstration, weapply a slowly varying magnetic field along the same di-rection ( z ) as the bias field, with 1 mHz frequency and 0.5nT amplitude. Since the total acquisition time for eachscan is 10 s, the oscillating magnetic field is effectivelyDC within this sampling window, and changes the twosplitting frequencies by a few tens of mHz. Figure 2(a)shows the measured frequencies ∆ ν , under the influ-ence of the oscillating magnetic field, both of which dis-play an evident 1 mHz modulation. The ratio between∆ ν , for each measurement is calculated and shown inFig. 2(b). Compared with Fig. 2(a), there is no appar-ent modulation of the frequency ratio. Based on mea-surements over 10 hours, the averaged value of ∆ ν / ∆ ν (a) N o r m a li ze d F r e qu e n c y r a ti o -12 -9 -6 -3 0 3 6 9 12 Magnetic field gradient d B z /d y (nT/cm) Bias magnetic field B z (nT) (b) N o r m a li ze d F r e qu e n c y r a ti o acetonitrile ( J CH ) + wateracetonitrile (2 J CH ) + wateracetonitrile ( J CH + J CH ) -180 -120 -60 0 60 120 1800.9880.9920.9961.0001.0041.0081.012 B A -12 -9 -6 -3 0 3 6 9 120.99250.99500.99751.00001.00251.00501.0075 D A acetonitrile ( J CH ) + wateracetonitrile (2 J CH ) + water acetonitrile ( J CH + J CH ) FIG. 3. (color online). Comparison of the normalized fre-quency ratios between a single-species comagnetometer (redstar, ∆ ν / ∆ ν ) and two dual-species reference comagnetome-ters (black square, ν h / ∆ ν , black circle, ν h / ∆ ν , discussed inthe text). All the data are taken with the same acetonitrile-2- C (with ∼
1% water) and are normalized with the the-oretical values at zero magnetic field gradient. (a) The nor-malized frequency ratio as a function of the gradient d B z / d y ,with constant bias magnetic field B z = 80 nT. The solid linesare the linearly fitted curves. (b) The normalized frequencyratios as a function of bias magnetic fields B z , with constantgradient d B z / d y = -3 nT/cm. The solid line is the fittedcurve based on B − z . is 0.70088(4). By using γ h = 42 . γ c = 10 . C, i.e., σ ( H) = 31ppm, σ ( C) = 185 ppm [33, 34], the theoretical valueis 0.70092. Besides this, the third-order Zeeman effectmodifies the frequency ratio at the level of 10 − based onthe current experimental parameters (see Appendix B).However, systematic effects related to such a differencecan be suppressed in precision measurements by employ-ing field-reversal methods [22].It has been demonstrated, both experimentally andtheoretically, that for a dual-species comagnetometerthe spin-precession frequency ratio is a function of themagnetic field gradient. A thorough investigation of magnetic-field-induced systematic effects can be foundin Ref. [25]. Although their analysis is based on a gas-phase comagnetometer, many of the conclusions are alsovalid for a liquid-state comagnetometer. Here, we focuson the shifts in the spin-precession frequency ratio dueto the first-order gradient, typically, the gradient of thebias magnetic field along the vertical direction ( y ), i.e.,d B z / d y . If there exist temperature gradients, differentspin ensembles can experience different thermal diffusionrates, which in turn causes gradients in the concentrationof the ensembles [25]. Thus, a first-order magnetic fieldgradient can introduce an additional component in thefrequency ratio, which has the form G ∆ /B z , where G is the first-order magnetic field gradient, ∆ is the sepa-ration of the centers of the ensemble-averaged positionof the spins. Considering that our sample is placed at asmall distance ( ∼ ∼ ◦ C, there is a large temperaturegradient along the vertical direction ( ∼
25 K/cm).In order to determine the sensitivity to magnetic fieldgradients, we compare our single-species comagnetome-ter to two dual-species reference comagnetometers. Thedual-species comagnetometers are based on the sameacetonitrile-2- C, but use one of the splittings, ∆ ν or∆ ν , together with the precession frequency of H inresidual water present in the sample ( ∼ H can be written as ν h = γ h B z .Therefore, for the two reference comagnetometers, themeasured spin-precession frequency ratios are ν h / ∆ ν = γ h / ( γ h + γ c ) and ν h / ∆ ν = 2 γ h / ( γ h + 3 γ c ), respectively.Figure 3(a) shows the spin-precession frequency ra-tios for the three comagnetometers as a function of thefirst-order magnetic field gradient d B z / d y . In order tocompare the results at the same level, the measuredspin-precession frequency ratios of each comagnetome-ter are normalized to the corresponding theoretical val-ues at zero magnetic field gradient. For the two ref-erence comagnetometers, the normalized frequency ra-tios are both linear in the magnetic field gradient, withslopes of 6 . × − cm/nT ( ν h / ∆ ν , black squares)and 6 . × − cm/nT ( ν h / ∆ ν , black circles), re-spectively. Such a linear dependence is also observed inRef. [26], in which a mixture of pentane and hexafluo-robenzene is used to realize a liquid-state comagnetome-ter. The slopes of the normalized frequency ratios for thetwo reference comagnetometers are nearly identical sincethey are based on the same sample. For the single-speciescomagnetometer, the results display a negligible lineardependence with the magnetic field gradient (∆ ν / ∆ ν ,red stars). Fitting the results with a linear function givesa slope of − . × − cm/nT, which is at least anorder of magnitude smaller than the dual-species refer-ence comagnetometers and, in fact, consistent with zero.The residual nonlinear dependence could be attributedto higher order effects of the gradient on the precessionfrequencies, which could introduce broadening and shiftof the resonance lines [25].Figure 3(b) shows the spin-precession frequency ratios B D A nega t i v e A B A -0.010 -0.005 0.000 0.005 0.010020406080 B A (a)(b) no. of measurement
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450 500 ( ) ( ) / ()() - + - O cc u rr e n ce O cc u rr e n ce O cc u rr e n ce ( ) ( ) -0.01 0.010.00 ( ) B D A ( ) FIG. 4. (color online). Demonstration of the sensitivity and accuracy of a possible measurement of a spin-gravity coupling. (a)The measured frequency ratios R ± = ∆ ν ( ± ) / ∆ ν ( ± ) (top) and the calculated difference in ∆ ν / ∆ ν based on the measuredresults of each consecutive { + B z , − B z } (bottom). (b) Histograms of the results (∆ ν / ∆ ν ) − (left), (∆ ν / ∆ ν ) + (middle), andthe difference (right). The numbers in the brackets are the standard error of the mean. The solid lines are the fitted curvesusing a Gaussian function, which indicate that the measurement results are consistent with normal distributions. as a function of the bias magnetic field. We apply a con-stant gradient d B z / d y = -3 nT/cm through the gradientcoils. For the dual-species comagnetometer, the resultsare fit to the inverse of the bias magnetic field amplitude, B − z , in agreement with the G ∆ /B z form of gradientdependence described above. Under the same condition,there is no apparent dependence of the frequency ratioon B − z for the single-species comagnetometer.We also apply first-order magnetic field gradients along x and z directions, i.e., d B z / d x and d B z / d z . Under theseconditions, the frequency ratios measured with the ref-erence comagnetometers similarly show no linear depen-dence on the first-order magnetic field gradient. Sincethere are negligible temperature gradients along x and z , the first-order gradient does not change the frequencyratio up to the second order of the gradient strength, ifthe Larmor frequency is much larger than the diffusionrate across the cell ( D/R , D is the diffusion constant,and R is the cell radius) [25, 35]. This situation is wellsatisfied in our experiment, considering that the diffusionconstants for acetonitrile and water are both on the orderof 10 − cm / s, the radius of the tube is ∼ C and H nuclei ofacetonitrile-2- C can be parameterized as modificationsof the spin-precession frequencies (see Appendix A)∆ ν ( ± ) = ( γ h + γ c ) B z ± ( − χ n + χ p ) g cos φ (cid:126) , (1)∆ ν ( ± ) = ( γ h + 3 γ c ) B z ± ( − χ n + 12 χ p ) g cos φ (cid:126) . (2)Here, ± refers to reversing the magnetic field direction, χ n and χ p are the gyrogravitational ratios of the neutron(from C) and proton (from H), respectively, g is ac-celeration due to gravity, and φ is the angle between themagnetic field and the Earth’s gravitational field [36–38].We construct the ratio R ± ≡ ∆ ν ( ± ) / ∆ ν ( ± ). The dif-ference in the ratio obtained by reversing the magneticfield direction is∆ R ≡ R − −R + ≈ γ h + 3 γ c γ h + γ c (cid:20) (5 χ p + 4 χ n ) g cos φ µ N B z (cid:21) , (3)where µ N is the nuclear magneton.Due to the current system configuration, the angle φ is fixed at 90 ◦ and can not be changed. Thus, the con-tribution from the Earth’s gravitational field is zero andcan not be measured directly with our current system.However, we can still reverse the magnetic field direc-tion, and record the variations in ∆ R , which demon-strates the achievable sensitivity for a measurement of thespin-gravity coupling. Considering this, we reverse themagnetic field direction for each measurement scan, i.e.,+ B z , − B z , + B z , − B z , ... . Each consecutive { + B z , − B z } is taken as a group, for which ∆ R is calculated. We per-form 1024 continuous measurements ( ∼
15 hours), whichcan be divided into 512 groups of { + B z , − B z } . The mea-sured frequency ratios R ± and the corresponding differ-ence ∆ R are shown in Fig. 4(a), respectively, includingthe histograms for these results, see Fig. 4(b). Based onthese measurements, we find that ∆ R = (1 ± stat ) × − .This uncertainty level indicates that for the current sys-tem, (5 χ p + 4 χ n ) could be measured at a level of 10 − g cm, which probes the spin-dependent gravitational en-ergy of a linear combination of the proton and neutronat a level of 10 − eV. This is comparable to the moststringent existing constraint on the spin-gravity couplingof protons [22].The measurement uncertainty for the current systemis statistics-limited based on the signal-to-noise ratio ofa single scan ( ∼ en-hancement of the polarization. This will enable a searchfor spin-gravity couplings of nuclei several orders of mag-nitude more sensitive than existing limits. Moreover, wecan also take advantage of high-sensitivity commercialatomic magnetometers, such as those from QuSpin Inc.,which could make a new single-species comagnetometermore compact and easier to rotate. Another advantagefor this comagnetometer is that, by using different kinds of molecular samples, one can realize comagnetometersto search for spin-gravity couplings using various combi-nations of protons and neutrons.In conclusion, we have demonstrated a new single-species liquid-state comagnetometer. We have shownexperimentally that the magnetic field gradient-inducedsystematic effects are significantly suppressed with asingle-species comagnetometer as compared to a comag-netometer based on overlapping ensembles of differentspecies. We have introduced a proof-of-principle exper-iment for a spin-gravity coupling measurement. Basedon the current sensitivity, our system is already com-parable to the most sensitive system for measuring thecoupling of proton spins with Earth’s gravitational field.We have outlined the next steps for improving our co-magnetometer based on parahydrogen-induced polariza-tion and compact atomic magnetometers. These im-provements could facilitate the development of low-cost,high-precision, and robust table-top systems for long-term measurements of exotic spin-dependent interactions[3].This research was supported by the DFG KoselleckProgram and the Heising-Simons and Simons Founda-tions, the European Research Council under the Euro-pean Union’s Horizon 2020 Research and Innovative Pro-gramme under Grant agreement No. 695405 (T. W., J.W. B., and D. B.), and by the National Science Foun-dation under Grant No. PHY-1707875 (D. F. J. K.).Correspondence and requests for materials should be ad-dressed to T. W. ([email protected]). Appendix A: Spin-gravity coupling
This section presents detailed derivations of the equa-tions (1)-(3) in the main text.The Hamiltonian containing scalar, Zeeman and spin-gravity couplings can be written as [20, 29] H = H J + H Z + H G = (cid:126) (cid:88) j ; k>j J jk I j · I k + (cid:126) (cid:88) i γ i I i · B + (cid:88) i χ i I i · g . (S1)Here, H J is the scalar couplings of different nuclear spins, H Z is the Zeeman interaction, H G is the contributionfrom spin-gravity couplings, J jk is the scalar couplingbetween spins I j and I k , γ i is the gyromagnetic ratio ofspin I i , B is the bias magnetic field, χ i is the gyrogravi-tational ratio for the i th spin, and g is the gravitationalacceleration due to Earth, which is the dominant gravita-tional field in a laboratory environment. The spin-gravitycoupling has the same form as the Zeeman interaction,and can be regarded as a quasi-magnetic field with astrength of χ i g cos φ/ (cid:126) γ i , where φ is the angle betweenthe bias magnetic field and the Earth’s gravitational field. If the bias magnetic field direction is reversed, this quasi-magnetic field changes its relative sign, and thus can beextracted by monitoring the variations of the Larmor pre-cession frequencies.Consider the case of acetonitrile-2- C, which is a CH system with three equivalent protons. In the ab-sence of the Zeeman interaction and spin-gravity cou-pling, the unperturbed state | f, m f , k (cid:105) has energy [29] E ( f, k, s ) = (cid:126) J/ f ( f + 1) − k ( k + 1) − s ( s + 1)] . (S2)Here, k = (cid:80) i k i is the quantum number of the sum ofthe three equivalent proton spins, which has the valuesof 1/2 and 3/2, s is the quantum number of the Cspin and is 1/2, f is the quantum number of the totalspin angular momentum, and has the values f = 0 , k = 1 / , s = 1 /
2, and f = 1 , k = 3 / , s =1 /
2. Therefore, the zero-field J -coupling spectrum ofacetonitrile-2- C consist of two resonance lines, with oneat E (1 , / , / − E (0 , / , /
2) = J and the other oneat E (2 , / , / − E (1 , / , /
2) = 2 J . In the limit where the Zeeman energies (and spin-gravity energy) are much smaller than the scalar cou-plings, we can use the first-order perturbation theory tocalculate the shift in energy levels (higher-order effectsare considered in the next section). The eigenstates arestill those of the unperturbed scalar Hamiltonian. Wecan write the shifts in the energy levels due to Zeemaninteraction and spin-gravity coupling as [29]∆ E ( f, m f , k, s ) = (cid:104) f m f | ( H Z + H G ) | f m f (cid:105) = (cid:104) f m f | [ (cid:126) B z ( γ h k z + γ c s z ) + g cos φ ( χ h k z + χ c s z )] | f m f (cid:105) = (cid:88) m k ,m s (cid:104) ksm k m s | f m f (cid:105) [ (cid:126) B z ( γ h m k + γ c m s ) + g cos φ ( χ h m k + χ c m s )] . (S3)Here, (cid:104) ksm k m s | f m f (cid:105) is the Clebsch-Gordan coefficient. γ h and γ c are the gyromagnetic ratios for H (proton)and C, respectively. By using the selection rules of themagnetic dipole transitions [29], i.e., ∆ f = 0 , ± , ∆ m f = ± , ∆ k = 0, under the influence of magnetic field, thespectrum around J splits into two peaks, while the spec-trum around 2 J splits into six, which are shown in Fig.1 in the main text.We now calculate the form of the two frequencies ∆ ν , that we employ to realize the comagnetometer. ∆ ν isthe frequency difference of the transitions | , , / (cid:105) →| , , / (cid:105) and | , , / (cid:105) → | , − , / (cid:105) , and ∆ ν isthe frequency difference of the transitions | , , / (cid:105) →| , , / (cid:105) and | , − , / (cid:105) → | , − , / (cid:105) . Based onEq. (S3), the two frequencies ∆ ν , can be written as∆ ν ( ± ) = ( γ h + γ c ) B z ± ( χ c + χ h ) g cos φ (cid:126) , ∆ ν ( ± ) = ( γ h + 3 γ c ) B z ± ( 32 χ c + 12 χ h ) g cos φ (cid:126) . (S4)In Eqs. (S4), ± refers to reversing the magnetic field di-rection.The next step is to express the gyrogravitational ratiosof χ c and χ h in terms of the coupling constants for theproton χ p and neutron χ n . Based on the nuclear shellmodel, for odd-A nuclei, the nuclear spin I is entirely dueto the orbital motion and the intrinsic spin of the valencenucleon [36–38]. For H, there is only a proton, whosestate is 1 s / . The C nucleus has a valence neutron,whose state is 2 p / . As do most theoretical models, weassume that there is no contribution from orbital angularmomentum to the spin-gravity coupling [38], and thus χ c and χ h can be rewritten as χ h = (cid:104) S p · I h (cid:105) I h ( I h + 1) χ p = χ p ,χ c = (cid:104) S n · I c (cid:105) I c ( I c + 1) χ n = − χ n , (S5) Here, S p and S n are the valence proton and neutronspins, I h and I c are the total angular momentum of Hand C, respectively. The quantum number of the or-bital angular momentum for the valence proton of H is0, and for the valence neutron of C, it is 1. Replacing χ h and χ c in Eqs. (S4), we obtain Eqs. (1) and (2) in themain text.Based on these, the spin-precession frequency ratio R ≡ ∆ ν / ∆ ν has the form R = γ h + 3 γ c γ h + γ c ) (cid:34) − χ n + χ p ) g cos φ (cid:126) ( γ h +3 γ c ) B z − χ n + χ p ) g cos φ (cid:126) ( γ h + γ c ) B z (cid:35) . (S6)Considering the fact that γ c,h B z (cid:29) gχ n,p , and replacing γ h,c with g h,c µ N / (cid:126) , where g h ≈ . H,and g c ≈ . C, µ N is the nuclearmagneton, Eq. (S6) can be rewritten as R ≈ γ h + 3 γ c γ h + γ c ) (cid:20) − (5 χ p + 4 χ n ) g cos φ µ N B z (cid:21) . (S7)Therefore, by reversing the magnetic field, ∆ R ≡ R + −R − can be calculated, which has the form as Eq. (3)shown in the main text. Appendix B: Higher-order Zeeman effects
This section shows the energy shifts of the magneticsublevels due to higher-order Zeeman effects.Based on the previous work [31], for the CH systemwith three equivalent protons, under the influence of amagnetic field, the energy of a sublevel can be expressedas E ( f, m f , k, s ) / (cid:126) = m f γ h B z − J/ s (cid:113) ( k + 1 / J + 2 m f J ( γ c − γ h ) B z + ( γ c − γ h ) B z , (S8)where all the symbols have the same definitions as in Eqs.(S2) and (S3). Equation (S8) is valid for arbitrary fields B z , and is the same as Eq. (S2) when B z = 0.As mentioned before, ∆ ν is the frequency difference ofthe transitions | , , / (cid:105) → | , , / (cid:105) and | , , / (cid:105) →| , − , / (cid:105) . Based on Eq. (S8), the energies of thesestates can be calculated as E (0 , , / , / / (cid:126) = − J − (cid:112) J + ( γ c − γ h ) B z ,E (1 , , / , / / (cid:126) = J γ c + γ h ) B z ,E (1 , − , / , / / (cid:126) = J −
12 ( γ c + γ h ) B z , (S9) Since | , , / (cid:105) and | , − , / (cid:105) are streched states, thereare no higher-order Zeeman effects for these states.Based on Eq. (S9), and the definition of ∆ ν , we canhave ∆ ν = ( γ c + γ h ) B z , which is linear with the mag-netic field B z .We can also calculate ∆ ν , which is the frequencydifference of the transitions | , , / (cid:105) → | , , / (cid:105) and | , − , / (cid:105) → | , − , / (cid:105) . Similarly, the energy levelsfor these states are E (1 , − , / , / / (cid:126) = − γ h B z − J − (cid:112) J − J ( γ c − γ h ) B z + ( γ c − γ h ) B z ,E (2 , − , / , / / (cid:126) = 3 J −
12 ( γ c + 3 γ h ) B z ,E (1 , , / , / / (cid:126) = γ h B z − J − (cid:112) J + 2 J ( γ c − γ h ) B z + ( γ c − γ h ) B z ,E (2 , , / , / / (cid:126) = 3 J γ c + 3 γ h ) B z . (S10)Since | , , / (cid:105) and | , − , / (cid:105) are streched states, thereare no higher-order Zeeman effects for these states. Byperforming a Taylor expansion for the square root in Eq. (S10) up to the third order in the small parameter x =( γ c − γ h ) B z / J , the energy levels for states | , − , / (cid:105) and | , , / (cid:105) can be written as E (1 , − , / , / / (cid:126) = − J − (5 γ h − γ c ) B z − γ c − γ h ) B z J − γ c − γ h ) B z J ,E (1 , , / , / / (cid:126) = − J γ h − γ c ) B z − γ c − γ h ) B z J + 3( γ c − γ h ) B z J . (S11)Based on these, ∆ ν is calculated as B z (3 γ c + γ h ) / γ c − γ h ) B z / J . We find that the even-order Zeemaneffects shift the energy levels along the same direction,which in turn have no contributions in ∆ ν . 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