Asymmetric Heading Errors and its Suppression in Atomic Magnetometers
AAsymmetric Heading Errors and its Suppression in Atomic Magnetometers
Yue Chang,
1, 2, ∗ Yu-Hao Guo,
1, 2
Shuang-Ai Wan,
1, 2 and Jie Qin
1, 2, † Beijing Automation Control Equipment Institute, Beijing 100074, China Quantum Technology R & D Center of China Aerospace Science and Industry Corporation, Beijing 100074, China
Precession frequencies measured by optically pumped scalar magnetometers are dependent on therelative angle between the sensor and the external magnetic field. This dependence is known to beinduced mainly by the nonlinear Zeeman effect and the orientation-dependent light shift, resultingin the so-called heading errors if the magnetic field orientation is not well known or is not stable.In this work, we find that the linear nuclear Zeeman effect has also a significant impact on theheading errors. It not only shifts the precession frequency but causes asymmetry: the heading errorfor sensors orienting in the upper-half plane with respect to the external field is different from thecase when the sensors work in the lower-half plane. This heading error also depends on the relativedirection of the probe laser to the driving magnetic field. With a left-handed circularly-polarizedpump laser, when the probe laser is parallel to the driving field, the angular dependence of theprecession frequency is smaller when the sensor is in the upper plane. Otherwise, when they areperpendicular to each other, the heading error is smaller when the sensor is in the lower plane.Furthermore, to suppress the heading error, we propose to utilize a small magnetic field alongthe propagation direction of the pump laser. By tuning the magnitude of this auxiliary field, theheading-error curve is flattened around different angles, which can increase the accuracy in practicewhen the magnetometer works around a certain orientation angle.
Optically-pumped magnetometers have achieved highsensitivity [1–3] and have been applied in a broad rangefrom archaeology and geophysics [4–7] to fundamentalphysics [8–10]. In scalar atomic magnetometers, the mag-nitude of the external magnetic field is determined bymeasuring the precession frequency of alkali-metal atoms.This frequency is dependent on the sensor’s orientation(the pump laser’s propagation direction) with respect tothe magnetic field, resulting in the so-called heading er-rors [11–14], which is one of the major sources of accu-racy degradation especially for magnetometers operatingin the geophysical range (20 − µ T).Heading errors in atomic magnetometers have beenstudied both theoretically and experimentally [11, 14–17]. It has been shown [17] that the main contributionsto the heading error are the nonlinear Zeeman (NLZ)effects [18, 19] and the light shift (LS) [20–22] that is ori-entation dependent. In many of these studies, apart froma small correction to the Larmor frequency, the interac-tion between the external magnetic field and the nuclearspins is neglected since the nuclear magneton µ N is about3 orders smaller than the Bohr magneton µ B . However,in the geophysical field range, the linear nuclear Zee-man (NuZ) splitting is comparable with the NLZ shift.In this letter, we theoretically and experimentally studythe heading error in atomic magnetometers by includingthe NuZ effect and find that it can significantly modifythe heading errors and bring asymmetry. The setup forour study is schematically shown in Fig. 1(a), where anatomic cell containing alkali-metal atoms and buffer gas(N ) is exposed to the external magnetic field (cid:126)B = B ˆ e z that is parallel or antiparallel to the z direction. Thecircularly-polarized pump laser, whose propagation di-rection together with (cid:126)B defines the xz plane, is tiltedby an angle θ with respect to the z direction. An os- cillating magnetic field perpendicular to the pump laseris generated by RF coils to induce atomic spin polariza-tions in the xy plane, which is reconstructed by mea-suring the optical rotation of a linearly-polarized probelaser. Without loss of generality, we assume the RF fieldis in the xz plane and the probe laser is propagating par-allel or perpendicular (along the y direction) to the RFfield since only their relative direction matters. For left-handed-circularly-polarized pump laser propagating withan angle θ ∈ [ − π/ , π/ B >
0, while the head-ing error for B < B > θ to − θ , and inverting the di-rection of the (cid:126)B is equivalent to inverting the propaga-tion direction of the pump laser or changing its circularpolarization from left/right-handed to right/left-handed.Therefore, in this letter, we take the z axis as the quan-tization axis and focus on the left-handed polarizationcase with the pump laser’s orientation angle θ ∈ [0 , π/ a r X i v : . [ qu a n t - ph ] F e b (a)(c) FineStructure
HyperfineStructure pumpprobe z P r o b e P u m p P u m p xzy Probe xy R F D r i v e RF Drive (b)
FIG. 1. (a) Schematic of an atomic magnetometer fixed ona rotatable table (the gray disc). Here, an atomic cell (yel-low cubic at the center) containing alkali-metal atoms andbuffer gas is pumped by a circularly-polarized laser (red ar-row) propagating in the xz plane with a tilted angle θ to the+ z axis. The external magnetic field (cid:126)B = B ˆ e z is along the+ z axis ( B >
0) or − z axis ( B < | F m (cid:105) S and the first excited states | F m (cid:105) P with F = a, b , and m themagnetic number. The probe laser is far detuned from theD1 transition. (c) Energy level spacing between two adjacentground-state Zeeman sublevels of the alkali-metal atom. Be-sides the Larmor frequency ω L ≡ µ eff | B | and the quantumbeat revival frequency ω rev ≡ µ B / ∆ S , there is the thirdterm ω NuZ ≡ g I µ N | B | coming from the linear NuZ splitting.It is this NuZ effect leading to different precession frequenciesin the a and b hyperfine manifolds, resulting in asymmetricheading errors for B > B < tion for the alkali-metal atom is [21, 24, 25]: ∂ t ρ = − i [ H, ρ ] + L P P ρ + L SP ρ + L SS ρ. (1)Here, the Hamiltonian H = H HF + H B + H LA + H D ,where H HF is the hyperfine interaction H HF = (cid:88) m ∆ S | am (cid:105) SS (cid:104) am | + ∆ | am (cid:105) P P (cid:104) am | + (∆ − ∆ P ) | bm (cid:105) P P (cid:104) bm | (2) with ∆ S (∆ P ) the hyperfine splitting in the ground states | F m (cid:105) S (excited states | F m (cid:105) P ), F = a , b , and ∆ the de-tuning of the pump laser (see Fig. 1(b)); H B depicts theinteraction btween the spins and the external magneticfield H B = g e µ B B z S z + g I µ N B z I z (3)with (cid:126)S ( (cid:126)I ) the electron (nuclear) spin operator and g e ( g I ) the electron (nuclear) g-factor of the alkali-metalatom; H LA is the light-atom interaction H LA = − E (cid:32) √ (cid:88) σ = ± d σ (cos θ + σ ) + d z sin θ (cid:33) (4)with E the electric field of the pump laser and (cid:126)d the dipole moment of the atom while d ± ≡− ( d x ± id y ) / √
2. Note that under the rotating-wave approximation, the dipole moment d ± hasonly matrix elements between | F m (cid:105) S and | F m ± (cid:105) P [26]. The coupling to the driving field H D = g e µ B B ( S x cos θ − S z sin θ ) cos ωt induces polarization inthe xy plane. In experiments, the precession frequencyis determined by the zero crossing ω of the in-phasepart in (cid:104) S x cos θ + S z sin θ (cid:105) when the probe laser is par-allel to the RF field or out-of-phase part in (cid:104) S y (cid:105) whenthe probe laser is perpendicular to the RF field. Apartfrom the coherent dynamics, the alkali-metal atom expe-riences excited-state mixture [24, 25] L P P and quenching[24, 25] L SP caused by collisions between alkali-metalatoms in excited states and buffer gas atoms, and dis-sipation [21, 24] L SS in the ground states induced bycollisions between alkali-metal atoms .Within the geophysical range, the interaction with themagnetic field (cid:126)B can be treated as a perturbation to thehyperfine states. To the second order of H B , the energy E ( a/b, m ) (apart from a constant independent of B ) ofthe ground-state Zeeman sublevel | a / b , m (cid:105) S is E ( a/b, m ) ≈ ( ± µ eff − g I µ N ) B m ∓ ω rev m , (5)where the effective magneton µ eff ≡ ( g S µ B + g I µ N ) / (2 I + 1) and the so-called quantum-beat revival frequency ω rev ≡ µ B / ∆ S . The last termin Eq. (5) is the lowest-order NLZ splitting [18, 19].With this NLZ effect, the energy spacing between twoadjacent Zeeman sublevels depends on the magneticquantum number m , as shown in Fig. 1(c), leading toheading errors: the population in each state | F m (cid:105) S changes when θ varies and thus the measured precessionfrequency is dependent on the orientation angle. Anothercontribution to heading errors is the LS [17] that shiftsthe energy of the state | F m (cid:105) S by an amount dependingon m and θ . Apart from the NLZ effect and LS, we findthat the NuZ effect also causes heading errors. Fig. 1(c)shows that for the same m , the adjacent-states’ energylevel spacings in the a and b manifolds are different FIG. 2. (a) Difference between the precession frequency ω and the Larmor frequency ω L as a function of the tilted an-gle θ when the probe laser propagates paralelly to the RFfield for B = 50 µ T (blue solid curve in the bottom) and B = − µ T (dotted-dash red curve). Their average value (cid:2) ω +0 ( θ ) + ω − ( θ ) (cid:3) / − ω L is shown in green dotted line, whichis also the precession frequency when only considering theNuZ effect (same line for ± µ T). For comparison, the con-tributions from only the NLZ effect or the LS are also shownin the upper part, which are symmetric for ± µ T, resultingin zero average values as shown in the grey dashed horizon-tal line. Here and after, solid line are for positive B whiledotted-dashed lines are for negative B . (b) Same as (a),but for the probe laser perpendicular to the RF field. (c)-(e)Heading errors ω ( θ ) − ω (0) for different values of B . Ex-perimental data is plotted in lines with dots ( B >
0) andstars ( B < because of the linear NuZ splitting ω NuZ ≡ g I µ N | B | .As θ varies, the populations in the a and b manifoldschange, which then shifts the precession frequency.In fact, within geophysical magnetic field range, thesplitting ω NuZ is comparable or even larger than ω rev and thus cannot be neglected. For instance, for Rb ina field B = 50 µ T, the former is about 171Hz while thelatter is about 18Hz.With the experimental condition: in a magnetic shield,a 4 mm Rb cell with 700Torr N is heated to 90 Cel-sius, the pump laser’s power is about 90 µ W with thedetuning ∆ = ∆ P , the probe laser propagating parallelto the driving field is about 0 . ω ( θ ) and the Larmor frequency ω L is plottedin the lower part of Figs. 2(a) for B = ± µ T. It showsthat: (1) the precession frequency is smaller than ω L ,since in the a manifold that has more population thanthe b manifold because of the optical pumping, the linearNuZ effect contributes − ω NuZ to the precession frequencyand this contribution is larger than the NLZ effect andLS; (2) the precession frequency ω +0 ( θ ) for B > ω − ( θ ) for B < m > m < θ increases, ω +0 ( θ )increases ( ω − ( θ ) decreases) because states with smaller m and states in the b manifolds get more populated andthese states have lager (smaller) precession frequencies;(4) the heading errors for B > B < θ and θ , (cid:12)(cid:12) ω +0 ( θ ) − ω +0 ( θ ) (cid:12)(cid:12) (cid:54) = (cid:12)(cid:12) ω − ( θ ) − ω − ( θ ) (cid:12)(cid:12) . (6)To understand the asymmetric heading error, inFig. 2(a) we plot the precession frequencies from threesources: the NLZ effect, the LS, and the NuZ. We seethat both the NLZ effect and the LS lead to symmet-ric heading errors for positive and negative B , and theyaverage to the Larmor frequency. However, when con-sidering only the NuZ Effect, the precession frequencyfor positive B and negative B are the same (greendotted line), and it is just the θ -dependent averagevalue (cid:0) ω +0 + ω − ( θ ) (cid:1) / − ω L . We can also prove ana-lytically that this asymmetry in the heading errors isinduced by the NuZ effect [23]: when the driving fre-quency ω is close to the Larmor frequency ω L , the in-phase part in (cid:104) S x (cid:105) and out-of-phase part in (cid:104) S y (cid:105) arefunctions of α a,bm (cid:2) ω L − σ a,b ω NuZ − (2 m + 1) ω rev − ω (cid:3) forpositive B , while for negative B they are the samefunctions of α a,bm (cid:2) − ω L + σ a,b ω NuZ − (2 m + 1) ω rev + ω (cid:3) ,where α a,bm are coefficients dependent on the manifold a , b , and the magnetic number m , but independent of ω and B , σ a = 1 and σ b = −
1. Therefore, one ac-quires ω +0 ( θ ) + ω − ( θ ) = 2 ω L when ignoring the linearNuZ splitting ω NuZ and the heading errors for positiveand negative B are symmetric. However, when includ-ing the NuZ effect, the heading error for negative B islarger. This conclusion changes when the probe light isperpendicular to the RF field. As shown in Fig. 2(b),the contributions from the NLZ effect or the LS are sym-metric as in the parallel case, but the NuZ effect causesdifferences, leading to smaller θ -dependence for B < ω ( θ ) − ω (0) when the probe laser is parallel to the RF fieldis shown in Fig. 2(c)-(e) for B = ± µ T, ± µ T, and ± µ T, respectively, which agree with the experimentaldata. With a larger magnetic field, the heading error islarger since both the NLZ and the linear NuZ splittingsbecome larger.
FIG. 3. (a) The relative precession frequency ω − ω L with auxiliary field B a = 0, 10, 20nT for B = ± µ T. (b) The headingerror ω ( θ ) − ω (0) and (c) the derivative ∂ θ ω as functions of B a and θ for B = 50 µ T. Considering the heading-error asymmetry that hasbeen shown to be induced by the NuZ effect, one canchoose the orientation direction in the upper/lower-halfplane to the external field (cid:126)B , the polarization of thepump laser, or the direction of the probe laser with re-spect to the driving field to take advantage of the smallerangular-dependent case. Nonetheless, the heading errorstill limits the accuracy of the scalar magnetometer ifthe orientation of the external magnetic field is not wellknown or not stable. For instance, Fig. 2(c) shows thatfrom θ = π/ π/
10, the precession-frequency varia-tion is about 14Hz, corresponding to 3nT. This headingerror is yet too large for some applications such as anaeromagnetic survey. A common method to suppress theheading error is to average the measurement output froma cell with a left-handed circularly polarized pump laserand a cell with a right-handed one [11, 17]. However, thistechnically requires perfect parameter matches betweenthe two cells, and because of the asymmetry, the headingerror can not be exactly canceled in this way, and in theaverage value it increases as the tilted angle θ increased(inset in Figs. 2(a) and (b)). Therefore, we propose an-other way to suppress the heading error by utilizing anauxiliary magnetic field (cid:126)B a along the propagation direc-tion of the pump laser. Its magnitude B a is a positiveconstant that is small (much smaller than B ) so that theNLZ and the NuZ effects induced by B a can be neglected.Consequently, the precession frequency ω ( θ, B a ) can beapproximated as ω ( θ, B a ) ≈ ω ( θ, B a = 0) + sign ( B ) µ eff B a cos θ. (7)It is followed by Eq. (7) that, for θ ∈ [0 , π/ | ∂ θ ω ( θ, B a > | is smaller than | ∂ θ ω ( θ, B a = 0) | as long as µ eff B a sin θ is not larger than 2 | ∂ θ ω ( θ ) | . Asa result, the heading errors for both cases with posi-tive and negative B can be reduced. Without loss ofgenerality, we consider left-handed circular polarization of the pump laser and the probe laser parallel to theRF field. With the auxiliary field B a = 0, 10, 20nT,the relative precession frequency ω − ω L is plotted inFig. 3(a) for B = ± µ T, while other parameters arethe same as in Fig. 2. In presence of the auxiliary field B a , ω increases when B >
0, while ω decreases when B <
0, which agree with Eq. (7). Correspondingly,the angular dependence becomes smaller. The head-ing error ω ( θ ) − ω ( θ = 0) is plotted in Fig. 3(b) for B = 50 µ T with B a from 0 to 20nT. As B a increasesuntil it reaches about 11 . ω ( θ ) − ω ( θ = 0), a monotonic function of θ ,becomes flatter. For instance, the precession frequencyvariance ω ( π/ − ω (0) is 17Hz when B a = 0, andit is reduced to 1 . B a = 11 . B a we find that ω ( θ ) can be smaller than ω (0) and there exists an angle θ at which the deriva-tive ∂ θ ω ( θ ) = 0. As shown in Fig. 3(c), θ is a mono-tonic function of B a . Therefore, one can tune the magni-tude of the auxiliary field B a , so that the heading errorcurve is flattened around a demanded angle θ . For ex-ample, by tuning B a to 13nT, the variance of ω ( θ ) for θ ∈ [ π/
5, 3 π/
10] becomes about 0 . B has a small shift∆ B (the NLZ effect and NuZ effect induced by ∆ B are negligible), the precession frequency ω ( B + ∆ B )is simply ω ( B ) + µ eff ∆ B . Since µ eff ∆ B is a constantindependent of θ and B a , the angular dependence of theprecession frequency remains the same. As a result, ourproposal for suppressing the heading error using the aux-iliary field B a does not require that B is well-known.We have presented a full analysis of the asymmetricheading errors in atomic magnetometers. We theoreti-cally prove that this asymmetry is induced by the NuZeffect. Our theoretical result acquired from numericallysolving the master equation agrees with our experimen-tal data. Based on our study, one can design the mag-netometer to have it work in the smaller heading errorregime. Moreover, we propose a scheme to largely re-duce the heading errors by utilizing a small magnetic fieldalong the propagation direction of the pump laser. Themagnitude of this auxiliary field can be tuned to flattenthe heading error curve around a desired angle, whichhas promising applications for magnetometers workingaround a certain orientation.The authors acknowledge support by the National Nat-ural Science Foundation of China Grants No.61627806and No.61903045. ∗ [email protected] † [email protected][1] I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V.Romalis, Nature (London) , 596 (2003).[2] D. Budker, Nature(London) , 574 (2003).[3] D. Budker and M. Romalis, Nat. Phys. , 227 (2007).[4] K. L. Kvamme, Remote sensing in archaeology: An ex-plicitly North American perspective , 205 (2006).[5] C. Gaffney, Archaeometry , 313 (2008).[6] M. A. Dang, H.B. and M. Romalis, Applied Physics Let-ters , 151110 (2010).[7] G. Vasilakis, J. M. Brown, T. W. Kornack, and M. V.Romalis, Phys. Rev. Lett. , 261801 (2009).[8] N. Fortson, P. Sandars, and S. Barr, Phys. Today ,33 (2003).[9] J. M. Amini, C. T. Munger, and H. Gould, Phys. Rev.A , 063416 (2007).[10] B. M. Roberts, V. A. Dzuba, and V. V. Flambaum,Annu. Rev. Nucl. Part. Sci. , 63 (2015).[11] T. Yabuzaki and T. Ogawa, Journal of Applied Physics , 1342 (1974), https://doi.org/10.1063/1.1663412.[12] E. Alexandrov, Physica Scripta , 27 (2003).[13] A. Ben-Kish and M. V. Romalis, Phys. Rev. Lett. ,193601 (2010).[14] G. Bao, A. Wickenbrock, S. Rochester, W. Zhang, andD. Budker, Phys. Rev. Lett. , 033202 (2018).[15] C. Hovde, B. Patton, O. Versolato, E. Corsini,S. Rochester, and D. Budker, in Defense + Commer-cial Sensing (2011).[16] S. Colombo, V. Dolgovskiy, T. Scholtes, Z. D. Gruji´c,V. Lebedev, and A. Weis, Applied Physics B: Lasersand Optics , 35 (2017).[17] G. Oelsner, V. Schultze, R. IJsselsteijn, F. Wittk¨amper,and R. Stolz, Phys. Rev. A , 013420 (2019).[18] K. Jensen, V. M. Acosta, J. M. Higbie, M. P. Ledbet-ter, S. M. Rochester, and D. Budker, Phys. Rev. A ,023406 (2009).[19] W. Chalupczak, A. Wojciechowski, S. Pustelny, andW. Gawlik, Phys. Rev. A , 023417 (2010).[20] W. Happer, Rev. Mod. Phys. , 169 (1972).[21] S. Appelt, A. B.-A. Baranga, C. J. Erickson, M. V. Ro-malis, A. R. Young, and W. Happer, Phys. Rev. A ,1412 (1998).[22] Y. Chang, Y.-H. Guo, and J. Qin, Phys. Rev. A ,063411 (2019).[23] See Supplemental Material at [] for demonstration of the system symmetry, the heading-error asymmetry in-dunced by the NuZ effect in details, and more experimen-tal data to compare with the theoretical result, which alsocontains Refs. [21,22,24-28].[24] W. Happer, Y.-Y. Jau, and T. Walker, Optically PumpedAtoms (Wiley, New York, 2010).[25] B. Lancor and T. G. Walker, Phys. Rev. A , 043417(2010).[26] D. F. Walls and G. J. Milburn, Quantum Optics ,SpringerLink: Springer e-Books (Springer Berlin, 2008).[27] C. Gardiner and P. Zoller,
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Supplemental Material:Asymmetric heading errors and its suppression in atomic magnetometers
In this supplemental material, we provide detailed proof of the system’s symmetry, explain analytically how thelinear nuclear Zeeman splitting induces asymmetry in the heading errors, and show more experimental date to supportour theory.
SYMMETRIES IN THE SYSTEMMaster equation
The master equation for the alkali-metal atoms is ∂ t ρ = − i [ H, ρ ] + L P P ρ + L SP ρ + L SS ρ, (SM1)where the Hamiltonian H = H HF + H B + H LA + H D . The light-atom interaction [26] H LA = − E (cid:32) √ (cid:88) σ = ± d σ (cos θ + σ ) + d z sin θ (cid:33) = − Ω [(cos θ + 1) A + + (cos θ − A − + ( A + A − ) sin θ ] + H.c., (SM2)where the Rabi frequency Ω = E √ (cid:104) / (cid:107) d (cid:107) / (cid:105) , (SM3) A ± = |±(cid:105) SP (cid:104)±| , and A ± = |∓(cid:105) SP (cid:104)±| , with the notations of the fine states |±(cid:105) S ( P ) ≡ (cid:12)(cid:12) S / (cid:0) P / (cid:1) , ± (cid:11) . Thedissipation from the excited-state mixture resulting from collisions between alkali-metal atoms and buffer gas isdepicted by [24, 25] L P P ρ = γ Mix (cid:16) (cid:126)J P · ρ (cid:126)J P − (cid:110) ρ, (cid:126)J P · (cid:126)J P (cid:111)(cid:17) , (SM4)where γ Mix is the collision rate of excited atoms with the buffer gas, J P is the angular momentum of the excited atomsdefined as J Px = ( | + (cid:105) P P (cid:104)−| + |−(cid:105) P P (cid:104) + | ) / J y = ( | + (cid:105) P P (cid:104)−| − |−(cid:105)
P P (cid:104) + | ) / i , and J z = ( | + (cid:105) P P (cid:104) + | − |−(cid:105) P P (cid:104)−| ) / L SP ρ = Γ Q (cid:88) j =0 ± , ± A j ρA † j − (cid:110) ρ, A † j A j (cid:111) , (SM5)where Γ Q is the quenching rate. Here, the spontaneous decay is neglected since its rate is much smaller than Γ Q under our experimental condition (several hundreds Torr N ). Dissipation L SS ρ in the ground-state is [21, 24] L SS ρ = ( γ SD + γ SE ) (cid:16) (cid:126)S · ρ(cid:126)S − (cid:110) ρ, (cid:126)S · (cid:126)S (cid:111)(cid:17) + 2 γ SE (cid:104) S z (cid:105) ( S + ρS − − S − ρS + + { ρ, S z } )+2 γ SE (cid:104) S + (cid:105) (cid:18) S − ρS z − S z ρS − + { ρ, S − } (cid:19) + H.c., (SM6)where γ SD ( γ SE ) is the spin-destruction (exchange) rate. Linear response
The master equation (1) can be written as ∂ t ρ = ( L + L ) ρ , where L ρ = − i [ H HF + H B + H LA , ρ ] + L P P ρ + L SP ρ + L SS ρ (SM7)and L ρ = − i [ H D , ρ ] . (SM8)Since the RF field is weak, one can treat L as a perturbation and keep it to the first order [28]. As a result, thedensity matrix ρ = ρ + ρ (+)1 e iωt + ρ ( − )1 e − iωt , (SM9)where in the long-term limit, L ρ = 0, ( L ∓ iω ) ρ ( ± )1 + L ρ = 0 , (SM10)and ρ ( − )1 = ρ (+) † . Note that in L ρ (+)1 , the collission induced dissipation L SS ρ (+)1 is L SS ρ (+)1 = ( γ SD + γ SE ) (cid:16) (cid:126)S · ρ (+)1 (cid:126)S − (cid:110) ρ (+)1 , (cid:126)S · (cid:126)S (cid:111)(cid:17) + 2 γ SE Tr ( S z ρ ) (cid:16) S + ρ (+)1 S − − S − ρ (+)1 S + + (cid:110) ρ (+)1 , S z (cid:111)(cid:17) +2 γ SE (cid:68) S + ρ (+)1 (cid:69) (cid:18) S − ρ S z − S z ρ S − + { ρ , S − } (cid:19) +2 γ SE (cid:68) S − ρ (+)1 (cid:69) (cid:18) S z ρ S + − S + ρ S z + { ρ , S + } (cid:19) . (SM11)When the probe laser is parallel to the RF field, the precession frequency ω is determined by the zero-crossing ofof the in-phase part S in ( ω ) ≡ (cid:104) Tr (cid:16) ( S x cos θ − S z sin θ ) ρ (+)1 (cid:17)(cid:105) = 2Re (cid:104) Tr (cid:16) ( S x cos θ − S z sin θ ) ρ ( − )1 (cid:17)(cid:105) , while whenthe probe laser is perpendicular to the RF field, ω is determined by the zero-crossing of of the out-of-phase part S out ( ω ) ≡ (cid:104) Tr (cid:16) S y ρ (+)1 (cid:17)(cid:105) = − (cid:104) Tr (cid:16) S y ρ ( − )1 (cid:17)(cid:105) .To prove ω is an even function of the tilted angle θ , we perform a rotation of angle 2 θ along the y axis to theatomic system. Under this unitary transformation, the light-atom interaction H LA ( θ ) becomes H LA ( − θ ), the drivingterm H D ( θ ) becomes − H D ( − θ ), while other parts in the master equation are invariant. As a result, the first-orderdensity matrix ρ (+)1 = − ( L − iω ) − L ρ with positive frequency changes to − ρ (+)1 ( − θ ), and thus S in / out ( θ, ω ) = S in / out ( − θ, ω ). Therefore, the precession frequency ω is invariant when changing θ to − θ .Similarly, performing a rotation of angle π along the y axis and a Z transformation to the excited states so thateach excited state has an additional global phase π , the interaction to the external magnetic field H B ( B ) changesto H B ( − B ), and the light-matter interaction in Eq. (4) becomes H LA = − E (cid:32) √ (cid:88) σ = ± d − σ (cos θ + σ ) + d z sin θ (cid:33) , (SM12)i.e., the polarization of the pump light changes. The driving H D changes to − H D . Therefore, for the precessionfrequency ω that is determined by the zero crossings of the in-phase/out-of-phase part S in / out is invarient whenchanging the pump laser’s polarization and at the same time inverting the magnetic field B , or equivalently, changingthe polarization of the circularly polarized pump laser is the same as inverting the magnetic field B . NUCLEAR ZEEMAN EFFECT INDUCED ASYMMETRYAdiabatic elimination of the excited states
In the steady state ρ , the population in the excited states are negligible because their decay rates is much largerthan the Rabi frequency. Therefore, in the zero-order master equation ∂ t ρ = L ρ , we can adiabatic eliminate theexcited state and acquire an effective master equation in the ground-state subspace [22, 27]. For this purpose, werewrite the Lindblad operator L as L = L (0)0 + L (1)0 , where L (0)0 ρ = − i [ H hf + H B , ρ ] + L P P ρ + L SP ρ + L SS ρ (SM13)and L (1)0 ρ = − i [ H LA , ρ ] , (SM14)and define two projection operators P ρ = (cid:88) F m F m | F m (cid:105) SS (cid:104) F m | ρ | F m (cid:105) SS (cid:104) F m | (SM15)and Q = 1 − P . Consequently, we have ∂ t P ρ = PL (0)0 P ρ + PL (0)0 Q ρ + PL (1)0 Q ρ (SM16)and ∂ t Q ρ = Q L (0)0 Q ρ + Q L (1)0 P ρ + Q L (1)0 Q ρ. (SM17)The solution of Q ρ can be formally written as Q ρ ( t ) = (cid:90) t e Q (cid:16) L (0)0 + L (1)0 (cid:17) ( t − t (cid:48) ) Q L (1)0 P ρ ( t (cid:48) ) dt (cid:48) , (SM18)and thus to the second order, the motion equation for ρ ( g ) ≡ P ρ is ∂ t ρ ( g ) ( t ) ≈ PL (0)0 ρ ( g ) ( t ) + PL (1)0 (cid:90) t e Q L (0)0 ( t − t (cid:48) ) Q L (1)0 ρ ( g ) ( t (cid:48) ) dt (cid:48) + PL (0)0 (cid:90) t e Q L (0)0 ( t − t (cid:48) ) (cid:32) (cid:90) t − t (cid:48) dt (cid:48)(cid:48) e −Q L (0)0 t (cid:48)(cid:48) Q L (1)0 e Q L (0)0 t (cid:48)(cid:48) (cid:33) Q L (1)0 ρ ( g ) ( t (cid:48) ) dt (cid:48) = L eff ρ ( g ) ( t ) , (SM19)where the effective Lindblad operator L eff ρ ( g ) ≡ PL ρ ( g ) + PL Q L Q L Q L Q L ρ ( g ) − PL Q L Q L ρ ( g ) . (SM20)The imaginary part in the last two term in Eq. (SM20) of L eff gives the LS.The Lindblad operator L eff shown in Eq. (SM20) can be obtained numerically in the superspace. The dimensionof the effective master equation in the superspace is (4 I + 2) . In principle, one needs to solve (4 I + 2) nonlinearequations since the Zeeman sublevels are mixed due to collisions in the ground states and pumping-induced dissipationsas shown in the last two terms in Eq. (SM20). In the geophysical field range, these mixing rates are much smallerthan the Larmor frequency ω L under the usual experimental condition. Therefore, one can ignore the off-diagonalterms in the steady state ρ ( g )0 and consider only 4 I + 2 equations. This largely accelerates the numerical calculation.With ρ ( g )0 , the density matrix to the first order is acquired through replacing L by L eff in Eq. (SM10) as (cid:16) L eff ∓ iω (cid:17) ρ ( ± )1 + L ρ ( g )0 = 0 , (SM21)where the electron spin operators in L are in the ground-state subspace. Nuclear Zeeman effect
The driving Hamiltonian ∝ S x cos θ − S z sin θ . Since ρ has only diagonal terms, in [ S z , ρ ] only terms | F m (cid:105) SS (cid:104) F m | with F (cid:54) = F exists which has large energy ∆ S in the superspace. Thus, we can neglect the term S z sin θ in H D . Wecan further apply the rotating-wave approximation and ignore the term S z sin θ in S in ( ω ) since the mixing rate of | F m (cid:105) SS (cid:104) F m | and | F (cid:48) m (cid:105) SS (cid:104) F (cid:48) m ± | is much smaller that the Larmor frequency ω L .Under our experimental conditions, S in / out ( ω ) = 0 has two solutions around ± ω L , respectively. Here, we focus onthe solution ω around + ω L . From Eq. (SM21) we have S in ( ω ) = − (cid:20) Tr (cid:18) S x (cid:16) L eff ∓ iω (cid:17) − L ρ (cid:19)(cid:21) cos θ (SM22)and S out ( ω ) = ∓ (cid:20) Tr (cid:18) S y (cid:16) L eff ∓ iω (cid:17) − L ρ (cid:19)(cid:21) cos θ. (SM23)For B >
0, we take the minus signs in the parentheses in Eqs. (SM22) and (SM23). Thus, in L ρ , only terms | a, m (cid:105) SS (cid:104) a, m + 1 | and | b, m + 1 (cid:105) SS (cid:104) b, m | need to be taken into account. In these basis, the B -dependent diagonalterms of L eff are E ( a, m + 1) − E ( a, m ) ≈ ω L − ω NuZ − (2 m + 1) ω rev , (SM24) E ( b, m ) − E ( b, m + 1) ≈ ω L + ω NuZ − (2 m + 1) ω rev . (SM25)If we ignore the small modification of the hyperfine states resulting from the interaction to the external field (cid:126)B ,which is an appropriate approximation, S in / out ( ω ) is a function of α a,bm (cid:2) ω L − σ a,b ω NuZ − (2 m + 1) ω rev − ω (cid:3) , where α a,bm is a coefficient dependent on the manifold a , b , and the magnetic number m , but independent of ω and B , σ a = 1 and σ b = −
1. When B <
0, we take the plus signs in Eqs. (SM22) and (SM23). Similarly, in L ρ , onlyterms | a, m (cid:105) SS (cid:104) a, m + 1 | and | b, m + 1 (cid:105) SS (cid:104) b, m | need to be considered and S in / out ( ω ) is the same/opposite functionof α a,bm (cid:2) − ω L + σ a,b ω NuZ − (2 m + 1) ω rev + ω (cid:3) . Denoting the solution to S in / out ( ω ) = 0 for B > ω +0 , in / out and ω − , in / out for B <
0, we have ω +0 , in / out + ω − , in / out = 2 ω L (SM26)if the NuZ splitting ω NuZ is ignored. In this case, the heading errors for B > B < ω +0 , in / out ( θ ) − ω +0 , in / out ( θ ) = ω − , in / out ( θ ) − ω − , in / out ( θ ) . (SM27)However, the existence of ω NuZ does not only shifts (cid:16) ω +0 , in / out + ω − , in / out (cid:17) / − ω L from zero, but breaks the symmetry(SM27), as shown in Figs. (2) and (3). FIG. SM1. Schematic of the contributions S a ( ω ) (solid lines with larger amplitude) and S b ( ω ) (solid lines with smalleramplitude) to the in-phase signal S in ( ω ) and out-of-phase signal S out ( ω ) from the manifolds a and b , respectively for positive(a) or negative (b) B . The crossing with the dotted line determines the precession frequency. (c) Difference between theprecession frequency ω ± , in / out ( ω ± a ) (determined by the zero crossing of S in / out ( S a ) ) and the Larmor frequency ω L , as a functionof the tilted angle θ . Compared with ω ± a , the precession frequency ω ± , in is shifted downward (upward) by the contribution fromthe b -manifold S b , resulting in asymmetric heading errors. To further understand the emerging of this asymmetric heading error, we approximate S in / out ( ω ) ≈− g e µ B B (cid:2) S a ( ω ) ± S b ( ω ) (cid:3) cos θ with the plus sign for S in ( ω ) and minus sign for S out ( ω ), where S a ( ω ) = Re (cid:20) Tr (cid:18) S + (cid:16) L eff − sign ( B ) iω (cid:17) − L − ρ ( a )0 (cid:19)(cid:21) , (SM28)0 S b ( ω ) = Re (cid:20) Tr (cid:18) S − (cid:16) L eff − sign ( B ) iω (cid:17) − L ρ ( b )0 (cid:19)(cid:21) . (SM29)Here, L ± ρ = − i [ S ± , ρ ] and ρ ( a/b )0 is the density matrix projected in the a/b -manifold: ρ ( F = a,b )0 = (cid:80) mm (cid:48) | F m (cid:105) SS (cid:104) F m | ρ | F m (cid:48) (cid:105) SS (cid:104) F m (cid:48) | . The schematics of S a/b ( ω ) are shown in Figs. SM1(a) and SM1(b). If weignore the b -manifold, i.e., the precession frequency denoted as ω ± a ( θ ) is determined by the solution to the equation S a ( ω ) = 0, from the analysis in the last paragraph we have the solutions ω ± a ( θ ) for positive and negative B satisfy[ ω + a ( θ ) + ω − a ( θ )] / ω L − ω NuZ . Namely, the heading errors for B > B < S b ( ω ) from the b -manifold shifts the zero-crossings of S in ( ω ) ( S out ( ω )) tothe left (right) of ω ± a ( θ ). The larger the tilted angle θ is, the more the b -manifold is populated, thus the more theshift is. Therefore, compared to ω ± a ( θ ), the precession frequencies ω +0 , in ( θ ) and ω − , in ( θ ) are shifted downward and theheading error in ω +0 , in ( θ ) is smaller; while the precession frequencies ω +0 , out ( θ ) and ω − , out ( θ ) are shifted upward andthe heading error in ω − , out ( θ ) is smaller, as shown in Fig. SM1(c). Lower pump power
When the power of the pump laser is lower, the heading errors for both B > B < θ varies because of the smaller Rabifrequency. The precession frequencies determined by S in ( ω ) = 0 and the corresponding heading errors for the pumplaser’s power 55 µ W are shown in Fig. SM2, where the experimental data shown in lines with red circles ( B >
0) andblue stars ( B < ± ± ± µ T. These deviations can notbe determined accurately in our experiment, but they are small so that the NLZ effect and the NuZ effect inducedby them can be ignore. Therefore, the heading errors ω ( θ ) − ω (0) are the same as in the B = ± ± ± µ Tcases.
FIG. SM2. (a) Difference between the precession frequency ω ± and the Larmor frequency ω L as a function of the tilted angle θ when the probe laser propagates along the RF field’s direction. The solid curves are for B >
0, while the dotted-dash redcurves are for B <
0. (b)-(d) Heading errors ω ( θ ) − ω (0) for different values of B . Experimental data is plotted in dots( B >
0) and stars ( B <<