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Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers
<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers</title> <!--Generated on Wed Feb 19 04:08:20 2025 by LaTeXML (version 0.8.8) http://dlmf.nist.gov/LaTeXML/.--> <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script 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ltx_tag_ref">II </span>Theoretical Background</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S2.SS1" title="In II Theoretical Background ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">II.1 </span>Bell-Bloom optical pumping</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S2.SS2" title="In II Theoretical Background ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">II.2 </span>Heading error analysis</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S3" title="In Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">III </span>Experiment setup and measurement scheme</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S4" title="In Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">IV </span>Identifying and suppressing heading errors</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S4.SS1" title="In IV Identifying and suppressing heading errors ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">IV.1 </span>LZE-induced heading error</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S4.SS2" title="In IV Identifying and suppressing heading errors ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">IV.2 </span>NLZE-induced heading error and its suppression scheme</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S5" title="In Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">V </span>Conclusion</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S6" title="In Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">VI </span>Acknowledgments</span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <h1 class="ltx_title ltx_title_document">Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">S.-Q. Liu </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation">Department of Precision Machinery and Precision Instrumentation, Key Laboratory of Precision Scientific Instrumentation of Anhui Higher Education Institutes, University of Science and Technology of China, Hefei 230027, China </span></span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">X.-K. Wang </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation">Department of Precision Machinery and Precision Instrumentation, Key Laboratory of Precision Scientific Instrumentation of Anhui Higher Education Institutes, University of Science and Technology of China, Hefei 230027, China </span></span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">X.-D. Zhang </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation">College of Physics and Optoelectronic Engineering, Shenzhen University, Shenzhen 518060, China </span> <span class="ltx_contact ltx_role_affiliation">Beijing Computational Science Research Center, Beijing 100193, China </span></span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">W. Xiao </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_email"><a href="mailto:xiao%24%CB%99%24wei@pku.edu.cn">xiao$˙$wei@pku.edu.cn</a> </span> <span class="ltx_contact ltx_role_affiliation">State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Electronics, and Center for Quantum Information Technology, Peking University, Beijing 100871, China </span> <span class="ltx_contact ltx_role_affiliation">MIIT Key Laboratory of Complex-field Intelligent Sensing, Advanced Research Institute of Multidisciplinary Science, Beijing Institute of Technology, Beijing 100081, China </span></span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">D. Sheng </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_email"><a href="mailto:dsheng@ustc.edu.cn">dsheng@ustc.edu.cn</a> </span> <span class="ltx_contact ltx_role_affiliation">Department of Precision Machinery and Precision Instrumentation, Key Laboratory of Precision Scientific Instrumentation of Anhui Higher Education Institutes, University of Science and Technology of China, Hefei 230027, China </span> <span class="ltx_contact ltx_role_affiliation">Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China </span></span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id1.id1">Heading errors of atomic magnetometers refer to the dependence of measurement results on the sensor orientation with respect to the external magnetic field. There are three main sources of such errors: the light shift effect, the linear nuclear-spin Zeeman effect, and the nonlinear Zeeman effect. In this work, we suppress the former two effects by using the Bell-Bloom optical pumping method and probe the atomic signals while the pumping beam is off, and focus on the heading error induced by nonlinear Zeeman effect while the sensor operates in the geomagnetic field range. We demonstrate several schemes to suppress this remaining heading error within 1 nT using a single magnetometer or a comagnetometer. In the magnetometer system, two schemes are developed to average out the horizontal atomic polarization in space or in time, respectively. In the comagnetometer system, we combine the simultaneously measured Larmor frequencies of two different kinds of alkali atoms to either suppress the heading error or extract the orientation of the pumping beam relative to the bias field.</p> </div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">I </span>Introduction</h2> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.1">Atomic scalar magnetometers directly measure the magnitude of the bias magnetic field using the Zeeman effect. For sensors based on alkali vapors, the sensor noise has reached fT/Hz<sup class="ltx_sup" id="S1.p1.1.1"><span class="ltx_text ltx_font_italic" id="S1.p1.1.1.1">1/2</span></sup> level <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib1" title="">1</a>, <a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib2" title="">2</a>]</cite>. Together with the intrinsic stability of atom transitions, these sensors are widely applied in different research fields. For example, atomic scalar magnetometers have been used to calibrate fluxgates in space science <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib3" title="">3</a>, <a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib4" title="">4</a>]</cite>, enable out-of-shield biomagnetic field detections <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib5" title="">5</a>, <a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib6" title="">6</a>]</cite>, monitor the bias field fluctuations on the pT level for precision measurements <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib7" title="">7</a>, <a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib8" title="">8</a>]</cite>, and serve as the fundamental units in comagnetometers based on comparing atomic spin precession frequencies <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib9" title="">9</a>, <a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib10" title="">10</a>]</cite>. However, an important factor to limit the accuracy of the scalar magnetometer readout is its heading error, where the measurement results correlate with the relative angle between the sensor and the bias field. It is important to both identify and suppress such an error in practical applications.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.1">A common source of the heading error is the light shift effect <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib11" title="">11</a>, <a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib12" title="">12</a>]</cite>, which is from either real or virtual atomic transitions <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib13" title="">13</a>, <a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib14" title="">14</a>]</cite> due to atom-photon interactions. Light shifts due to virtual transitions can be decomposed to several components according to symmetries <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib15" title="">15</a>]</cite>, among which the vector part <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib16" title="">16</a>]</cite> is the most important one in experiments conducted in atomic cells filled with quenching gases. The vector light shift is due to the interaction between atoms with the circular polarization component of a detuned beam, and its effect is equivalent to an effective magnetic field along the beam propagation direction. Experimental methods to circumvent this problem include locking beams on resonance, employing atomic-alignment-based magnetometry using only linearly polarized beams <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib7" title="">7</a>, <a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib17" title="">17</a>]</cite>, and averaging the beam polarization out in time <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib18" title="">18</a>]</cite> or space <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib12" title="">12</a>]</cite>. In the past decade, all-optical free-induction-decay (FID) magnetometers were developed <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib19" title="">19</a>, <a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib20" title="">20</a>, <a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib5" title="">5</a>]</cite>, where the pumping and probe stages are separated in time so that the magnetometer works in the pulse mode and the light shift from pumping beams is absent when the signal is recorded.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.1">The other main source of the heading error is due to the non-zero nuclear spin. For alkali atoms, the linear Zeeman effect (LZE) of the nuclear spin makes a slight difference in the absolute value of atomic gyromagnetic ratios for atoms in different ground hyperfine states. This effect can induce heading errors if the sensor signal has contributions from both hyperfine states, and the ratio of the two contributions changes with the sensor orientation. This is a serious problem for certain classes of magnetometers <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib21" title="">21</a>]</cite>. For FID magnetometers, an experiment scheme based on dual orthogonal probe beams is developed to suppress this effect <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib22" title="">22</a>]</cite> by taking advantage of the opposite signs of atomic gyromagnetic ratios in the two ground hyperfine states. Another method based on a double-frequency fitting function is also proposed to solve this problem <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib21" title="">21</a>]</cite>. A more direct way to suppress this effect is to selectively polarize atoms onto the upper hyperfine state. This can be achieved using Bell-Bloom optical pumping method <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib23" title="">23</a>]</cite>, which is due to the facts that the Zeeman transition line width is relatively narrow compared with the separation between resonant frequencies of the two ground states, and only the upper hyperfine state has a dark state in optical pumping processes.</p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.1">Even alkali atoms are all pumped to the upper hyperfine state, the nonlinear Zeeman effect (NLZE) can still lead to an additional energy shift that is dependent on the specific Zeeman sublevel. In this way, the dependence of the measurement results on the sensor orientation remains. While experiment schemes to directly cancel the NLZE by light shifts <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib24" title="">24</a>]</cite> have been developed, the main strategy in the literature to suppress NLZE is based on symmetry considerations so that the net atomic orientation in the longitudinal direction is nulled. This has been implemented by either modulating the atomic polarization using electric-optical modulators <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib18" title="">18</a>]</cite> or making measurements based on atomic alignment <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib17" title="">17</a>]</cite>. Recently, a systematic study of this effect in FID magnetometers has been performed by Romalis’ group <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib22" title="">22</a>]</cite>, and a new method is brought up in that work, which extracts the sensor orientation from the ratio between the longitudinal and transverse atomic polarization, and corrects the NLZE effect according to the derived formula.</p> </div> <div class="ltx_para" id="S1.p5"> <p class="ltx_p" id="S1.p5.1">In this work, we study the heading error of a compact Rb FID magnetometer, where the signal is based on atomic orientations directly generated by an amplitude-modulated pumping beam. Besides identifying sources of the measured heading error, we also develop generic methods in two different systems to suppress the dominant NLZE-induced heading error, which is controlled to be below 1 nT when the bias field is in the geomagnetic field range. Following this introduction, Sec. II introduces theoretical background of this work, Sec. III describes the sensor setup and measurement scheme, Sec. IV focuses on the sources of heading errors and experiment schemes to suppress the main source, and Sec. V concludes the paper.</p> </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">II </span>Theoretical Background</h2> <div class="ltx_para" id="S2.p1"> <p class="ltx_p" id="S2.p1.3">In this section and the rest of the paper, we are considering a general case that the bias field is along the <math alttext="z" class="ltx_Math" display="inline" id="S2.p1.1.m1.1"><semantics id="S2.p1.1.m1.1a"><mi id="S2.p1.1.m1.1.1" xref="S2.p1.1.m1.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S2.p1.1.m1.1b"><ci id="S2.p1.1.m1.1.1.cmml" xref="S2.p1.1.m1.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.1.m1.1c">z</annotation><annotation encoding="application/x-llamapun" id="S2.p1.1.m1.1d">italic_z</annotation></semantics></math> direction, and the pumping beam is aligned in the <math alttext="xz" class="ltx_Math" display="inline" id="S2.p1.2.m2.1"><semantics id="S2.p1.2.m2.1a"><mrow id="S2.p1.2.m2.1.1" xref="S2.p1.2.m2.1.1.cmml"><mi id="S2.p1.2.m2.1.1.2" xref="S2.p1.2.m2.1.1.2.cmml">x</mi><mo id="S2.p1.2.m2.1.1.1" xref="S2.p1.2.m2.1.1.1.cmml"></mo><mi id="S2.p1.2.m2.1.1.3" xref="S2.p1.2.m2.1.1.3.cmml">z</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.2.m2.1b"><apply id="S2.p1.2.m2.1.1.cmml" xref="S2.p1.2.m2.1.1"><times id="S2.p1.2.m2.1.1.1.cmml" xref="S2.p1.2.m2.1.1.1"></times><ci id="S2.p1.2.m2.1.1.2.cmml" xref="S2.p1.2.m2.1.1.2">𝑥</ci><ci id="S2.p1.2.m2.1.1.3.cmml" xref="S2.p1.2.m2.1.1.3">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.2.m2.1c">xz</annotation><annotation encoding="application/x-llamapun" id="S2.p1.2.m2.1d">italic_x italic_z</annotation></semantics></math> plane with an angle <math alttext="\theta" class="ltx_Math" display="inline" id="S2.p1.3.m3.1"><semantics id="S2.p1.3.m3.1a"><mi id="S2.p1.3.m3.1.1" xref="S2.p1.3.m3.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S2.p1.3.m3.1b"><ci id="S2.p1.3.m3.1.1.cmml" xref="S2.p1.3.m3.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.3.m3.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S2.p1.3.m3.1d">italic_θ</annotation></semantics></math> relative to the bias field direction.</p> </div> <section class="ltx_subsection" id="S2.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">II.1 </span>Bell-Bloom optical pumping</h3> <div class="ltx_para" id="S2.SS1.p1"> <p class="ltx_p" id="S2.SS1.p1.1">As first pointed out by Bell and Bloom <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib23" title="">23</a>]</cite>, transverse atomic polarization can be directly generated by the synchronized optical pumping method. In this work, we focus on pumping beams that are turned on and off by acousto-optical modulators (AOMs). The electron spin polarization <math alttext="\bm{P}" class="ltx_Math" display="inline" id="S2.SS1.p1.1.m1.1"><semantics id="S2.SS1.p1.1.m1.1a"><mi id="S2.SS1.p1.1.m1.1.1" xref="S2.SS1.p1.1.m1.1.1.cmml">𝑷</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.1.m1.1b"><ci id="S2.SS1.p1.1.m1.1.1.cmml" xref="S2.SS1.p1.1.m1.1.1">𝑷</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.1.m1.1c">\bm{P}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.1.m1.1d">bold_italic_P</annotation></semantics></math> can be described by the Bloch equation</p> <table class="ltx_equation ltx_eqn_table" id="S2.E1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\frac{d\bm{P}}{dt}=-\gamma\mathbf{B}\times\bm{P}+R_{op}(t)(\bm{s}-\bm{P})-% \Gamma_{1,0}P_{z}\hat{z}-\Gamma_{2,0}(P_{x}\hat{x}+P_{y}\hat{y})," class="ltx_Math" 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italic_γ bold_B × bold_italic_P + italic_R start_POSTSUBSCRIPT italic_o italic_p end_POSTSUBSCRIPT ( italic_t ) ( bold_italic_s - bold_italic_P ) - roman_Γ start_POSTSUBSCRIPT 1 , 0 end_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT over^ start_ARG italic_z end_ARG - roman_Γ start_POSTSUBSCRIPT 2 , 0 end_POSTSUBSCRIPT ( italic_P start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT over^ start_ARG italic_x end_ARG + italic_P start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT over^ start_ARG italic_y end_ARG ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(1)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p1.6">where <math alttext="\bm{s}" class="ltx_Math" display="inline" id="S2.SS1.p1.2.m1.1"><semantics id="S2.SS1.p1.2.m1.1a"><mi id="S2.SS1.p1.2.m1.1.1" xref="S2.SS1.p1.2.m1.1.1.cmml">𝒔</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.2.m1.1b"><ci id="S2.SS1.p1.2.m1.1.1.cmml" xref="S2.SS1.p1.2.m1.1.1">𝒔</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.2.m1.1c">\bm{s}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.2.m1.1d">bold_italic_s</annotation></semantics></math> is the pumping beam photon spin <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib25" title="">25</a>]</cite>, <math alttext="\Gamma_{1(2),0}" class="ltx_Math" display="inline" id="S2.SS1.p1.3.m2.3"><semantics id="S2.SS1.p1.3.m2.3a"><msub id="S2.SS1.p1.3.m2.3.4" xref="S2.SS1.p1.3.m2.3.4.cmml"><mi id="S2.SS1.p1.3.m2.3.4.2" mathvariant="normal" xref="S2.SS1.p1.3.m2.3.4.2.cmml">Γ</mi><mrow id="S2.SS1.p1.3.m2.3.3.3.3" xref="S2.SS1.p1.3.m2.3.3.3.4.cmml"><mrow id="S2.SS1.p1.3.m2.3.3.3.3.1" xref="S2.SS1.p1.3.m2.3.3.3.3.1.cmml"><mn 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id="S2.SS1.p1.3.m2.3.3.3.4.cmml" xref="S2.SS1.p1.3.m2.3.3.3.3"><apply id="S2.SS1.p1.3.m2.3.3.3.3.1.cmml" xref="S2.SS1.p1.3.m2.3.3.3.3.1"><times id="S2.SS1.p1.3.m2.3.3.3.3.1.1.cmml" xref="S2.SS1.p1.3.m2.3.3.3.3.1.1"></times><cn id="S2.SS1.p1.3.m2.3.3.3.3.1.2.cmml" type="integer" xref="S2.SS1.p1.3.m2.3.3.3.3.1.2">1</cn><cn id="S2.SS1.p1.3.m2.1.1.1.1.cmml" type="integer" xref="S2.SS1.p1.3.m2.1.1.1.1">2</cn></apply><cn id="S2.SS1.p1.3.m2.2.2.2.2.cmml" type="integer" xref="S2.SS1.p1.3.m2.2.2.2.2">0</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.3.m2.3c">\Gamma_{1(2),0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.3.m2.3d">roman_Γ start_POSTSUBSCRIPT 1 ( 2 ) , 0 end_POSTSUBSCRIPT</annotation></semantics></math> is the longitudinal (transverse) depolarization rate in the absence of light, <math alttext="R_{op}(t)" class="ltx_Math" display="inline" id="S2.SS1.p1.4.m3.1"><semantics id="S2.SS1.p1.4.m3.1a"><mrow 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xref="S2.SS1.p1.4.m3.1.2"><times id="S2.SS1.p1.4.m3.1.2.1.cmml" xref="S2.SS1.p1.4.m3.1.2.1"></times><apply id="S2.SS1.p1.4.m3.1.2.2.cmml" xref="S2.SS1.p1.4.m3.1.2.2"><csymbol cd="ambiguous" id="S2.SS1.p1.4.m3.1.2.2.1.cmml" xref="S2.SS1.p1.4.m3.1.2.2">subscript</csymbol><ci id="S2.SS1.p1.4.m3.1.2.2.2.cmml" xref="S2.SS1.p1.4.m3.1.2.2.2">𝑅</ci><apply id="S2.SS1.p1.4.m3.1.2.2.3.cmml" xref="S2.SS1.p1.4.m3.1.2.2.3"><times id="S2.SS1.p1.4.m3.1.2.2.3.1.cmml" xref="S2.SS1.p1.4.m3.1.2.2.3.1"></times><ci id="S2.SS1.p1.4.m3.1.2.2.3.2.cmml" xref="S2.SS1.p1.4.m3.1.2.2.3.2">𝑜</ci><ci id="S2.SS1.p1.4.m3.1.2.2.3.3.cmml" xref="S2.SS1.p1.4.m3.1.2.2.3.3">𝑝</ci></apply></apply><ci id="S2.SS1.p1.4.m3.1.1.cmml" xref="S2.SS1.p1.4.m3.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.4.m3.1c">R_{op}(t)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.4.m3.1d">italic_R start_POSTSUBSCRIPT italic_o italic_p end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> is the real-time optical pumping rate with its time-averaged value as <math alttext="\langle R_{op}\rangle" class="ltx_Math" display="inline" id="S2.SS1.p1.5.m4.1"><semantics id="S2.SS1.p1.5.m4.1a"><mrow id="S2.SS1.p1.5.m4.1.1.1" xref="S2.SS1.p1.5.m4.1.1.2.cmml"><mo id="S2.SS1.p1.5.m4.1.1.1.2" stretchy="false" xref="S2.SS1.p1.5.m4.1.1.2.1.cmml">⟨</mo><msub id="S2.SS1.p1.5.m4.1.1.1.1" xref="S2.SS1.p1.5.m4.1.1.1.1.cmml"><mi id="S2.SS1.p1.5.m4.1.1.1.1.2" xref="S2.SS1.p1.5.m4.1.1.1.1.2.cmml">R</mi><mrow id="S2.SS1.p1.5.m4.1.1.1.1.3" xref="S2.SS1.p1.5.m4.1.1.1.1.3.cmml"><mi id="S2.SS1.p1.5.m4.1.1.1.1.3.2" xref="S2.SS1.p1.5.m4.1.1.1.1.3.2.cmml">o</mi><mo id="S2.SS1.p1.5.m4.1.1.1.1.3.1" xref="S2.SS1.p1.5.m4.1.1.1.1.3.1.cmml"></mo><mi id="S2.SS1.p1.5.m4.1.1.1.1.3.3" xref="S2.SS1.p1.5.m4.1.1.1.1.3.3.cmml">p</mi></mrow></msub><mo id="S2.SS1.p1.5.m4.1.1.1.3" stretchy="false" xref="S2.SS1.p1.5.m4.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.5.m4.1b"><apply id="S2.SS1.p1.5.m4.1.1.2.cmml" xref="S2.SS1.p1.5.m4.1.1.1"><csymbol cd="latexml" id="S2.SS1.p1.5.m4.1.1.2.1.cmml" xref="S2.SS1.p1.5.m4.1.1.1.2">delimited-⟨⟩</csymbol><apply id="S2.SS1.p1.5.m4.1.1.1.1.cmml" xref="S2.SS1.p1.5.m4.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.5.m4.1.1.1.1.1.cmml" xref="S2.SS1.p1.5.m4.1.1.1.1">subscript</csymbol><ci id="S2.SS1.p1.5.m4.1.1.1.1.2.cmml" xref="S2.SS1.p1.5.m4.1.1.1.1.2">𝑅</ci><apply id="S2.SS1.p1.5.m4.1.1.1.1.3.cmml" xref="S2.SS1.p1.5.m4.1.1.1.1.3"><times id="S2.SS1.p1.5.m4.1.1.1.1.3.1.cmml" xref="S2.SS1.p1.5.m4.1.1.1.1.3.1"></times><ci id="S2.SS1.p1.5.m4.1.1.1.1.3.2.cmml" xref="S2.SS1.p1.5.m4.1.1.1.1.3.2">𝑜</ci><ci id="S2.SS1.p1.5.m4.1.1.1.1.3.3.cmml" xref="S2.SS1.p1.5.m4.1.1.1.1.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.5.m4.1c">\langle R_{op}\rangle</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.5.m4.1d">⟨ italic_R start_POSTSUBSCRIPT italic_o italic_p end_POSTSUBSCRIPT ⟩</annotation></semantics></math>, and <math alttext="\gamma" class="ltx_Math" display="inline" id="S2.SS1.p1.6.m5.1"><semantics id="S2.SS1.p1.6.m5.1a"><mi id="S2.SS1.p1.6.m5.1.1" xref="S2.SS1.p1.6.m5.1.1.cmml">γ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.6.m5.1b"><ci id="S2.SS1.p1.6.m5.1.1.cmml" xref="S2.SS1.p1.6.m5.1.1">𝛾</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.6.m5.1c">\gamma</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.6.m5.1d">italic_γ</annotation></semantics></math> is the atomic gyromagnetic ratio.</p> </div> <div class="ltx_para" id="S2.SS1.p2"> <p class="ltx_p" id="S2.SS1.p2.4">In the large magnetic field limit, <math alttext="\gamma B\gg\langle R_{op}\rangle" class="ltx_Math" display="inline" id="S2.SS1.p2.1.m1.1"><semantics id="S2.SS1.p2.1.m1.1a"><mrow id="S2.SS1.p2.1.m1.1.1" xref="S2.SS1.p2.1.m1.1.1.cmml"><mrow id="S2.SS1.p2.1.m1.1.1.3" xref="S2.SS1.p2.1.m1.1.1.3.cmml"><mi id="S2.SS1.p2.1.m1.1.1.3.2" xref="S2.SS1.p2.1.m1.1.1.3.2.cmml">γ</mi><mo id="S2.SS1.p2.1.m1.1.1.3.1" xref="S2.SS1.p2.1.m1.1.1.3.1.cmml"></mo><mi id="S2.SS1.p2.1.m1.1.1.3.3" xref="S2.SS1.p2.1.m1.1.1.3.3.cmml">B</mi></mrow><mo id="S2.SS1.p2.1.m1.1.1.2" xref="S2.SS1.p2.1.m1.1.1.2.cmml">≫</mo><mrow id="S2.SS1.p2.1.m1.1.1.1.1" xref="S2.SS1.p2.1.m1.1.1.1.2.cmml"><mo id="S2.SS1.p2.1.m1.1.1.1.1.2" stretchy="false" xref="S2.SS1.p2.1.m1.1.1.1.2.1.cmml">⟨</mo><msub id="S2.SS1.p2.1.m1.1.1.1.1.1" xref="S2.SS1.p2.1.m1.1.1.1.1.1.cmml"><mi id="S2.SS1.p2.1.m1.1.1.1.1.1.2" xref="S2.SS1.p2.1.m1.1.1.1.1.1.2.cmml">R</mi><mrow id="S2.SS1.p2.1.m1.1.1.1.1.1.3" xref="S2.SS1.p2.1.m1.1.1.1.1.1.3.cmml"><mi id="S2.SS1.p2.1.m1.1.1.1.1.1.3.2" xref="S2.SS1.p2.1.m1.1.1.1.1.1.3.2.cmml">o</mi><mo id="S2.SS1.p2.1.m1.1.1.1.1.1.3.1" xref="S2.SS1.p2.1.m1.1.1.1.1.1.3.1.cmml"></mo><mi id="S2.SS1.p2.1.m1.1.1.1.1.1.3.3" xref="S2.SS1.p2.1.m1.1.1.1.1.1.3.3.cmml">p</mi></mrow></msub><mo id="S2.SS1.p2.1.m1.1.1.1.1.3" stretchy="false" xref="S2.SS1.p2.1.m1.1.1.1.2.1.cmml">⟩</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.1.m1.1b"><apply id="S2.SS1.p2.1.m1.1.1.cmml" xref="S2.SS1.p2.1.m1.1.1"><csymbol cd="latexml" id="S2.SS1.p2.1.m1.1.1.2.cmml" xref="S2.SS1.p2.1.m1.1.1.2">much-greater-than</csymbol><apply id="S2.SS1.p2.1.m1.1.1.3.cmml" xref="S2.SS1.p2.1.m1.1.1.3"><times id="S2.SS1.p2.1.m1.1.1.3.1.cmml" xref="S2.SS1.p2.1.m1.1.1.3.1"></times><ci id="S2.SS1.p2.1.m1.1.1.3.2.cmml" xref="S2.SS1.p2.1.m1.1.1.3.2">𝛾</ci><ci id="S2.SS1.p2.1.m1.1.1.3.3.cmml" xref="S2.SS1.p2.1.m1.1.1.3.3">𝐵</ci></apply><apply id="S2.SS1.p2.1.m1.1.1.1.2.cmml" xref="S2.SS1.p2.1.m1.1.1.1.1"><csymbol cd="latexml" id="S2.SS1.p2.1.m1.1.1.1.2.1.cmml" xref="S2.SS1.p2.1.m1.1.1.1.1.2">delimited-⟨⟩</csymbol><apply id="S2.SS1.p2.1.m1.1.1.1.1.1.cmml" xref="S2.SS1.p2.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p2.1.m1.1.1.1.1.1.1.cmml" xref="S2.SS1.p2.1.m1.1.1.1.1.1">subscript</csymbol><ci id="S2.SS1.p2.1.m1.1.1.1.1.1.2.cmml" xref="S2.SS1.p2.1.m1.1.1.1.1.1.2">𝑅</ci><apply id="S2.SS1.p2.1.m1.1.1.1.1.1.3.cmml" xref="S2.SS1.p2.1.m1.1.1.1.1.1.3"><times id="S2.SS1.p2.1.m1.1.1.1.1.1.3.1.cmml" xref="S2.SS1.p2.1.m1.1.1.1.1.1.3.1"></times><ci id="S2.SS1.p2.1.m1.1.1.1.1.1.3.2.cmml" xref="S2.SS1.p2.1.m1.1.1.1.1.1.3.2">𝑜</ci><ci id="S2.SS1.p2.1.m1.1.1.1.1.1.3.3.cmml" xref="S2.SS1.p2.1.m1.1.1.1.1.1.3.3">𝑝</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.1.m1.1c">\gamma B\gg\langle R_{op}\rangle</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.1.m1.1d">italic_γ italic_B ≫ ⟨ italic_R start_POSTSUBSCRIPT italic_o italic_p end_POSTSUBSCRIPT ⟩</annotation></semantics></math> and <math alttext="\Gamma_{1(2),0}" class="ltx_Math" display="inline" id="S2.SS1.p2.2.m2.3"><semantics id="S2.SS1.p2.2.m2.3a"><msub id="S2.SS1.p2.2.m2.3.4" xref="S2.SS1.p2.2.m2.3.4.cmml"><mi id="S2.SS1.p2.2.m2.3.4.2" mathvariant="normal" xref="S2.SS1.p2.2.m2.3.4.2.cmml">Γ</mi><mrow id="S2.SS1.p2.2.m2.3.3.3.3" xref="S2.SS1.p2.2.m2.3.3.3.4.cmml"><mrow id="S2.SS1.p2.2.m2.3.3.3.3.1" xref="S2.SS1.p2.2.m2.3.3.3.3.1.cmml"><mn id="S2.SS1.p2.2.m2.3.3.3.3.1.2" xref="S2.SS1.p2.2.m2.3.3.3.3.1.2.cmml">1</mn><mo id="S2.SS1.p2.2.m2.3.3.3.3.1.1" xref="S2.SS1.p2.2.m2.3.3.3.3.1.1.cmml"></mo><mrow id="S2.SS1.p2.2.m2.3.3.3.3.1.3.2" xref="S2.SS1.p2.2.m2.3.3.3.3.1.cmml"><mo id="S2.SS1.p2.2.m2.3.3.3.3.1.3.2.1" stretchy="false" xref="S2.SS1.p2.2.m2.3.3.3.3.1.cmml">(</mo><mn id="S2.SS1.p2.2.m2.1.1.1.1" xref="S2.SS1.p2.2.m2.1.1.1.1.cmml">2</mn><mo id="S2.SS1.p2.2.m2.3.3.3.3.1.3.2.2" stretchy="false" xref="S2.SS1.p2.2.m2.3.3.3.3.1.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p2.2.m2.3.3.3.3.2" xref="S2.SS1.p2.2.m2.3.3.3.4.cmml">,</mo><mn id="S2.SS1.p2.2.m2.2.2.2.2" xref="S2.SS1.p2.2.m2.2.2.2.2.cmml">0</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.2.m2.3b"><apply id="S2.SS1.p2.2.m2.3.4.cmml" xref="S2.SS1.p2.2.m2.3.4"><csymbol cd="ambiguous" id="S2.SS1.p2.2.m2.3.4.1.cmml" xref="S2.SS1.p2.2.m2.3.4">subscript</csymbol><ci id="S2.SS1.p2.2.m2.3.4.2.cmml" xref="S2.SS1.p2.2.m2.3.4.2">Γ</ci><list id="S2.SS1.p2.2.m2.3.3.3.4.cmml" xref="S2.SS1.p2.2.m2.3.3.3.3"><apply id="S2.SS1.p2.2.m2.3.3.3.3.1.cmml" xref="S2.SS1.p2.2.m2.3.3.3.3.1"><times id="S2.SS1.p2.2.m2.3.3.3.3.1.1.cmml" xref="S2.SS1.p2.2.m2.3.3.3.3.1.1"></times><cn id="S2.SS1.p2.2.m2.3.3.3.3.1.2.cmml" type="integer" xref="S2.SS1.p2.2.m2.3.3.3.3.1.2">1</cn><cn id="S2.SS1.p2.2.m2.1.1.1.1.cmml" type="integer" xref="S2.SS1.p2.2.m2.1.1.1.1">2</cn></apply><cn id="S2.SS1.p2.2.m2.2.2.2.2.cmml" type="integer" xref="S2.SS1.p2.2.m2.2.2.2.2">0</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.2.m2.3c">\Gamma_{1(2),0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.2.m2.3d">roman_Γ start_POSTSUBSCRIPT 1 ( 2 ) , 0 end_POSTSUBSCRIPT</annotation></semantics></math>, the steady state solution for longitudinal polarization <math alttext="P_{l}" class="ltx_Math" display="inline" id="S2.SS1.p2.3.m3.1"><semantics id="S2.SS1.p2.3.m3.1a"><msub id="S2.SS1.p2.3.m3.1.1" xref="S2.SS1.p2.3.m3.1.1.cmml"><mi id="S2.SS1.p2.3.m3.1.1.2" xref="S2.SS1.p2.3.m3.1.1.2.cmml">P</mi><mi id="S2.SS1.p2.3.m3.1.1.3" xref="S2.SS1.p2.3.m3.1.1.3.cmml">l</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.3.m3.1b"><apply id="S2.SS1.p2.3.m3.1.1.cmml" xref="S2.SS1.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS1.p2.3.m3.1.1.1.cmml" xref="S2.SS1.p2.3.m3.1.1">subscript</csymbol><ci id="S2.SS1.p2.3.m3.1.1.2.cmml" xref="S2.SS1.p2.3.m3.1.1.2">𝑃</ci><ci id="S2.SS1.p2.3.m3.1.1.3.cmml" xref="S2.SS1.p2.3.m3.1.1.3">𝑙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.3.m3.1c">P_{l}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.3.m3.1d">italic_P start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT</annotation></semantics></math> and transverse polarization <math alttext="P_{t}" class="ltx_Math" display="inline" id="S2.SS1.p2.4.m4.1"><semantics id="S2.SS1.p2.4.m4.1a"><msub id="S2.SS1.p2.4.m4.1.1" xref="S2.SS1.p2.4.m4.1.1.cmml"><mi id="S2.SS1.p2.4.m4.1.1.2" xref="S2.SS1.p2.4.m4.1.1.2.cmml">P</mi><mi id="S2.SS1.p2.4.m4.1.1.3" xref="S2.SS1.p2.4.m4.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.4.m4.1b"><apply id="S2.SS1.p2.4.m4.1.1.cmml" xref="S2.SS1.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS1.p2.4.m4.1.1.1.cmml" xref="S2.SS1.p2.4.m4.1.1">subscript</csymbol><ci id="S2.SS1.p2.4.m4.1.1.2.cmml" xref="S2.SS1.p2.4.m4.1.1.2">𝑃</ci><ci id="S2.SS1.p2.4.m4.1.1.3.cmml" xref="S2.SS1.p2.4.m4.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.4.m4.1c">P_{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.4.m4.1d">italic_P start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> can be expressed as:</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S6.EGx1"> <tbody id="S2.E2"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle P_{l}" class="ltx_Math" display="inline" id="S2.E2.m1.1"><semantics id="S2.E2.m1.1a"><msub id="S2.E2.m1.1.1" xref="S2.E2.m1.1.1.cmml"><mi id="S2.E2.m1.1.1.2" xref="S2.E2.m1.1.1.2.cmml">P</mi><mi id="S2.E2.m1.1.1.3" xref="S2.E2.m1.1.1.3.cmml">l</mi></msub><annotation-xml encoding="MathML-Content" id="S2.E2.m1.1b"><apply id="S2.E2.m1.1.1.cmml" xref="S2.E2.m1.1.1"><csymbol cd="ambiguous" id="S2.E2.m1.1.1.1.cmml" xref="S2.E2.m1.1.1">subscript</csymbol><ci id="S2.E2.m1.1.1.2.cmml" xref="S2.E2.m1.1.1.2">𝑃</ci><ci id="S2.E2.m1.1.1.3.cmml" xref="S2.E2.m1.1.1.3">𝑙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E2.m1.1c">\displaystyle P_{l}</annotation><annotation encoding="application/x-llamapun" id="S2.E2.m1.1d">italic_P start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S2.E2.m2.1"><semantics id="S2.E2.m2.1a"><mo id="S2.E2.m2.1.1" xref="S2.E2.m2.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S2.E2.m2.1b"><eq id="S2.E2.m2.1.1.cmml" xref="S2.E2.m2.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S2.E2.m2.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="S2.E2.m2.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\frac{\langle R_{op}\rangle\cos\theta}{\Gamma_{1,0}+\langle R_{op% }\rangle},~{}" class="ltx_Math" display="inline" id="S2.E2.m3.4"><semantics id="S2.E2.m3.4a"><mrow id="S2.E2.m3.4.5.2" xref="S2.E2.m3.4.4.cmml"><mstyle displaystyle="true" id="S2.E2.m3.4.4" xref="S2.E2.m3.4.4.cmml"><mfrac id="S2.E2.m3.4.4a" xref="S2.E2.m3.4.4.cmml"><mrow id="S2.E2.m3.1.1.1" xref="S2.E2.m3.1.1.1.cmml"><mrow id="S2.E2.m3.1.1.1.1.1" xref="S2.E2.m3.1.1.1.1.2.cmml"><mo id="S2.E2.m3.1.1.1.1.1.2" stretchy="false" xref="S2.E2.m3.1.1.1.1.2.1.cmml">⟨</mo><msub id="S2.E2.m3.1.1.1.1.1.1" xref="S2.E2.m3.1.1.1.1.1.1.cmml"><mi id="S2.E2.m3.1.1.1.1.1.1.2" xref="S2.E2.m3.1.1.1.1.1.1.2.cmml">R</mi><mrow id="S2.E2.m3.1.1.1.1.1.1.3" xref="S2.E2.m3.1.1.1.1.1.1.3.cmml"><mi id="S2.E2.m3.1.1.1.1.1.1.3.2" xref="S2.E2.m3.1.1.1.1.1.1.3.2.cmml">o</mi><mo id="S2.E2.m3.1.1.1.1.1.1.3.1" xref="S2.E2.m3.1.1.1.1.1.1.3.1.cmml"></mo><mi id="S2.E2.m3.1.1.1.1.1.1.3.3" xref="S2.E2.m3.1.1.1.1.1.1.3.3.cmml">p</mi></mrow></msub><mo id="S2.E2.m3.1.1.1.1.1.3" stretchy="false" xref="S2.E2.m3.1.1.1.1.2.1.cmml">⟩</mo></mrow><mo id="S2.E2.m3.1.1.1.2" lspace="0.167em" xref="S2.E2.m3.1.1.1.2.cmml"></mo><mrow id="S2.E2.m3.1.1.1.3" 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xref="S2.E2.m3.4.4.4.3.1.1.3.1"></times><ci id="S2.E2.m3.4.4.4.3.1.1.3.2.cmml" xref="S2.E2.m3.4.4.4.3.1.1.3.2">𝑜</ci><ci id="S2.E2.m3.4.4.4.3.1.1.3.3.cmml" xref="S2.E2.m3.4.4.4.3.1.1.3.3">𝑝</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E2.m3.4c">\displaystyle\frac{\langle R_{op}\rangle\cos\theta}{\Gamma_{1,0}+\langle R_{op% }\rangle},~{}</annotation><annotation encoding="application/x-llamapun" id="S2.E2.m3.4d">divide start_ARG ⟨ italic_R start_POSTSUBSCRIPT italic_o italic_p end_POSTSUBSCRIPT ⟩ roman_cos italic_θ end_ARG start_ARG roman_Γ start_POSTSUBSCRIPT 1 , 0 end_POSTSUBSCRIPT + ⟨ italic_R start_POSTSUBSCRIPT italic_o italic_p end_POSTSUBSCRIPT ⟩ end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2)</span></td> </tr></tbody> <tbody id="S2.E3"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle P_{t}" class="ltx_Math" display="inline" id="S2.E3.m1.1"><semantics id="S2.E3.m1.1a"><msub id="S2.E3.m1.1.1" xref="S2.E3.m1.1.1.cmml"><mi id="S2.E3.m1.1.1.2" xref="S2.E3.m1.1.1.2.cmml">P</mi><mi id="S2.E3.m1.1.1.3" xref="S2.E3.m1.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S2.E3.m1.1b"><apply id="S2.E3.m1.1.1.cmml" xref="S2.E3.m1.1.1"><csymbol cd="ambiguous" id="S2.E3.m1.1.1.1.cmml" xref="S2.E3.m1.1.1">subscript</csymbol><ci id="S2.E3.m1.1.1.2.cmml" xref="S2.E3.m1.1.1.2">𝑃</ci><ci id="S2.E3.m1.1.1.3.cmml" xref="S2.E3.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E3.m1.1c">\displaystyle P_{t}</annotation><annotation encoding="application/x-llamapun" id="S2.E3.m1.1d">italic_P start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S2.E3.m2.1"><semantics id="S2.E3.m2.1a"><mo id="S2.E3.m2.1.1" xref="S2.E3.m2.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S2.E3.m2.1b"><eq id="S2.E3.m2.1.1.cmml" xref="S2.E3.m2.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S2.E3.m2.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="S2.E3.m2.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\frac{\langle R_{op}\rangle\sin\theta}{\Gamma_{2,0}+\langle R_{op% }\rangle}~{}." class="ltx_Math" display="inline" id="S2.E3.m3.4"><semantics id="S2.E3.m3.4a"><mrow id="S2.E3.m3.4.5.2" xref="S2.E3.m3.4.4.cmml"><mstyle displaystyle="true" id="S2.E3.m3.4.4" xref="S2.E3.m3.4.4.cmml"><mfrac id="S2.E3.m3.4.4a" xref="S2.E3.m3.4.4.cmml"><mrow 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xref="S2.E3.m3.4.4.4.5">subscript</csymbol><ci id="S2.E3.m3.4.4.4.5.2.cmml" xref="S2.E3.m3.4.4.4.5.2">Γ</ci><list id="S2.E3.m3.3.3.3.2.2.3.cmml" xref="S2.E3.m3.3.3.3.2.2.4"><cn id="S2.E3.m3.2.2.2.1.1.1.cmml" type="integer" xref="S2.E3.m3.2.2.2.1.1.1">2</cn><cn id="S2.E3.m3.3.3.3.2.2.2.cmml" type="integer" xref="S2.E3.m3.3.3.3.2.2.2">0</cn></list></apply><apply id="S2.E3.m3.4.4.4.3.2.cmml" xref="S2.E3.m3.4.4.4.3.1"><csymbol cd="latexml" id="S2.E3.m3.4.4.4.3.2.1.cmml" xref="S2.E3.m3.4.4.4.3.1.2">delimited-⟨⟩</csymbol><apply id="S2.E3.m3.4.4.4.3.1.1.cmml" xref="S2.E3.m3.4.4.4.3.1.1"><csymbol cd="ambiguous" id="S2.E3.m3.4.4.4.3.1.1.1.cmml" xref="S2.E3.m3.4.4.4.3.1.1">subscript</csymbol><ci id="S2.E3.m3.4.4.4.3.1.1.2.cmml" xref="S2.E3.m3.4.4.4.3.1.1.2">𝑅</ci><apply id="S2.E3.m3.4.4.4.3.1.1.3.cmml" xref="S2.E3.m3.4.4.4.3.1.1.3"><times id="S2.E3.m3.4.4.4.3.1.1.3.1.cmml" xref="S2.E3.m3.4.4.4.3.1.1.3.1"></times><ci id="S2.E3.m3.4.4.4.3.1.1.3.2.cmml" xref="S2.E3.m3.4.4.4.3.1.1.3.2">𝑜</ci><ci id="S2.E3.m3.4.4.4.3.1.1.3.3.cmml" xref="S2.E3.m3.4.4.4.3.1.1.3.3">𝑝</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E3.m3.4c">\displaystyle\frac{\langle R_{op}\rangle\sin\theta}{\Gamma_{2,0}+\langle R_{op% }\rangle}~{}.</annotation><annotation encoding="application/x-llamapun" id="S2.E3.m3.4d">divide start_ARG ⟨ italic_R start_POSTSUBSCRIPT italic_o italic_p end_POSTSUBSCRIPT ⟩ roman_sin italic_θ end_ARG start_ARG roman_Γ start_POSTSUBSCRIPT 2 , 0 end_POSTSUBSCRIPT + ⟨ italic_R start_POSTSUBSCRIPT italic_o italic_p end_POSTSUBSCRIPT ⟩ end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(3)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p2.10">Generally speaking, <math alttext="\Gamma_{1,0}" class="ltx_Math" display="inline" id="S2.SS1.p2.5.m1.2"><semantics id="S2.SS1.p2.5.m1.2a"><msub id="S2.SS1.p2.5.m1.2.3" xref="S2.SS1.p2.5.m1.2.3.cmml"><mi id="S2.SS1.p2.5.m1.2.3.2" mathvariant="normal" xref="S2.SS1.p2.5.m1.2.3.2.cmml">Γ</mi><mrow id="S2.SS1.p2.5.m1.2.2.2.4" xref="S2.SS1.p2.5.m1.2.2.2.3.cmml"><mn id="S2.SS1.p2.5.m1.1.1.1.1" xref="S2.SS1.p2.5.m1.1.1.1.1.cmml">1</mn><mo id="S2.SS1.p2.5.m1.2.2.2.4.1" xref="S2.SS1.p2.5.m1.2.2.2.3.cmml">,</mo><mn id="S2.SS1.p2.5.m1.2.2.2.2" xref="S2.SS1.p2.5.m1.2.2.2.2.cmml">0</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.5.m1.2b"><apply id="S2.SS1.p2.5.m1.2.3.cmml" xref="S2.SS1.p2.5.m1.2.3"><csymbol cd="ambiguous" id="S2.SS1.p2.5.m1.2.3.1.cmml" xref="S2.SS1.p2.5.m1.2.3">subscript</csymbol><ci id="S2.SS1.p2.5.m1.2.3.2.cmml" xref="S2.SS1.p2.5.m1.2.3.2">Γ</ci><list id="S2.SS1.p2.5.m1.2.2.2.3.cmml" xref="S2.SS1.p2.5.m1.2.2.2.4"><cn id="S2.SS1.p2.5.m1.1.1.1.1.cmml" type="integer" xref="S2.SS1.p2.5.m1.1.1.1.1">1</cn><cn id="S2.SS1.p2.5.m1.2.2.2.2.cmml" type="integer" xref="S2.SS1.p2.5.m1.2.2.2.2">0</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.5.m1.2c">\Gamma_{1,0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.5.m1.2d">roman_Γ start_POSTSUBSCRIPT 1 , 0 end_POSTSUBSCRIPT</annotation></semantics></math> is not equal to <math alttext="\Gamma_{2,0}" class="ltx_Math" display="inline" id="S2.SS1.p2.6.m2.2"><semantics id="S2.SS1.p2.6.m2.2a"><msub id="S2.SS1.p2.6.m2.2.3" xref="S2.SS1.p2.6.m2.2.3.cmml"><mi id="S2.SS1.p2.6.m2.2.3.2" mathvariant="normal" xref="S2.SS1.p2.6.m2.2.3.2.cmml">Γ</mi><mrow id="S2.SS1.p2.6.m2.2.2.2.4" xref="S2.SS1.p2.6.m2.2.2.2.3.cmml"><mn id="S2.SS1.p2.6.m2.1.1.1.1" xref="S2.SS1.p2.6.m2.1.1.1.1.cmml">2</mn><mo id="S2.SS1.p2.6.m2.2.2.2.4.1" xref="S2.SS1.p2.6.m2.2.2.2.3.cmml">,</mo><mn id="S2.SS1.p2.6.m2.2.2.2.2" xref="S2.SS1.p2.6.m2.2.2.2.2.cmml">0</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.6.m2.2b"><apply id="S2.SS1.p2.6.m2.2.3.cmml" xref="S2.SS1.p2.6.m2.2.3"><csymbol cd="ambiguous" id="S2.SS1.p2.6.m2.2.3.1.cmml" xref="S2.SS1.p2.6.m2.2.3">subscript</csymbol><ci id="S2.SS1.p2.6.m2.2.3.2.cmml" xref="S2.SS1.p2.6.m2.2.3.2">Γ</ci><list id="S2.SS1.p2.6.m2.2.2.2.3.cmml" xref="S2.SS1.p2.6.m2.2.2.2.4"><cn id="S2.SS1.p2.6.m2.1.1.1.1.cmml" type="integer" xref="S2.SS1.p2.6.m2.1.1.1.1">2</cn><cn id="S2.SS1.p2.6.m2.2.2.2.2.cmml" type="integer" xref="S2.SS1.p2.6.m2.2.2.2.2">0</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.6.m2.2c">\Gamma_{2,0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.6.m2.2d">roman_Γ start_POSTSUBSCRIPT 2 , 0 end_POSTSUBSCRIPT</annotation></semantics></math> due to extra contribution to transverse depolarization such as the spin-exchange interactions. This means that the polarization vector <math alttext="\bm{P}" class="ltx_Math" display="inline" id="S2.SS1.p2.7.m3.1"><semantics id="S2.SS1.p2.7.m3.1a"><mi id="S2.SS1.p2.7.m3.1.1" xref="S2.SS1.p2.7.m3.1.1.cmml">𝑷</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.7.m3.1b"><ci id="S2.SS1.p2.7.m3.1.1.cmml" xref="S2.SS1.p2.7.m3.1.1">𝑷</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.7.m3.1c">\bm{P}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.7.m3.1d">bold_italic_P</annotation></semantics></math> is generally not aligned with the propagation direction of the pumping beam. However, when the optical pumping rate is much larger than <math alttext="\Gamma_{1,0}" class="ltx_Math" display="inline" id="S2.SS1.p2.8.m4.2"><semantics id="S2.SS1.p2.8.m4.2a"><msub id="S2.SS1.p2.8.m4.2.3" xref="S2.SS1.p2.8.m4.2.3.cmml"><mi id="S2.SS1.p2.8.m4.2.3.2" mathvariant="normal" xref="S2.SS1.p2.8.m4.2.3.2.cmml">Γ</mi><mrow id="S2.SS1.p2.8.m4.2.2.2.4" xref="S2.SS1.p2.8.m4.2.2.2.3.cmml"><mn id="S2.SS1.p2.8.m4.1.1.1.1" xref="S2.SS1.p2.8.m4.1.1.1.1.cmml">1</mn><mo id="S2.SS1.p2.8.m4.2.2.2.4.1" xref="S2.SS1.p2.8.m4.2.2.2.3.cmml">,</mo><mn id="S2.SS1.p2.8.m4.2.2.2.2" xref="S2.SS1.p2.8.m4.2.2.2.2.cmml">0</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.8.m4.2b"><apply id="S2.SS1.p2.8.m4.2.3.cmml" xref="S2.SS1.p2.8.m4.2.3"><csymbol cd="ambiguous" id="S2.SS1.p2.8.m4.2.3.1.cmml" xref="S2.SS1.p2.8.m4.2.3">subscript</csymbol><ci id="S2.SS1.p2.8.m4.2.3.2.cmml" xref="S2.SS1.p2.8.m4.2.3.2">Γ</ci><list id="S2.SS1.p2.8.m4.2.2.2.3.cmml" xref="S2.SS1.p2.8.m4.2.2.2.4"><cn id="S2.SS1.p2.8.m4.1.1.1.1.cmml" type="integer" xref="S2.SS1.p2.8.m4.1.1.1.1">1</cn><cn id="S2.SS1.p2.8.m4.2.2.2.2.cmml" type="integer" xref="S2.SS1.p2.8.m4.2.2.2.2">0</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.8.m4.2c">\Gamma_{1,0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.8.m4.2d">roman_Γ start_POSTSUBSCRIPT 1 , 0 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\Gamma_{2,0}" class="ltx_Math" display="inline" id="S2.SS1.p2.9.m5.2"><semantics id="S2.SS1.p2.9.m5.2a"><msub id="S2.SS1.p2.9.m5.2.3" xref="S2.SS1.p2.9.m5.2.3.cmml"><mi id="S2.SS1.p2.9.m5.2.3.2" mathvariant="normal" xref="S2.SS1.p2.9.m5.2.3.2.cmml">Γ</mi><mrow id="S2.SS1.p2.9.m5.2.2.2.4" xref="S2.SS1.p2.9.m5.2.2.2.3.cmml"><mn id="S2.SS1.p2.9.m5.1.1.1.1" xref="S2.SS1.p2.9.m5.1.1.1.1.cmml">2</mn><mo id="S2.SS1.p2.9.m5.2.2.2.4.1" xref="S2.SS1.p2.9.m5.2.2.2.3.cmml">,</mo><mn id="S2.SS1.p2.9.m5.2.2.2.2" xref="S2.SS1.p2.9.m5.2.2.2.2.cmml">0</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.9.m5.2b"><apply id="S2.SS1.p2.9.m5.2.3.cmml" xref="S2.SS1.p2.9.m5.2.3"><csymbol cd="ambiguous" id="S2.SS1.p2.9.m5.2.3.1.cmml" xref="S2.SS1.p2.9.m5.2.3">subscript</csymbol><ci id="S2.SS1.p2.9.m5.2.3.2.cmml" xref="S2.SS1.p2.9.m5.2.3.2">Γ</ci><list id="S2.SS1.p2.9.m5.2.2.2.3.cmml" xref="S2.SS1.p2.9.m5.2.2.2.4"><cn id="S2.SS1.p2.9.m5.1.1.1.1.cmml" type="integer" xref="S2.SS1.p2.9.m5.1.1.1.1">2</cn><cn id="S2.SS1.p2.9.m5.2.2.2.2.cmml" type="integer" xref="S2.SS1.p2.9.m5.2.2.2.2">0</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.9.m5.2c">\Gamma_{2,0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.9.m5.2d">roman_Γ start_POSTSUBSCRIPT 2 , 0 end_POSTSUBSCRIPT</annotation></semantics></math>, atomic polarization is relatively large and the direction of <math alttext="\bm{P}" class="ltx_Math" display="inline" id="S2.SS1.p2.10.m6.1"><semantics id="S2.SS1.p2.10.m6.1a"><mi id="S2.SS1.p2.10.m6.1.1" xref="S2.SS1.p2.10.m6.1.1.cmml">𝑷</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.10.m6.1b"><ci id="S2.SS1.p2.10.m6.1.1.cmml" xref="S2.SS1.p2.10.m6.1.1">𝑷</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.10.m6.1c">\bm{P}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.10.m6.1d">bold_italic_P</annotation></semantics></math> is close to the pumping beam direction, which is the high-polarization limit.</p> </div> </section> <section class="ltx_subsection" id="S2.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">II.2 </span>Heading error analysis</h3> <div class="ltx_para" id="S2.SS2.p1"> <p class="ltx_p" id="S2.SS2.p1.2">For ground state alkali atoms with a nuclear spin of <math alttext="I" class="ltx_Math" display="inline" id="S2.SS2.p1.1.m1.1"><semantics id="S2.SS2.p1.1.m1.1a"><mi id="S2.SS2.p1.1.m1.1.1" xref="S2.SS2.p1.1.m1.1.1.cmml">I</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.1.m1.1b"><ci id="S2.SS2.p1.1.m1.1.1.cmml" xref="S2.SS2.p1.1.m1.1.1">𝐼</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.1.m1.1c">I</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.1.m1.1d">italic_I</annotation></semantics></math> in a bias field of <math alttext="\mathbf{B}" class="ltx_Math" display="inline" id="S2.SS2.p1.2.m2.1"><semantics id="S2.SS2.p1.2.m2.1a"><mi id="S2.SS2.p1.2.m2.1.1" xref="S2.SS2.p1.2.m2.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.2.m2.1b"><ci id="S2.SS2.p1.2.m2.1.1.cmml" xref="S2.SS2.p1.2.m2.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.2.m2.1c">\mathbf{B}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.2.m2.1d">bold_B</annotation></semantics></math>, its Hamiltonian is expressed as</p> <table 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xref="S2.E4.m1.1.1.1.1.3.3.2.3">subscript</csymbol><ci id="S2.E4.m1.1.1.1.1.3.3.2.3.2.cmml" xref="S2.E4.m1.1.1.1.1.3.3.2.3.2">𝜇</ci><ci id="S2.E4.m1.1.1.1.1.3.3.2.3.3.cmml" xref="S2.E4.m1.1.1.1.1.3.3.2.3.3">𝐵</ci></apply><ci id="S2.E4.m1.1.1.1.1.3.3.2.4.cmml" xref="S2.E4.m1.1.1.1.1.3.3.2.4">𝑰</ci></apply><ci id="S2.E4.m1.1.1.1.1.3.3.3.cmml" xref="S2.E4.m1.1.1.1.1.3.3.3">𝐁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E4.m1.1c">H=A_{hf}\bm{I}\cdot\bm{S}+g_{s}\mu_{B}\bm{S}\cdot\mathbf{B}-g_{I}\mu_{B}\bm{I}% \cdot\mathbf{B}.</annotation><annotation encoding="application/x-llamapun" id="S2.E4.m1.1d">italic_H = italic_A start_POSTSUBSCRIPT italic_h italic_f end_POSTSUBSCRIPT bold_italic_I ⋅ bold_italic_S + italic_g start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT bold_italic_S ⋅ bold_B - italic_g start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT bold_italic_I ⋅ bold_B .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p1.7">Here, <math alttext="A_{hf}" class="ltx_Math" display="inline" id="S2.SS2.p1.3.m1.1"><semantics id="S2.SS2.p1.3.m1.1a"><msub id="S2.SS2.p1.3.m1.1.1" xref="S2.SS2.p1.3.m1.1.1.cmml"><mi id="S2.SS2.p1.3.m1.1.1.2" xref="S2.SS2.p1.3.m1.1.1.2.cmml">A</mi><mrow id="S2.SS2.p1.3.m1.1.1.3" xref="S2.SS2.p1.3.m1.1.1.3.cmml"><mi id="S2.SS2.p1.3.m1.1.1.3.2" xref="S2.SS2.p1.3.m1.1.1.3.2.cmml">h</mi><mo id="S2.SS2.p1.3.m1.1.1.3.1" xref="S2.SS2.p1.3.m1.1.1.3.1.cmml"></mo><mi id="S2.SS2.p1.3.m1.1.1.3.3" xref="S2.SS2.p1.3.m1.1.1.3.3.cmml">f</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.3.m1.1b"><apply id="S2.SS2.p1.3.m1.1.1.cmml" xref="S2.SS2.p1.3.m1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.3.m1.1.1.1.cmml" xref="S2.SS2.p1.3.m1.1.1">subscript</csymbol><ci id="S2.SS2.p1.3.m1.1.1.2.cmml" xref="S2.SS2.p1.3.m1.1.1.2">𝐴</ci><apply id="S2.SS2.p1.3.m1.1.1.3.cmml" xref="S2.SS2.p1.3.m1.1.1.3"><times id="S2.SS2.p1.3.m1.1.1.3.1.cmml" xref="S2.SS2.p1.3.m1.1.1.3.1"></times><ci id="S2.SS2.p1.3.m1.1.1.3.2.cmml" xref="S2.SS2.p1.3.m1.1.1.3.2">ℎ</ci><ci id="S2.SS2.p1.3.m1.1.1.3.3.cmml" xref="S2.SS2.p1.3.m1.1.1.3.3">𝑓</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.3.m1.1c">A_{hf}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.3.m1.1d">italic_A start_POSTSUBSCRIPT italic_h italic_f end_POSTSUBSCRIPT</annotation></semantics></math> is the ground state hyperfine constant, <math alttext="\bm{S}" class="ltx_Math" display="inline" id="S2.SS2.p1.4.m2.1"><semantics id="S2.SS2.p1.4.m2.1a"><mi id="S2.SS2.p1.4.m2.1.1" xref="S2.SS2.p1.4.m2.1.1.cmml">𝑺</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.4.m2.1b"><ci id="S2.SS2.p1.4.m2.1.1.cmml" xref="S2.SS2.p1.4.m2.1.1">𝑺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.4.m2.1c">\bm{S}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.4.m2.1d">bold_italic_S</annotation></semantics></math> is the electron spin, <math alttext="\mu_{B}" class="ltx_Math" display="inline" id="S2.SS2.p1.5.m3.1"><semantics id="S2.SS2.p1.5.m3.1a"><msub id="S2.SS2.p1.5.m3.1.1" xref="S2.SS2.p1.5.m3.1.1.cmml"><mi id="S2.SS2.p1.5.m3.1.1.2" xref="S2.SS2.p1.5.m3.1.1.2.cmml">μ</mi><mi id="S2.SS2.p1.5.m3.1.1.3" xref="S2.SS2.p1.5.m3.1.1.3.cmml">B</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.5.m3.1b"><apply id="S2.SS2.p1.5.m3.1.1.cmml" xref="S2.SS2.p1.5.m3.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.5.m3.1.1.1.cmml" xref="S2.SS2.p1.5.m3.1.1">subscript</csymbol><ci id="S2.SS2.p1.5.m3.1.1.2.cmml" xref="S2.SS2.p1.5.m3.1.1.2">𝜇</ci><ci id="S2.SS2.p1.5.m3.1.1.3.cmml" xref="S2.SS2.p1.5.m3.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.5.m3.1c">\mu_{B}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.5.m3.1d">italic_μ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT</annotation></semantics></math> is the Bohr magneton, and the nuclear magnetic moment is <math alttext="\bm{\mu}_{I}=g_{I}\mu_{B}\bm{I}" class="ltx_Math" display="inline" id="S2.SS2.p1.6.m4.1"><semantics id="S2.SS2.p1.6.m4.1a"><mrow id="S2.SS2.p1.6.m4.1.1" xref="S2.SS2.p1.6.m4.1.1.cmml"><msub id="S2.SS2.p1.6.m4.1.1.2" xref="S2.SS2.p1.6.m4.1.1.2.cmml"><mi id="S2.SS2.p1.6.m4.1.1.2.2" xref="S2.SS2.p1.6.m4.1.1.2.2.cmml">𝝁</mi><mi id="S2.SS2.p1.6.m4.1.1.2.3" xref="S2.SS2.p1.6.m4.1.1.2.3.cmml">I</mi></msub><mo id="S2.SS2.p1.6.m4.1.1.1" xref="S2.SS2.p1.6.m4.1.1.1.cmml">=</mo><mrow id="S2.SS2.p1.6.m4.1.1.3" xref="S2.SS2.p1.6.m4.1.1.3.cmml"><msub id="S2.SS2.p1.6.m4.1.1.3.2" xref="S2.SS2.p1.6.m4.1.1.3.2.cmml"><mi id="S2.SS2.p1.6.m4.1.1.3.2.2" xref="S2.SS2.p1.6.m4.1.1.3.2.2.cmml">g</mi><mi id="S2.SS2.p1.6.m4.1.1.3.2.3" xref="S2.SS2.p1.6.m4.1.1.3.2.3.cmml">I</mi></msub><mo id="S2.SS2.p1.6.m4.1.1.3.1" xref="S2.SS2.p1.6.m4.1.1.3.1.cmml"></mo><msub id="S2.SS2.p1.6.m4.1.1.3.3" xref="S2.SS2.p1.6.m4.1.1.3.3.cmml"><mi id="S2.SS2.p1.6.m4.1.1.3.3.2" xref="S2.SS2.p1.6.m4.1.1.3.3.2.cmml">μ</mi><mi id="S2.SS2.p1.6.m4.1.1.3.3.3" xref="S2.SS2.p1.6.m4.1.1.3.3.3.cmml">B</mi></msub><mo id="S2.SS2.p1.6.m4.1.1.3.1a" xref="S2.SS2.p1.6.m4.1.1.3.1.cmml"></mo><mi id="S2.SS2.p1.6.m4.1.1.3.4" xref="S2.SS2.p1.6.m4.1.1.3.4.cmml">𝑰</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.6.m4.1b"><apply id="S2.SS2.p1.6.m4.1.1.cmml" xref="S2.SS2.p1.6.m4.1.1"><eq id="S2.SS2.p1.6.m4.1.1.1.cmml" xref="S2.SS2.p1.6.m4.1.1.1"></eq><apply id="S2.SS2.p1.6.m4.1.1.2.cmml" xref="S2.SS2.p1.6.m4.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.p1.6.m4.1.1.2.1.cmml" xref="S2.SS2.p1.6.m4.1.1.2">subscript</csymbol><ci id="S2.SS2.p1.6.m4.1.1.2.2.cmml" xref="S2.SS2.p1.6.m4.1.1.2.2">𝝁</ci><ci id="S2.SS2.p1.6.m4.1.1.2.3.cmml" xref="S2.SS2.p1.6.m4.1.1.2.3">𝐼</ci></apply><apply id="S2.SS2.p1.6.m4.1.1.3.cmml" xref="S2.SS2.p1.6.m4.1.1.3"><times id="S2.SS2.p1.6.m4.1.1.3.1.cmml" xref="S2.SS2.p1.6.m4.1.1.3.1"></times><apply id="S2.SS2.p1.6.m4.1.1.3.2.cmml" xref="S2.SS2.p1.6.m4.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS2.p1.6.m4.1.1.3.2.1.cmml" xref="S2.SS2.p1.6.m4.1.1.3.2">subscript</csymbol><ci id="S2.SS2.p1.6.m4.1.1.3.2.2.cmml" xref="S2.SS2.p1.6.m4.1.1.3.2.2">𝑔</ci><ci id="S2.SS2.p1.6.m4.1.1.3.2.3.cmml" xref="S2.SS2.p1.6.m4.1.1.3.2.3">𝐼</ci></apply><apply id="S2.SS2.p1.6.m4.1.1.3.3.cmml" xref="S2.SS2.p1.6.m4.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS2.p1.6.m4.1.1.3.3.1.cmml" xref="S2.SS2.p1.6.m4.1.1.3.3">subscript</csymbol><ci id="S2.SS2.p1.6.m4.1.1.3.3.2.cmml" xref="S2.SS2.p1.6.m4.1.1.3.3.2">𝜇</ci><ci id="S2.SS2.p1.6.m4.1.1.3.3.3.cmml" xref="S2.SS2.p1.6.m4.1.1.3.3.3">𝐵</ci></apply><ci id="S2.SS2.p1.6.m4.1.1.3.4.cmml" xref="S2.SS2.p1.6.m4.1.1.3.4">𝑰</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.6.m4.1c">\bm{\mu}_{I}=g_{I}\mu_{B}\bm{I}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.6.m4.1d">bold_italic_μ start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT = italic_g start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT bold_italic_I</annotation></semantics></math> as in Ref. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib22" title="">22</a>]</cite>, which leads to an opposite sign of <math alttext="g_{I}" class="ltx_Math" display="inline" id="S2.SS2.p1.7.m5.1"><semantics id="S2.SS2.p1.7.m5.1a"><msub id="S2.SS2.p1.7.m5.1.1" xref="S2.SS2.p1.7.m5.1.1.cmml"><mi id="S2.SS2.p1.7.m5.1.1.2" xref="S2.SS2.p1.7.m5.1.1.2.cmml">g</mi><mi id="S2.SS2.p1.7.m5.1.1.3" xref="S2.SS2.p1.7.m5.1.1.3.cmml">I</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.7.m5.1b"><apply id="S2.SS2.p1.7.m5.1.1.cmml" xref="S2.SS2.p1.7.m5.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.7.m5.1.1.1.cmml" xref="S2.SS2.p1.7.m5.1.1">subscript</csymbol><ci id="S2.SS2.p1.7.m5.1.1.2.cmml" xref="S2.SS2.p1.7.m5.1.1.2">𝑔</ci><ci id="S2.SS2.p1.7.m5.1.1.3.cmml" xref="S2.SS2.p1.7.m5.1.1.3">𝐼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.7.m5.1c">g_{I}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.7.m5.1d">italic_g start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT</annotation></semantics></math> compared with that in Refs. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib26" title="">26</a>, <a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib27" title="">27</a>]</cite>. The resulted energy shift for non-stretched states, relative to the original level without considering the hyperfine and Zeeman effect, can be extracted from the Breit-Rabi formula <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib28" title="">28</a>, <a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib29" title="">29</a>]</cite></p> <table class="ltx_equation ltx_eqn_table" id="S2.E5"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="~{}E_{ns}=-\frac{\hbar\omega_{hf}}{2(2I+1)}-g_{I}m_{F}\mu_{B}\mathrm{B}\pm% \frac{\hbar\omega_{hf}}{2}\sqrt{1+\frac{4xm_{F}}{2I+1}+x^{2}}," class="ltx_Math" display="block" id="S2.E5.m1.2"><semantics id="S2.E5.m1.2a"><mrow id="S2.E5.m1.2.2.1" xref="S2.E5.m1.2.2.1.1.cmml"><mrow id="S2.E5.m1.2.2.1.1" xref="S2.E5.m1.2.2.1.1.cmml"><msub id="S2.E5.m1.2.2.1.1.2" xref="S2.E5.m1.2.2.1.1.2.cmml"><mi id="S2.E5.m1.2.2.1.1.2.2" xref="S2.E5.m1.2.2.1.1.2.2.cmml">E</mi><mrow id="S2.E5.m1.2.2.1.1.2.3" xref="S2.E5.m1.2.2.1.1.2.3.cmml"><mi id="S2.E5.m1.2.2.1.1.2.3.2" xref="S2.E5.m1.2.2.1.1.2.3.2.cmml">n</mi><mo id="S2.E5.m1.2.2.1.1.2.3.1" xref="S2.E5.m1.2.2.1.1.2.3.1.cmml"></mo><mi id="S2.E5.m1.2.2.1.1.2.3.3" xref="S2.E5.m1.2.2.1.1.2.3.3.cmml">s</mi></mrow></msub><mo id="S2.E5.m1.2.2.1.1.1" xref="S2.E5.m1.2.2.1.1.1.cmml">=</mo><mrow id="S2.E5.m1.2.2.1.1.3" xref="S2.E5.m1.2.2.1.1.3.cmml"><mrow id="S2.E5.m1.2.2.1.1.3.2" xref="S2.E5.m1.2.2.1.1.3.2.cmml"><mrow id="S2.E5.m1.2.2.1.1.3.2.2" xref="S2.E5.m1.2.2.1.1.3.2.2.cmml"><mo id="S2.E5.m1.2.2.1.1.3.2.2a" xref="S2.E5.m1.2.2.1.1.3.2.2.cmml">−</mo><mfrac id="S2.E5.m1.1.1" xref="S2.E5.m1.1.1.cmml"><mrow id="S2.E5.m1.1.1.3" xref="S2.E5.m1.1.1.3.cmml"><mi id="S2.E5.m1.1.1.3.2" mathvariant="normal" xref="S2.E5.m1.1.1.3.2.cmml">ℏ</mi><mo id="S2.E5.m1.1.1.3.1" xref="S2.E5.m1.1.1.3.1.cmml"></mo><msub 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xref="S2.E5.m1.1.1.1.1.1.1.2.1.cmml"></mo><mi id="S2.E5.m1.1.1.1.1.1.1.2.3" xref="S2.E5.m1.1.1.1.1.1.1.2.3.cmml">I</mi></mrow><mo id="S2.E5.m1.1.1.1.1.1.1.1" xref="S2.E5.m1.1.1.1.1.1.1.1.cmml">+</mo><mn id="S2.E5.m1.1.1.1.1.1.1.3" xref="S2.E5.m1.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S2.E5.m1.1.1.1.1.1.3" stretchy="false" xref="S2.E5.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mfrac></mrow><mo id="S2.E5.m1.2.2.1.1.3.2.1" xref="S2.E5.m1.2.2.1.1.3.2.1.cmml">−</mo><mrow id="S2.E5.m1.2.2.1.1.3.2.3" xref="S2.E5.m1.2.2.1.1.3.2.3.cmml"><msub id="S2.E5.m1.2.2.1.1.3.2.3.2" xref="S2.E5.m1.2.2.1.1.3.2.3.2.cmml"><mi id="S2.E5.m1.2.2.1.1.3.2.3.2.2" xref="S2.E5.m1.2.2.1.1.3.2.3.2.2.cmml">g</mi><mi id="S2.E5.m1.2.2.1.1.3.2.3.2.3" xref="S2.E5.m1.2.2.1.1.3.2.3.2.3.cmml">I</mi></msub><mo id="S2.E5.m1.2.2.1.1.3.2.3.1" xref="S2.E5.m1.2.2.1.1.3.2.3.1.cmml"></mo><msub id="S2.E5.m1.2.2.1.1.3.2.3.3" xref="S2.E5.m1.2.2.1.1.3.2.3.3.cmml"><mi id="S2.E5.m1.2.2.1.1.3.2.3.3.2" xref="S2.E5.m1.2.2.1.1.3.2.3.3.2.cmml">m</mi><mi id="S2.E5.m1.2.2.1.1.3.2.3.3.3" xref="S2.E5.m1.2.2.1.1.3.2.3.3.3.cmml">F</mi></msub><mo id="S2.E5.m1.2.2.1.1.3.2.3.1a" xref="S2.E5.m1.2.2.1.1.3.2.3.1.cmml"></mo><msub id="S2.E5.m1.2.2.1.1.3.2.3.4" xref="S2.E5.m1.2.2.1.1.3.2.3.4.cmml"><mi id="S2.E5.m1.2.2.1.1.3.2.3.4.2" xref="S2.E5.m1.2.2.1.1.3.2.3.4.2.cmml">μ</mi><mi id="S2.E5.m1.2.2.1.1.3.2.3.4.3" xref="S2.E5.m1.2.2.1.1.3.2.3.4.3.cmml">B</mi></msub><mo id="S2.E5.m1.2.2.1.1.3.2.3.1b" xref="S2.E5.m1.2.2.1.1.3.2.3.1.cmml"></mo><mi id="S2.E5.m1.2.2.1.1.3.2.3.5" mathvariant="normal" xref="S2.E5.m1.2.2.1.1.3.2.3.5.cmml">B</mi></mrow></mrow><mo id="S2.E5.m1.2.2.1.1.3.1" xref="S2.E5.m1.2.2.1.1.3.1.cmml">±</mo><mrow id="S2.E5.m1.2.2.1.1.3.3" xref="S2.E5.m1.2.2.1.1.3.3.cmml"><mfrac id="S2.E5.m1.2.2.1.1.3.3.2" xref="S2.E5.m1.2.2.1.1.3.3.2.cmml"><mrow id="S2.E5.m1.2.2.1.1.3.3.2.2" xref="S2.E5.m1.2.2.1.1.3.3.2.2.cmml"><mi id="S2.E5.m1.2.2.1.1.3.3.2.2.2" mathvariant="normal" 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id="S2.E5.m1.2.2.1.1.3.3.3.2.4.3.cmml" type="integer" xref="S2.E5.m1.2.2.1.1.3.3.3.2.4.3">2</cn></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E5.m1.2c">~{}E_{ns}=-\frac{\hbar\omega_{hf}}{2(2I+1)}-g_{I}m_{F}\mu_{B}\mathrm{B}\pm% \frac{\hbar\omega_{hf}}{2}\sqrt{1+\frac{4xm_{F}}{2I+1}+x^{2}},</annotation><annotation encoding="application/x-llamapun" id="S2.E5.m1.2d">italic_E start_POSTSUBSCRIPT italic_n italic_s end_POSTSUBSCRIPT = - divide start_ARG roman_ℏ italic_ω start_POSTSUBSCRIPT italic_h italic_f end_POSTSUBSCRIPT end_ARG start_ARG 2 ( 2 italic_I + 1 ) end_ARG - italic_g start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT roman_B ± divide start_ARG roman_ℏ italic_ω start_POSTSUBSCRIPT italic_h italic_f end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG square-root start_ARG 1 + divide start_ARG 4 italic_x italic_m start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT end_ARG start_ARG 2 italic_I + 1 end_ARG + italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(5)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p1.11">with <math alttext="\hbar\omega_{hf}" class="ltx_Math" display="inline" id="S2.SS2.p1.8.m1.1"><semantics id="S2.SS2.p1.8.m1.1a"><mrow id="S2.SS2.p1.8.m1.1.1" xref="S2.SS2.p1.8.m1.1.1.cmml"><mi id="S2.SS2.p1.8.m1.1.1.2" mathvariant="normal" xref="S2.SS2.p1.8.m1.1.1.2.cmml">ℏ</mi><mo id="S2.SS2.p1.8.m1.1.1.1" xref="S2.SS2.p1.8.m1.1.1.1.cmml"></mo><msub id="S2.SS2.p1.8.m1.1.1.3" xref="S2.SS2.p1.8.m1.1.1.3.cmml"><mi id="S2.SS2.p1.8.m1.1.1.3.2" xref="S2.SS2.p1.8.m1.1.1.3.2.cmml">ω</mi><mrow id="S2.SS2.p1.8.m1.1.1.3.3" xref="S2.SS2.p1.8.m1.1.1.3.3.cmml"><mi id="S2.SS2.p1.8.m1.1.1.3.3.2" xref="S2.SS2.p1.8.m1.1.1.3.3.2.cmml">h</mi><mo id="S2.SS2.p1.8.m1.1.1.3.3.1" xref="S2.SS2.p1.8.m1.1.1.3.3.1.cmml"></mo><mi id="S2.SS2.p1.8.m1.1.1.3.3.3" xref="S2.SS2.p1.8.m1.1.1.3.3.3.cmml">f</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.8.m1.1b"><apply id="S2.SS2.p1.8.m1.1.1.cmml" xref="S2.SS2.p1.8.m1.1.1"><times id="S2.SS2.p1.8.m1.1.1.1.cmml" xref="S2.SS2.p1.8.m1.1.1.1"></times><csymbol cd="latexml" id="S2.SS2.p1.8.m1.1.1.2.cmml" xref="S2.SS2.p1.8.m1.1.1.2">Planck-constant-over-2-pi</csymbol><apply id="S2.SS2.p1.8.m1.1.1.3.cmml" xref="S2.SS2.p1.8.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p1.8.m1.1.1.3.1.cmml" xref="S2.SS2.p1.8.m1.1.1.3">subscript</csymbol><ci id="S2.SS2.p1.8.m1.1.1.3.2.cmml" xref="S2.SS2.p1.8.m1.1.1.3.2">𝜔</ci><apply id="S2.SS2.p1.8.m1.1.1.3.3.cmml" xref="S2.SS2.p1.8.m1.1.1.3.3"><times id="S2.SS2.p1.8.m1.1.1.3.3.1.cmml" 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xref="S2.SS2.p1.9.m2.1.1.1.1"><divide id="S2.SS2.p1.9.m2.1.1.1.1.2.cmml" xref="S2.SS2.p1.9.m2.1.1.1.1.2"></divide><apply id="S2.SS2.p1.9.m2.1.1.1.1.1.cmml" xref="S2.SS2.p1.9.m2.1.1.1.1.1"><times id="S2.SS2.p1.9.m2.1.1.1.1.1.2.cmml" xref="S2.SS2.p1.9.m2.1.1.1.1.1.2"></times><apply id="S2.SS2.p1.9.m2.1.1.1.1.1.1.1.1.cmml" xref="S2.SS2.p1.9.m2.1.1.1.1.1.1.1"><plus id="S2.SS2.p1.9.m2.1.1.1.1.1.1.1.1.1.cmml" xref="S2.SS2.p1.9.m2.1.1.1.1.1.1.1.1.1"></plus><apply id="S2.SS2.p1.9.m2.1.1.1.1.1.1.1.1.2.cmml" xref="S2.SS2.p1.9.m2.1.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.p1.9.m2.1.1.1.1.1.1.1.1.2.1.cmml" xref="S2.SS2.p1.9.m2.1.1.1.1.1.1.1.1.2">subscript</csymbol><ci id="S2.SS2.p1.9.m2.1.1.1.1.1.1.1.1.2.2.cmml" xref="S2.SS2.p1.9.m2.1.1.1.1.1.1.1.1.2.2">𝑔</ci><ci id="S2.SS2.p1.9.m2.1.1.1.1.1.1.1.1.2.3.cmml" xref="S2.SS2.p1.9.m2.1.1.1.1.1.1.1.1.2.3">𝑠</ci></apply><apply id="S2.SS2.p1.9.m2.1.1.1.1.1.1.1.1.3.cmml" xref="S2.SS2.p1.9.m2.1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" 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id="S2.SS2.p1.9.m2.1.1.1.3.1.cmml" xref="S2.SS2.p1.9.m2.1.1.1.3">subscript</csymbol><ci id="S2.SS2.p1.9.m2.1.1.1.3.2.cmml" xref="S2.SS2.p1.9.m2.1.1.1.3.2">𝜔</ci><apply id="S2.SS2.p1.9.m2.1.1.1.3.3.cmml" xref="S2.SS2.p1.9.m2.1.1.1.3.3"><times id="S2.SS2.p1.9.m2.1.1.1.3.3.1.cmml" xref="S2.SS2.p1.9.m2.1.1.1.3.3.1"></times><ci id="S2.SS2.p1.9.m2.1.1.1.3.3.2.cmml" xref="S2.SS2.p1.9.m2.1.1.1.3.3.2">ℎ</ci><ci id="S2.SS2.p1.9.m2.1.1.1.3.3.3.cmml" xref="S2.SS2.p1.9.m2.1.1.1.3.3.3">𝑓</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.9.m2.1c">x=(g_{s}+g_{I})\mu_{B}\mathrm{B}/\hbar\omega_{hf}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.9.m2.1d">italic_x = ( italic_g start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT + italic_g start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT ) italic_μ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT roman_B / roman_ℏ italic_ω start_POSTSUBSCRIPT italic_h italic_f end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="\pm" class="ltx_Math" display="inline" id="S2.SS2.p1.10.m3.1"><semantics id="S2.SS2.p1.10.m3.1a"><mo id="S2.SS2.p1.10.m3.1.1" xref="S2.SS2.p1.10.m3.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.10.m3.1b"><csymbol cd="latexml" id="S2.SS2.p1.10.m3.1.1.cmml" xref="S2.SS2.p1.10.m3.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.10.m3.1c">\pm</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.10.m3.1d">±</annotation></semantics></math> refers to <math alttext="F=I\pm 1/2" class="ltx_Math" display="inline" id="S2.SS2.p1.11.m4.1"><semantics id="S2.SS2.p1.11.m4.1a"><mrow id="S2.SS2.p1.11.m4.1.1" xref="S2.SS2.p1.11.m4.1.1.cmml"><mi id="S2.SS2.p1.11.m4.1.1.2" xref="S2.SS2.p1.11.m4.1.1.2.cmml">F</mi><mo id="S2.SS2.p1.11.m4.1.1.1" xref="S2.SS2.p1.11.m4.1.1.1.cmml">=</mo><mrow id="S2.SS2.p1.11.m4.1.1.3" xref="S2.SS2.p1.11.m4.1.1.3.cmml"><mi id="S2.SS2.p1.11.m4.1.1.3.2" xref="S2.SS2.p1.11.m4.1.1.3.2.cmml">I</mi><mo id="S2.SS2.p1.11.m4.1.1.3.1" xref="S2.SS2.p1.11.m4.1.1.3.1.cmml">±</mo><mrow id="S2.SS2.p1.11.m4.1.1.3.3" xref="S2.SS2.p1.11.m4.1.1.3.3.cmml"><mn id="S2.SS2.p1.11.m4.1.1.3.3.2" xref="S2.SS2.p1.11.m4.1.1.3.3.2.cmml">1</mn><mo id="S2.SS2.p1.11.m4.1.1.3.3.1" xref="S2.SS2.p1.11.m4.1.1.3.3.1.cmml">/</mo><mn id="S2.SS2.p1.11.m4.1.1.3.3.3" xref="S2.SS2.p1.11.m4.1.1.3.3.3.cmml">2</mn></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.11.m4.1b"><apply id="S2.SS2.p1.11.m4.1.1.cmml" xref="S2.SS2.p1.11.m4.1.1"><eq id="S2.SS2.p1.11.m4.1.1.1.cmml" xref="S2.SS2.p1.11.m4.1.1.1"></eq><ci id="S2.SS2.p1.11.m4.1.1.2.cmml" xref="S2.SS2.p1.11.m4.1.1.2">𝐹</ci><apply id="S2.SS2.p1.11.m4.1.1.3.cmml" xref="S2.SS2.p1.11.m4.1.1.3"><csymbol cd="latexml" id="S2.SS2.p1.11.m4.1.1.3.1.cmml" xref="S2.SS2.p1.11.m4.1.1.3.1">plus-or-minus</csymbol><ci id="S2.SS2.p1.11.m4.1.1.3.2.cmml" xref="S2.SS2.p1.11.m4.1.1.3.2">𝐼</ci><apply id="S2.SS2.p1.11.m4.1.1.3.3.cmml" xref="S2.SS2.p1.11.m4.1.1.3.3"><divide id="S2.SS2.p1.11.m4.1.1.3.3.1.cmml" xref="S2.SS2.p1.11.m4.1.1.3.3.1"></divide><cn id="S2.SS2.p1.11.m4.1.1.3.3.2.cmml" type="integer" xref="S2.SS2.p1.11.m4.1.1.3.3.2">1</cn><cn id="S2.SS2.p1.11.m4.1.1.3.3.3.cmml" type="integer" xref="S2.SS2.p1.11.m4.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.11.m4.1c">F=I\pm 1/2</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.11.m4.1d">italic_F = italic_I ± 1 / 2</annotation></semantics></math>. The energies of two stretched states contain only linear Zeeman terms with the expression as</p> <table class="ltx_equation ltx_eqn_table" id="S2.E6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="~{}E_{s}=\frac{I}{2I+1}\hbar\omega_{hf}\pm\frac{g_{s}-2Ig_{I}}{2}\mu_{B}% \mathrm{B}," class="ltx_Math" display="block" id="S2.E6.m1.1"><semantics id="S2.E6.m1.1a"><mrow id="S2.E6.m1.1.1.1" xref="S2.E6.m1.1.1.1.1.cmml"><mrow id="S2.E6.m1.1.1.1.1" xref="S2.E6.m1.1.1.1.1.cmml"><msub id="S2.E6.m1.1.1.1.1.2" xref="S2.E6.m1.1.1.1.1.2.cmml"><mi id="S2.E6.m1.1.1.1.1.2.2" xref="S2.E6.m1.1.1.1.1.2.2.cmml">E</mi><mi id="S2.E6.m1.1.1.1.1.2.3" xref="S2.E6.m1.1.1.1.1.2.3.cmml">s</mi></msub><mo id="S2.E6.m1.1.1.1.1.1" xref="S2.E6.m1.1.1.1.1.1.cmml">=</mo><mrow id="S2.E6.m1.1.1.1.1.3" xref="S2.E6.m1.1.1.1.1.3.cmml"><mrow id="S2.E6.m1.1.1.1.1.3.2" xref="S2.E6.m1.1.1.1.1.3.2.cmml"><mfrac id="S2.E6.m1.1.1.1.1.3.2.2" xref="S2.E6.m1.1.1.1.1.3.2.2.cmml"><mi id="S2.E6.m1.1.1.1.1.3.2.2.2" xref="S2.E6.m1.1.1.1.1.3.2.2.2.cmml">I</mi><mrow id="S2.E6.m1.1.1.1.1.3.2.2.3" xref="S2.E6.m1.1.1.1.1.3.2.2.3.cmml"><mrow id="S2.E6.m1.1.1.1.1.3.2.2.3.2" xref="S2.E6.m1.1.1.1.1.3.2.2.3.2.cmml"><mn id="S2.E6.m1.1.1.1.1.3.2.2.3.2.2" xref="S2.E6.m1.1.1.1.1.3.2.2.3.2.2.cmml">2</mn><mo id="S2.E6.m1.1.1.1.1.3.2.2.3.2.1" xref="S2.E6.m1.1.1.1.1.3.2.2.3.2.1.cmml"></mo><mi id="S2.E6.m1.1.1.1.1.3.2.2.3.2.3" xref="S2.E6.m1.1.1.1.1.3.2.2.3.2.3.cmml">I</mi></mrow><mo id="S2.E6.m1.1.1.1.1.3.2.2.3.1" xref="S2.E6.m1.1.1.1.1.3.2.2.3.1.cmml">+</mo><mn id="S2.E6.m1.1.1.1.1.3.2.2.3.3" xref="S2.E6.m1.1.1.1.1.3.2.2.3.3.cmml">1</mn></mrow></mfrac><mo id="S2.E6.m1.1.1.1.1.3.2.1" xref="S2.E6.m1.1.1.1.1.3.2.1.cmml"></mo><mi id="S2.E6.m1.1.1.1.1.3.2.3" mathvariant="normal" xref="S2.E6.m1.1.1.1.1.3.2.3.cmml">ℏ</mi><mo id="S2.E6.m1.1.1.1.1.3.2.1a" xref="S2.E6.m1.1.1.1.1.3.2.1.cmml"></mo><msub id="S2.E6.m1.1.1.1.1.3.2.4" xref="S2.E6.m1.1.1.1.1.3.2.4.cmml"><mi id="S2.E6.m1.1.1.1.1.3.2.4.2" xref="S2.E6.m1.1.1.1.1.3.2.4.2.cmml">ω</mi><mrow id="S2.E6.m1.1.1.1.1.3.2.4.3" xref="S2.E6.m1.1.1.1.1.3.2.4.3.cmml"><mi id="S2.E6.m1.1.1.1.1.3.2.4.3.2" xref="S2.E6.m1.1.1.1.1.3.2.4.3.2.cmml">h</mi><mo id="S2.E6.m1.1.1.1.1.3.2.4.3.1" xref="S2.E6.m1.1.1.1.1.3.2.4.3.1.cmml"></mo><mi id="S2.E6.m1.1.1.1.1.3.2.4.3.3" xref="S2.E6.m1.1.1.1.1.3.2.4.3.3.cmml">f</mi></mrow></msub></mrow><mo id="S2.E6.m1.1.1.1.1.3.1" xref="S2.E6.m1.1.1.1.1.3.1.cmml">±</mo><mrow id="S2.E6.m1.1.1.1.1.3.3" xref="S2.E6.m1.1.1.1.1.3.3.cmml"><mfrac id="S2.E6.m1.1.1.1.1.3.3.2" xref="S2.E6.m1.1.1.1.1.3.3.2.cmml"><mrow id="S2.E6.m1.1.1.1.1.3.3.2.2" xref="S2.E6.m1.1.1.1.1.3.3.2.2.cmml"><msub id="S2.E6.m1.1.1.1.1.3.3.2.2.2" xref="S2.E6.m1.1.1.1.1.3.3.2.2.2.cmml"><mi id="S2.E6.m1.1.1.1.1.3.3.2.2.2.2" xref="S2.E6.m1.1.1.1.1.3.3.2.2.2.2.cmml">g</mi><mi id="S2.E6.m1.1.1.1.1.3.3.2.2.2.3" xref="S2.E6.m1.1.1.1.1.3.3.2.2.2.3.cmml">s</mi></msub><mo id="S2.E6.m1.1.1.1.1.3.3.2.2.1" xref="S2.E6.m1.1.1.1.1.3.3.2.2.1.cmml">−</mo><mrow id="S2.E6.m1.1.1.1.1.3.3.2.2.3" xref="S2.E6.m1.1.1.1.1.3.3.2.2.3.cmml"><mn id="S2.E6.m1.1.1.1.1.3.3.2.2.3.2" xref="S2.E6.m1.1.1.1.1.3.3.2.2.3.2.cmml">2</mn><mo id="S2.E6.m1.1.1.1.1.3.3.2.2.3.1" xref="S2.E6.m1.1.1.1.1.3.3.2.2.3.1.cmml"></mo><mi id="S2.E6.m1.1.1.1.1.3.3.2.2.3.3" xref="S2.E6.m1.1.1.1.1.3.3.2.2.3.3.cmml">I</mi><mo id="S2.E6.m1.1.1.1.1.3.3.2.2.3.1a" xref="S2.E6.m1.1.1.1.1.3.3.2.2.3.1.cmml"></mo><msub id="S2.E6.m1.1.1.1.1.3.3.2.2.3.4" xref="S2.E6.m1.1.1.1.1.3.3.2.2.3.4.cmml"><mi id="S2.E6.m1.1.1.1.1.3.3.2.2.3.4.2" xref="S2.E6.m1.1.1.1.1.3.3.2.2.3.4.2.cmml">g</mi><mi id="S2.E6.m1.1.1.1.1.3.3.2.2.3.4.3" xref="S2.E6.m1.1.1.1.1.3.3.2.2.3.4.3.cmml">I</mi></msub></mrow></mrow><mn id="S2.E6.m1.1.1.1.1.3.3.2.3" xref="S2.E6.m1.1.1.1.1.3.3.2.3.cmml">2</mn></mfrac><mo id="S2.E6.m1.1.1.1.1.3.3.1" xref="S2.E6.m1.1.1.1.1.3.3.1.cmml"></mo><msub id="S2.E6.m1.1.1.1.1.3.3.3" xref="S2.E6.m1.1.1.1.1.3.3.3.cmml"><mi id="S2.E6.m1.1.1.1.1.3.3.3.2" xref="S2.E6.m1.1.1.1.1.3.3.3.2.cmml">μ</mi><mi id="S2.E6.m1.1.1.1.1.3.3.3.3" xref="S2.E6.m1.1.1.1.1.3.3.3.3.cmml">B</mi></msub><mo id="S2.E6.m1.1.1.1.1.3.3.1a" xref="S2.E6.m1.1.1.1.1.3.3.1.cmml"></mo><mi id="S2.E6.m1.1.1.1.1.3.3.4" mathvariant="normal" xref="S2.E6.m1.1.1.1.1.3.3.4.cmml">B</mi></mrow></mrow></mrow><mo id="S2.E6.m1.1.1.1.2" xref="S2.E6.m1.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E6.m1.1b"><apply id="S2.E6.m1.1.1.1.1.cmml" xref="S2.E6.m1.1.1.1"><eq id="S2.E6.m1.1.1.1.1.1.cmml" xref="S2.E6.m1.1.1.1.1.1"></eq><apply id="S2.E6.m1.1.1.1.1.2.cmml" xref="S2.E6.m1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.E6.m1.1.1.1.1.2.1.cmml" xref="S2.E6.m1.1.1.1.1.2">subscript</csymbol><ci id="S2.E6.m1.1.1.1.1.2.2.cmml" xref="S2.E6.m1.1.1.1.1.2.2">𝐸</ci><ci id="S2.E6.m1.1.1.1.1.2.3.cmml" xref="S2.E6.m1.1.1.1.1.2.3">𝑠</ci></apply><apply id="S2.E6.m1.1.1.1.1.3.cmml" xref="S2.E6.m1.1.1.1.1.3"><csymbol cd="latexml" id="S2.E6.m1.1.1.1.1.3.1.cmml" xref="S2.E6.m1.1.1.1.1.3.1">plus-or-minus</csymbol><apply id="S2.E6.m1.1.1.1.1.3.2.cmml" xref="S2.E6.m1.1.1.1.1.3.2"><times id="S2.E6.m1.1.1.1.1.3.2.1.cmml" xref="S2.E6.m1.1.1.1.1.3.2.1"></times><apply id="S2.E6.m1.1.1.1.1.3.2.2.cmml" xref="S2.E6.m1.1.1.1.1.3.2.2"><divide id="S2.E6.m1.1.1.1.1.3.2.2.1.cmml" xref="S2.E6.m1.1.1.1.1.3.2.2"></divide><ci id="S2.E6.m1.1.1.1.1.3.2.2.2.cmml" xref="S2.E6.m1.1.1.1.1.3.2.2.2">𝐼</ci><apply id="S2.E6.m1.1.1.1.1.3.2.2.3.cmml" xref="S2.E6.m1.1.1.1.1.3.2.2.3"><plus id="S2.E6.m1.1.1.1.1.3.2.2.3.1.cmml" xref="S2.E6.m1.1.1.1.1.3.2.2.3.1"></plus><apply id="S2.E6.m1.1.1.1.1.3.2.2.3.2.cmml" xref="S2.E6.m1.1.1.1.1.3.2.2.3.2"><times id="S2.E6.m1.1.1.1.1.3.2.2.3.2.1.cmml" xref="S2.E6.m1.1.1.1.1.3.2.2.3.2.1"></times><cn id="S2.E6.m1.1.1.1.1.3.2.2.3.2.2.cmml" type="integer" xref="S2.E6.m1.1.1.1.1.3.2.2.3.2.2">2</cn><ci id="S2.E6.m1.1.1.1.1.3.2.2.3.2.3.cmml" xref="S2.E6.m1.1.1.1.1.3.2.2.3.2.3">𝐼</ci></apply><cn id="S2.E6.m1.1.1.1.1.3.2.2.3.3.cmml" type="integer" xref="S2.E6.m1.1.1.1.1.3.2.2.3.3">1</cn></apply></apply><csymbol cd="latexml" id="S2.E6.m1.1.1.1.1.3.2.3.cmml" xref="S2.E6.m1.1.1.1.1.3.2.3">Planck-constant-over-2-pi</csymbol><apply id="S2.E6.m1.1.1.1.1.3.2.4.cmml" xref="S2.E6.m1.1.1.1.1.3.2.4"><csymbol cd="ambiguous" id="S2.E6.m1.1.1.1.1.3.2.4.1.cmml" xref="S2.E6.m1.1.1.1.1.3.2.4">subscript</csymbol><ci id="S2.E6.m1.1.1.1.1.3.2.4.2.cmml" xref="S2.E6.m1.1.1.1.1.3.2.4.2">𝜔</ci><apply id="S2.E6.m1.1.1.1.1.3.2.4.3.cmml" xref="S2.E6.m1.1.1.1.1.3.2.4.3"><times id="S2.E6.m1.1.1.1.1.3.2.4.3.1.cmml" xref="S2.E6.m1.1.1.1.1.3.2.4.3.1"></times><ci id="S2.E6.m1.1.1.1.1.3.2.4.3.2.cmml" xref="S2.E6.m1.1.1.1.1.3.2.4.3.2">ℎ</ci><ci id="S2.E6.m1.1.1.1.1.3.2.4.3.3.cmml" xref="S2.E6.m1.1.1.1.1.3.2.4.3.3">𝑓</ci></apply></apply></apply><apply id="S2.E6.m1.1.1.1.1.3.3.cmml" xref="S2.E6.m1.1.1.1.1.3.3"><times id="S2.E6.m1.1.1.1.1.3.3.1.cmml" xref="S2.E6.m1.1.1.1.1.3.3.1"></times><apply id="S2.E6.m1.1.1.1.1.3.3.2.cmml" xref="S2.E6.m1.1.1.1.1.3.3.2"><divide id="S2.E6.m1.1.1.1.1.3.3.2.1.cmml" xref="S2.E6.m1.1.1.1.1.3.3.2"></divide><apply id="S2.E6.m1.1.1.1.1.3.3.2.2.cmml" xref="S2.E6.m1.1.1.1.1.3.3.2.2"><minus id="S2.E6.m1.1.1.1.1.3.3.2.2.1.cmml" xref="S2.E6.m1.1.1.1.1.3.3.2.2.1"></minus><apply id="S2.E6.m1.1.1.1.1.3.3.2.2.2.cmml" xref="S2.E6.m1.1.1.1.1.3.3.2.2.2"><csymbol cd="ambiguous" id="S2.E6.m1.1.1.1.1.3.3.2.2.2.1.cmml" xref="S2.E6.m1.1.1.1.1.3.3.2.2.2">subscript</csymbol><ci id="S2.E6.m1.1.1.1.1.3.3.2.2.2.2.cmml" xref="S2.E6.m1.1.1.1.1.3.3.2.2.2.2">𝑔</ci><ci id="S2.E6.m1.1.1.1.1.3.3.2.2.2.3.cmml" xref="S2.E6.m1.1.1.1.1.3.3.2.2.2.3">𝑠</ci></apply><apply id="S2.E6.m1.1.1.1.1.3.3.2.2.3.cmml" xref="S2.E6.m1.1.1.1.1.3.3.2.2.3"><times id="S2.E6.m1.1.1.1.1.3.3.2.2.3.1.cmml" xref="S2.E6.m1.1.1.1.1.3.3.2.2.3.1"></times><cn id="S2.E6.m1.1.1.1.1.3.3.2.2.3.2.cmml" type="integer" xref="S2.E6.m1.1.1.1.1.3.3.2.2.3.2">2</cn><ci id="S2.E6.m1.1.1.1.1.3.3.2.2.3.3.cmml" xref="S2.E6.m1.1.1.1.1.3.3.2.2.3.3">𝐼</ci><apply id="S2.E6.m1.1.1.1.1.3.3.2.2.3.4.cmml" xref="S2.E6.m1.1.1.1.1.3.3.2.2.3.4"><csymbol cd="ambiguous" id="S2.E6.m1.1.1.1.1.3.3.2.2.3.4.1.cmml" xref="S2.E6.m1.1.1.1.1.3.3.2.2.3.4">subscript</csymbol><ci id="S2.E6.m1.1.1.1.1.3.3.2.2.3.4.2.cmml" xref="S2.E6.m1.1.1.1.1.3.3.2.2.3.4.2">𝑔</ci><ci id="S2.E6.m1.1.1.1.1.3.3.2.2.3.4.3.cmml" xref="S2.E6.m1.1.1.1.1.3.3.2.2.3.4.3">𝐼</ci></apply></apply></apply><cn id="S2.E6.m1.1.1.1.1.3.3.2.3.cmml" type="integer" xref="S2.E6.m1.1.1.1.1.3.3.2.3">2</cn></apply><apply id="S2.E6.m1.1.1.1.1.3.3.3.cmml" xref="S2.E6.m1.1.1.1.1.3.3.3"><csymbol cd="ambiguous" id="S2.E6.m1.1.1.1.1.3.3.3.1.cmml" xref="S2.E6.m1.1.1.1.1.3.3.3">subscript</csymbol><ci id="S2.E6.m1.1.1.1.1.3.3.3.2.cmml" xref="S2.E6.m1.1.1.1.1.3.3.3.2">𝜇</ci><ci id="S2.E6.m1.1.1.1.1.3.3.3.3.cmml" xref="S2.E6.m1.1.1.1.1.3.3.3.3">𝐵</ci></apply><ci id="S2.E6.m1.1.1.1.1.3.3.4.cmml" xref="S2.E6.m1.1.1.1.1.3.3.4">B</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E6.m1.1c">~{}E_{s}=\frac{I}{2I+1}\hbar\omega_{hf}\pm\frac{g_{s}-2Ig_{I}}{2}\mu_{B}% \mathrm{B},</annotation><annotation encoding="application/x-llamapun" id="S2.E6.m1.1d">italic_E start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT = divide start_ARG italic_I end_ARG start_ARG 2 italic_I + 1 end_ARG roman_ℏ italic_ω start_POSTSUBSCRIPT italic_h italic_f end_POSTSUBSCRIPT ± divide start_ARG italic_g start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT - 2 italic_I italic_g start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG italic_μ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT roman_B ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(6)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p1.15">with <math alttext="\pm" class="ltx_Math" display="inline" id="S2.SS2.p1.12.m1.1"><semantics id="S2.SS2.p1.12.m1.1a"><mo id="S2.SS2.p1.12.m1.1.1" xref="S2.SS2.p1.12.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.12.m1.1b"><csymbol cd="latexml" id="S2.SS2.p1.12.m1.1.1.cmml" xref="S2.SS2.p1.12.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.12.m1.1c">\pm</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.12.m1.1d">±</annotation></semantics></math> refering to <math alttext="m_{F}=\pm(I+1/2)" class="ltx_Math" display="inline" id="S2.SS2.p1.13.m2.1"><semantics id="S2.SS2.p1.13.m2.1a"><mrow id="S2.SS2.p1.13.m2.1.1" xref="S2.SS2.p1.13.m2.1.1.cmml"><msub id="S2.SS2.p1.13.m2.1.1.3" xref="S2.SS2.p1.13.m2.1.1.3.cmml"><mi id="S2.SS2.p1.13.m2.1.1.3.2" xref="S2.SS2.p1.13.m2.1.1.3.2.cmml">m</mi><mi id="S2.SS2.p1.13.m2.1.1.3.3" xref="S2.SS2.p1.13.m2.1.1.3.3.cmml">F</mi></msub><mo id="S2.SS2.p1.13.m2.1.1.2" xref="S2.SS2.p1.13.m2.1.1.2.cmml">=</mo><mrow id="S2.SS2.p1.13.m2.1.1.1" xref="S2.SS2.p1.13.m2.1.1.1.cmml"><mo id="S2.SS2.p1.13.m2.1.1.1a" xref="S2.SS2.p1.13.m2.1.1.1.cmml">±</mo><mrow id="S2.SS2.p1.13.m2.1.1.1.1.1" xref="S2.SS2.p1.13.m2.1.1.1.1.1.1.cmml"><mo id="S2.SS2.p1.13.m2.1.1.1.1.1.2" stretchy="false" xref="S2.SS2.p1.13.m2.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS2.p1.13.m2.1.1.1.1.1.1" xref="S2.SS2.p1.13.m2.1.1.1.1.1.1.cmml"><mi id="S2.SS2.p1.13.m2.1.1.1.1.1.1.2" xref="S2.SS2.p1.13.m2.1.1.1.1.1.1.2.cmml">I</mi><mo id="S2.SS2.p1.13.m2.1.1.1.1.1.1.1" xref="S2.SS2.p1.13.m2.1.1.1.1.1.1.1.cmml">+</mo><mrow id="S2.SS2.p1.13.m2.1.1.1.1.1.1.3" xref="S2.SS2.p1.13.m2.1.1.1.1.1.1.3.cmml"><mn id="S2.SS2.p1.13.m2.1.1.1.1.1.1.3.2" xref="S2.SS2.p1.13.m2.1.1.1.1.1.1.3.2.cmml">1</mn><mo id="S2.SS2.p1.13.m2.1.1.1.1.1.1.3.1" xref="S2.SS2.p1.13.m2.1.1.1.1.1.1.3.1.cmml">/</mo><mn id="S2.SS2.p1.13.m2.1.1.1.1.1.1.3.3" xref="S2.SS2.p1.13.m2.1.1.1.1.1.1.3.3.cmml">2</mn></mrow></mrow><mo id="S2.SS2.p1.13.m2.1.1.1.1.1.3" stretchy="false" xref="S2.SS2.p1.13.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.13.m2.1b"><apply id="S2.SS2.p1.13.m2.1.1.cmml" xref="S2.SS2.p1.13.m2.1.1"><eq id="S2.SS2.p1.13.m2.1.1.2.cmml" xref="S2.SS2.p1.13.m2.1.1.2"></eq><apply id="S2.SS2.p1.13.m2.1.1.3.cmml" xref="S2.SS2.p1.13.m2.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p1.13.m2.1.1.3.1.cmml" xref="S2.SS2.p1.13.m2.1.1.3">subscript</csymbol><ci id="S2.SS2.p1.13.m2.1.1.3.2.cmml" xref="S2.SS2.p1.13.m2.1.1.3.2">𝑚</ci><ci id="S2.SS2.p1.13.m2.1.1.3.3.cmml" xref="S2.SS2.p1.13.m2.1.1.3.3">𝐹</ci></apply><apply id="S2.SS2.p1.13.m2.1.1.1.cmml" xref="S2.SS2.p1.13.m2.1.1.1"><csymbol cd="latexml" id="S2.SS2.p1.13.m2.1.1.1.2.cmml" xref="S2.SS2.p1.13.m2.1.1.1">plus-or-minus</csymbol><apply id="S2.SS2.p1.13.m2.1.1.1.1.1.1.cmml" xref="S2.SS2.p1.13.m2.1.1.1.1.1"><plus id="S2.SS2.p1.13.m2.1.1.1.1.1.1.1.cmml" xref="S2.SS2.p1.13.m2.1.1.1.1.1.1.1"></plus><ci id="S2.SS2.p1.13.m2.1.1.1.1.1.1.2.cmml" xref="S2.SS2.p1.13.m2.1.1.1.1.1.1.2">𝐼</ci><apply id="S2.SS2.p1.13.m2.1.1.1.1.1.1.3.cmml" xref="S2.SS2.p1.13.m2.1.1.1.1.1.1.3"><divide id="S2.SS2.p1.13.m2.1.1.1.1.1.1.3.1.cmml" xref="S2.SS2.p1.13.m2.1.1.1.1.1.1.3.1"></divide><cn id="S2.SS2.p1.13.m2.1.1.1.1.1.1.3.2.cmml" type="integer" xref="S2.SS2.p1.13.m2.1.1.1.1.1.1.3.2">1</cn><cn id="S2.SS2.p1.13.m2.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S2.SS2.p1.13.m2.1.1.1.1.1.1.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.13.m2.1c">m_{F}=\pm(I+1/2)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.13.m2.1d">italic_m start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT = ± ( italic_I + 1 / 2 )</annotation></semantics></math>. By keeping the Zeeman interaction terms up to the second order, we get the Zeeman splitting between any pair of <math alttext="|F,m_{F}\rangle" class="ltx_Math" display="inline" id="S2.SS2.p1.14.m3.2"><semantics id="S2.SS2.p1.14.m3.2a"><mrow id="S2.SS2.p1.14.m3.2.2.1" xref="S2.SS2.p1.14.m3.2.2.2.cmml"><mo id="S2.SS2.p1.14.m3.2.2.1.2" stretchy="false" xref="S2.SS2.p1.14.m3.2.2.2.1.cmml">|</mo><mrow id="S2.SS2.p1.14.m3.2.2.1.1.1" xref="S2.SS2.p1.14.m3.2.2.1.1.2.cmml"><mi id="S2.SS2.p1.14.m3.1.1" xref="S2.SS2.p1.14.m3.1.1.cmml">F</mi><mo id="S2.SS2.p1.14.m3.2.2.1.1.1.2" xref="S2.SS2.p1.14.m3.2.2.1.1.2.cmml">,</mo><msub id="S2.SS2.p1.14.m3.2.2.1.1.1.1" xref="S2.SS2.p1.14.m3.2.2.1.1.1.1.cmml"><mi id="S2.SS2.p1.14.m3.2.2.1.1.1.1.2" xref="S2.SS2.p1.14.m3.2.2.1.1.1.1.2.cmml">m</mi><mi id="S2.SS2.p1.14.m3.2.2.1.1.1.1.3" xref="S2.SS2.p1.14.m3.2.2.1.1.1.1.3.cmml">F</mi></msub></mrow><mo id="S2.SS2.p1.14.m3.2.2.1.3" stretchy="false" xref="S2.SS2.p1.14.m3.2.2.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.14.m3.2b"><apply id="S2.SS2.p1.14.m3.2.2.2.cmml" xref="S2.SS2.p1.14.m3.2.2.1"><csymbol cd="latexml" id="S2.SS2.p1.14.m3.2.2.2.1.cmml" xref="S2.SS2.p1.14.m3.2.2.1.2">ket</csymbol><list id="S2.SS2.p1.14.m3.2.2.1.1.2.cmml" xref="S2.SS2.p1.14.m3.2.2.1.1.1"><ci id="S2.SS2.p1.14.m3.1.1.cmml" xref="S2.SS2.p1.14.m3.1.1">𝐹</ci><apply id="S2.SS2.p1.14.m3.2.2.1.1.1.1.cmml" xref="S2.SS2.p1.14.m3.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.14.m3.2.2.1.1.1.1.1.cmml" xref="S2.SS2.p1.14.m3.2.2.1.1.1.1">subscript</csymbol><ci id="S2.SS2.p1.14.m3.2.2.1.1.1.1.2.cmml" xref="S2.SS2.p1.14.m3.2.2.1.1.1.1.2">𝑚</ci><ci id="S2.SS2.p1.14.m3.2.2.1.1.1.1.3.cmml" xref="S2.SS2.p1.14.m3.2.2.1.1.1.1.3">𝐹</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.14.m3.2c">|F,m_{F}\rangle</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.14.m3.2d">| italic_F , italic_m start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ⟩</annotation></semantics></math> and <math alttext="|F,m_{F}-1\rangle" class="ltx_Math" display="inline" id="S2.SS2.p1.15.m4.2"><semantics id="S2.SS2.p1.15.m4.2a"><mrow id="S2.SS2.p1.15.m4.2.2.1" xref="S2.SS2.p1.15.m4.2.2.2.cmml"><mo id="S2.SS2.p1.15.m4.2.2.1.2" stretchy="false" xref="S2.SS2.p1.15.m4.2.2.2.1.cmml">|</mo><mrow id="S2.SS2.p1.15.m4.2.2.1.1.1" xref="S2.SS2.p1.15.m4.2.2.1.1.2.cmml"><mi id="S2.SS2.p1.15.m4.1.1" xref="S2.SS2.p1.15.m4.1.1.cmml">F</mi><mo id="S2.SS2.p1.15.m4.2.2.1.1.1.2" xref="S2.SS2.p1.15.m4.2.2.1.1.2.cmml">,</mo><mrow id="S2.SS2.p1.15.m4.2.2.1.1.1.1" xref="S2.SS2.p1.15.m4.2.2.1.1.1.1.cmml"><msub id="S2.SS2.p1.15.m4.2.2.1.1.1.1.2" xref="S2.SS2.p1.15.m4.2.2.1.1.1.1.2.cmml"><mi id="S2.SS2.p1.15.m4.2.2.1.1.1.1.2.2" xref="S2.SS2.p1.15.m4.2.2.1.1.1.1.2.2.cmml">m</mi><mi id="S2.SS2.p1.15.m4.2.2.1.1.1.1.2.3" xref="S2.SS2.p1.15.m4.2.2.1.1.1.1.2.3.cmml">F</mi></msub><mo id="S2.SS2.p1.15.m4.2.2.1.1.1.1.1" xref="S2.SS2.p1.15.m4.2.2.1.1.1.1.1.cmml">−</mo><mn id="S2.SS2.p1.15.m4.2.2.1.1.1.1.3" xref="S2.SS2.p1.15.m4.2.2.1.1.1.1.3.cmml">1</mn></mrow></mrow><mo id="S2.SS2.p1.15.m4.2.2.1.3" stretchy="false" xref="S2.SS2.p1.15.m4.2.2.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.15.m4.2b"><apply id="S2.SS2.p1.15.m4.2.2.2.cmml" xref="S2.SS2.p1.15.m4.2.2.1"><csymbol cd="latexml" id="S2.SS2.p1.15.m4.2.2.2.1.cmml" xref="S2.SS2.p1.15.m4.2.2.1.2">ket</csymbol><list id="S2.SS2.p1.15.m4.2.2.1.1.2.cmml" xref="S2.SS2.p1.15.m4.2.2.1.1.1"><ci id="S2.SS2.p1.15.m4.1.1.cmml" xref="S2.SS2.p1.15.m4.1.1">𝐹</ci><apply id="S2.SS2.p1.15.m4.2.2.1.1.1.1.cmml" xref="S2.SS2.p1.15.m4.2.2.1.1.1.1"><minus id="S2.SS2.p1.15.m4.2.2.1.1.1.1.1.cmml" xref="S2.SS2.p1.15.m4.2.2.1.1.1.1.1"></minus><apply id="S2.SS2.p1.15.m4.2.2.1.1.1.1.2.cmml" xref="S2.SS2.p1.15.m4.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.p1.15.m4.2.2.1.1.1.1.2.1.cmml" xref="S2.SS2.p1.15.m4.2.2.1.1.1.1.2">subscript</csymbol><ci id="S2.SS2.p1.15.m4.2.2.1.1.1.1.2.2.cmml" xref="S2.SS2.p1.15.m4.2.2.1.1.1.1.2.2">𝑚</ci><ci id="S2.SS2.p1.15.m4.2.2.1.1.1.1.2.3.cmml" xref="S2.SS2.p1.15.m4.2.2.1.1.1.1.2.3">𝐹</ci></apply><cn id="S2.SS2.p1.15.m4.2.2.1.1.1.1.3.cmml" type="integer" xref="S2.SS2.p1.15.m4.2.2.1.1.1.1.3">1</cn></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.15.m4.2c">|F,m_{F}-1\rangle</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.15.m4.2d">| italic_F , italic_m start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT - 1 ⟩</annotation></semantics></math> from Eqs. (<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S2.E5" title="In II.2 Heading error analysis ‣ II Theoretical Background ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">5</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S2.E6" title="In II.2 Heading error analysis ‣ II Theoretical Background ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">6</span></a>) as</p> <table class="ltx_equation ltx_eqn_table" id="S2.E7"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="~{}\Delta E=(-g_{I}\mu_{B}\pm\mu_{eff})\mathrm{B}\mp\frac{\mu_{eff}^{2}\mathrm% {B}^{2}}{\hbar\omega_{hf}}(2m_{F}-1)," class="ltx_Math" display="block" id="S2.E7.m1.1"><semantics id="S2.E7.m1.1a"><mrow id="S2.E7.m1.1.1.1" xref="S2.E7.m1.1.1.1.1.cmml"><mrow id="S2.E7.m1.1.1.1.1" xref="S2.E7.m1.1.1.1.1.cmml"><mrow id="S2.E7.m1.1.1.1.1.4" xref="S2.E7.m1.1.1.1.1.4.cmml"><mi id="S2.E7.m1.1.1.1.1.4.2" mathvariant="normal" xref="S2.E7.m1.1.1.1.1.4.2.cmml">Δ</mi><mo id="S2.E7.m1.1.1.1.1.4.1" xref="S2.E7.m1.1.1.1.1.4.1.cmml"></mo><mi id="S2.E7.m1.1.1.1.1.4.3" xref="S2.E7.m1.1.1.1.1.4.3.cmml">E</mi></mrow><mo id="S2.E7.m1.1.1.1.1.3" xref="S2.E7.m1.1.1.1.1.3.cmml">=</mo><mrow id="S2.E7.m1.1.1.1.1.2" xref="S2.E7.m1.1.1.1.1.2.cmml"><mrow id="S2.E7.m1.1.1.1.1.1.1" xref="S2.E7.m1.1.1.1.1.1.1.cmml"><mrow id="S2.E7.m1.1.1.1.1.1.1.1.1" xref="S2.E7.m1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.E7.m1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.E7.m1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.E7.m1.1.1.1.1.1.1.1.1.1" xref="S2.E7.m1.1.1.1.1.1.1.1.1.1.cmml"><mrow id="S2.E7.m1.1.1.1.1.1.1.1.1.1.2" xref="S2.E7.m1.1.1.1.1.1.1.1.1.1.2.cmml"><mo id="S2.E7.m1.1.1.1.1.1.1.1.1.1.2a" xref="S2.E7.m1.1.1.1.1.1.1.1.1.1.2.cmml">−</mo><mrow id="S2.E7.m1.1.1.1.1.1.1.1.1.1.2.2" xref="S2.E7.m1.1.1.1.1.1.1.1.1.1.2.2.cmml"><msub id="S2.E7.m1.1.1.1.1.1.1.1.1.1.2.2.2" xref="S2.E7.m1.1.1.1.1.1.1.1.1.1.2.2.2.cmml"><mi id="S2.E7.m1.1.1.1.1.1.1.1.1.1.2.2.2.2" xref="S2.E7.m1.1.1.1.1.1.1.1.1.1.2.2.2.2.cmml">g</mi><mi id="S2.E7.m1.1.1.1.1.1.1.1.1.1.2.2.2.3" xref="S2.E7.m1.1.1.1.1.1.1.1.1.1.2.2.2.3.cmml">I</mi></msub><mo id="S2.E7.m1.1.1.1.1.1.1.1.1.1.2.2.1" xref="S2.E7.m1.1.1.1.1.1.1.1.1.1.2.2.1.cmml"></mo><msub id="S2.E7.m1.1.1.1.1.1.1.1.1.1.2.2.3" xref="S2.E7.m1.1.1.1.1.1.1.1.1.1.2.2.3.cmml"><mi id="S2.E7.m1.1.1.1.1.1.1.1.1.1.2.2.3.2" xref="S2.E7.m1.1.1.1.1.1.1.1.1.1.2.2.3.2.cmml">μ</mi><mi id="S2.E7.m1.1.1.1.1.1.1.1.1.1.2.2.3.3" xref="S2.E7.m1.1.1.1.1.1.1.1.1.1.2.2.3.3.cmml">B</mi></msub></mrow></mrow><mo id="S2.E7.m1.1.1.1.1.1.1.1.1.1.1" xref="S2.E7.m1.1.1.1.1.1.1.1.1.1.1.cmml">±</mo><msub id="S2.E7.m1.1.1.1.1.1.1.1.1.1.3" xref="S2.E7.m1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S2.E7.m1.1.1.1.1.1.1.1.1.1.3.2" xref="S2.E7.m1.1.1.1.1.1.1.1.1.1.3.2.cmml">μ</mi><mrow id="S2.E7.m1.1.1.1.1.1.1.1.1.1.3.3" xref="S2.E7.m1.1.1.1.1.1.1.1.1.1.3.3.cmml"><mi id="S2.E7.m1.1.1.1.1.1.1.1.1.1.3.3.2" xref="S2.E7.m1.1.1.1.1.1.1.1.1.1.3.3.2.cmml">e</mi><mo id="S2.E7.m1.1.1.1.1.1.1.1.1.1.3.3.1" xref="S2.E7.m1.1.1.1.1.1.1.1.1.1.3.3.1.cmml"></mo><mi id="S2.E7.m1.1.1.1.1.1.1.1.1.1.3.3.3" xref="S2.E7.m1.1.1.1.1.1.1.1.1.1.3.3.3.cmml">f</mi><mo id="S2.E7.m1.1.1.1.1.1.1.1.1.1.3.3.1a" xref="S2.E7.m1.1.1.1.1.1.1.1.1.1.3.3.1.cmml"></mo><mi id="S2.E7.m1.1.1.1.1.1.1.1.1.1.3.3.4" xref="S2.E7.m1.1.1.1.1.1.1.1.1.1.3.3.4.cmml">f</mi></mrow></msub></mrow><mo id="S2.E7.m1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.E7.m1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S2.E7.m1.1.1.1.1.1.1.2" xref="S2.E7.m1.1.1.1.1.1.1.2.cmml"></mo><mi id="S2.E7.m1.1.1.1.1.1.1.3" mathvariant="normal" xref="S2.E7.m1.1.1.1.1.1.1.3.cmml">B</mi></mrow><mo 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start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ± italic_μ start_POSTSUBSCRIPT italic_e italic_f italic_f end_POSTSUBSCRIPT ) roman_B ∓ divide start_ARG italic_μ start_POSTSUBSCRIPT italic_e italic_f italic_f end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_B start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG roman_ℏ italic_ω start_POSTSUBSCRIPT italic_h italic_f end_POSTSUBSCRIPT end_ARG ( 2 italic_m start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT - 1 ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(7)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p1.16">with <math alttext="\mu_{eff}=(g_{s}+g_{I})\mu_{B}/(2I+1)" class="ltx_Math" display="inline" id="S2.SS2.p1.16.m1.2"><semantics 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end_POSTSUBSCRIPT + italic_g start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT ) italic_μ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT / ( 2 italic_I + 1 )</annotation></semantics></math> as the effective atomic magnetic dipole moment.</p> </div> <div class="ltx_para" id="S2.SS2.p2"> <p class="ltx_p" id="S2.SS2.p2.3">The first term on the right side of Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S2.E7" title="In II.2 Heading error analysis ‣ II Theoretical Background ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">7</span></a>) corresponds to the aforementioned LZE, and it introduces an energy difference of 2<math alttext="g_{I}\mu_{B}" class="ltx_Math" display="inline" id="S2.SS2.p2.1.m1.1"><semantics id="S2.SS2.p2.1.m1.1a"><mrow id="S2.SS2.p2.1.m1.1.1" xref="S2.SS2.p2.1.m1.1.1.cmml"><msub id="S2.SS2.p2.1.m1.1.1.2" xref="S2.SS2.p2.1.m1.1.1.2.cmml"><mi id="S2.SS2.p2.1.m1.1.1.2.2" xref="S2.SS2.p2.1.m1.1.1.2.2.cmml">g</mi><mi id="S2.SS2.p2.1.m1.1.1.2.3" xref="S2.SS2.p2.1.m1.1.1.2.3.cmml">I</mi></msub><mo id="S2.SS2.p2.1.m1.1.1.1" xref="S2.SS2.p2.1.m1.1.1.1.cmml"></mo><msub id="S2.SS2.p2.1.m1.1.1.3" xref="S2.SS2.p2.1.m1.1.1.3.cmml"><mi id="S2.SS2.p2.1.m1.1.1.3.2" xref="S2.SS2.p2.1.m1.1.1.3.2.cmml">μ</mi><mi id="S2.SS2.p2.1.m1.1.1.3.3" xref="S2.SS2.p2.1.m1.1.1.3.3.cmml">B</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.1.m1.1b"><apply id="S2.SS2.p2.1.m1.1.1.cmml" xref="S2.SS2.p2.1.m1.1.1"><times id="S2.SS2.p2.1.m1.1.1.1.cmml" xref="S2.SS2.p2.1.m1.1.1.1"></times><apply id="S2.SS2.p2.1.m1.1.1.2.cmml" xref="S2.SS2.p2.1.m1.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.p2.1.m1.1.1.2.1.cmml" xref="S2.SS2.p2.1.m1.1.1.2">subscript</csymbol><ci id="S2.SS2.p2.1.m1.1.1.2.2.cmml" xref="S2.SS2.p2.1.m1.1.1.2.2">𝑔</ci><ci id="S2.SS2.p2.1.m1.1.1.2.3.cmml" xref="S2.SS2.p2.1.m1.1.1.2.3">𝐼</ci></apply><apply id="S2.SS2.p2.1.m1.1.1.3.cmml" xref="S2.SS2.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p2.1.m1.1.1.3.1.cmml" xref="S2.SS2.p2.1.m1.1.1.3">subscript</csymbol><ci id="S2.SS2.p2.1.m1.1.1.3.2.cmml" xref="S2.SS2.p2.1.m1.1.1.3.2">𝜇</ci><ci id="S2.SS2.p2.1.m1.1.1.3.3.cmml" xref="S2.SS2.p2.1.m1.1.1.3.3">𝐵</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.1.m1.1c">g_{I}\mu_{B}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.1.m1.1d">italic_g start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT</annotation></semantics></math>B between the Larmor frequency of atom spin in the upper ground hyperfine state and that in the lower one. The second term corresponds to the NLZE. For an optically pumped magnetometer based on <sup class="ltx_sup" id="S2.SS2.p2.3.1"><span class="ltx_text ltx_font_italic" id="S2.SS2.p2.3.1.1">87</span></sup>Rb atoms in upper ground hyperfine states and the high-polarization limit as discussed in the previous subsection, we follow the treatment in Ref. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib22" title="">22</a>]</cite> by assuming a spin temperature atomic population distribution, and the measured field with the NLZE induced heading error is related with the measured Larmor frequency <math alttext="\omega_{L}" class="ltx_Math" display="inline" id="S2.SS2.p2.3.m3.1"><semantics id="S2.SS2.p2.3.m3.1a"><msub id="S2.SS2.p2.3.m3.1.1" xref="S2.SS2.p2.3.m3.1.1.cmml"><mi id="S2.SS2.p2.3.m3.1.1.2" xref="S2.SS2.p2.3.m3.1.1.2.cmml">ω</mi><mi id="S2.SS2.p2.3.m3.1.1.3" xref="S2.SS2.p2.3.m3.1.1.3.cmml">L</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.3.m3.1b"><apply id="S2.SS2.p2.3.m3.1.1.cmml" xref="S2.SS2.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.3.m3.1.1.1.cmml" xref="S2.SS2.p2.3.m3.1.1">subscript</csymbol><ci id="S2.SS2.p2.3.m3.1.1.2.cmml" xref="S2.SS2.p2.3.m3.1.1.2">𝜔</ci><ci id="S2.SS2.p2.3.m3.1.1.3.cmml" xref="S2.SS2.p2.3.m3.1.1.3">𝐿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.3.m3.1c">\omega_{L}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.3.m3.1d">italic_ω start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math> as</p> <table class="ltx_equation ltx_eqn_table" id="S2.E8"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="~{}\mathrm{B}_{87}\approx\frac{4\hbar\omega_{L}}{(g_{s}-3g_{I})\mu_{B}}\left[1% +\frac{3\omega_{L}}{\omega_{hf}}\cos\theta\frac{P(7+P^{2})}{5+3P^{2}}\right]." 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xref="S2.E8.m1.2.2.3.3.3.3">2</cn></apply></apply></apply></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E8.m1.3c">~{}\mathrm{B}_{87}\approx\frac{4\hbar\omega_{L}}{(g_{s}-3g_{I})\mu_{B}}\left[1% +\frac{3\omega_{L}}{\omega_{hf}}\cos\theta\frac{P(7+P^{2})}{5+3P^{2}}\right].</annotation><annotation encoding="application/x-llamapun" id="S2.E8.m1.3d">roman_B start_POSTSUBSCRIPT 87 end_POSTSUBSCRIPT ≈ divide start_ARG 4 roman_ℏ italic_ω start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT end_ARG start_ARG ( italic_g start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT - 3 italic_g start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT ) italic_μ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT end_ARG [ 1 + divide start_ARG 3 italic_ω start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT end_ARG start_ARG italic_ω start_POSTSUBSCRIPT italic_h italic_f end_POSTSUBSCRIPT end_ARG roman_cos italic_θ divide start_ARG italic_P ( 7 + italic_P start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_ARG start_ARG 5 + 3 italic_P start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ] .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(8)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p2.5">Please note that the equation above is different from the one in Ref. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib22" title="">22</a>]</cite> because <math alttext="\theta" class="ltx_Math" display="inline" id="S2.SS2.p2.4.m1.1"><semantics id="S2.SS2.p2.4.m1.1a"><mi id="S2.SS2.p2.4.m1.1.1" xref="S2.SS2.p2.4.m1.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.4.m1.1b"><ci id="S2.SS2.p2.4.m1.1.1.cmml" xref="S2.SS2.p2.4.m1.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.4.m1.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.4.m1.1d">italic_θ</annotation></semantics></math> is defined differently. The result extracted from <sup class="ltx_sup" id="S2.SS2.p2.5.1"><span class="ltx_text ltx_font_italic" id="S2.SS2.p2.5.1.1">85</span></sup>Rb atoms treated in the same approximation is:</p> <table class="ltx_equation ltx_eqn_table" id="S2.E9"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="~{}\mathrm{B}_{85}\approx\frac{6\hbar\omega_{L}}{(g_{s}-5g_{I})\mu_{B}}\left[1% +\frac{3\omega_{L}}{\omega_{hf}}\cos\theta\frac{P(P^{4}+18P^{2}+21)}{3P^{4}+14% P^{2}+7}\right]." class="ltx_Math" display="block" id="S2.E9.m1.3"><semantics id="S2.E9.m1.3a"><mrow id="S2.E9.m1.3.3.1" xref="S2.E9.m1.3.3.1.1.cmml"><mrow id="S2.E9.m1.3.3.1.1" xref="S2.E9.m1.3.3.1.1.cmml"><msub id="S2.E9.m1.3.3.1.1.3" xref="S2.E9.m1.3.3.1.1.3.cmml"><mi id="S2.E9.m1.3.3.1.1.3.2" mathvariant="normal" xref="S2.E9.m1.3.3.1.1.3.2.cmml">B</mi><mn id="S2.E9.m1.3.3.1.1.3.3" xref="S2.E9.m1.3.3.1.1.3.3.cmml">85</mn></msub><mo id="S2.E9.m1.3.3.1.1.2" xref="S2.E9.m1.3.3.1.1.2.cmml">≈</mo><mrow id="S2.E9.m1.3.3.1.1.1" xref="S2.E9.m1.3.3.1.1.1.cmml"><mfrac id="S2.E9.m1.1.1" xref="S2.E9.m1.1.1.cmml"><mrow id="S2.E9.m1.1.1.3" xref="S2.E9.m1.1.1.3.cmml"><mn id="S2.E9.m1.1.1.3.2" xref="S2.E9.m1.1.1.3.2.cmml">6</mn><mo id="S2.E9.m1.1.1.3.1" xref="S2.E9.m1.1.1.3.1.cmml"></mo><mi id="S2.E9.m1.1.1.3.3" mathvariant="normal" xref="S2.E9.m1.1.1.3.3.cmml">ℏ</mi><mo id="S2.E9.m1.1.1.3.1a" xref="S2.E9.m1.1.1.3.1.cmml"></mo><msub id="S2.E9.m1.1.1.3.4" xref="S2.E9.m1.1.1.3.4.cmml"><mi id="S2.E9.m1.1.1.3.4.2" xref="S2.E9.m1.1.1.3.4.2.cmml">ω</mi><mi id="S2.E9.m1.1.1.3.4.3" xref="S2.E9.m1.1.1.3.4.3.cmml">L</mi></msub></mrow><mrow id="S2.E9.m1.1.1.1" xref="S2.E9.m1.1.1.1.cmml"><mrow id="S2.E9.m1.1.1.1.1.1" xref="S2.E9.m1.1.1.1.1.1.1.cmml"><mo id="S2.E9.m1.1.1.1.1.1.2" stretchy="false" xref="S2.E9.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.E9.m1.1.1.1.1.1.1" 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xref="S2.E9.m1.2.2.3.4">7</cn></apply></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E9.m1.3c">~{}\mathrm{B}_{85}\approx\frac{6\hbar\omega_{L}}{(g_{s}-5g_{I})\mu_{B}}\left[1% +\frac{3\omega_{L}}{\omega_{hf}}\cos\theta\frac{P(P^{4}+18P^{2}+21)}{3P^{4}+14% P^{2}+7}\right].</annotation><annotation encoding="application/x-llamapun" id="S2.E9.m1.3d">roman_B start_POSTSUBSCRIPT 85 end_POSTSUBSCRIPT ≈ divide start_ARG 6 roman_ℏ italic_ω start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT end_ARG start_ARG ( italic_g start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT - 5 italic_g start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT ) italic_μ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT end_ARG [ 1 + divide start_ARG 3 italic_ω start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT end_ARG start_ARG italic_ω start_POSTSUBSCRIPT italic_h italic_f end_POSTSUBSCRIPT end_ARG roman_cos italic_θ divide start_ARG italic_P ( italic_P start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT + 18 italic_P start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 21 ) end_ARG start_ARG 3 italic_P start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT + 14 italic_P start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 7 end_ARG ] .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(9)</span></td> </tr></tbody> </table> </div> </section> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">III </span>Experiment setup and measurement scheme</h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.7">The magnetometer sensor sits in the middle of five-layer mu-metal shields, with a bias field <math alttext="\mathrm{B}_{z}" class="ltx_Math" display="inline" id="S3.p1.1.m1.1"><semantics id="S3.p1.1.m1.1a"><msub id="S3.p1.1.m1.1.1" xref="S3.p1.1.m1.1.1.cmml"><mi id="S3.p1.1.m1.1.1.2" mathvariant="normal" xref="S3.p1.1.m1.1.1.2.cmml">B</mi><mi id="S3.p1.1.m1.1.1.3" xref="S3.p1.1.m1.1.1.3.cmml">z</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p1.1.m1.1b"><apply id="S3.p1.1.m1.1.1.cmml" xref="S3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.p1.1.m1.1.1.1.cmml" xref="S3.p1.1.m1.1.1">subscript</csymbol><ci id="S3.p1.1.m1.1.1.2.cmml" xref="S3.p1.1.m1.1.1.2">B</ci><ci id="S3.p1.1.m1.1.1.3.cmml" xref="S3.p1.1.m1.1.1.3">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.1.m1.1c">\mathrm{B}_{z}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.1.m1.1d">roman_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT</annotation></semantics></math> generated by solenoid coils inside the shields, as shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S3.F1" title="Figure 1 ‣ III Experiment setup and measurement scheme ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">1</span></a>(a). In order to minimize the magnetic field gradients, we have compensated field gradient components of <math alttext="\partial\mathrm{B}_{z}/\partial z" class="ltx_Math" display="inline" id="S3.p1.2.m2.1"><semantics id="S3.p1.2.m2.1a"><mrow id="S3.p1.2.m2.1.1" xref="S3.p1.2.m2.1.1.cmml"><mo id="S3.p1.2.m2.1.1.1" rspace="0em" xref="S3.p1.2.m2.1.1.1.cmml">∂</mo><mrow id="S3.p1.2.m2.1.1.2" xref="S3.p1.2.m2.1.1.2.cmml"><msub id="S3.p1.2.m2.1.1.2.2" xref="S3.p1.2.m2.1.1.2.2.cmml"><mi id="S3.p1.2.m2.1.1.2.2.2" mathvariant="normal" xref="S3.p1.2.m2.1.1.2.2.2.cmml">B</mi><mi id="S3.p1.2.m2.1.1.2.2.3" xref="S3.p1.2.m2.1.1.2.2.3.cmml">z</mi></msub><mo id="S3.p1.2.m2.1.1.2.1" xref="S3.p1.2.m2.1.1.2.1.cmml">/</mo><mrow id="S3.p1.2.m2.1.1.2.3" xref="S3.p1.2.m2.1.1.2.3.cmml"><mo id="S3.p1.2.m2.1.1.2.3.1" lspace="0em" rspace="0em" xref="S3.p1.2.m2.1.1.2.3.1.cmml">∂</mo><mi id="S3.p1.2.m2.1.1.2.3.2" xref="S3.p1.2.m2.1.1.2.3.2.cmml">z</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.2.m2.1b"><apply id="S3.p1.2.m2.1.1.cmml" xref="S3.p1.2.m2.1.1"><partialdiff id="S3.p1.2.m2.1.1.1.cmml" xref="S3.p1.2.m2.1.1.1"></partialdiff><apply id="S3.p1.2.m2.1.1.2.cmml" xref="S3.p1.2.m2.1.1.2"><divide id="S3.p1.2.m2.1.1.2.1.cmml" xref="S3.p1.2.m2.1.1.2.1"></divide><apply id="S3.p1.2.m2.1.1.2.2.cmml" xref="S3.p1.2.m2.1.1.2.2"><csymbol cd="ambiguous" id="S3.p1.2.m2.1.1.2.2.1.cmml" xref="S3.p1.2.m2.1.1.2.2">subscript</csymbol><ci id="S3.p1.2.m2.1.1.2.2.2.cmml" xref="S3.p1.2.m2.1.1.2.2.2">B</ci><ci id="S3.p1.2.m2.1.1.2.2.3.cmml" xref="S3.p1.2.m2.1.1.2.2.3">𝑧</ci></apply><apply id="S3.p1.2.m2.1.1.2.3.cmml" xref="S3.p1.2.m2.1.1.2.3"><partialdiff id="S3.p1.2.m2.1.1.2.3.1.cmml" xref="S3.p1.2.m2.1.1.2.3.1"></partialdiff><ci id="S3.p1.2.m2.1.1.2.3.2.cmml" xref="S3.p1.2.m2.1.1.2.3.2">𝑧</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.2.m2.1c">\partial\mathrm{B}_{z}/\partial z</annotation><annotation encoding="application/x-llamapun" id="S3.p1.2.m2.1d">∂ roman_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT / ∂ italic_z</annotation></semantics></math>, <math alttext="\partial^{2}\mathrm{B}_{z}/\partial z^{2}" class="ltx_Math" display="inline" id="S3.p1.3.m3.1"><semantics id="S3.p1.3.m3.1a"><mrow id="S3.p1.3.m3.1.1" xref="S3.p1.3.m3.1.1.cmml"><msup id="S3.p1.3.m3.1.1.1" xref="S3.p1.3.m3.1.1.1.cmml"><mo id="S3.p1.3.m3.1.1.1.2" xref="S3.p1.3.m3.1.1.1.2.cmml">∂</mo><mn id="S3.p1.3.m3.1.1.1.3" xref="S3.p1.3.m3.1.1.1.3.cmml">2</mn></msup><mrow id="S3.p1.3.m3.1.1.2" xref="S3.p1.3.m3.1.1.2.cmml"><msub id="S3.p1.3.m3.1.1.2.2" xref="S3.p1.3.m3.1.1.2.2.cmml"><mi id="S3.p1.3.m3.1.1.2.2.2" mathvariant="normal" xref="S3.p1.3.m3.1.1.2.2.2.cmml">B</mi><mi id="S3.p1.3.m3.1.1.2.2.3" xref="S3.p1.3.m3.1.1.2.2.3.cmml">z</mi></msub><mo id="S3.p1.3.m3.1.1.2.1" xref="S3.p1.3.m3.1.1.2.1.cmml">/</mo><mrow id="S3.p1.3.m3.1.1.2.3" xref="S3.p1.3.m3.1.1.2.3.cmml"><mo id="S3.p1.3.m3.1.1.2.3.1" lspace="0em" rspace="0em" xref="S3.p1.3.m3.1.1.2.3.1.cmml">∂</mo><msup id="S3.p1.3.m3.1.1.2.3.2" xref="S3.p1.3.m3.1.1.2.3.2.cmml"><mi id="S3.p1.3.m3.1.1.2.3.2.2" xref="S3.p1.3.m3.1.1.2.3.2.2.cmml">z</mi><mn id="S3.p1.3.m3.1.1.2.3.2.3" xref="S3.p1.3.m3.1.1.2.3.2.3.cmml">2</mn></msup></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.3.m3.1b"><apply id="S3.p1.3.m3.1.1.cmml" xref="S3.p1.3.m3.1.1"><apply id="S3.p1.3.m3.1.1.1.cmml" xref="S3.p1.3.m3.1.1.1"><csymbol cd="ambiguous" id="S3.p1.3.m3.1.1.1.1.cmml" xref="S3.p1.3.m3.1.1.1">superscript</csymbol><partialdiff id="S3.p1.3.m3.1.1.1.2.cmml" xref="S3.p1.3.m3.1.1.1.2"></partialdiff><cn id="S3.p1.3.m3.1.1.1.3.cmml" type="integer" xref="S3.p1.3.m3.1.1.1.3">2</cn></apply><apply id="S3.p1.3.m3.1.1.2.cmml" xref="S3.p1.3.m3.1.1.2"><divide id="S3.p1.3.m3.1.1.2.1.cmml" xref="S3.p1.3.m3.1.1.2.1"></divide><apply id="S3.p1.3.m3.1.1.2.2.cmml" xref="S3.p1.3.m3.1.1.2.2"><csymbol cd="ambiguous" id="S3.p1.3.m3.1.1.2.2.1.cmml" 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/ ∂ italic_z start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>, and <math alttext="\partial\mathrm{B}_{z}/\partial x" class="ltx_Math" display="inline" id="S3.p1.4.m4.1"><semantics id="S3.p1.4.m4.1a"><mrow id="S3.p1.4.m4.1.1" xref="S3.p1.4.m4.1.1.cmml"><mo id="S3.p1.4.m4.1.1.1" rspace="0em" xref="S3.p1.4.m4.1.1.1.cmml">∂</mo><mrow id="S3.p1.4.m4.1.1.2" xref="S3.p1.4.m4.1.1.2.cmml"><msub id="S3.p1.4.m4.1.1.2.2" xref="S3.p1.4.m4.1.1.2.2.cmml"><mi id="S3.p1.4.m4.1.1.2.2.2" mathvariant="normal" xref="S3.p1.4.m4.1.1.2.2.2.cmml">B</mi><mi id="S3.p1.4.m4.1.1.2.2.3" xref="S3.p1.4.m4.1.1.2.2.3.cmml">z</mi></msub><mo id="S3.p1.4.m4.1.1.2.1" xref="S3.p1.4.m4.1.1.2.1.cmml">/</mo><mrow id="S3.p1.4.m4.1.1.2.3" xref="S3.p1.4.m4.1.1.2.3.cmml"><mo id="S3.p1.4.m4.1.1.2.3.1" lspace="0em" rspace="0em" xref="S3.p1.4.m4.1.1.2.3.1.cmml">∂</mo><mi id="S3.p1.4.m4.1.1.2.3.2" xref="S3.p1.4.m4.1.1.2.3.2.cmml">x</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.4.m4.1b"><apply id="S3.p1.4.m4.1.1.cmml" xref="S3.p1.4.m4.1.1"><partialdiff id="S3.p1.4.m4.1.1.1.cmml" xref="S3.p1.4.m4.1.1.1"></partialdiff><apply id="S3.p1.4.m4.1.1.2.cmml" xref="S3.p1.4.m4.1.1.2"><divide id="S3.p1.4.m4.1.1.2.1.cmml" xref="S3.p1.4.m4.1.1.2.1"></divide><apply id="S3.p1.4.m4.1.1.2.2.cmml" xref="S3.p1.4.m4.1.1.2.2"><csymbol cd="ambiguous" id="S3.p1.4.m4.1.1.2.2.1.cmml" xref="S3.p1.4.m4.1.1.2.2">subscript</csymbol><ci id="S3.p1.4.m4.1.1.2.2.2.cmml" xref="S3.p1.4.m4.1.1.2.2.2">B</ci><ci id="S3.p1.4.m4.1.1.2.2.3.cmml" xref="S3.p1.4.m4.1.1.2.2.3">𝑧</ci></apply><apply id="S3.p1.4.m4.1.1.2.3.cmml" xref="S3.p1.4.m4.1.1.2.3"><partialdiff id="S3.p1.4.m4.1.1.2.3.1.cmml" xref="S3.p1.4.m4.1.1.2.3.1"></partialdiff><ci id="S3.p1.4.m4.1.1.2.3.2.cmml" xref="S3.p1.4.m4.1.1.2.3.2">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.4.m4.1c">\partial\mathrm{B}_{z}/\partial x</annotation><annotation encoding="application/x-llamapun" id="S3.p1.4.m4.1d">∂ roman_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT / ∂ italic_x</annotation></semantics></math>. In the end, the field gradient of <math alttext="\mathrm{B}_{z}" class="ltx_Math" display="inline" id="S3.p1.5.m5.1"><semantics id="S3.p1.5.m5.1a"><msub id="S3.p1.5.m5.1.1" xref="S3.p1.5.m5.1.1.cmml"><mi id="S3.p1.5.m5.1.1.2" mathvariant="normal" xref="S3.p1.5.m5.1.1.2.cmml">B</mi><mi id="S3.p1.5.m5.1.1.3" xref="S3.p1.5.m5.1.1.3.cmml">z</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p1.5.m5.1b"><apply id="S3.p1.5.m5.1.1.cmml" xref="S3.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S3.p1.5.m5.1.1.1.cmml" xref="S3.p1.5.m5.1.1">subscript</csymbol><ci id="S3.p1.5.m5.1.1.2.cmml" xref="S3.p1.5.m5.1.1.2">B</ci><ci id="S3.p1.5.m5.1.1.3.cmml" xref="S3.p1.5.m5.1.1.3">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.5.m5.1c">\mathrm{B}_{z}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.5.m5.1d">roman_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT</annotation></semantics></math> along <math alttext="x" class="ltx_Math" display="inline" id="S3.p1.6.m6.1"><semantics id="S3.p1.6.m6.1a"><mi id="S3.p1.6.m6.1.1" xref="S3.p1.6.m6.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.p1.6.m6.1b"><ci id="S3.p1.6.m6.1.1.cmml" xref="S3.p1.6.m6.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.6.m6.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.p1.6.m6.1d">italic_x</annotation></semantics></math> and <math alttext="z" class="ltx_Math" display="inline" id="S3.p1.7.m7.1"><semantics id="S3.p1.7.m7.1a"><mi id="S3.p1.7.m7.1.1" xref="S3.p1.7.m7.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S3.p1.7.m7.1b"><ci id="S3.p1.7.m7.1.1.cmml" xref="S3.p1.7.m7.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.7.m7.1c">z</annotation><annotation encoding="application/x-llamapun" id="S3.p1.7.m7.1d">italic_z</annotation></semantics></math> directions are both less than 0.1 nT/mm. The sensor is also connected with a high-precision rotation table outside the shields through an epoxy holder, so that the relative orientation between the pumping beam and the bias field direction can be precisely controlled.</p> </div> <figure class="ltx_figure" id="S3.F1"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="410" id="S3.F1.g1" src="x1.png" width="415"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S3.F1.2.1.1" style="font-size:90%;">Figure 1</span>: </span><span class="ltx_text" id="S3.F1.3.2" style="font-size:90%;"> Illustration of the experiment setup, with the inset as the top view of the 3D-printed optical platform for the scalar FID magnetometer.</span></figcaption> </figure> <div class="ltx_para" id="S3.p2"> <p class="ltx_p" id="S3.p2.4">Figure <a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S3.F1" title="Figure 1 ‣ III Experiment setup and measurement scheme ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">1</span></a>(b) shows the optical platform of the magnetometer, which is manufactured by high-precision 3D printing technology. To improve the signal-to-noise ratio of the magnetometer, a Herriott-cavity-assisted vapor cell <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib30" title="">30</a>]</cite> is used to increase the interaction length between light and atoms. This Herriott cavity consists of two identical cylindrical mirrors <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib31" title="">31</a>]</cite>, with a radius of curvature of 60 mm, a diameter of 12.7 mm, a thickness of 2.5 mm, a separation of 11.5 mm and a relative angle between symmetrical axes of 52.2<sup class="ltx_sup" id="S3.p2.4.1">∘</sup>. Such a cavity is bonded on a piece of silicon wafer and attached to the glass cell using the anodic bonding technique <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib32" title="">32</a>]</cite>. As discussed in Ref. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib30" title="">30</a>]</cite>, the direction of the beam that passes through the cavity is defined by the averaged beam orientation inside the cavity, which is along the connection of both cavity mirror centers. The atomic cell is filled with N<sub class="ltx_sub" id="S3.p2.4.2">2</sub> gas of 400 Torr and a droplet of <sup class="ltx_sup" id="S3.p2.4.3"><span class="ltx_text ltx_font_italic" id="S3.p2.4.3.1">87</span></sup>Rb atoms. This cell is normally operated at a temperature around 75<sup class="ltx_sup" id="S3.p2.4.4">∘</sup>C by running high-frequency ac currents through ceramic heaters.</p> </div> <div class="ltx_para" id="S3.p3"> <p class="ltx_p" id="S3.p3.2">The pumping and probe beams are independently generated by distributed-Bragg-reflector (DBR) laser diodes, and coupled to the sensor by polarization-maintaining fibers. To identify and suppress certain sources of heading error, we couple two pumping beams on resonance with Rb D1 transition into the sensor. These two beams are combined by a polarizing beam-splitter (PBS) so that they have orthogonal polarizations afterwards. They are circularly polarized after passing through a quarter-wave plate (<math alttext="\lambda/4" class="ltx_Math" display="inline" id="S3.p3.1.m1.1"><semantics id="S3.p3.1.m1.1a"><mrow id="S3.p3.1.m1.1.1" xref="S3.p3.1.m1.1.1.cmml"><mi id="S3.p3.1.m1.1.1.2" xref="S3.p3.1.m1.1.1.2.cmml">λ</mi><mo id="S3.p3.1.m1.1.1.1" xref="S3.p3.1.m1.1.1.1.cmml">/</mo><mn id="S3.p3.1.m1.1.1.3" xref="S3.p3.1.m1.1.1.3.cmml">4</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.1.m1.1b"><apply id="S3.p3.1.m1.1.1.cmml" xref="S3.p3.1.m1.1.1"><divide id="S3.p3.1.m1.1.1.1.cmml" xref="S3.p3.1.m1.1.1.1"></divide><ci id="S3.p3.1.m1.1.1.2.cmml" xref="S3.p3.1.m1.1.1.2">𝜆</ci><cn id="S3.p3.1.m1.1.1.3.cmml" type="integer" xref="S3.p3.1.m1.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.1.m1.1c">\lambda/4</annotation><annotation encoding="application/x-llamapun" id="S3.p3.1.m1.1d">italic_λ / 4</annotation></semantics></math>), and combined with the probe beam, which is 54 GHz blue detuned from the Rb D1 line, by a non-polarizing beam-splitter before entering the atomic cell. The transmitted probe beam is analyzed by a polarimeter, which consists of a half-wave plate (<math alttext="\lambda/2" class="ltx_Math" display="inline" id="S3.p3.2.m2.1"><semantics id="S3.p3.2.m2.1a"><mrow id="S3.p3.2.m2.1.1" xref="S3.p3.2.m2.1.1.cmml"><mi id="S3.p3.2.m2.1.1.2" xref="S3.p3.2.m2.1.1.2.cmml">λ</mi><mo id="S3.p3.2.m2.1.1.1" xref="S3.p3.2.m2.1.1.1.cmml">/</mo><mn id="S3.p3.2.m2.1.1.3" xref="S3.p3.2.m2.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.2.m2.1b"><apply id="S3.p3.2.m2.1.1.cmml" xref="S3.p3.2.m2.1.1"><divide id="S3.p3.2.m2.1.1.1.cmml" xref="S3.p3.2.m2.1.1.1"></divide><ci id="S3.p3.2.m2.1.1.2.cmml" xref="S3.p3.2.m2.1.1.2">𝜆</ci><cn id="S3.p3.2.m2.1.1.3.cmml" type="integer" xref="S3.p3.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.2.m2.1c">\lambda/2</annotation><annotation encoding="application/x-llamapun" id="S3.p3.2.m2.1d">italic_λ / 2</annotation></semantics></math>), a PBS and two photodiode detectors (PD).</p> </div> <div class="ltx_para" id="S3.p4"> <p class="ltx_p" id="S3.p4.2">The magnetometer operates in a pulsed pump-probe mode with a period of 16 ms, and the bandwidth of this magnetometer is 31.25 Hz limited by the Nyquist sampling theorem. In the first half period, one of the pumping beam is turned on with its power modulated at a frequency close to the atomic Larmor frequency by a fiber acoustic-optical modulator with a duty cycle of 20%. The time-averaged pumping beam power before entering the cell is 1.13 mW. In the next half period, the pumping beam is turned off, and the FID signal of atomic polarization is recorded. During the whole period, the probe beam is kept on with an input beam power of 0.7 mW. A typical signal is shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S3.F2" title="Figure 2 ‣ III Experiment setup and measurement scheme ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">2</span></a> (a), which includes contributions from both transverse (<math alttext="V_{t}(t)" class="ltx_Math" display="inline" id="S3.p4.1.m1.1"><semantics id="S3.p4.1.m1.1a"><mrow id="S3.p4.1.m1.1.2" xref="S3.p4.1.m1.1.2.cmml"><msub id="S3.p4.1.m1.1.2.2" xref="S3.p4.1.m1.1.2.2.cmml"><mi id="S3.p4.1.m1.1.2.2.2" xref="S3.p4.1.m1.1.2.2.2.cmml">V</mi><mi id="S3.p4.1.m1.1.2.2.3" xref="S3.p4.1.m1.1.2.2.3.cmml">t</mi></msub><mo id="S3.p4.1.m1.1.2.1" xref="S3.p4.1.m1.1.2.1.cmml"></mo><mrow id="S3.p4.1.m1.1.2.3.2" xref="S3.p4.1.m1.1.2.cmml"><mo id="S3.p4.1.m1.1.2.3.2.1" stretchy="false" xref="S3.p4.1.m1.1.2.cmml">(</mo><mi id="S3.p4.1.m1.1.1" xref="S3.p4.1.m1.1.1.cmml">t</mi><mo id="S3.p4.1.m1.1.2.3.2.2" stretchy="false" xref="S3.p4.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.1.m1.1b"><apply id="S3.p4.1.m1.1.2.cmml" xref="S3.p4.1.m1.1.2"><times id="S3.p4.1.m1.1.2.1.cmml" xref="S3.p4.1.m1.1.2.1"></times><apply id="S3.p4.1.m1.1.2.2.cmml" xref="S3.p4.1.m1.1.2.2"><csymbol cd="ambiguous" id="S3.p4.1.m1.1.2.2.1.cmml" xref="S3.p4.1.m1.1.2.2">subscript</csymbol><ci id="S3.p4.1.m1.1.2.2.2.cmml" xref="S3.p4.1.m1.1.2.2.2">𝑉</ci><ci id="S3.p4.1.m1.1.2.2.3.cmml" xref="S3.p4.1.m1.1.2.2.3">𝑡</ci></apply><ci id="S3.p4.1.m1.1.1.cmml" xref="S3.p4.1.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.1.m1.1c">V_{t}(t)</annotation><annotation encoding="application/x-llamapun" id="S3.p4.1.m1.1d">italic_V start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math>) and longitudinal (<math alttext="V_{l}(t)" class="ltx_Math" display="inline" id="S3.p4.2.m2.1"><semantics id="S3.p4.2.m2.1a"><mrow id="S3.p4.2.m2.1.2" xref="S3.p4.2.m2.1.2.cmml"><msub id="S3.p4.2.m2.1.2.2" xref="S3.p4.2.m2.1.2.2.cmml"><mi id="S3.p4.2.m2.1.2.2.2" xref="S3.p4.2.m2.1.2.2.2.cmml">V</mi><mi id="S3.p4.2.m2.1.2.2.3" xref="S3.p4.2.m2.1.2.2.3.cmml">l</mi></msub><mo id="S3.p4.2.m2.1.2.1" xref="S3.p4.2.m2.1.2.1.cmml"></mo><mrow id="S3.p4.2.m2.1.2.3.2" xref="S3.p4.2.m2.1.2.cmml"><mo id="S3.p4.2.m2.1.2.3.2.1" stretchy="false" xref="S3.p4.2.m2.1.2.cmml">(</mo><mi id="S3.p4.2.m2.1.1" xref="S3.p4.2.m2.1.1.cmml">t</mi><mo id="S3.p4.2.m2.1.2.3.2.2" stretchy="false" xref="S3.p4.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.2.m2.1b"><apply id="S3.p4.2.m2.1.2.cmml" xref="S3.p4.2.m2.1.2"><times id="S3.p4.2.m2.1.2.1.cmml" xref="S3.p4.2.m2.1.2.1"></times><apply id="S3.p4.2.m2.1.2.2.cmml" xref="S3.p4.2.m2.1.2.2"><csymbol cd="ambiguous" id="S3.p4.2.m2.1.2.2.1.cmml" xref="S3.p4.2.m2.1.2.2">subscript</csymbol><ci id="S3.p4.2.m2.1.2.2.2.cmml" xref="S3.p4.2.m2.1.2.2.2">𝑉</ci><ci id="S3.p4.2.m2.1.2.2.3.cmml" xref="S3.p4.2.m2.1.2.2.3">𝑙</ci></apply><ci id="S3.p4.2.m2.1.1.cmml" xref="S3.p4.2.m2.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.2.m2.1c">V_{l}(t)</annotation><annotation encoding="application/x-llamapun" id="S3.p4.2.m2.1d">italic_V start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math>) polarizations:</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S6.EGx2"> <tbody id="S3.E10"> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle V(t)" class="ltx_Math" display="inline" id="S3.Ex1.m1.1"><semantics id="S3.Ex1.m1.1a"><mrow id="S3.Ex1.m1.1.2" xref="S3.Ex1.m1.1.2.cmml"><mi id="S3.Ex1.m1.1.2.2" xref="S3.Ex1.m1.1.2.2.cmml">V</mi><mo id="S3.Ex1.m1.1.2.1" xref="S3.Ex1.m1.1.2.1.cmml"></mo><mrow id="S3.Ex1.m1.1.2.3.2" xref="S3.Ex1.m1.1.2.cmml"><mo id="S3.Ex1.m1.1.2.3.2.1" stretchy="false" xref="S3.Ex1.m1.1.2.cmml">(</mo><mi id="S3.Ex1.m1.1.1" xref="S3.Ex1.m1.1.1.cmml">t</mi><mo id="S3.Ex1.m1.1.2.3.2.2" stretchy="false" xref="S3.Ex1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex1.m1.1b"><apply id="S3.Ex1.m1.1.2.cmml" xref="S3.Ex1.m1.1.2"><times id="S3.Ex1.m1.1.2.1.cmml" xref="S3.Ex1.m1.1.2.1"></times><ci id="S3.Ex1.m1.1.2.2.cmml" xref="S3.Ex1.m1.1.2.2">𝑉</ci><ci id="S3.Ex1.m1.1.1.cmml" xref="S3.Ex1.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex1.m1.1c">\displaystyle V(t)</annotation><annotation encoding="application/x-llamapun" id="S3.Ex1.m1.1d">italic_V ( italic_t )</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S3.Ex1.m2.1"><semantics id="S3.Ex1.m2.1a"><mo id="S3.Ex1.m2.1.1" xref="S3.Ex1.m2.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S3.Ex1.m2.1b"><eq id="S3.Ex1.m2.1.1.cmml" xref="S3.Ex1.m2.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex1.m2.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="S3.Ex1.m2.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle V_{t}(t)+V_{l}(t)+V_{0}" class="ltx_Math" display="inline" id="S3.Ex1.m3.2"><semantics id="S3.Ex1.m3.2a"><mrow id="S3.Ex1.m3.2.3" xref="S3.Ex1.m3.2.3.cmml"><mrow id="S3.Ex1.m3.2.3.2" xref="S3.Ex1.m3.2.3.2.cmml"><msub id="S3.Ex1.m3.2.3.2.2" xref="S3.Ex1.m3.2.3.2.2.cmml"><mi id="S3.Ex1.m3.2.3.2.2.2" xref="S3.Ex1.m3.2.3.2.2.2.cmml">V</mi><mi id="S3.Ex1.m3.2.3.2.2.3" xref="S3.Ex1.m3.2.3.2.2.3.cmml">t</mi></msub><mo id="S3.Ex1.m3.2.3.2.1" xref="S3.Ex1.m3.2.3.2.1.cmml"></mo><mrow id="S3.Ex1.m3.2.3.2.3.2" xref="S3.Ex1.m3.2.3.2.cmml"><mo id="S3.Ex1.m3.2.3.2.3.2.1" stretchy="false" xref="S3.Ex1.m3.2.3.2.cmml">(</mo><mi id="S3.Ex1.m3.1.1" xref="S3.Ex1.m3.1.1.cmml">t</mi><mo id="S3.Ex1.m3.2.3.2.3.2.2" stretchy="false" xref="S3.Ex1.m3.2.3.2.cmml">)</mo></mrow></mrow><mo id="S3.Ex1.m3.2.3.1" xref="S3.Ex1.m3.2.3.1.cmml">+</mo><mrow id="S3.Ex1.m3.2.3.3" xref="S3.Ex1.m3.2.3.3.cmml"><msub id="S3.Ex1.m3.2.3.3.2" xref="S3.Ex1.m3.2.3.3.2.cmml"><mi id="S3.Ex1.m3.2.3.3.2.2" xref="S3.Ex1.m3.2.3.3.2.2.cmml">V</mi><mi id="S3.Ex1.m3.2.3.3.2.3" xref="S3.Ex1.m3.2.3.3.2.3.cmml">l</mi></msub><mo id="S3.Ex1.m3.2.3.3.1" xref="S3.Ex1.m3.2.3.3.1.cmml"></mo><mrow id="S3.Ex1.m3.2.3.3.3.2" xref="S3.Ex1.m3.2.3.3.cmml"><mo id="S3.Ex1.m3.2.3.3.3.2.1" stretchy="false" xref="S3.Ex1.m3.2.3.3.cmml">(</mo><mi id="S3.Ex1.m3.2.2" xref="S3.Ex1.m3.2.2.cmml">t</mi><mo id="S3.Ex1.m3.2.3.3.3.2.2" stretchy="false" xref="S3.Ex1.m3.2.3.3.cmml">)</mo></mrow></mrow><mo id="S3.Ex1.m3.2.3.1a" xref="S3.Ex1.m3.2.3.1.cmml">+</mo><msub id="S3.Ex1.m3.2.3.4" xref="S3.Ex1.m3.2.3.4.cmml"><mi id="S3.Ex1.m3.2.3.4.2" xref="S3.Ex1.m3.2.3.4.2.cmml">V</mi><mn id="S3.Ex1.m3.2.3.4.3" xref="S3.Ex1.m3.2.3.4.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex1.m3.2b"><apply id="S3.Ex1.m3.2.3.cmml" xref="S3.Ex1.m3.2.3"><plus id="S3.Ex1.m3.2.3.1.cmml" xref="S3.Ex1.m3.2.3.1"></plus><apply id="S3.Ex1.m3.2.3.2.cmml" xref="S3.Ex1.m3.2.3.2"><times id="S3.Ex1.m3.2.3.2.1.cmml" xref="S3.Ex1.m3.2.3.2.1"></times><apply id="S3.Ex1.m3.2.3.2.2.cmml" xref="S3.Ex1.m3.2.3.2.2"><csymbol cd="ambiguous" id="S3.Ex1.m3.2.3.2.2.1.cmml" xref="S3.Ex1.m3.2.3.2.2">subscript</csymbol><ci id="S3.Ex1.m3.2.3.2.2.2.cmml" xref="S3.Ex1.m3.2.3.2.2.2">𝑉</ci><ci id="S3.Ex1.m3.2.3.2.2.3.cmml" xref="S3.Ex1.m3.2.3.2.2.3">𝑡</ci></apply><ci id="S3.Ex1.m3.1.1.cmml" xref="S3.Ex1.m3.1.1">𝑡</ci></apply><apply id="S3.Ex1.m3.2.3.3.cmml" xref="S3.Ex1.m3.2.3.3"><times id="S3.Ex1.m3.2.3.3.1.cmml" xref="S3.Ex1.m3.2.3.3.1"></times><apply id="S3.Ex1.m3.2.3.3.2.cmml" xref="S3.Ex1.m3.2.3.3.2"><csymbol cd="ambiguous" id="S3.Ex1.m3.2.3.3.2.1.cmml" xref="S3.Ex1.m3.2.3.3.2">subscript</csymbol><ci id="S3.Ex1.m3.2.3.3.2.2.cmml" xref="S3.Ex1.m3.2.3.3.2.2">𝑉</ci><ci id="S3.Ex1.m3.2.3.3.2.3.cmml" xref="S3.Ex1.m3.2.3.3.2.3">𝑙</ci></apply><ci id="S3.Ex1.m3.2.2.cmml" xref="S3.Ex1.m3.2.2">𝑡</ci></apply><apply id="S3.Ex1.m3.2.3.4.cmml" xref="S3.Ex1.m3.2.3.4"><csymbol cd="ambiguous" id="S3.Ex1.m3.2.3.4.1.cmml" xref="S3.Ex1.m3.2.3.4">subscript</csymbol><ci id="S3.Ex1.m3.2.3.4.2.cmml" xref="S3.Ex1.m3.2.3.4.2">𝑉</ci><cn id="S3.Ex1.m3.2.3.4.3.cmml" type="integer" xref="S3.Ex1.m3.2.3.4.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex1.m3.2c">\displaystyle V_{t}(t)+V_{l}(t)+V_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex1.m3.2d">italic_V start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_t ) + italic_V start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_t ) + italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="2"><span class="ltx_tag ltx_tag_equation ltx_align_right">(10)</span></td> </tr> <tr class="ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S3.E10.m1.1"><semantics id="S3.E10.m1.1a"><mo id="S3.E10.m1.1.1" xref="S3.E10.m1.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S3.E10.m1.1b"><eq id="S3.E10.m1.1.1.cmml" xref="S3.E10.m1.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S3.E10.m1.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="S3.E10.m1.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math 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xref="S3.E10.m2.6.6.1.1.4.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E10.m2.6c">\displaystyle V_{t,0}\sin(\omega_{L}t+\phi)e^{{-t}/{T_{2}}}+V_{l,0}e^{{-t}/{T_% {1}}}+V_{0},</annotation><annotation encoding="application/x-llamapun" id="S3.E10.m2.6d">italic_V start_POSTSUBSCRIPT italic_t , 0 end_POSTSUBSCRIPT roman_sin ( italic_ω start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT italic_t + italic_ϕ ) italic_e start_POSTSUPERSCRIPT - italic_t / italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT + italic_V start_POSTSUBSCRIPT italic_l , 0 end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT - italic_t / italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT + italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </tbody> </table> <p class="ltx_p" id="S3.p4.8">where <math alttext="V_{t,0}" class="ltx_Math" display="inline" id="S3.p4.3.m1.2"><semantics id="S3.p4.3.m1.2a"><msub id="S3.p4.3.m1.2.3" xref="S3.p4.3.m1.2.3.cmml"><mi id="S3.p4.3.m1.2.3.2" xref="S3.p4.3.m1.2.3.2.cmml">V</mi><mrow id="S3.p4.3.m1.2.2.2.4" xref="S3.p4.3.m1.2.2.2.3.cmml"><mi id="S3.p4.3.m1.1.1.1.1" xref="S3.p4.3.m1.1.1.1.1.cmml">t</mi><mo id="S3.p4.3.m1.2.2.2.4.1" xref="S3.p4.3.m1.2.2.2.3.cmml">,</mo><mn id="S3.p4.3.m1.2.2.2.2" xref="S3.p4.3.m1.2.2.2.2.cmml">0</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.p4.3.m1.2b"><apply id="S3.p4.3.m1.2.3.cmml" xref="S3.p4.3.m1.2.3"><csymbol cd="ambiguous" id="S3.p4.3.m1.2.3.1.cmml" xref="S3.p4.3.m1.2.3">subscript</csymbol><ci id="S3.p4.3.m1.2.3.2.cmml" xref="S3.p4.3.m1.2.3.2">𝑉</ci><list id="S3.p4.3.m1.2.2.2.3.cmml" xref="S3.p4.3.m1.2.2.2.4"><ci id="S3.p4.3.m1.1.1.1.1.cmml" xref="S3.p4.3.m1.1.1.1.1">𝑡</ci><cn id="S3.p4.3.m1.2.2.2.2.cmml" type="integer" xref="S3.p4.3.m1.2.2.2.2">0</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.3.m1.2c">V_{t,0}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.3.m1.2d">italic_V start_POSTSUBSCRIPT italic_t , 0 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="V_{l,0}" class="ltx_Math" display="inline" id="S3.p4.4.m2.2"><semantics id="S3.p4.4.m2.2a"><msub id="S3.p4.4.m2.2.3" xref="S3.p4.4.m2.2.3.cmml"><mi id="S3.p4.4.m2.2.3.2" xref="S3.p4.4.m2.2.3.2.cmml">V</mi><mrow id="S3.p4.4.m2.2.2.2.4" xref="S3.p4.4.m2.2.2.2.3.cmml"><mi id="S3.p4.4.m2.1.1.1.1" xref="S3.p4.4.m2.1.1.1.1.cmml">l</mi><mo id="S3.p4.4.m2.2.2.2.4.1" xref="S3.p4.4.m2.2.2.2.3.cmml">,</mo><mn id="S3.p4.4.m2.2.2.2.2" xref="S3.p4.4.m2.2.2.2.2.cmml">0</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.p4.4.m2.2b"><apply id="S3.p4.4.m2.2.3.cmml" xref="S3.p4.4.m2.2.3"><csymbol cd="ambiguous" id="S3.p4.4.m2.2.3.1.cmml" xref="S3.p4.4.m2.2.3">subscript</csymbol><ci id="S3.p4.4.m2.2.3.2.cmml" xref="S3.p4.4.m2.2.3.2">𝑉</ci><list id="S3.p4.4.m2.2.2.2.3.cmml" xref="S3.p4.4.m2.2.2.2.4"><ci id="S3.p4.4.m2.1.1.1.1.cmml" xref="S3.p4.4.m2.1.1.1.1">𝑙</ci><cn id="S3.p4.4.m2.2.2.2.2.cmml" type="integer" xref="S3.p4.4.m2.2.2.2.2">0</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.4.m2.2c">V_{l,0}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.4.m2.2d">italic_V start_POSTSUBSCRIPT italic_l , 0 end_POSTSUBSCRIPT</annotation></semantics></math> corresponds to the initial amplitude of the transverse and longitudinal signal, respectively. The free parameters <math alttext="T_{1}" class="ltx_Math" display="inline" id="S3.p4.5.m3.1"><semantics id="S3.p4.5.m3.1a"><msub id="S3.p4.5.m3.1.1" xref="S3.p4.5.m3.1.1.cmml"><mi id="S3.p4.5.m3.1.1.2" xref="S3.p4.5.m3.1.1.2.cmml">T</mi><mn id="S3.p4.5.m3.1.1.3" xref="S3.p4.5.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.p4.5.m3.1b"><apply id="S3.p4.5.m3.1.1.cmml" xref="S3.p4.5.m3.1.1"><csymbol cd="ambiguous" id="S3.p4.5.m3.1.1.1.cmml" xref="S3.p4.5.m3.1.1">subscript</csymbol><ci id="S3.p4.5.m3.1.1.2.cmml" xref="S3.p4.5.m3.1.1.2">𝑇</ci><cn id="S3.p4.5.m3.1.1.3.cmml" type="integer" xref="S3.p4.5.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.5.m3.1c">T_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.5.m3.1d">italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="T_{2}" class="ltx_Math" display="inline" id="S3.p4.6.m4.1"><semantics id="S3.p4.6.m4.1a"><msub id="S3.p4.6.m4.1.1" xref="S3.p4.6.m4.1.1.cmml"><mi id="S3.p4.6.m4.1.1.2" xref="S3.p4.6.m4.1.1.2.cmml">T</mi><mn id="S3.p4.6.m4.1.1.3" xref="S3.p4.6.m4.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.p4.6.m4.1b"><apply id="S3.p4.6.m4.1.1.cmml" xref="S3.p4.6.m4.1.1"><csymbol cd="ambiguous" id="S3.p4.6.m4.1.1.1.cmml" xref="S3.p4.6.m4.1.1">subscript</csymbol><ci id="S3.p4.6.m4.1.1.2.cmml" xref="S3.p4.6.m4.1.1.2">𝑇</ci><cn id="S3.p4.6.m4.1.1.3.cmml" type="integer" xref="S3.p4.6.m4.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.6.m4.1c">T_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.6.m4.1d">italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> represent the decay time of the longitudinal and transverse atomic polarization, and <math alttext="V_{0}" class="ltx_Math" display="inline" id="S3.p4.7.m5.1"><semantics id="S3.p4.7.m5.1a"><msub id="S3.p4.7.m5.1.1" xref="S3.p4.7.m5.1.1.cmml"><mi id="S3.p4.7.m5.1.1.2" xref="S3.p4.7.m5.1.1.2.cmml">V</mi><mn id="S3.p4.7.m5.1.1.3" xref="S3.p4.7.m5.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S3.p4.7.m5.1b"><apply id="S3.p4.7.m5.1.1.cmml" xref="S3.p4.7.m5.1.1"><csymbol cd="ambiguous" id="S3.p4.7.m5.1.1.1.cmml" xref="S3.p4.7.m5.1.1">subscript</csymbol><ci id="S3.p4.7.m5.1.1.2.cmml" xref="S3.p4.7.m5.1.1.2">𝑉</ci><cn id="S3.p4.7.m5.1.1.3.cmml" type="integer" xref="S3.p4.7.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.7.m5.1c">V_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.7.m5.1d">italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> represents the offset voltage from the electronics. We can also separate the signals from these two polarization components by adding a high-pass or low-pass filter to the signal. In addition, we have also checked the atomic diffusion effects by assigning two free parameters to the transverse polarization decay time <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib33" title="">33</a>]</cite>, and found no difference in the extracted <math alttext="\omega_{L}" class="ltx_Math" display="inline" id="S3.p4.8.m6.1"><semantics id="S3.p4.8.m6.1a"><msub id="S3.p4.8.m6.1.1" xref="S3.p4.8.m6.1.1.cmml"><mi id="S3.p4.8.m6.1.1.2" xref="S3.p4.8.m6.1.1.2.cmml">ω</mi><mi id="S3.p4.8.m6.1.1.3" xref="S3.p4.8.m6.1.1.3.cmml">L</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p4.8.m6.1b"><apply id="S3.p4.8.m6.1.1.cmml" xref="S3.p4.8.m6.1.1"><csymbol cd="ambiguous" id="S3.p4.8.m6.1.1.1.cmml" xref="S3.p4.8.m6.1.1">subscript</csymbol><ci id="S3.p4.8.m6.1.1.2.cmml" xref="S3.p4.8.m6.1.1.2">𝜔</ci><ci id="S3.p4.8.m6.1.1.3.cmml" xref="S3.p4.8.m6.1.1.3">𝐿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.8.m6.1c">\omega_{L}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.8.m6.1d">italic_ω start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math>. Therefore, we neglect the atomic diffusion effect in this work.</p> </div> <figure class="ltx_figure" id="S3.F2"><img alt="Refer to caption" class="ltx_graphics ltx_img_portrait" height="676" id="S3.F2.g1" src="x2.png" width="415"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S3.F2.15.7.1" style="font-size:90%;">Figure 2</span>: </span><span class="ltx_text" id="S3.F2.12.6" style="font-size:90%;"> (a) Measured FID signals at B=10 <math alttext="\mu" class="ltx_Math" display="inline" id="S3.F2.7.1.m1.1"><semantics id="S3.F2.7.1.m1.1b"><mi id="S3.F2.7.1.m1.1.1" xref="S3.F2.7.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.F2.7.1.m1.1c"><ci id="S3.F2.7.1.m1.1.1.cmml" xref="S3.F2.7.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.7.1.m1.1d">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.F2.7.1.m1.1e">italic_μ</annotation></semantics></math>T and <math alttext="\theta" class="ltx_Math" display="inline" id="S3.F2.8.2.m2.1"><semantics id="S3.F2.8.2.m2.1b"><mi id="S3.F2.8.2.m2.1.1" xref="S3.F2.8.2.m2.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S3.F2.8.2.m2.1c"><ci id="S3.F2.8.2.m2.1.1.cmml" xref="S3.F2.8.2.m2.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.8.2.m2.1d">\theta</annotation><annotation encoding="application/x-llamapun" id="S3.F2.8.2.m2.1e">italic_θ</annotation></semantics></math>=45<sup class="ltx_sup" id="S3.F2.12.6.1">∘</sup>, with the white dash line showing the signal after a low-pass filter and the inset showing the signal after a high-pass filter. (b) The initial amplitude of transverse signals (<math alttext="V_{t,0}" class="ltx_Math" display="inline" id="S3.F2.10.4.m4.2"><semantics id="S3.F2.10.4.m4.2b"><msub id="S3.F2.10.4.m4.2.3" xref="S3.F2.10.4.m4.2.3.cmml"><mi id="S3.F2.10.4.m4.2.3.2" xref="S3.F2.10.4.m4.2.3.2.cmml">V</mi><mrow id="S3.F2.10.4.m4.2.2.2.4" xref="S3.F2.10.4.m4.2.2.2.3.cmml"><mi id="S3.F2.10.4.m4.1.1.1.1" xref="S3.F2.10.4.m4.1.1.1.1.cmml">t</mi><mo id="S3.F2.10.4.m4.2.2.2.4.1" xref="S3.F2.10.4.m4.2.2.2.3.cmml">,</mo><mn id="S3.F2.10.4.m4.2.2.2.2" xref="S3.F2.10.4.m4.2.2.2.2.cmml">0</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.F2.10.4.m4.2c"><apply id="S3.F2.10.4.m4.2.3.cmml" xref="S3.F2.10.4.m4.2.3"><csymbol cd="ambiguous" id="S3.F2.10.4.m4.2.3.1.cmml" xref="S3.F2.10.4.m4.2.3">subscript</csymbol><ci id="S3.F2.10.4.m4.2.3.2.cmml" xref="S3.F2.10.4.m4.2.3.2">𝑉</ci><list id="S3.F2.10.4.m4.2.2.2.3.cmml" xref="S3.F2.10.4.m4.2.2.2.4"><ci id="S3.F2.10.4.m4.1.1.1.1.cmml" xref="S3.F2.10.4.m4.1.1.1.1">𝑡</ci><cn id="S3.F2.10.4.m4.2.2.2.2.cmml" type="integer" xref="S3.F2.10.4.m4.2.2.2.2">0</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.10.4.m4.2d">V_{t,0}</annotation><annotation encoding="application/x-llamapun" id="S3.F2.10.4.m4.2e">italic_V start_POSTSUBSCRIPT italic_t , 0 end_POSTSUBSCRIPT</annotation></semantics></math>) as a function of <math alttext="\theta" class="ltx_Math" display="inline" id="S3.F2.11.5.m5.1"><semantics id="S3.F2.11.5.m5.1b"><mi id="S3.F2.11.5.m5.1.1" xref="S3.F2.11.5.m5.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S3.F2.11.5.m5.1c"><ci id="S3.F2.11.5.m5.1.1.cmml" xref="S3.F2.11.5.m5.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.11.5.m5.1d">\theta</annotation><annotation encoding="application/x-llamapun" id="S3.F2.11.5.m5.1e">italic_θ</annotation></semantics></math>, with the dotted line showing the <math alttext="\sin^{2}{\theta}" class="ltx_Math" display="inline" id="S3.F2.12.6.m6.1"><semantics id="S3.F2.12.6.m6.1b"><mrow id="S3.F2.12.6.m6.1.1" xref="S3.F2.12.6.m6.1.1.cmml"><msup id="S3.F2.12.6.m6.1.1.1" xref="S3.F2.12.6.m6.1.1.1.cmml"><mi id="S3.F2.12.6.m6.1.1.1.2" xref="S3.F2.12.6.m6.1.1.1.2.cmml">sin</mi><mn id="S3.F2.12.6.m6.1.1.1.3" xref="S3.F2.12.6.m6.1.1.1.3.cmml">2</mn></msup><mo id="S3.F2.12.6.m6.1.1b" lspace="0.167em" xref="S3.F2.12.6.m6.1.1.cmml"></mo><mi id="S3.F2.12.6.m6.1.1.2" xref="S3.F2.12.6.m6.1.1.2.cmml">θ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.F2.12.6.m6.1c"><apply id="S3.F2.12.6.m6.1.1.cmml" xref="S3.F2.12.6.m6.1.1"><apply id="S3.F2.12.6.m6.1.1.1.cmml" xref="S3.F2.12.6.m6.1.1.1"><csymbol cd="ambiguous" id="S3.F2.12.6.m6.1.1.1.1.cmml" xref="S3.F2.12.6.m6.1.1.1">superscript</csymbol><sin id="S3.F2.12.6.m6.1.1.1.2.cmml" xref="S3.F2.12.6.m6.1.1.1.2"></sin><cn id="S3.F2.12.6.m6.1.1.1.3.cmml" type="integer" xref="S3.F2.12.6.m6.1.1.1.3">2</cn></apply><ci id="S3.F2.12.6.m6.1.1.2.cmml" xref="S3.F2.12.6.m6.1.1.2">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.12.6.m6.1d">\sin^{2}{\theta}</annotation><annotation encoding="application/x-llamapun" id="S3.F2.12.6.m6.1e">roman_sin start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_θ</annotation></semantics></math> relation.</span></figcaption> </figure> <div class="ltx_para" id="S3.p5"> <p class="ltx_p" id="S3.p5.3">In the experiment, we focus on the transverse part of the FID signal, and get the fitting results of <math alttext="\omega_{L}" class="ltx_Math" display="inline" id="S3.p5.1.m1.1"><semantics id="S3.p5.1.m1.1a"><msub id="S3.p5.1.m1.1.1" xref="S3.p5.1.m1.1.1.cmml"><mi id="S3.p5.1.m1.1.1.2" xref="S3.p5.1.m1.1.1.2.cmml">ω</mi><mi id="S3.p5.1.m1.1.1.3" xref="S3.p5.1.m1.1.1.3.cmml">L</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p5.1.m1.1b"><apply id="S3.p5.1.m1.1.1.cmml" xref="S3.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S3.p5.1.m1.1.1.1.cmml" xref="S3.p5.1.m1.1.1">subscript</csymbol><ci id="S3.p5.1.m1.1.1.2.cmml" xref="S3.p5.1.m1.1.1.2">𝜔</ci><ci id="S3.p5.1.m1.1.1.3.cmml" xref="S3.p5.1.m1.1.1.3">𝐿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.1.m1.1c">\omega_{L}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.1.m1.1d">italic_ω start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math>. The magnetic field magnitude <math alttext="\mathrm{B}" class="ltx_Math" display="inline" id="S3.p5.2.m2.1"><semantics id="S3.p5.2.m2.1a"><mi id="S3.p5.2.m2.1.1" mathvariant="normal" xref="S3.p5.2.m2.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S3.p5.2.m2.1b"><ci id="S3.p5.2.m2.1.1.cmml" xref="S3.p5.2.m2.1.1">B</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.2.m2.1c">\mathrm{B}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.2.m2.1d">roman_B</annotation></semantics></math> is initially extracted from <math alttext="\omega_{L}" class="ltx_Math" display="inline" id="S3.p5.3.m3.1"><semantics id="S3.p5.3.m3.1a"><msub id="S3.p5.3.m3.1.1" xref="S3.p5.3.m3.1.1.cmml"><mi id="S3.p5.3.m3.1.1.2" xref="S3.p5.3.m3.1.1.2.cmml">ω</mi><mi id="S3.p5.3.m3.1.1.3" xref="S3.p5.3.m3.1.1.3.cmml">L</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p5.3.m3.1b"><apply id="S3.p5.3.m3.1.1.cmml" xref="S3.p5.3.m3.1.1"><csymbol cd="ambiguous" id="S3.p5.3.m3.1.1.1.cmml" xref="S3.p5.3.m3.1.1">subscript</csymbol><ci id="S3.p5.3.m3.1.1.2.cmml" xref="S3.p5.3.m3.1.1.2">𝜔</ci><ci id="S3.p5.3.m3.1.1.3.cmml" xref="S3.p5.3.m3.1.1.3">𝐿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.3.m3.1c">\omega_{L}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.3.m3.1d">italic_ω start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math> using the linear term in Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S2.E7" title="In II.2 Heading error analysis ‣ II Theoretical Background ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">7</span></a>),</p> <table class="ltx_equation ltx_eqn_table" id="S3.E11"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="~{}\mathrm{B}\approx\hbar\omega_{L}/(\mu_{eff}-g_{I}\mu_{B})." class="ltx_Math" display="block" id="S3.E11.m1.1"><semantics id="S3.E11.m1.1a"><mrow id="S3.E11.m1.1.1.1" xref="S3.E11.m1.1.1.1.1.cmml"><mrow id="S3.E11.m1.1.1.1.1" xref="S3.E11.m1.1.1.1.1.cmml"><mi id="S3.E11.m1.1.1.1.1.3" mathvariant="normal" xref="S3.E11.m1.1.1.1.1.3.cmml">B</mi><mo id="S3.E11.m1.1.1.1.1.2" xref="S3.E11.m1.1.1.1.1.2.cmml">≈</mo><mrow id="S3.E11.m1.1.1.1.1.1" 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xref="S3.E11.m1.1.1.1.1.1.1.1.1.3.2.2">𝑔</ci><ci id="S3.E11.m1.1.1.1.1.1.1.1.1.3.2.3.cmml" xref="S3.E11.m1.1.1.1.1.1.1.1.1.3.2.3">𝐼</ci></apply><apply id="S3.E11.m1.1.1.1.1.1.1.1.1.3.3.cmml" xref="S3.E11.m1.1.1.1.1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S3.E11.m1.1.1.1.1.1.1.1.1.3.3.1.cmml" xref="S3.E11.m1.1.1.1.1.1.1.1.1.3.3">subscript</csymbol><ci id="S3.E11.m1.1.1.1.1.1.1.1.1.3.3.2.cmml" xref="S3.E11.m1.1.1.1.1.1.1.1.1.3.3.2">𝜇</ci><ci id="S3.E11.m1.1.1.1.1.1.1.1.1.3.3.3.cmml" xref="S3.E11.m1.1.1.1.1.1.1.1.1.3.3.3">𝐵</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E11.m1.1c">~{}\mathrm{B}\approx\hbar\omega_{L}/(\mu_{eff}-g_{I}\mu_{B}).</annotation><annotation encoding="application/x-llamapun" id="S3.E11.m1.1d">roman_B ≈ roman_ℏ italic_ω start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT / ( italic_μ start_POSTSUBSCRIPT italic_e italic_f italic_f end_POSTSUBSCRIPT - italic_g start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(11)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p5.12">For an arbitrary <math alttext="\theta" class="ltx_Math" display="inline" id="S3.p5.4.m1.1"><semantics id="S3.p5.4.m1.1a"><mi id="S3.p5.4.m1.1.1" xref="S3.p5.4.m1.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S3.p5.4.m1.1b"><ci id="S3.p5.4.m1.1.1.cmml" xref="S3.p5.4.m1.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.4.m1.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S3.p5.4.m1.1d">italic_θ</annotation></semantics></math>, the atomic transverse polarization can be expressed as <math alttext="P_{t}(t)=P_{0}\sin\theta\cos(\omega t)e^{-t/T_{2}}" class="ltx_Math" 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xref="S3.p5.5.m2.3.3.1.5.3.2.3.2.cmml">T</mi><mn id="S3.p5.5.m2.3.3.1.5.3.2.3.3" xref="S3.p5.5.m2.3.3.1.5.3.2.3.3.cmml">2</mn></msub></mrow></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.5.m2.3b"><apply id="S3.p5.5.m2.3.3.cmml" xref="S3.p5.5.m2.3.3"><eq id="S3.p5.5.m2.3.3.2.cmml" xref="S3.p5.5.m2.3.3.2"></eq><apply id="S3.p5.5.m2.3.3.3.cmml" xref="S3.p5.5.m2.3.3.3"><times id="S3.p5.5.m2.3.3.3.1.cmml" xref="S3.p5.5.m2.3.3.3.1"></times><apply id="S3.p5.5.m2.3.3.3.2.cmml" xref="S3.p5.5.m2.3.3.3.2"><csymbol cd="ambiguous" id="S3.p5.5.m2.3.3.3.2.1.cmml" xref="S3.p5.5.m2.3.3.3.2">subscript</csymbol><ci id="S3.p5.5.m2.3.3.3.2.2.cmml" xref="S3.p5.5.m2.3.3.3.2.2">𝑃</ci><ci id="S3.p5.5.m2.3.3.3.2.3.cmml" xref="S3.p5.5.m2.3.3.3.2.3">𝑡</ci></apply><ci id="S3.p5.5.m2.1.1.cmml" xref="S3.p5.5.m2.1.1">𝑡</ci></apply><apply id="S3.p5.5.m2.3.3.1.cmml" xref="S3.p5.5.m2.3.3.1"><times id="S3.p5.5.m2.3.3.1.2.cmml" xref="S3.p5.5.m2.3.3.1.2"></times><apply 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id="S3.p5.5.m2.3.3.1.5.cmml" xref="S3.p5.5.m2.3.3.1.5"><csymbol cd="ambiguous" id="S3.p5.5.m2.3.3.1.5.1.cmml" xref="S3.p5.5.m2.3.3.1.5">superscript</csymbol><ci id="S3.p5.5.m2.3.3.1.5.2.cmml" xref="S3.p5.5.m2.3.3.1.5.2">𝑒</ci><apply id="S3.p5.5.m2.3.3.1.5.3.cmml" xref="S3.p5.5.m2.3.3.1.5.3"><minus id="S3.p5.5.m2.3.3.1.5.3.1.cmml" xref="S3.p5.5.m2.3.3.1.5.3"></minus><apply id="S3.p5.5.m2.3.3.1.5.3.2.cmml" xref="S3.p5.5.m2.3.3.1.5.3.2"><divide id="S3.p5.5.m2.3.3.1.5.3.2.1.cmml" xref="S3.p5.5.m2.3.3.1.5.3.2.1"></divide><ci id="S3.p5.5.m2.3.3.1.5.3.2.2.cmml" xref="S3.p5.5.m2.3.3.1.5.3.2.2">𝑡</ci><apply id="S3.p5.5.m2.3.3.1.5.3.2.3.cmml" xref="S3.p5.5.m2.3.3.1.5.3.2.3"><csymbol cd="ambiguous" id="S3.p5.5.m2.3.3.1.5.3.2.3.1.cmml" xref="S3.p5.5.m2.3.3.1.5.3.2.3">subscript</csymbol><ci id="S3.p5.5.m2.3.3.1.5.3.2.3.2.cmml" xref="S3.p5.5.m2.3.3.1.5.3.2.3.2">𝑇</ci><cn id="S3.p5.5.m2.3.3.1.5.3.2.3.3.cmml" type="integer" xref="S3.p5.5.m2.3.3.1.5.3.2.3.3">2</cn></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.5.m2.3c">P_{t}(t)=P_{0}\sin\theta\cos(\omega t)e^{-t/T_{2}}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.5.m2.3d">italic_P start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_t ) = italic_P start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT roman_sin italic_θ roman_cos ( italic_ω italic_t ) italic_e start_POSTSUPERSCRIPT - italic_t / italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> in the high-polarization limit, where <math alttext="P_{0}=\sqrt{P_{t,0}^{2}+P_{l,0}^{2}}" class="ltx_Math" display="inline" id="S3.p5.6.m3.4"><semantics id="S3.p5.6.m3.4a"><mrow id="S3.p5.6.m3.4.5" xref="S3.p5.6.m3.4.5.cmml"><msub id="S3.p5.6.m3.4.5.2" xref="S3.p5.6.m3.4.5.2.cmml"><mi id="S3.p5.6.m3.4.5.2.2" xref="S3.p5.6.m3.4.5.2.2.cmml">P</mi><mn id="S3.p5.6.m3.4.5.2.3" xref="S3.p5.6.m3.4.5.2.3.cmml">0</mn></msub><mo id="S3.p5.6.m3.4.5.1" xref="S3.p5.6.m3.4.5.1.cmml">=</mo><msqrt id="S3.p5.6.m3.4.4" xref="S3.p5.6.m3.4.4.cmml"><mrow id="S3.p5.6.m3.4.4.4" xref="S3.p5.6.m3.4.4.4.cmml"><msubsup id="S3.p5.6.m3.4.4.4.6" xref="S3.p5.6.m3.4.4.4.6.cmml"><mi id="S3.p5.6.m3.4.4.4.6.2.2" xref="S3.p5.6.m3.4.4.4.6.2.2.cmml">P</mi><mrow id="S3.p5.6.m3.2.2.2.2.2.4" xref="S3.p5.6.m3.2.2.2.2.2.3.cmml"><mi id="S3.p5.6.m3.1.1.1.1.1.1" xref="S3.p5.6.m3.1.1.1.1.1.1.cmml">t</mi><mo id="S3.p5.6.m3.2.2.2.2.2.4.1" xref="S3.p5.6.m3.2.2.2.2.2.3.cmml">,</mo><mn id="S3.p5.6.m3.2.2.2.2.2.2" xref="S3.p5.6.m3.2.2.2.2.2.2.cmml">0</mn></mrow><mn id="S3.p5.6.m3.4.4.4.6.3" xref="S3.p5.6.m3.4.4.4.6.3.cmml">2</mn></msubsup><mo id="S3.p5.6.m3.4.4.4.5" xref="S3.p5.6.m3.4.4.4.5.cmml">+</mo><msubsup id="S3.p5.6.m3.4.4.4.7" xref="S3.p5.6.m3.4.4.4.7.cmml"><mi id="S3.p5.6.m3.4.4.4.7.2.2" xref="S3.p5.6.m3.4.4.4.7.2.2.cmml">P</mi><mrow id="S3.p5.6.m3.4.4.4.4.2.4" xref="S3.p5.6.m3.4.4.4.4.2.3.cmml"><mi id="S3.p5.6.m3.3.3.3.3.1.1" xref="S3.p5.6.m3.3.3.3.3.1.1.cmml">l</mi><mo id="S3.p5.6.m3.4.4.4.4.2.4.1" xref="S3.p5.6.m3.4.4.4.4.2.3.cmml">,</mo><mn id="S3.p5.6.m3.4.4.4.4.2.2" xref="S3.p5.6.m3.4.4.4.4.2.2.cmml">0</mn></mrow><mn id="S3.p5.6.m3.4.4.4.7.3" xref="S3.p5.6.m3.4.4.4.7.3.cmml">2</mn></msubsup></mrow></msqrt></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.6.m3.4b"><apply id="S3.p5.6.m3.4.5.cmml" xref="S3.p5.6.m3.4.5"><eq id="S3.p5.6.m3.4.5.1.cmml" xref="S3.p5.6.m3.4.5.1"></eq><apply id="S3.p5.6.m3.4.5.2.cmml" xref="S3.p5.6.m3.4.5.2"><csymbol cd="ambiguous" id="S3.p5.6.m3.4.5.2.1.cmml" xref="S3.p5.6.m3.4.5.2">subscript</csymbol><ci id="S3.p5.6.m3.4.5.2.2.cmml" xref="S3.p5.6.m3.4.5.2.2">𝑃</ci><cn id="S3.p5.6.m3.4.5.2.3.cmml" type="integer" xref="S3.p5.6.m3.4.5.2.3">0</cn></apply><apply id="S3.p5.6.m3.4.4.cmml" xref="S3.p5.6.m3.4.4"><root id="S3.p5.6.m3.4.4a.cmml" xref="S3.p5.6.m3.4.4"></root><apply id="S3.p5.6.m3.4.4.4.cmml" xref="S3.p5.6.m3.4.4.4"><plus id="S3.p5.6.m3.4.4.4.5.cmml" xref="S3.p5.6.m3.4.4.4.5"></plus><apply id="S3.p5.6.m3.4.4.4.6.cmml" xref="S3.p5.6.m3.4.4.4.6"><csymbol cd="ambiguous" id="S3.p5.6.m3.4.4.4.6.1.cmml" xref="S3.p5.6.m3.4.4.4.6">superscript</csymbol><apply id="S3.p5.6.m3.4.4.4.6.2.cmml" xref="S3.p5.6.m3.4.4.4.6"><csymbol cd="ambiguous" id="S3.p5.6.m3.4.4.4.6.2.1.cmml" xref="S3.p5.6.m3.4.4.4.6">subscript</csymbol><ci id="S3.p5.6.m3.4.4.4.6.2.2.cmml" xref="S3.p5.6.m3.4.4.4.6.2.2">𝑃</ci><list id="S3.p5.6.m3.2.2.2.2.2.3.cmml" xref="S3.p5.6.m3.2.2.2.2.2.4"><ci id="S3.p5.6.m3.1.1.1.1.1.1.cmml" xref="S3.p5.6.m3.1.1.1.1.1.1">𝑡</ci><cn id="S3.p5.6.m3.2.2.2.2.2.2.cmml" type="integer" xref="S3.p5.6.m3.2.2.2.2.2.2">0</cn></list></apply><cn id="S3.p5.6.m3.4.4.4.6.3.cmml" type="integer" xref="S3.p5.6.m3.4.4.4.6.3">2</cn></apply><apply id="S3.p5.6.m3.4.4.4.7.cmml" xref="S3.p5.6.m3.4.4.4.7"><csymbol cd="ambiguous" id="S3.p5.6.m3.4.4.4.7.1.cmml" xref="S3.p5.6.m3.4.4.4.7">superscript</csymbol><apply id="S3.p5.6.m3.4.4.4.7.2.cmml" xref="S3.p5.6.m3.4.4.4.7"><csymbol cd="ambiguous" id="S3.p5.6.m3.4.4.4.7.2.1.cmml" xref="S3.p5.6.m3.4.4.4.7">subscript</csymbol><ci id="S3.p5.6.m3.4.4.4.7.2.2.cmml" xref="S3.p5.6.m3.4.4.4.7.2.2">𝑃</ci><list id="S3.p5.6.m3.4.4.4.4.2.3.cmml" xref="S3.p5.6.m3.4.4.4.4.2.4"><ci id="S3.p5.6.m3.3.3.3.3.1.1.cmml" xref="S3.p5.6.m3.3.3.3.3.1.1">𝑙</ci><cn id="S3.p5.6.m3.4.4.4.4.2.2.cmml" type="integer" xref="S3.p5.6.m3.4.4.4.4.2.2">0</cn></list></apply><cn id="S3.p5.6.m3.4.4.4.7.3.cmml" type="integer" xref="S3.p5.6.m3.4.4.4.7.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.6.m3.4c">P_{0}=\sqrt{P_{t,0}^{2}+P_{l,0}^{2}}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.6.m3.4d">italic_P start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = square-root start_ARG italic_P start_POSTSUBSCRIPT italic_t , 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_P start_POSTSUBSCRIPT italic_l , 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG</annotation></semantics></math> as shown in Eqs. (<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S2.E2" title="In II.1 Bell-Bloom optical pumping ‣ II Theoretical Background ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">2</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S2.E3" title="In II.1 Bell-Bloom optical pumping ‣ II Theoretical Background ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">3</span></a>). The recorded transverse signal <math alttext="V_{t}(t)" class="ltx_Math" display="inline" id="S3.p5.7.m4.1"><semantics id="S3.p5.7.m4.1a"><mrow id="S3.p5.7.m4.1.2" xref="S3.p5.7.m4.1.2.cmml"><msub id="S3.p5.7.m4.1.2.2" xref="S3.p5.7.m4.1.2.2.cmml"><mi id="S3.p5.7.m4.1.2.2.2" xref="S3.p5.7.m4.1.2.2.2.cmml">V</mi><mi id="S3.p5.7.m4.1.2.2.3" xref="S3.p5.7.m4.1.2.2.3.cmml">t</mi></msub><mo id="S3.p5.7.m4.1.2.1" xref="S3.p5.7.m4.1.2.1.cmml"></mo><mrow id="S3.p5.7.m4.1.2.3.2" xref="S3.p5.7.m4.1.2.cmml"><mo id="S3.p5.7.m4.1.2.3.2.1" stretchy="false" xref="S3.p5.7.m4.1.2.cmml">(</mo><mi id="S3.p5.7.m4.1.1" xref="S3.p5.7.m4.1.1.cmml">t</mi><mo id="S3.p5.7.m4.1.2.3.2.2" stretchy="false" xref="S3.p5.7.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.7.m4.1b"><apply id="S3.p5.7.m4.1.2.cmml" xref="S3.p5.7.m4.1.2"><times id="S3.p5.7.m4.1.2.1.cmml" xref="S3.p5.7.m4.1.2.1"></times><apply id="S3.p5.7.m4.1.2.2.cmml" xref="S3.p5.7.m4.1.2.2"><csymbol cd="ambiguous" id="S3.p5.7.m4.1.2.2.1.cmml" xref="S3.p5.7.m4.1.2.2">subscript</csymbol><ci id="S3.p5.7.m4.1.2.2.2.cmml" xref="S3.p5.7.m4.1.2.2.2">𝑉</ci><ci id="S3.p5.7.m4.1.2.2.3.cmml" xref="S3.p5.7.m4.1.2.2.3">𝑡</ci></apply><ci id="S3.p5.7.m4.1.1.cmml" xref="S3.p5.7.m4.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.7.m4.1c">V_{t}(t)</annotation><annotation encoding="application/x-llamapun" id="S3.p5.7.m4.1d">italic_V start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_t )</annotation></semantics></math> is proportional to the projection of the transverse polarization on the probe beam, which is <math alttext="V_{t}(t)\propto P_{t}(t)\sin\theta=P_{0}\sin^{2}\theta\cos(\omega t)e^{-t/T_{2}}" class="ltx_Math" display="inline" id="S3.p5.8.m5.4"><semantics id="S3.p5.8.m5.4a"><mrow id="S3.p5.8.m5.4.4" xref="S3.p5.8.m5.4.4.cmml"><mrow id="S3.p5.8.m5.4.4.3" xref="S3.p5.8.m5.4.4.3.cmml"><msub id="S3.p5.8.m5.4.4.3.2" xref="S3.p5.8.m5.4.4.3.2.cmml"><mi id="S3.p5.8.m5.4.4.3.2.2" xref="S3.p5.8.m5.4.4.3.2.2.cmml">V</mi><mi id="S3.p5.8.m5.4.4.3.2.3" xref="S3.p5.8.m5.4.4.3.2.3.cmml">t</mi></msub><mo id="S3.p5.8.m5.4.4.3.1" xref="S3.p5.8.m5.4.4.3.1.cmml"></mo><mrow id="S3.p5.8.m5.4.4.3.3.2" xref="S3.p5.8.m5.4.4.3.cmml"><mo id="S3.p5.8.m5.4.4.3.3.2.1" stretchy="false" xref="S3.p5.8.m5.4.4.3.cmml">(</mo><mi id="S3.p5.8.m5.1.1" xref="S3.p5.8.m5.1.1.cmml">t</mi><mo id="S3.p5.8.m5.4.4.3.3.2.2" stretchy="false" xref="S3.p5.8.m5.4.4.3.cmml">)</mo></mrow></mrow><mo id="S3.p5.8.m5.4.4.4" xref="S3.p5.8.m5.4.4.4.cmml">∝</mo><mrow id="S3.p5.8.m5.4.4.5" xref="S3.p5.8.m5.4.4.5.cmml"><msub id="S3.p5.8.m5.4.4.5.2" xref="S3.p5.8.m5.4.4.5.2.cmml"><mi id="S3.p5.8.m5.4.4.5.2.2" xref="S3.p5.8.m5.4.4.5.2.2.cmml">P</mi><mi id="S3.p5.8.m5.4.4.5.2.3" xref="S3.p5.8.m5.4.4.5.2.3.cmml">t</mi></msub><mo id="S3.p5.8.m5.4.4.5.1" xref="S3.p5.8.m5.4.4.5.1.cmml"></mo><mrow id="S3.p5.8.m5.4.4.5.3.2" xref="S3.p5.8.m5.4.4.5.cmml"><mo id="S3.p5.8.m5.4.4.5.3.2.1" 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italic_e start_POSTSUPERSCRIPT - italic_t / italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>. Therefore, the initial amplitude of the transverse signal (<math alttext="V_{t,0}(\theta)" class="ltx_Math" display="inline" id="S3.p5.9.m6.3"><semantics id="S3.p5.9.m6.3a"><mrow id="S3.p5.9.m6.3.4" xref="S3.p5.9.m6.3.4.cmml"><msub id="S3.p5.9.m6.3.4.2" xref="S3.p5.9.m6.3.4.2.cmml"><mi id="S3.p5.9.m6.3.4.2.2" xref="S3.p5.9.m6.3.4.2.2.cmml">V</mi><mrow id="S3.p5.9.m6.2.2.2.4" xref="S3.p5.9.m6.2.2.2.3.cmml"><mi id="S3.p5.9.m6.1.1.1.1" xref="S3.p5.9.m6.1.1.1.1.cmml">t</mi><mo id="S3.p5.9.m6.2.2.2.4.1" xref="S3.p5.9.m6.2.2.2.3.cmml">,</mo><mn id="S3.p5.9.m6.2.2.2.2" xref="S3.p5.9.m6.2.2.2.2.cmml">0</mn></mrow></msub><mo id="S3.p5.9.m6.3.4.1" xref="S3.p5.9.m6.3.4.1.cmml"></mo><mrow id="S3.p5.9.m6.3.4.3.2" xref="S3.p5.9.m6.3.4.cmml"><mo id="S3.p5.9.m6.3.4.3.2.1" stretchy="false" xref="S3.p5.9.m6.3.4.cmml">(</mo><mi id="S3.p5.9.m6.3.3" xref="S3.p5.9.m6.3.3.cmml">θ</mi><mo id="S3.p5.9.m6.3.4.3.2.2" stretchy="false" xref="S3.p5.9.m6.3.4.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.9.m6.3b"><apply id="S3.p5.9.m6.3.4.cmml" xref="S3.p5.9.m6.3.4"><times id="S3.p5.9.m6.3.4.1.cmml" xref="S3.p5.9.m6.3.4.1"></times><apply id="S3.p5.9.m6.3.4.2.cmml" xref="S3.p5.9.m6.3.4.2"><csymbol cd="ambiguous" id="S3.p5.9.m6.3.4.2.1.cmml" xref="S3.p5.9.m6.3.4.2">subscript</csymbol><ci id="S3.p5.9.m6.3.4.2.2.cmml" xref="S3.p5.9.m6.3.4.2.2">𝑉</ci><list id="S3.p5.9.m6.2.2.2.3.cmml" xref="S3.p5.9.m6.2.2.2.4"><ci id="S3.p5.9.m6.1.1.1.1.cmml" xref="S3.p5.9.m6.1.1.1.1">𝑡</ci><cn id="S3.p5.9.m6.2.2.2.2.cmml" type="integer" xref="S3.p5.9.m6.2.2.2.2">0</cn></list></apply><ci id="S3.p5.9.m6.3.3.cmml" xref="S3.p5.9.m6.3.3">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.9.m6.3c">V_{t,0}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S3.p5.9.m6.3d">italic_V start_POSTSUBSCRIPT italic_t , 0 end_POSTSUBSCRIPT ( italic_θ )</annotation></semantics></math>) follows a <math alttext="\sin^{2}(\theta)" class="ltx_Math" display="inline" id="S3.p5.10.m7.2"><semantics id="S3.p5.10.m7.2a"><mrow id="S3.p5.10.m7.2.2.1" xref="S3.p5.10.m7.2.2.2.cmml"><msup id="S3.p5.10.m7.2.2.1.1" xref="S3.p5.10.m7.2.2.1.1.cmml"><mi id="S3.p5.10.m7.2.2.1.1.2" xref="S3.p5.10.m7.2.2.1.1.2.cmml">sin</mi><mn id="S3.p5.10.m7.2.2.1.1.3" xref="S3.p5.10.m7.2.2.1.1.3.cmml">2</mn></msup><mo id="S3.p5.10.m7.2.2.1a" xref="S3.p5.10.m7.2.2.2.cmml"></mo><mrow id="S3.p5.10.m7.2.2.1.2" xref="S3.p5.10.m7.2.2.2.cmml"><mo id="S3.p5.10.m7.2.2.1.2.1" stretchy="false" xref="S3.p5.10.m7.2.2.2.cmml">(</mo><mi id="S3.p5.10.m7.1.1" xref="S3.p5.10.m7.1.1.cmml">θ</mi><mo id="S3.p5.10.m7.2.2.1.2.2" stretchy="false" xref="S3.p5.10.m7.2.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.10.m7.2b"><apply id="S3.p5.10.m7.2.2.2.cmml" xref="S3.p5.10.m7.2.2.1"><apply id="S3.p5.10.m7.2.2.1.1.cmml" xref="S3.p5.10.m7.2.2.1.1"><csymbol cd="ambiguous" id="S3.p5.10.m7.2.2.1.1.1.cmml" xref="S3.p5.10.m7.2.2.1.1">superscript</csymbol><sin id="S3.p5.10.m7.2.2.1.1.2.cmml" xref="S3.p5.10.m7.2.2.1.1.2"></sin><cn id="S3.p5.10.m7.2.2.1.1.3.cmml" type="integer" xref="S3.p5.10.m7.2.2.1.1.3">2</cn></apply><ci id="S3.p5.10.m7.1.1.cmml" xref="S3.p5.10.m7.1.1">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.10.m7.2c">\sin^{2}(\theta)</annotation><annotation encoding="application/x-llamapun" id="S3.p5.10.m7.2d">roman_sin start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ )</annotation></semantics></math> function in the high-polarization limit, which is confirmed by the results in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S3.F2" title="Figure 2 ‣ III Experiment setup and measurement scheme ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">2</span></a>(b). This indicates a detection dead zone when <math alttext="\theta" class="ltx_Math" display="inline" id="S3.p5.11.m8.1"><semantics id="S3.p5.11.m8.1a"><mi id="S3.p5.11.m8.1.1" xref="S3.p5.11.m8.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S3.p5.11.m8.1b"><ci id="S3.p5.11.m8.1.1.cmml" xref="S3.p5.11.m8.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.11.m8.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S3.p5.11.m8.1d">italic_θ</annotation></semantics></math> is close to zero, and the measurement in this work is limited in the range of <math alttext="25^{\circ}\leq\theta\leq 155^{\circ}" class="ltx_Math" display="inline" id="S3.p5.12.m9.1"><semantics id="S3.p5.12.m9.1a"><mrow id="S3.p5.12.m9.1.1" xref="S3.p5.12.m9.1.1.cmml"><msup id="S3.p5.12.m9.1.1.2" xref="S3.p5.12.m9.1.1.2.cmml"><mn id="S3.p5.12.m9.1.1.2.2" xref="S3.p5.12.m9.1.1.2.2.cmml">25</mn><mo id="S3.p5.12.m9.1.1.2.3" xref="S3.p5.12.m9.1.1.2.3.cmml">∘</mo></msup><mo id="S3.p5.12.m9.1.1.3" xref="S3.p5.12.m9.1.1.3.cmml">≤</mo><mi id="S3.p5.12.m9.1.1.4" xref="S3.p5.12.m9.1.1.4.cmml">θ</mi><mo id="S3.p5.12.m9.1.1.5" xref="S3.p5.12.m9.1.1.5.cmml">≤</mo><msup id="S3.p5.12.m9.1.1.6" xref="S3.p5.12.m9.1.1.6.cmml"><mn id="S3.p5.12.m9.1.1.6.2" xref="S3.p5.12.m9.1.1.6.2.cmml">155</mn><mo id="S3.p5.12.m9.1.1.6.3" xref="S3.p5.12.m9.1.1.6.3.cmml">∘</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.12.m9.1b"><apply id="S3.p5.12.m9.1.1.cmml" xref="S3.p5.12.m9.1.1"><and id="S3.p5.12.m9.1.1a.cmml" xref="S3.p5.12.m9.1.1"></and><apply id="S3.p5.12.m9.1.1b.cmml" xref="S3.p5.12.m9.1.1"><leq id="S3.p5.12.m9.1.1.3.cmml" xref="S3.p5.12.m9.1.1.3"></leq><apply id="S3.p5.12.m9.1.1.2.cmml" xref="S3.p5.12.m9.1.1.2"><csymbol cd="ambiguous" id="S3.p5.12.m9.1.1.2.1.cmml" xref="S3.p5.12.m9.1.1.2">superscript</csymbol><cn id="S3.p5.12.m9.1.1.2.2.cmml" type="integer" xref="S3.p5.12.m9.1.1.2.2">25</cn><compose id="S3.p5.12.m9.1.1.2.3.cmml" xref="S3.p5.12.m9.1.1.2.3"></compose></apply><ci id="S3.p5.12.m9.1.1.4.cmml" xref="S3.p5.12.m9.1.1.4">𝜃</ci></apply><apply id="S3.p5.12.m9.1.1c.cmml" xref="S3.p5.12.m9.1.1"><leq id="S3.p5.12.m9.1.1.5.cmml" xref="S3.p5.12.m9.1.1.5"></leq><share href="https://arxiv.org/html/2502.13414v1#S3.p5.12.m9.1.1.4.cmml" id="S3.p5.12.m9.1.1d.cmml" xref="S3.p5.12.m9.1.1"></share><apply id="S3.p5.12.m9.1.1.6.cmml" xref="S3.p5.12.m9.1.1.6"><csymbol cd="ambiguous" id="S3.p5.12.m9.1.1.6.1.cmml" xref="S3.p5.12.m9.1.1.6">superscript</csymbol><cn id="S3.p5.12.m9.1.1.6.2.cmml" type="integer" xref="S3.p5.12.m9.1.1.6.2">155</cn><compose id="S3.p5.12.m9.1.1.6.3.cmml" xref="S3.p5.12.m9.1.1.6.3"></compose></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.12.m9.1c">25^{\circ}\leq\theta\leq 155^{\circ}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.12.m9.1d">25 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT ≤ italic_θ ≤ 155 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.p6"> <p class="ltx_p" id="S3.p6.6">Two independent methods are developed to calibrate the atomic polarization <math alttext="P" class="ltx_Math" display="inline" id="S3.p6.1.m1.1"><semantics id="S3.p6.1.m1.1a"><mi id="S3.p6.1.m1.1.1" xref="S3.p6.1.m1.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.p6.1.m1.1b"><ci id="S3.p6.1.m1.1.1.cmml" xref="S3.p6.1.m1.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.1.m1.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.p6.1.m1.1d">italic_P</annotation></semantics></math> in this system. The first one is to extract the polarization from the pumping beam transmission. In the high atomic polarization limit, if the input beam has an intensity of <math alttext="I_{i}" class="ltx_Math" display="inline" id="S3.p6.2.m2.1"><semantics id="S3.p6.2.m2.1a"><msub id="S3.p6.2.m2.1.1" xref="S3.p6.2.m2.1.1.cmml"><mi id="S3.p6.2.m2.1.1.2" xref="S3.p6.2.m2.1.1.2.cmml">I</mi><mi id="S3.p6.2.m2.1.1.3" xref="S3.p6.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p6.2.m2.1b"><apply id="S3.p6.2.m2.1.1.cmml" xref="S3.p6.2.m2.1.1"><csymbol cd="ambiguous" id="S3.p6.2.m2.1.1.1.cmml" xref="S3.p6.2.m2.1.1">subscript</csymbol><ci id="S3.p6.2.m2.1.1.2.cmml" xref="S3.p6.2.m2.1.1.2">𝐼</ci><ci id="S3.p6.2.m2.1.1.3.cmml" xref="S3.p6.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.2.m2.1c">I_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.p6.2.m2.1d">italic_I start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and a photon spin of <math alttext="\boldsymbol{s}" class="ltx_Math" display="inline" id="S3.p6.3.m3.1"><semantics id="S3.p6.3.m3.1a"><mi id="S3.p6.3.m3.1.1" xref="S3.p6.3.m3.1.1.cmml">𝒔</mi><annotation-xml encoding="MathML-Content" id="S3.p6.3.m3.1b"><ci id="S3.p6.3.m3.1.1.cmml" xref="S3.p6.3.m3.1.1">𝒔</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.3.m3.1c">\boldsymbol{s}</annotation><annotation encoding="application/x-llamapun" id="S3.p6.3.m3.1d">bold_italic_s</annotation></semantics></math> (<math alttext="|s|=1" class="ltx_Math" display="inline" id="S3.p6.4.m4.1"><semantics id="S3.p6.4.m4.1a"><mrow id="S3.p6.4.m4.1.2" xref="S3.p6.4.m4.1.2.cmml"><mrow id="S3.p6.4.m4.1.2.2.2" xref="S3.p6.4.m4.1.2.2.1.cmml"><mo id="S3.p6.4.m4.1.2.2.2.1" stretchy="false" xref="S3.p6.4.m4.1.2.2.1.1.cmml">|</mo><mi id="S3.p6.4.m4.1.1" xref="S3.p6.4.m4.1.1.cmml">s</mi><mo id="S3.p6.4.m4.1.2.2.2.2" stretchy="false" xref="S3.p6.4.m4.1.2.2.1.1.cmml">|</mo></mrow><mo id="S3.p6.4.m4.1.2.1" xref="S3.p6.4.m4.1.2.1.cmml">=</mo><mn id="S3.p6.4.m4.1.2.3" xref="S3.p6.4.m4.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p6.4.m4.1b"><apply id="S3.p6.4.m4.1.2.cmml" xref="S3.p6.4.m4.1.2"><eq id="S3.p6.4.m4.1.2.1.cmml" xref="S3.p6.4.m4.1.2.1"></eq><apply id="S3.p6.4.m4.1.2.2.1.cmml" xref="S3.p6.4.m4.1.2.2.2"><abs id="S3.p6.4.m4.1.2.2.1.1.cmml" xref="S3.p6.4.m4.1.2.2.2.1"></abs><ci id="S3.p6.4.m4.1.1.cmml" xref="S3.p6.4.m4.1.1">𝑠</ci></apply><cn id="S3.p6.4.m4.1.2.3.cmml" type="integer" xref="S3.p6.4.m4.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.4.m4.1c">|s|=1</annotation><annotation encoding="application/x-llamapun" id="S3.p6.4.m4.1d">| italic_s | = 1</annotation></semantics></math>), the transmitted beam intensity <math alttext="I_{t}" class="ltx_Math" display="inline" id="S3.p6.5.m5.1"><semantics id="S3.p6.5.m5.1a"><msub id="S3.p6.5.m5.1.1" xref="S3.p6.5.m5.1.1.cmml"><mi id="S3.p6.5.m5.1.1.2" xref="S3.p6.5.m5.1.1.2.cmml">I</mi><mi id="S3.p6.5.m5.1.1.3" xref="S3.p6.5.m5.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p6.5.m5.1b"><apply id="S3.p6.5.m5.1.1.cmml" xref="S3.p6.5.m5.1.1"><csymbol cd="ambiguous" id="S3.p6.5.m5.1.1.1.cmml" xref="S3.p6.5.m5.1.1">subscript</csymbol><ci id="S3.p6.5.m5.1.1.2.cmml" xref="S3.p6.5.m5.1.1.2">𝐼</ci><ci id="S3.p6.5.m5.1.1.3.cmml" xref="S3.p6.5.m5.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.5.m5.1c">I_{t}</annotation><annotation encoding="application/x-llamapun" id="S3.p6.5.m5.1d">italic_I start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> is dependent on <math alttext="P" class="ltx_Math" display="inline" id="S3.p6.6.m6.1"><semantics id="S3.p6.6.m6.1a"><mi id="S3.p6.6.m6.1.1" xref="S3.p6.6.m6.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S3.p6.6.m6.1b"><ci id="S3.p6.6.m6.1.1.cmml" xref="S3.p6.6.m6.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.6.m6.1c">P</annotation><annotation encoding="application/x-llamapun" id="S3.p6.6.m6.1d">italic_P</annotation></semantics></math> according to the relation</p> <table class="ltx_equation ltx_eqn_table" id="S3.E12"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="I_{t}=I_{i}e^{-\mathrm{OD}|s-P|}," class="ltx_Math" display="block" id="S3.E12.m1.2"><semantics id="S3.E12.m1.2a"><mrow id="S3.E12.m1.2.2.1" xref="S3.E12.m1.2.2.1.1.cmml"><mrow id="S3.E12.m1.2.2.1.1" xref="S3.E12.m1.2.2.1.1.cmml"><msub id="S3.E12.m1.2.2.1.1.2" xref="S3.E12.m1.2.2.1.1.2.cmml"><mi id="S3.E12.m1.2.2.1.1.2.2" xref="S3.E12.m1.2.2.1.1.2.2.cmml">I</mi><mi id="S3.E12.m1.2.2.1.1.2.3" xref="S3.E12.m1.2.2.1.1.2.3.cmml">t</mi></msub><mo id="S3.E12.m1.2.2.1.1.1" xref="S3.E12.m1.2.2.1.1.1.cmml">=</mo><mrow id="S3.E12.m1.2.2.1.1.3" xref="S3.E12.m1.2.2.1.1.3.cmml"><msub id="S3.E12.m1.2.2.1.1.3.2" xref="S3.E12.m1.2.2.1.1.3.2.cmml"><mi id="S3.E12.m1.2.2.1.1.3.2.2" xref="S3.E12.m1.2.2.1.1.3.2.2.cmml">I</mi><mi id="S3.E12.m1.2.2.1.1.3.2.3" xref="S3.E12.m1.2.2.1.1.3.2.3.cmml">i</mi></msub><mo id="S3.E12.m1.2.2.1.1.3.1" xref="S3.E12.m1.2.2.1.1.3.1.cmml"></mo><msup id="S3.E12.m1.2.2.1.1.3.3" xref="S3.E12.m1.2.2.1.1.3.3.cmml"><mi id="S3.E12.m1.2.2.1.1.3.3.2" xref="S3.E12.m1.2.2.1.1.3.3.2.cmml">e</mi><mrow id="S3.E12.m1.1.1.1" xref="S3.E12.m1.1.1.1.cmml"><mo id="S3.E12.m1.1.1.1a" xref="S3.E12.m1.1.1.1.cmml">−</mo><mrow id="S3.E12.m1.1.1.1.1" xref="S3.E12.m1.1.1.1.1.cmml"><mi id="S3.E12.m1.1.1.1.1.3" xref="S3.E12.m1.1.1.1.1.3.cmml">OD</mi><mo id="S3.E12.m1.1.1.1.1.2" xref="S3.E12.m1.1.1.1.1.2.cmml"></mo><mrow id="S3.E12.m1.1.1.1.1.1.1" xref="S3.E12.m1.1.1.1.1.1.2.cmml"><mo id="S3.E12.m1.1.1.1.1.1.1.2" stretchy="false" xref="S3.E12.m1.1.1.1.1.1.2.1.cmml">|</mo><mrow id="S3.E12.m1.1.1.1.1.1.1.1" xref="S3.E12.m1.1.1.1.1.1.1.1.cmml"><mi 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xref="S3.E12.m1.1.1.1.1.3">OD</ci><apply id="S3.E12.m1.1.1.1.1.1.2.cmml" xref="S3.E12.m1.1.1.1.1.1.1"><abs id="S3.E12.m1.1.1.1.1.1.2.1.cmml" xref="S3.E12.m1.1.1.1.1.1.1.2"></abs><apply id="S3.E12.m1.1.1.1.1.1.1.1.cmml" xref="S3.E12.m1.1.1.1.1.1.1.1"><minus id="S3.E12.m1.1.1.1.1.1.1.1.1.cmml" xref="S3.E12.m1.1.1.1.1.1.1.1.1"></minus><ci id="S3.E12.m1.1.1.1.1.1.1.1.2.cmml" xref="S3.E12.m1.1.1.1.1.1.1.1.2">𝑠</ci><ci id="S3.E12.m1.1.1.1.1.1.1.1.3.cmml" xref="S3.E12.m1.1.1.1.1.1.1.1.3">𝑃</ci></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E12.m1.2c">I_{t}=I_{i}e^{-\mathrm{OD}|s-P|},</annotation><annotation encoding="application/x-llamapun" id="S3.E12.m1.2d">italic_I start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = italic_I start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT - roman_OD | italic_s - italic_P | end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(12)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p6.16">where OD is the optical depth on resonance. The second method is to vary the input pumping beam power, measure the corresponding <math alttext="V_{t,0}" class="ltx_Math" display="inline" id="S3.p6.7.m1.2"><semantics id="S3.p6.7.m1.2a"><msub id="S3.p6.7.m1.2.3" xref="S3.p6.7.m1.2.3.cmml"><mi id="S3.p6.7.m1.2.3.2" xref="S3.p6.7.m1.2.3.2.cmml">V</mi><mrow id="S3.p6.7.m1.2.2.2.4" xref="S3.p6.7.m1.2.2.2.3.cmml"><mi id="S3.p6.7.m1.1.1.1.1" xref="S3.p6.7.m1.1.1.1.1.cmml">t</mi><mo id="S3.p6.7.m1.2.2.2.4.1" xref="S3.p6.7.m1.2.2.2.3.cmml">,</mo><mn id="S3.p6.7.m1.2.2.2.2" xref="S3.p6.7.m1.2.2.2.2.cmml">0</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.p6.7.m1.2b"><apply id="S3.p6.7.m1.2.3.cmml" xref="S3.p6.7.m1.2.3"><csymbol cd="ambiguous" id="S3.p6.7.m1.2.3.1.cmml" xref="S3.p6.7.m1.2.3">subscript</csymbol><ci id="S3.p6.7.m1.2.3.2.cmml" xref="S3.p6.7.m1.2.3.2">𝑉</ci><list id="S3.p6.7.m1.2.2.2.3.cmml" xref="S3.p6.7.m1.2.2.2.4"><ci id="S3.p6.7.m1.1.1.1.1.cmml" xref="S3.p6.7.m1.1.1.1.1">𝑡</ci><cn id="S3.p6.7.m1.2.2.2.2.cmml" type="integer" xref="S3.p6.7.m1.2.2.2.2">0</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.7.m1.2c">V_{t,0}</annotation><annotation encoding="application/x-llamapun" id="S3.p6.7.m1.2d">italic_V start_POSTSUBSCRIPT italic_t , 0 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="V_{l,0}" class="ltx_Math" display="inline" id="S3.p6.8.m2.2"><semantics id="S3.p6.8.m2.2a"><msub id="S3.p6.8.m2.2.3" xref="S3.p6.8.m2.2.3.cmml"><mi id="S3.p6.8.m2.2.3.2" xref="S3.p6.8.m2.2.3.2.cmml">V</mi><mrow id="S3.p6.8.m2.2.2.2.4" xref="S3.p6.8.m2.2.2.2.3.cmml"><mi id="S3.p6.8.m2.1.1.1.1" xref="S3.p6.8.m2.1.1.1.1.cmml">l</mi><mo id="S3.p6.8.m2.2.2.2.4.1" xref="S3.p6.8.m2.2.2.2.3.cmml">,</mo><mn id="S3.p6.8.m2.2.2.2.2" xref="S3.p6.8.m2.2.2.2.2.cmml">0</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.p6.8.m2.2b"><apply id="S3.p6.8.m2.2.3.cmml" xref="S3.p6.8.m2.2.3"><csymbol cd="ambiguous" id="S3.p6.8.m2.2.3.1.cmml" xref="S3.p6.8.m2.2.3">subscript</csymbol><ci id="S3.p6.8.m2.2.3.2.cmml" xref="S3.p6.8.m2.2.3.2">𝑉</ci><list id="S3.p6.8.m2.2.2.2.3.cmml" xref="S3.p6.8.m2.2.2.2.4"><ci id="S3.p6.8.m2.1.1.1.1.cmml" xref="S3.p6.8.m2.1.1.1.1">𝑙</ci><cn id="S3.p6.8.m2.2.2.2.2.cmml" type="integer" xref="S3.p6.8.m2.2.2.2.2">0</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.8.m2.2c">V_{l,0}</annotation><annotation encoding="application/x-llamapun" id="S3.p6.8.m2.2d">italic_V start_POSTSUBSCRIPT italic_l , 0 end_POSTSUBSCRIPT</annotation></semantics></math> in Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S3.E10" title="In III Experiment setup and measurement scheme ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">10</span></a>), and extract the atomic polarization by fitting the results of these initial amplitudes using Eqs. (<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S2.E2" title="In II.1 Bell-Bloom optical pumping ‣ II Theoretical Background ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">2</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S2.E3" title="In II.1 Bell-Bloom optical pumping ‣ II Theoretical Background ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">3</span></a>). Atomic polarizations calibrated by the above two ways are denoted as <math alttext="P_{1}" class="ltx_Math" display="inline" id="S3.p6.9.m3.1"><semantics id="S3.p6.9.m3.1a"><msub id="S3.p6.9.m3.1.1" xref="S3.p6.9.m3.1.1.cmml"><mi id="S3.p6.9.m3.1.1.2" xref="S3.p6.9.m3.1.1.2.cmml">P</mi><mn id="S3.p6.9.m3.1.1.3" xref="S3.p6.9.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.p6.9.m3.1b"><apply id="S3.p6.9.m3.1.1.cmml" xref="S3.p6.9.m3.1.1"><csymbol cd="ambiguous" id="S3.p6.9.m3.1.1.1.cmml" xref="S3.p6.9.m3.1.1">subscript</csymbol><ci id="S3.p6.9.m3.1.1.2.cmml" xref="S3.p6.9.m3.1.1.2">𝑃</ci><cn id="S3.p6.9.m3.1.1.3.cmml" type="integer" xref="S3.p6.9.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.9.m3.1c">P_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.p6.9.m3.1d">italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="P_{2}" class="ltx_Math" display="inline" id="S3.p6.10.m4.1"><semantics id="S3.p6.10.m4.1a"><msub id="S3.p6.10.m4.1.1" xref="S3.p6.10.m4.1.1.cmml"><mi id="S3.p6.10.m4.1.1.2" xref="S3.p6.10.m4.1.1.2.cmml">P</mi><mn id="S3.p6.10.m4.1.1.3" xref="S3.p6.10.m4.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.p6.10.m4.1b"><apply id="S3.p6.10.m4.1.1.cmml" xref="S3.p6.10.m4.1.1"><csymbol cd="ambiguous" id="S3.p6.10.m4.1.1.1.cmml" xref="S3.p6.10.m4.1.1">subscript</csymbol><ci id="S3.p6.10.m4.1.1.2.cmml" xref="S3.p6.10.m4.1.1.2">𝑃</ci><cn id="S3.p6.10.m4.1.1.3.cmml" type="integer" xref="S3.p6.10.m4.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.10.m4.1c">P_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.p6.10.m4.1d">italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, respectively. For a <sup class="ltx_sup" id="S3.p6.16.1"><span class="ltx_text ltx_font_italic" id="S3.p6.16.1.1">87</span></sup>Rb atomic magnetometer working at 75 <sup class="ltx_sup" id="S3.p6.16.2">∘</sup>C, the results of <math alttext="P_{1}" class="ltx_Math" display="inline" id="S3.p6.13.m7.1"><semantics id="S3.p6.13.m7.1a"><msub id="S3.p6.13.m7.1.1" xref="S3.p6.13.m7.1.1.cmml"><mi id="S3.p6.13.m7.1.1.2" xref="S3.p6.13.m7.1.1.2.cmml">P</mi><mn id="S3.p6.13.m7.1.1.3" xref="S3.p6.13.m7.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.p6.13.m7.1b"><apply id="S3.p6.13.m7.1.1.cmml" xref="S3.p6.13.m7.1.1"><csymbol cd="ambiguous" id="S3.p6.13.m7.1.1.1.cmml" xref="S3.p6.13.m7.1.1">subscript</csymbol><ci id="S3.p6.13.m7.1.1.2.cmml" xref="S3.p6.13.m7.1.1.2">𝑃</ci><cn id="S3.p6.13.m7.1.1.3.cmml" type="integer" xref="S3.p6.13.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.13.m7.1c">P_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.p6.13.m7.1d">italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="P_{2}" class="ltx_Math" display="inline" id="S3.p6.14.m8.1"><semantics id="S3.p6.14.m8.1a"><msub id="S3.p6.14.m8.1.1" xref="S3.p6.14.m8.1.1.cmml"><mi id="S3.p6.14.m8.1.1.2" xref="S3.p6.14.m8.1.1.2.cmml">P</mi><mn id="S3.p6.14.m8.1.1.3" xref="S3.p6.14.m8.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.p6.14.m8.1b"><apply id="S3.p6.14.m8.1.1.cmml" xref="S3.p6.14.m8.1.1"><csymbol cd="ambiguous" id="S3.p6.14.m8.1.1.1.cmml" xref="S3.p6.14.m8.1.1">subscript</csymbol><ci id="S3.p6.14.m8.1.1.2.cmml" xref="S3.p6.14.m8.1.1.2">𝑃</ci><cn id="S3.p6.14.m8.1.1.3.cmml" type="integer" xref="S3.p6.14.m8.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.14.m8.1c">P_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.p6.14.m8.1d">italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> are listed in Tab. <a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S3.T1" title="Table 1 ‣ III Experiment setup and measurement scheme ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">1</span></a>. It can be concluded that, for the experimental conditions in this work, the results from both calibrations agree with each other within 5%, and are all in the high polarization regime, which is consistent with the initial assumption. In addition, the difference of the field magnitude calculated from Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S2.E8" title="In II.2 Heading error analysis ‣ II Theoretical Background ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">8</span></a>) using <math alttext="P_{1}" class="ltx_Math" display="inline" id="S3.p6.15.m9.1"><semantics id="S3.p6.15.m9.1a"><msub id="S3.p6.15.m9.1.1" xref="S3.p6.15.m9.1.1.cmml"><mi id="S3.p6.15.m9.1.1.2" xref="S3.p6.15.m9.1.1.2.cmml">P</mi><mn id="S3.p6.15.m9.1.1.3" xref="S3.p6.15.m9.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.p6.15.m9.1b"><apply id="S3.p6.15.m9.1.1.cmml" xref="S3.p6.15.m9.1.1"><csymbol cd="ambiguous" id="S3.p6.15.m9.1.1.1.cmml" xref="S3.p6.15.m9.1.1">subscript</csymbol><ci id="S3.p6.15.m9.1.1.2.cmml" xref="S3.p6.15.m9.1.1.2">𝑃</ci><cn id="S3.p6.15.m9.1.1.3.cmml" type="integer" xref="S3.p6.15.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.15.m9.1c">P_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.p6.15.m9.1d">italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> or <math alttext="P_{2}" class="ltx_Math" display="inline" id="S3.p6.16.m10.1"><semantics id="S3.p6.16.m10.1a"><msub id="S3.p6.16.m10.1.1" xref="S3.p6.16.m10.1.1.cmml"><mi id="S3.p6.16.m10.1.1.2" xref="S3.p6.16.m10.1.1.2.cmml">P</mi><mn id="S3.p6.16.m10.1.1.3" xref="S3.p6.16.m10.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.p6.16.m10.1b"><apply id="S3.p6.16.m10.1.1.cmml" xref="S3.p6.16.m10.1.1"><csymbol cd="ambiguous" id="S3.p6.16.m10.1.1.1.cmml" xref="S3.p6.16.m10.1.1">subscript</csymbol><ci id="S3.p6.16.m10.1.1.2.cmml" xref="S3.p6.16.m10.1.1.2">𝑃</ci><cn id="S3.p6.16.m10.1.1.3.cmml" type="integer" xref="S3.p6.16.m10.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.16.m10.1c">P_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.p6.16.m10.1d">italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, is on the order of 0.1 nT, which is negligible.</p> </div> <figure class="ltx_table" id="S3.T1"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="S3.T1.19.5.1" style="font-size:90%;">Table 1</span>: </span><span class="ltx_text" id="S3.T1.8.4" style="font-size:90%;">Atomic polarizations calibrated by two different methods (<math alttext="P_{1}" class="ltx_Math" display="inline" id="S3.T1.5.1.m1.1"><semantics id="S3.T1.5.1.m1.1b"><msub id="S3.T1.5.1.m1.1.1" xref="S3.T1.5.1.m1.1.1.cmml"><mi id="S3.T1.5.1.m1.1.1.2" xref="S3.T1.5.1.m1.1.1.2.cmml">P</mi><mn id="S3.T1.5.1.m1.1.1.3" xref="S3.T1.5.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.T1.5.1.m1.1c"><apply id="S3.T1.5.1.m1.1.1.cmml" xref="S3.T1.5.1.m1.1.1"><csymbol cd="ambiguous" id="S3.T1.5.1.m1.1.1.1.cmml" xref="S3.T1.5.1.m1.1.1">subscript</csymbol><ci id="S3.T1.5.1.m1.1.1.2.cmml" xref="S3.T1.5.1.m1.1.1.2">𝑃</ci><cn id="S3.T1.5.1.m1.1.1.3.cmml" type="integer" xref="S3.T1.5.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.5.1.m1.1d">P_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.T1.5.1.m1.1e">italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="P_{2}" class="ltx_Math" display="inline" id="S3.T1.6.2.m2.1"><semantics id="S3.T1.6.2.m2.1b"><msub id="S3.T1.6.2.m2.1.1" xref="S3.T1.6.2.m2.1.1.cmml"><mi id="S3.T1.6.2.m2.1.1.2" xref="S3.T1.6.2.m2.1.1.2.cmml">P</mi><mn id="S3.T1.6.2.m2.1.1.3" xref="S3.T1.6.2.m2.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.T1.6.2.m2.1c"><apply id="S3.T1.6.2.m2.1.1.cmml" xref="S3.T1.6.2.m2.1.1"><csymbol cd="ambiguous" id="S3.T1.6.2.m2.1.1.1.cmml" xref="S3.T1.6.2.m2.1.1">subscript</csymbol><ci id="S3.T1.6.2.m2.1.1.2.cmml" xref="S3.T1.6.2.m2.1.1.2">𝑃</ci><cn id="S3.T1.6.2.m2.1.1.3.cmml" type="integer" xref="S3.T1.6.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.6.2.m2.1d">P_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.T1.6.2.m2.1e">italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> defined in the main text) for a <sup class="ltx_sup" id="S3.T1.8.4.1"><span class="ltx_text ltx_font_italic" id="S3.T1.8.4.1.1">87</span></sup>Rb atomic magnetometer working at 75 <sup class="ltx_sup" id="S3.T1.8.4.2">∘</sup>C pumped by an input beam power with a 20% modulation duty cycle and a time-averaged power of 1.13 mW.</span></figcaption> <table class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="S3.T1.15"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S3.T1.13.5"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S3.T1.9.1.1"><math alttext="|90^{\circ}-\theta|" class="ltx_Math" display="inline" id="S3.T1.9.1.1.m1.1"><semantics id="S3.T1.9.1.1.m1.1a"><mrow id="S3.T1.9.1.1.m1.1.1.1" xref="S3.T1.9.1.1.m1.1.1.2.cmml"><mo id="S3.T1.9.1.1.m1.1.1.1.2" stretchy="false" xref="S3.T1.9.1.1.m1.1.1.2.1.cmml">|</mo><mrow id="S3.T1.9.1.1.m1.1.1.1.1" xref="S3.T1.9.1.1.m1.1.1.1.1.cmml"><msup id="S3.T1.9.1.1.m1.1.1.1.1.2" xref="S3.T1.9.1.1.m1.1.1.1.1.2.cmml"><mn id="S3.T1.9.1.1.m1.1.1.1.1.2.2" xref="S3.T1.9.1.1.m1.1.1.1.1.2.2.cmml">90</mn><mo id="S3.T1.9.1.1.m1.1.1.1.1.2.3" xref="S3.T1.9.1.1.m1.1.1.1.1.2.3.cmml">∘</mo></msup><mo id="S3.T1.9.1.1.m1.1.1.1.1.1" xref="S3.T1.9.1.1.m1.1.1.1.1.1.cmml">−</mo><mi id="S3.T1.9.1.1.m1.1.1.1.1.3" xref="S3.T1.9.1.1.m1.1.1.1.1.3.cmml">θ</mi></mrow><mo id="S3.T1.9.1.1.m1.1.1.1.3" stretchy="false" xref="S3.T1.9.1.1.m1.1.1.2.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.T1.9.1.1.m1.1b"><apply id="S3.T1.9.1.1.m1.1.1.2.cmml" xref="S3.T1.9.1.1.m1.1.1.1"><abs 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ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="S3.T1.10.2.2">0<sup class="ltx_sup" id="S3.T1.10.2.2.1">∘</sup> </th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="S3.T1.11.3.3">25<sup class="ltx_sup" id="S3.T1.11.3.3.1">∘</sup> </th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="S3.T1.12.4.4">45<sup class="ltx_sup" id="S3.T1.12.4.4.1">∘</sup> </th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="S3.T1.13.5.5">65<sup class="ltx_sup" id="S3.T1.13.5.5.1">∘</sup> </th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S3.T1.14.6"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S3.T1.14.6.1"><math alttext="|P_{1}|" class="ltx_Math" display="inline" id="S3.T1.14.6.1.m1.1"><semantics id="S3.T1.14.6.1.m1.1a"><mrow id="S3.T1.14.6.1.m1.1.1.1" xref="S3.T1.14.6.1.m1.1.1.2.cmml"><mo id="S3.T1.14.6.1.m1.1.1.1.2" stretchy="false" xref="S3.T1.14.6.1.m1.1.1.2.1.cmml">|</mo><msub id="S3.T1.14.6.1.m1.1.1.1.1" xref="S3.T1.14.6.1.m1.1.1.1.1.cmml"><mi id="S3.T1.14.6.1.m1.1.1.1.1.2" xref="S3.T1.14.6.1.m1.1.1.1.1.2.cmml">P</mi><mn id="S3.T1.14.6.1.m1.1.1.1.1.3" xref="S3.T1.14.6.1.m1.1.1.1.1.3.cmml">1</mn></msub><mo id="S3.T1.14.6.1.m1.1.1.1.3" stretchy="false" xref="S3.T1.14.6.1.m1.1.1.2.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.T1.14.6.1.m1.1b"><apply id="S3.T1.14.6.1.m1.1.1.2.cmml" xref="S3.T1.14.6.1.m1.1.1.1"><abs id="S3.T1.14.6.1.m1.1.1.2.1.cmml" xref="S3.T1.14.6.1.m1.1.1.1.2"></abs><apply id="S3.T1.14.6.1.m1.1.1.1.1.cmml" xref="S3.T1.14.6.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S3.T1.14.6.1.m1.1.1.1.1.1.cmml" xref="S3.T1.14.6.1.m1.1.1.1.1">subscript</csymbol><ci id="S3.T1.14.6.1.m1.1.1.1.1.2.cmml" xref="S3.T1.14.6.1.m1.1.1.1.1.2">𝑃</ci><cn id="S3.T1.14.6.1.m1.1.1.1.1.3.cmml" type="integer" xref="S3.T1.14.6.1.m1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.14.6.1.m1.1c">|P_{1}|</annotation><annotation encoding="application/x-llamapun" id="S3.T1.14.6.1.m1.1d">| italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT |</annotation></semantics></math></th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T1.14.6.2">0.92</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T1.14.6.3">0.93</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T1.14.6.4">0.95</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T1.14.6.5">0.98</td> </tr> <tr class="ltx_tr" id="S3.T1.15.7"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_b ltx_border_l ltx_border_r ltx_border_t" id="S3.T1.15.7.1"><math alttext="|P_{2}|" class="ltx_Math" display="inline" id="S3.T1.15.7.1.m1.1"><semantics id="S3.T1.15.7.1.m1.1a"><mrow id="S3.T1.15.7.1.m1.1.1.1" 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xref="S3.T1.15.7.1.m1.1.1.1.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.15.7.1.m1.1c">|P_{2}|</annotation><annotation encoding="application/x-llamapun" id="S3.T1.15.7.1.m1.1d">| italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT |</annotation></semantics></math></th> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S3.T1.15.7.2">0.84</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S3.T1.15.7.3">0.87</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S3.T1.15.7.4">0.92</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S3.T1.15.7.5">0.94</td> </tr> </tbody> </table> </figure> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">IV </span>Identifying and suppressing heading errors</h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.1">In this paper, the statistical error of the magnetometer measurements is well below 1 nT, which is neglected in the rest of the paper. The measurement uncertainty is dominated by heading errors discussed in the following parts.</p> </div> <section class="ltx_subsection" id="S4.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">IV.1 </span>LZE-induced heading error</h3> <div class="ltx_para" id="S4.SS1.p1"> <p class="ltx_p" id="S4.SS1.p1.1">We measure the sensor heading error by comparing the fitted result B(<math alttext="\theta" class="ltx_Math" display="inline" id="S4.SS1.p1.1.m1.1"><semantics id="S4.SS1.p1.1.m1.1a"><mi id="S4.SS1.p1.1.m1.1.1" xref="S4.SS1.p1.1.m1.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.1.m1.1b"><ci id="S4.SS1.p1.1.m1.1.1.cmml" xref="S4.SS1.p1.1.m1.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.1.m1.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.1.m1.1d">italic_θ</annotation></semantics></math>) from Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S3.E11" title="In III Experiment setup and measurement scheme ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">11</span></a>) at different sensor orientations. The two main sources of heading errors are LZE and NLZE, mentioned in Sec. <a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S2.SS2" title="II.2 Heading error analysis ‣ II Theoretical Background ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">II.2</span></a>. According to Eqs. (<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S2.E7" title="In II.2 Heading error analysis ‣ II Theoretical Background ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">7</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S2.E8" title="In II.2 Heading error analysis ‣ II Theoretical Background ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">8</span></a>), LZE-induced heading error is independent of the sign of atomic polarization, while it is the opposite case for the NLZE-induced heading error. Therefore, we can separate these two effects by comparing the results with opposite atomic orientations.</p> </div> <div class="ltx_para" id="S4.SS1.p2"> <p class="ltx_p" id="S4.SS1.p2.6">Figure <a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S4.F3" title="Figure 3 ‣ IV.1 LZE-induced heading error ‣ IV Identifying and suppressing heading errors ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">3</span></a> shows the measured field magnitude as a function of the sensor orientation when the bias field is around 50 <math alttext="\mu" class="ltx_Math" display="inline" id="S4.SS1.p2.1.m1.1"><semantics id="S4.SS1.p2.1.m1.1a"><mi id="S4.SS1.p2.1.m1.1.1" xref="S4.SS1.p2.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.1.m1.1b"><ci id="S4.SS1.p2.1.m1.1.1.cmml" xref="S4.SS1.p2.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.1.m1.1d">italic_μ</annotation></semantics></math>T. We eliminate the NLZE induced heading error by taking the average of the measured results with pumping beams 1 and 2 in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S3.F1" title="Figure 1 ‣ III Experiment setup and measurement scheme ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">1</span></a>, which have opposite circular polarizations. 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From these measurements, we can conclude that the LZE-induced heading error is within 0.5 nT in the measured range of <math alttext="\theta" class="ltx_Math" display="inline" id="S4.SS1.p2.6.m6.1"><semantics id="S4.SS1.p2.6.m6.1a"><mi id="S4.SS1.p2.6.m6.1.1" xref="S4.SS1.p2.6.m6.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.6.m6.1b"><ci id="S4.SS1.p2.6.m6.1.1.cmml" xref="S4.SS1.p2.6.m6.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.6.m6.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.6.m6.1d">italic_θ</annotation></semantics></math>.</p> </div> <figure class="ltx_figure" id="S4.F3"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="336" id="S4.F3.g1" src="x3.png" width="415"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F3.4.2.1" style="font-size:90%;">Figure 3</span>: </span><span class="ltx_text" id="S4.F3.2.1" style="font-size:90%;"> Experiment results of heading errors using opposite circularly polarized pumping beams at a bias field around 50 <math alttext="\mu" class="ltx_Math" display="inline" id="S4.F3.2.1.m1.1"><semantics id="S4.F3.2.1.m1.1b"><mi id="S4.F3.2.1.m1.1.1" xref="S4.F3.2.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S4.F3.2.1.m1.1c"><ci id="S4.F3.2.1.m1.1.1.cmml" xref="S4.F3.2.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.2.1.m1.1d">\mu</annotation><annotation encoding="application/x-llamapun" id="S4.F3.2.1.m1.1e">italic_μ</annotation></semantics></math>T.</span></figcaption> </figure> <div class="ltx_para" id="S4.SS1.p3"> <p class="ltx_p" id="S4.SS1.p3.5">While the pumping beam modulation frequency <math alttext="\omega_{m}" class="ltx_Math" display="inline" id="S4.SS1.p3.1.m1.1"><semantics id="S4.SS1.p3.1.m1.1a"><msub id="S4.SS1.p3.1.m1.1.1" xref="S4.SS1.p3.1.m1.1.1.cmml"><mi id="S4.SS1.p3.1.m1.1.1.2" 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heading error ‣ IV Identifying and suppressing heading errors ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">4</span></a>. The beating signal in the time domain comes from the difference in linear Zeeman shifts (2<math alttext="g_{I}\mu_{B}" class="ltx_Math" display="inline" id="S4.SS1.p3.5.m5.1"><semantics id="S4.SS1.p3.5.m5.1a"><mrow id="S4.SS1.p3.5.m5.1.1" xref="S4.SS1.p3.5.m5.1.1.cmml"><msub id="S4.SS1.p3.5.m5.1.1.2" xref="S4.SS1.p3.5.m5.1.1.2.cmml"><mi id="S4.SS1.p3.5.m5.1.1.2.2" xref="S4.SS1.p3.5.m5.1.1.2.2.cmml">g</mi><mi id="S4.SS1.p3.5.m5.1.1.2.3" xref="S4.SS1.p3.5.m5.1.1.2.3.cmml">I</mi></msub><mo id="S4.SS1.p3.5.m5.1.1.1" xref="S4.SS1.p3.5.m5.1.1.1.cmml"></mo><msub id="S4.SS1.p3.5.m5.1.1.3" xref="S4.SS1.p3.5.m5.1.1.3.cmml"><mi id="S4.SS1.p3.5.m5.1.1.3.2" xref="S4.SS1.p3.5.m5.1.1.3.2.cmml">μ</mi><mi id="S4.SS1.p3.5.m5.1.1.3.3" xref="S4.SS1.p3.5.m5.1.1.3.3.cmml">B</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.5.m5.1b"><apply id="S4.SS1.p3.5.m5.1.1.cmml" xref="S4.SS1.p3.5.m5.1.1"><times id="S4.SS1.p3.5.m5.1.1.1.cmml" xref="S4.SS1.p3.5.m5.1.1.1"></times><apply id="S4.SS1.p3.5.m5.1.1.2.cmml" xref="S4.SS1.p3.5.m5.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p3.5.m5.1.1.2.1.cmml" xref="S4.SS1.p3.5.m5.1.1.2">subscript</csymbol><ci id="S4.SS1.p3.5.m5.1.1.2.2.cmml" xref="S4.SS1.p3.5.m5.1.1.2.2">𝑔</ci><ci id="S4.SS1.p3.5.m5.1.1.2.3.cmml" xref="S4.SS1.p3.5.m5.1.1.2.3">𝐼</ci></apply><apply id="S4.SS1.p3.5.m5.1.1.3.cmml" xref="S4.SS1.p3.5.m5.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p3.5.m5.1.1.3.1.cmml" xref="S4.SS1.p3.5.m5.1.1.3">subscript</csymbol><ci id="S4.SS1.p3.5.m5.1.1.3.2.cmml" xref="S4.SS1.p3.5.m5.1.1.3.2">𝜇</ci><ci id="S4.SS1.p3.5.m5.1.1.3.3.cmml" xref="S4.SS1.p3.5.m5.1.1.3.3">𝐵</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.5.m5.1c">g_{I}\mu_{B}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.5.m5.1d">italic_g start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT</annotation></semantics></math>B) between two hyperfine states as shown in Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S2.E7" title="In II.2 Heading error analysis ‣ II Theoretical Background ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">7</span></a>). We have tried to use two different models to fit the data in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S4.F4" title="Figure 4 ‣ IV.1 LZE-induced heading error ‣ IV Identifying and suppressing heading errors ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">4</span></a>, one is Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S3.E10" title="In III Experiment setup and measurement scheme ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">10</span></a>) with a single frequency parameter and the other one is a double-frequency fitting model. The frequency extracted from the single-frequency model agrees with the lower-frequency parameter extracted with the double-frequency model within 1 Hz. This is due to the fact that, even in this extreme case, atoms are still dominantly distributed at the upper hyperfine state which has a unique dark state for optical pumping.</p> </div> <figure class="ltx_figure" id="S4.F4"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="321" id="S4.F4.g1" src="x4.png" width="415"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F4.11.5.1" style="font-size:90%;">Figure 4</span>: </span><span class="ltx_text" id="S4.F4.8.4" style="font-size:90%;"> Experiment results of <sup class="ltx_sup" id="S4.F4.8.4.1"><span class="ltx_text ltx_font_italic" id="S4.F4.8.4.1.1">87</span></sup>Rb FID signals at B=50 <math alttext="\mu" class="ltx_Math" display="inline" id="S4.F4.6.2.m2.1"><semantics id="S4.F4.6.2.m2.1b"><mi id="S4.F4.6.2.m2.1.1" xref="S4.F4.6.2.m2.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S4.F4.6.2.m2.1c"><ci id="S4.F4.6.2.m2.1.1.cmml" xref="S4.F4.6.2.m2.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.6.2.m2.1d">\mu</annotation><annotation encoding="application/x-llamapun" id="S4.F4.6.2.m2.1e">italic_μ</annotation></semantics></math>T, with the pumping-beam power modulation frequency close to the Larmor frequency of <math alttext="F=1" class="ltx_Math" display="inline" id="S4.F4.7.3.m3.1"><semantics id="S4.F4.7.3.m3.1b"><mrow id="S4.F4.7.3.m3.1.1" xref="S4.F4.7.3.m3.1.1.cmml"><mi id="S4.F4.7.3.m3.1.1.2" xref="S4.F4.7.3.m3.1.1.2.cmml">F</mi><mo id="S4.F4.7.3.m3.1.1.1" xref="S4.F4.7.3.m3.1.1.1.cmml">=</mo><mn id="S4.F4.7.3.m3.1.1.3" xref="S4.F4.7.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F4.7.3.m3.1c"><apply id="S4.F4.7.3.m3.1.1.cmml" xref="S4.F4.7.3.m3.1.1"><eq id="S4.F4.7.3.m3.1.1.1.cmml" xref="S4.F4.7.3.m3.1.1.1"></eq><ci id="S4.F4.7.3.m3.1.1.2.cmml" xref="S4.F4.7.3.m3.1.1.2">𝐹</ci><cn id="S4.F4.7.3.m3.1.1.3.cmml" type="integer" xref="S4.F4.7.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.7.3.m3.1d">F=1</annotation><annotation encoding="application/x-llamapun" id="S4.F4.7.3.m3.1e">italic_F = 1</annotation></semantics></math> ground states, which is about 1.4 kHz above the Larmor frequency of <math alttext="F=2" class="ltx_Math" display="inline" id="S4.F4.8.4.m4.1"><semantics id="S4.F4.8.4.m4.1b"><mrow id="S4.F4.8.4.m4.1.1" xref="S4.F4.8.4.m4.1.1.cmml"><mi id="S4.F4.8.4.m4.1.1.2" xref="S4.F4.8.4.m4.1.1.2.cmml">F</mi><mo id="S4.F4.8.4.m4.1.1.1" xref="S4.F4.8.4.m4.1.1.1.cmml">=</mo><mn id="S4.F4.8.4.m4.1.1.3" xref="S4.F4.8.4.m4.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F4.8.4.m4.1c"><apply id="S4.F4.8.4.m4.1.1.cmml" xref="S4.F4.8.4.m4.1.1"><eq id="S4.F4.8.4.m4.1.1.1.cmml" xref="S4.F4.8.4.m4.1.1.1"></eq><ci id="S4.F4.8.4.m4.1.1.2.cmml" xref="S4.F4.8.4.m4.1.1.2">𝐹</ci><cn id="S4.F4.8.4.m4.1.1.3.cmml" type="integer" xref="S4.F4.8.4.m4.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.8.4.m4.1d">F=2</annotation><annotation encoding="application/x-llamapun" id="S4.F4.8.4.m4.1e">italic_F = 2</annotation></semantics></math> ground states. The upper inset shows the fitting result using Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S3.E10" title="In III Experiment setup and measurement scheme ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">10</span></a>), while the lower inset shows the amplitude spectral density (ASD) of the measurement.</span></figcaption> </figure> </section> <section class="ltx_subsection" id="S4.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">IV.2 </span>NLZE-induced heading error and its suppression scheme</h3> <div class="ltx_para" id="S4.SS2.p1"> <p class="ltx_p" id="S4.SS2.p1.2">In the rest of the paper, we use the parameter <math alttext="\Delta\mathrm{B}" class="ltx_Math" display="inline" id="S4.SS2.p1.1.m1.1"><semantics id="S4.SS2.p1.1.m1.1a"><mrow id="S4.SS2.p1.1.m1.1.1" xref="S4.SS2.p1.1.m1.1.1.cmml"><mi id="S4.SS2.p1.1.m1.1.1.2" mathvariant="normal" 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ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(13)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.p1.6">The measurement results of <math alttext="\Delta\mathrm{B}" class="ltx_Math" display="inline" id="S4.SS2.p1.3.m1.1"><semantics id="S4.SS2.p1.3.m1.1a"><mrow id="S4.SS2.p1.3.m1.1.1" xref="S4.SS2.p1.3.m1.1.1.cmml"><mi id="S4.SS2.p1.3.m1.1.1.2" mathvariant="normal" xref="S4.SS2.p1.3.m1.1.1.2.cmml">Δ</mi><mo id="S4.SS2.p1.3.m1.1.1.1" xref="S4.SS2.p1.3.m1.1.1.1.cmml"></mo><mi id="S4.SS2.p1.3.m1.1.1.3" mathvariant="normal" xref="S4.SS2.p1.3.m1.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.3.m1.1b"><apply id="S4.SS2.p1.3.m1.1.1.cmml" xref="S4.SS2.p1.3.m1.1.1"><times id="S4.SS2.p1.3.m1.1.1.1.cmml" xref="S4.SS2.p1.3.m1.1.1.1"></times><ci id="S4.SS2.p1.3.m1.1.1.2.cmml" xref="S4.SS2.p1.3.m1.1.1.2">Δ</ci><ci id="S4.SS2.p1.3.m1.1.1.3.cmml" xref="S4.SS2.p1.3.m1.1.1.3">B</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.3.m1.1c">\Delta\mathrm{B}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.3.m1.1d">roman_Δ roman_B</annotation></semantics></math> of a <sup class="ltx_sup" id="S4.SS2.p1.6.1"><span class="ltx_text ltx_font_italic" id="S4.SS2.p1.6.1.1">87</span></sup>Rb magnetometer as a function of <math alttext="\theta" class="ltx_Math" display="inline" id="S4.SS2.p1.5.m3.1"><semantics id="S4.SS2.p1.5.m3.1a"><mi id="S4.SS2.p1.5.m3.1.1" xref="S4.SS2.p1.5.m3.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.5.m3.1b"><ci id="S4.SS2.p1.5.m3.1.1.cmml" xref="S4.SS2.p1.5.m3.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.5.m3.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.5.m3.1d">italic_θ</annotation></semantics></math> in the field range of 10 to 50 <math alttext="\mu" class="ltx_Math" display="inline" id="S4.SS2.p1.6.m4.1"><semantics id="S4.SS2.p1.6.m4.1a"><mi id="S4.SS2.p1.6.m4.1.1" xref="S4.SS2.p1.6.m4.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.6.m4.1b"><ci id="S4.SS2.p1.6.m4.1.1.cmml" xref="S4.SS2.p1.6.m4.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.6.m4.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.6.m4.1d">italic_μ</annotation></semantics></math>T are shown in the Fig. <a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S4.F5" title="Figure 5 ‣ IV.2 NLZE-induced heading error and its suppression scheme ‣ IV Identifying and suppressing heading errors ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">5</span></a>(a). The measured experiment data points agree well with the predictions using Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S2.E8" title="In II.2 Heading error analysis ‣ II Theoretical Background ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">8</span></a>) and calibrated atomic polarizations, which also confirms that atom populations follow the spin-temperature distribution in a magnetometer using a Bell-Bloom optical pumping method.</p> </div> <figure class="ltx_figure" id="S4.F5"><img alt="Refer to caption" class="ltx_graphics ltx_img_portrait" height="656" id="S4.F5.g1" src="x5.png" width="415"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F5.8.4.1" style="font-size:90%;">Figure 5</span>: </span><span class="ltx_text" id="S4.F5.6.3" style="font-size:90%;"> (a) Measured heading errors as a function of <math alttext="\theta" class="ltx_Math" display="inline" id="S4.F5.4.1.m1.1"><semantics id="S4.F5.4.1.m1.1b"><mi id="S4.F5.4.1.m1.1.1" xref="S4.F5.4.1.m1.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S4.F5.4.1.m1.1c"><ci id="S4.F5.4.1.m1.1.1.cmml" xref="S4.F5.4.1.m1.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.4.1.m1.1d">\theta</annotation><annotation encoding="application/x-llamapun" id="S4.F5.4.1.m1.1e">italic_θ</annotation></semantics></math> in the field range of 10 <math alttext="\sim" class="ltx_Math" display="inline" id="S4.F5.5.2.m2.1"><semantics id="S4.F5.5.2.m2.1b"><mo id="S4.F5.5.2.m2.1.1" xref="S4.F5.5.2.m2.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S4.F5.5.2.m2.1c"><csymbol cd="latexml" id="S4.F5.5.2.m2.1.1.cmml" xref="S4.F5.5.2.m2.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.5.2.m2.1d">\sim</annotation><annotation encoding="application/x-llamapun" id="S4.F5.5.2.m2.1e">∼</annotation></semantics></math> 50 <math alttext="\mu" class="ltx_Math" display="inline" id="S4.F5.6.3.m3.1"><semantics id="S4.F5.6.3.m3.1b"><mi id="S4.F5.6.3.m3.1.1" xref="S4.F5.6.3.m3.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S4.F5.6.3.m3.1c"><ci id="S4.F5.6.3.m3.1.1.cmml" xref="S4.F5.6.3.m3.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.6.3.m3.1d">\mu</annotation><annotation encoding="application/x-llamapun" id="S4.F5.6.3.m3.1e">italic_μ</annotation></semantics></math>T (solid markers) and theoretical prediction results using Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S2.E8" title="In II.2 Heading error analysis ‣ II Theoretical Background ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">8</span></a>) (dashed lines). (b) The remaining heading error after averaging out atomic longitudinal polarization in time or space.</span></figcaption> </figure> <div class="ltx_para" id="S4.SS2.p2"> <p class="ltx_p" id="S4.SS2.p2.1">In the following, we focus on suppressing the heading error either directly or indirectly. Here, the direct methods refer to suppressing the heading error without using the measurement results, while the indirect methods suppress the heading error by using a comagnetometer system and combining multiple atomic precession frequencies, which are the most direct and accurate parameters extracted from the FID magnetometer.</p> </div> <div class="ltx_para" id="S4.SS2.p3"> <p class="ltx_p" id="S4.SS2.p3.2">For the direct methods, we try to suppress the NLZE-induced heading error by eliminating the horizontal atomic polarization, as indicated in Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S2.E8" title="In II.2 Heading error analysis ‣ II Theoretical Background ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">8</span></a>). This can be achieved by averaging out the horizontal atomic polarization either in time or space. In the former case, both pumping beam 1 and 2 in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S3.F1" title="Figure 1 ‣ III Experiment setup and measurement scheme ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">1</span></a> are used, where the pumping beam 2 is turned on with the same modulation frequency and duty cycle as beam 1, except a <math alttext="\pi" class="ltx_Math" display="inline" id="S4.SS2.p3.1.m1.1"><semantics id="S4.SS2.p3.1.m1.1a"><mi id="S4.SS2.p3.1.m1.1.1" xref="S4.SS2.p3.1.m1.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.1.m1.1b"><ci id="S4.SS2.p3.1.m1.1.1.cmml" xref="S4.SS2.p3.1.m1.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.1.m1.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.1.m1.1d">italic_π</annotation></semantics></math> phase delay. 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(<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S3.E10" title="In III Experiment setup and measurement scheme ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">10</span></a>) is kept nulled. In this way, the transverse atomic polarization is built up constructively by the two pumping beams while the longitudinal atomic polarization are averaged out. The squared data points in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S4.F5" title="Figure 5 ‣ IV.2 NLZE-induced heading error and its suppression scheme ‣ IV Identifying and suppressing heading errors ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">5</span></a>(b) confirm that this scheme helps to suppress the heading error within 1 nT over the sensor orientations used in this work. In free-space experiments, this scheme can also be realized by modulating the pumping beam polarization with an electro-optic modulator.</p> </div> <div class="ltx_para" id="S4.SS2.p4"> <p class="ltx_p" id="S4.SS2.p4.1">To average out the longitudinal atomic polarization in space, a half-wave plate is added in the middle of the Herriott cavity <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib12" title="">12</a>]</cite>. In this way, the pumping beam polarization is flipped each time it passes the wave plate, and the measurement result is effectively the average of the results from two atomic ensembles pumped by beams with opposite polarizations. The triangle data points in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S4.F5" title="Figure 5 ‣ IV.2 NLZE-induced heading error and its suppression scheme ‣ IV Identifying and suppressing heading errors ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">5</span></a>(b) show that this space-average scheme achieves similar performance on suppressing the heading error as the time-average one. It needs to be noted that, in practice, the quality of the pumping beam polarization degrades as it reflects on the mirror surfaces and passes through the wave plate, which may contribute to the imperfect suppression result.</p> </div> <div class="ltx_para" id="S4.SS2.p5"> <p class="ltx_p" id="S4.SS2.p5.6">To demonstrate the indirect methods to suppress the NLZE-induced heading error, we make use of a Rb isotope comagnetometer. 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italic_ω start_POSTSUBSCRIPT italic_L , italic_i end_POSTSUBSCRIPT + italic_b ( italic_P ) start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ω start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_L , italic_i end_POSTSUBSCRIPT roman_cos italic_θ .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(14)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.p5.4">Here, <math alttext="i" class="ltx_Math" display="inline" id="S4.SS2.p5.1.m1.1"><semantics id="S4.SS2.p5.1.m1.1a"><mi id="S4.SS2.p5.1.m1.1.1" xref="S4.SS2.p5.1.m1.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.1.m1.1b"><ci id="S4.SS2.p5.1.m1.1.1.cmml" xref="S4.SS2.p5.1.m1.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.1.m1.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.1.m1.1d">italic_i</annotation></semantics></math> denotes the isotope <sup class="ltx_sup" id="S4.SS2.p5.4.1"><span class="ltx_text ltx_font_italic" id="S4.SS2.p5.4.1.1">i</span></sup>Rb, and <math alttext="a" class="ltx_Math" display="inline" id="S4.SS2.p5.3.m3.1"><semantics id="S4.SS2.p5.3.m3.1a"><mi id="S4.SS2.p5.3.m3.1.1" xref="S4.SS2.p5.3.m3.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.3.m3.1b"><ci id="S4.SS2.p5.3.m3.1.1.cmml" xref="S4.SS2.p5.3.m3.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.3.m3.1c">a</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.3.m3.1d">italic_a</annotation></semantics></math> and <math alttext="b" class="ltx_Math" display="inline" id="S4.SS2.p5.4.m4.1"><semantics id="S4.SS2.p5.4.m4.1a"><mi id="S4.SS2.p5.4.m4.1.1" xref="S4.SS2.p5.4.m4.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.4.m4.1b"><ci id="S4.SS2.p5.4.m4.1.1.cmml" xref="S4.SS2.p5.4.m4.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.4.m4.1c">b</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.4.m4.1d">italic_b</annotation></semantics></math> represents the coefficients for linear and nonlinear terms in Eqs. (<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S2.E8" title="In II.2 Heading error analysis ‣ II Theoretical Background ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">8</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S2.E9" title="In II.2 Heading error analysis ‣ II Theoretical Background ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">9</span></a>). 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id="S4.E15.m1.7c">~{}\mathrm{B}=\frac{c(P)a_{87}\omega_{L,87}-a_{85}\omega_{L,85}}{c(P)-1},</annotation><annotation encoding="application/x-llamapun" id="S4.E15.m1.7d">roman_B = divide start_ARG italic_c ( italic_P ) italic_a start_POSTSUBSCRIPT 87 end_POSTSUBSCRIPT italic_ω start_POSTSUBSCRIPT italic_L , 87 end_POSTSUBSCRIPT - italic_a start_POSTSUBSCRIPT 85 end_POSTSUBSCRIPT italic_ω start_POSTSUBSCRIPT italic_L , 85 end_POSTSUBSCRIPT end_ARG start_ARG italic_c ( italic_P ) - 1 end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(15)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.p5.5">where <math alttext="c(P)=b(P)_{85}\omega^{2}_{L,85}/b(P)_{87}\omega^{2}_{L,87}=\Delta\mathrm{B}_{8% 5}/\Delta\mathrm{B}_{87}" class="ltx_Math" display="inline" id="S4.SS2.p5.5.m1.7"><semantics 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xref="S4.SS2.p5.5.m1.7.8.6.1"></times><apply id="S4.SS2.p5.5.m1.7.8.6.2.cmml" xref="S4.SS2.p5.5.m1.7.8.6.2"><divide id="S4.SS2.p5.5.m1.7.8.6.2.1.cmml" xref="S4.SS2.p5.5.m1.7.8.6.2.1"></divide><apply id="S4.SS2.p5.5.m1.7.8.6.2.2.cmml" xref="S4.SS2.p5.5.m1.7.8.6.2.2"><times id="S4.SS2.p5.5.m1.7.8.6.2.2.1.cmml" xref="S4.SS2.p5.5.m1.7.8.6.2.2.1"></times><ci id="S4.SS2.p5.5.m1.7.8.6.2.2.2.cmml" xref="S4.SS2.p5.5.m1.7.8.6.2.2.2">Δ</ci><apply id="S4.SS2.p5.5.m1.7.8.6.2.2.3.cmml" xref="S4.SS2.p5.5.m1.7.8.6.2.2.3"><csymbol cd="ambiguous" id="S4.SS2.p5.5.m1.7.8.6.2.2.3.1.cmml" xref="S4.SS2.p5.5.m1.7.8.6.2.2.3">subscript</csymbol><ci id="S4.SS2.p5.5.m1.7.8.6.2.2.3.2.cmml" xref="S4.SS2.p5.5.m1.7.8.6.2.2.3.2">B</ci><cn id="S4.SS2.p5.5.m1.7.8.6.2.2.3.3.cmml" type="integer" xref="S4.SS2.p5.5.m1.7.8.6.2.2.3.3">85</cn></apply></apply><ci id="S4.SS2.p5.5.m1.7.8.6.2.3.cmml" xref="S4.SS2.p5.5.m1.7.8.6.2.3">Δ</ci></apply><apply id="S4.SS2.p5.5.m1.7.8.6.3.cmml" xref="S4.SS2.p5.5.m1.7.8.6.3"><csymbol cd="ambiguous" id="S4.SS2.p5.5.m1.7.8.6.3.1.cmml" xref="S4.SS2.p5.5.m1.7.8.6.3">subscript</csymbol><ci id="S4.SS2.p5.5.m1.7.8.6.3.2.cmml" xref="S4.SS2.p5.5.m1.7.8.6.3.2">B</ci><cn id="S4.SS2.p5.5.m1.7.8.6.3.3.cmml" type="integer" xref="S4.SS2.p5.5.m1.7.8.6.3.3">87</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.5.m1.7c">c(P)=b(P)_{85}\omega^{2}_{L,85}/b(P)_{87}\omega^{2}_{L,87}=\Delta\mathrm{B}_{8% 5}/\Delta\mathrm{B}_{87}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.5.m1.7d">italic_c ( italic_P ) = italic_b ( italic_P ) start_POSTSUBSCRIPT 85 end_POSTSUBSCRIPT italic_ω start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_L , 85 end_POSTSUBSCRIPT / italic_b ( italic_P ) start_POSTSUBSCRIPT 87 end_POSTSUBSCRIPT italic_ω start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_L , 87 end_POSTSUBSCRIPT = roman_Δ roman_B start_POSTSUBSCRIPT 85 end_POSTSUBSCRIPT / roman_Δ roman_B start_POSTSUBSCRIPT 87 end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS2.p6"> <p class="ltx_p" id="S4.SS2.p6.10">This scheme is implemented by using a cell consisting of Rb atoms with natural abundance, while all other conditions are same as the previous <sup class="ltx_sup" id="S4.SS2.p6.10.1"><span class="ltx_text ltx_font_italic" id="S4.SS2.p6.10.1.1">87</span></sup>Rb cell. Due to the pressure broadening effect, the same pumping and probe beams work well for both isotopes. The pumping beam is amplitude modulated at a frequency of <math alttext="\omega_{t}" class="ltx_Math" display="inline" id="S4.SS2.p6.2.m2.1"><semantics id="S4.SS2.p6.2.m2.1a"><msub id="S4.SS2.p6.2.m2.1.1" xref="S4.SS2.p6.2.m2.1.1.cmml"><mi id="S4.SS2.p6.2.m2.1.1.2" xref="S4.SS2.p6.2.m2.1.1.2.cmml">ω</mi><mi id="S4.SS2.p6.2.m2.1.1.3" xref="S4.SS2.p6.2.m2.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p6.2.m2.1b"><apply id="S4.SS2.p6.2.m2.1.1.cmml" xref="S4.SS2.p6.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.p6.2.m2.1.1.1.cmml" xref="S4.SS2.p6.2.m2.1.1">subscript</csymbol><ci id="S4.SS2.p6.2.m2.1.1.2.cmml" xref="S4.SS2.p6.2.m2.1.1.2">𝜔</ci><ci id="S4.SS2.p6.2.m2.1.1.3.cmml" xref="S4.SS2.p6.2.m2.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p6.2.m2.1c">\omega_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p6.2.m2.1d">italic_ω start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>, which is approximately equal to <math alttext="\omega_{L,87}/3" class="ltx_Math" display="inline" id="S4.SS2.p6.3.m3.2"><semantics id="S4.SS2.p6.3.m3.2a"><mrow id="S4.SS2.p6.3.m3.2.3" xref="S4.SS2.p6.3.m3.2.3.cmml"><msub id="S4.SS2.p6.3.m3.2.3.2" xref="S4.SS2.p6.3.m3.2.3.2.cmml"><mi id="S4.SS2.p6.3.m3.2.3.2.2" xref="S4.SS2.p6.3.m3.2.3.2.2.cmml">ω</mi><mrow id="S4.SS2.p6.3.m3.2.2.2.4" xref="S4.SS2.p6.3.m3.2.2.2.3.cmml"><mi id="S4.SS2.p6.3.m3.1.1.1.1" xref="S4.SS2.p6.3.m3.1.1.1.1.cmml">L</mi><mo id="S4.SS2.p6.3.m3.2.2.2.4.1" xref="S4.SS2.p6.3.m3.2.2.2.3.cmml">,</mo><mn id="S4.SS2.p6.3.m3.2.2.2.2" xref="S4.SS2.p6.3.m3.2.2.2.2.cmml">87</mn></mrow></msub><mo id="S4.SS2.p6.3.m3.2.3.1" xref="S4.SS2.p6.3.m3.2.3.1.cmml">/</mo><mn id="S4.SS2.p6.3.m3.2.3.3" xref="S4.SS2.p6.3.m3.2.3.3.cmml">3</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p6.3.m3.2b"><apply id="S4.SS2.p6.3.m3.2.3.cmml" xref="S4.SS2.p6.3.m3.2.3"><divide id="S4.SS2.p6.3.m3.2.3.1.cmml" xref="S4.SS2.p6.3.m3.2.3.1"></divide><apply id="S4.SS2.p6.3.m3.2.3.2.cmml" xref="S4.SS2.p6.3.m3.2.3.2"><csymbol cd="ambiguous" id="S4.SS2.p6.3.m3.2.3.2.1.cmml" xref="S4.SS2.p6.3.m3.2.3.2">subscript</csymbol><ci id="S4.SS2.p6.3.m3.2.3.2.2.cmml" xref="S4.SS2.p6.3.m3.2.3.2.2">𝜔</ci><list id="S4.SS2.p6.3.m3.2.2.2.3.cmml" xref="S4.SS2.p6.3.m3.2.2.2.4"><ci id="S4.SS2.p6.3.m3.1.1.1.1.cmml" xref="S4.SS2.p6.3.m3.1.1.1.1">𝐿</ci><cn id="S4.SS2.p6.3.m3.2.2.2.2.cmml" type="integer" xref="S4.SS2.p6.3.m3.2.2.2.2">87</cn></list></apply><cn id="S4.SS2.p6.3.m3.2.3.3.cmml" type="integer" xref="S4.SS2.p6.3.m3.2.3.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p6.3.m3.2c">\omega_{L,87}/3</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p6.3.m3.2d">italic_ω start_POSTSUBSCRIPT italic_L , 87 end_POSTSUBSCRIPT / 3</annotation></semantics></math> and <math alttext="\omega_{L,85}/2" class="ltx_Math" display="inline" id="S4.SS2.p6.4.m4.2"><semantics id="S4.SS2.p6.4.m4.2a"><mrow id="S4.SS2.p6.4.m4.2.3" xref="S4.SS2.p6.4.m4.2.3.cmml"><msub id="S4.SS2.p6.4.m4.2.3.2" xref="S4.SS2.p6.4.m4.2.3.2.cmml"><mi id="S4.SS2.p6.4.m4.2.3.2.2" xref="S4.SS2.p6.4.m4.2.3.2.2.cmml">ω</mi><mrow id="S4.SS2.p6.4.m4.2.2.2.4" xref="S4.SS2.p6.4.m4.2.2.2.3.cmml"><mi id="S4.SS2.p6.4.m4.1.1.1.1" xref="S4.SS2.p6.4.m4.1.1.1.1.cmml">L</mi><mo id="S4.SS2.p6.4.m4.2.2.2.4.1" xref="S4.SS2.p6.4.m4.2.2.2.3.cmml">,</mo><mn id="S4.SS2.p6.4.m4.2.2.2.2" xref="S4.SS2.p6.4.m4.2.2.2.2.cmml">85</mn></mrow></msub><mo id="S4.SS2.p6.4.m4.2.3.1" xref="S4.SS2.p6.4.m4.2.3.1.cmml">/</mo><mn id="S4.SS2.p6.4.m4.2.3.3" xref="S4.SS2.p6.4.m4.2.3.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p6.4.m4.2b"><apply id="S4.SS2.p6.4.m4.2.3.cmml" xref="S4.SS2.p6.4.m4.2.3"><divide id="S4.SS2.p6.4.m4.2.3.1.cmml" xref="S4.SS2.p6.4.m4.2.3.1"></divide><apply id="S4.SS2.p6.4.m4.2.3.2.cmml" xref="S4.SS2.p6.4.m4.2.3.2"><csymbol cd="ambiguous" id="S4.SS2.p6.4.m4.2.3.2.1.cmml" xref="S4.SS2.p6.4.m4.2.3.2">subscript</csymbol><ci id="S4.SS2.p6.4.m4.2.3.2.2.cmml" xref="S4.SS2.p6.4.m4.2.3.2.2">𝜔</ci><list id="S4.SS2.p6.4.m4.2.2.2.3.cmml" xref="S4.SS2.p6.4.m4.2.2.2.4"><ci id="S4.SS2.p6.4.m4.1.1.1.1.cmml" xref="S4.SS2.p6.4.m4.1.1.1.1">𝐿</ci><cn id="S4.SS2.p6.4.m4.2.2.2.2.cmml" type="integer" xref="S4.SS2.p6.4.m4.2.2.2.2">85</cn></list></apply><cn id="S4.SS2.p6.4.m4.2.3.3.cmml" type="integer" xref="S4.SS2.p6.4.m4.2.3.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p6.4.m4.2c">\omega_{L,85}/2</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p6.4.m4.2d">italic_ω start_POSTSUBSCRIPT italic_L , 85 end_POSTSUBSCRIPT / 2</annotation></semantics></math>, and the modulation duty cycle is 10%. As shown in Tab. <a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S4.T2" title="Table 2 ‣ IV.2 NLZE-induced heading error and its suppression scheme ‣ IV Identifying and suppressing heading errors ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">2</span></a>, the atomic polarization of <sup class="ltx_sup" id="S4.SS2.p6.10.2"><span class="ltx_text ltx_font_italic" id="S4.SS2.p6.10.2.1">85</span></sup>Rb is larger than <sup class="ltx_sup" id="S4.SS2.p6.10.3"><span class="ltx_text ltx_font_italic" id="S4.SS2.p6.10.3.1">87</span></sup>Rb due to a larger effective duty cycle for the isotope with a smaller gyromagnetic ratio. In addition, the atomic polarization decreases as <math alttext="\theta" class="ltx_Math" display="inline" id="S4.SS2.p6.7.m7.1"><semantics id="S4.SS2.p6.7.m7.1a"><mi id="S4.SS2.p6.7.m7.1.1" xref="S4.SS2.p6.7.m7.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p6.7.m7.1b"><ci id="S4.SS2.p6.7.m7.1.1.cmml" xref="S4.SS2.p6.7.m7.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p6.7.m7.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p6.7.m7.1d">italic_θ</annotation></semantics></math> gets close to 90<sup class="ltx_sup" id="S4.SS2.p6.10.4">∘</sup>, and the minimum atomic polarization is below 0.75 for both isotopes at <math alttext="\theta=90^{\circ}" class="ltx_Math" display="inline" id="S4.SS2.p6.9.m9.1"><semantics id="S4.SS2.p6.9.m9.1a"><mrow id="S4.SS2.p6.9.m9.1.1" xref="S4.SS2.p6.9.m9.1.1.cmml"><mi id="S4.SS2.p6.9.m9.1.1.2" xref="S4.SS2.p6.9.m9.1.1.2.cmml">θ</mi><mo id="S4.SS2.p6.9.m9.1.1.1" xref="S4.SS2.p6.9.m9.1.1.1.cmml">=</mo><msup id="S4.SS2.p6.9.m9.1.1.3" xref="S4.SS2.p6.9.m9.1.1.3.cmml"><mn id="S4.SS2.p6.9.m9.1.1.3.2" xref="S4.SS2.p6.9.m9.1.1.3.2.cmml">90</mn><mo id="S4.SS2.p6.9.m9.1.1.3.3" xref="S4.SS2.p6.9.m9.1.1.3.3.cmml">∘</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p6.9.m9.1b"><apply id="S4.SS2.p6.9.m9.1.1.cmml" xref="S4.SS2.p6.9.m9.1.1"><eq id="S4.SS2.p6.9.m9.1.1.1.cmml" xref="S4.SS2.p6.9.m9.1.1.1"></eq><ci id="S4.SS2.p6.9.m9.1.1.2.cmml" xref="S4.SS2.p6.9.m9.1.1.2">𝜃</ci><apply id="S4.SS2.p6.9.m9.1.1.3.cmml" xref="S4.SS2.p6.9.m9.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p6.9.m9.1.1.3.1.cmml" xref="S4.SS2.p6.9.m9.1.1.3">superscript</csymbol><cn id="S4.SS2.p6.9.m9.1.1.3.2.cmml" type="integer" xref="S4.SS2.p6.9.m9.1.1.3.2">90</cn><compose id="S4.SS2.p6.9.m9.1.1.3.3.cmml" xref="S4.SS2.p6.9.m9.1.1.3.3"></compose></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p6.9.m9.1c">\theta=90^{\circ}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p6.9.m9.1d">italic_θ = 90 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT</annotation></semantics></math>. Figure <a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S4.F6" title="Figure 6 ‣ IV.2 NLZE-induced heading error and its suppression scheme ‣ IV Identifying and suppressing heading errors ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">6</span></a>(a) shows the comparison between experimental data for each isotope and theoretical predictions based on Eqs. (<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S2.E8" title="In II.2 Heading error analysis ‣ II Theoretical Background ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">8</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S2.E9" title="In II.2 Heading error analysis ‣ II Theoretical Background ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">9</span></a>). Here the discrepancy between them when <math alttext="\cos\theta" class="ltx_Math" display="inline" id="S4.SS2.p6.10.m10.1"><semantics id="S4.SS2.p6.10.m10.1a"><mrow id="S4.SS2.p6.10.m10.1.1" xref="S4.SS2.p6.10.m10.1.1.cmml"><mi id="S4.SS2.p6.10.m10.1.1.1" xref="S4.SS2.p6.10.m10.1.1.1.cmml">cos</mi><mo id="S4.SS2.p6.10.m10.1.1a" lspace="0.167em" xref="S4.SS2.p6.10.m10.1.1.cmml"></mo><mi id="S4.SS2.p6.10.m10.1.1.2" xref="S4.SS2.p6.10.m10.1.1.2.cmml">θ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p6.10.m10.1b"><apply id="S4.SS2.p6.10.m10.1.1.cmml" xref="S4.SS2.p6.10.m10.1.1"><cos id="S4.SS2.p6.10.m10.1.1.1.cmml" xref="S4.SS2.p6.10.m10.1.1.1"></cos><ci id="S4.SS2.p6.10.m10.1.1.2.cmml" xref="S4.SS2.p6.10.m10.1.1.2">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p6.10.m10.1c">\cos\theta</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p6.10.m10.1d">roman_cos italic_θ</annotation></semantics></math> is small may be attributed to the relatively low atomic polarization in these cases where the feasibility of Eqs. (<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S2.E8" title="In II.2 Heading error analysis ‣ II Theoretical Background ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">8</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S2.E9" title="In II.2 Heading error analysis ‣ II Theoretical Background ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">9</span></a>) decrease.</p> </div> <figure class="ltx_table" id="S4.T2"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="S4.T2.12.2.1" style="font-size:90%;">Table 2</span>: </span><span class="ltx_text" id="S4.T2.2.1" style="font-size:90%;">Absolute values of atomic polarizations and <math alttext="c(P)" class="ltx_Math" display="inline" id="S4.T2.2.1.m1.1"><semantics id="S4.T2.2.1.m1.1b"><mrow id="S4.T2.2.1.m1.1.2" xref="S4.T2.2.1.m1.1.2.cmml"><mi id="S4.T2.2.1.m1.1.2.2" xref="S4.T2.2.1.m1.1.2.2.cmml">c</mi><mo id="S4.T2.2.1.m1.1.2.1" xref="S4.T2.2.1.m1.1.2.1.cmml"></mo><mrow id="S4.T2.2.1.m1.1.2.3.2" xref="S4.T2.2.1.m1.1.2.cmml"><mo id="S4.T2.2.1.m1.1.2.3.2.1" stretchy="false" xref="S4.T2.2.1.m1.1.2.cmml">(</mo><mi id="S4.T2.2.1.m1.1.1" xref="S4.T2.2.1.m1.1.1.cmml">P</mi><mo id="S4.T2.2.1.m1.1.2.3.2.2" stretchy="false" xref="S4.T2.2.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.2.1.m1.1c"><apply id="S4.T2.2.1.m1.1.2.cmml" xref="S4.T2.2.1.m1.1.2"><times id="S4.T2.2.1.m1.1.2.1.cmml" xref="S4.T2.2.1.m1.1.2.1"></times><ci id="S4.T2.2.1.m1.1.2.2.cmml" xref="S4.T2.2.1.m1.1.2.2">𝑐</ci><ci id="S4.T2.2.1.m1.1.1.cmml" xref="S4.T2.2.1.m1.1.1">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.2.1.m1.1d">c(P)</annotation><annotation encoding="application/x-llamapun" id="S4.T2.2.1.m1.1e">italic_c ( italic_P )</annotation></semantics></math> for simultaneously optical pumping both Rb isotopes.</span></figcaption> <table class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="S4.T2.10"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S4.T2.7.5"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_l ltx_border_r ltx_border_t" id="S4.T2.3.1.1"><math alttext="|90^{\circ}-\theta|" class="ltx_Math" display="inline" id="S4.T2.3.1.1.m1.1"><semantics id="S4.T2.3.1.1.m1.1a"><mrow id="S4.T2.3.1.1.m1.1.1.1" xref="S4.T2.3.1.1.m1.1.1.2.cmml"><mo id="S4.T2.3.1.1.m1.1.1.1.2" stretchy="false" xref="S4.T2.3.1.1.m1.1.1.2.1.cmml">|</mo><mrow id="S4.T2.3.1.1.m1.1.1.1.1" xref="S4.T2.3.1.1.m1.1.1.1.1.cmml"><msup id="S4.T2.3.1.1.m1.1.1.1.1.2" xref="S4.T2.3.1.1.m1.1.1.1.1.2.cmml"><mn id="S4.T2.3.1.1.m1.1.1.1.1.2.2" xref="S4.T2.3.1.1.m1.1.1.1.1.2.2.cmml">90</mn><mo id="S4.T2.3.1.1.m1.1.1.1.1.2.3" xref="S4.T2.3.1.1.m1.1.1.1.1.2.3.cmml">∘</mo></msup><mo id="S4.T2.3.1.1.m1.1.1.1.1.1" xref="S4.T2.3.1.1.m1.1.1.1.1.1.cmml">−</mo><mi id="S4.T2.3.1.1.m1.1.1.1.1.3" xref="S4.T2.3.1.1.m1.1.1.1.1.3.cmml">θ</mi></mrow><mo id="S4.T2.3.1.1.m1.1.1.1.3" stretchy="false" xref="S4.T2.3.1.1.m1.1.1.2.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.3.1.1.m1.1b"><apply id="S4.T2.3.1.1.m1.1.1.2.cmml" xref="S4.T2.3.1.1.m1.1.1.1"><abs id="S4.T2.3.1.1.m1.1.1.2.1.cmml" xref="S4.T2.3.1.1.m1.1.1.1.2"></abs><apply id="S4.T2.3.1.1.m1.1.1.1.1.cmml" xref="S4.T2.3.1.1.m1.1.1.1.1"><minus id="S4.T2.3.1.1.m1.1.1.1.1.1.cmml" xref="S4.T2.3.1.1.m1.1.1.1.1.1"></minus><apply id="S4.T2.3.1.1.m1.1.1.1.1.2.cmml" xref="S4.T2.3.1.1.m1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.T2.3.1.1.m1.1.1.1.1.2.1.cmml" xref="S4.T2.3.1.1.m1.1.1.1.1.2">superscript</csymbol><cn id="S4.T2.3.1.1.m1.1.1.1.1.2.2.cmml" type="integer" xref="S4.T2.3.1.1.m1.1.1.1.1.2.2">90</cn><compose id="S4.T2.3.1.1.m1.1.1.1.1.2.3.cmml" xref="S4.T2.3.1.1.m1.1.1.1.1.2.3"></compose></apply><ci id="S4.T2.3.1.1.m1.1.1.1.1.3.cmml" xref="S4.T2.3.1.1.m1.1.1.1.1.3">𝜃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.3.1.1.m1.1c">|90^{\circ}-\theta|</annotation><annotation encoding="application/x-llamapun" id="S4.T2.3.1.1.m1.1d">| 90 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT - italic_θ |</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="S4.T2.4.2.2">0<sup class="ltx_sup" id="S4.T2.4.2.2.1">∘</sup> </th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="S4.T2.5.3.3">25<sup class="ltx_sup" id="S4.T2.5.3.3.1">∘</sup> </th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="S4.T2.6.4.4">45<sup class="ltx_sup" id="S4.T2.6.4.4.1">∘</sup> </th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="S4.T2.7.5.5">65<sup class="ltx_sup" id="S4.T2.7.5.5.1">∘</sup> </th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S4.T2.8.6"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t" id="S4.T2.8.6.1"> <sup class="ltx_sup" id="S4.T2.8.6.1.1"><span class="ltx_text ltx_font_italic" id="S4.T2.8.6.1.1.1">85</span></sup>Rb polarization</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.8.6.2">0.72</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.8.6.3">0.76</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.8.6.4">0.84</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.8.6.5">0.93</td> </tr> <tr class="ltx_tr" id="S4.T2.9.7"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t" id="S4.T2.9.7.1"> <sup class="ltx_sup" id="S4.T2.9.7.1.1"><span class="ltx_text ltx_font_italic" id="S4.T2.9.7.1.1.1">87</span></sup>Rb polarization</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.9.7.2">0.57</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.9.7.3">0.63</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.9.7.4">0.79</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.9.7.5">0.92</td> </tr> <tr class="ltx_tr" id="S4.T2.10.8"> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_l ltx_border_r ltx_border_t" id="S4.T2.10.8.1"><math alttext="c(P)" class="ltx_Math" display="inline" id="S4.T2.10.8.1.m1.1"><semantics id="S4.T2.10.8.1.m1.1a"><mrow id="S4.T2.10.8.1.m1.1.2" xref="S4.T2.10.8.1.m1.1.2.cmml"><mi id="S4.T2.10.8.1.m1.1.2.2" xref="S4.T2.10.8.1.m1.1.2.2.cmml">c</mi><mo id="S4.T2.10.8.1.m1.1.2.1" xref="S4.T2.10.8.1.m1.1.2.1.cmml"></mo><mrow id="S4.T2.10.8.1.m1.1.2.3.2" xref="S4.T2.10.8.1.m1.1.2.cmml"><mo id="S4.T2.10.8.1.m1.1.2.3.2.1" stretchy="false" xref="S4.T2.10.8.1.m1.1.2.cmml">(</mo><mi id="S4.T2.10.8.1.m1.1.1" xref="S4.T2.10.8.1.m1.1.1.cmml">P</mi><mo id="S4.T2.10.8.1.m1.1.2.3.2.2" stretchy="false" xref="S4.T2.10.8.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.10.8.1.m1.1b"><apply id="S4.T2.10.8.1.m1.1.2.cmml" xref="S4.T2.10.8.1.m1.1.2"><times id="S4.T2.10.8.1.m1.1.2.1.cmml" xref="S4.T2.10.8.1.m1.1.2.1"></times><ci id="S4.T2.10.8.1.m1.1.2.2.cmml" xref="S4.T2.10.8.1.m1.1.2.2">𝑐</ci><ci id="S4.T2.10.8.1.m1.1.1.cmml" xref="S4.T2.10.8.1.m1.1.1">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.10.8.1.m1.1c">c(P)</annotation><annotation encoding="application/x-llamapun" id="S4.T2.10.8.1.m1.1d">italic_c ( italic_P )</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T2.10.8.2">3.16</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T2.10.8.3">3.01</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T2.10.8.4">2.70</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T2.10.8.5">2.57</td> </tr> </tbody> </table> </figure> <figure class="ltx_figure" id="S4.F6"><img alt="Refer to caption" class="ltx_graphics ltx_img_portrait" height="663" id="S4.F6.g1" src="x6.png" width="415"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F6.14.7.1" style="font-size:90%;">Figure 6</span>: </span><span class="ltx_text" id="S4.F6.12.6" style="font-size:90%;">(a) Experiment results of heading errors simultaneously measured by two Rb isotopes at B=50 <math alttext="\mu" class="ltx_Math" display="inline" id="S4.F6.7.1.m1.1"><semantics id="S4.F6.7.1.m1.1b"><mi id="S4.F6.7.1.m1.1.1" xref="S4.F6.7.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S4.F6.7.1.m1.1c"><ci id="S4.F6.7.1.m1.1.1.cmml" xref="S4.F6.7.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F6.7.1.m1.1d">\mu</annotation><annotation encoding="application/x-llamapun" id="S4.F6.7.1.m1.1e">italic_μ</annotation></semantics></math>T, compared with theoretical predictions based on Eqs. (<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S2.E8" title="In II.2 Heading error analysis ‣ II Theoretical Background ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">8</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S2.E9" title="In II.2 Heading error analysis ‣ II Theoretical Background ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">9</span></a>). The inset shows the remaining heading error when the value of <math alttext="c(P)" class="ltx_Math" display="inline" id="S4.F6.8.2.m2.1"><semantics id="S4.F6.8.2.m2.1b"><mrow id="S4.F6.8.2.m2.1.2" xref="S4.F6.8.2.m2.1.2.cmml"><mi id="S4.F6.8.2.m2.1.2.2" xref="S4.F6.8.2.m2.1.2.2.cmml">c</mi><mo id="S4.F6.8.2.m2.1.2.1" xref="S4.F6.8.2.m2.1.2.1.cmml"></mo><mrow id="S4.F6.8.2.m2.1.2.3.2" xref="S4.F6.8.2.m2.1.2.cmml"><mo id="S4.F6.8.2.m2.1.2.3.2.1" stretchy="false" xref="S4.F6.8.2.m2.1.2.cmml">(</mo><mi id="S4.F6.8.2.m2.1.1" xref="S4.F6.8.2.m2.1.1.cmml">P</mi><mo id="S4.F6.8.2.m2.1.2.3.2.2" stretchy="false" xref="S4.F6.8.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.F6.8.2.m2.1c"><apply id="S4.F6.8.2.m2.1.2.cmml" xref="S4.F6.8.2.m2.1.2"><times id="S4.F6.8.2.m2.1.2.1.cmml" xref="S4.F6.8.2.m2.1.2.1"></times><ci id="S4.F6.8.2.m2.1.2.2.cmml" xref="S4.F6.8.2.m2.1.2.2">𝑐</ci><ci id="S4.F6.8.2.m2.1.1.cmml" xref="S4.F6.8.2.m2.1.1">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F6.8.2.m2.1d">c(P)</annotation><annotation encoding="application/x-llamapun" id="S4.F6.8.2.m2.1e">italic_c ( italic_P )</annotation></semantics></math> at <math alttext="\theta=25^{\circ}" class="ltx_Math" display="inline" id="S4.F6.9.3.m3.1"><semantics id="S4.F6.9.3.m3.1b"><mrow id="S4.F6.9.3.m3.1.1" xref="S4.F6.9.3.m3.1.1.cmml"><mi id="S4.F6.9.3.m3.1.1.2" xref="S4.F6.9.3.m3.1.1.2.cmml">θ</mi><mo id="S4.F6.9.3.m3.1.1.1" xref="S4.F6.9.3.m3.1.1.1.cmml">=</mo><msup id="S4.F6.9.3.m3.1.1.3" xref="S4.F6.9.3.m3.1.1.3.cmml"><mn id="S4.F6.9.3.m3.1.1.3.2" xref="S4.F6.9.3.m3.1.1.3.2.cmml">25</mn><mo id="S4.F6.9.3.m3.1.1.3.3" xref="S4.F6.9.3.m3.1.1.3.3.cmml">∘</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.F6.9.3.m3.1c"><apply id="S4.F6.9.3.m3.1.1.cmml" xref="S4.F6.9.3.m3.1.1"><eq id="S4.F6.9.3.m3.1.1.1.cmml" xref="S4.F6.9.3.m3.1.1.1"></eq><ci id="S4.F6.9.3.m3.1.1.2.cmml" xref="S4.F6.9.3.m3.1.1.2">𝜃</ci><apply id="S4.F6.9.3.m3.1.1.3.cmml" xref="S4.F6.9.3.m3.1.1.3"><csymbol cd="ambiguous" id="S4.F6.9.3.m3.1.1.3.1.cmml" xref="S4.F6.9.3.m3.1.1.3">superscript</csymbol><cn id="S4.F6.9.3.m3.1.1.3.2.cmml" type="integer" xref="S4.F6.9.3.m3.1.1.3.2">25</cn><compose id="S4.F6.9.3.m3.1.1.3.3.cmml" xref="S4.F6.9.3.m3.1.1.3.3"></compose></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F6.9.3.m3.1d">\theta=25^{\circ}</annotation><annotation encoding="application/x-llamapun" id="S4.F6.9.3.m3.1e">italic_θ = 25 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT</annotation></semantics></math> (<math alttext="\bar{c}" class="ltx_Math" display="inline" id="S4.F6.10.4.m4.1"><semantics id="S4.F6.10.4.m4.1b"><mover accent="true" id="S4.F6.10.4.m4.1.1" xref="S4.F6.10.4.m4.1.1.cmml"><mi id="S4.F6.10.4.m4.1.1.2" xref="S4.F6.10.4.m4.1.1.2.cmml">c</mi><mo id="S4.F6.10.4.m4.1.1.1" xref="S4.F6.10.4.m4.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S4.F6.10.4.m4.1c"><apply id="S4.F6.10.4.m4.1.1.cmml" xref="S4.F6.10.4.m4.1.1"><ci id="S4.F6.10.4.m4.1.1.1.cmml" xref="S4.F6.10.4.m4.1.1.1">¯</ci><ci id="S4.F6.10.4.m4.1.1.2.cmml" xref="S4.F6.10.4.m4.1.1.2">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F6.10.4.m4.1d">\bar{c}</annotation><annotation encoding="application/x-llamapun" id="S4.F6.10.4.m4.1e">over¯ start_ARG italic_c end_ARG</annotation></semantics></math>=2.57) is used is used in Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S4.E15" title="In IV.2 NLZE-induced heading error and its suppression scheme ‣ IV Identifying and suppressing heading errors ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">15</span></a>). (b) Experiment results of <math alttext="(R(\theta)/R_{0}-1)/\omega_{L,85}" class="ltx_Math" display="inline" id="S4.F6.11.5.m5.4"><semantics id="S4.F6.11.5.m5.4b"><mrow id="S4.F6.11.5.m5.4.4" xref="S4.F6.11.5.m5.4.4.cmml"><mrow id="S4.F6.11.5.m5.4.4.1.1" xref="S4.F6.11.5.m5.4.4.1.1.1.cmml"><mo id="S4.F6.11.5.m5.4.4.1.1.2" stretchy="false" xref="S4.F6.11.5.m5.4.4.1.1.1.cmml">(</mo><mrow id="S4.F6.11.5.m5.4.4.1.1.1" xref="S4.F6.11.5.m5.4.4.1.1.1.cmml"><mrow id="S4.F6.11.5.m5.4.4.1.1.1.2" xref="S4.F6.11.5.m5.4.4.1.1.1.2.cmml"><mrow id="S4.F6.11.5.m5.4.4.1.1.1.2.2" xref="S4.F6.11.5.m5.4.4.1.1.1.2.2.cmml"><mi id="S4.F6.11.5.m5.4.4.1.1.1.2.2.2" xref="S4.F6.11.5.m5.4.4.1.1.1.2.2.2.cmml">R</mi><mo id="S4.F6.11.5.m5.4.4.1.1.1.2.2.1" xref="S4.F6.11.5.m5.4.4.1.1.1.2.2.1.cmml"></mo><mrow id="S4.F6.11.5.m5.4.4.1.1.1.2.2.3.2" xref="S4.F6.11.5.m5.4.4.1.1.1.2.2.cmml"><mo id="S4.F6.11.5.m5.4.4.1.1.1.2.2.3.2.1" stretchy="false" xref="S4.F6.11.5.m5.4.4.1.1.1.2.2.cmml">(</mo><mi id="S4.F6.11.5.m5.3.3" xref="S4.F6.11.5.m5.3.3.cmml">θ</mi><mo id="S4.F6.11.5.m5.4.4.1.1.1.2.2.3.2.2" stretchy="false" xref="S4.F6.11.5.m5.4.4.1.1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.F6.11.5.m5.4.4.1.1.1.2.1" xref="S4.F6.11.5.m5.4.4.1.1.1.2.1.cmml">/</mo><msub id="S4.F6.11.5.m5.4.4.1.1.1.2.3" xref="S4.F6.11.5.m5.4.4.1.1.1.2.3.cmml"><mi id="S4.F6.11.5.m5.4.4.1.1.1.2.3.2" xref="S4.F6.11.5.m5.4.4.1.1.1.2.3.2.cmml">R</mi><mn id="S4.F6.11.5.m5.4.4.1.1.1.2.3.3" xref="S4.F6.11.5.m5.4.4.1.1.1.2.3.3.cmml">0</mn></msub></mrow><mo id="S4.F6.11.5.m5.4.4.1.1.1.1" xref="S4.F6.11.5.m5.4.4.1.1.1.1.cmml">−</mo><mn id="S4.F6.11.5.m5.4.4.1.1.1.3" xref="S4.F6.11.5.m5.4.4.1.1.1.3.cmml">1</mn></mrow><mo id="S4.F6.11.5.m5.4.4.1.1.3" stretchy="false" xref="S4.F6.11.5.m5.4.4.1.1.1.cmml">)</mo></mrow><mo id="S4.F6.11.5.m5.4.4.2" xref="S4.F6.11.5.m5.4.4.2.cmml">/</mo><msub id="S4.F6.11.5.m5.4.4.3" xref="S4.F6.11.5.m5.4.4.3.cmml"><mi id="S4.F6.11.5.m5.4.4.3.2" xref="S4.F6.11.5.m5.4.4.3.2.cmml">ω</mi><mrow id="S4.F6.11.5.m5.2.2.2.4" xref="S4.F6.11.5.m5.2.2.2.3.cmml"><mi id="S4.F6.11.5.m5.1.1.1.1" xref="S4.F6.11.5.m5.1.1.1.1.cmml">L</mi><mo id="S4.F6.11.5.m5.2.2.2.4.1" xref="S4.F6.11.5.m5.2.2.2.3.cmml">,</mo><mn id="S4.F6.11.5.m5.2.2.2.2" xref="S4.F6.11.5.m5.2.2.2.2.cmml">85</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.F6.11.5.m5.4c"><apply id="S4.F6.11.5.m5.4.4.cmml" xref="S4.F6.11.5.m5.4.4"><divide id="S4.F6.11.5.m5.4.4.2.cmml" xref="S4.F6.11.5.m5.4.4.2"></divide><apply id="S4.F6.11.5.m5.4.4.1.1.1.cmml" xref="S4.F6.11.5.m5.4.4.1.1"><minus id="S4.F6.11.5.m5.4.4.1.1.1.1.cmml" xref="S4.F6.11.5.m5.4.4.1.1.1.1"></minus><apply id="S4.F6.11.5.m5.4.4.1.1.1.2.cmml" xref="S4.F6.11.5.m5.4.4.1.1.1.2"><divide id="S4.F6.11.5.m5.4.4.1.1.1.2.1.cmml" xref="S4.F6.11.5.m5.4.4.1.1.1.2.1"></divide><apply id="S4.F6.11.5.m5.4.4.1.1.1.2.2.cmml" xref="S4.F6.11.5.m5.4.4.1.1.1.2.2"><times id="S4.F6.11.5.m5.4.4.1.1.1.2.2.1.cmml" xref="S4.F6.11.5.m5.4.4.1.1.1.2.2.1"></times><ci id="S4.F6.11.5.m5.4.4.1.1.1.2.2.2.cmml" xref="S4.F6.11.5.m5.4.4.1.1.1.2.2.2">𝑅</ci><ci id="S4.F6.11.5.m5.3.3.cmml" xref="S4.F6.11.5.m5.3.3">𝜃</ci></apply><apply id="S4.F6.11.5.m5.4.4.1.1.1.2.3.cmml" xref="S4.F6.11.5.m5.4.4.1.1.1.2.3"><csymbol cd="ambiguous" id="S4.F6.11.5.m5.4.4.1.1.1.2.3.1.cmml" xref="S4.F6.11.5.m5.4.4.1.1.1.2.3">subscript</csymbol><ci id="S4.F6.11.5.m5.4.4.1.1.1.2.3.2.cmml" xref="S4.F6.11.5.m5.4.4.1.1.1.2.3.2">𝑅</ci><cn id="S4.F6.11.5.m5.4.4.1.1.1.2.3.3.cmml" type="integer" xref="S4.F6.11.5.m5.4.4.1.1.1.2.3.3">0</cn></apply></apply><cn id="S4.F6.11.5.m5.4.4.1.1.1.3.cmml" type="integer" xref="S4.F6.11.5.m5.4.4.1.1.1.3">1</cn></apply><apply id="S4.F6.11.5.m5.4.4.3.cmml" xref="S4.F6.11.5.m5.4.4.3"><csymbol cd="ambiguous" id="S4.F6.11.5.m5.4.4.3.1.cmml" xref="S4.F6.11.5.m5.4.4.3">subscript</csymbol><ci id="S4.F6.11.5.m5.4.4.3.2.cmml" xref="S4.F6.11.5.m5.4.4.3.2">𝜔</ci><list id="S4.F6.11.5.m5.2.2.2.3.cmml" xref="S4.F6.11.5.m5.2.2.2.4"><ci id="S4.F6.11.5.m5.1.1.1.1.cmml" xref="S4.F6.11.5.m5.1.1.1.1">𝐿</ci><cn id="S4.F6.11.5.m5.2.2.2.2.cmml" type="integer" xref="S4.F6.11.5.m5.2.2.2.2">85</cn></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F6.11.5.m5.4d">(R(\theta)/R_{0}-1)/\omega_{L,85}</annotation><annotation encoding="application/x-llamapun" id="S4.F6.11.5.m5.4e">( italic_R ( italic_θ ) / italic_R start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - 1 ) / italic_ω start_POSTSUBSCRIPT italic_L , 85 end_POSTSUBSCRIPT</annotation></semantics></math> at different <math alttext="\cos\theta" class="ltx_Math" display="inline" id="S4.F6.12.6.m6.1"><semantics id="S4.F6.12.6.m6.1b"><mrow id="S4.F6.12.6.m6.1.1" xref="S4.F6.12.6.m6.1.1.cmml"><mi id="S4.F6.12.6.m6.1.1.1" xref="S4.F6.12.6.m6.1.1.1.cmml">cos</mi><mo id="S4.F6.12.6.m6.1.1b" lspace="0.167em" xref="S4.F6.12.6.m6.1.1.cmml"></mo><mi id="S4.F6.12.6.m6.1.1.2" xref="S4.F6.12.6.m6.1.1.2.cmml">θ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.F6.12.6.m6.1c"><apply id="S4.F6.12.6.m6.1.1.cmml" xref="S4.F6.12.6.m6.1.1"><cos id="S4.F6.12.6.m6.1.1.1.cmml" xref="S4.F6.12.6.m6.1.1.1"></cos><ci id="S4.F6.12.6.m6.1.1.2.cmml" xref="S4.F6.12.6.m6.1.1.2">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F6.12.6.m6.1d">\cos\theta</annotation><annotation encoding="application/x-llamapun" id="S4.F6.12.6.m6.1e">roman_cos italic_θ</annotation></semantics></math> under the same experiment conditions as plot (a), where the dash line shows a linear fitting with a coefficient of determination of 0.997.</span></figcaption> </figure> <div class="ltx_para" id="S4.SS2.p7"> <p class="ltx_p" id="S4.SS2.p7.18">Since the polarization <math alttext="P" class="ltx_Math" display="inline" id="S4.SS2.p7.1.m1.1"><semantics id="S4.SS2.p7.1.m1.1a"><mi id="S4.SS2.p7.1.m1.1.1" xref="S4.SS2.p7.1.m1.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p7.1.m1.1b"><ci id="S4.SS2.p7.1.m1.1.1.cmml" xref="S4.SS2.p7.1.m1.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p7.1.m1.1c">P</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p7.1.m1.1d">italic_P</annotation></semantics></math> changes with <math alttext="\theta" class="ltx_Math" display="inline" id="S4.SS2.p7.2.m2.1"><semantics id="S4.SS2.p7.2.m2.1a"><mi id="S4.SS2.p7.2.m2.1.1" xref="S4.SS2.p7.2.m2.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p7.2.m2.1b"><ci id="S4.SS2.p7.2.m2.1.1.cmml" xref="S4.SS2.p7.2.m2.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p7.2.m2.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p7.2.m2.1d">italic_θ</annotation></semantics></math> in the experiment, the value of <math alttext="c(P)" class="ltx_Math" display="inline" id="S4.SS2.p7.3.m3.1"><semantics id="S4.SS2.p7.3.m3.1a"><mrow id="S4.SS2.p7.3.m3.1.2" xref="S4.SS2.p7.3.m3.1.2.cmml"><mi id="S4.SS2.p7.3.m3.1.2.2" xref="S4.SS2.p7.3.m3.1.2.2.cmml">c</mi><mo id="S4.SS2.p7.3.m3.1.2.1" xref="S4.SS2.p7.3.m3.1.2.1.cmml"></mo><mrow id="S4.SS2.p7.3.m3.1.2.3.2" xref="S4.SS2.p7.3.m3.1.2.cmml"><mo id="S4.SS2.p7.3.m3.1.2.3.2.1" stretchy="false" xref="S4.SS2.p7.3.m3.1.2.cmml">(</mo><mi id="S4.SS2.p7.3.m3.1.1" xref="S4.SS2.p7.3.m3.1.1.cmml">P</mi><mo id="S4.SS2.p7.3.m3.1.2.3.2.2" stretchy="false" xref="S4.SS2.p7.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p7.3.m3.1b"><apply id="S4.SS2.p7.3.m3.1.2.cmml" xref="S4.SS2.p7.3.m3.1.2"><times id="S4.SS2.p7.3.m3.1.2.1.cmml" xref="S4.SS2.p7.3.m3.1.2.1"></times><ci id="S4.SS2.p7.3.m3.1.2.2.cmml" xref="S4.SS2.p7.3.m3.1.2.2">𝑐</ci><ci id="S4.SS2.p7.3.m3.1.1.cmml" xref="S4.SS2.p7.3.m3.1.1">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p7.3.m3.1c">c(P)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p7.3.m3.1d">italic_c ( italic_P )</annotation></semantics></math> is actually depending on <math alttext="\theta" class="ltx_Math" display="inline" id="S4.SS2.p7.4.m4.1"><semantics id="S4.SS2.p7.4.m4.1a"><mi id="S4.SS2.p7.4.m4.1.1" xref="S4.SS2.p7.4.m4.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p7.4.m4.1b"><ci id="S4.SS2.p7.4.m4.1.1.cmml" xref="S4.SS2.p7.4.m4.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p7.4.m4.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p7.4.m4.1d">italic_θ</annotation></semantics></math>, as listed in Tab. <a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S4.T2" title="Table 2 ‣ IV.2 NLZE-induced heading error and its suppression scheme ‣ IV Identifying and suppressing heading errors ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">2</span></a>. When the value of <math alttext="\theta" class="ltx_Math" display="inline" id="S4.SS2.p7.5.m5.1"><semantics id="S4.SS2.p7.5.m5.1a"><mi id="S4.SS2.p7.5.m5.1.1" xref="S4.SS2.p7.5.m5.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p7.5.m5.1b"><ci id="S4.SS2.p7.5.m5.1.1.cmml" xref="S4.SS2.p7.5.m5.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p7.5.m5.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p7.5.m5.1d">italic_θ</annotation></semantics></math> is unknown, we need to use a constant <math alttext="\bar{c}" class="ltx_Math" display="inline" id="S4.SS2.p7.6.m6.1"><semantics id="S4.SS2.p7.6.m6.1a"><mover accent="true" id="S4.SS2.p7.6.m6.1.1" xref="S4.SS2.p7.6.m6.1.1.cmml"><mi id="S4.SS2.p7.6.m6.1.1.2" xref="S4.SS2.p7.6.m6.1.1.2.cmml">c</mi><mo id="S4.SS2.p7.6.m6.1.1.1" xref="S4.SS2.p7.6.m6.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S4.SS2.p7.6.m6.1b"><apply id="S4.SS2.p7.6.m6.1.1.cmml" xref="S4.SS2.p7.6.m6.1.1"><ci id="S4.SS2.p7.6.m6.1.1.1.cmml" xref="S4.SS2.p7.6.m6.1.1.1">¯</ci><ci id="S4.SS2.p7.6.m6.1.1.2.cmml" xref="S4.SS2.p7.6.m6.1.1.2">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p7.6.m6.1c">\bar{c}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p7.6.m6.1d">over¯ start_ARG italic_c end_ARG</annotation></semantics></math> as the approximation of <math alttext="c(P)" class="ltx_Math" display="inline" id="S4.SS2.p7.7.m7.1"><semantics id="S4.SS2.p7.7.m7.1a"><mrow id="S4.SS2.p7.7.m7.1.2" xref="S4.SS2.p7.7.m7.1.2.cmml"><mi id="S4.SS2.p7.7.m7.1.2.2" xref="S4.SS2.p7.7.m7.1.2.2.cmml">c</mi><mo id="S4.SS2.p7.7.m7.1.2.1" xref="S4.SS2.p7.7.m7.1.2.1.cmml"></mo><mrow id="S4.SS2.p7.7.m7.1.2.3.2" xref="S4.SS2.p7.7.m7.1.2.cmml"><mo id="S4.SS2.p7.7.m7.1.2.3.2.1" stretchy="false" xref="S4.SS2.p7.7.m7.1.2.cmml">(</mo><mi id="S4.SS2.p7.7.m7.1.1" xref="S4.SS2.p7.7.m7.1.1.cmml">P</mi><mo id="S4.SS2.p7.7.m7.1.2.3.2.2" stretchy="false" xref="S4.SS2.p7.7.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p7.7.m7.1b"><apply id="S4.SS2.p7.7.m7.1.2.cmml" xref="S4.SS2.p7.7.m7.1.2"><times id="S4.SS2.p7.7.m7.1.2.1.cmml" xref="S4.SS2.p7.7.m7.1.2.1"></times><ci id="S4.SS2.p7.7.m7.1.2.2.cmml" xref="S4.SS2.p7.7.m7.1.2.2">𝑐</ci><ci id="S4.SS2.p7.7.m7.1.1.cmml" xref="S4.SS2.p7.7.m7.1.1">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p7.7.m7.1c">c(P)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p7.7.m7.1d">italic_c ( italic_P )</annotation></semantics></math> in Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S4.E15" title="In IV.2 NLZE-induced heading error and its suppression scheme ‣ IV Identifying and suppressing heading errors ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">15</span></a>) to minimize the NLZE for the whole experiment range of <math alttext="\theta" class="ltx_Math" display="inline" id="S4.SS2.p7.8.m8.1"><semantics id="S4.SS2.p7.8.m8.1a"><mi id="S4.SS2.p7.8.m8.1.1" xref="S4.SS2.p7.8.m8.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p7.8.m8.1b"><ci id="S4.SS2.p7.8.m8.1.1.cmml" xref="S4.SS2.p7.8.m8.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p7.8.m8.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p7.8.m8.1d">italic_θ</annotation></semantics></math>. In practice, we find that <math alttext="\bar{c}" class="ltx_Math" display="inline" id="S4.SS2.p7.9.m9.1"><semantics id="S4.SS2.p7.9.m9.1a"><mover accent="true" id="S4.SS2.p7.9.m9.1.1" xref="S4.SS2.p7.9.m9.1.1.cmml"><mi id="S4.SS2.p7.9.m9.1.1.2" xref="S4.SS2.p7.9.m9.1.1.2.cmml">c</mi><mo id="S4.SS2.p7.9.m9.1.1.1" xref="S4.SS2.p7.9.m9.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S4.SS2.p7.9.m9.1b"><apply id="S4.SS2.p7.9.m9.1.1.cmml" xref="S4.SS2.p7.9.m9.1.1"><ci id="S4.SS2.p7.9.m9.1.1.1.cmml" xref="S4.SS2.p7.9.m9.1.1.1">¯</ci><ci id="S4.SS2.p7.9.m9.1.1.2.cmml" xref="S4.SS2.p7.9.m9.1.1.2">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p7.9.m9.1c">\bar{c}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p7.9.m9.1d">over¯ start_ARG italic_c end_ARG</annotation></semantics></math> can be chosen near the value of <math alttext="c(P)" class="ltx_Math" display="inline" id="S4.SS2.p7.10.m10.1"><semantics id="S4.SS2.p7.10.m10.1a"><mrow id="S4.SS2.p7.10.m10.1.2" xref="S4.SS2.p7.10.m10.1.2.cmml"><mi id="S4.SS2.p7.10.m10.1.2.2" xref="S4.SS2.p7.10.m10.1.2.2.cmml">c</mi><mo id="S4.SS2.p7.10.m10.1.2.1" xref="S4.SS2.p7.10.m10.1.2.1.cmml"></mo><mrow id="S4.SS2.p7.10.m10.1.2.3.2" xref="S4.SS2.p7.10.m10.1.2.cmml"><mo id="S4.SS2.p7.10.m10.1.2.3.2.1" stretchy="false" xref="S4.SS2.p7.10.m10.1.2.cmml">(</mo><mi id="S4.SS2.p7.10.m10.1.1" xref="S4.SS2.p7.10.m10.1.1.cmml">P</mi><mo id="S4.SS2.p7.10.m10.1.2.3.2.2" stretchy="false" xref="S4.SS2.p7.10.m10.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p7.10.m10.1b"><apply id="S4.SS2.p7.10.m10.1.2.cmml" xref="S4.SS2.p7.10.m10.1.2"><times id="S4.SS2.p7.10.m10.1.2.1.cmml" xref="S4.SS2.p7.10.m10.1.2.1"></times><ci id="S4.SS2.p7.10.m10.1.2.2.cmml" xref="S4.SS2.p7.10.m10.1.2.2">𝑐</ci><ci id="S4.SS2.p7.10.m10.1.1.cmml" xref="S4.SS2.p7.10.m10.1.1">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p7.10.m10.1c">c(P)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p7.10.m10.1d">italic_c ( italic_P )</annotation></semantics></math> at <math alttext="\theta" class="ltx_Math" display="inline" id="S4.SS2.p7.11.m11.1"><semantics id="S4.SS2.p7.11.m11.1a"><mi id="S4.SS2.p7.11.m11.1.1" xref="S4.SS2.p7.11.m11.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p7.11.m11.1b"><ci id="S4.SS2.p7.11.m11.1.1.cmml" xref="S4.SS2.p7.11.m11.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p7.11.m11.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p7.11.m11.1d">italic_θ</annotation></semantics></math> farthest away from 90<sup class="ltx_sup" id="S4.SS2.p7.18.1">∘</sup>. For example, in our experiment, the heading error can be suppressed within 1 nT when <math alttext="\bar{c}" class="ltx_Math" display="inline" id="S4.SS2.p7.13.m13.1"><semantics id="S4.SS2.p7.13.m13.1a"><mover accent="true" id="S4.SS2.p7.13.m13.1.1" xref="S4.SS2.p7.13.m13.1.1.cmml"><mi id="S4.SS2.p7.13.m13.1.1.2" xref="S4.SS2.p7.13.m13.1.1.2.cmml">c</mi><mo id="S4.SS2.p7.13.m13.1.1.1" xref="S4.SS2.p7.13.m13.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S4.SS2.p7.13.m13.1b"><apply id="S4.SS2.p7.13.m13.1.1.cmml" xref="S4.SS2.p7.13.m13.1.1"><ci id="S4.SS2.p7.13.m13.1.1.1.cmml" xref="S4.SS2.p7.13.m13.1.1.1">¯</ci><ci id="S4.SS2.p7.13.m13.1.1.2.cmml" xref="S4.SS2.p7.13.m13.1.1.2">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p7.13.m13.1c">\bar{c}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p7.13.m13.1d">over¯ start_ARG italic_c end_ARG</annotation></semantics></math> is chosen in the range of 2.40 to 2.62, and a case with <math alttext="\bar{c}" class="ltx_Math" display="inline" id="S4.SS2.p7.14.m14.1"><semantics id="S4.SS2.p7.14.m14.1a"><mover accent="true" id="S4.SS2.p7.14.m14.1.1" xref="S4.SS2.p7.14.m14.1.1.cmml"><mi id="S4.SS2.p7.14.m14.1.1.2" xref="S4.SS2.p7.14.m14.1.1.2.cmml">c</mi><mo id="S4.SS2.p7.14.m14.1.1.1" xref="S4.SS2.p7.14.m14.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S4.SS2.p7.14.m14.1b"><apply id="S4.SS2.p7.14.m14.1.1.cmml" xref="S4.SS2.p7.14.m14.1.1"><ci id="S4.SS2.p7.14.m14.1.1.1.cmml" xref="S4.SS2.p7.14.m14.1.1.1">¯</ci><ci id="S4.SS2.p7.14.m14.1.1.2.cmml" xref="S4.SS2.p7.14.m14.1.1.2">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p7.14.m14.1c">\bar{c}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p7.14.m14.1d">over¯ start_ARG italic_c end_ARG</annotation></semantics></math> equal to <math alttext="c(P)" class="ltx_Math" display="inline" id="S4.SS2.p7.15.m15.1"><semantics id="S4.SS2.p7.15.m15.1a"><mrow id="S4.SS2.p7.15.m15.1.2" xref="S4.SS2.p7.15.m15.1.2.cmml"><mi id="S4.SS2.p7.15.m15.1.2.2" xref="S4.SS2.p7.15.m15.1.2.2.cmml">c</mi><mo id="S4.SS2.p7.15.m15.1.2.1" xref="S4.SS2.p7.15.m15.1.2.1.cmml"></mo><mrow id="S4.SS2.p7.15.m15.1.2.3.2" xref="S4.SS2.p7.15.m15.1.2.cmml"><mo id="S4.SS2.p7.15.m15.1.2.3.2.1" stretchy="false" xref="S4.SS2.p7.15.m15.1.2.cmml">(</mo><mi id="S4.SS2.p7.15.m15.1.1" xref="S4.SS2.p7.15.m15.1.1.cmml">P</mi><mo id="S4.SS2.p7.15.m15.1.2.3.2.2" stretchy="false" xref="S4.SS2.p7.15.m15.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p7.15.m15.1b"><apply id="S4.SS2.p7.15.m15.1.2.cmml" xref="S4.SS2.p7.15.m15.1.2"><times id="S4.SS2.p7.15.m15.1.2.1.cmml" xref="S4.SS2.p7.15.m15.1.2.1"></times><ci id="S4.SS2.p7.15.m15.1.2.2.cmml" xref="S4.SS2.p7.15.m15.1.2.2">𝑐</ci><ci id="S4.SS2.p7.15.m15.1.1.cmml" xref="S4.SS2.p7.15.m15.1.1">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p7.15.m15.1c">c(P)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p7.15.m15.1d">italic_c ( italic_P )</annotation></semantics></math> at <math alttext="\theta" class="ltx_Math" display="inline" id="S4.SS2.p7.16.m16.1"><semantics id="S4.SS2.p7.16.m16.1a"><mi id="S4.SS2.p7.16.m16.1.1" xref="S4.SS2.p7.16.m16.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p7.16.m16.1b"><ci id="S4.SS2.p7.16.m16.1.1.cmml" xref="S4.SS2.p7.16.m16.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p7.16.m16.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p7.16.m16.1d">italic_θ</annotation></semantics></math>=25<sup class="ltx_sup" id="S4.SS2.p7.18.2">∘</sup> (<math alttext="\bar{c}" class="ltx_Math" display="inline" id="S4.SS2.p7.18.m18.1"><semantics id="S4.SS2.p7.18.m18.1a"><mover accent="true" id="S4.SS2.p7.18.m18.1.1" xref="S4.SS2.p7.18.m18.1.1.cmml"><mi id="S4.SS2.p7.18.m18.1.1.2" xref="S4.SS2.p7.18.m18.1.1.2.cmml">c</mi><mo id="S4.SS2.p7.18.m18.1.1.1" xref="S4.SS2.p7.18.m18.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S4.SS2.p7.18.m18.1b"><apply id="S4.SS2.p7.18.m18.1.1.cmml" xref="S4.SS2.p7.18.m18.1.1"><ci id="S4.SS2.p7.18.m18.1.1.1.cmml" xref="S4.SS2.p7.18.m18.1.1.1">¯</ci><ci id="S4.SS2.p7.18.m18.1.1.2.cmml" xref="S4.SS2.p7.18.m18.1.1.2">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p7.18.m18.1c">\bar{c}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p7.18.m18.1d">over¯ start_ARG italic_c end_ARG</annotation></semantics></math>= 2.57) is demonstrated in the inset of Fig. <a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S4.F6" title="Figure 6 ‣ IV.2 NLZE-induced heading error and its suppression scheme ‣ IV Identifying and suppressing heading errors ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">6</span></a>(a).</p> </div> <div class="ltx_para" id="S4.SS2.p8"> <p class="ltx_p" id="S4.SS2.p8.3">The common readout from the Rb-isotope comagnetometer is the frequency ratio <math alttext="R=\omega_{L,87}/\omega_{L,85}" class="ltx_Math" display="inline" id="S4.SS2.p8.1.m1.4"><semantics id="S4.SS2.p8.1.m1.4a"><mrow id="S4.SS2.p8.1.m1.4.5" xref="S4.SS2.p8.1.m1.4.5.cmml"><mi id="S4.SS2.p8.1.m1.4.5.2" xref="S4.SS2.p8.1.m1.4.5.2.cmml">R</mi><mo id="S4.SS2.p8.1.m1.4.5.1" xref="S4.SS2.p8.1.m1.4.5.1.cmml">=</mo><mrow id="S4.SS2.p8.1.m1.4.5.3" xref="S4.SS2.p8.1.m1.4.5.3.cmml"><msub id="S4.SS2.p8.1.m1.4.5.3.2" xref="S4.SS2.p8.1.m1.4.5.3.2.cmml"><mi id="S4.SS2.p8.1.m1.4.5.3.2.2" xref="S4.SS2.p8.1.m1.4.5.3.2.2.cmml">ω</mi><mrow id="S4.SS2.p8.1.m1.2.2.2.4" xref="S4.SS2.p8.1.m1.2.2.2.3.cmml"><mi id="S4.SS2.p8.1.m1.1.1.1.1" xref="S4.SS2.p8.1.m1.1.1.1.1.cmml">L</mi><mo id="S4.SS2.p8.1.m1.2.2.2.4.1" xref="S4.SS2.p8.1.m1.2.2.2.3.cmml">,</mo><mn id="S4.SS2.p8.1.m1.2.2.2.2" xref="S4.SS2.p8.1.m1.2.2.2.2.cmml">87</mn></mrow></msub><mo id="S4.SS2.p8.1.m1.4.5.3.1" xref="S4.SS2.p8.1.m1.4.5.3.1.cmml">/</mo><msub id="S4.SS2.p8.1.m1.4.5.3.3" xref="S4.SS2.p8.1.m1.4.5.3.3.cmml"><mi id="S4.SS2.p8.1.m1.4.5.3.3.2" xref="S4.SS2.p8.1.m1.4.5.3.3.2.cmml">ω</mi><mrow id="S4.SS2.p8.1.m1.4.4.2.4" xref="S4.SS2.p8.1.m1.4.4.2.3.cmml"><mi id="S4.SS2.p8.1.m1.3.3.1.1" xref="S4.SS2.p8.1.m1.3.3.1.1.cmml">L</mi><mo id="S4.SS2.p8.1.m1.4.4.2.4.1" xref="S4.SS2.p8.1.m1.4.4.2.3.cmml">,</mo><mn id="S4.SS2.p8.1.m1.4.4.2.2" xref="S4.SS2.p8.1.m1.4.4.2.2.cmml">85</mn></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p8.1.m1.4b"><apply id="S4.SS2.p8.1.m1.4.5.cmml" xref="S4.SS2.p8.1.m1.4.5"><eq id="S4.SS2.p8.1.m1.4.5.1.cmml" xref="S4.SS2.p8.1.m1.4.5.1"></eq><ci id="S4.SS2.p8.1.m1.4.5.2.cmml" xref="S4.SS2.p8.1.m1.4.5.2">𝑅</ci><apply id="S4.SS2.p8.1.m1.4.5.3.cmml" xref="S4.SS2.p8.1.m1.4.5.3"><divide id="S4.SS2.p8.1.m1.4.5.3.1.cmml" xref="S4.SS2.p8.1.m1.4.5.3.1"></divide><apply id="S4.SS2.p8.1.m1.4.5.3.2.cmml" xref="S4.SS2.p8.1.m1.4.5.3.2"><csymbol cd="ambiguous" id="S4.SS2.p8.1.m1.4.5.3.2.1.cmml" xref="S4.SS2.p8.1.m1.4.5.3.2">subscript</csymbol><ci id="S4.SS2.p8.1.m1.4.5.3.2.2.cmml" xref="S4.SS2.p8.1.m1.4.5.3.2.2">𝜔</ci><list id="S4.SS2.p8.1.m1.2.2.2.3.cmml" xref="S4.SS2.p8.1.m1.2.2.2.4"><ci id="S4.SS2.p8.1.m1.1.1.1.1.cmml" xref="S4.SS2.p8.1.m1.1.1.1.1">𝐿</ci><cn id="S4.SS2.p8.1.m1.2.2.2.2.cmml" type="integer" xref="S4.SS2.p8.1.m1.2.2.2.2">87</cn></list></apply><apply id="S4.SS2.p8.1.m1.4.5.3.3.cmml" xref="S4.SS2.p8.1.m1.4.5.3.3"><csymbol cd="ambiguous" id="S4.SS2.p8.1.m1.4.5.3.3.1.cmml" xref="S4.SS2.p8.1.m1.4.5.3.3">subscript</csymbol><ci id="S4.SS2.p8.1.m1.4.5.3.3.2.cmml" xref="S4.SS2.p8.1.m1.4.5.3.3.2">𝜔</ci><list id="S4.SS2.p8.1.m1.4.4.2.3.cmml" xref="S4.SS2.p8.1.m1.4.4.2.4"><ci id="S4.SS2.p8.1.m1.3.3.1.1.cmml" xref="S4.SS2.p8.1.m1.3.3.1.1">𝐿</ci><cn id="S4.SS2.p8.1.m1.4.4.2.2.cmml" type="integer" xref="S4.SS2.p8.1.m1.4.4.2.2">85</cn></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p8.1.m1.4c">R=\omega_{L,87}/\omega_{L,85}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p8.1.m1.4d">italic_R = italic_ω start_POSTSUBSCRIPT italic_L , 87 end_POSTSUBSCRIPT / italic_ω start_POSTSUBSCRIPT italic_L , 85 end_POSTSUBSCRIPT</annotation></semantics></math>. According to Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S4.E14" title="In IV.2 NLZE-induced heading error and its suppression scheme ‣ IV Identifying and suppressing heading errors ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">14</span></a>), <math alttext="R" class="ltx_Math" display="inline" id="S4.SS2.p8.2.m2.1"><semantics id="S4.SS2.p8.2.m2.1a"><mi id="S4.SS2.p8.2.m2.1.1" xref="S4.SS2.p8.2.m2.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p8.2.m2.1b"><ci id="S4.SS2.p8.2.m2.1.1.cmml" xref="S4.SS2.p8.2.m2.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p8.2.m2.1c">R</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p8.2.m2.1d">italic_R</annotation></semantics></math> is dependent on <math alttext="\theta" class="ltx_Math" display="inline" id="S4.SS2.p8.3.m3.1"><semantics id="S4.SS2.p8.3.m3.1a"><mi id="S4.SS2.p8.3.m3.1.1" xref="S4.SS2.p8.3.m3.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p8.3.m3.1b"><ci id="S4.SS2.p8.3.m3.1.1.cmml" xref="S4.SS2.p8.3.m3.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p8.3.m3.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p8.3.m3.1d">italic_θ</annotation></semantics></math>, and in the high-polarization limit, this relation can be expressed as</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S6.EGx3"> <tbody id="S4.E16"> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle~{}\cos\theta" class="ltx_Math" display="inline" id="S4.Ex2.m1.1"><semantics id="S4.Ex2.m1.1a"><mrow id="S4.Ex2.m1.1.1" xref="S4.Ex2.m1.1.1.cmml"><mi id="S4.Ex2.m1.1.1.1" xref="S4.Ex2.m1.1.1.1.cmml">cos</mi><mo id="S4.Ex2.m1.1.1a" lspace="0.167em" xref="S4.Ex2.m1.1.1.cmml"></mo><mi id="S4.Ex2.m1.1.1.2" xref="S4.Ex2.m1.1.1.2.cmml">θ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex2.m1.1b"><apply id="S4.Ex2.m1.1.1.cmml" xref="S4.Ex2.m1.1.1"><cos id="S4.Ex2.m1.1.1.1.cmml" xref="S4.Ex2.m1.1.1.1"></cos><ci id="S4.Ex2.m1.1.1.2.cmml" xref="S4.Ex2.m1.1.1.2">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex2.m1.1c">\displaystyle~{}\cos\theta</annotation><annotation encoding="application/x-llamapun" id="S4.Ex2.m1.1d">roman_cos italic_θ</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S4.Ex2.m2.1"><semantics id="S4.Ex2.m2.1a"><mo id="S4.Ex2.m2.1.1" xref="S4.Ex2.m2.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S4.Ex2.m2.1b"><eq id="S4.Ex2.m2.1.1.cmml" xref="S4.Ex2.m2.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex2.m2.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="S4.Ex2.m2.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\frac{a_{85}}{b(P)_{85}-R^{2}(\theta)b(P)_{87}}\frac{R(\theta)/R_% {0}-1}{\omega_{L,85}}" class="ltx_Math" display="inline" id="S4.Ex2.m3.6"><semantics id="S4.Ex2.m3.6a"><mrow id="S4.Ex2.m3.6.7" xref="S4.Ex2.m3.6.7.cmml"><mstyle displaystyle="true" id="S4.Ex2.m3.3.3" xref="S4.Ex2.m3.3.3.cmml"><mfrac id="S4.Ex2.m3.3.3a" xref="S4.Ex2.m3.3.3.cmml"><msub id="S4.Ex2.m3.3.3.5" xref="S4.Ex2.m3.3.3.5.cmml"><mi id="S4.Ex2.m3.3.3.5.2" xref="S4.Ex2.m3.3.3.5.2.cmml">a</mi><mn id="S4.Ex2.m3.3.3.5.3" xref="S4.Ex2.m3.3.3.5.3.cmml">85</mn></msub><mrow id="S4.Ex2.m3.3.3.3" xref="S4.Ex2.m3.3.3.3.cmml"><mrow id="S4.Ex2.m3.3.3.3.5" xref="S4.Ex2.m3.3.3.3.5.cmml"><mi id="S4.Ex2.m3.3.3.3.5.2" xref="S4.Ex2.m3.3.3.3.5.2.cmml">b</mi><mo id="S4.Ex2.m3.3.3.3.5.1" 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xref="S4.Ex2.m3.4.4.1"><minus id="S4.Ex2.m3.4.4.1.2.cmml" xref="S4.Ex2.m3.4.4.1.2"></minus><apply id="S4.Ex2.m3.4.4.1.3.cmml" xref="S4.Ex2.m3.4.4.1.3"><divide id="S4.Ex2.m3.4.4.1.3.1.cmml" xref="S4.Ex2.m3.4.4.1.3.1"></divide><apply id="S4.Ex2.m3.4.4.1.3.2.cmml" xref="S4.Ex2.m3.4.4.1.3.2"><times id="S4.Ex2.m3.4.4.1.3.2.1.cmml" xref="S4.Ex2.m3.4.4.1.3.2.1"></times><ci id="S4.Ex2.m3.4.4.1.3.2.2.cmml" xref="S4.Ex2.m3.4.4.1.3.2.2">𝑅</ci><ci id="S4.Ex2.m3.4.4.1.1.cmml" xref="S4.Ex2.m3.4.4.1.1">𝜃</ci></apply><apply id="S4.Ex2.m3.4.4.1.3.3.cmml" xref="S4.Ex2.m3.4.4.1.3.3"><csymbol cd="ambiguous" id="S4.Ex2.m3.4.4.1.3.3.1.cmml" xref="S4.Ex2.m3.4.4.1.3.3">subscript</csymbol><ci id="S4.Ex2.m3.4.4.1.3.3.2.cmml" xref="S4.Ex2.m3.4.4.1.3.3.2">𝑅</ci><cn id="S4.Ex2.m3.4.4.1.3.3.3.cmml" type="integer" xref="S4.Ex2.m3.4.4.1.3.3.3">0</cn></apply></apply><cn id="S4.Ex2.m3.4.4.1.4.cmml" type="integer" xref="S4.Ex2.m3.4.4.1.4">1</cn></apply><apply id="S4.Ex2.m3.6.6.3.cmml" xref="S4.Ex2.m3.6.6.3"><csymbol cd="ambiguous" id="S4.Ex2.m3.6.6.3.3.cmml" xref="S4.Ex2.m3.6.6.3">subscript</csymbol><ci id="S4.Ex2.m3.6.6.3.4.cmml" xref="S4.Ex2.m3.6.6.3.4">𝜔</ci><list id="S4.Ex2.m3.6.6.3.2.2.3.cmml" xref="S4.Ex2.m3.6.6.3.2.2.4"><ci id="S4.Ex2.m3.5.5.2.1.1.1.cmml" xref="S4.Ex2.m3.5.5.2.1.1.1">𝐿</ci><cn id="S4.Ex2.m3.6.6.3.2.2.2.cmml" type="integer" xref="S4.Ex2.m3.6.6.3.2.2.2">85</cn></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex2.m3.6c">\displaystyle\frac{a_{85}}{b(P)_{85}-R^{2}(\theta)b(P)_{87}}\frac{R(\theta)/R_% {0}-1}{\omega_{L,85}}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex2.m3.6d">divide start_ARG italic_a start_POSTSUBSCRIPT 85 end_POSTSUBSCRIPT end_ARG start_ARG italic_b ( italic_P ) start_POSTSUBSCRIPT 85 end_POSTSUBSCRIPT - italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_θ ) italic_b ( italic_P ) start_POSTSUBSCRIPT 87 end_POSTSUBSCRIPT end_ARG divide start_ARG italic_R ( italic_θ ) / italic_R start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - 1 end_ARG start_ARG italic_ω start_POSTSUBSCRIPT italic_L , 85 end_POSTSUBSCRIPT end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="2"><span class="ltx_tag ltx_tag_equation ltx_align_right">(16)</span></td> </tr> <tr class="ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S4.E16.m1.1"><semantics id="S4.E16.m1.1a"><mo id="S4.E16.m1.1.1" xref="S4.E16.m1.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S4.E16.m1.1b"><eq id="S4.E16.m1.1.1.cmml" xref="S4.E16.m1.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S4.E16.m1.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="S4.E16.m1.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle k(P)\frac{R(\theta)/R_{0}-1}{\omega_{L,85}}," class="ltx_Math" display="inline" id="S4.E16.m2.5"><semantics id="S4.E16.m2.5a"><mrow id="S4.E16.m2.5.5.1" xref="S4.E16.m2.5.5.1.1.cmml"><mrow id="S4.E16.m2.5.5.1.1" xref="S4.E16.m2.5.5.1.1.cmml"><mi id="S4.E16.m2.5.5.1.1.2" xref="S4.E16.m2.5.5.1.1.2.cmml">k</mi><mo id="S4.E16.m2.5.5.1.1.1" xref="S4.E16.m2.5.5.1.1.1.cmml"></mo><mrow id="S4.E16.m2.5.5.1.1.3.2" xref="S4.E16.m2.5.5.1.1.cmml"><mo id="S4.E16.m2.5.5.1.1.3.2.1" stretchy="false" xref="S4.E16.m2.5.5.1.1.cmml">(</mo><mi id="S4.E16.m2.4.4" xref="S4.E16.m2.4.4.cmml">P</mi><mo id="S4.E16.m2.5.5.1.1.3.2.2" stretchy="false" xref="S4.E16.m2.5.5.1.1.cmml">)</mo></mrow><mo id="S4.E16.m2.5.5.1.1.1a" xref="S4.E16.m2.5.5.1.1.1.cmml"></mo><mstyle displaystyle="true" id="S4.E16.m2.3.3" xref="S4.E16.m2.3.3.cmml"><mfrac id="S4.E16.m2.3.3a" xref="S4.E16.m2.3.3.cmml"><mrow id="S4.E16.m2.1.1.1" xref="S4.E16.m2.1.1.1.cmml"><mrow id="S4.E16.m2.1.1.1.3" xref="S4.E16.m2.1.1.1.3.cmml"><mrow id="S4.E16.m2.1.1.1.3.2" xref="S4.E16.m2.1.1.1.3.2.cmml"><mi id="S4.E16.m2.1.1.1.3.2.2" xref="S4.E16.m2.1.1.1.3.2.2.cmml">R</mi><mo id="S4.E16.m2.1.1.1.3.2.1" xref="S4.E16.m2.1.1.1.3.2.1.cmml"></mo><mrow id="S4.E16.m2.1.1.1.3.2.3.2" xref="S4.E16.m2.1.1.1.3.2.cmml"><mo id="S4.E16.m2.1.1.1.3.2.3.2.1" stretchy="false" xref="S4.E16.m2.1.1.1.3.2.cmml">(</mo><mi id="S4.E16.m2.1.1.1.1" xref="S4.E16.m2.1.1.1.1.cmml">θ</mi><mo id="S4.E16.m2.1.1.1.3.2.3.2.2" stretchy="false" xref="S4.E16.m2.1.1.1.3.2.cmml">)</mo></mrow></mrow><mo id="S4.E16.m2.1.1.1.3.1" xref="S4.E16.m2.1.1.1.3.1.cmml">/</mo><msub id="S4.E16.m2.1.1.1.3.3" xref="S4.E16.m2.1.1.1.3.3.cmml"><mi id="S4.E16.m2.1.1.1.3.3.2" xref="S4.E16.m2.1.1.1.3.3.2.cmml">R</mi><mn id="S4.E16.m2.1.1.1.3.3.3" xref="S4.E16.m2.1.1.1.3.3.3.cmml">0</mn></msub></mrow><mo id="S4.E16.m2.1.1.1.2" xref="S4.E16.m2.1.1.1.2.cmml">−</mo><mn id="S4.E16.m2.1.1.1.4" xref="S4.E16.m2.1.1.1.4.cmml">1</mn></mrow><msub id="S4.E16.m2.3.3.3" xref="S4.E16.m2.3.3.3.cmml"><mi id="S4.E16.m2.3.3.3.4" xref="S4.E16.m2.3.3.3.4.cmml">ω</mi><mrow id="S4.E16.m2.3.3.3.2.2.4" xref="S4.E16.m2.3.3.3.2.2.3.cmml"><mi id="S4.E16.m2.2.2.2.1.1.1" xref="S4.E16.m2.2.2.2.1.1.1.cmml">L</mi><mo id="S4.E16.m2.3.3.3.2.2.4.1" xref="S4.E16.m2.3.3.3.2.2.3.cmml">,</mo><mn id="S4.E16.m2.3.3.3.2.2.2" xref="S4.E16.m2.3.3.3.2.2.2.cmml">85</mn></mrow></msub></mfrac></mstyle></mrow><mo id="S4.E16.m2.5.5.1.2" xref="S4.E16.m2.5.5.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E16.m2.5b"><apply id="S4.E16.m2.5.5.1.1.cmml" xref="S4.E16.m2.5.5.1"><times id="S4.E16.m2.5.5.1.1.1.cmml" xref="S4.E16.m2.5.5.1.1.1"></times><ci id="S4.E16.m2.5.5.1.1.2.cmml" xref="S4.E16.m2.5.5.1.1.2">𝑘</ci><ci id="S4.E16.m2.4.4.cmml" xref="S4.E16.m2.4.4">𝑃</ci><apply id="S4.E16.m2.3.3.cmml" xref="S4.E16.m2.3.3"><divide id="S4.E16.m2.3.3.4.cmml" xref="S4.E16.m2.3.3"></divide><apply id="S4.E16.m2.1.1.1.cmml" xref="S4.E16.m2.1.1.1"><minus id="S4.E16.m2.1.1.1.2.cmml" xref="S4.E16.m2.1.1.1.2"></minus><apply id="S4.E16.m2.1.1.1.3.cmml" xref="S4.E16.m2.1.1.1.3"><divide id="S4.E16.m2.1.1.1.3.1.cmml" xref="S4.E16.m2.1.1.1.3.1"></divide><apply id="S4.E16.m2.1.1.1.3.2.cmml" xref="S4.E16.m2.1.1.1.3.2"><times id="S4.E16.m2.1.1.1.3.2.1.cmml" xref="S4.E16.m2.1.1.1.3.2.1"></times><ci id="S4.E16.m2.1.1.1.3.2.2.cmml" xref="S4.E16.m2.1.1.1.3.2.2">𝑅</ci><ci id="S4.E16.m2.1.1.1.1.cmml" xref="S4.E16.m2.1.1.1.1">𝜃</ci></apply><apply id="S4.E16.m2.1.1.1.3.3.cmml" xref="S4.E16.m2.1.1.1.3.3"><csymbol cd="ambiguous" id="S4.E16.m2.1.1.1.3.3.1.cmml" xref="S4.E16.m2.1.1.1.3.3">subscript</csymbol><ci id="S4.E16.m2.1.1.1.3.3.2.cmml" xref="S4.E16.m2.1.1.1.3.3.2">𝑅</ci><cn id="S4.E16.m2.1.1.1.3.3.3.cmml" type="integer" xref="S4.E16.m2.1.1.1.3.3.3">0</cn></apply></apply><cn id="S4.E16.m2.1.1.1.4.cmml" type="integer" xref="S4.E16.m2.1.1.1.4">1</cn></apply><apply id="S4.E16.m2.3.3.3.cmml" xref="S4.E16.m2.3.3.3"><csymbol cd="ambiguous" id="S4.E16.m2.3.3.3.3.cmml" xref="S4.E16.m2.3.3.3">subscript</csymbol><ci id="S4.E16.m2.3.3.3.4.cmml" xref="S4.E16.m2.3.3.3.4">𝜔</ci><list id="S4.E16.m2.3.3.3.2.2.3.cmml" xref="S4.E16.m2.3.3.3.2.2.4"><ci id="S4.E16.m2.2.2.2.1.1.1.cmml" xref="S4.E16.m2.2.2.2.1.1.1">𝐿</ci><cn id="S4.E16.m2.3.3.3.2.2.2.cmml" type="integer" xref="S4.E16.m2.3.3.3.2.2.2">85</cn></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E16.m2.5c">\displaystyle k(P)\frac{R(\theta)/R_{0}-1}{\omega_{L,85}},</annotation><annotation encoding="application/x-llamapun" id="S4.E16.m2.5d">italic_k ( italic_P ) divide start_ARG italic_R ( italic_θ ) / italic_R start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - 1 end_ARG start_ARG italic_ω start_POSTSUBSCRIPT italic_L , 85 end_POSTSUBSCRIPT end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </tbody> </table> <p class="ltx_p" id="S4.SS2.p8.7">where <math alttext="R_{0}=a_{85}/a_{87}=3(g_{s}-3g_{I,87})/2(g_{s}-5g_{I,85})=1.49886" class="ltx_Math" display="inline" id="S4.SS2.p8.4.m1.6"><semantics id="S4.SS2.p8.4.m1.6a"><mrow id="S4.SS2.p8.4.m1.6.6" xref="S4.SS2.p8.4.m1.6.6.cmml"><msub id="S4.SS2.p8.4.m1.6.6.4" xref="S4.SS2.p8.4.m1.6.6.4.cmml"><mi id="S4.SS2.p8.4.m1.6.6.4.2" xref="S4.SS2.p8.4.m1.6.6.4.2.cmml">R</mi><mn id="S4.SS2.p8.4.m1.6.6.4.3" xref="S4.SS2.p8.4.m1.6.6.4.3.cmml">0</mn></msub><mo id="S4.SS2.p8.4.m1.6.6.5" xref="S4.SS2.p8.4.m1.6.6.5.cmml">=</mo><mrow id="S4.SS2.p8.4.m1.6.6.6" xref="S4.SS2.p8.4.m1.6.6.6.cmml"><msub id="S4.SS2.p8.4.m1.6.6.6.2" xref="S4.SS2.p8.4.m1.6.6.6.2.cmml"><mi id="S4.SS2.p8.4.m1.6.6.6.2.2" xref="S4.SS2.p8.4.m1.6.6.6.2.2.cmml">a</mi><mn id="S4.SS2.p8.4.m1.6.6.6.2.3" xref="S4.SS2.p8.4.m1.6.6.6.2.3.cmml">85</mn></msub><mo id="S4.SS2.p8.4.m1.6.6.6.1" xref="S4.SS2.p8.4.m1.6.6.6.1.cmml">/</mo><msub 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id="S4.SS2.p8.4.m1.6.6.2.2.1.1.3.2.cmml" type="integer" xref="S4.SS2.p8.4.m1.6.6.2.2.1.1.3.2">5</cn><apply id="S4.SS2.p8.4.m1.6.6.2.2.1.1.3.3.cmml" xref="S4.SS2.p8.4.m1.6.6.2.2.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS2.p8.4.m1.6.6.2.2.1.1.3.3.1.cmml" xref="S4.SS2.p8.4.m1.6.6.2.2.1.1.3.3">subscript</csymbol><ci id="S4.SS2.p8.4.m1.6.6.2.2.1.1.3.3.2.cmml" xref="S4.SS2.p8.4.m1.6.6.2.2.1.1.3.3.2">𝑔</ci><list id="S4.SS2.p8.4.m1.4.4.2.3.cmml" xref="S4.SS2.p8.4.m1.4.4.2.4"><ci id="S4.SS2.p8.4.m1.3.3.1.1.cmml" xref="S4.SS2.p8.4.m1.3.3.1.1">𝐼</ci><cn id="S4.SS2.p8.4.m1.4.4.2.2.cmml" type="integer" xref="S4.SS2.p8.4.m1.4.4.2.2">85</cn></list></apply></apply></apply></apply></apply><apply id="S4.SS2.p8.4.m1.6.6e.cmml" xref="S4.SS2.p8.4.m1.6.6"><eq id="S4.SS2.p8.4.m1.6.6.8.cmml" xref="S4.SS2.p8.4.m1.6.6.8"></eq><share href="https://arxiv.org/html/2502.13414v1#S4.SS2.p8.4.m1.6.6.2.cmml" id="S4.SS2.p8.4.m1.6.6f.cmml" xref="S4.SS2.p8.4.m1.6.6"></share><cn id="S4.SS2.p8.4.m1.6.6.9.cmml" type="float" xref="S4.SS2.p8.4.m1.6.6.9">1.49886</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p8.4.m1.6c">R_{0}=a_{85}/a_{87}=3(g_{s}-3g_{I,87})/2(g_{s}-5g_{I,85})=1.49886</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p8.4.m1.6d">italic_R start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT 85 end_POSTSUBSCRIPT / italic_a start_POSTSUBSCRIPT 87 end_POSTSUBSCRIPT = 3 ( italic_g start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT - 3 italic_g start_POSTSUBSCRIPT italic_I , 87 end_POSTSUBSCRIPT ) / 2 ( italic_g start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT - 5 italic_g start_POSTSUBSCRIPT italic_I , 85 end_POSTSUBSCRIPT ) = 1.49886</annotation></semantics></math>. It should be noted that <math alttext="k(P)" class="ltx_Math" display="inline" id="S4.SS2.p8.5.m2.1"><semantics id="S4.SS2.p8.5.m2.1a"><mrow id="S4.SS2.p8.5.m2.1.2" xref="S4.SS2.p8.5.m2.1.2.cmml"><mi id="S4.SS2.p8.5.m2.1.2.2" xref="S4.SS2.p8.5.m2.1.2.2.cmml">k</mi><mo id="S4.SS2.p8.5.m2.1.2.1" xref="S4.SS2.p8.5.m2.1.2.1.cmml"></mo><mrow id="S4.SS2.p8.5.m2.1.2.3.2" xref="S4.SS2.p8.5.m2.1.2.cmml"><mo id="S4.SS2.p8.5.m2.1.2.3.2.1" stretchy="false" xref="S4.SS2.p8.5.m2.1.2.cmml">(</mo><mi id="S4.SS2.p8.5.m2.1.1" xref="S4.SS2.p8.5.m2.1.1.cmml">P</mi><mo id="S4.SS2.p8.5.m2.1.2.3.2.2" stretchy="false" xref="S4.SS2.p8.5.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p8.5.m2.1b"><apply id="S4.SS2.p8.5.m2.1.2.cmml" xref="S4.SS2.p8.5.m2.1.2"><times id="S4.SS2.p8.5.m2.1.2.1.cmml" xref="S4.SS2.p8.5.m2.1.2.1"></times><ci id="S4.SS2.p8.5.m2.1.2.2.cmml" xref="S4.SS2.p8.5.m2.1.2.2">𝑘</ci><ci id="S4.SS2.p8.5.m2.1.1.cmml" xref="S4.SS2.p8.5.m2.1.1">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p8.5.m2.1c">k(P)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p8.5.m2.1d">italic_k ( italic_P )</annotation></semantics></math> is independent of the bias field magnitude, and its value at the high-polarization limit is 6.37 <math alttext="\times 10^{9}" class="ltx_Math" display="inline" id="S4.SS2.p8.6.m3.1"><semantics id="S4.SS2.p8.6.m3.1a"><mrow id="S4.SS2.p8.6.m3.1.1" xref="S4.SS2.p8.6.m3.1.1.cmml"><mi id="S4.SS2.p8.6.m3.1.1.2" xref="S4.SS2.p8.6.m3.1.1.2.cmml"></mi><mo id="S4.SS2.p8.6.m3.1.1.1" lspace="0.222em" rspace="0.222em" xref="S4.SS2.p8.6.m3.1.1.1.cmml">×</mo><msup id="S4.SS2.p8.6.m3.1.1.3" xref="S4.SS2.p8.6.m3.1.1.3.cmml"><mn id="S4.SS2.p8.6.m3.1.1.3.2" xref="S4.SS2.p8.6.m3.1.1.3.2.cmml">10</mn><mn id="S4.SS2.p8.6.m3.1.1.3.3" xref="S4.SS2.p8.6.m3.1.1.3.3.cmml">9</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p8.6.m3.1b"><apply id="S4.SS2.p8.6.m3.1.1.cmml" xref="S4.SS2.p8.6.m3.1.1"><times id="S4.SS2.p8.6.m3.1.1.1.cmml" xref="S4.SS2.p8.6.m3.1.1.1"></times><csymbol cd="latexml" id="S4.SS2.p8.6.m3.1.1.2.cmml" xref="S4.SS2.p8.6.m3.1.1.2">absent</csymbol><apply id="S4.SS2.p8.6.m3.1.1.3.cmml" xref="S4.SS2.p8.6.m3.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p8.6.m3.1.1.3.1.cmml" xref="S4.SS2.p8.6.m3.1.1.3">superscript</csymbol><cn id="S4.SS2.p8.6.m3.1.1.3.2.cmml" type="integer" xref="S4.SS2.p8.6.m3.1.1.3.2">10</cn><cn id="S4.SS2.p8.6.m3.1.1.3.3.cmml" type="integer" xref="S4.SS2.p8.6.m3.1.1.3.3">9</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p8.6.m3.1c">\times 10^{9}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p8.6.m3.1d">× 10 start_POSTSUPERSCRIPT 9 end_POSTSUPERSCRIPT</annotation></semantics></math> s<sup class="ltx_sup" id="S4.SS2.p8.7.1"><span class="ltx_text ltx_font_italic" id="S4.SS2.p8.7.1.1">-1</span></sup>.</p> </div> <div class="ltx_para" id="S4.SS2.p9"> <p class="ltx_p" id="S4.SS2.p9.11">Figure <a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S4.F6" title="Figure 6 ‣ IV.2 NLZE-induced heading error and its suppression scheme ‣ IV Identifying and suppressing heading errors ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">6</span></a>(b) shows the experiment results of <math alttext="(R(\theta)/R_{0}-1)/\omega_{L,85}" class="ltx_Math" display="inline" id="S4.SS2.p9.1.m1.4"><semantics id="S4.SS2.p9.1.m1.4a"><mrow id="S4.SS2.p9.1.m1.4.4" xref="S4.SS2.p9.1.m1.4.4.cmml"><mrow id="S4.SS2.p9.1.m1.4.4.1.1" xref="S4.SS2.p9.1.m1.4.4.1.1.1.cmml"><mo id="S4.SS2.p9.1.m1.4.4.1.1.2" stretchy="false" xref="S4.SS2.p9.1.m1.4.4.1.1.1.cmml">(</mo><mrow id="S4.SS2.p9.1.m1.4.4.1.1.1" xref="S4.SS2.p9.1.m1.4.4.1.1.1.cmml"><mrow id="S4.SS2.p9.1.m1.4.4.1.1.1.2" 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xref="S4.SS2.p9.1.m1.4.4.1.1.1.2.3.2">𝑅</ci><cn id="S4.SS2.p9.1.m1.4.4.1.1.1.2.3.3.cmml" type="integer" xref="S4.SS2.p9.1.m1.4.4.1.1.1.2.3.3">0</cn></apply></apply><cn id="S4.SS2.p9.1.m1.4.4.1.1.1.3.cmml" type="integer" xref="S4.SS2.p9.1.m1.4.4.1.1.1.3">1</cn></apply><apply id="S4.SS2.p9.1.m1.4.4.3.cmml" xref="S4.SS2.p9.1.m1.4.4.3"><csymbol cd="ambiguous" id="S4.SS2.p9.1.m1.4.4.3.1.cmml" xref="S4.SS2.p9.1.m1.4.4.3">subscript</csymbol><ci id="S4.SS2.p9.1.m1.4.4.3.2.cmml" xref="S4.SS2.p9.1.m1.4.4.3.2">𝜔</ci><list id="S4.SS2.p9.1.m1.2.2.2.3.cmml" xref="S4.SS2.p9.1.m1.2.2.2.4"><ci id="S4.SS2.p9.1.m1.1.1.1.1.cmml" xref="S4.SS2.p9.1.m1.1.1.1.1">𝐿</ci><cn id="S4.SS2.p9.1.m1.2.2.2.2.cmml" type="integer" xref="S4.SS2.p9.1.m1.2.2.2.2">85</cn></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p9.1.m1.4c">(R(\theta)/R_{0}-1)/\omega_{L,85}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p9.1.m1.4d">( italic_R ( italic_θ ) / italic_R start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - 1 ) / italic_ω start_POSTSUBSCRIPT italic_L , 85 end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="\cos\theta" class="ltx_Math" display="inline" id="S4.SS2.p9.2.m2.1"><semantics id="S4.SS2.p9.2.m2.1a"><mrow id="S4.SS2.p9.2.m2.1.1" xref="S4.SS2.p9.2.m2.1.1.cmml"><mi id="S4.SS2.p9.2.m2.1.1.1" xref="S4.SS2.p9.2.m2.1.1.1.cmml">cos</mi><mo id="S4.SS2.p9.2.m2.1.1a" lspace="0.167em" xref="S4.SS2.p9.2.m2.1.1.cmml"></mo><mi id="S4.SS2.p9.2.m2.1.1.2" xref="S4.SS2.p9.2.m2.1.1.2.cmml">θ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p9.2.m2.1b"><apply id="S4.SS2.p9.2.m2.1.1.cmml" xref="S4.SS2.p9.2.m2.1.1"><cos id="S4.SS2.p9.2.m2.1.1.1.cmml" xref="S4.SS2.p9.2.m2.1.1.1"></cos><ci id="S4.SS2.p9.2.m2.1.1.2.cmml" xref="S4.SS2.p9.2.m2.1.1.2">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p9.2.m2.1c">\cos\theta</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p9.2.m2.1d">roman_cos italic_θ</annotation></semantics></math> at a bias field around 50 <math alttext="\mu" class="ltx_Math" display="inline" id="S4.SS2.p9.3.m3.1"><semantics id="S4.SS2.p9.3.m3.1a"><mi id="S4.SS2.p9.3.m3.1.1" xref="S4.SS2.p9.3.m3.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p9.3.m3.1b"><ci id="S4.SS2.p9.3.m3.1.1.cmml" xref="S4.SS2.p9.3.m3.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p9.3.m3.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p9.3.m3.1d">italic_μ</annotation></semantics></math>T. The experiment results can be well fitted by a linear relation, which is due to the high atomic polarization at relatively large values of <math alttext="\cos\theta" class="ltx_Math" display="inline" id="S4.SS2.p9.4.m4.1"><semantics id="S4.SS2.p9.4.m4.1a"><mrow id="S4.SS2.p9.4.m4.1.1" xref="S4.SS2.p9.4.m4.1.1.cmml"><mi id="S4.SS2.p9.4.m4.1.1.1" xref="S4.SS2.p9.4.m4.1.1.1.cmml">cos</mi><mo id="S4.SS2.p9.4.m4.1.1a" lspace="0.167em" xref="S4.SS2.p9.4.m4.1.1.cmml"></mo><mi id="S4.SS2.p9.4.m4.1.1.2" xref="S4.SS2.p9.4.m4.1.1.2.cmml">θ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p9.4.m4.1b"><apply id="S4.SS2.p9.4.m4.1.1.cmml" xref="S4.SS2.p9.4.m4.1.1"><cos id="S4.SS2.p9.4.m4.1.1.1.cmml" xref="S4.SS2.p9.4.m4.1.1.1"></cos><ci id="S4.SS2.p9.4.m4.1.1.2.cmml" xref="S4.SS2.p9.4.m4.1.1.2">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p9.4.m4.1c">\cos\theta</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p9.4.m4.1d">roman_cos italic_θ</annotation></semantics></math> as shown in Tab. <a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S4.T2" title="Table 2 ‣ IV.2 NLZE-induced heading error and its suppression scheme ‣ IV Identifying and suppressing heading errors ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">2</span></a>, and the slope is extracted from the linear fitting as <math alttext="(5.98\pm 0.15)\times 10^{9}" class="ltx_Math" display="inline" id="S4.SS2.p9.5.m5.1"><semantics id="S4.SS2.p9.5.m5.1a"><mrow id="S4.SS2.p9.5.m5.1.1" xref="S4.SS2.p9.5.m5.1.1.cmml"><mrow id="S4.SS2.p9.5.m5.1.1.1.1" xref="S4.SS2.p9.5.m5.1.1.1.1.1.cmml"><mo id="S4.SS2.p9.5.m5.1.1.1.1.2" stretchy="false" xref="S4.SS2.p9.5.m5.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.p9.5.m5.1.1.1.1.1" xref="S4.SS2.p9.5.m5.1.1.1.1.1.cmml"><mn id="S4.SS2.p9.5.m5.1.1.1.1.1.2" xref="S4.SS2.p9.5.m5.1.1.1.1.1.2.cmml">5.98</mn><mo id="S4.SS2.p9.5.m5.1.1.1.1.1.1" xref="S4.SS2.p9.5.m5.1.1.1.1.1.1.cmml">±</mo><mn id="S4.SS2.p9.5.m5.1.1.1.1.1.3" xref="S4.SS2.p9.5.m5.1.1.1.1.1.3.cmml">0.15</mn></mrow><mo id="S4.SS2.p9.5.m5.1.1.1.1.3" rspace="0.055em" stretchy="false" xref="S4.SS2.p9.5.m5.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.SS2.p9.5.m5.1.1.2" rspace="0.222em" xref="S4.SS2.p9.5.m5.1.1.2.cmml">×</mo><msup id="S4.SS2.p9.5.m5.1.1.3" xref="S4.SS2.p9.5.m5.1.1.3.cmml"><mn id="S4.SS2.p9.5.m5.1.1.3.2" xref="S4.SS2.p9.5.m5.1.1.3.2.cmml">10</mn><mn id="S4.SS2.p9.5.m5.1.1.3.3" xref="S4.SS2.p9.5.m5.1.1.3.3.cmml">9</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p9.5.m5.1b"><apply id="S4.SS2.p9.5.m5.1.1.cmml" xref="S4.SS2.p9.5.m5.1.1"><times id="S4.SS2.p9.5.m5.1.1.2.cmml" xref="S4.SS2.p9.5.m5.1.1.2"></times><apply id="S4.SS2.p9.5.m5.1.1.1.1.1.cmml" xref="S4.SS2.p9.5.m5.1.1.1.1"><csymbol cd="latexml" id="S4.SS2.p9.5.m5.1.1.1.1.1.1.cmml" xref="S4.SS2.p9.5.m5.1.1.1.1.1.1">plus-or-minus</csymbol><cn id="S4.SS2.p9.5.m5.1.1.1.1.1.2.cmml" type="float" xref="S4.SS2.p9.5.m5.1.1.1.1.1.2">5.98</cn><cn id="S4.SS2.p9.5.m5.1.1.1.1.1.3.cmml" type="float" xref="S4.SS2.p9.5.m5.1.1.1.1.1.3">0.15</cn></apply><apply id="S4.SS2.p9.5.m5.1.1.3.cmml" xref="S4.SS2.p9.5.m5.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p9.5.m5.1.1.3.1.cmml" xref="S4.SS2.p9.5.m5.1.1.3">superscript</csymbol><cn id="S4.SS2.p9.5.m5.1.1.3.2.cmml" type="integer" xref="S4.SS2.p9.5.m5.1.1.3.2">10</cn><cn id="S4.SS2.p9.5.m5.1.1.3.3.cmml" type="integer" xref="S4.SS2.p9.5.m5.1.1.3.3">9</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p9.5.m5.1c">(5.98\pm 0.15)\times 10^{9}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p9.5.m5.1d">( 5.98 ± 0.15 ) × 10 start_POSTSUPERSCRIPT 9 end_POSTSUPERSCRIPT</annotation></semantics></math> s<sup class="ltx_sup" id="S4.SS2.p9.11.1"><span class="ltx_text ltx_font_italic" id="S4.SS2.p9.11.1.1">-1</span></sup>, which agrees with the fore-mentioned high-polarization limit value of <math alttext="k(P)" class="ltx_Math" display="inline" id="S4.SS2.p9.7.m7.1"><semantics id="S4.SS2.p9.7.m7.1a"><mrow id="S4.SS2.p9.7.m7.1.2" xref="S4.SS2.p9.7.m7.1.2.cmml"><mi id="S4.SS2.p9.7.m7.1.2.2" xref="S4.SS2.p9.7.m7.1.2.2.cmml">k</mi><mo id="S4.SS2.p9.7.m7.1.2.1" xref="S4.SS2.p9.7.m7.1.2.1.cmml"></mo><mrow id="S4.SS2.p9.7.m7.1.2.3.2" xref="S4.SS2.p9.7.m7.1.2.cmml"><mo id="S4.SS2.p9.7.m7.1.2.3.2.1" stretchy="false" xref="S4.SS2.p9.7.m7.1.2.cmml">(</mo><mi id="S4.SS2.p9.7.m7.1.1" xref="S4.SS2.p9.7.m7.1.1.cmml">P</mi><mo id="S4.SS2.p9.7.m7.1.2.3.2.2" stretchy="false" xref="S4.SS2.p9.7.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p9.7.m7.1b"><apply id="S4.SS2.p9.7.m7.1.2.cmml" xref="S4.SS2.p9.7.m7.1.2"><times id="S4.SS2.p9.7.m7.1.2.1.cmml" xref="S4.SS2.p9.7.m7.1.2.1"></times><ci id="S4.SS2.p9.7.m7.1.2.2.cmml" xref="S4.SS2.p9.7.m7.1.2.2">𝑘</ci><ci id="S4.SS2.p9.7.m7.1.1.cmml" xref="S4.SS2.p9.7.m7.1.1">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p9.7.m7.1c">k(P)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p9.7.m7.1d">italic_k ( italic_P )</annotation></semantics></math>. In practice, we can extract the value of <math alttext="\cos\theta" class="ltx_Math" display="inline" id="S4.SS2.p9.8.m8.1"><semantics id="S4.SS2.p9.8.m8.1a"><mrow id="S4.SS2.p9.8.m8.1.1" xref="S4.SS2.p9.8.m8.1.1.cmml"><mi id="S4.SS2.p9.8.m8.1.1.1" xref="S4.SS2.p9.8.m8.1.1.1.cmml">cos</mi><mo id="S4.SS2.p9.8.m8.1.1a" lspace="0.167em" xref="S4.SS2.p9.8.m8.1.1.cmml"></mo><mi id="S4.SS2.p9.8.m8.1.1.2" xref="S4.SS2.p9.8.m8.1.1.2.cmml">θ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p9.8.m8.1b"><apply id="S4.SS2.p9.8.m8.1.1.cmml" xref="S4.SS2.p9.8.m8.1.1"><cos id="S4.SS2.p9.8.m8.1.1.1.cmml" xref="S4.SS2.p9.8.m8.1.1.1"></cos><ci id="S4.SS2.p9.8.m8.1.1.2.cmml" xref="S4.SS2.p9.8.m8.1.1.2">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p9.8.m8.1c">\cos\theta</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p9.8.m8.1d">roman_cos italic_θ</annotation></semantics></math> using the calibrated <math alttext="k(P)" class="ltx_Math" display="inline" id="S4.SS2.p9.9.m9.1"><semantics id="S4.SS2.p9.9.m9.1a"><mrow id="S4.SS2.p9.9.m9.1.2" xref="S4.SS2.p9.9.m9.1.2.cmml"><mi id="S4.SS2.p9.9.m9.1.2.2" xref="S4.SS2.p9.9.m9.1.2.2.cmml">k</mi><mo id="S4.SS2.p9.9.m9.1.2.1" xref="S4.SS2.p9.9.m9.1.2.1.cmml"></mo><mrow id="S4.SS2.p9.9.m9.1.2.3.2" xref="S4.SS2.p9.9.m9.1.2.cmml"><mo id="S4.SS2.p9.9.m9.1.2.3.2.1" stretchy="false" xref="S4.SS2.p9.9.m9.1.2.cmml">(</mo><mi id="S4.SS2.p9.9.m9.1.1" xref="S4.SS2.p9.9.m9.1.1.cmml">P</mi><mo id="S4.SS2.p9.9.m9.1.2.3.2.2" stretchy="false" xref="S4.SS2.p9.9.m9.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p9.9.m9.1b"><apply id="S4.SS2.p9.9.m9.1.2.cmml" xref="S4.SS2.p9.9.m9.1.2"><times id="S4.SS2.p9.9.m9.1.2.1.cmml" xref="S4.SS2.p9.9.m9.1.2.1"></times><ci id="S4.SS2.p9.9.m9.1.2.2.cmml" xref="S4.SS2.p9.9.m9.1.2.2">𝑘</ci><ci id="S4.SS2.p9.9.m9.1.1.cmml" xref="S4.SS2.p9.9.m9.1.1">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p9.9.m9.1c">k(P)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p9.9.m9.1d">italic_k ( italic_P )</annotation></semantics></math> and measured frequency ratio <math alttext="R(\theta)" class="ltx_Math" display="inline" id="S4.SS2.p9.10.m10.1"><semantics id="S4.SS2.p9.10.m10.1a"><mrow id="S4.SS2.p9.10.m10.1.2" xref="S4.SS2.p9.10.m10.1.2.cmml"><mi id="S4.SS2.p9.10.m10.1.2.2" xref="S4.SS2.p9.10.m10.1.2.2.cmml">R</mi><mo id="S4.SS2.p9.10.m10.1.2.1" xref="S4.SS2.p9.10.m10.1.2.1.cmml"></mo><mrow id="S4.SS2.p9.10.m10.1.2.3.2" xref="S4.SS2.p9.10.m10.1.2.cmml"><mo id="S4.SS2.p9.10.m10.1.2.3.2.1" stretchy="false" xref="S4.SS2.p9.10.m10.1.2.cmml">(</mo><mi id="S4.SS2.p9.10.m10.1.1" xref="S4.SS2.p9.10.m10.1.1.cmml">θ</mi><mo id="S4.SS2.p9.10.m10.1.2.3.2.2" stretchy="false" xref="S4.SS2.p9.10.m10.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p9.10.m10.1b"><apply id="S4.SS2.p9.10.m10.1.2.cmml" xref="S4.SS2.p9.10.m10.1.2"><times id="S4.SS2.p9.10.m10.1.2.1.cmml" xref="S4.SS2.p9.10.m10.1.2.1"></times><ci id="S4.SS2.p9.10.m10.1.2.2.cmml" xref="S4.SS2.p9.10.m10.1.2.2">𝑅</ci><ci id="S4.SS2.p9.10.m10.1.1.cmml" xref="S4.SS2.p9.10.m10.1.1">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p9.10.m10.1c">R(\theta)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p9.10.m10.1d">italic_R ( italic_θ )</annotation></semantics></math>, and correct the heading error using Eqs. (<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S2.E8" title="In II.2 Heading error analysis ‣ II Theoretical Background ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">8</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#S2.E9" title="In II.2 Heading error analysis ‣ II Theoretical Background ‣ Suppression of heading errors in Bell-Bloom optically pumped free-induction-decay alkali-metal atomic magnetometers"><span class="ltx_text ltx_ref_tag">9</span></a>). Compared with the way to extract <math alttext="\cos\theta" class="ltx_Math" display="inline" id="S4.SS2.p9.11.m11.1"><semantics id="S4.SS2.p9.11.m11.1a"><mrow id="S4.SS2.p9.11.m11.1.1" xref="S4.SS2.p9.11.m11.1.1.cmml"><mi id="S4.SS2.p9.11.m11.1.1.1" xref="S4.SS2.p9.11.m11.1.1.1.cmml">cos</mi><mo id="S4.SS2.p9.11.m11.1.1a" lspace="0.167em" xref="S4.SS2.p9.11.m11.1.1.cmml"></mo><mi id="S4.SS2.p9.11.m11.1.1.2" xref="S4.SS2.p9.11.m11.1.1.2.cmml">θ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p9.11.m11.1b"><apply id="S4.SS2.p9.11.m11.1.1.cmml" xref="S4.SS2.p9.11.m11.1.1"><cos id="S4.SS2.p9.11.m11.1.1.1.cmml" xref="S4.SS2.p9.11.m11.1.1.1"></cos><ci id="S4.SS2.p9.11.m11.1.1.2.cmml" xref="S4.SS2.p9.11.m11.1.1.2">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p9.11.m11.1c">\cos\theta</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p9.11.m11.1d">roman_cos italic_θ</annotation></semantics></math> using the amplitude information of longitudinal and transverse signals in Ref. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib22" title="">22</a>]</cite>, the method here relies only on the frequency information.</p> </div> </section> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">V </span>Conclusion</h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.2">In conclusion, we have studied different sources of heading errors in a Rb FID magnetometer. By employing the Bell-Bloom optical pumping method, the heading error induced by the nuclear spin LZE is almost eliminated, and the NLZE-induced error is identified as the main source in this system. To address this problem, we have developed several different suppression methods, which have been demonstrated to keep the heading error under 1 nT at a bias field of 50 <math alttext="\mu" class="ltx_Math" display="inline" id="S5.p1.1.m1.1"><semantics id="S5.p1.1.m1.1a"><mi id="S5.p1.1.m1.1.1" xref="S5.p1.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S5.p1.1.m1.1b"><ci id="S5.p1.1.m1.1.1.cmml" xref="S5.p1.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S5.p1.1.m1.1d">italic_μ</annotation></semantics></math>T over a sensor orientation range of <math alttext="25^{\circ}\leq\theta\leq 155^{\circ}" class="ltx_Math" display="inline" id="S5.p1.2.m2.1"><semantics id="S5.p1.2.m2.1a"><mrow id="S5.p1.2.m2.1.1" xref="S5.p1.2.m2.1.1.cmml"><msup id="S5.p1.2.m2.1.1.2" xref="S5.p1.2.m2.1.1.2.cmml"><mn id="S5.p1.2.m2.1.1.2.2" xref="S5.p1.2.m2.1.1.2.2.cmml">25</mn><mo id="S5.p1.2.m2.1.1.2.3" xref="S5.p1.2.m2.1.1.2.3.cmml">∘</mo></msup><mo id="S5.p1.2.m2.1.1.3" xref="S5.p1.2.m2.1.1.3.cmml">≤</mo><mi id="S5.p1.2.m2.1.1.4" xref="S5.p1.2.m2.1.1.4.cmml">θ</mi><mo id="S5.p1.2.m2.1.1.5" xref="S5.p1.2.m2.1.1.5.cmml">≤</mo><msup id="S5.p1.2.m2.1.1.6" xref="S5.p1.2.m2.1.1.6.cmml"><mn id="S5.p1.2.m2.1.1.6.2" xref="S5.p1.2.m2.1.1.6.2.cmml">155</mn><mo id="S5.p1.2.m2.1.1.6.3" xref="S5.p1.2.m2.1.1.6.3.cmml">∘</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.2.m2.1b"><apply id="S5.p1.2.m2.1.1.cmml" xref="S5.p1.2.m2.1.1"><and id="S5.p1.2.m2.1.1a.cmml" xref="S5.p1.2.m2.1.1"></and><apply id="S5.p1.2.m2.1.1b.cmml" xref="S5.p1.2.m2.1.1"><leq id="S5.p1.2.m2.1.1.3.cmml" xref="S5.p1.2.m2.1.1.3"></leq><apply id="S5.p1.2.m2.1.1.2.cmml" xref="S5.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S5.p1.2.m2.1.1.2.1.cmml" xref="S5.p1.2.m2.1.1.2">superscript</csymbol><cn id="S5.p1.2.m2.1.1.2.2.cmml" type="integer" xref="S5.p1.2.m2.1.1.2.2">25</cn><compose id="S5.p1.2.m2.1.1.2.3.cmml" xref="S5.p1.2.m2.1.1.2.3"></compose></apply><ci id="S5.p1.2.m2.1.1.4.cmml" xref="S5.p1.2.m2.1.1.4">𝜃</ci></apply><apply id="S5.p1.2.m2.1.1c.cmml" xref="S5.p1.2.m2.1.1"><leq id="S5.p1.2.m2.1.1.5.cmml" xref="S5.p1.2.m2.1.1.5"></leq><share href="https://arxiv.org/html/2502.13414v1#S5.p1.2.m2.1.1.4.cmml" id="S5.p1.2.m2.1.1d.cmml" xref="S5.p1.2.m2.1.1"></share><apply id="S5.p1.2.m2.1.1.6.cmml" xref="S5.p1.2.m2.1.1.6"><csymbol cd="ambiguous" id="S5.p1.2.m2.1.1.6.1.cmml" xref="S5.p1.2.m2.1.1.6">superscript</csymbol><cn id="S5.p1.2.m2.1.1.6.2.cmml" type="integer" xref="S5.p1.2.m2.1.1.6.2">155</cn><compose id="S5.p1.2.m2.1.1.6.3.cmml" xref="S5.p1.2.m2.1.1.6.3"></compose></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.2.m2.1c">25^{\circ}\leq\theta\leq 155^{\circ}</annotation><annotation encoding="application/x-llamapun" id="S5.p1.2.m2.1d">25 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT ≤ italic_θ ≤ 155 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT</annotation></semantics></math>. The schemes based on a single magnetometer rely on averaging out the atomic longitudinal polarization, and have the advantages of maintaining the signal magnitude and correcting the heading error automatically. The schemes based on a comagnetometer do not require additional hardware costs in the sensor head, and only need the information of the recorded precession frequencies. In the following work, we plan to continue the efforts in this paper and focus on understanding and reducing the residual heading error. We also hope to apply the scalar magnetometers studied in this work to precision measurements. Though preliminary results for this application are promising <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib34" title="">34</a>]</cite>, special attentions are required for systematic effects besides heading errors <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib7" title="">7</a>, <a class="ltx_ref" href="https://arxiv.org/html/2502.13414v1#bib.bib8" title="">8</a>]</cite>. Moreover, we want to further develop a vector atomic magnetometer.</p> </div> </section> <section class="ltx_section" id="S6"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">VI </span>Acknowledgments</h2> <div class="ltx_para" id="S6.p1"> <p class="ltx_p" id="S6.p1.1">This work was partially carried out at the University of Science and Technology of China (USTC) Center for Micro and Nanoscale Research and Fabrication. 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