Open Access
Add to BibSonomy
Abstract
In this paper, we present an analysis of the amplitude variations of the opto-magnetic resonance absorption signals obtained in a single-beam magnetometer driven by radially or azimuthally polarized light (RPL/APL). It is shown that optically polarized atoms driven by cylindrical vector beams obtained only the alignment of atomic multipole moments but not the orientation, which is in good agreement with our simulation and experimental results. In comparison with the plane polarized pump light fields, cylindrical vector beams with much more complete electric vector polarization distribution in the transverse plane, make it unlikely to create the “emptying state “ (no-atom populated) among the ground-state Zeeman sublevels for any possible orientation of the applied static magnetic field. These characteristics of the RPL/APL lead to generally smaller atomic population difference and lower response intensity of the transmitted signal. The tensor decomposition of atomic polarized states and the evolution of atomic multipole moments with the sweeping radio frequency (RF) field offer the way to show the magnetic orientation sensitivity of the radially or azimuthally polarized probe light, which possess similar profiles as that of the linearly polarized light, only with a constant phase lag of about π/2 and obvious amplitude differences.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The opto-magnetic resonance (OMR) mechanism is a common atomic physical process that combines the optical pumping and radio frequency (RF) field magnetic resonance, which was first proposed and implemented by Alfred Kastler in the 1950s [1]. Atomic magnetometers based on the OMR technique have proven to be reliable and competitive [2–4], and show a vigorous application prospect in biomedical imaging [5], physical constant measurement [6] and ultracold assembly [7]. Most practical or theoretical OMR models focus on plane waves, which rely on atomic orientation or alignment prepared by optical pumping with the circular or linear polarized light, respectively. Furthermore, specified polarization states of optically driven atoms can be selectively prepared by changing the electric vector polarization direction or properties of pumped light, which has been verified by numerous theoretical and experimental researches [8–10].
To optimize and reduce the fundamental issue “dead zone” of traditional optically pumped atomic magnetometers, which is an inherent feature of the vector or tensor interactions for optical pumping, shown as the vanished response to certain orientations of the applied magnetic field, various excellent designs were proposed and tried, such as employing additional vapor cells and reference beams [11,12], schemed modulations for the light polarization [13–15], and the combination of unpolarized light and spatially varying microwave fields based on the symmetry of atomic transition hyperfine structure [16], etc. This naturally arouses our attention about the variation trend of response sensitivity with the respect to different orientations of the applied static magnetic field when utilizing cylindrical vector beams with more complex polarization spatial distribution as pump/probe light fields. Therefore, it is also one of the focuses of our research to explore and explain the special magnetic orientations when the OMR absorption signal is almost unresponsive to the applied static magnetic field, namely the so-called “magic angle”, including the typical plane light field and vector light field. Here, we use the RPL/APL, which possesses the spatially symmetrical electric vector polarization distribution in the transverse plane, as the pump/probe field for the ${}^{87}\textrm{Rb}$ warm atomic vapor single-beam magnetometer layout.
In this paper, we present a theoretical model to investigate the absorption signals obtained in the OMR magnetometer layout using the RPL/APL field, which is the same pumping beam polarizing the vapor atoms, also used to monitor the oscillations by measuring its transmitted power. The multipole moment tensor representation for polarized atoms is directly related to the ground-state sublevels atomic population and coherence. Depolarization of optically pumped atoms is essentially a magnetic resonance process involving the RF field and static magnetic field. We utilize the solutions of Liouville equations to show the orientation dependence and line shapes of the OMR signals, and offer a clear mathematical explanation for the appearance of the “magic angle”. The simulation results are in good agreements with the experimental data. Our work would be expected to point out a feasible optimization direction for the better magnetic angular response sensitivity modulation of specific spatially polarized beams to the vector magnetic field.
2. Optically polarized atoms model driven by vector beams
The atomic population distribution among ground-state sublevels pumped by the RPL/APL field is the basic premise for our discussion of the response strength of the OMR absorption signal relative to the applied static magnetic fields, which has been done by our previous work [17]. The density matrix ρ of optically polarized atoms among ground-state sublevels reveals the atomic population polarization with the orientation dependence of the static magnetic field. For the RPL/APL pumping field, the maximum population difference (0.25, 0.5, 0.25) corresponds to the atomic population polarization of optically driven ground-state (F = 1) atoms among the Zeeman sublevels, when the applied direction of static magnetic field is parallel to the light propagation axis [17], and the ground-state atomic density matrix ρ can be expressed as
Because the electric dipole transitions involved in weak pumping conditions only couple to the atomic multipole moments ${m_{k,q}}$ with orders k = 1 and k = 2, the equilibrium state $M_{k,q}^{ini}$ implied by Eq. (1) are shown as following
As the geometry shown in Fig. 1, the light field propagation vector is along the positive direction of the y-axis, the orientation of the total static magnetic field is confined in the y-z plane, and the RF field is set on the x-axis perpendicular to the former two. The rotating frame can be obtained by a series of operations, following a rotation of ${\pi / 2}$ around z axis, a rotation of θ around -x axis and a rotation of ωt around the new quantization axis p’, respectively. The rotation transformation for matrix elements in the laboratory coordinate frame can be expressed as
where the Wigner D function is
Fig. 2. The schematic structure of the experimental apparatus. PBS, polarization beam splitter; ISO, optical isolator; PD, photodiode.
These transformations build a new operator matrix ${M_{rot}}$ in the rotating frame, representing the alignment of atomic multipole moments.
Due to the time dependent exponential factor ${\textrm{e}^{ {\pm} n\textrm{i}\omega t}}$ related to the RF field frequency ω, the transverse multiple moments ${m^{\prime}_{k,q \ne 0}}$ would relax to zero and the equilibrium state for the atomic ensemble only possesses the longitudinal component,
In the new rotating frame, the evolution of multipole moments for our ensemble follows the Liouville equation,
The transmitted light signal is proportional to α, when the power of pump light field is low enough to make the upper excited states population negligible. The measurement of the detected transmitted light signal can be defined in units of the equilibrium alignment driven by optical pumping, which is actually normalized to the longitudinal alignment [19,20] and can be written as
The dimensionless result of Eq. (10) contains the time independent DC signal, which can be captured by a photodiode and displays a characteristic absorptive line shape. Our focus is on the detected response of the transmitted light to the orientation of the static magnetic field, and therefore the amplitude components with magnetic angular dependence for the DC signal is
This amplitude component is proportional to the absolute value of the absorption peak depth for transmitted light intensity line shapes, as the RF field sweeps around the resonant frequency ${\omega _0}$ with an appropriate period under a specific orientation θ of the static magnetic field.
3. Experimental result simulation & analysis
3.1 Setup of the single-beam OMR magnetometer
Our experimental setup (shown in Fig. 2) adopts the classical laser-pumped single-beam OMR magnetometer layout, which is based on the macro magnetic moment generated by linear Zeeman effect under specific external magnetic fields, using the optically pumping process and sweeping RF field to measure the magnitude and direction of the static magnetic field. The light source is Toptica's DL100 external cavity tuned semiconductor laser, whose working central wavelength is set to 794.985 nm near the D1 line of ${}^{87}\textrm{Rb}\textrm{ }({5{S_{{1 / 2}}}\textrm{, }F = 2 \to 5{P_{{1 / 2}}}\textrm{, }F^{\prime} = 1} )$. We chose the S-waveplate from Altechna & Wophotonics to obtain the RPL/APL. This super-structured spatial polarization converter can convert incident linearly polarized light into a radial or azimuthal polarization distribution. In order to match the working aperture of the S-waveplate and increase the area of reaction cross section between pump light and atoms, we expand the initial beam spot diameter to about 8 mm.
The sensing atomic species is pure ${}^{87}\textrm{Rb}$, loaded in a paraffin-coated cylindrical vapour cell ($\Phi 35\textrm{ }\textrm{mm} \times 50\textrm{ }\textrm{mm}$). A close-wound solenoid controlled by a DC stabilized power supply is arranged along the light propagation axis (y) to prepare the longitudinal magnetic field, while the Helmholtz coils arranged in the x-axis direction produce the transverse magnetic field. Therefore, the orientation of total static magnetic field required for the Zeeman splitting of atomic energy levels is adjustable in the y-z plane perpendicular to the axis (x) of RF coils installation. The applied value (5000 nT) of total static magnetic field corresponds to a Larmor frequency of ${}^{87}\textrm{Rb}$ atomic magnetic moments’ precession around the quantization axis, which is about 35 kHz. A pair of RF coils is connected to the function signal generator, which provides a voltage signal with adjustable amplitude and waveform, and controls the sweeping rate to achieve atomic population depolarization among the ground state Zeeman sublevels. A highly sensitive Photodiode (PD) is used to detect the power of the transmitted light and convert it into voltage. The ${}^{87}\textrm{Rb}$ cell is placed in a five-layers permalloy magnetic shielding cylinder to suppress the ambient magnetic field to dozens of nG; the operating temperature of the atomic vapour is heated and maintained at the temperature of around 42°C.
3.2 Angular dependence of the OMR signal absorption depth
In order to avoid the transient nutation effect [21] near the resonance point, we set a sufficiently long sweeping period of about 100 ms for the RF field, and limit its driving power to 2 mVrms. Since the installation axis (x) of the RF coils is always perpendicular to the total magnetic field, the effective angular momentum projections, corresponding to rotate directions of the RF field ω acting on the alignment of atomic multipole moments, reach the maximum value and remain unchanged. This allows us to adjust the orientation of the total static magnetic field in the y-z plane without additional consideration of angular dependences of the depolarization efficiency of the RF field.
For the RPL/APL as the pump-probe light field, we define the response of the transmitted light signal to the static magnetic field as the absolute value of the peak depth of the absorption line shape. This allows us to discuss more purely the dependence of the probe light absorption on the orientation of the total static magnetic field with a specific intensity. By changing the angle θ (shown in Fig. 1) of the total static magnetic field, we can get the following results,
The discrete points in Fig. 3(a) are our experimental data, and the solid lines are the curves fitted by the theoretical model proposed in the Section 2 of this paper. Due to the consistency of the atomic population distribution among the ground-state magnetic sublevels for specific magnetic field orientations [17], response strengths of the transmitted OMR signal exhibits very similar line patterns for the RPL/APL pump-probe field. Although the cylindrical vector beams usually have more abundant spatial polarization distribution in the reaction cross section, there are still special directions of the total static magnetic field for which the transmitted absorption signal does not respond. It should be mentioned that the transmitted OMR signal for the RPL pump-probe field shows the weak absorption line shape when $\theta \to {\pi / 2}$, whereas the APL does not. And the slight differences in the absolute value of response amplitudes between the radially and azimuthally polarized probe light are mainly determined by the facts that the intensity distribution of our experimental beam along the y-axis direction is slightly larger than that of the x-axis within the transverse plane.
Fig. 3. (a) The line shape fitting for the absorption peak depth of the OMR signals driven by the RPL/ APL pump-probe field; (b) the angular dependence for transmitted signal amplitude dominated by the ${h_{DC}}$ component.
These special static magnetic orientations correspond to the atomic population unpolarized states prepared by the RPL/APL pump field. From the fitting results to the experimental data, it can be seen that the DC signal part dominates the whole line patterns, with minor corrections from harmonic signal parts. The static magnetic angular dependence of OMR transmitted signal amplitude is mainly determined by the DC component shown by Eq. (11). It is not difficult to find from Fig. 3(b) that when the directions of the total static magnetic field are parallel to the propagation axis of the optical field, the peak depth of the resonance absorption signal presents a maximum value, that is, the static magnetic responses in these directions are the strongest. In contrast, when the total static magnetic field is set near the plane of the polarization distribution of the electric vector of the light field, the amplitude of OMR absorption signal is much smaller. Moreover, the change in this response strength is not monotonic with respect to the orientation θ of the resultant static magnetic field. The “magic angle” $\theta = \arccos \left( {\sqrt {{1 / 3}} } \right) \approx 54.7^\circ$ implies that the ground-state atoms driven by the RPL/APL pumping process are uniformly distributed among the magnetic sublevels, with no atomic population difference. In this particular case, the optical pumping does not prepare the polarization of the atomic states. Furthermore, the depolarization of the RF field acting on the ground-state magnetic sublevels cannot change non-polarized scenarios of the atomic states. The so-called “dead zone” also emerges in the RPL/APL pump-probe layout, where the OMR signal vanishes.
In order to compare the influences of the light fields with different polarization distributions on line shapes and amplitudes of the OMR absorption signal under the pump-probe structure, we also give the relevant experimental data and fitting results of linearly polarized light (LPL) and ${\sigma ^ + }$ circularly polarized light (CPL), as shown in Fig. 4(a) and (b). The obvious difference mainly reflects that the LPL exhibits more non-responsive positions relative to the CPL. For resonant CPL, the ground-state atoms obtain the magnetic vector moments $M_{1,q}^{ini} = {\left( {\begin{array}{{ccc}} 0&{{1 / {2\sqrt 2 }}}&0 \end{array}} \right)^T}$ when the total static magnetic field is parallel to the light propagation axis y. And the DC signal component $\left( {h_{DC}^{\textrm{CPL}} = {{3{{\cos }^2}\theta } / {2\sqrt 2 }}} \right)$ for the CPL indicates the extreme values of response strength, which just correspond the maximum position of the atomic population difference ${\theta _{\max }} = 0\textrm{ or }\pi $, and the minimum ${\theta _{\min }} = {\pi / 2}$, as shown in Fig. 4(d). In contrast, the ground-state atoms driven by the LPL pump fields, obtain the magnetic tensor moments $M_{2,q}^{ini} = {\left( {\begin{array}{{ccccc}} 0&0&{{1 / {\sqrt 6 }}}&0&0 \end{array}} \right)^T}$ symmetrically distributed along the light polarization direction, when the direction of the total static magnetic field is parallel to the light polarization axis z. Similarly, from the angular dependence shown in Fig. 4(c), it is easy to find the “magic angle” ${\theta _{\min }} = \arccos \left( {\sqrt {{2 / 3}} } \right) \approx 35.3^\circ$ with no signal response by the DC signal component $h_{DC}^{\textrm{LPL}} = {{{{({2 - 3{{\cos }^2}\theta } )}^2}} / 4}$.
Fig. 4. The line shape fitting for the absorption peak depth of the OMR signals driven by the (a) linear polarized light (LPL) and (b) circular polarized light (CPL) pump-probe field; the static magnetic angular dependence for transmitted signal amplitude dominated by the (c) $h_{DC}^{\textrm{LPL}}$ and (d) $h_{DC}^{\textrm{CPL}}$ component.
All these conclusions mentioned above are consistent with the experimental observations, which confirms the reliability of our theoretical model. It can be clearly stated that the response strength of OMR signals driven by plane polarized light fields to the static magnetic field is overall stronger than that of the RPL/APL. Since the RPL/APL tends to have more complex polarization distributions in the plane perpendicular to the propagation vector, the atomic population among ground-state Zeeman sublevels prepared by its pumping is unlikely to create the “emptying state” for any possible orientation of the resultant static magnetic field. This means that the atomic population difference prepared by the RPL/APL is generally smaller than that of the plane wave field (LPL or CPL), which directly leads to the same trend in the amplitude variation of the OMR absorption signal.
4. Summary and outlook
In conclusion, we study and analyze the OMR model of the pump-probe RPL/APL (single-beam) with the participation of RF fields. Based on the ground-state atomic population prepared by the RPL/APL pump field under different directions of the applied static magnetic field, we can define the atomic polarization states prepared by cylindrical vector beams. Through the tensor expression of the atomic multipole moments held by polarized atoms driven by the RPL/APL field, we make it clear that the RPL/APL pump field can only create the atomic alignment, not atomic orientation. By solving Liouville equations for the evolution of atomic multipole moments in a specific rotating frame, we can obtain the magnetic angular dependence factor of the OMR absorption signal.
Through the model fitting of the experimental data, we have basically confirmed that the single-beam model of the RPL/APL under a classical OMR magnetometer structure has no obvious advantages over the plane-wave light field in terms of the responsive angular range or the transmitted signal amplitude. Furthermore, the curve illustrating static magnetic angular dependence for the transmitted RPL/APL signal, possess similar profiles as that of the LPL, only with a constant phase lag about ${\pi / 2}$ and obvious amplitude differences. It is hoped that our results can provide prospective ideas for exploring the opto-magnetic resonance model of the light-atom interaction of vector beams with more complex polarization distributions in the reaction cross section.
Funding
National Natural Science Foundation of China (12274366).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. A. Kastler, “Optical Methods of Atomic Orientation and of Magnetic Resonance*,” J. Opt. Soc. Am. 47(6), 460–465 (1957). [CrossRef]
2. G. Lieb, T. Jager, A. Palacios-Laloy, and H. Gilles, “All-optical isotropic scalar 4He magnetometer based on atomic alignment,” Rev. Sci. Instrum. 90(7), 075104 (2019). [CrossRef]
3. S. J. Ingleby, C. O’Dwyer, P. F. Griffin, A. S. Arnold, and E. Riis, “Orientational effects on the amplitude and phase of polarimeter signals in double-resonance atomic magnetometry,” Phys. Rev. A 96(1), 013429 (2017). [CrossRef]
4. Z. Liang, X. X. Geng, P. L. Qi, K. Jin, G. Q. Yang, G. X. Li, and G. M. Huang, “Laser-detected magnetic resonance spectra dressed by a radio-frequency field,” Opt. Express 29(21), 33197–33209 (2021). [CrossRef]
5. R. Zhang, W. Xiao, Y. Ding, Y. Feng, X. Peng, L. Shen, C. Sun, T. Wu, Y. Wu, Y. Yang, Z. Zheng, X. Zhang, J. Chen, and H. Guo, “Recording brain activities in unshielded Earth’s field with optically pumped atomic magnetometers,” Sci. Adv. 6(24), eaba8792 (2020). [CrossRef]
6. C. Abel, S. Afach, N. J. Ayres, C. A. Baker, G. Ban, G. Bison, K. Bodek, V. Bondar, M. Burghoff, E. Chanel, Z. Chowdhuri, P. J. Chiu, B. Clement, C. B. Crawford, M. Daum, S. Emmenegger, L. Ferraris-Bouchez, M. Fertl, P. Flaux, B. Franke, A. Fratangelo, P. Geltenbort, K. Green, W. C. Griffith, M. van der Grinten, Z. D. Grujic, P. G. Harris, L. Hayen, W. Heil, R. Henneck, V. Helaine, N. Hild, Z. Hodge, M. Horras, P. Iaydjiev, S. N. Ivanov, M. Kasprzak, Y. Kermaidic, K. Kirch, A. Knecht, P. Knowles, H. C. Koch, P. A. Koss, S. Komposch, A. Kozela, A. Kraft, J. Krempel, M. Kuzniak, B. Lauss, T. Lefort, Y. Lemiere, A. Leredde, P. Mohanmurthy, A. Mtchedlishvili, M. Musgrave, O. Naviliat-Cuncic, D. Pais, F. M. Piegsa, E. Pierre, G. Pignol, C. Plonka-Spehr, P. N. Prashanth, G. Quemener, M. Rawlik, D. Rebreyend, I. Rienacker, D. Ries, S. Roccia, G. Rogel, D. Rozpedzik, A. Schnabel, P. Schmidt-Wellenburg, N. Severijns, D. Shiers, R. Tavakoli Dinani, J. A. Thorne, R. Virot, J. Voigt, A. Weis, E. Wursten, G. Wyszynski, J. Zejma, J. Zenner, and G. Zsigmond, “Measurement of the Permanent Electric Dipole Moment of the Neutron,” Phys. Rev. Lett. 124(8), 081803 (2020). [CrossRef]
7. H. Yang, X. Y. Wang, Z. Su, J. Cao, D. C. Zhang, J. Rui, B. Zhao, C. L. Bai, and J. W. Pan, “Evidence for the association of triatomic molecules in ultracold 23 Na 40 K + 40 K mixtures,” Nature 602(7896), 229–233 (2022). [CrossRef]
8. F. Beato, E. Belorizky, E. Labyt, M. Le Prado, and A. Palacios-Laloy, “Theory of a 4He parametric-resonance magnetometer based on atomic alignment,” Phys. Rev. A 98(5), 053431 (2018). [CrossRef]
9. G. Di Domenico, G. Bison, S. Groeger, P. Knowles, A. S. Pazgalev, M. Rebetez, H. Saudan, and A. Weis, “Experimental study of laser-detected magnetic resonance based on atomic alignment,” Phys. Rev. A 74(6), 063415 (2006). [CrossRef]
10. F. Bouchard, H. Larocque, A. M. Yao, C. Travis, I. De Leon, A. Rubano, E. Karimi, G. L. Oppo, and R. W. Boyd, “Polarization Shaping for Control of Nonlinear Propagation,” Phys. Rev. Lett. 117(23), 233903 (2016). [CrossRef]
11. M. K. Plante, D. L. MacFarlane, D. D. McGregor, R. E. Slocum, W. M. Sampson, and A. W. Brown, “Generalized theory of double-resonance optical pumping of 4He,” Phys. Rev. A 82(1), 013837 (2010). [CrossRef]
12. B. Chéron, H. Gilles, and J. Hamel, “Spatial frequency isotropy of an optically pumped 4He magnetometer,” Eur. Phys. J. AP 13(2), 143–145 (2001). [CrossRef]
13. A. Ben-Kish and M. V. Romalis, “Dead-zone-free atomic magnetometry with simultaneous excitation of orientation and alignment resonances,” Phys. Rev. Lett. 105(19), 193601 (2010). [CrossRef]
14. G. Bevilacqua and E. Breschi, “Magneto-optic spectroscopy with linearly polarized modulated light: Theory and experiment,” Phys. Rev. A 89(6), 062507 (2014). [CrossRef]
15. H. Wang, T. Wu, H. Wang, X. Peng, and H. Guo, “Magneto-optical spectroscopy with arbitrarily polarized intensity-modulated light in 4He atoms,” Phys. Rev. A 101(6), 063408 (2020). [CrossRef]
16. E. B. Aleksandrov, A. K. Vershovskiĭ, and A. S. Pazgalev, “Magnetometer based on a pair of symmetric transitions in the 87Rb hyperfine structure,” Tech. Phys. 51(7), 919–923 (2006). [CrossRef]
17. (In submission) Y. Ren, and Z. Wang, “Atomic population polarization pumped by radially or azimuthally polarized vector beams,” 2022.
18. A. Weis, G. Bison, and A. S. Pazgalev, “Theory of double resonance magnetometers based on atomic alignment,” Phys. Rev. A 74(3), 033401 (2006). [CrossRef]
19. E. Breschi and A. Weis, “Ground-state Hanle effect based on atomic alignment,” Phys. Rev. A 86(5), 053427 (2012). [CrossRef]
20. H. Wang, T. Wu, W. Xiao, H. Wang, X. Peng, and H. Guo, “Dual-Mode Dead-Zone-Free Double-Resonance Alignment-Based Magnetometer,” Phys. Rev. Applied 15(2), 024033 (2021). [CrossRef]
21. G. Jin, Y. Xu, and Z. Wang, “Transient evolution of optical magnetic resonance in rubidium vapor,” Opt. Express 27(5), 7087–7097 (2019). [CrossRef]
