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Abstract
The magnitude of the electron spin-relaxation rate Rrel of the atomic ensemble directly affects the sensitivity of the spin-exchange relaxation-free (SERF) atomic magnetometer (AM). The rapid and in-situ characterization of Rrel is of great importance. In this work, a fast extraction method of Rrel is proposed with a measurement period shorten to 0.5 s, merely detecting the transient response of SERF AM to a transverse DC excitation magnetic field after switching off the pump beam. In contrast to the conventional methods based on the measurement of the magnetic resonance linewidth, this method circumvents the involvement of optical pumping rate, and enables monitoring Rrel under arbitrary polarization, which is expected to improve the authenticity of Rrel measurement in a more convenient way.
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1. Introduction
The study on the interaction between light and atoms has revealed many novel quantum phenomena [1–4]. And numerous quantum devices can be designed by exploiting their exotic properties [5–7]. In the field of extremely weak magnetic field measurement, optically pumped atomic magnetometers (OPAMs) working in the spin-exchange relaxation-free (SERF) regime are the most sensitive magnetometer so far, which have been widely applied in fundamental physics [8,9] and biomagnetism detecting [10,11]. Its core sensing device is the atomic vapor cell, in which an ensemble of alkali-metal atoms is polarized by optical pumping [14]. The sensitivity of the SERF AM increases as the spin coherence lifetime of atomic ensemble becomes longer [15]. Accordingly, the precise determination of the electron spin-relaxation rate Rrel is significant for optimizing the content of components in the atomic vapor cell and improving the performance of SERF AMs [16].
The conventional measurement method of electron spin-relaxation rate in the SERF regime is commonly based on the detection of magnetic resonance linewidth [17], which are often obtained by utilizing the synchronous pumping technique [14] or the RF magnetic field excitation technique [18]. Since the relaxation introduced by the optical pumping is not stripped out of the linewidth information, the measurements of Rrel in SERF AMs working under the constant pumping mode are often operated under the condition of low polarization to reduce the measurement error. Furthermore, achieving high frequency resolution of magnetic resonance curves is time-consuming [19,20], always on the order of several minutes [20], and the long-term measurements bring about low-frequency drift and detection errors. Therefore, a high-efficient and accurate measurement scheme for Rrel without being restricted by the limit of low polarization is urgently needed.
Detecting the free induction decay (FID) is a typical extraction method for transverse relaxation rate, which is commonly used in nuclear magnetic resonance (NMR) devices owing to the long relaxation time of nuclei (up to tens of seconds) [21,22]. In contrast, the polarization lifetime of electron spin in high density atomic ensemble is significantly shorter due to the rapid spin-exchange collisions [23–25], which restricts the measurement of FID signals in traditional AMs. However, owing to the elimination of the spin-exchange relaxation, the polarization lifetime of the atomic ensemble in SERF regime is much longer [14,26], reaching the order of 10 ms [19,27], which raises the possibility for detecting the FID signals in SERF AMs. In order to observe an obvious time-domain oscillation signal, a feasible way is to appropriately increase the transverse DC excitation magnetic field [28,29]. However, the significant spin-exchange relaxation introduced by large magnetic field should be avoided, since its rate is quadratic in the magnetic field under the low magnetic field conditions [14]. Unsatisfactorily, the damped oscillation of transient response in the SERF regime is affected by the polarization-dependent slowing-down factor $q$ [30], which makes it no longer suitable to directly fit the global decay process of the time-domain signal. Consequently, in order to extract Rrel by the same process as NMR, intuition suggests that the strategy is that the low polarization limit must also be required, so that $q$ can be treated as constant during the decay of polarization [29,31,32].
In this work, we demonstrate a method for extracting Rrel of alkali-metal atoms based on the transient response of SERF AM under the external excitation of a static transverse magnetic field of nT magnitude after switching off the pump beam. And to confirm the validity, an elaborate experiment is conducted simultaneously. Compared with the traditional methods, the method proposed here has many superiorities presented as follows: First, it is universal to implement whatever the polarization is. Second, the extraction of Rrel can be realized by only one measurement within half-second. Moreover, the influence of optical pumping on the determination result is successfully circumvented.
2. Principle
The alkali-metal vapor (usually K, Rb, Cs) cell is the core component of AM. When illuminated by the resonant $\sigma ^+$ or $\sigma ^-$ laser of D1 line, the atomic ensemble absorbs the spin angular momentum transferred from photons, a resultant spin polarization comes into being by the depopulation pumping [33], so that it lies along the pumping direction. If we terminate the optical pumping after the atomic spins arrive at an equilibrium state, the atomic ensemble will undergo relaxation processes until completely depolarized.
In the regular AMs, the spin-exchange collisions between alkali-metal atoms often dominates the relaxation, whose rate is given by $\Gamma_{\textrm{SE}}=T_{\textrm{SE}}^{-1}=n\bar {v}\sigma_{\textrm{SE}}$, where $T_{\textrm{SE}}$ is the spin-exchange time, $n$ is the density of alkali-metal atoms, $\bar {v}$ is the average thermal velocity, and $\sigma_{\textrm{SE}}\simeq 2\times 10^{-14}\;{{\textrm{cm}}^2}$ is the spin-exchange cross section [18]. Counterintuitively, when operating in the SERF regime where $\Gamma_{\textrm{SE}}$ can achieve 10$^{4}$ times larger than the Larmor frequency [19,34], the spin-exchange decoherence can be suppressed. In the low polarization limit, spin-exchange relaxation rate $R_{\textrm{SE}}^e$ in the SERF regime is quadratic in the total magnetic field and vanishes for zero magnetic field [2]. Subsequently, the total electron spin-relaxation rate Rrel is described as follows [18]:
Here, $R_{\textrm{p}}$ is the absorption rate of photons from the probe beam, which can be minimized by detuning the frequency, $R_{\textrm{wall}}$ is the relaxation rate due to collision with the wall of vapor cell, which is always neglected due to the high pressure of buffer gas, and $R_{\textrm{SD}}^e$ is the relaxation rate due to spin-destruction collisions of the alkali-metal atoms with each other and with buffer-gas atoms, which have much smaller cross-sections compared to spin-exchange but become the predominant role in such case.
In the SERF regime, the evolution dynamics of the electronic spin vector $\boldsymbol {S}$ of the entire alkali ensemble is well-described by the Bloch equation [34]
Fig. 1. Pump and relaxation process. (a) Time sequential of measurement. (b) Polarization process of alkali metal atoms in the absence of external magnetic field. (c) Precession and relaxation process of electron spin polarization excited by $y$-direction DC magnetic field $B_y$ in the $x$-$z$ plane after turning off the pump beam.
3. Simulation and experiment
Taking into account the polarization dependence of $q$ in the SERF regime, for potassium atoms with nuclear spin I=3/2, $q(P)$ is described as [6,28]
with $P=2\langle S \rangle$. As $P$ gradually attenuates with time during the free precession decay process, the $P$-dependent $q$ is essentially time-dependent, and this effect will cause distortion of the transient response. To further investigate the effects of $q$ on the transient response in the second stage, a targeted numerical simulation was performed with parameters $R_{\textrm{rel}}=100\,s^{-1}$, $B_x=B_z=0\;\textrm{nT}$ and $B_y=7\;\textrm{nT}$. As presented in Fig. 2(a), if $q$ can be treated as constant and time-independent, the decay rate $R_{\textrm{rel}}/q$ and oscillation frequency $f=\frac {\gamma ^e\!|\boldsymbol {B}|}{2\!\pi \!q}$ of the response signal are both invariant, noting that this happens only under the condition of low polarization limit. Under more general circumstances, taking the initial spin polarization $S_0=1/4$ as an example, Fig. 2(b) shows that $-R_{\textrm{rel}}/q$ increases rapidly and $f$ decreases quickly within the first $200\;\textrm{ms}$ of the evolution, resulting in the decay mitigation and frequency chirp of the temporal signal as shown in Fig. 2(a). What deserves our special attention is that, as shown in the insets Fig. 2(c) and Fig. 2(d), after evolving to $200\;\textrm{ms}$, $-R_{\textrm{rel}}/q$ eventually reaches its limit value and $f$ also tends to be stable. This means that, compared with the initial state, the polarization decays significantly and approaches zero. Therefore, this situation can be considered to satisfy the low polarization limit. Fortunately, this is exactly what we can take advantage of to extract Rrel, provided that the signal-to-noise ratio (SNR) of the transient response in this period is high enough.Fig. 2. Theoretical analysis of the effect of nuclear slowing-down factor $q$ on the transient response during oscillatory decay. (a) Transverse transient response of the electron spins to a DC excitation magnetic field along the $y$-axis between 0-200 ms. The blue solid line and magenta dashed line indicate that $q$ is constant and time-dependent respectively. The light-blue dotted line represents the attenuation envelope of the total spin polarization in the time-dependent $q$ condition. (b) Variation of $-R_{\textrm{rel}}/q$ and oscillation frequency $f$ with time during oscillation decay of transient response, with their detailed zoom-in between 200-400 ms in insets (c) and (d).
In order to verify the feasibility of this idea, we implement an experiment based on a SERF AM. The experimental setup is shown in Fig. 3. The core sensing unit in the AM is a spherical glass vapor cell with diameter of 28 mm, containing a droplet of potassium, 50 Torr nitrogen as quenching gas and 1.4 atm $^4\;\textrm{He}$ as buffer gas. The vapor cell is placed in a boron nitride ceramic oven heated by a set of twisted-pair wire heater driven by alternating current with frequency of 100 kHz. The temperature of the oven is monitored by the nonmagnetic platinum resistor placed on the wall of oven. The oven is encased by a nonmagnetic poly-ether-ether-ketone (PEEK) vacuum chamber wrapped by water-cooling tubes. The vacuum chamber is enclosed by a shielding system which is consisted by a four-layer outer $\mu$-metal cylindric shield and an inner ferrite barrel, which attenuates the residual magnetic field to less than 0.6 nT for every axis. Meanwhile, a set of three-axis coils is utilized to further compensate the residual magnetic field and generate the required magnetic field in the measurement.
Fig. 3. Experimental setup. PBS, polarization beam splitter; AOM, acoustic optical modulator; HWP, half-wave plate; QWP, quarter-wave plate; PEM, photo-elastic modulator; PD, photodetector.
The potassium atomic ensemble is polarized by the pump beam emitted by a distributed bragg reflector (DBR) laser with tapered amplifier (TAL801-770, Uniquanta), and whose wavelength is tuned to the center of potassium’s D1 line. The pump beam is expanded to 15 mm in diameter by the beam expander, and converted into circular polarized beam after passing through a polarizer and a quarter wave plate. An acoustic optical modulator (AOM) (G&H, 3080-125) is used to switch on and off the pump beam with switching time less than 50 ns. The probe beam is emitted by a DBR laser (DBR801-770, Uniquanta), whose wavelength is tuned to 0.26 nm away from the D1 line of potassium. The probe beam is linearly polarized and its diameter is 3 mm. The probe beam is modulated by a photo-elastic modulator (PEM) (PEM-100, HINDS) with a modulation frequency of 50 kHz, and an amplitude of 0.08 rad. After passing through the analyzer, the probe beam is received by the photodetector (S5980, Hamamatsu). And the signal is demodulated by the lock-in amplifier (HF2LI, Zurich Instruments). A function generator connects to the $y$-axis coil, and the other one outputs the modulation current for the AOM controller. A homemade LabVIEW program controls the two function generators and the DAQ system to synchronize the pump beam, bias magnetic field and the data acquirement.
In the experiment, after the magnetic field compensation and sufficient spin polarization period (longer than 5 s), the pump beam is shut off, and a transverse DC excitation magnetic field is generated along $y$-axis. Simultaneously, a half-second signal response of the SERF AM is recorded by the DAQ system.
4. Experimental results and discussion
After switching off the pump beam, a transverse DC excitation magnetic field is applied on the $y$-axis ($B_{y}$) synchronously, and this moment is defined as the starting point of the following evolution. And then the polarization starts to freely precess and gradually decays to zero. Under the condition that the pump beam power and the DC excitation magnetic field were set to 9 mW and 7 nT respectively, the transient response between 280 ms and 450 ms is shown in Fig. 4(a), with a sampling rate of 899 S/s after demodulated by the lock-in. The SNR of the transient response, which is defined as the ratio of peak-to-peak amplitude of the last cycle in the region of interest to the root-mean-square of the signal acquired without excitation magnetic field, is better than 6.5 dB. Since the polarization at this time has been greatly attenuated, $q$ can be regarded as the low polarization limit of 6 when fitting it by Eq. (4). As shown by the grey line, the fitted result with a root mean square error of 0.019 confirms the rationality of the $q$-constant approximation. In order to avoid the imprecise positioning of the peak points caused by sparse sampling with a time interval of $\sim$1.1 ms, an interpolation operation for the acquired transient response is performed, and the peak points of the interpolated oscillatory response signal are highlighted by the red cross. After a linear fit to the natural logarithm values of all peak points presented in Fig. 4(b), it can be found that these points show satisfactory linearity, the goodness of the fitting is 0.9998. Essentially, the fitting curve satisfies the following linear form:
and Rrel is implied in the slope value. Specifically, the relaxation rate calculated from Fig. 4(b) is 117.6 $s^{-1}$.Fig. 4. Extraction of Rrel from the experimental data of transient response. (a) The transient response between 280 and 450 ms after synchronously switching off the pump light and switching on the transverse DC excitation magnetic field. The red cross symbols represent the peak points of the oscillatory response signal, the gray line is the fitting result, and blue dots are experimental data. (b) Linear fitting results after taking the natural logarithm of the peak points, the relaxation rate information can be directly obtained from the slope value.
In the damped oscillation stage, the effect of optical pumping is successfully circumvented. To validate the measurement ability under arbitrary spin polarization, we measured Rrel of different pump beam powers while under the same excitation magnetic field of 7 nT. As shown in Fig. 5, the mean value of the measured data taken from 9 mW to 15 mW is 117.4 $s^{-1}$ with a small standard deviation of 1.2 $s^{-1}$, i.e., the coefficient of variation is 0.01, indicating that the pumping effect has negligible influence on Rrel measurement, regardless of the pumping power.
Fig. 5. Relationship between Rrel and the pump beam power under the same excitation magnetic field of 7 nT. The measured data is shown as blue dots, and the average value of Rrel is shown as green line. The light green area represents the $2\%$ error margin of the mean value.
Compared with the methods of measuring the linewidth of magnetic resonance, the method based on the reliable part of the tail of the transient response after switching off the pump beam makes the limit of low spin polarization no longer an essential prerequisite for relaxation rate extraction, as well as improving the efficiency of monitoring the working status of spin ensemble. It is worth noting that, the fluctuation of the power and frequency of probe laser during the short measuring period may introduce accidental errors. Thus, a series of measurement for more than 5 times ought to be performed to confirm the average spin-relaxation rate of the atomic ensemble. Nevertheless, the measurement duration is shorter than the measurement of linewidth of magnetic resonance which takes several minutes [20].
5. Conclusion
In conclusion, an ingenious and highly efficient measurement method for the spin-relaxation rate of optically pumped alkali-metal atomic ensemble in the SERF regime is proposed. By highlighting the polarization dependence of the slowing-down factor, its influence on the time-varying characteristics of the transient response under universal conditions has been analyzed in detail. Furthermore, the method is experimentally tested, which confirmed the validity of the fast measurement scheme at arbitrary spin polarization. Hence, this work would be useful not only for providing a practical alternative but also for clarifying the physics essentials in the relaxation process, shedding more light on the involved dynamics of electron spin polarization. Meanwhile, potential applications are foreseen for atomic magnetometers, as well as atomic spin gyroscopes.
Funding
National Key Research and Development Program of China (No. 2018YFB2002405); National Natural Science Foundation of China (No. 61903013).
Acknowledgments
We gratefully acknowledge assistance in the construction of the experimental apparatus from Zixuan Wang, Jiahe Zhang and Bowen Sun.
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. W. E. Bell and A. L. Bloom, “Optical detection of magnetic resonance in alkali metal vapor,” Phys. Rev. 107(6), 1559–1565 (1957). [CrossRef]
2. W. Happer and A. C. Tam, “Effect of rapid spin exchange on the magnetic-resonance spectrum of alkali vapors,” Phys. Rev. A 16(5), 1877–1891 (1977). [CrossRef]
3. A.-C. Ji, Q. Sun, X. C. Xie, and W. M. Liu, “Josephson effect for photons in two weakly linked microcavities,” Phys. Rev. Lett. 102(2), 023602 (2009). [CrossRef]
4. R. Qi, X.-L. Yu, Z. B. Li, and W. M. Liu, “Non-abelian Josephson effect between two f = 2 spinor bose-einstein condensates in double optical traps,” Phys. Rev. Lett. 102(18), 185301 (2009). [CrossRef]
5. I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis, “A subfemtotesla multichannel atomic magnetometer,” Nature 422(6932), 596–599 (2003). [CrossRef]
6. I. M. Savukov and M. V. Romalis, “Effects of spin-exchange collisions in a high-density alkali-metal vapor in low magnetic fields,” Phys. Rev. A 71(2), 023405 (2005). [CrossRef]
7. H. B. Dang, A. C. Maloof, and M. V. Romalis, “Ultrahigh sensitivity magnetic field and magnetization measurements with an atomic magnetometer,” Appl. Phys. Lett. 97(15), 151110 (2010). [CrossRef]
8. M. Safronova, D. Budker, D. DeMille, D. F. J. Kimball, A. Derevianko, and C. W. Clark, “Search for new physics with atoms and molecules,” Rev. Mod. Phys. 90(2), 025008 (2018). [CrossRef]
9. T. Wang, D. F. J. Kimball, A. O. Sushkov, D. Aybas, J. W. Blanchard, G. Centers, S. R. O. Kelley, A. Wickenbrock, J. Fang, and D. Budker, “Application of spin-exchange relaxation-free magnetometry to the cosmic axion spin precession experiment,” Phys. Dark Universe 19, 27–35 (2018). [CrossRef]
10. S. Taue, Y. Sugihara, T. Kobayashi, S. Ichihara, K. Ishikawa, and N. Mizutani, “Development of a highly sensitive optically pumped atomic magnetometer for biomagnetic field measurements: A phantom study,” IEEE Trans. Magn. 46(9), 3635–3638 (2010). [CrossRef]
11. E. Boto, N. Holmes, J. Leggett, G. Roberts, V. Shah, S. S. Meyer, L. D. Mu noz, K. J. Mullinger, T. M. Tierney, S. Bestmann, G. R. Barnes, R. Bowtell, and M. J. Brookes, “Moving magnetoencephalography towards real-world applications with a wearable system,” Nature 555(7698), 657–661 (2018). [CrossRef]
12. V. Mathé, F. Lévêque, P.-E. Mathé, C. Chevallier, and Y. Pons, “Soil anomaly mapping using a caesium magnetometer: Limits in the low magnetic amplitude case,” J. Appl. Geophys. 58(3), 202–217 (2006). [CrossRef]
13. L. Lu, J. Shang, Z. Pan, and Y. Ji, “Chip-scale SERF atomic magnetometer without magnetic shield,” in 2017 IEEE 67th Electronic Components and Technology Conference (ECTC) (IEEE, 2017).
14. J. C. Allred, R. N. Lyman, T. W. Kornack, and M. V. Romalis, “High-sensitivity atomic magnetometer unaffected by spin-exchange relaxation,” Phys. Rev. Lett. 89(13), 130801 (2002). [CrossRef]
15. S. J. Seltzer, “Developments in alkali-metal atomic magnetometry,” Ph.D. thesis, Princeton University (2008).
16. J. Liu, L. Jiang, Y. Liang, F. Liu, and W. Quan, “A fast measurement for relaxation rates and fermi-contact fields in spin-exchange relaxation-free comagnetometers,” IEEE Trans. Instrum. Meas. 69(10), 7805–7812 (2020). [CrossRef]
17. W. Happer and H. Tang, “Spin-exchange shift and narrowing of magnetic resonance lines in optically pumped alkali vapors,” Phys. Rev. Lett. 31(5), 273–276 (1973). [CrossRef]
18. M. P. Ledbetter, I. M. Savukov, V. M. Acosta, D. Budker, and M. V. Romalis, “Spin-exchange-relaxation-free magnetometry with Cs vapor,” Phys. Rev. A 77(3), 033408 (2008). [CrossRef]
19. J. Lu, Z. Qian, and J. Fang, “A fast determination method for transverse relaxation of spin-exchange-relaxation-free magnetometer,” Rev. Sci. Instrum. 86(4), 043104 (2015). [CrossRef]
20. X. Zhang, J. Qin, Y. Wang, and C. Chen, “A fast identification on the spin-exchange relaxation-free regime of atomic magnetometer exploiting measurement on gyromagnetic ratio,” IEEE Trans. Instrum. Meas. 68(4), 1157–1164 (2019). [CrossRef]
21. J. Keeler, Understanding NMR Spectroscopy, 2nd Edition (John Wiley & Sons Inc, 2010).
22. D. Meyer and M. Larsen, “Nuclear magnetic resonance gyro for inertial navigation,” Gyroscopy Navig. 5(2), 75–82 (2014). [CrossRef]
23. N. Castagna and A. Weis, “Measurement of longitudinal and transverse spin relaxation rates using the ground-state hanle effect,” Phys. Rev. A 84(5), 053421 (2011). [CrossRef]
24. R. Li, F. N. Baynes, A. N. Luiten, and C. Perrella, “Continuous high-sensitivity and high-bandwidth atomic magnetometer,” Phys. Rev. Appl. 14(6), 064067 (2020). [CrossRef]
25. A. Horsley, G.-X. Du, M. Pellaton, C. Affolderbach, G. Mileti, and P. Treutlein, “Imaging of relaxation times and microwave field strength in a microfabricated vapor cell,” Phys. Rev. A 88(6), 063407 (2013). [CrossRef]
26. V. Shah, S. Knappe, P. D. D. Schwindt, and J. Kitching, “Subpicotesla atomic magnetometry with a microfabricated vapour cell,” Nat. Photonics 1(11), 649–652 (2007). [CrossRef]
27. X.-J. Liu, M. Ding, Y. Li, Y.-H. Hu, W. Jin, and J.-C. Fang, “Transverse relaxation determination based on light polarization modulation for spin-exchange relaxation free atomic magnetometer,” Chin. Phys. B 27(7), 073201 (2018). [CrossRef]
28. J. Zhao, M. Ding, J. Lu, K. Yang, D. Ma, H. Yao, B. Han, and G. Liu, “Determination of spin polarization in spin-exchange relaxation-free atomic magnetometer using transient response,” IEEE Trans. Instrum. Meas. 69(3), 845–852 (2020). [CrossRef]
29. J. Tang, Y. Yin, Y. Zhai, B. Zhou, B. Han, H. Yang, and G. Liu, “Transient dynamics of atomic spin in the spin-exchange-relaxation-free regime,” Opt. Express 29(6), 8333 (2021). [CrossRef]
30. S. Appelt, A. B.-A. Baranga, C. J. Erickson, M. V. Romalis, A. R. Young, and W. Happer, “Theory of spin-exchange optical pumping of 3 He and 129 Xe,” Phys. Rev. A 58(2), 1412–1439 (1998). [CrossRef]
31. A. Gusarov, D. Levron, A. B.-A. Baranga, E. Paperno, and R. Shuker, “An all-optical scalar and vector spin-exchange relaxation-free magnetometer employing on–off pump modulation,” J. Appl. Phys. 109(7), 07E507 (2011). [CrossRef]
32. J.-Q. Fu, P.-C. Du, Q. Zhou, and R.-Q. Wang, “Spin dynamics of the potassium magnetometer in spin-exchange relaxation free regime,” Chin. Phys. B 25(1), 010302 (2016). [CrossRef]
33. W. Happer and W. A. Van Wijngaarden, “An optical pumping primer,” Hyperfine Interact. 38(1-4), 435–470 (1987). [CrossRef]
34. S. Seltzer and M. Romalis, “Unshielded three-axis vector operation of a spin-exchange-relaxation-free atomic magnetometer,” Appl. Phys. Lett. 85(20), 4804–4806 (2004). [CrossRef]
