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Abstract
The transient dynamics of atomic spins under oscillating and static magnetic fields have been studied in the spin-exchange relaxation-free (SERF) regime with a dual-beam configuration. The spin-relaxation rate can be accurately measured by detecting the transient response signal of the free induction decay (FID) process within several milliseconds. Leveraging this convenient method for measuring a large relaxation rate in a small cell volume, the dependence of the spin-relaxation rate on the probe intensity and ambient magnetic field was studied in the limit of low spin polarization. Moreover, by theoretical analysis of the dynamic evolution of the Rabi oscillation generated by a consecutive oscillating field and a small static magnetic field, we experimentally demonstrate that the amplitude of the Rabi oscillation is affected by the amplitude of the oscillating field in the SERF regime. According to the retrieved frequency of the FID signal and amplitude of relevant Rabi oscillation, the coil constants were 75.55 ± 0.78~nT/mA, 151.5 ± 0.9~nT/mA, and 116.6 ± 0.3~nT/mA along the x-, y-, and z-axes, respectively.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Detection of extremely weak magnetic-fields using atomic magnetometers has received extensive attention [1]. Romalis et al. first verified that a $\rm {K}$ magnetometer achieves extremely high sensitivity in the spin-exchange relaxation-free (SERF) regime [2]. Spin-relaxation, caused by spin-exchange collisions among polarized alkali atoms, is diminished in the SERF regime, thus increasing the coherence time [3,4]. Moreover, SERF magnetometers are the typical type of magnetometer with sufficiently high sensitivity ($\mathrm {fT}/\sqrt {\mathrm {Hz}}$) and sufficiently small size for biomagnetic applications, including measurement of the magnetic fields from the heart, magnetocardiography [5–7], and magnetoencephalography of the brain [8,9]. The sensitivity of magnetometers is closely related to the spin-relaxation rate ${R_{{\rm {rel}}}}$ in a negative correlation. Thereby, the precise determination of ${R_{{\rm {rel}}}}$ plays a crucial role in optimizing the gas components in alkali cells and boosting the performance of magnetometers [10].
The general approach to measuring ${R_{{\rm {rel}}}}$ is to detect the magnetic resonance linewidth with a pump-probe configuration, which leverages synchronous pumping or RF magnetic field excitation, and is fitted by a Lorentzian lineshape [2,11,12]. Both techniques delay the duration of the measurement to achieve high-frequency resolution of the magnetic resonance curve [13]. Consequently, there is a drift error during a long test period [14]. In addition, ${R_{{\rm {rel}}}}$ is accurately extracted from both techniques, as the resonance frequency is significantly larger than the magnetic resonance linewidth, ${\omega _0} \gg \Delta \omega$. Otherwise, the negative frequency response signal centered at $- {\omega _0}$ cannot be ignored [15], particularly in smaller volume cells. The quasi-Lorentzian curve cannot be exploited to measure ${R_{{\rm {rel}}}}$ precisely. Measuring the free induction decay (FID) signals is a common approach for extracting ${R_{{\rm {rel}}}}$ used in nuclear magnetic resonance gyroscopes (NMRGs), owing to the long nuclear spin coherence time [16]. By contrast, SERF magnetometers have shorter relaxation times of only tens of milliseconds, which limits the measurement of FID signals [17]. A simple method involves detecting the transient signals of a single-beam magnetometer in the SERF regime [18]. However, the accuracy of measurement is highly dependent on the pumping intensity in this method. Moreover, the response signal is weak due to the small detection pumping intensity, which results in a determination error. To enhance the signal, it is necessary to increase the pumping intensity, and it takes a long time to measure the polarization-dependent nuclear slowing-down factor $q$. Another similar method is to investigate the role of pumping and relaxation rate on the transient dynamics of the system via measurements of nonlinear magneto-optical rotation signals at different light intensities [19], where a high-intensity magnetic field and a lower working temperature are not suitable for the SERF device.
SERF magnetometers typically operate in an extremely low-magnetic field environment to restrain the spin-exchange relaxation for femtotesla-level field detection [20]. In addition to reducing the ambient magnetic field with a magnetic shield, coils are used for precise compensation of the residual magnetic field. Theoretically, the coil constants are calculated by magnetic simulation based on the turns and shape of the coils [21], which are conventionally calibrated using a fluxgate magnetometer mounted at a central location. However, the accuracy is significantly affected by the installation deviation and the performance of the fluxgate. $In~situ$ calibration of coil constants based on the characteristics of the sensor is a reliable method. One approach is to detect the spin precession frequency signal of hyperpolarized $^{3}\rm {He}$ and $^{21}\rm {Ne}$ with an ultrasensitive atomic spin co-magnetometer [22,23]. Another method is based on measuring the $\pi {\rm {/2}}$ pulse duration in an NMRG [24]. However, polarized nuclei with a long coherence time are crucial for both methods, which limits their application in SERF magnetometers. Another method of calibrating the coil constant is to measure the frequency of the damped oscillation using a SERF magnetometer [18]. However, the coil constant along the direction of the pumping beam cannot be measured.
In this study, we investigated the transient dynamics of a polarized atomic ensemble under oscillating and static magnetic fields using a dual-beam configuration in the SERF regime. Under the limitation of low-polarization, the transient response signal of the FID process can be detected by applying an oscillating field for a moment after turning off the pump light, and the entire measurement process lasts milliseconds. Moreover, spin-exchange relaxation can be precisely measured by analyzing this signal. We theoretically and experimentally demonstrate the dynamic evolution of Rabi oscillation generated by applying a consecutive oscillating field and a static magnetic field, particularly in the SERF regime with the limit of a small ambient magnetic field. The coil constants were calibrated using the amplitude and frequency of the transient response signal. This method can effectively realizes accurate and fast measurement of the larger relaxation rate of miniaturized alkali cells, and it is convenient to calibrate the coil constants.
2. Theory
The dynamics of a polarized atomic ensemble can be described by the evolution of the density matrix equation [2,20]. However, the description can be significantly simplified when the spin-exchange rate is significantly faster than the precession in the magnetic field. Generally, the density matrix is expressed using the Bloch equation. [11]. The initial equilibrium spin polarization ${P_0} = {R_{{\rm {op}}}}/({R_{{\rm {op}}}} + {R_{{\rm {rel}}}})$ can be achieved instantaneously when the pumping beam is turned on, where ${R_{{\rm {op}}}}$ is the pumping rate. A static field ${B_0}$ was applied parallel to the pumping beam along the $z$-axis. Then, the pumping beam was turned off, and the Bloch equation under an ambient magnetic field ${\bf {B}}$ can be expressed as
Fig. 1. Dynamic evolution of Rabi oscillation with a continuous-wave oscillating field ${B_x}\cos ( {\omega t} )$ and a static magnetic field $B_0$. (a): The blue spiral curve shows a trajectory of ${\bf {P}}$ during one transition. The red vortex line represents the projection of the polarization trajectory on the $x-y$ plane before arriving the point M. (b): Several blue envelopes show the Rabi oscillation of $P_y$, and the frequency of the red envelope profile is only influenced by the amplitude of the oscillating field $B_x$.
If the oscillating field is turned on for a moment and then turned off, the direction of the equilibrium spin polarization $P_0$ is deflected and projected on the $x-y$ plane, as shown in Fig. 2(a). The spin polarization precesses around $B_0$, yielding an oscillating decay transverse y-component
where $P_0^x$ and $P_0^y$ are the projections of $P_0$ on the $x$- and $y$-axes, respectively, ${\omega _0} = {\gamma ^{\rm {e}}}{B_0}/q$. $P_y$ is considered as the FID signal. As shown in Fig. 2(a), the polarization ${\bf {P}}$ exhibits a decaying precession motion around the $z$-axis due to the effect of relaxation during the FID process. The frequency of the decaying precession is a constant value that is influenced by the static magnetic field $B_0$. The profile line of the FID signal with a single exponential curve contains information on the spin-relaxation rate ${R_{{\rm {rel}}}}$ in Fig. 2(b).Fig. 2. Dynamic evolution of the FID signal with a limited duration oscillating field ${B_x}\cos ( {\omega t} )$ and a static magnetic field $B_0$. (a) The spiral curve shows a trajectory of ${\bf {P}}$, where the yellow part represents Rabi oscillation with an oscillating field turning on, and the blue part represents FID signal with an oscillating field turning off. The red vortex line represents the projection of the polarization trajectory on the $x-y$ plane during the FID process. (b): The blue damped oscillation curve depicts the FID signal of $P_y$. The frequency of this signal can be fitted by Eq. (5) yielding the numerical value of $B_0$.
3. Experiment
A schematic of the experimental setup used to study transient atom spin dynamics is shown in Fig. 3. A miniaturized borosilicate cubic cell with an inner diameter of $3 \times 3 \times 3~{\rm {m}}{{\rm {m}}^3}$, containing a droplet of $^{87}\rm {Rb}$ metal and 1064 Torr of $\rm {N_2}$ buffer gas, was used. Full width at half maximum $\Gamma _{{\rm {D}}}$ of the optical absorption profile of this cell is 24.5 GHz, the center wavelength of the ${{\rm {D}}_1}$ line is 794.9984 nm, and the center wavelength of the ${{\rm {D}}_2}$ line is 780.2561 nm. The optical depth (OD) is 6.38 for the pump beam, and was measured by spectral profiles at 423 K. This cell was placed at the center of a magnetic shield comprising four layers of $\mu$-metal, which provides a residual magnetic field below 1 nT with a quasi-static shielding factor of approximately 75000. A triaxial coil with uniform field was constructed to compensate the residual magnetic field and provide external fields [25]. The coil constants for the saddle coil are 77.142 nT/mA ($x$-axis, with 10 turns) and 152.75 nT/mA ($y$-axis,with 20 turns), and that of the Lee-Whiting coil is 119.404 nT/mA ($z$-axis), as measured by a fluxgate [26]. The cell was heated by a wire heater using a high-frequency (200 kHz) alternating current inside a boron nitride ceramic oven. The temperature, which was monitored by a PT 1000 resistor, was stable at 423 K with a fluctuation of 0.01 K.
Fig. 3. Schematic of the experimental setup for measurements. A circularly polarized laser beam at $^{87}\rm {Rb}$ ${{\rm {D}}_1}$ resonance line optical pumps the rubidium atoms along the $z$-direction, and the transmitted light intensity is monitored by PD0. A linearly polarized probe beam, which is blue detuned from the ${{\rm {D}}_2}$ line, propagates along the $y$-direction, passing sequentially through the Rb cell, a polarizing beam splitter, and two subtraction photodiodes (PD+ and PD-). LP denotes a linearly polarized plate; QWP is a quarter-wave plate; PD is a photodiode; PBS is a polarized beam splitter; DAQ is the data acquisition system. AOM is used to turn the pump beam on/off when measuring the transient response signal.
Two distributed feedback lasers were coupled to the setup using a polarization-maintaining fiber with a collimating lens. A circularly polarized laser beam at $^{87}\rm {Rb}$ ${{\rm {D}}_1}$ optical beam pumped atoms along the $z$-direction. An acoustic optical modulator (AOM) was manipulated to turn the pumping beam on and off with a closing time of less than 1 ${\rm {\upmu s}}$. A linearly polarized probe beam tuned to the ${{\rm {D}}_2}$ line propagated along the $y$-direction, passing sequentially through the Rb cell, a polarizing beam splitter, and two subtraction photodiodes (PD+ and PD-). The diameter $d$ of the pump and probe beam is 2.7 mm, and their linewidths are on the order of 1 MHz. A polarization beam splitter and noise eater combination was used to control the probe and pump laser intensity, which changed by less than 0.1$\%$ during the experiment. The extinction ratio of the two beams passing through the fibers was larger than 40 dB. In the experiment, all data were collected using DAQs for subsequent data processing.
4. Results and discussion
To study the transient atom spin dynamics in the SERF regime. The coils were used to compensate ambient field inside the magnetic shield and generate a static field $B_0$ along the $z$-axis. The dark state is formed by turning off the pumping beam using the AOM after the equilibrium spin polarization $P_0$ is established. If the oscillating field is applied in the total dark state, the transient response signal of the Rabi oscillation can be obtained by detecting the optical rotation of light, as shown in Fig. 4(a). We can leverage this signal to measure the spin-relaxation rate and calibrate the radial coil constant. However, the oscillating field is only turned on for a certain period of time and then turned off, and the FID signal can be detected, as shown in Fig. 4(b). The spin-relaxation rate can be studied using the decay component of the damped oscillation curve, and the frequency of this signal can be used to calibrate the axial coil constant along the $z$-axis.
Fig. 4. Transient atomic spin dynamics under different oscillating fields in the SERF regime. The static field $B_0$ is 50 nT, while the amplitude $B_x$ of an oscillating field is 3.16 nT. The temperature is 423 K with the density $n_{{\rm {Rb}}} \approx 1 \times {10^{14}}$ ${\rm {cm}}^{-3}$ (a): The transient response of the Rabi oscillation. (b): The transient response of the FID process.
4.1 Spin-relaxation rate
The extraction of ${R_{{\rm {rel}}}}$ was quickly realized by fitting the transient response of the FID process in milliseconds. The damped oscillation curve is influenced by $q$ according to Eq. (5).
The initial $q$ was determined by the intensity of the pumping beam. If the pump intensity is strong, $q$ becomes larger as the polarization decreases over time, so that the frequency and decay rate of the damped oscillation curve are gradually smaller in the dark state. This phenomenon results in a signal distortion. If the pump intensity is weak, then $q$ is approximately 6 ($P \approx 0$). As shown in Fig. 5, when the pump intensity is 19.88 ${\rm {mW}}/{\rm {c}}{{\rm {m}}^2}$ with $q=5.67$ measured using the fast Fourier transform analysis result of the transient response [14,27], the fitting cure gradually deviated from the measured data with a constant $q$. The fitting curve is consistent with the experimental data, with a constant $q=6$ when the pump intensity is weak.Fig. 5. The dots are the measured FID signals at different pump intensities, and the solid lines are fitted by Eq. (5) with different constant $q$.
To accurately measure the relaxation rate, a low pump intensity and polarization should be adopted, and the signal intensity should also become smaller. In addition, the transient response of the spin polarization was measured by the optical rotation angle of the linearly polarized probe beam rooting from the cell. The optical rotation of linearly polarized light can be written as [11]
Fig. 6. Optical rotation obtained through dividing the response signal by the probe intensity, taking into account absorption $\exp ( - {\rm {OD}})$ of the probe beam. The optical rotation gets smaller with increasing probe intensity due to the pumping rate of the probe beam ${R_{{\rm {pr}}}}$.
Under optimal optical parameters, Fig. 7 shows several FID signals with different static magnetic fields $B_0$. As shown in Fig. 7, the frequency has an increment as the field $B_0$ increases, and the damping of these signals due to the spin-relaxation rate $R_{\rm {rel}}$ are also different. To find the dependence of the spin-relaxation rate $R_{\rm {rel}}$ and static magnetic field $B_0$, we obtain the spin-relaxation rate $R_{\rm {rel}}$ under different static magnetic fields $B_0$ using Eq. (5). The tendency of $R_{\rm {rel}}$ for different static magnetic fields $B_0$ is shown in Fig. 8. Considering spin-exchange relaxation, the spin-relaxation rate $R_{\rm {rel}}$ is expressed as [2,11,28]:
Fig. 7. The measured FID signals at different static magnetic fields $B_0$ fitted with Eq. (5). In order to attain the obvious response signal, the oscillating fields last different times.
Fig. 8. Relationship between the spin-relaxation rate $R_{\rm {rel}}$ and static magnetic field $B_0$ fitted with Eq. (9), which gives ${{R_{{\rm {SE}}}}}$ = 73.163 kHz at 423 K.
Fig. 9. The optical pumping rate of probe beam ${R_{{\rm {pr}}}}$ as a function of intensity $I$ at different static magnetic field $B_0$. The experimental data is described by ${R_{{\rm {pr}}}} = \eta I$, which gives $\eta = 1.28 \pm 0.15~{\rm {Hz}} \cdot {\rm {c}}{{\rm {m}}^2}{\rm {/mW}}$.
4.2 Magnetic field calibration
The performance of high-sensitivity magnetometers is highly dependent on the precision of the coil constants, which can be calibrated using transient response signals. The strength of the magnetic field generated by the coil is linearly related to the applied current. For the axial coil ($z$-axis), the relationship between the frequency $\omega$ of the FID signal and the applied current strength $I$ is described as $\omega = {\gamma ^{\rm {e}}}{k_0}{I}/q$, where $k_0$ is the axial coil constant. The dependence of static field $B_0$ on applied current $I$ is shown in Fig. 10 yielding the ${k_0} = 116.6 \pm 0.3~{\rm {nT/mA}}$.
Fig. 10. Dependence of the static field $B_0$ on applied current in the $z$-axis. The data based on Eq. (5) is linear fitted with ${k_0} = 116.6 \pm 0.3~{\rm {nT/mA}}$.
The radial coil constant can be calibrated using the Rabi oscillation, which is changed by the amplitude $B_x$ of the oscillating field. As shown in Fig. 11(a), it can be seen that not only the frequency of the envelope profile of the Rabi oscillation becomes smaller, but also this signal amplitude becomes weaker as the amplitude $B_x$ decreases. Furthermore, the method of finding the first peak of the Rabi oscillation to measure the $\pi /2$ pulse duration is unreliable with a small amplitude $B_x$, because the signal decays fast owing to large relaxation in a small-volume cell. Although the polarization ${\bf {P}}$ at point B is tilted into the $x-y$ plane, it is also smaller than the projection of ${\bf {P}}$ on the $x-y$ plane at point A, as shown in Fig. 11(b). Fortunately, we can obtain the value of $B_x$ by fitting Rabi oscillation according to Eq. (4). The measured value of $B_x$ will be more accurate in the SERF regime, because the transverse relaxation time is approximately equal to the longitudinal relaxation time, which is consistent with the condition of the theoretical derivation above. When the alkali vapor density is sufficiently high, the spin-exchange rate ${{R_{{\rm {SE}}}}}$ is much larger than the Larmor precession frequency, and the alkali atoms operate in the SERF regime, where the magnetic field must be less than approximately hundreds of nT. Therefore, the amplitude $B_x$ corresponding to ${\omega _x}$ and static magnetic field $B_0$ corresponding to ${\omega _0}$ are finite. The Rabi oscillation in the SERF regime is depicted in Fig. 12(a). There is no other envelope for Rabi oscillation under a small ambient magnetic field in the SERF regime. This is because the $\pi$ pulse duration of generating the second envelope is longer than the relaxation time depicted in Fig. 12(b). The amplitude of the Rabi oscillation increases with increasing $B_x$, because the deflection of the polarization ${\bf {P}}$ increases, resulting in an increase in the projection of ${\bf {P}}$ on $x-y$ plane. According to this theory, the radial coil constant can be calibrated under a weak current ( Fig. 13). The radial coil constant $k_x$ can be expressed as ${k_x}I = {B_x} = 2q{\omega _x}{\rm {/}}{\gamma ^{\rm {e}}}$. The dependence of amplitude the $B_x$ on the applied current $I$ is shown in Fig. 14, yielding ${k_x} = 75.55 \pm 0.78~{\rm {nT/mA}}$, similarly, ${k_y} = 151.5 \pm 0.9~{\rm {nT/mA}}$.
Fig. 11. (a) The frequency and amplitude of envelope profile based on Eq. (4) at different amplitude $B_x$ with a static magnetic field $B_0 = 1967~ {\rm {nT}}$ at 383 K in the non-SERF regime. (b) Schematic depiction of the difference between the time at maximum peak with $\pi {\rm {/}}2$ pulse duration at the first envelope of Rabi oscillation shown in the bottom of picture (a).
Fig. 12. (a) The Rabi oscillation fitted by Eq. (4) with $B_0= 72~{\rm {nT}}$ at 423 K. The half period of the envelope profile is 37 ms with the amplitude $B_x =5.8{\rm {nT}}$, while the relaxation time is 7.1 ms. (b) A description of the time evolution of the Rabi oscillation with a small ambient magnetic field in the SERF regime. The polarization ${\bf {P}}$ drastically decays to zero due to larger relaxation within an envelope time.
Fig. 13. (a) description of the dynamic process of the Rabi oscillation at different amplitude $B_x$ at the same time. The spiral curves are the trajectory of polarization ${\bf {P}}$, and the vortex cures are the projection of polarization ${\bf {P}}$ on the $x-y$ plane. (b) The Rabi oscillations at different amplitude $B_x$ with $B_0= 72~{\rm {nT}}$ at 423 K.
Fig. 14. Relationship between the amplitude $B_x$ of an oscillating field with a applied current in the $x$-axis. The data based on Eq. (4) is linear fit yielding ${k_x} = 75.55 \pm 0.78~{\rm {nT/mA}}$, similarly, ${k_y} = 151.5 \pm 0.9~{\rm {nT/mA}}$.
5. Conclusion
In this study, the spin dynamics of polarized atoms under an oscillating field and a static magnetic field in the SERF regime were studied. The FID signal is generated by applying an oscillating field for a moment under a static magnetic field, and the spin-relaxation rate ${R_{{\rm {rel}}}}$ can be accurately measured by fitting this signal in several milliseconds. This is a feasible method for measuring the large relaxation rate in a small cell volume. We also analyzed the relationship between the probe intensity and ambient magnetic field with the spin-relaxation rate ${R_{{\rm {rel}}}}$. Based on the dependence of the FID frequency on the applied current, the axial coil ($z$ axis) can be calibrated $in ~situ$. By applying a consecutive oscillating field and static magnetic field, the dynamic evolution of the Rabi oscillation can be detected by a linearly polarized probe beam. We theoretically and experimentally demonstrate that the amplitude of the Rabi oscillation is affected by the amplitude of the oscillating field in the SERF regime. The phenomenon of only one envelope of Rabi oscillation is explained. According to this method, the radial coil ($x$- and $y$-axes) constant can also be measured $in ~situ$.
Funding
Key Research and Development Program of Zhejiang, China under Grant (2020C01037); National Natural Science Foundation of China (51925501, 62003022); Beijing Municipal Natural Science Foundation (1222025, 4191002).
Disclosures
The authors declare no conflicts of interest related to this article.
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.
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