1. Introduction

Atomic spin comagnetometers using optically pumped alkali metal atoms to polarize the noble gas atoms and detect their precession have undergone remarkable development [1–3]. A long coherence time of the optically polarized alkali metal atoms contributes to improve the sensitivity of the comagnetometers, which can be realized by operating the alkali metal atoms in the spin-exchange relaxation-free (SERF) regime [4]. Benefiting from the ultrahigh sensitivity, the SERF comagnetometer has attracted widespread attention and found significant applications in the fundamental physics experiments, such as the tests of Lorentz and CPT symmetries [5,6] and the searches for exotic spin-dependent interactions [7,8]. Furthermore, the SERF comagnetometer is regarded as a sensitive gyroscope for applying in next-generation inertial navigation [9,10].

The SERF comagnetometer was first presented by Romalis group based on the K-$^3$He coupled ensemble in 2005 [1]. Afterwards, in order to improve the sensitivity of the comagnetometers, $^{21}$Ne atoms were widely used due to the smaller gyromagnetic ratio than $^3$He atoms [5,11,12]. Moreover, with the development of the hybrid optical pumping [13–15], the low-noise magnetic shield [16,17] and the closed-loop control of compensation point in the SERF comagnetometer [18], our group has realized a rotation sensitivity of 2.1$\times$10$^{-8}$ rad s$^{-1}$Hz$^{-1/2}$ and a bias drift of $10^{-2}$ deg/h in a K-Rb-$^{21}$Ne comagnetometer [19–21]. Nonetheless, the incompletely suppressed spin-exchange relaxation caused by the Fermi-contact field of the electron spins has been proved to exist in the K-Rb-$^{21}$Ne comagnetometer, which is an important limiting factor to improve the sensitivity of the K-Rb-$^{21}$Ne comagnetometer and has yet been solved [22,23]. In the recent years, the spin-exchange relaxation in a finite magnetic field has been suppressed by the pulse modulation technique in the atomic magnetometer and the nuclear magnetic resonance (NMR) comagnetometer [24–28]. In addition, the pulse modulation technique has been adopted in the NMR comagnetometer by Walker group to suppress the fluctuation from the longitudinal polarization and promote the long-term stability of the comagnetometer [29]. However, as a particularly promising strategy for improving the performance of the atomic comagnetometer, the pulse modulation technique has not been introduced in the SERF comagnetometer. The lack of the specific pulse response model limits the application of the pulse modulation technique in the SERF comagnetometer. Therefore, the investigation on pulse response of the SERF comagnetometer is of great significance.

In this paper, we study the pulse response of a SERF comagnetometer considering the dynamics of the coupled spin ensemble of alkali metal and noble gas. The pulse response model describing the response along $x$-, $y$- and $z$-axis of the comagnetometer during and after the pulse is established for the first time. In order to verify the model, a novel three-beam comagnetometer is created with a circularly and two linearly polarized lasers to measure the pulse response in three axes of the comagnetometer simultaneously and independently. The pulse responses with small and large excitations are studied in detail in the comagnetometer. The theory and method presented here lay a foundation for the application of the pulse modulation technology in the SERF comagnetometer.

2. Theoretical model of the pulse response

The dynamics of the coupled spin ensemble of alkali metal and noble gas in the SERF comagnetometer can be described by a set of Bloch equations [1]. In order to obtain homogeneous polarization of the electron spins, the hybrid optical pumping technique is adopted in the K-Rb-$^{21}$Ne comagnetometer. In general, the number density of the K atoms is much smaller than that of the Rb atoms, so that the dynamics of the hybrid alkali-metal atoms are dominated by the Rb atoms [30]. The evolutions of the electron-spin polarization ${{\bf {P}}^e}$ and the nuclear-spin polarization ${{\bf {P}}^n}$ can be described as follows,

Here the first term represents the precession of the electron spins and nuclear spins. ${\gamma _e} = 28{\rm {Hz/nT}}$ and ${\gamma _n} = 3.36 \times {10^{ - 3}}{\rm {Hz/nT}}$ are the gyromagnetic ratios of the electron spins and nuclear spins, respectively. $Q$ is the slowing-down factor, which depends on the nuclear spin $I$ and the polarization ${{\bf {P}}^e}$ [4]. ${\bf {B}}$ is the ambient magnetic field. ${\rm {\lambda }}{M^e}{P^e}$ and ${\rm {\lambda }}{M^n}{P^n}$ are the Fermi-contact fields of the electron spins and nuclear spins. $\lambda {\rm {\ =\ }}{{8\pi {\kappa _0}} \mathord {\left /{\vphantom {{8\pi {\kappa _0}} 3}} \right. } 3}$ is the geometrical factor, where ${\kappa _0}$ is the enhancement factor. ${{M^e}}$ and ${{M^n}}$ are the magnetizations of electron spins and nuclear spins corresponding to full spin polarization. ${\bf {L}}$ is the light shift from the pump and probe lasers. The second term accounts for the evolution of rotation. ${\bf {\Omega }}$ is the input inertial rotation. The third term describes the relaxation rates of the electron spins and nuclear spins. $R_{tot}^e$ and $R_{tot}^n$ are the total relaxation rates of electron spins and nuclear spins, which are defined as $R_{tot}^e = {R_p} + {R_m} + R_{se}^{en} + R_{sd}^e$ and $R_{tot}^n = R_{se}^{ne} + R_{sd}^n$, respectively. ${{R_p}}$ and ${{R_m}}$ are the pumping rates of pump and porbe light, while ${{{s}^p}}$ and ${{{s}^m}}$ represent the degree of circular polarization. ${R_{se}^{en}}$ is the spin-exchange rate from nuclear spins to electron spins and ${R_{se}^{ne}}$ is the spin-exchange rate from electron spins to nuclear spins. $R_{sd}^e$ and $R_{sd}^n$ are the spin-destruction rates of electron spins and nuclear spins, respectively.

Based on the Bloch equations, the pulse response of the comagnetometer is systematically investigated. Considering that the magnetic fields experienced by the coupled ensemble during and after the pulse are different, the pulse response can be divided into the responses during and after the pulse to study. The pulsed magnetic field is usually applied in a direction perpendicular to the pump laser [24–27]. Without loss of generality, the pulse field ${B_y}$ is applied along the $y$-axis of the comagnetometer. In order to operate the comagnetometer in a self-compensating regime, where the slow changes in the ambient magnetic field can be automatically compensated by the nuclear magnetization, the longitudinal magnetic field is set to the compensation magnetic field ${B_z} = - \lambda {M^e}P_z^{e0} - \lambda {M^n}P_z^{n0}$. Here $P_z^{e0}$ and $P_z^{n0}$ are the equilibrium polarizations of the electron spins and nuclear spins before the pulse, respectively. Since the light shift ${{L_z}}$ can be zeroed by tuning the frequency of the laser to atomic resonance line or can be compensated by the longitudinal magnetic field [31,32], the value of ${{L_z}}$ is set to zero in the following analysis.

We first analyze the response in the pulse duration $t_1$. The dynamics of the electron spins and nuclear spins can be deemed as decoupled when operating the comagnetometer in a large external magnetic field [33]. Consequently, the responses of the electron spins and nuclear spins are almost independent during the pulse. In addition, the alkali metal atoms in the comagnetometer can be regarded as an integrated in-situ magnetometer to read out the effective magnetic field produced by the nuclear spins. Therefore, the response of the comagnetometer during the pulse can be understood as the response of the electron spins and the response of the nuclear spins read out by the electron spins. The behavior of the nuclear spins during the pulse can be obtained by solving the Bloch equation of the nuclear spins in Eq. (1). Since the longitudinal magnetic field ${B_z} \gg \left | {\lambda {M^e}P_z^{e0}} \right |$, the effective magnetic field experienced by the nuclear spins during the pulse can be approximated as $\sqrt {{B_z}^2 + {B_y}^2}$. The three components of the nuclear-spin polarization in the $x$-, $y$- and $z$-axis of the comagnetometer during the pulse are obtained and shown below,

According to Eq. (2), the magnetic fields generated by the nuclear spins through the Fermi-contact interaction in the three axes of the comagnetometer are $B_n^x = \lambda {M^n}P_x^n\left ( {{t_1}} \right )$, $B_n^y = \lambda {M^n}P_y^n\left ( {{t_1}} \right )$ and $B_n^z = \lambda {M^n}P_z^n\left ( {{t_1}} \right )$, so the effective magnetic fields experienced by the electron spins are $B_n^x$, ${B_y^t}={B_y} + B_n^y$ and ${B_z^t}={B_z} + B_n^z$ during the pulse. The total relaxation rate of the electron spins during the pulse is defined as $R_{tot}^{ep} = R_{tot}^e + {R_{se}^p}$. ${R_{se}^p}$ is the spin exchange relaxation of the electron spins caused by the pulse, which can be calculated according to the Ref. [23]. For simplicity, we define ${{R_{tot}^{ep}} \mathord {\left / {\vphantom {{R_{tot}^{ep}} Q}} \right. } Q}$ as $\tilde R_{tot}^{ep}$, ${{{\gamma _e}} \mathord {\left / {\vphantom {{{\gamma _e}} Q}} \right. } Q}$ as ${\tilde \gamma _e}$ and ${{{R_p}} \mathord {\left / {\vphantom {{{R_p}} Q}} \right. } Q}$ as ${\tilde R_p}$. Since the electron spins are not operated in SERF regime with the effect of the large pulsed magnetic field, the value of the $Q$ is $2I+1$, where $I$ is the nuclear spin of the alkali metal atoms. In addition, since $R_{se}^{en} \ll {R_p}$, the relaxation rate ${R_{se}^{en}}$ can be safely ignored in this period. As a result, the responses in the three axes of the comagnetometer in the pulse duration $t_1$ can be solved and shown in Eq. (3). Here ${D_1}\!=\!\left [ {{{\left ( {{\gamma _e}B_1} \right )}^2}\!+\! R{{_{tot}^{ep}}^2}} \right ]$, ${D_2}\!=\!\left [ {{\gamma _e}^2R_{tot}^{ep}\left ( {B{{_n^x}^2}\!+\!B{{_z^t}^2}\!+\!B{{_y^t}^2}} \right )\!+\!R{{_{tot}^{ep}}^3}} \right ]$ and $B_1 = \sqrt {{B_e}^2\!+\!{B_y}^2}$. The tipping angles of the nuclear spins with the small and large excitations are different, leading to the difference of the magnetic fields experienced by the electron spins. Thus, the dynamics of the responses with the small and large excitations are discussed respectively. For the small excitation, the tipping angle of nuclear spins is so small that the transverse components of the magnetic fields generated by the nuclear spins are approximately 0. In this case, only the electron spins oscillate during the pulse, which is expressed in the first term in Eq. (3). The second term in Eq. (3) is the steady-state value of the electron spins during the pulse in the case of the small excitation. Different from the response with small excitation, the response consists of two oscillations with different frequencies for the large excitation. The faster oscillation is response of the electron spins to the pulse, which is shown in the first term in Eq. (3). The second term in Eq. (3) is the oscillation of the nuclear spins reading by the electron spins, whose frequency is much slower than that of the electron spins.

After the pulse, the electron spins and nuclear spins precess back to the $z$-axis. Since the electron spins are directly polarized via the pump laser, the electron spins are rapidly polarized along the $z$-axis when the pulse disappears, and the process usually lasts for tens of milliseconds. Whereas the time of the nuclear spins back to $z$-axis is much longer, which is on the order of seconds. For convenience, the response after the pulse is divided into the fast oscillation process with duration of $t_2$ and the slow oscillation process with duration of $t_3$. During the fast oscillation process, the nuclear spins can be considered to remain in the state at the end of the pulse. Thus, the magnetic fields experienced by the electron spins along the $x$-, $y$- and $z$-axis are ${B_n^x}$, ${B_n^y}$ and ${B_z^t}$. The initial values of the electron-spin polarization in this period along the three axes are ${P_x^{ep}}$, ${P_y^{ep}}$ and ${P_z^{ep}}$, which depend on the tipping angles of the nuclear spins and the electron spins during the pulse shown in Eq. (2) and Eq. (3). Here the slowing-down factor $Q$ is a function of the electron-spin polarization. In order to obtain the analytical solution to describe the response in $t_2$, the value of Q is treated as a constant to represent the average effect of the varied slowing-down factor in the $t_2$. The responses of the fast oscillation process are solved and shown as follows,

Here ${D_3} = {\gamma _e}R_{tot}^e{B_2}^3$ and ${B_2} = \sqrt {B{{_n^x}^2} + B{{_n^y}^2} + B{{_z^t}^2}}$. For the small pulse excitation, the magnetic fields ${B_n^x}$ and ${B_n^y}$ are about 0 owing to a small tipping angle of the nuclear spins during the pulse. Hence the second terms of $P_x^e\left ( {{t_2}} \right )$ and $P_y^e\left ( {{t_2}} \right )$ in Eq. (4) are 0 and $P_z^e\left ( {{t_2}} \right )$ is ${e^{ - \tilde R_{tot}^e{t_2}}}\left ( {P_z^{ep} - P_z^{e0}} \right ) + P_z^{e0}$, which implies that the transverse components of the electron-spin polarization decay to 0 and the longitudinal polarization recovers to $P_z^{e0}$ in $t_2$. However, with the increase of the tipping angle of the nuclear spins during the pulse, the influence of the magnetic fields produced by the nuclear spins on the behavior of the electron spins in $t_2$ increases. For the large pulse excitation, the electron spins precess in the combined magnetic field of the longitudinal magnetic field and the magnetic fields produced by the nuclear spins, resulting in nonzero transverse components of the electron-spin polarization when the electron spins reach the steady state in $t_2$.

Then, the nuclear spins rotate back along the $z$-axis by interacting with the electron spins in the period $t_3$. In this period, the electron spins and nuclear spins precess in the longitudinal magnetic field and the Fermi-contact field generated by each other. For the small pulse excitation, the longitudinal polarization components of electron spins and nuclear spins and the slowing-down factor $Q$ during this period are regarded as constants to linearize the Bloch equations [34], and the response of the comagnetometer in the period $t_3$ is obtained in Eq. (5) by solving the linearized Bloch equations. Therein, ${\omega _e} = {\tilde \gamma _e}\left ( {{B_z} + {B_n}} \right )$ and ${\omega _n} = {\gamma _n}\left ( {{B_z} + {B_e}} \right )$ are the precession frequencies of the electron spins and nuclear spins, respectively. However, for the large pulse excitation, the transverse components of the response in $t_3$ no longer monotonously decay and the longitudinal polarization components of electron spins and nuclear spins cannot be regarded as constants. In this case, the dynamics of the comagnetometer is much more complicated and hard to describe by analytical solution. The dynamics of the comagnetometer for this case are studied and analysed experimentally.

In order to illustrate the pulse response of the comagnetometer more clearly, the evolution of the coupled ensemble following a ${\pi \mathord {\left / {\vphantom {\pi 2}} \right. } 2}$ pulse excitation of the electron spins is intuitively explained and shown in Fig. 1. When the pulse is input along the $y$-axis, the electron spins respond to the pulse and are tipped along the $x$-axis while the nuclear spins deviate from the $z$-axis a extremely small angle, as shown in Fig. 1(a), corresponding to the dynamics described in Eq. (3). After the pulse, there are the fast oscillation process with duration of tens of milliseconds and the slow oscillation process with duration of several seconds. The fast oscillation process is shown in Fig. 1(b), which is described by Eq. (4). The electron spins rapidly precess back to the $z$-axis with the effect of the pump laser. Meanwhile, the nuclear spins remain almost in the state at the end of the pulse. Then, the electron spins and nuclear spins precess back to the $z$-axis in the longitudinal magnetic field and the Fermi-contact field generated by each other in the slow oscillation process, as shown in Fig. 1(c). The dynamics in the slow oscillation process is described in Eq. (5). After the fast and slow oscillation processes, the electron-spin polarization and the nuclear-spin polarization recover to their equilibrium polarizatuions before the pulse.

 

Fig. 1. Intuitive explanation of the pulse response of the coupled ensemble. (a) In the pulse duration $t_1$, the electron spins are tipped along the $x$ axis, the nuclear spins deviate from the $z$ axis a extreme small angle. (b) The electron spins precess rapidly back to the $z$ axis with the effect of the pump laser after the pulse in $t_2$. (c) The coupled ensemble of the electron spins and the nuclear spins rotate back to the $z$ axis.

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3. Experimental setup

The pulse response of the coupled ensemble is investigated in a novel three-beam SERF comagnetometer, which is created with a circularly polarized and two linearly polarized lasers and shown in Fig. 2. An 8-mm-diameter spherical GE180 aluminosilicate glass cell is in the center of the comagnetometer. The cell contains a droplet of natural abundance of Rb atoms and a small amount of K atoms, 60 torr ${N_2}$ for quenching and 2 atm $^{21}$Ne noble gas (70% isotope enriched) for providing the nuclear spins. A circularly polarized laser (labeled Laser1 in Fig. 2) is used to pump the alkali metal and noble gas along $z$-axis and probe the electron-spin polarization in $z$-axis at the same time. The pump laser is generated by a distributed Bragg reflector (DBR) diode laser and expanded by a set of beam expander (BE) to illuminate the whole cell. The wavelength of the pump laser is centered on the K D1 resonance line to polarize the K atoms. The Rb atoms are polarized by spin-exchange collisions with the K atoms, and they hyperpolarize the $^{21}$Ne atoms together. Meanwhile, the electron-spin polarization in $z$-axis is measured by the absorption of the circularly polarized laser [35], which is detected by the photodetector (PD1). Two linearly polarized lasers (labeled Laser2 and Laser3 in Fig. 2) are formed by splitting the laser emitted from distributed feedback (DFB) diode laser through a beam splitter, and propagate along $x$- and $y$-axis to measure the electron-spin polarization in $x$- and $y$-axis, respectively. Different from the detecting of the circularly polarized laser, the linearly polarized lasers sense the electron-spin polarization using Faraday effect [36]. The polarizations in $x$- and $y$-axis are extracted from the difference signals by the balanced photodetectors such as the combination of PD2 and PD3. The output signals of the comagnetometer are recorded by a data acquisition(DAQ) system. Therefore, the pulse response along the $x$-, $y$- and $z$-axis can be detected in the novel three-beam SERF comagnetometer simultaneously and independently. In addition, the power fluctuations of the pump and probe laser are suppressed by a homemade laser intensity controller.

 

Fig. 2. Schematic of the three-beam SERF comagnetometer. Laser1: the circularly polarized laser propagates along $z$-axis to pump the alkali metal and noble gas and detect the electron-spin polarization in $z$-axis. Laser2: the linearly polarized laser propagates along $x$-axis to detect the electron-spin polarization in $x$-axis. Laser3: the linearly polarized laser propagates along $y$-axis to detect the electron-spin polarization in $y$-axis. BE, beam expander; P, linear polarizer; LVCR, liquid crystal variable retarder; GT, Glan-Taylor polarizer; M, reflection mirror; PBS, polarizing beamsplitter; PD, photodetector; DAQ, data acquisition system;

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The high number density and low magnetic field are two essential conditions to operate the alkali metal atoms in the SERF regime [1]. On the one hand, the cell is placed in a boron nitride ceramic oven and heated to 170 $^\circ \mathrm {C}$ using a twisted-pair winding resistor, which is driven by a homemade 199-kHz ac electrical heater for suppressing the magnetic noise [37]. At this cell temperature, the number density of Rb atoms is about $2.60 \times {10^{14}}$ cm$^{-3}$ and the density ratio of K to Rb is approximately 1:190. On the other hand, the oven with cell is surrounded by the magnetic shielding system, which consists of two layers of high-permeability $\mu$-metal cylindrical magnetic shields and an inner MnZn ferrite barrel, for shielding the environment magnetic field. The three-axis magnetic coil system driven by the function generator is used to cancel the residual magnetic fields and produce the pulsed magnetic field inside the magnetic shielding system.

4. Results and discussion

The coupled ensemble of the K-Rb-$^{21}$Ne comagnetometer is stably polarized with the continuous pumping along the $z$ axis. The residual magnetic fields are canceled and the compensation point is found to be 82 nT. The responses in the $x$-, $y$- and $z$-axis of the comagnetometer following a pulsed magnetic field with amplitude of 249 nT and duration of 2 ms are investigated, which corresponds to the case of the small pulse excitation. The pulsed magnetic field is input on the $y$-axis of the comagnetometer. The responses of the comagnetometer in the pulse duration $t_1$ are measured and shown in Fig. 3(a), and the effect of the pulsed magnetic field on the nuclear-spin polarization in this period is simulated by Eq. (2) and shown in Fig. 3(b). The transverse components of the nuclear-spin polarization are magnified and shown in the inset, which shows that the change of the nuclear-spin polarization is on the order of 10$^{-4}$. Thus, the variation of the nuclear spins is so small that the effect of the nuclear spins on the responses during the pulse shown in Fig. 3(a) can be neglected. In Fig. 3(a), the responses in the $x$- and $y$-axis of the comagnetometer are plotted on the left $Y$-axis bottom $X$-axis and the response in the $z$-axis of the comagnetometer is plotted on the right $Y$-axis bottom $X$-axis. The experimental data are fitted by Eq. (3). Here the second terms of Eq. (3) is regarded as a constant. The fitting curves are represented as the solid lines, whose R-squares are 99.99% with 95% confidence bounds. The tipping angle of electron spins increases with the increase of pulse duration. When the pulse duration is 1.07 ms, the tipping angle of the electron spins is $2\pi$. Moreover, the attainable maximum amplitudes of the responses decay exponentially with the increase of pulse duration, which is caused by the relaxation of the electron spins predicted by Eq. (3).

 

Fig. 3. The responses of the comagnetometer and the simulation of the nuclear-spin polarization during the pulse in the case of the small excitation. (a) The responses of the comagnetometer in the three axes of the comagnetometer during the pulse. The solid lines are fitting curves based on Eq. (3). (b) Simulation of the variation of the nuclear-spin polarization in the comagnetometer during the pulse based on Eq. (2).

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After the small pulse excitation, the responses consist of the fast and slow oscillation processes. The fast oscillation process with duration of $t_2$ of the comagnetometer is measured and shown in Fig. 4. The solid lines are the fitting curves based on Eq. (4). As we can see, the components of the electron-spin polarization in the $x$- and $y$-axis of the comagnetometer oscillate and decay to about 0, and the polarization in the $z$-axis of the comagnetometer exponentially increases and gradually approaches the equilibrium polarization before the pulse. The experimental results verify the theory for the small pulse excitation described in Eq. (4).

 

Fig. 4. The fast oscillation process of the comagnetometer in the three axes of the comagnetometer after the pulse in Fig. 3. The solid lines are fitting curves based on Eq. (4).

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Then, the slow oscillation process with duration of $t_3$ are recorded and shown in Fig. 5. The solid lines are the fitting curves based on Eq. (5). The experimental data show that the electron-spin polarization in the $z$-axis of the comagnetometer returns to the equilibrium polarization and hardly fluctuates during the process, which verifies the hypothesis of Eq. (5) that the longitudinal polarizations of the electron spins and the nuclear spins and the slowing-down factor $Q$ can be regarded as constants for the small pulse excitation. Meanwhile, the electron-spin polarization in $x$- and $y$-axis decay to 0, which is in good agreement with Eq. (5).

 

Fig. 5. The slow oscillation process of the comagnetometer in the three axes of the comagnetometer after the pulse in Fig. 3. The solid lines are fitting curves based on Eq. (5).

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Moreover, in order to study the pulse response with large excitation, the response of the K-Rb-$^{21}$Ne comagnetometer following a pulsed magnetic field with amplitude of 249 nT and duration of 3 s is investigated. The responses in the $x$-, $y$- and $z$-axis of the comagnetometer during the pulse are shown in Fig. 6(a). As predicted by Eq. (3), the pulse response with large excitation includes the fast oscillation term and the slow oscillation term, whose dynamics are determined by the electron spins and nuclear spins, respectively. The response for the first 6 ms of Fig. 6(a) corresponds to the fast oscillation term, which is enlarged and shown in Fig. 6(b). The dynamics of the electron spins in Fig. 6(b) is similar with that in in Fig. 3(a). With the increase of the pulse duration, the electron-spin polarization reaches equilibrium with the effects of the pulsed magnetic field and the magnetic field produced by the nuclear spins. Subsequently, the nuclear spins oscillate with the effect of the pulsed magnetic field. The precession of the nuclear spins is read by the electron spin serving as the in-situ magnetometer and show in Fig. 6(a). The experimental data in Fig. 6(a) and Fig. 6(b) are well fitted by Eq. (3).

 

Fig. 6. The responses during the pulse with the large pulse excitation. The solid lines are the fitting curves based on Eq. (3). (a) The responses in the three axes of the comagnetometer during the pulse. (b) The magnification of the first 6 ms in Fig. 6(a).

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Similar with the response with the small pulse excitation, there are also the fast oscillation process and the slow oscillation process after the large pulse excitation. The fast oscillation process is shown in Fig. 7. Compared with the responses in Fig. 4, the dynamics of electron spins are more obviously affected by magnetic fields produced by the nuclear spins. The electron spins precess with the effects of the longitudinal magnetic field and the magnetic field generated by the nuclear spin in this period. In Fig. 4, the tipping angle of the nuclear spins is so small that the electron spins precess around the magnetic field along $z$-aixs. Whereas, in Fig. 7, the tipping angle of the nuclear spins is so large that the electron spins precess around the combined magnetic field of the longitudinal magnetic field and the magnetic field produced by the nuclear spins through the Fermi-contact interaction. Therefore, the steady-state value of the electron-spin polarization in Fig. 7 deviates far from the equilibrium polarization before the pulse. The responses in Fig. 7 are fitted by Eq. (4). There is a little deviation between the experimental results and the fitting curve, which is caused by the varied slowing-down factor $Q$ but has little effect on the overall trend of the response.

 

Fig. 7. The responses of the fast oscillation process after the large pulse excitation. The solid lines are fitting curves based on Eq. (4).

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Then, the slow oscillation process after the large pulse excitation is shown in Fig. 8, and the novel dynamics of the coupled ensemble are observed here. The response in the $y$-axis of the comagnetometer is similar with that in the $x$-axis in this period. For simplicity and clarity, the evolutions of the transverse components of the polarization of the coupled ensemble are reflected by the response in the $x$-axis in Fig. 8. Compared with the response shown in Fig. 5, where the transverse components of the electron-spin polarization decay monotonically to the equilibrium polarization, the response show in Fig. 8 is much more complicated. For describing the evolution more clearly, the response in Fig. 8 can be divided into three periods: P1, P2 and P3. In the P1 period, the oscillation amplitude of response in the transverse component increases with time while that of the longitudinal component decreases with time. This is because after the large pulse excitation, the longitudinal component of the nuclear-spin polarization is reversed along the negative $z$-axis. With the effect of the longitudinal magnetic field and the polarizing by the electron spins, the transverse components of the nuclear-spin polarization increase with time, so that the electron spins experience the transverse magnetic fields increasing with time through the Fermi-contact interaction during P1 period. Then, a damped oscillation is observed in the P2 and P3 periods, which is caused by the precessing of the nuclear spins back to $z$-axis under the influence of the longitudinal magnetic field and the interaction with electron spins. We find the precession frequency of the transverse component in period P3 is significantly slower than that in the other two periods. This is because that in the P1 and P2 periods, the dynamics of nuclear spins and electron spins are decoupled due to the large angle between electron spins and nuclear spins, the precession frequency of the nuclear spins is ${\gamma _n}{B_z}$. Nevertheless, the angle between electron spins and nuclear spins is small enough that the dynamics of nuclear spins and electron spins are coupled in the P3 period. The evolution of the electron spins and nuclear spins in the P3 period is similar with that in Fig. 5, and the precession frequency of the nuclear spins can be represented as $\nu _{n}$ shown in Eq. (5), which is smaller than the value of the ${\gamma _n}{B_z}$. Consequently, the precession frequency in the P3 period is obviously slowed down. Finally, the transverse component of the electron-spin polarization decays to 0 and the longitudinal component of the electron-spin polarization return to the equilibrium polarization before the pulse.

 

Fig. 8. The responses of the slow oscillation process after the large pulse excitation.

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From the above discussion, the validity of the pulse response model of the SERF comagnetometer is verified by the experimental results of the three-beam comagnetometer with the small and large excitation. For the small excitation, the nuclear spins are approximately in equilibrium so that only the electron spins oscillate during the pulse and the dynamics of the coupled ensemble is determined by that of the electron spins. The result is suitable for the response when the tipping angle of the electron spins is in the range of 0 to $2\pi$, which is a typical operating range of the electron spins in the pulse modulation method [24]. Hence the pulse response model lays a solid foundation for the application of the pulse modulation technology in the SERF comagnetometer. For the large excitation, the response includes two oscillations with different frequencies, which are determined by the electron spins and the nuclear spins, respectively. The novel dynamics of the coupled ensemble are observed after the large pulse excitation that the oscillation amplitude of response after the pulse first increases and then decreases, and the oscillation frequency is slowed down when the coupled ensemble approach the equilibrium. The novel dynamics of great significance for the investigation on the dynamics of coupled ensemble in the SERF comagnetometer.

5. Conclusion

In conclusion, we have investigated the magnetic pulse response of the optically pumped comagnetometer operated in the SERF regime. We have established the pulse response model for the first time to describe the evolution of the coupled spin ensemble of alkali metal and noble gas in $x$-, $y$- and $z$-axis of the comagnetometer during and after the pulse. According to the model, the pulse responses with small and large excitations have been studied experimentally in a three-beam SERF comagnetometer, which has been created with a circularly and two linearly polarized lasers to detect the pulse response in the three axes of the comagnetometer simultaneously and independently. The results indicate that for the response to the small excitation, the nuclear spins are approximately in equilibrium that the dynamics of the coupled spin ensemble is determined by that of the electron spins. Whereas the response to the large excitation and both responses after the pulse include a fast and a slow oscillation, which are dominated by the electron spins and nuclear spins, respectively. Novel dynamics have been observed when the nuclear spins are tipped far away from the equilibrium. The theory and method presented here not only facilitate the investigation on the dynamics of the optically pumped coupled spin ensemble, but also pave the way for the application of the pulse modulation technology in the SERF comagnetometer to suppress the spin-exchange relaxation of the electron spins cuased by the Fermi-contact field and realize the closed-loop control of the electron-spin polarization.