Optimized gas pressure of an Rb vapor cell in a single-beam SERF magnetometer

Yaxiang Wang; Ge Jin; Junjian Tang; Weiyong Zhou; Weiyong Zhou; Bangcheng Han; Bangcheng Han; BinQuan Zhou; BinQuan Zhou; BinQuan Zhou; Tao Shi; Tao Shi

doi:10.1364/OE.447456

1. Introduction

It has been experimentally discovered that a sufficient spin-exchange collision in alkali metal vapor can slow the Larmor frequency and narrow the magnetic resonance linewidth [1] proportionally to the density of the alkali metal vapor. In 1977, Happer and Tam [2] theoretically determined how spin-exchange affects magnetic resonance in alkali metal vapor. It was shown that spin-exchange broadening could be eliminated at a high alkali metal density of $10^{13}\;\textrm{cm}^{-3}$ and a low magnetic field. In 2002, Romalis et al. [3] first demonstrated the operation of a K magnetometer in a spin-exchange-relaxation-free (SERF) regime, achieving a sensitivity of $10~\textrm{fT}/\sqrt {\textrm{Hz}}$. After a few years, SERF magnetometers achieved a sensitivity of $0.16~\textrm{fT}/\sqrt {\textrm{Hz}}$ [4], surpassing that of superconducting quantum interference device (SQUID) magnetometers. Ultrasensitive magnetometers in the femtotesla range enable a number of new applications that were previously accessible only for SQUID magnetometers. Compared with a SQUID, which is expensively operated in liquid helium, SERF magnetometers are non-cryogenically cooled, commercially available, compact, and flexible [5,6]. Tens to hundreds of sensors can be easily placed around a subject to create a spatial map of the magnetic field. With these advantages, SERF magnetometers have been applied in magnetoencephalography (MEG) [7,8], magnetocardiography (MCG) [9], high-precision fundamental measurements [10], and other areas.

Unlike the conventional two-beam SERF magnetometer, a single-beam SERF magnetometer [11] uses a pump beam to polarize atoms, and the intensity of the light transmitted through the cell is used to detect the polarization of atoms. To achieve high-sensitivity performance of a single-beam magnetometer, two approaches should be considered simultaneously. One is narrowing the magnetic linewidth and increasing the transverse coherence time $T_{2}$, and the other is improving the signal-to-noise ratio, which maximizes the magnetometer response and suppresses the system noise. In the SERF regime, spin-exchange relaxation is eliminated, and the most important mechanism for spin relaxation is spin destruction collision between alkali atoms or with the buffer and quenching gases. When the cell contains only alkali atoms, i.e., $\textrm{Rb}$ vapor, at 423 $\textrm{K}$, without a buffer gas to inhibit the motion, the effect of wall depolarization becomes more pronounced in smaller cells, such as cells with dimensions of $3\times 3\times 3~\textrm{mm}^{3}$. The magnetic linewidth is at least $\Delta v=37~\textrm{kHz}$, such a broad linewidth can hinder high-sensitivity magnetometer performance. One method for suppressing wall depolarization is the use of buffer gas. Adding a buffer gas to the cell suppresses the wall collision rate but increases the spin destruction rate, so it is necessary to find the optimal buffer gas pressure. To improve the signal-to-noise ratio, the pumping light intensity is optimized, and the magnetic field is modulated, which shifts the detected signal to a higher frequency $\Omega _{\textrm{mod}}$ without the need for an external Faraday modulator or similar techniques. In this study, we constructed seven $^{87}$Rb vapor cells containing different N$_2$ pressures. We used the same technique as that presented in Ref. [12], measured the pressure broadening of the $D_{1}$ line, and determined the pressure of N$_2$ in the cells. We also measured magnetic linewidths using the same method as that described in Ref. [13] and the sensitivity as described in Ref. [14] in seven cells. By comparing the magnetic linewidths and magnetometer sensitivities of seven cells, we theoretically and experimentally demonstrated the optimal gas pressure, optimal pumping light intensity, and modulation parameter in Pyrex cells (inner dimension $3\times 3\times 3~\textrm{mm}^{3}$).

2. Theoretical analysis

In the SERF regime, the evolution of alkali metal atom spin is described by the Bloch equation [13]:

(1)$${{d{\boldsymbol{P}} \over {dt}}} = \gamma {\boldsymbol{P}} \times {\boldsymbol{B}} + {1 \over q}\left( {{R_{\text{OP}}}\left( {s\boldsymbol{\hat{z}} - {\boldsymbol{P}}} \right) - {R_{\text{SD}}}{\boldsymbol{P}}} \right),$$

where the gyromagnetic ratio $\gamma =\gamma ^{\textrm{e}}/(2I+1)$, $\gamma ^{\textrm{e}}$ is the gyromagnetic ratio of the bare electron, $q$ is the slow down factor, $\boldsymbol {P}$ is the atomic polarization, $s=1$ represents polarization of the pumped light along the $z$-axis, $R_{\textrm{OP}}$ is the optical pumping rate, and $R_{\textrm{SD}}$ is the spin destruction rate. When a cell only contains $\textrm{Rb}$ vapor and nitrogen gas in the SERF regime, the spin exchange rate is neglected. Thus, $R_{\textrm{SD}}$ is contributed to by three terms, as presented previously [15]: the Rb-Rb spin destruction rate $R_{\textrm{SD}}^{\textrm{Rb}}=n_{\textrm{Rb}} \bar {v}_{\textrm{Rb}}\sigma _{\textrm{Rb-Rb}}$, the Rb-N$_2$ spin destruction rate $R_{\textrm{SD}}^{\textrm {N}_{2}}=n_{\textrm {N}_{2}}\bar {v}_{\textrm {N}_{2}}\sigma _{\textrm {Rb}\textrm{-}\textrm {N}_{2}}$, and the wall collision rate $R_{\textrm{wall}}$, where $\sigma$ is the binary collision cross section coefficient, $n_{\textrm{N}_{2}}$ is the density of Rb vapor [16], $\bar {v}$ is the relative thermal velocity, and $R_{\textrm{wall}}$ is relaxation rate due to diffusion to the cell walls [17].

The pump beam intensity is attenuated by the absorption of the alkali vapor; therefore, a position-dependent distribution of the pumping rate is achieved by using the Lambert-W function [18,19]:

(2)$$R_{\text{OP}}(z)=R_{\text{SD}}W[\frac{R_{\text{OP}}(0)}{R_{\text{SD}}}\exp({-}n_{\textrm{Rb}}\sigma(v)z+\frac{R_{\text{OP}}(0)}{R_{\text{SD}}})],$$

$R_{\textrm{OP}}(0)$ is the pump rate at the entrance window of the vapor cell, which is expressed as

(3)$$R_{\text{OP}}(0)=\kappa \frac{I_{0}cr_{e}f}{hvA}\frac{\Gamma_{L} / 2}{\left(v-v_{0}\right)^{2}+\left(\Gamma_{L} / 2\right)^{2}},$$

where $I_{0}$ is the pump beam power, $A$ is the pump beam cross section, $\kappa$ is a revised factor when simply considered the spatially dependent effect of a Gaussian beam [20], $c$ is the speed of light, $r_{e}$ is the classic electron radius, $f$ is the oscillator strength (roughly 1/3 for line $D_{1}$ and 2/3 for line $D_{2}$), $h$ is the Planck constant, and $\Gamma _{L}$ is the full width at half maximum of the optical transition of the frequency $v_{0}$, which is proportional to the gas pressure.

Considering the non-uniform distribution of the pump rate, an average pumping rate can be expressed by summing $R_{\textrm{OP}}(z)$ along the $z$-axis and dividing the result by the cell length $L$, and the numerical solution of the average pumping rate was simplified to the following form:

(4)$$\overline{R_{\text{OP}}}=\frac{1}{L}\sum_{i=0}^{n}R_{\text{SD}}W[\frac{R_{\text{OP}}(0)}{R_{\text{SD}}}\exp({-}n_{\textrm{Rb}}\sigma(v)i\Delta z+\frac{R_{\text{OP}}(0)}{R_{\text{SD}}})]\Delta z,$$

where $n$ is a number of segments in cell length $L$, and $\Delta z$ is the segment length.

The cell only contained $^{87}$Rb and nitrogen gas. The atomic polarization relaxation was mainly contributed to by two terms: the optical pumping rate $R_{\textrm{OP}}$ and the spin destruction rate $R_{\textrm{SD}}$, namely,

(5)$$R_{\text{rel}}=\overline{R_{\text{OP}}}+n_{\text{Rb}} \bar{v}_{\text{Rb}}\sigma_{\text{Rb-Rb}}+n_{\text{N}_{2}}\bar{v}_{\text{N}_{2}}\sigma_{\text{Rb-N}_{2}}+3q D_{0}\left(T_{0}\right)\left(\frac{T}{T_{0}}\right)^{3 / 2}\left(\frac{n_{\text{amg}}}{n_{\text{N}_{2}}}\right)\left(\frac{\pi}{L}\right)^{2},$$

where $D_{0}$ is the diffusion constant of Rb within the nitrogen gas, and $L$ is the cubic cell length. $n_{\textrm{amg}}=2.46\times 10^{19}/\textrm{cm}^{3}$ is density of gas at ${p_0}$ = 1 amg and ${T_0}$ = 333.15 K. For $^{87}$Rb vapor, the slow down factor $q=(6+2P^{2})/(1+2P^{2})$ [21].

We measured the spin relaxation using a zero-field resonance technique. After the magnetic field in the shield was nullified and only a static magnetic field $B_{y}$ along the $y$-axis was applied, a dispersive Lorentzian function, given by Eq. (1), can be simplified to the following form [13,22]:

(6)$$P_{x}=\frac{-\Omega_{y} R_{\text{OP}}}{\Omega_{y}^{2}+(R_{\text{OP}}+R_{\text{SD}})^{2}},$$

where $\Omega _{y}=\gamma _{e} B_{y}$. The half-width of the Lorentzian dispersive curve is the spin relaxation rate $R_{\textrm{rel}}=R_{\textrm{OP}}+R_{\textrm{SD}}$.

We now address the solution of the Bloch equation in x-mode. The components of $\mathbf {B}$ are $B_{x}=B_{0}+B_{1}\cos \left (\omega t\right )+B_{\textrm{mod}}\cos \left (\Omega _{\textrm{mod}}t\right )$, $B_{y}=0$, and $B_{z}=0$. 74$B_{0}$ is the static magnetic field, $B_{1}\cos \left (\omega t\right )$ is the field to be measured, and $B_{\textrm{mod}}\cos \left (\Omega _{\textrm{mod}}t\right )$ is a modulation field along the $x$-axis. When the pumping light is propagating along the $z$-axis. The transverse component of the polarization can be written in the complex form $\tilde {P}=P_{y}+i P_ {z}$, and then the transverse Bloch equation is given in the form of

(7)$$\frac{d \tilde{P}}{d t}={-}i\left(\omega_{0}+\omega_{1} \cos (\omega t)+\omega_{\text{mod} } \cos \left(\Omega_{\text{mod }} t\right)\right) \tilde{P}-\frac{\tilde{P}}{\tau}+\frac{i P_{0}}{\tau},$$

where $\omega _{0}=\gamma B_{0}$, $\omega _{1}=\gamma B_{1}$, $\omega _{\textrm{mod }}=\gamma B_{\textrm{mod }}$, and $\tau =q/\left (R_{\textrm{OP}}+R_{\textrm{SD}}\right )$.

We used a previously described method [23,24] to solve Eq. (7) with test solution $\tilde {P}\left (t\right )$

(8)$$\tilde{P}\left(t\right)=\lambda e^{{-}t / \tau} e^{{-}i \omega_{0} t} e^{{-}i\beta_{1}\sin (\omega t)} e^{{-}i\beta_{\text{mod}} \sin \left(\Omega_{\text{mod }} t\right)},\\$$

where $\beta _{1}=\omega _{1}/\omega$, and $\beta _{\textrm{mod}}=\omega _{\textrm{mod}}/\Omega _{\textrm{mod }}$. $\tilde {P}(t)$ is then simplified as

(9)$$\tilde{P}(t) =i P_{0} \sum_{m={-}\infty}^{\infty} \sum_{n={-}\infty}^{\infty} \sum_{p={-}\infty}^{\infty} \sum_{q={-}\infty}^{\infty} J_{m}\left(\beta_{1}\right) J_{n}\left(\beta_{1}\right) J_{p}\left(\beta_{\text{mod}}\right) J_{q}\left(\beta_{\text{mod}}\right) \frac{e^{i(m-n)\omega t} e^{i(p-q) \Omega_{\text{mod}} t}}{1+i\left(\omega_{0}+m \omega+p \Omega_{\text{mod}}\right) \tau}.$$

When $\omega _{0}=0$, $\tau \omega \ll 1$, and the second-order component of $\tau \omega$ is negligible. In particular, $P_{z}$ at the zero- and first-order modulation frequency $\Omega _{\textrm{mod}}$ is in the form of

(10)$$P_{z} \approx P_{0} J_{0}\left(\beta_{\textrm{mod}}\right)^{2}-4 P_{0} J_{0}(\beta_{1})J_{1}\left(\beta_{1}\right) J_{0}\left(\beta_{\text{mod}}\right) J_{1}\left(\beta_{\text{mod}}\right) \cos (\omega t) \sin \left(\Omega_{\text{mod}}t\right)\frac{\tau }{{{\tau ^2}{\omega ^2} + 1}},$$

where $J_{0}$ and $J_{1}$ are Bessel functions of the first kind.

After performing a proper approximation, the polarization $P_{z}$ is given in the form of

(11)$$P_{z} \approx P_{0}\left(1-\frac{\beta_{\textrm{mod}}^{2}}{2}\right)- P_{0} \beta_{\textrm{mod}} \left(-\gamma B_{1} \cos (\omega t)\right) \sin \left(\Omega_{\textrm{mod}}t\right)\frac{\tau }{{{\tau ^2}{\omega ^2} + 1}}.$$

When the circularly polarized pump beam passes through the cell, the transmitted light intensity implies information about $P_{z}$. Therefore, the response of the x-mode magnetometer is directly obtained by monitoring the pump intensity behind the cell, which can be expressed as [25]

(12)$$S \approx KI_{0} e^{{-}n \sigma(v)l\left[1-\left(P_{z}\right)\right]}=K I_{0} e^{{-}n \sigma(v) l\left(1-P_{z}^{0}\right)} e^{n \sigma(v) l P_{z}^{t}},$$

where $K$ is a conversion factor between the light intensity and voltage, and $I_{0}$ is the pumped beam intensity. $P_{z}^{0}$ and $P_{z}^{t}$ are the time-independent and time-dependent components of $P_{z}$. Because $P_{z}^{t}$ is much smaller than 1, we can expand Eq. (12) to the first order in $P_{z}^{t}$:

(13)$$S \approx K I_{0} e^{{-}n \sigma(v) l\left(1-P_{z}^{0}\right)}\left[1+n\sigma(v) l P_{z}^{t}\right].$$

The magnetometer response signal consisted a direct current (DC) component $S_{0}$ and an alternating current (AC) component $S(t)$. Demodulated at a frequency $\Omega _{\textrm{mod}}$ and passed through a low-pass filter with a bandwidth of 200 Hz, the response signal only contained the AC component $S (t)$. We defined $x=n\sigma (v)l$, and $S(t)$ is reduced to

(14)$$S (t)= K I_{0} e^{{-}x} x e^{x P_{0}\left(1-\frac{\beta_{\textrm{mod}}^{2}}{2}\right)} P_{0} \beta_{\textrm{mod}}\gamma B_{1} \cos (\omega t)\frac{\tau }{{{\tau ^2}{\omega ^2} + 1}}.$$

3. Experimental results and discussion

3.1 Experimental setup and procedure

A schematic of the experimental setup used in studies of the spin relaxation rate and sensitivity is shown in Fig. 1. Seven cells containing $^{87}$Rb and different pressures of nitrogen were used in the experiment. The cell was placed in a boron nitride ceramic oven and heated by a resistive heater using AC power at 200 kHz. The temperature of the cell was maintained at 150$^{\circ }$C to obtain a sufficient signal-to-noise ratio. The temperature of the cell was stable to 0.01$^{\circ }$C and varied less than 1$^{\circ }$C across the cell. A PT1000 resistor pasted on the inner wall of the oven provided a real-time monitor of the cell temperature, so that the density of $^{87}$Rb was $n_{\textrm {Rb}}=1.2\times 10^{14}~\textrm{cm}^{-3}$ [16]. The cell, oven, and three-axis Helmholtz coils were placed in a four-layer permalloy magnetic shield. Three-axis Helmholtz coils were separately driven by function generators that zeroed the residual magnetic field inside the shield before operation. The circularly polarized pump beam propagated along the $z$-axis and was tuned on the resonance line $D_{1}$ (the exact tuning was chosen to minimize light shift). The transmitted intensity of the pump beam was monitored by photodetectors PD0.

Fig. 1. Schematic of the experimental setup for measurements of the spin relaxation rate and sensitivity. A circularly polarized pump beam propagating along the $z$-axis was used to polarize rubidium atoms, and the transmitted light intensity was monitored by a photodetector PD0. A linearly polarized probe beam was tuned on the off-resonance line $D_{2}$ propagating along the $x$-axis. The optical rotation of the probe beam was detected by a pair of photodiodes PD1 and PD2. LP denotes a linearly polarized plate, QWP denotes a quarter-wave plate, PD denotes a photodiode, PBS denotes a polarized beam splitter, LIA denotes a lock-in amplifier, and the shutter is used to turn off the probe light when measuring the sensitivity in single-beam-configuration magnetometer.

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To investigate the spin relaxation rate by the zero-field resonance, the linearly polarized probe beam along the $x$-axis was detuned by 100 GHz (where the signal was a maximum) from the $D_{2}$ resonance line, and the probe light intensity was $7~\textrm{mW}/\textrm{cm}^{2}$. The pumping beam and probe beam were about 2.7 mm in diameter. Circular birefringence of the medium proportional to $P_{x}$ rotated the polarization of the probe beam, which was analyzed after the cell with a balanced polarimeter consisting of two photodetectors, PD1 and PD2. The optical rotation of the probe beam was measured as a function of the static field $B_{y}$.

To study the sensitivity of the x-mode magnetometer in a single-beam configuration, we turned off the probe beam by a shutter, as shown in Fig. 1. The intensity variation of the circularly polarized pumping light transmitted through the vapor cell was detected by a photodiode, which sent the signal to the lock-in amplifier. An analog adder applied the modulation field and calibration field to a triaxial coil system synchronously. A transimpedance amplifier amplified the photocurrent induced by the photodiode PD0. A commercial computer-controlled digital lock-in amplifier was used as a demodulator at a frequency of $\Omega _{\textrm{mod}}$ to obtain the response of the x-mode magnetometer, as shown in Eq. (12), for an applied weak magnetic field.

3.2 Optical depth and density of nitrogen gas

We used the optical absorption profile of line $D_{1}$ to determine the nitrogen pressure in seven cells. The principle of operation and the experimental setup were similar those described in a previous publication [12]. We obtained the full linewidths $\Gamma$ and shifts $\delta$ of line $D_{1}$ at different gas pressures of N$_2$. According to previously reported experimental data [12,26], N$_2$ contributed a 17.8 GHz/amg width to the $D_{1}$ line at 273 K, and the width was independent of temperature. Thus, we can determine the density of N$_2$ in our cells. The linewidths, optical depths, and densities of N$_2$ are listed in Table 1.

Table 1. Full widths at half maximum (HWHM) $\Gamma$, densities of nitrogen $n_{\textrm {N}_{2}}$, and optical depths (OD) in different cells. Temperature of 150$^{\circ }$C.

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3.3 Zero-field resonance measurements of spin relaxation rate

For a typical cell, e.g., No. 3, we set the pumping light intensity to different values and measured the magnetic optical rotation of the probe beam ($7~\textrm{mW}/\textrm{cm}^{2}$) by sweeping $B_{y}$ from $-80$ nT to $80$ nT. The data for different pumping intensities are well described by a Lorentzian curve, as shown in Fig. 2. The half-width at half maximum (HWHM) of the Lorentzian curve was $\Delta v= R_{\textrm{rel}}/2\pi$. We plotted the HWHM $\Delta v$ of cell No. 3 (triangles) as a function of pumping light intensity.

Fig. 2. Optical rotation signal of probe beam as a function of $B_{y}$ at five different pumping light intensities for cell No. 3. The HWHM of the Lorentzian curve corresponded to the spin relaxation rate $R_{\textrm{rel}}$. For these data, the probe intensity was $7~\textrm{mW}/\textrm{cm}^{2}$.

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The same process described above was repeated, and we measured the spin relaxation rates $R_{\textrm{rel}}$ of other cells at different pumping light intensities, as shown in Fig. 3. The intercepts and slopes of the curves in Fig. 3 correspond to the $R_{\textrm{SD}}$ and the pump rate $\overline {R_{\textrm{OP}}}$, respectively.

Fig. 3. Measurement of the spin relaxation rate $R_{\textrm{rel}}$ as a function of the incident intensity of the pump beam in seven Rb vapor cells under different nitrogen gas pressures.

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For a given value of the pumping light intensity, we are interested in which spin relaxation rate $R_{\textrm{rel}}$ of the cell is a minimum. We selected the experimental data shown in Fig. 3 and plotted the $R_{\textrm{rel}}$ values for all seven cells as functions of the nitrogen pressure at four pumping light intensities, as shown in Fig. 4(a), (b), (c), and (d), respectively. The N$_2$ pressure unit was amg. In Fig. 4(a), dots represent the experimental $R_{\textrm{SD}}$ data. To ensure agreement with the experimental data, we varied the parameter values, such as the cross section, from previously published values, as shown in Table 2. The numerical calculation using Eq. (5) illustrated that the minimum spin destruction rate $R_{\textrm{SD}}^{\textrm{cal}}=130~\textrm{Hz}$, corresponding to a minimum N$_2$ pressure of $1.26~ \textrm{amg}$. This can be compared to the experimental data which showed that the minimum spin destruction rate was $R_{\textrm{SD}}^{\textrm{m}}=121~\textrm{Hz}$, corresponding to an N$_2$ pressure of $1.37~ \textrm{amg}$.

Fig. 4. Measurement of the spin relaxation rate $R_{\textrm{rel}}$ as a function of the density of N$_2$ in seven Rb vapor cells at four pumping light intensities. The pumping light intensity was (a) $I=0\;\textrm{mW}/\textrm{cm}^{2}$, (b) $I=4\;\textrm{mW}/\textrm{cm}^{2}$, (c) $I=11\;\textrm{mW}/\textrm{cm}^{2}$, and (d) $26\;\textrm{mW}/\textrm{cm}^{2}$. The dots represent experimental data, and the solid lines served to guide the eye.

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Table 2. Parameters for all calculations.

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As the pumping light intensity increased, the spin relaxation rate became larger, and the pressure of N$_2$ corresponding to the minimum spin relaxation rate $R_{\textrm{rel}}$ also changed. When the light intensity was $4~\textrm{mW}/\textrm{cm}^{2}$, the experimental data in Fig. 4(b) showed that the optimal pressure of N$_2$ was $1.75~\textrm{amg}$, corresponding to a minimum spin relaxation rate $R_{\textrm{rel}}^{\textrm{m}}=213~\textrm{Hz}$, which was consistent with the numerical calculation of the spin relaxation rate $R_{\textrm{rel}}^{\textrm{cal}}=211~\textrm{Hz}$. When the pumping light intensity was $11~\textrm{mW}/\textrm{cm}^{2}$, experimental data in Fig. 4(c) shows that the optimal pressure of N$_2$ was $2.43~\textrm{amg}$, which corresponded to the minimum spin relaxation rate $R_{\textrm{rel}}^{\textrm{m}}=459~\textrm{Hz}$, and the numerically calculated spin relaxation rate was $R_{\textrm{rel}}^{\textrm{cal}}=468~\textrm{Hz}$. When the pumping light intensity was $26~\textrm{mW}/\textrm{cm}^{2}$, Fig. 4(d) shows that optimal pressure of N$_2$ was $2.82~\textrm{amg}$, which corresponded to a minimum spin relaxation rate of $R_{\textrm{rel}}^{\textrm{m}}=826~\textrm{Hz}$, and the numerically calculated spin relaxation rate was $R_{\textrm{rel}}^{\textrm{m}}=922~\textrm{Hz}$. When the pump intensity was below $5~\textrm{mW}/\textrm{cm}^{2}$, the spin relaxation rates $R_{\textrm{rel}}$ in each cell varied nonlinearly with the pumping light intensity, which is described by Eq. (4). The numerical calculation result was reasonably consistent with the experimental data in Fig. 4, but some disagreements were found from the uncertainty of the parameters, such as the spin destruction cross section, diffusion constant, and gas pressure, which were determined by the optical absorption of the $D_{1}$ line. At a high pumping intensity ($I\geqslant 5~\textrm{mW}/\textrm{cm}^{2}$), $R_{\textrm{rel}}$ was approximately proportional to the pumping light intensity. Our experimental data for the pump rates at low gas pressures, such as cells No. 1 and No. 2, were approximately a factor of $2$ lower than the pumping rates calculated from Eq. (4). This discrepancy mainly occurred because in the average model (Eq. (4)), the spatially dependent effect of the Gaussian beam and magnetometer response are neglected [20], and the pumping light intensity was simply modified by a factor of $\kappa =0.55$ when we calculated $R_{\textrm{OP}}(0)$.

To estimate the pump rate when we numerically simulated the magnetometer response, we used the approximation $\overline {R_{\textrm{OP}}}=\eta I_{\textrm{pump}}$. Overlaying the high-power data in Fig. 3 is a linear fit based on Eq. (5), with $\overline {R_{\textrm{OP}}}=\eta I_{\textrm{pump}}$ yielding the $\eta$ values of the seven cells. We plotted slopes of the pump rate $\eta$ versus the pressure of N$_2$, as shown in Fig. 5. $\eta$ is inversely proportional to the pressure of N$_2$. In addition, the alkali density calculated from our experimental data was approximately a factor of $0.7$ lower than the density of saturated Rb vapor at the corresponding temperature. The same phenomenon was found previously [29], and it is common in Pyrex cells, which is likely due to the slow reaction between alkali atoms with the glass walls.

Fig. 5. Measurement of pump rate slope $\eta$ as a function of N$_2$ pressure.

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3.4 Sensitivity

We used the basic idea from Eq. (14) to determine the optimal pumping light intensity, which was selected to obtain the maximized signal. The basic idea was as follows:

After nullifying the static magnetic field in the shield, we applied a small calibrating magnetic field $B_{1}$ and a modulation field $B_{\textrm{mod}}$ in the $x$-direction. The amplitude of the calibration magnetic field was 100 pTrms, and the frequency was 33.7 Hz, which generated a sinusoidal signal at the output of the lock-in amplifier, whose amplitude $V_{1}$ is described by Eq. (14). The amplitude of the modulated magnetic field was 80 nT at a frequency of 773 Hz to ensure $\beta _{\textrm{mod}}=0.47$ (where the signal was maximized) [30]. Next, we maintained the oscillating magnetic field, but we varied the pumping light intensity at different values. Our measurement results are shown in Fig. 6(a). The response of the magnetometer was enhanced with the increase in the pumping light intensity. When the light intensity was larger than $15~\textrm{mW}/\textrm{cm}^{2}$, the magnetometer response signal of every cell approached a plateau. Thus, a large pumping light intensity only caused a small variation of the magnetometer response signal $S(t)$ but sharply increased the Schottky noise, which was proportional to the root of the total intensity of light transmitted through the cell [31]. The signal-to-noise ratio (S/N) can be approximately written in a simple form:

(15)$$S/N \propto \frac{{S(t)}}{{\sqrt {{I_0}} }} \approx k\sqrt {{I_0}} {e^{ - x}}x{e^{x{P_0}\left( {1 - \frac{{\beta _{{\text{mod}}}^2}}{2}} \right)}}{P_0}{\beta _{{\text{mod}}}}\gamma {B_1}\frac{\tau }{{{\tau ^2}{\omega ^2} + 1}},$$

where $S(t)$ is the magnetometer response signal, $I_{0}$ is the pumping light intensity, and $k$ is the calibration factor.

Fig. 6. (a) Measurement of single-beam magnetometer response amplitudes $S(t)$ as a function of pumping light intensity in seven cells under different nitrogen pressures. (b) Measurement of magnetometer signal-to-noise ratio $S/N$ as a function of the incident intensity of the pump beam in seven cells under different nitrogen pressures.

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We divided the amplitude shown in Fig. 6(a) by the square root of the pumping light intensity, and we plotted S/N as a function of the pumping light intensity, as shown in Fig. 6(b). The plateau in Fig. 6(a) corresponded to the maximum S/N shown in Fig. 6(b). The S/N values of cells No. 3, No. 4, No. 5, and No. 6 were much closer to each other, and the value of No. 4 was the maximum. Because large optical depths decreased the transmitted light significantly, the S/N of No. 1 was much smaller than those of the other cells.

We also used Eq. (14) to maximize S/N values based on our experimental data. The S/N obtained by numerical simulation (see Eq. (15)) are shown in Fig. 7. The curves in Fig. 7 illustrated the same result as that shown in Fig. 6(b). Thus, the optimal pumping light intensity was around $11~\textrm{mW}/\textrm{cm}^{2}$ when we measured the magnetic field sensitivity.

Fig. 7. Numerical simulation of magnetometer signal-to-noise ratio $S/N$ as a function of the incident intensity of the pumping beam in seven cells under different nitrogen pressures.

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According to spin projection noise [3,13], the fundamental sensitivity per unit bandwidth was

(16)$$\delta B=\frac{1}{\gamma \sqrt{n_{\textrm{Rb}}\tau Vt}},$$

where $n_{\textrm {Rb}}$ is the density of Rb atoms, $\gamma =g_{s}\mu _{B}/q$ is the gyromagnetic ratio, $\tau =q/R_{\textrm{rel}}$ is the transverse spin relaxation time, $q$ is the slow down factor, $V$ is the cell volume, and $t$ is the measurement time. The sensitivity is inversely proportional to the spin relaxation rate. We combined Eq. (16) and the spin relaxation rates shown in Fig. 4(c), and we speculated that the sensitivity of No. 6 was the maximum corresponding to the minimum spin relaxation rate in the seven cells. However, from the experimental sensitivity data (Fig. 8), we observed that the sensitivity of No. 4 was the highest $8.89~\textrm{fT}/\sqrt {\textrm{Hz}}$, but the spin relaxation rate $R_{\textrm{rel}}=460~\textrm{Hz}$ was larger than that of No. 6. The reason for this atypical behavior was somewhat subtle, so we go into some detail below.

Fig. 8. Sensitivities, or the magnetic noise spectra, of the seven cells as functions of the measurement frequency based on calibration peaks with amplitudes of 100 pTrms applied at 33.7 Hz. For these data, the pumping light intensity was $11~\textrm{mW}/\textrm{cm}^{2}$. These data were obtained by recording the response of the magnetometer for about 500 s, and the sampling rate was 1000 Hz.

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The optical depths of the seven cells were large under our experimental conditions, which required a high laser power to serve as the pump and probe light in the single-beam magnetometer. A high pumping light intensity increased the magnetometer response signal, as shown in Fig. 6, which significantly limited the effect of the technical noise. Thus, we could improve the S/N of the magnetometer by increasing the pumping light intensity. Even when the pumping light intensity was set to $40~\textrm{mW}/\textrm{cm}^{2}$, the sensitivity due to spin-projection noise was smaller than $5~\textrm{fT}/\sqrt {\textrm{Hz}}$. Consequently, for a single-beam magnetometer, S/N became a dominant factor, instead of the spin-projection noise, which limited the magnetic field sensitivity. The results in Fig. 6(b) and Fig. 7 showed that the sensitivity matched up well with S/N. A high S/N corresponded to a high magnetic field sensitivity. For example, the highest sensitivity occurred for cell No. 4, with a value of $8.89~\textrm{fT}/\sqrt {\textrm{Hz}}$, corresponding to the highest S/N at a pumping light intensity $I_{0}=11~\textrm{mW}/\textrm{cm}^{2}$. In contrast, the lowest sensitivity occurred for cell No. 1, with a value of $18.89~\textrm{fT}/\sqrt {\textrm{Hz}}$, corresponding the lowest S/N at a pumping light intensity of $I_{0}=11~\textrm{mW}/\textrm{cm}^{2}$. The signal-to-noise ratios of the seven cells were highest when the pumping light intensity was in the interval of 9–15 $\textrm{mW}/\textrm{cm}^{2}$. The values of the magnetic field sensitivity $\delta B_{\textrm{scn}}$ of cells No. 3, No. 4, No. 5, and No. 6 were much closer.

As mentioned above, we found that the optimal pumping light intensity was in the interval of 9–15 $\textrm{mW}/\textrm{cm}^{2}$ corresponding to an optimal N$_2$ pressure ranging from 1.2 to 2.2 amg.

The magnetometer frequency response was measured by applying a known oscillating $B_{x}$ field at several frequencies. The frequency response depended on the pump rate and spin relaxation rate, which can be well described by a first-order low-pass filter [14,30,32,33]. A fit for the expected amplitude response function is

(17)$$\frac{A}{\sqrt{(f-B)^{2}+f_{\text{3dB}}^{2}}},$$

where $f$ is the frequency, and $f_{\textrm{3dB}}$ is the 3-dB bandwidth. We measured the bandwidths of the seven cells at a pumping light intensity of $I=11~\textrm{mW}/\textrm{cm}^{2}$ (the exact intensity was selected to obtain the maximized signal), as shown in Fig. 9. We found that the bandwidths of the seven cells corresponded to the spin relaxation rates. For example, the maximized bandwidth of cell No. 1 was 143 Hz, corresponding to the maximized spin relaxation rate of 650 Hz. The minimized bandwidth of No. 6 was 104 Hz, corresponding to the minimized spin relaxation rate of 450 Hz. The bandwidth may be increased further by increasing the spin relaxation rate. Both increasing the pumping light intensity and the gas pressure could increase the bandwidth of the magnetometer when we implemented them in the MEG experiment. The comparison of the bandwidths and sensitivities of cells No. 4 and No. 7 showed that the high pressure in the cell could be a better method to increase the bandwidth without significantly sacrificing the sensitivity.

Fig. 9. Normalized frequency responses of the seven cells operated in an x-mode magnetometer as a function of frequency. The maximized bandwidth of No. 1 was 143 Hz, corresponding to the maximized spin relaxation rate of 650 Hz. The minimized bandwidth of No. 6 was 104 Hz, corresponding to the minimized spin relaxation rate of 450 Hz. For these data, the pumping light intensity was $11~\textrm{mW}/\textrm{cm}^{2}$.

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4. Conclusion

In this study, we systematically analyzed the optimal nitrogen pressure of a single-beam magnetometer operated in the SERF regime to achieve a high magnetic sensitivity. We measured the spin relaxation rates of cells under different nitrogen pressures by the zero-field resonance method. The spin relaxation rate is a key parameter to limit the magnetic sensitivity. However, in a single-beam magnetometer, the pumping light intensity becomes another key factor in the improvement of the magnetometer performance. Based on our theoretical analysis and experimental results, we determined that the optimal pumping light intensity was in the interval of 9–15 $\textrm{mW}/\textrm{cm}^{2}$ and the N$_2$ optimal pressure was in the interval of 1.2 to 2.2 amg. A sensitivity of $8.89~\textrm{fT}/\sqrt {\textrm{Hz}}$ was achieved using the optimal parameters in the experiments, thus significantly improving on previous sensitivities and opening the possibility for further miniaturization of such cells. The analytical approach and experimental methods developed in this paper can be easily adapted to another alkali atom by modifying the coefficients in Eq. (14).

Funding

National Key Research and Development Program of China (2018YFB2002405); Key Research and Development Program of Zhejiang Province (2020C01037).

Disclosures

The authors declare that there are 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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Number of cells	No. 1	No. 2	No. 3	No. 4	No. 5	No. 6	No. 7
$Γ$ (GHz)	10.31	16.82	24.51	28.27	38.82	43.28	50.26
$n_{N_{2}}$ (amg)	0.58	0.94	1.38	1.59	2.18	2.43	2.82
OD (at 423 K)	14.64	9.12	6.38	5.22	4.11	3.69	3.03

Parameters	Value or expression	This work
Spin-destruction cross section of Rb-Rb: $σ_{Rb-Rb}^{SD}$	$1.6 \times 10^{- 17} {cm}^{2}$ [3]	$(1.8 \pm 0.5) \times 10^{- 17} {cm}^{2}$
Spin-exchange cross section of Rb-Rb: $σ_{Rb-Rb}^{SE}$	$1.9 \times 10^{- 14} {cm}^{2}$ [3]	$1.9 \times 10^{- 14} {cm}^{2}$
Spin-destruction cross section of Rb-N $_{2}$ : $σ_{{Rb-N}_{2}}^{SD}$	$1.5 \times 10^{- 22} {cm}^{2}$ [27]	$(1.7 \pm 0.2) \times 10^{- 22} {cm}^{2}$
Diffusion coefficient of Rb in N $_{2}$ : $D (T)$	$0.16 \times (\frac{T}{333 K})^{\frac{3}{2}} (\frac{n_{amg}}{n_{N_{2}}}) {cm}^{2} / s$ [28]	$(0.16 \pm 0.03) \times (\frac{T}{333 K})^{\frac{3}{2}} (\frac{n_{amg}}{n_{N_{2}}}) {cm}^{2} / s$
Saturated density of Rb at temperature (Klevin): $n_{Rb}$	$\frac{1}{T} 10^{21.866 + 4.857 - 4215 / T} / {cm}^{3}$ [16]	$0.7 \times \frac{1}{T} 10^{21.866 + 4.857 - 4215 / T} / {cm}^{3}$
Pressure broadening width of Rb $D_{1}$ line in N $_{2}$ : $Γ^{N_{2}}$	$17.8 \pm 0.2 GHz / amg$ [12]	$17.8 \pm 0.2 GHz / amg$

Number of cells	No. 1	No. 2	No. 3	No. 4	No. 5	No. 6	No. 7
$Γ$ (GHz)	10.31	16.82	24.51	28.27	38.82	43.28	50.26
$n_{N_{2}}$ (amg)	0.58	0.94	1.38	1.59	2.18	2.43	2.82
OD (at 423 K)	14.64	9.12	6.38	5.22	4.11	3.69	3.03

Parameters	Value or expression	This work
Spin-destruction cross section of Rb-Rb: $σ_{Rb-Rb}^{SD}$	$1.6 \times 10^{- 17} {cm}^{2}$ [3]	$(1.8 \pm 0.5) \times 10^{- 17} {cm}^{2}$
Spin-exchange cross section of Rb-Rb: $σ_{Rb-Rb}^{SE}$	$1.9 \times 10^{- 14} {cm}^{2}$ [3]	$1.9 \times 10^{- 14} {cm}^{2}$
Spin-destruction cross section of Rb-N $_{2}$ : $σ_{{Rb-N}_{2}}^{SD}$	$1.5 \times 10^{- 22} {cm}^{2}$ [27]	$(1.7 \pm 0.2) \times 10^{- 22} {cm}^{2}$
Diffusion coefficient of Rb in N $_{2}$ : $D (T)$	$0.16 \times (\frac{T}{333 K})^{\frac{3}{2}} (\frac{n_{amg}}{n_{N_{2}}}) {cm}^{2} / s$ [28]	$(0.16 \pm 0.03) \times (\frac{T}{333 K})^{\frac{3}{2}} (\frac{n_{amg}}{n_{N_{2}}}) {cm}^{2} / s$
Saturated density of Rb at temperature (Klevin): $n_{Rb}$	$\frac{1}{T} 10^{21.866 + 4.857 - 4215 / T} / {cm}^{3}$ [16]	$0.7 \times \frac{1}{T} 10^{21.866 + 4.857 - 4215 / T} / {cm}^{3}$
Pressure broadening width of Rb $D_{1}$ line in N $_{2}$ : $Γ^{N_{2}}$	$17.8 \pm 0.2 GHz / amg$ [12]	$17.8 \pm 0.2 GHz / amg$

Optimized gas pressure of an Rb vapor cell in a single-beam SERF magnetometer

Abstract

1. Introduction

2. Theoretical analysis

3. Experimental results and discussion

3.1 Experimental setup and procedure

3.2 Optical depth and density of nitrogen gas

3.3 Zero-field resonance measurements of spin relaxation rate

3.4 Sensitivity

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (9)

Tables (2)

Equations (17)

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