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Abstract
In order to investigate the influence of pump beam truncation on the transverse relaxation time of 129Xe nuclear spins in a nuclear magnetic resonance gyroscope, we measured the transverse relaxation rate as a function of pump laser power and vapor cell temperature for two ovens with different optical access sizes. The experimental results conform qualitatively to a theoretical model based on magnetic field gradient induced by polarization gradient of Rb atoms. It is found that the non-uniform power distribution in the beam cross-section led to remarkable relaxation, especially for the 129Xe nuclear spins in a large-size vapor cell. To reduce the polarization gradient, the spatial distribution of the pump laser power in the cell should be as homogenous as possible. These results are of significance to the design of a high precision nuclear magnetic resonance gyroscope.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Atomic sensors based on vapor cells have gained wide applications [1]. For example, atomic magnetometers with sensitivity below 1 fT/Hz1/2 have been demonstrated [2,3], which can be used in searching for new physics [4]. A nuclear magnetic resonance gyroscope (NMRG) with a physics package volume of 10 cm3 and an angular random walk of 0.005 deg/h1/2 has been demonstrated [5,6], which may cause a revolution in inertial field. These atomic sensors usually utilize lasers to manipulate and probe the quantum states of the atomic ensembles sealed in vapor cells. One problem in this process is the inhomogeneity of the spin polarization in space since it is difficult to obtain fully uniform beam profile. As a result, the performance of the atomic sensors such as signal amplitude, measurement accuracy could be degraded, as shown in recent studies on the beam inhomogeneity effect in atomic clocks, atomic magnetometers and NMRGs [7]. Here we are interested in the beam inhomogeneity effect in the NMRG.
The vapor cell in a typical NMRG contains both alkali-metal atoms and noble gas atoms. The nuclear spins of the noble gas atoms are used as inertial sensing unit, while the alkali-metal atoms are used as a medium to polarize and probe the state of nuclear spins. The influence of pump beam on spin polarization homogeneity has been studied in [7], which shows the 3-D distribution of electronic and nuclear spin polarization clearly. The study gives a very useful guide to optimize the pump beam diameter. But the sensitivity of the NMRG depends not only on spin polarization, but also on the transvers relaxation time [8]. The inhomogeneity of alkali-metal spin polarization will lead to magnetic field gradient, which could reduce the transverse relaxation time of nuclear spins [9]. The relaxation rate is proportional to the fourth-power of vapor cell size [9], so the effect is especially remarkable in NMRGs with large-size cells. Previous studies were mainly focused on the polarization gradient in the laser propagation direction caused by the light attenuation, while the polarization gradient due to laser beam inhomogeneity in cross-section has not been paid enough attention.
In this study, we investigated the influence of pump beam truncation on the relaxation time of 129Xe spins. We measured the relaxation time as a function of pump laser intensity and vapor cell temperature. A model was built to fit the experimental data, which shows the beam inhomogeneity will produce magnetic field gradient and reduces the transverse relaxation time. To address this issue, it had better make the power distribution of the pump laser in the cell as homogenous as possible.
2. Theoretical analysis
A configuration of a typical NMRG is shown in Fig. 1 [5,8]. The magnetic shield saturates the outer magnetic field and prevents the unwanted magnetic interference from the environment for the spins. At least two sets of coils provide the static magnetic field in the z axis and driving field in the x axis. A vapor cell is placed in an oven at the center of the magnetic field shield, which contains a droplet of isotope enriched alkali metal 87Rb, and gases such as N2, 129Xe and 131Xe. The cell can be heated by an AC current heater so as to vaporize the alkali metal. A pump laser which propagates through a linear polarizer and a 1/4 wave-plate is used to polarize the Rb atoms. A probe laser which propagates through a linear polarizer transverses the cell and impinges on a Wollaston prism and a balanced detector, so as to detect the polarization projection of the Rb atoms in the x axis.
The gyromagnetic ratio of the 131Xe nuclear spin is approximately one-third of that of the 129Xe nuclear spin and the 131Xe nuclear spin has strong quadrupolar interaction with the cell wall in a vapor cell without Rb hydrid coating [10,11]. Thus the 131Xe nuclear spins are not strongly affected by the magnetic field gradients due to the pump beam inhomogeneity. Therefore, here we only concern the influence of the pump beam inhomogeneity on the transverse relaxation time of 129Xe nuclear spins.
The evolution of nuclear spin polarization ${{\boldsymbol P}^n}$ can be described by the following Bloch Equation [7],
At constant temperature and pump laser parameters, the Rb polarization ${{\boldsymbol P}_{Rb}}$ in an NMRG is nearly a constant,
where ${P_{Rb}}$ is the magnitude of ${{\boldsymbol P}_{Rb}}$, and ${{\boldsymbol e}_z}$ is the unit vector in the z axis.Assuming at $t = 0$, ${{\boldsymbol P}^n} = P_{y0}^n{{\boldsymbol e}_y}$. At a constant magnetic field ${\boldsymbol B = }{B_0}{{\boldsymbol e}_z}$, the solution of Eq. (1) is,
Through recording the time evolution of $P_y^n$, we can obtain its decaying rate $R_{SE}^{} + \Gamma _2^n$. The spin exchange rate $R_{SE}^{}$ can be expressed as [12],
where $k_{SE}^{vdW}$ is the spin-exchange coefficient via van der Waals molecules, $k_{SE}^B \approx 2 \times {10^{ - 16}}$ cm3/s is the spin-exchange coefficient due to binary collision, and ${n_{Rb}}$ is the atomic number density of Rb vapor. According to [12], $k_{SE}^{vdW} = 1/\left( {\frac{{{n_{Xe}}}}{{{\eta_{Xe}}}} + \frac{{{n_{{N_2}}}}}{{{\eta_{{N_2}}}}}} \right)$, where ${n_{Xe}}$ and ${n_{{N_2}}}$ are the atomic number density of Xe and N2, ${\eta _{Xe}}$ = 5230 s−1, ${\eta _{{N_2}}}$ = 5700 s−1.In our case, the transverse relaxation rate $\Gamma _2^n$ can be denoted as,
where $\Gamma _{\nabla B}^n$ is the contribution from magnetic field gradient.According to [9], for a spherical cell at the operating condition of a Xe NMRG, $\Gamma _{\nabla B}^n$ can be expressed as,
where $R$ is the radius of the vapor cell, ${B_z}$ is the z component of magnetic field, and $D$ is the diffusion coefficient of 129Xe atoms. This equation is referred to a spherical cell, but we can use it as an estimate for a cubic cell by replacing R with a half of the side length.In a system with cylindrical symmetry, $\nabla {B_z}$ can be expressed as,
The polarized Rb spins produce magnetic field sensed by the Xe atoms, which can be expressed as [13,14],
where ${g_S} \approx 2$, ${\kappa _0} \approx 518$ is the enhancement factor for Rb-Xe spin interaction [15], ${\mu _0}$ is the permeability of free space, ${\mu _B}$ is the Bohr magneton.In our case, we can express magnetic field in the z direction as,
where ${B_0}$ is the magnetic field independent of Rb polarization, and $\alpha ={-} \frac{2}{3}{\kappa _0}{\mu _0}{g_S}{\mu _B}$.Assuming the Rb atoms are homogenously distributed in the vapor cell and the gradient of ${B_0}$ is negligible, we have
The Rb polarization ${P_{Rb}}$ can be expressed as [16,17],
where ${R_{SD}}$ is the relaxation rate of Rb spins and ${R_p}$ can be expressed as [16], where ${\Phi _0} = \frac{{{I_p}}}{{h{\nu _l}S}}$ is the photon flux across the cell, which is proportional to the power ${I_p}$ of the pump laser, $h$ is the Planck constant, ${\nu _l}$ is the pump laser frequency, $S = \pi {R^2}$ is the cross-section of the cell, ${\sigma _0} = {r_e}c{f_{D1}}$ is the total absorption cross section of Rb atoms, ${r_e}$ is the classic radius of the electron, $c$ is light speed in the vacuum, ${f_{D1}} \approx 1/3$ for the Rb D1 line. We use a Lorentzian function $L({{\nu_l} - {\nu_0}} )= \frac{{\Delta \Gamma /2}}{{{{({\Delta \Gamma /2} )}^2} + {{({{\nu_l} - {\nu_0}} )}^2}}}$ to approximately describe the absorption profile of Rb atoms, here $\Delta \Gamma $ is the linewidth of the Lorentzian profile, and ${\nu _0}$ is the center frequency of the absorption line.In fact, a laser with homogenous intensity distribution is difficult to obtain. Along the propagation direction, the laser is absorbed so its intensity decreases as it propagates [17]. And the cross-section distribution of a laser beam is typically a Gaussian profile. These factors make the polarization of Rb atoms inhomogeneous in the cell, although atoms diffusing across the vapor cell may smear out the inhomogeneity to some extent. The polarization distribution of Rb spins has been obtained in [7] with a simulation based on finite element method. According to their simulation, when the pump beam diameter or aperture is much smaller than the cell size, Rb polarization is extremely inhomogeneous spatially. The polarization gradient will produce a magnetic field gradient sensed by the Xe atoms. Next we will use an approximation to obtain an analytical expression.
With a linear estimate, the average radial magnetic field gradient can be expressed as,
and the longitudinal magnetic field gradient can be expressed as, where d is the side length of the cubic cell, ${B_{Rb}}(0 )$ is the magnetic field produced by the polarized Rb at the center-front side of cell and ${B_{Rb}}(d )$ at the center-back side of the cell.Substituting Eqs. (14) and (15) into Eq. (8), we have
From Eq. (9), we know ${B_{Rb}}$ is proportional to ${n_{Rb}}{P_{Rb}}$. So we can express ${|{\nabla {B_z}} |^2}$ as,3. Experimental results and discussion
The experiment was carried out on a cubic cell with 10 mm internal side length. It contains 2 Torr of 129Xe gas and 300 Torr of N2. The pump beam waist diameter is about 15 mm with a Gaussian profile. In order to compare the relaxation rate variation under different pump beam, we use two ovens with different hole-size to heat the vapor cell. Oven A has a hole with diameter 10 mm and oven B 15 mm. So oven A stops laser outside of the entrance hole and oven B does not. With each oven, we change the temperature of the vapor cell, and then measure the equivalent transverse relaxation time $T_2^\ast $ as a function of pump laser power with the free induction decay method.
From the above analysis, we know $T_2^\ast $ can be modeled as,
where $\Gamma _1^n$ is determined by the cell and nearly independent of temperature and pump laser power, $R_{SE}^{}$ depends on ${n_{Rb}}$ which varies strongly with temperature [18,19], $\Gamma _{\nabla B}^n$ depends on ${n_{Rb}}$ and the pump laser power.At each specific temperature, we use the following expression
to fit the experimental data, where u denotes laser power and f denotes $1/T_2^\ast $.Comparing Eq. (18) with Eq. (19), we have $a = R_{SE}^{} + \Gamma _1^n$, $b = \sqrt {\frac{\beta }{{{d^2}}}{\alpha ^2}n_{Rb}^2}$ and $c = \kappa {R_{SD}}$, here $\kappa$ is a constant for each oven. Since $R_{SE}^{} = ({k_{SE}^{vdW} + k_{SE}^B} ){n_{Rb}}$, we can further fit the data set $a \sim {n_{Rb}}$ with a linear function and the slope gives $({k_{SE}^{vdW} + k_{SE}^B} )$, the intercept gives $\Gamma _1^n$. We can check the validity of above model by fitting the data set $b \sim {n_{Rb}}$ with a proportional function, as b is proportional to ${n_{Rb}}$ in theoretical model above.
We measured the pump laser power after the linear polarizer in Fig. 1. Note it is not the power at the cell face, since there are propagation loss, reflection loss, and divergence loss. But now that the loss rate is a constant for a specific oven, these losses can be modeled in $\kappa$. Experimental data and corresponding fitting results are shown in Figs. 2 and 3. The fitting coefficients a, b and c are listed in Table 1. As shown in Table 1 and Fig. 3(b), the coefficient c obtained at three different temperatures shows only a little change and the fitting curve of b vs. Rb density is nearly a proportional function, implying the fitting model is a good approximation to the theoretical analysis. The fitting curves in Fig. 3(a) give $k_{SE}^{vdW} + k_{SE}^B$ and $\Gamma _1^n$, which are listed in Table 2. The fitting error of data by oven A in Fig. 3(a) is somewhat large. It may be due to variation of polarization gradient in the z axis with laser power.
Fig. 2. Measured relaxation rates as a function of pump power at different temperatures and corresponding fitting curves when using oven A (a) and oven B (b). In the legend, “Exp.” denotes experimental data and “Fit.” denotes fitting data.
Fig. 3. Fitting curves of (a) a vs. Rb density and (b) b vs. Rb density for ovens A and B. In the legend, the A and B after “Fit.” and “Exp.” denote oven A and B respectively.
Table 1. Fitting coefficients of data.a
Table 2. Spin exchange coefficients and longitudinal relaxation rates obtained with ovens A and B.
According to Table 2, $k_{SE}^{vdW} + k_{SE}^B$ and $\Gamma _1^n$ of the same cell obtained from the experiments with oven A are larger than those with oven B. It may be from a new “relaxation” mechanism as follows. In the vapor cell, the area without pump laser has a low Rb polarization. When Xe atoms diffuse in this area, their nuclear polarization will decrease. As a result, the apparent spin “relaxation” of nuclear spins is quickened further. Moreover, oven A has a small volume and the heating band is closer to the cell. The residual magnetic field and accompanying magnetic gradient may be larger, which may also contribute to spin relaxation. But as a whole, pump laser with a homogenous cross-section transmitting the cell shows lower relaxation. Note both $k_{SE}^{vdW} + k_{SE}^B$ and $\Gamma _1^n$ are temperature dependent, but it has not too much influence on the measured data of $k_{SE}^{vdW} + k_{SE}^B$ and $\Gamma _1^n$ in our case [19].
We can make a theoretical estimate for the relaxation rate due to polarization gradient. At ${n_{Rb}} = 1 \times {10^{13}}$ atoms/cm3 and ${P_{Rb}} = 0.5$, we have ${B_{Rb}}(0 )= 40.2$ nT according to Eq. (9). With the cell gas components, we have $D \approx 0.58$ cm2/s. For 129Xe, we know $\gamma _n^{} = 2\pi \times 11.86$ Hz/µT [21]. With $R = 0.5$ cm and $d = 1.0$ cm, we obtain $\Gamma _{\nabla B}^n = 0.18 \sim 0.35$ s−1. While in Fig. 2(a), the relaxation rate due to polarization gradient is approximately 0.07 s−1. In fact, the theoretical estimate is an overestimate. According to the simulation results in [7], the magnetic field gradient should be several times less than the estimate based on Eq. (17), since diffusion will reduce the Rb polarization gradient to much extent. In addition, using the Eq. (6) of the spherical cell and Eq. (7) of the cylindrical cell to estimate our cubic cell also leads to errors. The Gaussian beam will distort due to truncation by the optical access hole, making the light power distribution more complicated [22]. For the above reasons, the estimated $\Gamma _{\nabla B}^n$ is only consistent with the experimental result qualitatively. However, it proves that the transversal beam inhomogeneity leads to transverse relaxation.
As a comparison, we measured the transverse relaxation time of two spherical cells. The cells have a diameter of approximately 9 mm, which is smaller than the diameter of the oven optical access. We found the transverse relaxation time was much less dependent on pump laser power, as shown in Fig. 4. The slight change with temperature is due to factors such as residual polarization gradient of Rb atoms.
Fig. 4. Measured relaxation rate as a function of pump power for cells S1 and S2 using Oven A at 105°C. The Cells S1 and S2 are spherical with a radius of 5 mm.
In order to reduce $|{\partial {B_z}/\partial z} |$, optical pumping from both sides of the cell can be used [23]. To reduce $|{\partial {B_z}/\partial \rho } |$, the power in the cell cross-section should also be as homogenous as possible. In the design of an NMRG, we could use a large-size vapor cell to reduce the angular random walk, but from Eq. (6), we know the relaxation rate due to magnetic field gradient is proportional to the fourth power of the cell size [24]. As a result, the NMRG with a larger cell demands a more homogenous laser beam.
4. Conclusions
In summary, we measured the relaxation rate of the 129Xe nuclear spins as a function of temperature and pump laser power for two ovens with different optical access size. The inhomogeneity of the laser beam at the cell induces remarkable relaxation. The experimental results conform to the theory model based on magnetic field gradient induced by polarization gradient qualitatively. The relaxation rate due to magnetic field gradient is proportional to the fourth power of the cell size, so this relaxation is especially severe in an NMRG with larger cell. In order to reduce the transversal polarization gradient, the pump laser should have a uniform power distribution transversally.
Funding
National Natural Science Foundation of China (61671458); Natural Science Foundation of Hunan Province (2018JJ3608); Research Project of National University of Defense Technology (ZK17-02-04).
Acknowledgments
The authors would like to thank Dr. Xiang Zhan and Prof. Hui Luo for checking the writing, expressions, and data analysis of the paper.
Disclosures
The authors declare no conflicts of interest.
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