Electric quadrupole shifts of the precession frequencies of ¹³¹Xe atoms in rectangular cells

A noble-gas comagnetometer consists of two kinds of atoms, whose nuclear spin precession frequencies are inter-compared to cancel any drifts and noises due to changes in the bias magnetic field. In a hybrid system mixed with alkali atoms, noble-gas atoms are polarized Bouchiat et al. (1960); Walker and Happer (1997) and probed Grover (1978) through their interactions with alkali atoms. Such systems have been widely used in precision measurements of both fundamental physics Bear et al. (2000); Rosenberry and Chupp (2001); Tullney et al. (2013); Bulatowicz et al. (2013); Korver et al. (2015); Limes et al. (2018) and inertial sensing Donley and Kitching (2013); Walker and Larsen (2016).

The coupling between the nuclear magnetic dipole moment $\bm{\mu}$ and the magnetic field $\mathbf{B}$, under the Hamiltonian

$$H_{D}=-\bm{\mu}\cdot\mathbf{B},$$

(1)

determines the magnetic-dipole precession frequency $\omega=\gamma{B}$, where $\gamma$ is the nuclear gyromagnetic ratio. In an ideal comagnetometer, two isotopes of both nuclear spin 1/2, denoted as $a$ and $b$, occupy the same spatial region. Consequently, the ratio of their precession frequencies,

$${\omega_{a}}/{\omega_{b}}={\gamma_{a}}/{\gamma_{b}},$$

(2)

is independent of the external field. There are indeed two and only two stable noble-gas isotopes with nuclear spin 1/2: ³He and ¹²⁹Xe. A ³He-¹²⁹Xe comagnetometer system mixed with alkali atoms suffers from a systematic effect due to the large difference between helium and xenon in their atomic sizes and collision properties with the alkali atoms. Consequently, ³He and ¹²⁹Xe atoms experience different effective magnetic fields generated by the polarized alkali atoms Schaefer et al. (1989); Romalis and Cates (1998); Ma et al. (2011). Various schemes have been developed to solve this problem Sheng et al. (2014); Korver et al. (2015); Limes et al. (2018), but the strong modulation fields introduced in these methods present potential problems on the stability of the system.

Besides ³He, the stable isotope ¹³¹Xe ($I$=3/2) can also be combined with ¹²⁹Xe to form a comagnetometer. Here, the two xenon isotopes have nearly identical collisional properties with alkali atoms Bulatowicz et al. (2013), thus suppressing the aforementioned systematic effect. However, with a nuclear spin of $3/2$, ¹³¹Xe has not only a nuclear magnetic dipole moment, but also a nuclear electric quadrupole moment that couples to the external electric field gradient, during the dwelling time ($\sim 10^{-11}$ s) of the atoms on the cell surfaces Kwon et al. (1981); Wu et al. (1987, 1990).

Figure 1(a) shows two coordinate systems that are set up in this paper: the $xyz$ system defined by the principal axes of the rectangular cell, and the $x^{\prime}y^{\prime}z^{\prime}$ system, where the $z^{\prime}$ axis is the quantization axis. When the two systems coincide with each other, the Hamiltonian for the electric quadrupole interaction is Slichter (1990)

$$H_{Q0}=\frac{eQ}{4I(2I-1)}[V_{zz}(3\bm{I}_{z}^{2}-\bm{I}^{2})+(V_{xx}-V_{yy})(\bm{I}_{x}^{2}-\bm{I}_{y}^{2})],$$

(3)

where $Q$ is the electric quadrupole moment of the atom, and $V_{ii}$ is the second-order derivative of the electric potential along the $i$ direction. Because the interaction happens only when the atoms dwell on the cell surfaces, experimentally observed quadrupole effects in ¹³¹Xe precession spectrum are time-averaged, and can be described by an effective Hamiltonian which is equal to $H_{Q0}$ in Eq. (3) times a small coefficient $p$ Kwon et al. (1981). By defining $q=V_{zz}/e$ and the asymmetry parameter $\eta=(V_{xx}-V_{yy})/{V_{zz}}$, Eq. (3) becomes:

$$H_{Q0}=\frac{pe^{2}qQ}{4I(2I-1)}[3\bm{I}_{z}^{2}-\bm{I}^{2}+\eta(\bm{I}_{x}^{2}-\bm{I}_{y}^{2})].$$

(4)

The combination of the magnetic dipole and electric quadrupole interactions leads to a triplet structure in the precession spectrum of ¹³¹Xe. To recover the ideal comagnetometer working condition in Eq. (2) for ¹³¹Xe atoms, it is important to understand and control the quadrupole splittings among the triplet in the precession spectrum. For an arbitrary magnetic field orientation, $H_{Q0}$ is transformed into $H_{Q}$ using the rotational properties of the angular momentum Sakurai and Napolitano (2013). In the case of strict $x-y$ symmetry, $\eta=0$, $H_{Q}$ shows a 90^∘-rotation symmetry around the $z$ axis Venema et al. (1992); Donley et al. (2009). In practice, however, the 90^∘-rotation symmetry of the quadrupole interaction is often degraded by imperfect cell conditions, for example, the addition of a cell stem and non-uniform deposition of alkali atoms on the cell surfaces. This asymmetry leads to additional frequency shifts, and complicates the combination method used to extract the magnetic-dipole precession frequency of ¹³¹Xe Bulatowicz et al. (2013).

In this work, we examine the comagnetometer system of ¹²⁹Xe-¹³¹Xe mixed with Rb atoms in rectangular cells with a square cross section, and focus on understanding the quadrupole shifts and systematics of the precession spectrum of ¹³¹Xe when the asymmetry parameter $\eta$ in the quadrupole interaction can not be neglected. Following this introduction, Sec. II presents the theoretical formulation of the quadrupole splittings, Sec. III describes the experiment apparatus and methods, Sec. IV shows the measurement results, and Sec. V concludes the paper.

When the $xyz$ coordinate coincides with $x^{\prime}y^{\prime}z^{\prime}$ coordinate, the quadrupole interaction Hamiltonian for ¹³¹Xe atoms, previously expressed in Eq. (4), can be written in the matrix form

$\displaystyle~{}H_{Q0}=\frac{pe^{2}qQ}{12}\left(\begin{array}[]{cccc}3&0&\sqrt{3}\eta&0\\ 0&-3&0&\sqrt{3}\eta\\ \sqrt{3}\eta&0&-3&0\\ 0&\sqrt{3}\eta&0&3\\ \end{array}\right).$

(9)

For an arbitrary $\mathbf{B}$, characterized by its zenith angle $\psi$ and azimuth angle $\phi$ shown in Fig. 1(a), the $xyz$ and $x^{\prime}y^{\prime}z^{\prime}$ coordinates are connected by a single rotation around the $R$ axis, which is perpendicular to the $zz^{\prime}$ plane. Such a rotation can also be described by three successive Euler rotations shown in Fig. 1(b), with the Euler angles of $(\phi,\psi,-\phi)$. Then the interaction Hamiltonian in Eq. (9) is transformed as

$$~{}H_{Q}=\mathcal{D}_{\frac{3}{2}}(\phi,\psi,-\phi)H_{Q0}\mathcal{D}^{-1}_{\frac{3}{2}}(\phi,\psi,-\phi),$$

(10)

where $\mathcal{D}_{j}$ is the Wigner D matrix for the angular momentum $\bm{j}$.

The complete Hamitonian for ¹³¹Xe, including both the magnetic dipole and electric quadrupole interactions, is

	$\displaystyle H=\hbar\omega\left(\begin{array}[]{cccc}\frac{3}{2}&0&0&0\\ 0&\frac{1}{2}&0&0\\ 0&0&-\frac{1}{2}&0\\ 0&0&0&-\frac{3}{2}\\ \end{array}\right)+\frac{pe^{2}qQ}{12}\left(\begin{array}[]{cccc}a&b&c&0\\ b^{*}&-a&0&c\\ c^{*}&0&-a&-b\\ 0&c^{*}&-b^{*}&a\\ \end{array}\right)~{},$		(19)
	$\displaystyle a=\frac{3}{2}\left[3\cos^{2}(\psi)-1+\eta\sin^{2}(\psi)\cos(2\phi)\right],$		(20)
	$\displaystyle b=\sqrt{3}\sin(\psi)e^{-i\phi}\left\{3\cos(\psi)-\eta[\cos(\psi)\cos(2\phi)+i\sin(2\phi)]\right\},$		(21)
	$\displaystyle c=\frac{\sqrt{3}}{2}e^{-2i\phi}\left\{3\sin^{2}(\psi)+\eta\left[\left(\cos^{2}(\psi)+1\right)\cos(2\phi)+2i\cos(\psi)\sin(2\phi)\right]\right\}.$		(22)

To simplify the expression in the above equations, we introduce a parameter $\omega_{q}=pe^{2}qQ/2\hbar$. For experimental conditions of ¹³¹Xe atoms studied in this paper, $\omega_{q}/2\pi\sim$50 mHz, and the magnetic-dipole precession frequency $\omega/2\pi\sim$3 Hz. Therefore, we can treat the quadrupole interaction as a perturbation to the dipole interaction. In the second order approximation of quadrupole interactions, the eigenvalues of $H$ in Eq. (19) are

	$\displaystyle E_{1}=\frac{3}{2}\hbar\omega+\frac{\hbar\omega_{q}}{6}a+\frac{\hbar\omega_{q}^{2}}{36\omega}\left(|b|^{2}+\frac{|c|^{2}}{2}\right),$		(23)
	$\displaystyle E_{2}=\frac{1}{2}\hbar\omega-\frac{\hbar\omega_{q}}{6}a-\frac{\hbar\omega_{q}^{2}}{36\omega}\left(|b|^{2}-\frac{|c|^{2}}{2}\right),$		(24)
	$\displaystyle E_{3}=-\frac{1}{2}\hbar\omega-\frac{\hbar\omega_{q}}{6}a+\frac{\hbar\omega_{q}^{2}}{36\omega}\left(|b|^{2}-\frac{|c|^{2}}{2}\right),$		(25)
	$\displaystyle E_{4}=-\frac{3}{2}\hbar\omega+\frac{\hbar\omega_{q}}{6}a-\frac{\hbar\omega_{q}^{2}}{36\omega}\left(|b|^{2}+\frac{|c|^{2}}{2}\right).$		(26)

This leads to the three observed transition lines in the precession spectrum of ¹³¹Xe atoms:

	$\displaystyle\hbar\omega_{+}=E_{1}-E_{2}=\hbar\omega+\frac{\hbar\omega_{q}}{3}a+\frac{\hbar\omega_{q}^{2}}{18\omega}|b|^{2},$		(27)
	$\displaystyle\hbar\omega_{0}=E_{2}-E_{3}=\hbar\omega-\frac{\hbar\omega_{q}^{2}}{18\omega}\left(|b|^{2}-\frac{|c|^{2}}{2}\right),$		(28)
	$\displaystyle\hbar\omega_{-}=E_{3}-E_{4}=\hbar\omega-\frac{\hbar\omega_{q}}{3}a+\frac{\hbar\omega_{q}^{2}}{18\omega}|b|^{2}.$		(29)

We define the difference between $\omega_{+}$ and $\omega_{-}$ in the equations above as the quadrupole splitting $\Omega$:

	$\displaystyle~{}\Omega$	$\displaystyle=$	$\displaystyle\omega_{+}-\omega_{-}=\frac{2}{3}\omega_{q}a$		(30)
		$\displaystyle=$	$\displaystyle\omega_{q}[3\cos^{2}(\psi)-1+\eta\sin^{2}(\psi)\cos(2\phi)].$

Note that, when $\eta=0$, $\Omega$ is only dependent on the zenith angle $\psi$ between $\mathbf{B}$ and the $z$ axis in the second order approximation of the quadrupole interactions, same as the cases of cylindrical and spherical cells Kwon et al. (1981); Wu et al. (1987). Another indication from Eq. (30) is that the dependence of $\Omega$ on $\eta$ can be removed by combining the results from two different magnetic field orientations or choosing $\phi=\pi/4$:

	$\displaystyle~{}\frac{\Omega(\psi,\phi)+\Omega(\psi,\frac{\pi}{2}+\phi)}{2}$	$\displaystyle=$	$\displaystyle\Omega(\psi,\pi/4)$		(31)
		$\displaystyle=$	$\displaystyle\omega_{q}\left[3\cos^{2}(\psi)-1\right].$

In the experiment, we use rectangular cells made of Pyrex glass (Fig. 2(a)). Each cell has an inner dimension of 9 mm$\times$4 mm$\times$4 mm. The cell is loaded with Rb atoms of natural abundances and the following gases: 15 Torr of ¹³¹Xe, 4 Torr of ¹²⁹Xe, 5 Torr of H₂, and 400 Torr of N₂. The cell sits inside an oven, heated by ac currents with a frequency of 70 kHz through twisted resistance wires, and placed inside a five-layer cylindrical mu-metal magnetic shields. We align the long axis of the cell with the longitudinal axis of the shields ($z$ axis in Fig. 2(b)). The magnetic fields inside the shields are controlled by three sets of coils, with a set of solenoid coils for the longitudinal field, and two sets of cosine-theta coils for the transverse field Venema (1994).

A circularly polarized laser beam passes through the cell along the $z$ axis. With a beam diameter of 1 cm and a beam intensity of 50 mW/cm², this pump beam is on resonance with the D1 transitions of both Rb isotopes, whose full linewidths are pressure broadened to 9 GHz Romalis et al. (1997). A linearly polarized probe beam 5 GHz red detuned from the Rb D1 line, with a beam diameter of 2 mm and a beam power of 3 mW, passes through the cell along the $x$ axis. The pump and probe beams are generated from two separate distributed-Bragg-reflector laser diodes. We detect the Rb magnetometer signals by analyzing the probe beam polarization with a polarimeter, which consists of a photo-elastic modulator (PEM) modulating the light polarization at a frequency of 50 kHz, a set of cross-polarizers (one before the cell and one after the cell in Fig. 2), and a photodiode detector. The detector signal is demodulated by a lock-in amplifier.

Each experiment cycle lasts 180 s, consisting of three stages. In the first stage, the polarized Rb atoms exchange polarization with the Xe atoms over 60 seconds under a bias field $\mathbf{B}_{0}=$9 mG in the $z$ direction. At the beginning of the second stage, a short $\pi/2$ pulse is applied to simultaneously tip the polarizations of both ¹²⁹Xe and ¹³¹Xe to the $xy$ plane. Following this pulse, the bias magnetic field is adjusted to $\mathbf{B}_{1}$, with $|\mathbf{B}_{1}|=|\mathbf{B}_{0}|$, whose direction is varied as to study the effect of the magnetic field orientation on Xe precession spectrum. Xe atoms precess around $\mathbf{B}_{1}$ for 90 seconds while their spins are monitored by the probe beam through the Rb magnetometer. In the third stage, preparing for the next experiment cycle, the bias field is changed back to the $\mathbf{B}_{0}$, and a 30 s long field gradient pulse ($dB_{z}/dz$) is applied to depolarize completely Xe atoms. The pump and probe beams are kept on throughout the cycle.

Figure 3(a) shows a typical plot of the Xe precession signal recorded during the second stage of an experimental cycle at the cell temperature of 383 K. The Fourier transform of the experimental data shows four peaks in Fig. 3(b). While the Larmor frequency of ¹²⁹Xe is at 10.5 Hz, the other three peaks near 3.1 Hz form the quadrupole-split spectrum of ¹³¹Xe.

We use four exponential-decay-oscillation functions to fit the data:

$$~{}y=\sum_{i=1}^{4}a_{i}e^{-(t-t_{0})/\tau_{i}}\sin[2\pi f_{i}(t-t_{0}+\varphi_{i})]+b,$$

(32)

The typical fitting error is 1 $\mu$Hz for the ¹²⁹Xe precession frequency, and are 4 $\mu$Hz and 15 $\mu$Hz for the central peak and sidebands of ¹³¹Xe atoms, respectively. The quadrupole splittings of ¹³¹Xe atoms are extracted from the fitting results.

We first characterize the properties of the quadrupole interactions between the ¹³¹Xe atoms and the cell surfaces. As discussed in Ref. Wu et al. (1987), the dependence of the quadrupole splittings on the cell wall temperature is described by the relation:

$$~{}\Omega\propto e^{-E_{a}/k_{B}T},$$

(33)

where $E_{a}$ is the adsorption potential of the cell walls acting on the ¹³¹Xe atoms. Figure 4(a) shows the measured temperature dependence of $\Omega$ when the magnetic field is along the $z$ axis. Using Eq. (33), we extract the adsorption energy $E_{a}=-0.14\pm 0.01$ eV. This result is consistent with previously measured activation energy of cured Pyrex cells filled with H₂ gas Wu et al. (1990). Figure 4(b) shows that the normalized results of $\Omega(T)/\Omega(T_{0}=353~{}K)$ for two different magnetic field directions agree with each other, confirming the expectation that the relation between $\Omega$ and the direction of the bias field is independent of the cell temperature. We then fix the cell temperature at $T=383$ K for the rest of the study.

To probe the dependence of the ¹³¹Xe quadrupole splittings on the magnetic field, we performed measurements on two atomic cells, which have the same conditions except the distribution of Rb atoms inside the cells. Cell #1 has Rb droplets accumulated on one of its inner surfaces, while most the Rb atoms of cell #2 are chased into the cell tip. For each cell, we study three cases: $\mathbf{B}_{1}$ in the second stage of the experiment stays in the $xz$ plane ($\phi=0^{\circ}$), $yz$ plane ($\phi=90^{\circ}$), and the plane of $\phi=45^{\circ}$. In each case, we scan the zenith angle $\psi$ of $\mathbf{B}_{1}$ using the field coils.

Figure 5 shows the experimental data of cell #1, together with the fitting lines using Eq. (30), where we leave $\omega_{q}$, $\eta$, and the offset of $\psi$ ($\psi_{0}$) as free parameters. The data for the case of $\phi=45^{\circ}$ gives a fitting result of $\eta\cos\phi=-0.05\pm 0.02$, consistent with the prediction of Eq. (31) that the quadrupole splitting is insensitive to $\eta$ in this case. Data sets of $\phi=0^{\circ}$ and $90^{\circ}$ show clear evidences of asymmetrical quadrupole interactions. Using the combined fitting results of both data sets, we extracted $\omega_{q}/2\pi=50.4\pm 0.3$ mHz and $|\eta|=0.62\pm 0.1$ for cell #1. There are two possible origins for this surprisingly large asymmetry parameter in cell #1. One is the geometric asymmetry due to the cell stem, and the other is that there are several Rb droplets accumulated on one of inner surfaces Volk et al. (1979). To quantify each effect, we performed similar measurements on cell #2, and obtained $\omega_{q}/2\pi=62.4\pm 0.1$ mHz and $|\eta|=0.22\pm 0.02$. This suggests that Rb deposition on the cell surfaces can be a significant factor, possibly even the dominant one for the asymmetry parameter.

As discussed in the introduction, it is often required in precision measurements to extrapolate the magnetic-dipole precession frequency ($\omega$) of ¹³¹Xe from the measured $\omega_{+}$, $\omega_{0}$, and $\omega_{-}$ in Eqs. (27-29). Compared with $\omega_{\pm}$, $\omega_{0}$ is closer to $\omega$ because it contains only a quadratic term of $\omega_{q}$. Two other combinations have been used in the literature:

	$\displaystyle\omega_{1}=\frac{\omega_{+}+\omega_{-}}{2}=\omega+\frac{\omega_{q}^{2}}{6\omega}\sin^{2}(\psi)\left\{\left[3-\eta\cos(2\phi)\right]^{2}\cos^{2}(\psi)+\eta^{2}\sin^{2}(2\phi)\right\}$		(34)
	$\displaystyle\omega_{2}=\frac{\omega_{+}+\omega_{-}+2\omega_{0}}{4}=\omega+\frac{\omega_{q}^{2}}{96\omega}\left\{\left[3\sin^{2}(\psi)+\eta(\cos^{2}(\psi)+1)\cos(2\phi)\right]^{2}+4\eta^{2}\cos^{2}(\psi)\sin^{2}(2\phi)\right\}$		(35)

It is good practice in experiments to align the magnetic field direction as close as possible along the $z$ axis so that $\psi$ is near zero. Consequently, $\omega_{0,1,2}$ are simplified to:

	$\displaystyle\begin{split}\omega_{0}=\omega-\frac{\omega_{q}^{2}}{\omega}\left\{\left[\frac{3}{2}-\frac{5}{4}\eta\cos(2\phi)+\frac{\eta^{2}}{6}\right]\psi^{2}\right.\\ \left.-\frac{\eta^{2}}{12}-\frac{3}{16}\psi^{4}\right\},\end{split}$		(36)
	$\displaystyle\omega_{1}=\omega+\frac{\omega_{q}^{2}}{6\omega}\psi^{2}[9-6\eta\cos(2\phi)+\eta^{2}],$		(37)
	$\displaystyle\omega_{2}=\omega+\frac{\omega_{q}^{2}}{96\omega}[4\eta^{2}+12\eta\psi^{2}\cos(2\phi)+9\psi^{4}].$		(38)

In past works, the x-y asymmetry properties were often neglected Kwon et al. (1981); Wu et al. (1987); Majumder et al. (1990); Venema et al. (1992); Donley et al. (2009), as if the parameter $\eta$ were assumed to be zero. Under such an assumption, $\omega_{2}$ is preferred because it deviates from $\omega$ only by a small term of $\psi^{4}$. However, as demonstrated in this paper, $\eta$ should not be considered zero, as its size can be significantly larger than $\sin\psi$. In the two cells examined in this work, $\psi=0.01$, much less than the $\eta$ of each cell. Taking cell #1 for example ($|\eta|=0.6$, $\omega_{q}/2\pi=50$ mHz, $\psi=0.01$, and $\omega/2\pi=3$ Hz), the asymmetrical quadrupole interaction adds a frequency shift of 12 $\mu$Hz to $\omega_{2}$ and 24 $\mu$Hz to $\omega_{0}$. A $1\%$ change of $\eta$ during an experiment run, possibly caused by a redistribution of Rb deposition inside the cell, would lead to a shift of 250 nHz to $\omega_{2}$ and 500 nHz to $\omega_{0}$. $\omega_{1}$ can be a much better choice because its asymmetrical quadrupole term is proportional to $\psi^{2}$. In the example of cell #1, the asymmetrical quadrupole shift is only 100 nHz, two orders of magnitude smaller than those of $\omega_{0}$ and $\omega_{2}$.

In summary, we have studied the electric quadrupole splittings of ¹³¹Xe atoms in rectangular cells with non-zero asymmetry parameters, which introduce additional quadrupole shifts to the ¹³¹Xe precession frequencies. These effects are caused not only by the asymmetries in the geometric properties, but also in the surface properties of the cells. In particular, we find that non-uniform Rb covering of the cell can induce a significantly nonzero $\eta$ parameter. These effects are expected to be present even if the cells are in cylindrical, spherical or cubic shapes.

In certain types of experiments where additional modulation parameters are available, for example the spin-gravity experiment Venema et al. (1992), the quadrupole shifts introduced by the asymmetry parameter can be cancelled by comparing successive runs of flipped parameters. However, it is an important systematic effect when the accuracy of the ¹³¹Xe magnetic-dipole precession frequency is required, such as extracting the absolute value of $\gamma_{\mathrm{Xe129}}/\gamma_{\mathrm{Xe131}}$.

In addition to measuring $\eta$ and correcting the asymmetric quadrupole shifts with a high precision as demonstrated in this paper, one can suppress the asymmetric quadrupole shifts as follows. First, one can increase the bias field strength, thus reducing $\omega_{q}/\omega$. Second, one can reduce $\eta$, for example, by reducing the stem size (or even removing the stem Limes et al. (2019)) and by chasing excess Rb into the stem. Third, one can use $\omega_{1}$ in Eq. (37) to approximate $\omega$ Bulatowicz et al. (2013), which takes advantage of the small factor $\psi^{2}$ in the expression.

This work was supported by Natural Science Foundation of China (Grant No.11774329 and 11974329) and Key Research Program of Frontier Sciences, CAS (NO. XDB21010200).