Dipolar and scalar $^{3}$He-$^{129}$Xe frequency shifts in stemless cells

TL;DR

This study investigates how cell shape affects nuclear spin frequency shifts in a $^{3}$He-$^{129}$Xe comagnetometer, identifying an optimal aspect ratio to cancel dipolar effects and measuring a scalar collisional frequency shift consistent with theory.

Contribution

It demonstrates how to control dipolar frequency shifts via cell geometry and accurately measures the scalar $^{3}$He-$^{129}$Xe collisional frequency shift.

Findings

Optimal aspect ratio cancels dipolar effects in cylindrical cells.

Measured scalar collisional frequency shift with enhancement factor $oxed{-0.011 extpm0.001}$.

Results agree with theoretical predictions.

Abstract

We study nuclear spin frequency shifts in a $^{3}$He-$^{129}$Xe comagnetometer caused by spin polarization of $^{3}$He. We use stemless cylindrical cells to systematically vary the cell geometry and separately measure the cell shape-dependent and shape-independent frequency shifts. We find that a certain aspect ratio for a cylindrical cell cancels the dipolar effects of $^3$He magnetization in the regime of fast spin diffusion. Using this control we observe a scalar $^{3}$He-$^{129}$Xe collisional frequency shift characterized by an enhancement factor $\kappa_{\text{HeXe}} = -0.011\pm0.001$ in excellent agreement with theoretical calculation.

Tables

Table 1. Table 1: Relative size of the slopes ( B d / μ 0 ​ M z He ) / ( h / d ) superscript 𝐵 𝑑 subscript 𝜇 0 subscript superscript 𝑀 He 𝑧 ℎ 𝑑 (B^{d}/\mu_{0}M^{\text{He}}_{z})/(h/d) from fits to Fig.  3 data, scaled to the 129 Xe B 0 ∥ B_{0}\parallel case.

	129Xe field	129Xe theory	3He field	3He theory
$B_0\parallel$	$1$	$1$	$1.50\pm 0.02$	$3/2$
$B_0⟂$	$-0.44\pm 0.03$	$-1/2$	$-0.72\pm 0.03$	$-3/4$

Equations

$${⟨B_i^d⟩}_V={\mu }_0\left[M_i-n_i{M}_i+\frac{2}{3}(\kappa -1)M_i\right].$$

$$\frac{{⟨{\mathbf{B}}^{\text{He}}⟩}_V}{{\mu }_0}=\left(\frac{n_z}{2}-\frac{1}{6}\right){\mathbf{M}}^{\text{He}}+\frac{3}{2}\left(\frac{1}{3}-n_z\right)M_z^{\text{He}}\hat{z}.$$

$$\kappa =-\frac{3\pi }{{\mu }_0{\gamma }_1{\gamma }_2\hslash }\int 4\pi r^{2}g(r)J(r)dr$$

Full text

Nuclear spin comagnetometers Aleksandrov et al. (1983); Lamoreaux et al. (1986) are used in a number of precision fundamental physics experiments Safronova et al. (2018), such as searches for new long-range spin-dependent forces Bulatowicz et al. (2013); Tullney et al. (2013); Glenday et al. (2008); Venema et al. (1992) and tests of CP, CPT and Lorentz symmetries Bear et al. (2000); Rosenberry and Chupp (2001); Sachdeva et al. (2019). They are also used for inertial rotation sensing Donley et al. (2009); Donley (2010); Larsen and Bulatowicz (2012); Walker and Larsen (2016); Buchs et al. (2018), and magnetometry Bulatowicz and Larsen (2012). Measurements of nuclear spin precession frequencies allow nHz level frequency resolution because of long nuclear spin coherence times, as well as good accuracy and long-term stability Limes et al. (2018a); Sheng et al. (2014).

However, one unavoidable source of frequency shifts in nuclear magnetic resonance experiments is due to magnetic dipolar interactions between the spins. Local dipolar interactions are averaged to zero in a gas or liquid due to fast isotropic tumbling, but distant dipolar interactions can lead to complex spin dynamics Warren et al. (1993); Nacher and Stoltz (1995); Jeener (1999); Ledbetter and Romalis (2002); Zänker et al. (2008). Unlike most previous studies, we measure the dipolar frequency shifts in the regime where the time scale of atomic diffusion across the whole sample is much faster than both the time scale of long-range dipolar interactions and of the transverse spin relaxation. In this regime, used in most precision co-magnetometer experiments Sheng et al. (2014), the frequency shifts depend only on the shape of the cell containing the atoms. A similar fast-diffusion regime was previously studied by NMR in nanopores Baugh et al. (2001).

Nuclear spin dipolar interactions cause systematic frequency shifts in comagnetometer precision measurements Allmendinger et al. (2014a); Terrano et al. (2018) and are subject of some controversy Romalis et al. (2014); Allmendinger et al. (2014b). Here we use an anodic bonding batch fabrication process Liew et al. (2004) to make a series of stemless cylindrical cells that contain 3He and 129Xe, as well as 87Rb and N2. Well-defined cell shapes allow precision comparison with a simple theory for dipolar frequency shifts that we develop based on magnetometric demagnetizing factors Moskowitz and Torre (1966). We find that for a certain aspect ratio of the cylindrical cell the dipolar frequency shifts can be eliminated. An optimal and well-defined cylindrical geometry can improve the stability of nuclear-spin comagnetometers used for fundamental physics experiments Bulatowicz et al. (2013); Glenday et al. (2008); Tullney et al. (2013); Venema et al. (1992); Rosenberry and Chupp (2001); Allmendinger et al. (2014a); Sachdeva et al. (2019). It also can be used in NMR metrology applications instead of spherical cells that are hard to fabricate without stems Maul et al. (2016).

Excellent control of long-range dipolar fields also allows us to resolve a small scalar frequency shift between 3He and 129Xe nuclear spins mediated by second-order electron interactions. Such through-space $J~{}{\bf I}_{1}{\bf\cdot}{\bf I}_{2}$ coupling in van der Waals molecules has been theoretically studied in several systems F.R. Salisbury and Harris (1998); Pecul (2000); Bagno and Saielli (2003); Vaara et al. (2013) and was only observed experimentally between 129Xe and 1H in a liquid mixture of Xe and pentane Ledbetter et al. (2012). Here we report the first observation of $J$-coupling between atomic spins in a gas mixture Limes et al. (2018b). Simple atomic structure allows quantitative comparison with a precision first-principles calculation of the 3He-129Xe scalar spin coupling strength Vaara and Romalis (2019). We observe a weak temperature dependence of the scalar enhancement factor $\kappa$, in excellent agreement with calculations. This coupling has been previously assumed to be zero and not considered in precision measurements based on nuclear spin co-magnetometers.

The miniature 6 mm3 vapor cells are made using an anodically bonded glass-Si-glass construction Liew et al. (2004) in a custom-built system able to fabricate cells containing isotopically enriched alkali metals and noble gases. The Si wafer is 2 mm nominal thickness with a $7\times 7$ array of machined holes with a diameter $d=2.005\pm 0.005$ mm. We polished one side of a wafer at a small angle to obtain cell height variations from 1.666 mm to 1.988 mm across the wafer. The wafer is baked under high vacuum inside the fabrication system to remove contaminants, which is crucial to obtain long wall spin relaxation times, 300 s for 129Xe and 4 hours for 3He. We close the cells with 0.2 mm thick aluminosilicate glass SD-2 that has low 3He permeability Dellis et al. (2016). After anodically bonding glass on one side of the wafer, we distill $99.9\%$ isotopically pure 87Rb metal and bond the second glass in an atmosphere of 80 torr N2, 6.5 torr 129Xe, and 1400 torr of 3He.

The experimental apparatus is shown in Fig. 1a. We optically pump 87Rb to polarize nuclear spins by spin-exchange and detect their precession with an in-situ 87Rb magnetometer Sheng et al. (2014); Limes et al. (2018a). This detection yields a high signal-to-noise ratio because Rb experiences enhanced nuclear spin dipolar magnetic field due to electron-nuclear Fermi-contact interaction Schaefer et al. (1989); Romalis and Cates (1998); Ma et al. (2011). The pulse-train magnetometer described in Limes et al. (2018a) is adapted here for use with cells with a single optical axis by using counter-propagating pump and probe beams. After an initial pump period to polarize the nuclear spins along a $\hat{z}$ bias magnetic field $B_{0}\approx 5$ mG, the polarization of the on-resonant 795 nm pump laser is switched between $\sigma^{+}$ and $\sigma^{-}$ light with an electro-optic modulator (EOM) at 13 kHz. Simultaneously a train of 3 $\mu$s magnetic field $\pi$ pulses are applied along $\hat{y}$ to flip the 87Rb polarization back and forth along $\hat{z}$, suppressing spin-exchange relaxation Limes et al. (2018a). 87Rb polarization projection on $\hat{z}$ is detected with paramagnetic Faraday rotation of an off-resonant probe beam passing through the cell to a balanced polarimeter. An acousto-optic modulator turns on the probe laser only during the $\pi$ pulses to avoid unnecessary probe broadening during pumping intervals. An additional $B_{y}$ field causes an advancement or retardation of the 87Rb polarization phase during the $\pi$ pulse flip, see Fig. 1b). The polarimeter signal is sent to a lock-in amplifier referenced to half the EOM frequency and gives a lock-in output proportional to $B_{y}$. The pulse-train magnetometer has a sensitivity of 300 fT/$\sqrt{\text{Hz}}$ in our miniature cells.

We can also operate the co-magnetometer with $B_{0}$ perpendicular to the cell’s axis. To polarize the nuclear spins we apply fast $\pi_{\pm y}$ pulses with $\sigma^{+}$/$\sigma^{-}$ pumping to create a time-averaged Rb polarization along $B_{0}$ parallel to $\hat{x}$. The detection scheme remains the same because the $\pi$ pulses decouple Rb spin precession due to $B_{0}$ field Limes et al. (2018a).

Each run starts by first polarizing 3He for several hours along $B_{0}$ and then creating a small 129Xe polarization. The pulse-train 87Rb magnetometer is then turned on and we apply a tipping pulse for the nuclear spins calculated to leave a certain percentage of $M_{\text{He}}$ along $B_{0}$ and place 129Xe magnetization in the transverse plane. Noble-gas precession signals are recorded in a Ramsey-style sequence for about 100 s, with two detection periods separated by a dark period that has a rotating, two-axis decoupling pulse train applied Limes et al. (2018a). This decoupling pulse train removes noble-gas frequency shifts due to 87Rb back-polarization and nulls Bloch-Siegert shifts from the pulse train. Each detection period is fit to two decaying sine waves to extract the phases with which the noble gases enter and leave the dark period. Knowing the number of cycles elapsed during the dark period, we find the in-the-dark free-precession frequencies of the noble gases $\omega_{\text{He}}$ and $\omega_{\text{Xe}}$. We divide $\omega_{\text{He}}$ and $\omega_{\text{Xe}}$ by their gyromagnetic ratios $\gamma_{\text{He}}$, $\gamma_{\text{Xe}}$ Makulski (2015) to find the total effective field experienced by 3He and 129Xe, $B_{0}+B_{\text{He}}^{d}$ and $B_{0}+B_{\text{Xe}}^{d}$, respectively.

After a Ramsey measurement we place $M_{\text{He}}$ along or against $B_{0}$ by using dumping feedback Alem et al. (2013), and repeat the sequence for many $M_{\text{He}}$ values and tipping angles to find the slopes $B^{d}/\mu_{0}M^{\text{He}}_{z}$ for 3He and 129Xe. To avoid systematic errors from $B_{0}$ drift, chemical shifts, and remnant 129Xe polarization projection onto $B_{0}$ we use identical tipping pulses while alternating the sign of the initial $M_{\text{He}}$ projection along $B_{0}$. $M_{\text{He}}$ is found by comparing the amplitude of the fitted 3He signal to the 87Rb magnetometer response from a known external magnetic field. The 3He signal measured by Rb atoms is given by $B_{\rm Rb}=2\mu_{0}\kappa_{\rm RbHe}M_{\rm He}/3$ Romalis and Cates (1998). We neglect a 1% correction from the long-range dipolar effect of $M_{\text{He}}$ on 87Rb. In Fig. 2 we plot the effective dipolar field experienced by 3He and 129Xe due to $M_{z}^{\text{He}}$ for a cell with $h=1.718$ mm, $d=2.005$ mm ($h/d=0.857$) at 120*∘*C. We make the measurements with $B_{0}$ parallel and perpendicular to the optical axis of the cylindrical stemless cell.

In our cells the diffusion time across the cell is much shorter than the period of spin precession in dipolar fields and the spin relaxation time. In this regime each spin species has a uniform nuclear magnetization $\bf M$ inside the cell, unlike previous studies of long-range dipolar fields R.Bowtell et al. (1990); Warren et al. (1993). The spin precession frequencies are determined by the volume average magnetic field inside the cell, which can be calculated using the magnetometric demagnetizing factors for the average $\bf H$ field in a uniformly-magnetized sample, $\left\langle H_{i}\right\rangle_{V}=-n_{i}M_{i}$, $(i=x,y,z$) Moskowitz and Torre (1966). Analytical expressions for $n_{i}(\gamma)$ for a cylinder, where $\gamma=h/d$ is the height over diameter ratio, are given in Joseph (1966); Chen et al. (2006). The demagnetizing factors satisfy $n_{x}+n_{y}+n_{z}=1$. For a cylinder with $\gamma=0.9065,$ $n_{i}=1/3$ (the same as for a sphere)—for this aspect ratio the dipolar effect is zeroed out for all orientations of $B_{0}$ with respect to the cylindrical axis.

The classical average magnetic field needs to be corrected by separating out the scalar contact term $2\mu_{0}\delta^{3}({\bf r}){\bf m}/3$ of the classical dipolar field for non-interacting point dipoles $\bf m$ Vlassenbroek et al. (1996). For real particles, the contact interaction can be enhanced or suppressed relative to the classical value Romalis et al. (2014). It is enhanced for electron-nuclear spin interactions between alkali metal and noble gas spins, $\kappa_{\rm RbHe}=5.6,\kappa_{\rm RbXe}=500$ Schaefer et al. (1989); Romalis and Cates (1998); Ma et al. (2011). For nuclear spins the direct contact interaction is zero, but indirect second-order electron-mediated $J$-coupling can induce a finite effect. The scalar spin-spin interaction can be generally parametrized by $\kappa$, where $\kappa=1$ would correspond to the classical result Schaefer et al. (1989); Heckman et al. (2003). We write

[TABLE]

Bloch equations can be used to describe spin precession in the presence of a constant bias field and small nuclear-spin dipolar fields. In our case only 3He has a significant magnetization. The dipole field experienced by 129Xe from 3He can be written as $\left\langle B^{\text{Xe}}_{z}\right\rangle_{V}=\mu_{0}(1/3-n_{z}+2\kappa_{\text{HeXe}}/3)M_{z}^{\text{He}}$, where the $z$ axis is defined by the magnetic field direction. The rotating components of the 3He magnetization do not have a net effect on 129Xe precession frequency to first order in $M_{z}^{\text{He}}$. From the linearity of data in Fig. 2 one can see that the second-order Bloch-Siegert shift from the rotating components is negligible in our case.

In contrast, the 3He precession frequency is affected by the secular co-rotating components of the 3He dipolar field but is not affected by the scalar contact interaction with 3He because scalar interactions are not observable between like spins. One can write Vlassenbroek et al. (1996); Warren et al. (1995):

[TABLE]

The first term on the right hand side does not generate any frequency shift because it gives $\left\langle{\bf B}^{\text{He}}\right\rangle_{V}$ parallel to ${\bf M}^{\text{He}}$. The effective dipolar field responsible for a 3He frequency shift is given by the second term. It is 3/2 times larger than for 129Xe and both are proportional to the $M_{z}^{\text{He}}$ projection.

In Fig. 3 we plot the slope of the dipolar field $B^{d}/\mu_{0}M^{\text{He}}_{z}$ for a number of cells as a function of the cell aspect ratio $h/d$, e.g. the slopes of the line fits in the top panel of Fig. 2 are shown at $h/d=0.857$. The lines through the 3He data in Fig. 3 represent first-principles calculations of the dipolar fields using the theoretical expression for $n_{z}(\gamma)$ for a cylinder. For Xe data the lines $B^{d}_{\text{Xe}}/\mu_{0}M^{\text{He}}$ have a vertical offset representing the scalar frequency shift $\kappa_{\text{HeXe}}=-0.011\pm 0.001$, where the error is determined by the $M_{z}^{\text{He}}$ calibration uncertainty.

The accuracy of the dipolar field model in cells with well-defined geometry can also be checked by fitting the data points in Fig. 3. The relative size of the long-range dipolar frequency shifts for the four cases are related to each other by simple ratios shown in Table 1, which are applicable for any cell with uniaxial symmetry Moskowitz and Torre (1966). The 3He data cross zero and the 129Xe data intersect each other at $h/d=0.911\pm 0.001$, while $n_{z}(\gamma)=1/3$ for a cylinder with $h/d=0.9065.$ The $0.5\%$ discrepancy is likely due to a slightly larger effective diameter of the holes drilled in the Si wafer, which have a surface roughness of about $10~{}\mu$m and were measured with pin gauges that determine the minimum diameter.

The existence of a finite $\kappa_{\text{HeXe}}$ implies that the cell aspect ratio required to cancel the nuclear magnetization effects on the frequency ratio in the comagnetometer is different from the condition $n_{i}=1/3$. The optimal cell aspect ratio depends on the orientation of $B_{0}$ with respect to the cylindrical axis. In the bottom panel of Fig. 2 we show the comagnetometer frequency ratio $f_{r}-\gamma_{r}\equiv\omega_{\text{He}}/\omega_{\text{Xe}}-\gamma_{\text{He}}/\gamma_{\text{Xe}}$ for the cell with $h/d=0.857$. We find $f_{r}$ is almost independent of $\mu_{0}M^{\text{He}}$ for $B_{0}$ parallel to the cylinder axis for this particular cell.

We measured the temperature dependence of $\kappa_{\text{HeXe}}$ using a cell with $h/d=0.905$. In Fig. 4 we plot $3B^{d}_{\text{}}/2\mu_{0}M^{\text{He}}$ which is equal to $\kappa_{\text{HeXe}}$ for 129Xe in the absence of dipolar field. As expected, the small remaining dipolar field for 3He is independent of temperature. We use the 3He data and the relative slope from Table 1 to correct the 129Xe data for the remaining dipolar field. Relatively weak temperature dependence of $\kappa_{\text{HeXe}}$ implies that dipolar and scalar frequency shifts can cancel each other over a significant temperature range by proper choice of the cell shape and orientation.

The indirect spin-spin $J$ couplings for noble gases was calculated recently from first-principles for 3He-129Xe Vaara and Romalis (2019), and previously for 129Xe-131Xe Vaara et al. (2013). For freely moving spins in gases and liquids it is more convenient to parameterize the interaction in terms of $\kappa$ Heckman et al. (2003); Ledbetter et al. (2012):

[TABLE]

where $g(r)$ is the radial intermolecular distribution function. Ref. Vaara and Romalis (2019) gives $\kappa_{\text{HeXe}}=-0.0105\pm 0.0015$ at a temperature of 120 *∘*C, in excellent agreement with our experimental value. A temperature dependence is also predicted as $\kappa_{\text{HeXe}}=-0.00453-1.51\times 10^{-5}T$ with $T$ in K. While small, this temperature trend is in good agreement with our data.

In conclusion, we have used stemless cells with well-controlled geometry to investigate dipolar and scalar frequency shifts in nuclear spin co-magnetometers. We developed a simple model for dipolar magnetic fields based on magnetometric demagnetizing factors that applies in the regime of fast diffusions. It is in excellent agreement with our measurements. We observed a small scalar interaction between 3He and 129Xe spins and find it to be in very good agreement with recent first-principles calculation Vaara and Romalis (2019). The presence of scalar spin-spin coupling implies that the optimal shape of a co-magnetometer cell is slightly different from a sphere. We show that a cylindrical cell with a certain aspect ratio and orientation relative to the bias field can cancel both dipolar and scalar frequency shifts. Thus, one can eliminate an intrinsic source of frequency shifts present in all nuclear spin co-magnetometer experiments.

This work was funded by DARPA, NSF, and Academy of Finland.