Measurements of diffusion coefficients for rubidium--inert gas mixtures using coherent scattering from optically pumped population gratings

TL;DR

The paper reports room-temperature diffusion coefficients $D(T,p)$ for trace rubidium in six inert buffer gases (He, Ne, N$_2$, Ar, Kr, Xe) using a coherent-population-grating technique formed by two laser beams intersecting at a small angle $\theta$. The grating decay, measured via phase-matched coherent scattering, scales as $D(T,p)(k\theta)^2$, enabling extraction of $D$ over $p$ from $7{,}000$ to $90{,}000\ \text{Pa}$ and extrapolation to standard pressure $p_0=101{,}325\ \text{Pa}$; transit-time and residual-light corrections are applied. Quantum, classical, and semiclassical diffusion theories based on accurate interatomic potentials are used for comparison; after corrections, the quantum calculations agree with experiment across all gases, yielding the reported values ($D$ in cm$^2$/s): He $0.33(5)$, Ne $0.214(14)$, N$_2$ $0.132(7)$, Ar $0.123(9)$, Kr $0.093(9)$, Xe $0.073(4)$. The work informs magnetometer optimization, spin-based imaging, collision-model validation, and the development of pressure standards, highlighting the value of a single-technique, cross-gas benchmark for diffusion in alkali–noble-gas systems.

Abstract

We present comprehensive determinations of the diffusion coefficients $D$ at $T=24\,\degree$C for trace amounts of naturally abundant Rb atoms in inert, naturally abundant He, Ne, N$_2$, Ar, Kr, and Xe buffer gases using a single measurement technique. We establish a spatially periodic population grating in the Rb sample using two laser beams that intersect at a small angle $θ$ of a few milliradians. The atomic population grating decays exponentially in time due to diffusive motion induced by momentum-changing elastic collisions between Rb and buffer gas atoms or molecules, and is monitored by observing the scattered field from a read-out beam. We distinguish the contribution of diffusion from other collisional processes by measuring the characteristic $θ^2$ dependence of the decay rate. We also measure the systematic dependence of the decay rate on the buffer gas pressure over a range of $7\,000$ Pa to $90\,000$ Pa. In this manner, we obtain diffusion coefficients at standard atmospheric pressure of $101\,325$ Pa and at a temperature of 24.0(5)~$^\circ$C. We obtain weighted averages of $0.33(5)$ cm$^2$/s, $0.214(14)$ cm$^2$/s, $0.132(7)$ cm$^2$/s, $0.123(9)$ cm$^2$/s, $0.093(9)$ cm$^2$/s, and $0.073(4)$ cm$^2$/s for Rb in He, Ne, N$_2$, Ar, Kr, and Xe, respectively. We compare this data with diffusion coefficients obtained using quantum, classical, and semi-classical theoretical methods based on the most accurate interatomic interaction potentials from the literature. Our computed diffusion coefficients based on the quantum theory agree with the experimental determinations when systematic effects are taken into account. Our measurements and modeling are relevant to the optimization of magnetometers, imaging using spin-polarized noble gases, tests of collision models based on interatomic potentials, and the development of pressure sensors.

Figures (12)

• Figure 1: (color online) a) Diagram of the optical polarization grating formed by two perpendicular linear polarized laser beams with wavevectors $\vec{k}_1$ and $\vec{k}_2$ intersecting at a small angle $\theta$. One laser is polarized in the plane, the other is polarized out of plane. The polarization grating forms along $\vec{k}_1 - \vec{k}_2$ and is indicated by red and white stripes. An excitation pulse is applied along both directions creating a spatial population grating in each $m$-level of the Rb sample along the same direction. A read-out pulse is applied along $\vec{k_2}$ with variable time delay after the excitation pulse, inducing a coherent scattered signal from the grating in direction $\vec{k_1}$. Fields applied at the time of the excitation pulse are shown in red, while those present at the time of the read-out pulse are shown in purple. All beams have the same spatial profile, but the read-out and signal beams are shown narrower so that they do not obscure the excitation beams. b) Timing diagram for the experiment showing the laser pulses applied along directions $\vec{k}_1$ and $\vec{k}_2$ as well as the intensity of the detected fields. The detector records light incident along direction $\vec{k}_1$. The integrated signal pulse at each read-out delay is indicated by the dotted purple line.
• Figure 2: (Color online) Simulation of the population distribution after a 50 $\mu$s excitation pulse in the ground-state magnetic sublevels quantized along ${\vec{k}_1}-{\vec{k}_2}$ of $^{85}$Rb $F = 2$ (panel a), $F = 3$ (b) and of $^{87}$Rb $F = 1$ (c), $F = 2$ (d) as functions of position along $\vec{k}_1-\vec{k}_2$ for one period of the grating. We have assumed a 200 Torr N$_2$ buffer gas and an average light intensity of 120 mW/cm$^2$. The distribution for each level is shown in terms of its deviation $\Delta$ from a uniform population in each $m_F$ sublevel for each $F$ state (i.e. a population of 1/(2$F$+1)). After the pulse, 96 % of the $^{85}$Rb population is pumped into the $F=2$ ground state and 4 % into the $F = 3$ state. For $^{87}$Rb the percentages are 87 % and 13 % for $F = 1$ and 2, respectively. For all four panels, the two linearly polarized laser beams intersect at angle $\theta=3$ mrad and have a laser frequency that coincides with the $F=3 \rightarrow F'=4$ transition for a $^{85}$Rb atom in vacuum.
• Figure 3: (Color online)"Weighted" differential cross sections $(1-\cos\Theta){\rm d}\sigma/{\rm d}\Omega$ at a collision energy of $k_{\rm B}\times 300$ K as functions of polar scattering angle $\Theta$ for systems $^{87}{\rm Rb}+X$, where $X={^4{\rm He}}$, $^{20}$Ne, $^{14}$N$_2$, $^{40}$Ar, $^{84}$Kr, and $^{132}$Xe. The red curves are obtained from quantum scattering calculations using the most-accurate electronic potentials by Ref. Klos. Blue curves are obtained with quantum scattering calculations using a purely repulsive potential derived from the repulsive wall of the potential for this system (See text for details). Data for the various systems have the same vertical scale but are displaced by unequal amounts for clarity. Dashed black lines correspond to the zero values for the relevant data.
• Figure 4: (Color online) Thermalized diffusion coefficients as functions of temperature $T$ for $^{87}$Rb+$^{14}$N$_2$ and at the standard atmospheric pressure of $p=101\,325$ Pa. The solid black curve with small standard-uncertainty error bars correspond to our quantum simulations as well as the (nearly) indistinguishable classical simulations from Ref. Medvedev2018. The dashed black curve corresponds to a quantum simulation using the purely repulsive potential constructed from our most-accurate electronic potential. The blue curve corresponds to the semi-classical, analytical estimate. Two overlapping red markers with standard uncertainties just below $T=300$ K are from Ref. Diffusion and our current measurements. Green markers with standard uncertainties and author labels at higher temperatures correspond to measurements found in Table \ref{['tab:compare']}. For publications that did not supply an uncertainty budget we assume standard uncertainty 1 in the last significant digit.
• Figure 5: (Color online) a) Sketch of gas handling apparatus including the glass manifold. Gate valves are represented by circles with crosses. A capacitance manometer ("Cap. Manom." in the figure) measures pressures in the central chamber between 0.1 Torr and 1000 Torr, while an ion gauge measures pressures from $10^{-9}$ Torr to $10^{-4}$ Torr. b) Detailed sketch of the glass manifold with the valve connecting the manifold to the central chamber shown in panel a). The cylindrical out-of-plane heated duct (red circle) helps maintain a constant rubidium vapor density in the experimental cell.